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abstract: 'Heavy Ion Collisions ($HIC$) represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation. In this report we present a selection of new reaction observables in dissipative collisions particularly sensitive to the symmetry term of the nuclear Equation of State ($Iso-EoS$). We will first discuss the Isospin Equilibration Dynamics. At low energies this manifests via the recently observed Dynamical Dipole Radiation, due to a collective neutron-proton oscillation with the symmetry term acting as a restoring force. At higher beam energies Iso-EoS effects will be seen in Imbalance Ratio Measurements, in particular from the correlations with the total kinetic energy loss. For fragmentation reactions in central events we suggest to look at the coupling between isospin distillation and radial flow. In Neck Fragmentation reactions important $Iso-EoS$ information can be obtained from the correlation between isospin content and alignement. The high density symmetry term can be probed from isospin effects on heavy ion reactions at relativistic energies (few $AGeV$ range). Rather isospin sensitive observables are proposed from nucleon/cluster emissions, collective flows and meson production. The possibility to shed light on the controversial neutron/proton effective mass splitting in asymmetric matter is also suggested. A large symmetry repulsion at high baryon density will also lead to an “earlier” hadron-deconfinement transition in n-rich matter. A suitable treatment of the isovector interaction in the partonic $EoS$ appears very relevant.'
author:
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M. Di Toro,$^{1,2}$ V. Baran,$^3$ M. Colonna,$^2$ G. Ferini ,$^{2}$\
T. Gaitanos,$^4$ V. Giordano,$^{1,2}$ V. Greco,$^{1,2}$ Liu Bo,$^5$ M. Zielinska-Pfabe,$^6$\
S. Plumari,$^{1,7}$ V. Prassa,$^8$ C. Rizzo,$^{1,2}$ J. Rizzo,$^{1,2}$ and H. H. Wolter,$^9$\
\
$^1$Dipartimento di Fisica e Astronomia dell’Università, Catania, Italy\
$^2$INFN, Laboratori Nazionali del Sud, Catania, Italy\
$^3$Dept. of Theoretical Physics, Bucharest Univ. and NIPNE-HH, Romania\
$^4$ Institute for Theoretical Physics, Giessen University, Germany\
$^5$ IHEP, Chinese Academy of Science, Beijing, China\
$^6$ Smith College, Northampton, Mass., USA\
$^7$ INFN, Sezione di Catania, Italy\
$^8$ Physics Department, Aristotles Univ.of Thessaloniki, Grece\
$^9$ Department für Physik, Unversität Munchen, Garching, Germany
title: ' Isospin Dynamics in Heavy Ion Collisions: from Coulomb Barrier to Quark Gluon Plasma'
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Introduction: The Elusive Symmetry Term of the EoS
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The symmetry energy $E_{sym}$ appears in the energy density $\epsilon(\rho,\rho_3) \equiv \epsilon(\rho)+\rho E_{sym} (\rho_3/\rho)^2
+ O(\rho_3/\rho)^4 +..$, expressed in terms of total ($\rho=\rho_p+\rho_n$) and isospin ($\rho_3=\rho_p-\rho_n$) densities. The symmetry term gets a kinetic contribution directly from basic Pauli correlations and a potential part from the highly controversial isospin dependence of the effective interactions. Both at sub-saturation and supra-saturation densities, predictions based of the existing many-body techniques diverge rather widely, see [@fuchswci; @fantoni08].
We recall that the knowledge of the EoS of asymmetric matter is very important at low densities ( e.g. neutron skins, pigmy resonances, nuclear structure at the drip lines, neutron distillation in fragmentation, neutron star formation and crust) as well as at high densities ( e.g. neutron star mass-radius relation, cooling, hybrid structure, transition to a deconfined phase, formation of black holes). Several observables which are sensitive to the Iso-EoS and testable experimentally, have been suggested [@colonnaPRC57; @Isospin01; @baranPR; @wcineck; @WCI_betty; @baoPR08]. We take advantage of new opportunities in theory (development of rather reliable microscopic transport codes for $HIC$) and in experiments (availability of very asymmetric radioactive beams, improved possibility of measuring event-by-event correlations) to present new results that are constraining the existing effective interaction models. We will discuss dissipative collisions in a wide range of beam energies, from just above the Coulomb barrier up to the $AGeV$ range. Isospin effects on the chiral/deconfinement transition at high baryon density will be also discussed. Low to Fermi energies will bring information on the symmetry term around (below) normal density, while intermediate energies will probe high density regions. The transport codes are based on mean field theories, with correlations included via hard nucleon-nucleon elastic and inelastic collisions and via stochastic forces, selfconsistently evaluated from the mean phase-space trajectory, see [@baranPR; @guarneraPLB373; @colonnaNPA642; @chomazPR]. Stochasticity is essential in order to get distributions as well as to allow the growth of dynamical instabilities.
Relativistic collisions are described via a fully covariant transport approach, related to an effective field exchange model, where the relevant degrees of freedom of the nuclear dynamics are accounted for [@baranPR; @liubo02; @theo04; @santini05; @ferini05; @ferini06]. We will have a propagation of particles suitably dressed by self-energies that will influence collective flows and in medium nucleon-nucleon inelastic cross sections. The construction of an $Hadron-EoS$ at high baryon and isospin densities will finally allow the possibility of developing a model of a hadron-deconfinement transition at high density for an asymmetric matter [@ditoro_dec]. The problem of a correct treatment of the isospin in a effective partonic $E0S$ will be stressed.
We will always test the sensitivity of our simulation results to different choices of the density and momentum dependence of the Isovector part of the Equation of State ($Iso-EoS$). In the non-relativistic frame the potential part of the symmetry energy, $C(\rho)$, [@baranPR]: $$\frac{E_{sym}}{A}=\frac{E_{sym}}{A}(kin)+\frac{E_{sym}}{A}(pot)\equiv
\frac{\epsilon_F}{3} + \frac{C(\rho)}{2\rho_0}\rho$$ is tested by employing two different density parametrizations, Isovector Equation of State (Iso-Eos) [@colonnaPRC57; @bar02], of the mean field: i) $\frac{C(\rho)}{\rho_0}=482-1638 \rho$, $(MeV fm^{3})$, for “Asysoft” EoS: ${E_{sym}/{A}}(pot)$ has a weak density dependence close to the saturation, with an almost flat behavior below $\rho_0$ and even decreasing at suprasaturation; ii) a constant coefficient, $C=32 MeV$, for the “Asystiff” EoS choice: the interaction part of the symmetry term displays a linear dependence with the density, i.e. with a faster decrease at lower densities and much stiffer above saturation. The isoscalar section of the EoS is the same in both cases, fixed requiring that the saturation properties of symmetric nuclear matter with a compressibility around $220MeV$ are reproduced.
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Isospin Equilibration
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The Prompt Dipole $\gamma$-Ray Emission
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The possibility of an entrance channel bremsstrahlung dipole radiation due to an initial different N/Z distribution was suggested at the beginning of the nineties [@ChomazNPA563; @BortignonNPA583]. After several experimental evidences, in fusion as well as in deep-inelastic reactions, [@PierrouPRC71; @medea] and refs. therein, we have now a good understanding of the process and stimulating new perspectives from the use of radioactive beams.
During the charge equilibration process taking place in the first stages of dissipative reactions between colliding ions with different N/Z ratios, a large amplitude dipole collective motion develops in the composite dinuclear system, the so-called Dynamical Dipole mode. This collective dipole gives rise to a prompt $\gamma $-ray emission which depends: i) on the absolute value of the intial dipole moment $$\begin{aligned}
&&D(t= 0)= \frac{NZ}{A} \left|{R_{Z}}(t=0)- {R_{N}}(t=0)\right| = \nonumber \\
&&\frac{R_{P}+R_{T}}{A}Z_{P}Z_{T}\left| (\frac{N}{Z})_{T}-(\frac{N}{Z})_{P}
\right|,
\label{indip}\end{aligned}$$ being ${R_{Z}}= \frac {\Sigma_i x_i(p)}{Z}$ and ${R_{N}}=\frac {\Sigma_i x_i(n)}{N} $ the center of mass of protons and of neutrons respectively, while R$_{P}$ and R$_{T}$ are the projectile and target radii; ii) on the fusion/deep-inelastic dynamics; iii) on the symmetry term, below saturation, that is acting as a restoring force.
A detailed description is obtained in mean field transport approaches, [@BrinkNPA372; @BaranPRL87]. We can follow the time evolution of the dipole moment in the $r$-space, $D(t)= \frac{NZ}{A} ({R_{Z}}- {R_{N}})$ and in $p-$space, $DK(t)=(\frac{P_{p}}{Z}-\frac{P_{n}}{N})$, with $P_{p}$ ($P_{n}$) center of mass in momentum space for protons (neutrons), just the canonically conjugate momentum of the $D(t)$ coordinate, i.e. as operators $[D(t),DK(t)]=i\hbar$. A nice “spiral-correlation” clearly denotes the collective nature of the mode, see Fig.1.
We can directly apply a bremsstrahlung approach, to the dipole evolution given from the Landau-Vlasov transport [@BaranPRL87], to estimate the “prompt” photon emission probability ($E_{\gamma}= \hbar \omega$): $$\frac{dP}{dE_{\gamma}}= \frac{2 e^2}{3\pi \hbar c^3 E_{\gamma}}
|D''(\omega)|^{2} \label{brems},$$ where $D''(\omega)$ is the Fourier transform of the dipole acceleration $D''(t)$. We remark that in this way it is possible to evaluate, in [*absolute*]{} values, the corresponding pre-equilibrium photon emission.
We must add a couple of comments of interest for the experimental selection of the Dynamical Dipole: i) The centroid is always shifted to lower energies (large deformation of the dinucleus); ii) A clear angular anisotropy should be present since the prompt mode has a definite axis of oscillation (on the reaction plane) at variance with the statistical $GDR$. In a recent experiment the prompt dipole radiation has been investigated with a $4 \pi$ gamma detector. A strong dipole-like photon angular distribution $(\theta_\gamma)=W_0[1+a_2P_2(cos \theta_\gamma)]$, $\theta_\gamma$ being the angle between the emitted photon and the beam axis, has been observed, with the $a_2$ parameter close to $-1$, see [@medea].
At higher beam energies we expect a decrease of the direct dipole radiation for two main reasons both due to the increasing importance of hard NN collisions: i) a larger fast nucleon emission that will equilibrate the isospin during the dipole oscillation; ii) a larger damping of the collective mode due to $np$ collisions.
The use of unstable neutron rich projectiles would largely increase the effect, due to the possibility of larger entrance channel asymmetries [@ditoro_kaz07]. In order to suggest proposals for the new $RIB$ facility $Spiral~2$, [@lewrio] we have studied fusion events in the reaction $^{132}Sn+^{58}Ni$ at $10AMeV$, [@ditoro_kaz07; @spiral2]. We espect a $Monster$ Dynamical Dipole, the initial dipole moment $D(t=0)$ being of the order of 50fm, about two times the largest values probed so far, allowing a detailed study of the symmetry potential, below saturation, responsible of the restoring force of the dipole oscillation and even of the damping, via the fast neutron emission.
In Figure 1 we report some global informations concerning the dipole mode in entrance channel.
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In the Left-Upper panel we have the time evolution of the dipole moment $D(t)$ for the “132” system at $b=4fm$. We notice the large amplitude of the first oscillation but also the delayed dynamics for the Asystiff EOS related to a weaker isovector restoring force. The phase space correlation (spiraling) between $D(t)$ and $DK(t)$, is reported in Fig.1 (Left-Lower). It nicely points out a collective behavior which initiates very early, with a dipole moment still close to the touching configuration value reported above. This can be explained by the fast formation of a well developed neck mean field which sustains the collective dipole oscillation in the dinuclear configuration. The role of a large charge asymmetry between the two colliding nuclei can be seen from Fig.1 (Right Panels), where we show the analogous dipole phase space trajectories for the stable $^{124}Sn+^{58}Ni$ system at the same value of impact parameter and energy. A clear reduction of the collective behavior is evidenced.
![Left Panel, Exotic “132” system. Power spectra of the dipole acceleration at $b=4$fm (in $c^2$ units). Right Panel: Corresponding results for the stable “124” system. Solid lines correspond to Asysoft EoS, the dashed to Asystiff EoS.[]{data-label="yield1"}](erice2.eps)
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In Fig. 2(Left Panel) we report the power spectrum, $\mid D''(\omega) \mid^2$ in semicentral “132” reactions, for different $Iso-EoS$ choices. The gamma multiplicity is simply related to it, see Eq.(\[brems\]). The corresponding results for the stable “124” system are drawn in the Right Panel. As expected from the larger initial charge asymmetry, we clearly see an increase of the Prompt Dipole Emission for the exotic n-rich beam. Such entrance channel effect will be enhanced, allowing a better observation of the Iso-EoS dependence. In fact from Fig.2 we see also other isospin effects.
A detailed analysis can be performed just using a simple damped oscillator model for the dipole moment $D(t)=D(t_0) e^{i (\omega_0+i/\tau) t}$, where $D(t_0)$ is the value at the onset of the collective dinuclear response, $\omega_0$ the frequency and $\tau$ the damping rate.The power spectrum of the dipole acceleration is given by
$$\mid D''(\omega)\mid ^2 =\frac{(\omega_0^2+{1/\tau}^2)^2 {D(t_0)}^2}
{(\omega-\omega_0)^2 + {1/\tau}^2}
\label{power}$$
which from Eq.(\[brems\]) leads to a total yield proportional to $\omega_0 \tau (\omega_0^2+{1/\tau}^2){D(t_0)}^2 \simeq
\omega_0^3 \tau {D(t_0)}^2$ since $\omega_0 \tau >1$. We remind that in the Asystiff case we have a weaker restoring force for the dynamical dipole in the dilute “neck” region, where the symmetry energy is smaller [@baranPR]. This is reflected in lower values of the centroids as well as in reduced total yields, as shown in Fig.2. The sensitivity of $\omega_0$ to the stiffness of the symmetry energy will be amplified by the increase of $D(t_0)$ when we use exotic, more asymmetric beams.
The prompt dipole radiation angular distribution is the result of the interplay between the collective oscillation life-time and the dinuclear rotation. In this sense we expect also a sensitivity to the $Iso-Eos$ of the anusotropy, in particular for high spin event selections, [@dipang08].
In the Asysoft choice we expect also larger widths of the “resonance” due to the larger fast neutron emission. We note the opposite effect of the Asy-stiffness on neutron vs proton emissions. The latter point is important even for the possibility of an independent test just measuring the $N/Z$ of the pre-equilibrium nucleon emission, [@pfabe_iwm].
Isospin Equilibration at the Fermi Energies
-------------------------------------------
In this energy range the doorway state mechanism of the Dynamical Dipole will disappear and so we can study a direct isospin transport in binary events. This can be discussed in a compact way by means of the chemical potentials for protons and neutrons as a function of density $\rho$ and isospin $I$ [@isotr05]. The $p/n$ currents can be expressed as $${\bf j}_{p/n} = D^{\rho}_{p/n}{\bf \nabla} \rho - D^{I}_{p/n}{\bf \nabla} I$$ with $D^{\rho}_{p/n}$ the drift, and $D^{I}_{p/n}$ the diffusion coefficients for transport, which are given explicitely in ref. [@isotr05]. Of interest here are the differences of currents between protons and neutrons which have a simple relation to the density dependence of the symmetry energy $$\begin{aligned}
D^{\rho}_{n} - D^{\rho}_{p} & \propto & 4 I \frac{\partial E_{sym}}
{\partial \rho} \,
, \nonumber\\
D^{I}_{n} - D^{I}_{p} & \propto & 4 \rho E_{sym} \, .
\label{trcoeff}\end{aligned}$$ Thus the isospin transport due to density gradients, i.e. isospin migration, depends on the slope of the symmetry energy, or the symmetry pressure, while the transport due to isospin concentration gradients, i.e. isospin diffusion, depends on the absolute value of the symmetry energy. We can discuss the asymmetries of the various parts of the reaction system (gas, PLF/TLF’s , and in the case of ternary events, IMF’s ). In particular, we study the so-called Imbalance Ratio [@imbalance], which is defined as $$R^x_{P,T} = \frac{2(x^M-x^{eq})}{(x^H-x^L)}~,
\label{imb_rat}$$ with $x^{eq}=\frac{1}{2}(x^H+x^L)$. Here, $x$ is an isospin sensitive quantity that has to be investigated with respect to equilibration. In this work we consider primarily the asymmetry $\beta=(N-Z)/(N+Z)$, but also other quantities, such as isoscaling coefficients, ratios of production of light fragments, etc, can be of interest [@WCI_betty]. The indices $H$ and $L$ refer to the symmetric reaction between the heavy ($n$-rich) and the light ($n$-poor) systems, while $M$ refers to the mixed reaction. $P,T$ denote the rapidity region, in which this quantity is measured, in particular the PLF and TLF rapidity regions. Clearly, this ratio is $\pm1$ in the projectile and target regions, respectively, for complete transparency, and oppositely for complete rebound, while it is zero for complete equilibration.
![ Left Panel.Imbalance ratios for $Sn + Sn$ collisions for incident energies of 50 (left) and 35 $AMeV$ (right) as a function of the impact parameter. Signatures of the curves: iso-EoS stiff (solid lines), soft (dashed lines); MD interaction (circles), MI interaction (squares); projectile rapidity ( full symbols, upper curves ), target rapidity ( open symbols, lower curves ). Right Panel. Imbalance ratios as a function of relative energy loss for both beam energies. Upper: Separately for stiff (solid) and soft (dashed) iso-EoS, and for MD (circles and squares) and MI (diamonds and triangles) interactions, in the projectile region (full symbols) and the target region (open symbols). Lower: Quadratic fit to all points for the stiff (solid), resp. soft (dashed) iso-EoS.[]{data-label="imb_eloss"}](erice3a.eps "fig:"){width="7.0cm"} ![ Left Panel.Imbalance ratios for $Sn + Sn$ collisions for incident energies of 50 (left) and 35 $AMeV$ (right) as a function of the impact parameter. Signatures of the curves: iso-EoS stiff (solid lines), soft (dashed lines); MD interaction (circles), MI interaction (squares); projectile rapidity ( full symbols, upper curves ), target rapidity ( open symbols, lower curves ). Right Panel. Imbalance ratios as a function of relative energy loss for both beam energies. Upper: Separately for stiff (solid) and soft (dashed) iso-EoS, and for MD (circles and squares) and MI (diamonds and triangles) interactions, in the projectile region (full symbols) and the target region (open symbols). Lower: Quadratic fit to all points for the stiff (solid), resp. soft (dashed) iso-EoS.[]{data-label="imb_eloss"}](erice3b.eps "fig:"){width="7.5cm"}
In a simple model we can show that the imbalance ratio mainly depends on two quantities: the strength of the symmetry energy and the interaction time between the two reaction partners. Let us take, for instance, the asymmetry $\beta$ of the PLF (or TLF) as the quantity $x$. As a first order approximation, in the mixed reaction this quantity relaxes towards its complete equilibration value, $\beta_{eq} = (\beta_H + \beta_L)/2$ as $$\label{dif_new}
\beta^M_{P,T} = \beta^{eq} + (\beta^{H,L} - \beta^{eq})~e^{-t/\tau},$$ where $t$ is the time elapsed while the reaction partners are interacting (interaction time) and the damping $\tau$ is mainly connected to the strength of the symmetry energy. Inserting this expression into Eq.(\[imb\_rat\]), one obtains $ R^{\beta}_{P,T} = \pm e^{-t/\tau}$ for the PLF and TLF regions, respectively. Hence the imbalance ratio can be considered as a good observable to trace back the strength of the symmetry energy from the reaction dynamics provided a suitable selection of the interaction time is performed. The centrality dependence of the Imbalance Ratio, for (Sn,Sn) collisions, has been investigated in experiments [@tsang92] as well as in theory [@isotr05; @BALi]. We propose here a new analysis which appears experimentally more selective. The interaction time certainly influences the amount of isospin equilibration, see Eq.(6) and refs. [@isotr05; @isotr07]. Longer interaction times should be correlated to a larger dissipation. It is then natural to look at the correlation between the imbalance ratio and the total kinetic energy loss. In this way we can also better disentangle dynamical effects of the isoscalar and isovector part of the EoS, see [@isotr07].
It is seen in Fig.3 that the curves for the [*asy-soft*]{} EoS (dashed) are generally lower in the projectile region (and oppositely for the target region), i.e. show more equilibration, than those for the [*asy-stiff*]{} EoS. In order to emphasize this trend we have, in the lower panel of the figure, collected together all the values for the stiff (circles) and the soft (squares) iso-EoS, and fitted them by a quadratic curve. It is seen that this fit gives a good representation of the trend of the results.
The difference between the curves for the stiff and soft iso-EOS in the lower panel then isolates the influence of the iso-EoS from kinematical effects associated with the interaction time. It is seen, that there is a systematic effect of the symmetry energy of the order of about 20 percent, which should be measurable. The correlation suggested in Fig.3 should represent a general feature of isospin diffusion, and it would be of great interest to verify experimentally.
![Left Panel.The N/Z of the liquid (left) and of the gas (right) phase is displayed as a function of the system initial N/Z. Full lines and symbols refer to the asystiff parameterization. Dashed lines and open symbols are for asysoft. Right Panel. The fragment N/Z (see text) as a function of the kinetic energy. Left: Asystiff; Right: Asysoft. []{data-label="iso_kin"}](erice4a.eps "fig:"){width="7.0cm"} 0.5cm ![Left Panel.The N/Z of the liquid (left) and of the gas (right) phase is displayed as a function of the system initial N/Z. Full lines and symbols refer to the asystiff parameterization. Dashed lines and open symbols are for asysoft. Right Panel. The fragment N/Z (see text) as a function of the kinetic energy. Left: Asystiff; Right: Asysoft. []{data-label="iso_kin"}](erice4b.eps "fig:"){width="7.0cm"}
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Isospin Distillation with Radial Flow
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In central collisions at 30-50 MeV/A, where the full disassembly of the system into many fragments is observed, one can study specifically properties of liquid-gas phase transitions occurring in asymmetric matter [@mue95; @bao197; @BaranNPA632; @chomazPR; @baranPR]. For instance, in neutron-rich matter, phase co-existence leads to a different asymmetry in the liquid and gaseous phase: fragments (liquid) appear more symmetric with respect to the initial matter, while light particles (gas) are more neutron-rich. The amplitude of this effect depends on specific properties of the isovector part of the nuclear interaction, namely on the value and the derivative of the symmetry energy at low density. This investigation is interesting in a more general context: In heavy ion collisions the dilute phase appears during the expansion of the interacting matter. Thus we study effects of the coupling of expansion, fragmentation and distillation in a two-component (neutron-proton) system [@col07] . We focus on central collisions, $b = 2~fm$, considering symmetric reactions between systems having three different initial asymmetry: $^{112}Sn + ^{112}Sn,^{124}Sn + ^{124}Sn,
^{132}Sn + ^{132}Sn,$ with $(N/Z)_{in}$ = 1.24,1.48,1.64, respectively. The considered beam energy is 50 MeV/A. 1200 events have been run for each reaction and for each of the two symmetry energies adopted (asy-soft and asystiff, see before) [@col07]. The average N/Z of emitted nucleons (gas phase) and Intermediate Mass Fragments ($IMF$) is presented in Fig.4 (Left Panel) as a function of the initial $(N/Z)_{in}$ of the three colliding systems. One observes a clear Isospin-Distillation effect, i.e. the gas phase (right) more neutron-rich than the IMF’s (left). This is particular evident in the Asysoft case due to the larger value of the symmetry energy at low density [@baranPR].
In central collisions, after the initial collisional shock, the system expands and breaks up into many pieces, due to the development of volume (spinodal) and surface instabilities. The formation of a bubble-like configuration is observed, where the initial fragments are located [@colonna_npa742].
Fragmentation originates from the break-up of a composite source that expands with a given velocity field. Since neutrons and protons experience different forces, one may expect a different radial flow for the two species. In this case, the N/Z composition of the source would not be uniform, but would depend on the radial distance from the center or mass or, equivalently, on the local velocity. This trend should then be reflected in a clear correlation between isospin content and kinetic energy of the formed $IMF$’s, [@col07].
This observable is plotted in Fig.4 (Right Panel) for the three reactions. The behaviour observed is rather sensitive to the Iso-EoS. For the proton-rich system, the N/Z decreases with the fragment kinetic energy, expecially in the Asystiff case (left), where the symmetry energy is relatively small at low density. In this case, the Coulomb repulsion pushes the protons towards the surface of the system. Hence, more symmetric fragments acquire larger velocity. The decreasing trend is less pronounced in the Asysoft case (right) because Coulomb effects on protons are counterbalanced by the larger attraction of the symmetry potential. In systems with higher initial asymmetry, the decreasing trend is inversed, due to the larger neutron repulsion in neutron-rich systems.
In conclusion, this analysis reveals the existence of significant, EoS-dependent correlations between the $N/Z$ and the kinetic energy of IMF’s produced in central collisions.
-0.5cm ![Correlation between $N/Z$ of $IMF$ and $alignement$ in ternary events of the $^{124}Sn+^{64}Ni$ reaction at $35~AMeV$. $Left~Panel$. Exp. results: points correspond to fast formed $IMF$s (Viola-violation selection); histogram for all $IMF$s at mid-rapidity (including statistical emissions). $Right~Panel$. Simulation results: squares, asysoft; circles, asystiff.[]{data-label="nzphi"}](erice5a.eps "fig:")
(0,0) (6.,140.)
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Isospin Dynamics in Neck Fragmentation at Fermi Energies
========================================================
It is now quite well established that the largest part of the reaction cross section for dissipative collisions at Fermi energies goes through the [*Neck Fragmentation*]{} channel, with $IMF$s directly produced in the interacting zone in semiperipheral collisions on very short time scales [@colonnaNPA589; @wcineck]. We can predict interesting isospin transport effects for this new fragmentation mechanism since clusters are formed still in a dilute asymmetric matter but always in contact with the regions of the projectile-like and target-like remnants almost at normal densities. As discussed in Sect.2.2 in presence of density gradients the isospin transport is mainly ruled by drift coefficients and so we expect a larger neutron flow to the neck clusters for a stiffer symmetry energy around saturation, [@baranPR; @baranPRC72]. The isospin dynamics can be directly extracted from correlations between $N/Z$, $alignement$ and emission times of the $IMF$s. The alignment between $PLF-IMF$ and $PLF-TLF$ directions represents a very convincing evidence of the dynamical origin of the mid-rapidity fragments produced on short time scales [@baranNPA730]. The form of the $\Phi_{plane}$ distributions (centroid and width) can give a direct information on the fragmentation mechanism [@dynfiss05]. Recent calculations confirm that the light fragments are emitted first, a general feature expected for that rupture mechanism [@liontiPLB625]. The same conclusion can be derived from [*direct*]{} emission time measurements based on deviations from Viola systematics observed in event-by-event velocity correlations between $IMF$s and the $PLF/TLF$ residues [@baranNPA730; @dynfiss05; @velcorr04]. We can figure out a continuous transition from fast produced fragments via neck instabilities to clusters formed in a dynamical fission of the projectile(target) residues up to the evaporated ones (statistical fission). Along this line it would be even possible to disentangle the effects of volume and shape instabilities. A neutron enrichment of the overlap (“neck”) region is expected, due to the neutron migration from higher (spectator) to lower (neck) density regions, directly related to the slope of the symmetry energy [@liontiPLB625].
A very nice new analysis has been performed on the $Sn+Ni$ data at $35~AMeV$ by the Chimera Collab.,[@defilposter], see Fig.\[nzphi\] left panel. A strong correlation between neutron enrichement and alignement (when the short emission time selection is enforced) is seen, that can be reproduced only with a stiff behavior of the symmetry energy, Fig.3 right panel (for primary fragments) [@baran08]. This represents a clear evidence in favor of a relatively large slope (symmetry pressure) around saturation. We note a recent confirmation from structure data, i.e. from monopole resonances in Sn-isotopes [@garg_prl07].
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Isospin Effects at High Baryon Density: Effective Mass Splitting and Collective Flows
=====================================================================================
![132Sn+124Sn at 400AMeV, central coll.Isospin content of nucleon and light ion emissions vs $p_t$ (upper) and kinetic energy (lower). Upper Panel: Asysoft; Lower Panel: Asystiff.[]{data-label="fastratios"}](erice6a.eps "fig:"){width="8.5cm"} 0.2cm ![132Sn+124Sn at 400AMeV, central coll.Isospin content of nucleon and light ion emissions vs $p_t$ (upper) and kinetic energy (lower). Upper Panel: Asysoft; Lower Panel: Asystiff.[]{data-label="fastratios"}](erice6b.eps "fig:"){width="8.5cm"}
The problem of Momentum Dependence in the Isovector channel ($Iso-MD$) is still very controversial and it would be extremely important to get more definite experimental information, see the recent refs. [@BaoNPA735; @ditoroAIP05; @rizzoPRC72]. Exotic Beams at intermediate energies are of interest in order to have high momentum particles and to test regions of high baryon (isoscalar) and isospin (isovector) density during the reaction dynamics. Our transport code has been implemented with a $BGBD-like$ [@GalePRC41; @BombaciNPA583] mean field with a different $(n,p)$ momentum dependence, see [@ditoroAIP05; @rizzoPRC72]. This will allow to follow the dynamical effect of opposite n/p effective mass splitting while keeping the same density dependence of the symmetry energy [@isotr07].
We present here some preliminary results for reactions induced by $^{132}Sn$ beams on $^{124}Sn$ targets at $400AMeV$ [@vale08]. For central collisions in the interacting zone we can reach baryon densities about $1.7-1.8 \rho_0$ in a transient time of the order of 15-20 fm/c. The system is quickly expanding and the Freeze-Out time is around 50fm/c.
In Fig.6 we present the $(n/p)$ and $^3H/^3He$ yield ratios at freeze-out, for two choices of Asy-stiffness and mass splitting, vs. transverse momentum (upper curves) and kinetic energy (lower curves). In this way we can separate particle emissions from sources at different densities. We note two interesting features: i) the curves are crossing at $p_t \simeq p_{projectile}= 2.13 fm^{-1}$; ii) the effect is not much dependent on the stiffness of the symmetry term. The crossing nicely corresponds to a source at baryon density $\rho \simeq 1.6 \rho_0$, [@ditoroAIP05; @rizzoPRC72]. These data seem to be suitable to disentangle $Iso-MD$ effects.
Collective flows are very good candidates since they are expected to be very sensitive to the momentum dependence of the mean field, see [@DanielNPA673; @baranPR]. The transverse flow, $V_1(y,p_t)=\langle \frac{p_x}{p_t} \rangle$, provides information on the anisotropy of nucleon emission on the reaction plane. Very important for the reaction dynamics is the elliptic flow, $V_2(y,p_t)=\langle \frac{p_x^2-p_y^2}{p_t^2} \rangle$. The sign of $V_2$ indicates the azimuthal anisotropy of emission: on the reaction plane ($V_2>0$) or out-of-plane ($squeeze-out,~V_2<0$) [@DanielNPA673]. We have then tested the $Iso-MD$ of the fields just evaluating the $Difference$ of neutron/proton transverse and elliptic flows $
V^{(n-p)}_{1,2} (y,p_t) \equiv V^n_{1,2}(y,p_t) - V^p_{1,2}(y,p_t)
$ at various rapidities and transverse momenta in semicentral ($b/b_{max}=0.5$) $^{197}Au+^{197}Au$ collisons at $400AMeV$. For the nucleon elliptic flows the mass splitting effect is evident at all rapidities, and nicely increasing at larger rapidities and transverse momenta, with more neutron flow when $m_n^*<m_p^*$.
![ Transverse momentum dependence of the difference between proton and neutron $V_2$ flows, at mid-rapidity, in a semi-central reaction Au+Au at 400AMeV.[]{data-label="v2dif"}](erice7.eps){width="7.0cm"}
-0.3cm
From Fig.7 we clearly see how at mid-rapidity the mass splitting effects are more evident for higher tranverse momentum selections, i.e. for high density sources. In particular the elliptic flow difference becomes negative when $m_n^*<m_p^*$, revealing a faster neutron emission and so more neutron squeeze out (more spectator shadowing). In correspondance the proton flow is more negative (more proton squeeeze out) when $m_n^*>m_p^*$. It is however difficult to draw definite conclusions only from proton data. The measurement of n/p flow differences appears essential. Due to the difficulties in measuring neutrons, our suggestion is to measure the difference between light isobar flows, like $^3H$ vs. $^3He$ and so on. We expect to still see effective mass splitting effects.
Isospin Effects on Meson Production in Relativistic Heavy Ion Collisions
========================================================================
The phenomenology of isospin effects on heavy ion reactions at intermediate energies (few $AGeV$ range) is extremely rich and can allow a “direct” study of the covariant structure of the isovector interaction in a high density hadron medium. We work within a relativistic transport frame, beyond the cascade picture, consistently derived from effective Lagrangians, where isospin effects are accounted for in the mean field and collision terms. We show that rather sensitive observables are provided by the pion/kaon production ($\pi^-/\pi^+$, $K^0/K^+$ yields). Relevant non-equilibrium effects are stressed.
An effective Lagrangian approach to the hadron interacting system is extended to the isospin degree of freedom: within the same frame equilibrium properties ($EoS$, [@qhd]) and transport dynamics can be consistently derived. Within a covariant picture of the nuclear mean field, for the description of the symmetry energy at saturation ($a_{4}$ parameter of the Weizsäecker mass formula) (a) only the Lorentz vector $\rho$ mesonic field, and (b) both, the vector $\rho$ (repulsive) and scalar $\delta$ (attractive) effective fields are included. In the latter case a rather intuitive form of the Symmetry Energy can be obtained [@liubo02; @theo04] $$E_{sym} = \frac{1}{6} \frac{k_{F}^{2}}{E_{F}} +
\frac{1}{2}
\left[ f_{\rho} - f_{\delta}\left( \frac{m^{*}}{E_{F}} \right)^{2}
\right] \rho_{B}.
\label{esym3}
\quad .$$ The competition between scalar and vector fields leads to a stiffer symmetry term at high density [@liubo02; @baranPR].
We present here observable effects in the dynamics of heavy ion collisions. We focus our attention on the isospin content of meson production. The starting point is a simple phenomenological version of the Non-Linear (with respect to the iso-scalar, Lorentz scalar $\sigma$ field) effective nucleon-boson field theory, the Quantum-Hadro-Dynamics [@qhd]. According to this picture the presence of the hadronic medium leads to effective masses and momenta $M^{*}=M+\Sigma_{s}$, $k^{*\mu}=k^{\mu}-\Sigma^{\mu}$, with $\Sigma_{s},~\Sigma^{\mu}$ scalar and vector self-energies. For asymmetric matter the self-energies are different for protons and neutrons, depending on the isovector meson contributions. We will call the corresponding models as $NL\rho$ and $NL\rho\delta$, respectively, and just $NL$ the case without isovector interactions. For the more general $NL\rho\delta$ case the self-energies of protons and neutrons read: $$\begin{aligned}
\Sigma_{s}(p,n) = - f_{\sigma}\sigma(\rho_{s}) \pm f_{\delta}\rho_{s3},
\nonumber \\
\Sigma^{\mu}(p,n) = f_{\omega}j^{\mu} \mp f_{\rho}j^{\mu}_{3},
\label{selfen}\end{aligned}$$ (upper signs for neutrons), where $\rho_{s}=\rho_{sp}+\rho_{sn},~
j^{\alpha}=j^{\alpha}_{p}+j^{\alpha}_{n},\rho_{s3}=\rho_{sp}-\rho_{sn},
~j^{\alpha}_{3}=j^{\alpha}_{p}-j^{\alpha}_{n}$ are the total and isospin scalar densities and currents and $f_{\sigma,\omega,\rho,\delta}$ are the coupling constants of the various mesonic fields. $\sigma(\rho_{s})$ is the solution of the non linear equation for the $\sigma$ field [@liubo02; @baranPR]. From the form of the scalar self-energies we note that in n-rich environment the neutron effective masses are definitely below the proton ones.
For the description of heavy ion collisions we solve the covariant transport equation of the Boltzmann type within the Relativistic Landau Vlasov ($RLV$) method, using phase-space Gaussian test particles [@FuchsNPA589], and applying a Monte-Carlo procedure for the hard hadron collisions. The collision term includes elastic and inelastic processes involving the production/absorption of the $\Delta(1232 MeV)$ and $N^{*}(1440
MeV)$ resonances as well as their decays into pion channels, [@ferini05].
Kaon production has been proven to be a reliable observable for the high density $EoS$ in the isoscalar sector [@FuchsPPNP56; @HartPRL96]. Here we show that the $K^{0,+}$ production (in particular the $K^0/K^+$ yield ratio) can be also used to probe the isovector part of the $EoS$, [@ferini06; @Pra07].
Using our $RMF$ transport approach we analyze pion and kaon production in central $^{197}Au+^{197}Au$ collisions in the $0.8-1.8~AGeV$ beam energy range, comparing models giving the same “soft” $EoS$ for symmetric matter and with different effective field choices for $E_{sym}$. Fig. \[kaon1\] reports the temporal evolution of $\Delta^{\pm,0,++}$ resonances, pions ($\pi^{\pm,0}$) and kaons ($K^{+,0}$) for central Au+Au collisions at $1AGeV$.
 -0.3cm
-1.0cm \[kaon1\]
It is clear that, while the pion yield freezes out at times of the order of $50 fm/c$, i.e. at the final stage of the reaction (and at low densities), kaon production occurs within the very early (compression) stage, and the yield saturates at around $20 fm/c$. From Fig. \[kaon1\] we see that the pion results are weakly dependent on the isospin part of the nuclear mean field. However, a slight increase (decrease) in the $\pi^{-}$ ($\pi^{+}$) multiplicity is observed when going from the $NL$ to the $NL\rho$ and then to the $NL\rho\delta$ model, i.e. increasing the vector contribution $f_\rho$ in the isovector channel. This trend is more pronounced for kaons, see the right panel, due to the high density selection of the source and the proximity to the production threshold. Consistently, as shown in the insert, larger effects are expected for early emitted kaons, reflecting the early $N/Z$ of the system.
When isovector fields are included the symmetry potential energy in neutron-rich matter is repulsive for neutrons and attractive for protons. In a $HIC$ this leads to a fast, pre-equilibrium, emission of neutrons. Such a $mean~field$ mechanism, often referred to as isospin fractionation [@baranPR], is responsible for a reduction of the neutron to proton ratio during the high density phase, with direct consequences on particle production in inelastic $NN$ collisions. $Threshold$ effects represent a more subtle point. The energy conservation in a hadron collision in general has to be formulated in terms of the canonical momenta, i.e. for a reaction $1+2 \rightarrow 3+4$ as $
s_{in} = (k_1^\mu + k_2^\mu)^2 = (k_3^\mu + k_4^\mu)^2 = s_{out}.
$ Since hadrons are propagating with effective (kinetic) momenta and masses, an equivalent relation should be formulated starting from the effective in-medium quantities $k^{*\mu}=k^\mu-\Sigma^\mu$ and $m^*=m+\Sigma_s$, where $\Sigma_s$ and $\Sigma^\mu$ are the scalar and vector self-energies, Eqs.(\[selfen\]). The self-energy contributions will influence the particle production at the level of thresholds as well as of the phase space available in the final channel. In fact the [*threshold*]{} effect is dominant and consequently the results are nicely sensitive to the covariant structure of the isovector fields. At each beam energy we see an increase of the $\pi^-/\pi^+$ and $K^{0}/K^{+}$ yield ratios with the models $NL \rightarrow DDF \rightarrow NL\rho \rightarrow NL\rho\delta$. The effect is larger for the $K^{0}/K^{+}$ compared to the $\pi^-/\pi^+$ ratio. This is due to the subthreshold production and to the fact that the isospin effect enters twice in the two-step production of kaons, see [@ferini06]. Interestingly the Iso-$EoS$ effect for pions is increasing at lower energies, when approaching the production threshold.
We have to note that in a previous study of kaon production in excited nuclear matter the dependence of the $K^{0}/K^{+}$ yield ratio on the effective isovector interaction appears much larger (see Fig.8 of ref.[@ferini05]). The point is that in the non-equilibrium case of a heavy ion collision the asymmetry of the source where kaons are produced is in fact reduced by the $n \rightarrow p$ “transformation”, due to the favored $nn \rightarrow p\Delta^-$ processes. This effect is almost absent at equilibrium due to the inverse transitions, see Fig.3 of ref.[@ferini05]. Moreover in infinite nuclear matter even the fast neutron emission is not present. This result clearly shows that chemical equilibrium models can lead to uncorrect results when used for transient states of an $open$ system.
On the Transition to a Mixed Hadron-Quark Phase at High Baryon and Isospin Density
==================================================================================
The possibility of the transition to a mixed hadron-quark phase, at high baryon and isospin density, is finally suggested. Some signatures could come from an expected “neutron trapping” effect.
![$^{238}U+^{238}U$, $1~AGeV$, semicentral. Correlation between density, temperature (black values), momentum thermalization (3-D plots), inside a cubic cell, 2.5 $fm$ wide, in the center of mass of the system.[]{data-label="figUU"}](erice9.ps)
In order to check the possibility of observing some precursor signals of a new physics even in collisions of stable nuclei at intermediate energies we have performed some event simulations for the collision of very heavy, neutron-rich, elements. We have chosen the reaction $^{238}U+^{238}U$ (average proton fraction $Z/A=0.39$) at $1~AGeV$ and semicentral impact parameter $b=7~fm$ just to increase the neutron excess in the interacting region. In Fig. \[figUU\] we report the evolution of momentum distribution and baryon density in a space cell located in the c.m. of the system. We see that after about $10~fm/c$ a local equilibration is achieved. We have a unique Fermi distribution and from a simple fit we can evaluate the local temperature (black numbers in MeV). We note that a rather exotic nuclear matter is formed in a transient time of the order of $10~fm/c$, with baryon density around $3-4\rho_0$, temperature $50-60~MeV$, energy density $500~MeV~fm^{-3}$ and proton fraction between $0.35$ and $0.40$, likely inside the estimated mixed phase region.
In fact we can study the isospin dependence of the transition densities [@ditoro_dec]. The structure of the mixed phase is obtained by imposing the Gibbs conditions [@Landaustat] for chemical potentials and pressure and by requiring the conservation of the total baryon and isospin densities $$\begin{aligned}
\label{gibbs}
&&\mu_B^{(H)} = \mu_B^{(Q)}\, ,~~
\mu_3^{(H)} = \mu_3^{(Q)} \, , \nonumber \\
&&P^{(H)}(T,\mu_{B,3}^{(H)}) = P^{(Q)} (T,\mu_{B,3}^{(Q)})\, ,\nonumber \\
&&\rho_B=(1-\chi)\rho_B^H+\chi\rho_B^Q \, , \nonumber \\
&&\rho_3=(1-\chi)\rho_3^H+\chi\rho_3^Q\, , \end{aligned}$$ where $\chi$ is the fraction of quark matter in the mixed phase.
-0.3cm
-1.0cm \[rhodelta\]
1.0cm
In this way we get the $binodal$ surface which gives the phase coexistence region in the $(T,\rho_B,\rho_3)$ space. For a fixed value of the conserved charge $\rho_3$ we will study the boundaries of the mixed phase region in the $(T,\rho_B)$ plane. In the hadronic phase the charge chemical potential is given by $
\mu_3 = 2 E_{sym}(\rho_B) \frac{\rho_3}{\rho_B}\, .
$ Thus, we expect critical densities rather sensitive to the isovector channel in the hadronic $EoS$.
In Fig. \[rhodelta\] we show the crossing density $\rho_{cr}$ separating nuclear matter from the quark-nucleon mixed phase, as a function of the proton fraction $Z/A$. We can see the effect of the $\delta$-coupling towards an $earlier$ crossing due to the larger symmetry repulsion at high baryon densities. In the same figure we report the paths in the $(\rho,Z/A)$ plane followed in the c.m. region during the collision of the n-rich $^{132}$Sn+$^{132}$Sn system, at different energies. At $300~AMeV$ we are just reaching the border of the mixed phase, and we are well inside it at $1~AGeV$.
We can expect a [*neutron trapping*]{} effect, supported by statistical fluctuations as well as by a symmetry energy difference in the two phases. In fact while in the hadron phase we have a large neutron potential repulsion (in particular in the $NL\rho\delta$ case), in the quark phase we only have the much smaller kinetic contribution. Observables related to such neutron “trapping” could be an inversion in the trend of the formation of neutron rich fragments and/or of the $\pi^-/\pi^+$, $K^0/K^+$ yield ratios for reaction products coming from high density regions, i.e. with large transverse momenta.
-0.5cm \[MuMd\]
0.5cm [*Isospin in Effective Partonic Models*]{}
From the above discussion it appears extremely important to include the Isospin degree of freedom in any effective approach to the QCD dynamics. This can be easily performed in a two-flavor $NJL$ model [@NJL] where the isospin asymmetry can be included in a flavor-mixing picture [@frank03] via a Gap Equation like $M_i=m_i-4G_1 \Phi_i-4G_2 \Phi_j$, $i \not= j,(u,d)$ , where the $\Phi_{u,d}=<\bar u u>,<\bar d d>$ are the two (negative) condensates and $m_{u,d}=m$ the (equal) current masses. Introducing explicitily a flavor mixing, i.e. the dependence of the constituent mass of a given flavor to both condensate, via $G_1=(1-\alpha) G_0, G_2= \alpha G_0$ we have the coupled equations $$\begin{aligned}
M_u=m - 4 G_0 \Phi_u - 4 \alpha G_0 (\Phi_u - \Phi_d), \nonumber \\
M_d=m - 4G_0 \Phi_u + 4 (1-\alpha) G_0 (\Phi_u - \Phi_d).
\label{mix}\end{aligned}$$ For $\alpha=1/2$ we have back the usual NJL ($M_u=M_d$), while small/large mixing is for $\alpha \Rightarrow 0$/$\alpha \Rightarrow 1$ respectively.
In neutron rich matter $\mid \Phi_d \mid$ decreases more rapidly due to the larger $\rho_d$ and so $(\Phi_u -\Phi_d)<0$. In the “realistic” small mixing case, see also [@frank03; @shao06], we will get a definite $M_u>M_d$ splitting at high baryon density (before the chiral restoration). This expectation is nicely confirmed by a full calculation [@plum_tesi] of the coupled gap equations with standard parameters (same as in ref.[@frank03]). The results are shown in Fig.\[MuMd\].
All that represents a more fundamental confirmation of the $m^*_p>m^*_n$ choice in the hadron phase, as suggested by the effective $QHD$ model with the isovector scalar $\delta$ coupling, see before and [@liubo02]. However this can represent just a very first step towards a more complete treatment of isovector interactions in effective partonic models, of large interest for the discussion of the phase transition at high densities. We can easily see that the mass splitting effect is not changing much the symmetry energy in the quark phase addressed before. However confinement is still missing in these mean field models. Stimulating new perspectives are open.
-1.0cm
Perspectives
============
We have shown that [*violent*]{} collisions of n-rich heavy ions from low to relativistic energies can bring new information on the isovector part of the in-medium interaction, qualitatively different from equilibrium $EoS$ properties. We have presented quantitative results in a wide range of beam energies. At low energies we see isospin effects on the dissipation in fusion and deep inelastic collisions, at Fermi and Intermediate energies the Iso-EoS sensitivity of the isospin dynamics in fragment reactions and in collective flows.
We have shown that meson production in n-rich heavy ions collisions at intermediate energies can bring new information on the isovector part of the in-medium interaction at high baryon densities. Important non-equilibrium effects for particle production are stressed. Finally the possibility of observing precursor signals of the phase transition to a mixed hadron-quark matter at high baryon density is suggested.
10 years after the first studies of Iso-EoS effects on reactions [@colonnaPRC57] the symmetry energy case is still open. The results presented in this “Isospin Journey” appear very promising for the possibility of exciting new data from dissipative collisions with radioactive beams.
1.0cm [**Aknowledgements**]{}
We warmly thank A.Drago and A.Lavagno for the collaboration on the mixed hadron-quark phase transition at high baryon and isospin density.
One of authors, V. B. thanks for warm hospitality at Laboratori Nazionali del Sud, INFN. This work was supported in part by the Romanian Ministery for Education and Research under the contracts PNII, No. ID-946/2007.
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|
---
author:
- |
Stefan Goedecker,\
Max-Planck Institute for Solid State Research,\
Stuttgart, Germany\
goedeck@prr.mpi-stuttgart.mpg.de
title: Linear Scaling Electronic Structure Methods
---
Methods exhibiting linear scaling with respect to the size of the system, so called O(N) methods, are an essential tool for the calculation of the electronic structure of large systems containing many atoms. They are based on algorithms which take advantage of the decay properties of the density matrix. In this article the physical decay properties of the density matrix will first be studied for both metals and insulators. Several strategies to construct O(N) algorithms will then be presented and critically examined. Some issues which are relevant only for self-consistent O(N) methods, such as the calculation of the Hartree potential and mixing issues, will also be discussed. Finally some typical applications of O(N) methods are briefly described.
Introduction
============
The exact quantum mechanical equations for many-electron systems are highly intricate. Any attempt to solve these equations analytically for real systems is doomed to fail. Numerical methods such as Configuration Interaction based methods (McWeeny 1989, Fulde 1995) or Quantum Monte Carlo methods (Hammond 1994, Nightingale 1998) can in principle solve these many-electron equations but because of the extremely high numerical effort required, their applicability is rather limited in practice.
The bulk of all practical applications is therefore done within various independent-electron approximations such as the Hartree-Fock method (A. Szabo and N. Ostlund 1982), Density Functional methods (R. Parr and W. Yang 1989) or Tight Binding methods (C. Goringe [*et al.,*]{} 1997a, Majewski and Vogl 1986). A comparison of the strength of different methods together with a selection of some interesting applications is given by Wimmer (1996). Even these approximate quantum mechanical equations are still fairly complicated and in general not solvable by analytical methods. Finding efficient algorithms to solve the many-electron problem numerically within any of these approximations is imperative for the applicability of quantum mechanics to physics as well as to chemistry and materials science. Due to efforts in the past satisfactory algorithms are now available and computational electronic structure methods are making very important contributions to our understanding of matter at the microscopic level. The 1998 Nobel prize for W. Kohn and J. Pople is a landmark sign of the importance of this approach. Since computational electronic structure methods are used over a very wide spectrum of applications is it hard to summarize their use. A hint of their versatility can be obtained by looking at the fraction of theoretical articles in several solid state and chemistry journals where computational electronic structure methods are used. As can be seen from Table 1 this fraction varies between 11 and 59 % , being 27 % in the two largest journals in solid state physics (PRB) and chemical physics (JCP).
Table 1: The importance of computational electronic structure methods as measured by the number of publications. Listed are the total number of publications, the number of theoretical publications and the number of publications using computational electronic structure methods in the period from January 1997 to September 1988. To determine whether a paper belongs into this last category, it has to contain either of the key words “electronic structure calculation", “ab initio calculation", “tight binding calculation" or “density functional calculation". I thank Dr Wanitschek for providing me with these data.
Journal total theoretical computational
-------------------------------------------- ------- ------------- ---------------
Physical Review B (Condensed Matter) 7924 4629 1263
Journal of chemical physics 4066 2764 754
Theochem 905 856 502
Chemical physics letters 2319 1209 490
International journal of quantum chemistry 609 582 271
Physical review letters 4558 2834 250
Journal of physics: Condensed matter 1774 927 220
Due to the constant increase in computer power and due to algorithmic improvements the importance of computational methods is growing further. Whereas computational methods nowadays mainly supplement experimentally obtained information, they are expected to increasingly supersede this information.
This article will concentrate on recently developed methods that allow us to calculate the total energy within various independent-electron methods for large systems. Practically all physical observables can be obtained from the total energy, for instance in the form of derivatives with respect to certain external parameters. The reason why large systems containing many atoms are accessible with these algorithms is their linear scaling with respect to the number of atoms. In principle linear scaling should also be obtainable for true many-electron methods. For the MP2 method such an algorithms has indeed recently been reported (Ayala and Scuseria, 1998).
Traditional electronic structure algorithms calculate eigenstates associated with discrete energy levels. The reason for this is probably historical since the prediction of these experimentally observed levels was the first big success of quantum mechanics. The disadvantage of this approach is that it leads to a diagonalization problem which has a cubic scaling in the computational effort. Direct diagonalization (Press [*et al.,*]{} 1986), which was the standard approach in the early days of the computational electronic structure era, has a cubic scaling with respect to the size of the Hamiltonian matrix, i.e. with respect to the number of basis functions $M_b$. Iterative diagonalization schemes (Saad 1996), preconditioned conjugate gradient minimizations (Teter [*et al.,*]{} 1989, Štich [*et al.,*]{} 1989, Payne, [*et al.,*]{} 1992 ) and the Car-Parrinello method (Car and Parrinello 1985) for molecular dynamics simulations were a big algorithmic progress because of their improved scaling behavior. Their scaling was not any more proportional to the cube of the the number of basis functions but grew only like $M_b \log(M_b)$ if plane waves were used as a basis set. Nevertheless these methods still have a cubic scaling with respect to the number of atoms $N_{at}$, which comes from the orthogonality requirement of the wavefunctions. The reason why this orthogonalization step scales cubically can easily be seen. As the system grows, each wavefunction extends over a larger volume and has therefore to be represented by a larger basis set resulting in a longer vector. At the same time there are more such wavefunctions and each wavefunction has to be orthogonalized to all the others. Thus there are 3 factors that grow linearly, resulting in the postulated cubic behavior. The computer time $T_{CPU}$ required to do the calculation is thus given by $$\label{cpu3}
T_{CPU} = c_3 N_{at}^3 \: ,$$ where $c_3$ is a prefactor. It has to be pointed out that Equation (\[cpu3\]) gives only the asymptotic scaling behavior. Within Density Functional and Hartree Fock calculations there are other terms with a lower scaling which dominate for system sizes of less than a few hundred atoms due to their large prefactor. In the case of plane wave type calculations the Fast Fourier transformations necessary for the application of the potential to the wavefunctions consume most of the computational time for small systems, in the case of calculations using Gaussian type orbitals (Hehre 1996) it is the calculation of the Hartree potential. This cubic scaling is a major bottleneck nowadays since in many problems of practical interest one has to do electronic structure calculations for systems containing many (a few hundred or more) atoms. Evidently, cubic scaling means that if one doubles the number of atoms in the systems the required computer time will increase by a factor of eight. By enlarging the system one therefore rapidly reaches the limits of the most powerful computers. So called O(N) or low complexity algorithms are therefore a logical next step of algorithmic progress since they exhibit linear scaling with respect to the number of atoms $$\label{cpu1}
T_{CPU} = c_1 N_{at} \: .$$ These methods offer thus the potential to calculate very large systems. The prefactors $c_1$ and $c_3$ depend on the approximation used for the many-electron problem. For a Density Functional calculation with a large basis set the prefactors are of course much larger than for a Tight Binding calculation, where the number of degrees of freedom per atom is much smaller. The prefactor $c_1$ depends also on what O(N) method is used, but in general the prefactor $c_1$ is always larger than $c_3$ assuming that the same independent-electron approximation is used both in the traditional and O(N) version. There is therefore a so called cross over point. For system sizes smaller than the cross over point the traditional cubic scaling algorithms are faster, for larger systems the O(N) methods win. Tight Binding calculations are an ideal test emvironment for O(N) algorithms. Because of their rather small memory and CPU requirements one can easily treat systems comprising of a very large number of atoms and venture into regions beyond the cross over point. Contrary to what one might naively think, the importance of O(N) algorithms will also increase as computers get faster. Whereas at present it is difficult to access the cross over region situated at some 100 atoms using the Density Functional framework, this will be easy with faster computers and O(N) algorithms will be the algorithms of choice.
Even though O(N) algorithms contain many aspects of mathematics and computer science they have nevertheless deep roots in physics. Obtaining linear scaling is not possible by purely mathematical tricks but it is based on the understanding of the concept of locality in quantum mechanics. Conversely, the need of constructing O(N) algorithms was also an incentive to investigate locality questions more deeply, and has thus lead to a better understanding of this very fundamental concept. An algorithmic description of electronic structure in local terms can give a justification of the well established concepts of bonds and lone electron pairs in empirical chemistry.
Since O(N) algorithms are based on a certain subdivision of a big system into smaller subsystems, techniques developed in this context might also be helpful in reaching another important goal for treating large systems, namely combining electronic structure methods of different accuracy such as empirical Tight Binding and Density Functional theory in a single system.
Locality in Quantum Mechanics {#general}
=============================
Locality in Quantum Mechanics means that the properties of a certain observation region comprising one or a few atoms are only weakly influenced by factors that are spatially far away form this observation region. This fundamental characteristic of insulators is well established within independent-electron theories (Heine 1980) and it can even be carried over into the many-electron framework (Kohn 1964).
Traditional chemistry is based on local concepts. Covalently bonded materials are described in terms of bonds and lone electron pairs. It is standard textbook knowledge that the properties of a bond are mainly determined by its immediate neighborhood. The decisive factors are what type of atoms and how many of them (the coordination number) are surrounding it. Second nearest neighbors and other more distant atoms have a very small influence. As an example let us look at the total energy of a hydrocarbon chain molecule $C_n H_{2n+2}$. In this case each $C H_2$ subunit is from an energetical point of view practically an independent unit. As one adds one $C H_2$ subunit, the energy increases by an amount which is nearly independent of the chain length. Already the insertion of a $C H_2$ subunit into the smallest chain $C_2 H_6$ gives an energy gain which agrees within $10^{-4}$ a.u. with the asymptotic value of the insertion energy for very long chains. This means that the electrons belonging to this inserted subunit already do not see any more the end of the chain for very short chain lengths. This example is a drastic illustration of a principle sometimes termed “nearsightedness” (Kohn 1996). In other insulating materials the influence of the neighboring atoms decays slower. An example is shown in Figure (\[sidecay\]), where the total energy per silicon atom is plotted as a function of the size of its crystalline environment.
( 8.,4.5) (-4.,-2.0)
Even in metallic systems, where the elementary bond concept is not any more valid, locality still exists. This is supported by the well known fact, that the total charge density in a metal is given with reasonable accuracy by the superposition of the atomic charge densities. Since atomic charge densities decay rapidly, this implies that the charge density at the midpoint of two neighboring atoms is mainly determined by the two closest atoms and very little by other more distant atoms. Another related example is given by V. Heine (Heine 1980) who points out, that the magnetic moment of an iron atom, which is embedded in an iron-aluminum alloy differs by less than 5 % from the value for pure iron if the atoms are locally surrounded by only eight aluminum atoms.
This locality is not at all reflected in standard electronic structure calculations which are based on eigenorbitals extending over the whole system, making both the interpretation of the results more difficult and requiring unnecessary computational effort. The simplistic bond concepts of empirical chemistry are certainly not adequate for electronic structure calculations aiming at high accuracy. Nevertheless one might hope to incorporate some more general locality concepts into electronic structure calculation to make them both more intuitive and efficient. In the following we will therefore carefully examine the range of interactions in quantum mechanical systems.
Self-consistent electronic structure methods require essentially two steps. The calculation of the potential from the electronic charge distribution and the determination of the wavefunction for a given potential. In non-self-consistent calculations such as Tight Binding calculations, the first step is not needed.
The calculation of the potential consists usually of two parts, the exchange correlation potential, and the Coulomb potential. The exchange correlation potential is a purely local expression in Density Functional Theory and can therefore be calculated with linear scaling. In the Hartree Fock scheme one might first think that the exchange part is non-local, but a more profound examination reveals (section \[coulomb\]) that it is local even in this case. The Coulomb potential on the other hand is very long range and needs proper treatment. A naive evaluation of the potential $U$ arising from a charge distribution $\rho$ by subdividing space into subvolumes $\Delta V$ and summing over these subvolumes, $$U( {\bf r}_i ) = \sum_j \frac{\rho({\bf r}_j)}{|{\bf r}_i-{\bf r}_j|} \Delta V \: ,$$ would result in a quadratic scaling since both indices $i$ and $j$ have too run over all grid points in the system. The Coulomb problem actually arises not only in the context of electronic structure calculations but also in classical calculations of coulombic and gravitational systems such as galaxies of stars. Much effort has therefore been invested in this computational problem and several algorithms are known which solve the problem with linear scaling. These methods will be described in section \[coulomb\].
The more interesting and more difficult part is to assess the role of locality for a given external potential. The appropriate quantity to study this property is the density matrix. The one-particle density matrix $F$ completely specifies our quantum mechanical system within the independent electron approximation and all quantities of interest can easily be calculated from it. The central quantities in any electronic structure calculation, the kinetic energy $E_{kin}$, the potential energy $E_{pot}$ and the electronic charge density $\rho$ are given by $$\begin{aligned}
E_{kin} & = & - \frac{1}{2} \int \left. \nabla^2_{\bf r} F({\bf r},{\bf r}') \right|_{{\bf r}={\bf r}'} d{\bf r}' \\
E_{pot} & = & \int F({\bf r}',{\bf r}') U({\bf r}') d{\bf r}' \\
\rho({\bf r}) & = & F({\bf r},{\bf r}) \:,\end{aligned}$$ where $U({\bf r}')$ is the potential. A related quantity which will frequently be used throughout the article is the band structure energy $E_{BS}$ defined as $$\label{ebsdef}
E_{BS} = E_{kin} + E_{pot}$$ and the grand potential $$\label{omdef}
\Omega = E_{BS} - \mu N_{el} \: ,$$ where $\mu$ is the chemical potential and $N_{el}$ the number of electrons. Subtracting $\mu N_{el}$ from $E_{BS}$ leaves $\Omega$ invariant under a constant potential offset. If one applies the shift ( $U({\bf r}) \rightarrow U({\bf r}) + const$) the potential energy will increase by $N_{el} \: const$. In order to conserve the total number of electrons, $\mu$ also has to be shifted ($\mu \rightarrow \mu + const$) and thus $\Omega$ remains constant.
Discretizing the Hamiltonian $H$ which is the sum of the kinetic and potential energy as well as $F$ with respect to a finite orthogonal basis $\phi_i({\bf r})$, $i=1, ... , M_b$ one obtains $$\begin{aligned}
H_{i,j} & = & \int \phi^*_i({\bf r})
\left( - \frac{1}{2} \nabla^2_{\bf r} + U({\bf r}) \right) \phi_j({\bf r}) d{\bf r} \\
F_{i,j} & = & \int \int \phi^*_i({\bf r}) F({\bf r},{\bf r}') \phi_j({\bf r}') d{\bf r} d{\bf r}'
\label{fijdef}\end{aligned}$$ and the expressions for the central quantities become $$\begin{aligned}
E_{BS} & = & Tr [F H] \label{ebstrace} \\
\Omega & = & Tr [F \: (H - \mu I) ] \label{omegatrace} \end{aligned}$$ $$\label{rhotrace}
\rho({\bf r}) = \sum_{i,j} F_{i,j} \: \phi_i({\bf r}) \phi_j({\bf r}) \: ,$$ where $Tr$ denotes the trace. It follows from Equation (\[rhotrace\]) that the total number of electrons $N_{el}$ in the system is given by $$\label{nel}
N_{el} = Tr [F] \: .$$
Evaluating the traces using the eigenfunctions $\Psi_n$ of the Hamiltonian one obtains immediately the well known expressions for $N_{el}$, $E_{BS}$, $\Omega$ and $\rho$ within the context of conventional calculations which are based on diagonalization. Denoting the eigenvalues associated with the eigenfunctions $\Psi_n$ by $\epsilon_n$ one obtains $$\begin{aligned}
N_{el} & = & \sum_n f(\epsilon_n) \label{diagnel} \\
E_{BS} & = & \sum_n f(\epsilon_n) \: \epsilon_n \label{diagebs} \\
\Omega & = & \sum_n (f(\epsilon_n)-\mu) \: \epsilon_n =
\sum_n f(\epsilon_n) \: \epsilon_n - \mu N_{el} \label{diaomega} \\
\rho({\bf r}) & = & \sum_n f(\epsilon_n) \: \Psi^*_n({\bf r}) \Psi_n({\bf r}) \label{diagrho} \: .\end{aligned}$$ The function $f$ is the the Fermi distribution $$f(\epsilon) = \frac{1}{1+\exp (\frac{\epsilon-\mu}{k_B T})} \: ,$$ where $k_B$ is Boltzmann’s constant and $T$ the temperature. If we talk about temperature in this article, we always mean the electronic temperature since we are not considering the motion of the ionic degrees of freedom which might be associated with a different ionic temperature. In the expressions (\[diagnel\]), (\[diagebs\]), (\[diaomega\]), and (\[diagrho\]), as well as in the remainder of the whole article, we will use the convention, that all the subscripts indexing eigenvalues and eigenfunctions are combined orbital and spin indices, i.e. that we can put at most one electron in each orbital. This will eliminate bothering factors of 2. The usually relevant case of an unpolarized spin restricted system can always easily be obtained by cutting into half all sums over these indices and multiplying by 2.
In terms of the Hamiltonian $H$ the density matrix is defined as the following matrix functional $$\label{foe}
F = f(H) \:.$$ Since $F$ is a matrix function of $H$ it has the same eigenfunctions $\Psi_n$ as H $$\begin{aligned}
H \Psi_n & = & \epsilon_n \Psi_n \label{hevcs} \\
F \Psi_n & = & f(\epsilon_n) \Psi_n \label{fevcs} \: .\end{aligned}$$ The density matrix can consequently be written as $$\begin{aligned}
\label{fdens}
F({\bf r},{\bf r}') = \sum_n f(\epsilon_n) \Psi^*_n({\bf r}) \Psi_n({\bf r}') \: ,\end{aligned}$$ where $n$ runs over all the eigenstates of the Hamiltonian. From the functional form of the Fermi distribution it follows that the eigenvalues $f(\epsilon_n)$ are always in the interval \[0:1\]. At zero temperature the density matrix of an insulating system containing $N_{el}$ electrons will have $N_{el}$ eigenvalues of value one, all others being zero. Thus the density matrix does not have full rank, but only rank $N_{el}$. Hence we can write it as $$\label{denst0}
F({\bf r},{\bf r}') = \sum_{n=occ} \Psi^*_n({\bf r}) \Psi_n({\bf r}') \: ,$$ where $n$ runs now only over the $N_{el}$ occupied states. It is easy to see that $F({\bf r},{\bf r}')$ is a projection operator in this case $$\label{denst0proj}
\int F({\bf r},{\bf r}'') F({\bf r}'',{\bf r}') d{\bf r}'' = F({\bf r},{\bf r}') \: .$$
A new set of $N_{el}$ eigenfunctions $\Psi^{new}_n({\bf r})$ can be obtained by any unitary transformation of all the $N_{el}$ degenerate eigenfunctions $\Psi_n({\bf r})$ associated with eigenvalues one, $$\Psi^{new}_n({\bf r}) = \sum_{m=occ} U_{n,m} \Psi_m({\bf r}) \: ,$$ where $U$ is a unitary $N_{el}$ by $N_{el}$ matrix. In the case of a crystalline periodic solids such a transformation can be used to generate the localized Wannier functions (Blount) from the extended eigenfunctions $\Psi_n$. We will refer to any set of orthogonal exponentially localized orbitals which can be used to represent the density matrix according to Equation (\[denst0\]) as Wannier functions. How to construct an optimally localized set of Wannier functions by the minimization of the total spread $\sum_n <r^2>_n -<{\bf r}>^2_n$ in a crystalline periodic solid has recently been shown by Marzari and Vanderbilt (1997). It has been well known in the chemistry community (Chalvet 1976) that sets of maximally localized orbitals give excellent insight into the bonding properties of systems. In addition to the spread criterion used by Marzari [*et al.*]{} there are still other criteria in common use in the chemistry community. They are all in a certain sense arbitrary, but usually lead to the same interpretation of the bonding properties. Figure \[water\] shows the four Wannier functions for the water molecule.
( 8.,9.9) (-4.,-5.25)
The density matrix $F({\bf r},{\bf r}')$ is a diagonally dominant operator, whose off-diagonal elements decay with increasing distance from the diagonal. The exact decay behavior depends on the material. We will derive the decay properties within the theoretical framework of the description of periodic crystalline solids. For a periodic solid the density matrix is given by $$\begin{aligned}
\label{ftrans}
F({\bf r},{\bf r}') & = & \sum_n \frac{V}{(2 \pi)^3} \int_{BZ} d{\bf k} \:
f(\epsilon_n({\bf k})) \: \Psi^*_{n,{\bf k}}({\bf r}) \Psi_{n,{\bf k}}({\bf r}') \\
& = & \sum_{n} \frac{V}{(2 \pi)^3} \int_{BZ} \: d{\bf k} \:
f(\epsilon_n({\bf k})) \: u^*_{n,{\bf k}}({\bf r}) u_{n,{\bf k}}({\bf r}')
e^{i{\bf k}({\bf r}'-{\bf r})} \nonumber \: ,\end{aligned}$$ where $\Psi_{n,{\bf k}}({\bf r}) = u_{n,{\bf k}}({\bf r}) e^{i{\bf k}({\bf r})} $ are the Bloch functions associated with the wave vector ${\bf k}$ and band index $n$. The integral is taken over the Brillouin zone (BZ) and $V$ is the volume of the real space primitive cell.
The Wannier functions $W_n$ of the $n$-th band in an insulating crystal are defined in the usual way $$W_n({\bf r}-{\bf R}) = \frac{V}{(2 \pi)^3} \int_{BZ} d{\bf k} \:
e^{- i {\bf k} {\bf R}} \: \Psi_{n,{\bf k}}({\bf r}) \: .$$ The Wannier functions are not uniquely defined. One can construct a different set of Bloch functions by multiplying them with a phase factor, $\Psi_{n,{\bf k}}({\bf r}) \leftarrow e^{i \omega({\bf k})} \Psi_{n,{\bf k}}({\bf r})$, where $\omega({\bf k})$ is an arbitrary function. This will obviously modify the Wannier functions. Further ambiguities arise in the case of degenerate bands (Blount). Because of these ambiguities in the construction of the Wannier functions it is advantageous to work with the density matrix where any phase factors cancel (Equation (\[ftrans\])) and where degeneracies do not cause any problems since one sums over all the occupied bands.
We will first discuss the decay properties of the density matrix in metallic systems. In this discussion we will assume that metals behave essentially like jellium and that exact results for jellium can be carried over to real metals.
The decay properties of the density matrix of a metallic system at zero temperature are well known (March). Because the integral in Equation (\[ftrans\]) contains a discontinuity in the metallic case, the density matrix decays only algebraically with respect to the distance between ${\bf r}$ and ${\bf r}'$. The decay is given by $$\label{march}
F({\bf r},{\bf r}') \propto k_F
\frac{\cos (k_F |{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|^2} \: ,$$ where the Fermi wave vector $k_F$ is related to the valence electron density by $ \frac{N_{el}}{V} = \frac{k_F^3}{3 \pi^2}$ in a non-spin-polarized system.
Introducing a finite electronic temperature $T$ in a metal leads to a drastic change in this decay behavior. Instead of an algebraic decay one has a much faster exponential decay. As shown independently by Goedecker (1998a) and Ismail-Beigi and Arias (1998), the decay at low temperatures is then given by $$F({\bf r},{\bf r}') \propto k_F
\frac{\cos (k_F |{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|^2} \:
\exp \left( - c \frac{k_B T}{k_F} |{\bf r}-{\bf r}'| \right) \: ,$$ where $c$ is a constant on the order of 1. We thus find oscillatory behavior with an exponentially damped amplitude. The decay rate depends linearly on temperature and the oscillatory part is described by the wave vector $k_F$. The related correlation function at finite temperature exhibits the same temperature dependence of the decay rate with respect to temperature (Landau and Lifshitz, 1980). In an insulator finite temperature plays no role as long as the thermal energy $k_B T$ is much smaller than the gap, which is usually fulfilled.
Let us next discuss the important case of an insulator with a band gap $\epsilon_{gap}$ at zero temperature. We will first present some numerical results, then we will put forward some arguments to explain the qualitative features of the density matrix and finally discuss in a more quantitative way the factors which determine the exact decay rate.
Numerical calculations of the density matrix or the related Wannier functions show an oscillatory behavior with a decaying amplitude. There is exactly one node per primitive cell and logarithmic plots of the amplitude clearly reveal an exponential decay. In the case of alkanes the decay of the density matrix calculated by the Hartree-Fock method has been studied and plotted on a logarithmic scale by Maslem [*et al.*]{} (1997). Interestingly, the decay depends also on the basis set used. Small low quality basis sets lead to a larger band gap and consequently to a faster decay of the density matrix. In the case of silicon, treated by Density Functional theory, logarithmic plots revealing the exponential decay of the Wannier functions have also been done both for grid based basis sets (Goedecker unpublished) and atomic basis sets (Stephan 1998). Within the Tight Binding method the decay of the density matrix has also been studied numerically for crystalline and liquid carbon systems by Goedecker (1995) and for fullerenes by Itoh [*et al.*]{} (1996).
Let us now make plausible the exponential decay of the density matrix. The demonstration is based on the fact, that one can express the Fourier components $\epsilon_n({\bf R})$ of the band energy $\epsilon_n({\bf k})$ through the Wannier functions $W_n({\bf r})$ $$\label{enkfour}
\epsilon_n({\bf R}) =
\frac{V}{(2 \pi)^3} \int_{BZ} \epsilon_n({\bf k}) e^{-i{\bf k}{\bf R}} \: d{\bf k} =
\frac{(2 \pi)^3}{V} \int_{space} W^*_n({\bf r}') H W_n({\bf r}'-{\bf R}) \: d{\bf r}' \: ,$$ where ${\bf R}$ is a Bravais lattice vector. Now it is known, that the band energy $\epsilon_n({\bf k})$ is an analytic function (Blount). This is actually not surprising. The first and second derivatives of the band-structure have physical meaning since they are related to the electron velocity and effective mass. So it is to be expected that higher derivatives exist as well. Since the Fourier transform of an analytic function decays faster than algebraically (See Appendix) there exists a decay constant $\gamma$ and a normalization constant $C$ such that $$\label{decins}
C e^{-\gamma R} \geq \epsilon_n({\bf R})
= \frac{1}{V} \int_{space} W^*_n({\bf r}') H W_n({\bf r}'-{\bf R}) \: d{\bf r}'$$ It is reasonable to expect that $H W_n({\bf r})$ will behave similarly as $W_n({\bf r})$. In particular we expect $W_n({\bf r})$ to be small whenever $H W_n({\bf r})$ is small. So we will just drop $H$ in Equation (\[decins\]). In addition we will define this modified integral not only for lattice vectors ${\bf R}$ but for arbitrary vectors ${\bf r}$ to obtain. $$\label{decinsul}
C e^{-\gamma r} \geq
\frac{1}{V} \int_{space} W^*_n({\bf r}') W_n({\bf r}'-{\bf r}) \: d{\bf r}'$$ If Equation (\[decinsul\]) holds, then one can use the mean value theorem to show that $$\begin{aligned}
C e^{-\gamma r} & \geq &
\frac{1}{V} \int_{space}
W^*_n({\bf r}') W_n({\bf r}'-{\bf r}) \: d{\bf r}' \nonumber \\
& = & \frac{1}{V} \sum_{{\bf R}'} \int_{cell}
W^*_n({\bf r}'-{\bf R}') W_n({\bf r}'-{\bf R}'-{\bf r}) \: d{\bf r}' \nonumber \\
& = & \sum_{{\bf R}'}
W^*_n({\bf s}({\bf r})-{\bf R}') W_n({\bf s}({\bf r})-{\bf R}'-{\bf r}) \: d{\bf r}' \nonumber \\
& = & F({\bf s}({\bf r}),{\bf s}({\bf r})-{\bf r}) \label{transins}\end{aligned}$$ where the mean value ${\bf s}({\bf r})$ is a vector within the primitive cell. Assuming that the density matrix has the same order of magnitude within each cell one can neglect the dependence of ${\bf s}$ on ${\bf r}$ to obtain the final result $$C e^{-\gamma r} \geq F({\bf s},{\bf s}-{\bf r})$$
The numerically observed nodal structure of the density matrix can be motivated in a very similar way. Because of the orthogonality of the Wannier functions we have $$0 = \int_{space} W^*_n({\bf r}') W_n({\bf r}'-{\bf R}) \: d{\bf r}'$$ for any non-zero lattice vector ${\bf R}$. Doing the same sequence of transformation as in Equation (\[transins\]) one obtains $$\label{nodpre}
0 = F({\bf s}({\bf R}),{\bf s}({\bf R})-{\bf R})$$ So there has to be one node in each cell. The numerically calculated nodal structure for a 1-dimensional model insulator is shown in Figure \[nodes\].
( 8.,6.5) (-1.5,-1.5)
The next step is to examine in a more quantitative way which factors determine the rate of this exponential decay for an insulator with a band gap $\epsilon_{gap}$ at zero temperature.
Cloizeaux (1964) proved the exponential decay behavior of the zero temperature density matrix, which is a projection operator. Considering the extension of the band energy $\epsilon_n({\bf k})$ into the complex ${\bf k}$ plane he found that the minimal distance of the branch points of $\epsilon_n({\bf k})$ from the real axis determines the decay behavior. For the Wannier functions, which are closely related to the density matrix by Equation (\[denst0\]), Kohn (1959) proved the same decay behavior in the case of a one-dimensional model crystal. In a later publication Kohn (1993) claims that this distance to the real ${\bf k}$ axis should be related to the square root of the gap. Even though he did not present a derivation of this result, it was widely accepted to be generally valid. Ismail-Beigi and Arias (1998) have however shown that Kohn’s claim is not generally valid. They demonstrated that in the Tight Binding limit the square root behavior can be found under certain circumstances, but that different behaviors can be found as well. In the weak binding limit, where the band-structure can be obtained by perturbation theory from the band structure of the free electron gas, they showed that the dependence is actually linear.
$$\label{insulator}
F({\bf r},{\bf r}') \propto \exp( - \gamma |{\bf r}-{\bf r}'| )
\hspace{1cm} where \:
\gamma = c \: \epsilon_{gap} \: a$$
The lattice constant is denoted by $a$, and $c$ is an unknown constant of the order of 1.
The dependence of the decay rate on the size of the band gap is a rather surprising relation. After all it follows from Equation (\[ftrans\]) that only the properties of the occupied bands enter into the calculation of the density matrix, whereas the size of the gap is not directly related to the occupied states. In the following we will give an intuitive explanation of the factors determining the decay rate. This explanation will again be based on Equation (\[enkfour\]) relating the bandstructure to the decay properties of the density matrix. As is known from complex analysis, the distance of the singularities from the real axis is comparable to the length over which one has very strong variations along the real axis of a complex function. Now, the long range decay properties of a Fourier transform are exactly determined by the length $\Delta k$ of such a region of strongest variation (See Appendix). One thus regains Cloizeaux’s result that the decay rate is proportional to the distance of singularities from the real axis. Let us now explain the behavior found in the weak binding limit by Ismail-Beigi and Arias. In the weak binding limit the effective mass establishes the connection between the gap and the important features of the occupied bands. The effective mass for the $n$-th band at the point ${\bf k}_0$ is defined as (Kittel 1963) $$\frac{1}{m} = 1 + \frac{2}{3} \sum_{m \neq n}
\frac{ | \int \Psi^*_{n,{\bf k}_0}({\bf r}) \nabla \Psi_{m,{\bf k}_0}({\bf r}) d{\bf r} |^2 }
{\epsilon_n({\bf k}_0)-\epsilon_m({\bf k}_0)}$$ Since we are only interested in order of magnitudes, we have here averaged over the diagonal elements of the effective mass tensor in order to obtain a effective mass which is a scalar quantity. In the case of the weak binding limit, a gap will open up at the boundaries of the Brioullin zone and this gap will be small. The effective mass is therefore small and proportional to $a ^2 \epsilon_{gap}$, where we have assumed that the dipole matrix elements $ \int \Psi^*_{n,{\bf k}_0}({\bf r}) \nabla \Psi_{i,{\bf k}_0}({\bf r}) d{\bf r}$ are on the order of $\frac{1}{a}$. The band-structure near the boundaries of the Brioullin zone is then given by $$\label{bseff}
\frac{1}{2 m} (\Delta k )^2 \propto \frac{1}{a^2 \epsilon_{gap}} (\Delta k )^2$$ where $\Delta k$ is the distance from the boundary, neglecting directional effects. Since the effective mass is small, the curvature of the band-structure is large in this region. Hence this region is just the region with the strongest variation. As is well known (Ashcroft and Mermin 1976), the perturbation theory arguments leading to Equation (\[bseff\]) are valid within an energy range of the order of $\epsilon_{gap}$. It then follows from Equation (\[bseff\]) that the corresponding range of $\Delta k$ is $\epsilon_{gap} \: a$, confirming the linear decay of the density matrix with respect to the size of the gap, i.e. $\gamma = c \: \epsilon_{gap} \: a $.
Let us next show how a square root like behavior $\gamma = c \: \sqrt{ \epsilon_{gap} }$ can arise for real crystals with a big gap. In this case the effective mass is of the order of one at all stationary points ${\bf k}_0$ in the Brioullin zone. Assuming that it is then of the order of one over the whole Brioullin zone, the region of largest variation is just the Brillouin zone itself. The decay constant is therefore simply related to the lattice constant $a$. $$\label{insulator2}
\gamma = c \frac{1}{a}$$ In order to get the square root dependence of the decay constant $\gamma$, one has to assume that $$\label{harrison}
\epsilon_{gap} = C_{gap} \frac{1}{a^2}$$ where $C_{gap}$ is a constant which is not or only weakly dependent on the material. Such a behavior has indeed been observed for certain classes of materials, where the tight binding limit is the most appropriate one, such as ionic crystals (Harrison 1980), but with a non-negligible variation of $C_{gap}$ across different materials. A square root behavior of $\gamma$ can therefore be expected if one varies the lattice constant for a certain material, but the decay constants for different materials that happen to have the same gap are not necessarily comparable.
In practice the distinction between the Tight Binding and weak binding case may not always be clear. Unless the region of strongest variation is really a very small fraction of the whole Brioullin zone, all the prefactors which were neglected in these considerations might be important enough to blur out differences. The importance of these prefactors can also be seen from the fairly strong directional dependence of the decay rate. Ismail-Beigi and Arias (1998) found such a strong directional dependence in numerical tests to confirm the linear dependence of the decay constant on the size of the gap (Figure \[beigi\]). Stephan [*et al.*]{} (1998) found the same behavior during Tight Binding studies of carbon. So a statement in an old paper by Kohn (1964), namely that the decay length of the Wannier functions is of the order of the interatomic spacing, is for practical purposes probably in many cases the best available characterization of localization.
( 8.,5.0) (-1.5,-3.)
As a numerical illustration of this surprising result, that only a small part of the Brillouin zone where one has the strongest variation determines the decay behavior of the Wannier functions, we compared the decay behavior of carbon in the diamond structure with “syntheticum” in the same structure. The artificial element “syntheticum” was computer generated within the Tight Binding context in such a way that its top part of the conduction band as well as the gap is nearly identical to real carbon, whereas the lower part of the valence band is drastically different as shown in Figure \[syndos\]. More precisely, Carbon was characterized by the parameters of Goodwin (1991) and to obtain syntheticum $\epsilon_{s}$ was modified from -5.16331 to -1.16331 and $V_{s s \sigma}$ was modified from -4.43338 to -2.43338.
( 8.,5.5) (-1.5,-1.)
Figure \[syndec\] shows the decay behavior of the density matrix. As one can see the decay behavior is very similar in both cases. We note that not only the gap is similar but also the effective mass since the density of states at the top of the valence band has the same behavior in both materials.
( 8.,5.5) (-1.5,-1.)
All the above arguments apply to simple and mainly periodic materials. Advanced electronic structure calculations however frequently study materials which are not in this class. The localization properties of such materials have not yet been studied systematically and so there is some incertitude about which orbitals are localized and to what extent (Kohn 1995). If the localization properties are unknown one should better not impose any localization constraints. In this case some of the discussed O(N) techniques still give a quadratic scaling, which also allows us to gain computational efficiency compared to the traditional cubically scaling algorithms.
Basic strategies for O(N) scaling {#four}
=================================
Most O(N) algorithms are built around the density matrix or its representation in terms of Wannier functions and take advantage of its decay properties. To obtain linear scaling one has to cut off the exponentially decaying quantities when they are small enough. This introduces the concept of a localization region. Only inside this localization region the quantity is calculated, outside it is assumed to vanish. For simplicity the localization region is usually taken to be a sphere, even though the optimal shape might be different (Stephan 1998). In the Tight Binding context the boundary of the localization region can either be defined by a geometric distance criterion or in terms of the number of “hops”, i.e. the number of steps one has to do along bonds connecting neighboring atoms to reach this boundary (Voter [*et al.,*]{} 1996). Different localization regions generally have significant overlaps. The localization regions thus do not form a partition of the computational volume and one atom in general belongs to several localization regions.
In a numerical calculation the density operator $F(r,r')$ is discretized with respect to a basis. The basis set has to be chosen such that the matrix elements $F_{i,j}$ reflect the decay properties of the operator $F(r,r')$. This will obviously only be the case if the basis set consists of localized functions, such as atom centered Gaussian type basis functions. Sets of orthonormal basis functions usually facilitate the calculations. Unfortunately all currently used localized basis sets are non-orthogonal. In the context of the orthogonal Tight Binding scheme (Goringe [*et al.,*]{} 1997a, Majewski and Vogl 1986) one just assumes the existence of an basis set which is both atom centered and orthogonal. Since only the parameterized Hamiltonian matrix elements enter in the calculation, there is no need to explicitly ever construct such a basis set. In the following sections, we will follow this practice and assume in all relevant parts that we are dealing with such a localized orthogonal basis set. The non-orthogonal case will be discussed in section \[nonorthog\]. Whenever we refer from now on to a localization region, we actually mean the subset of all basis functions which are contained within this spatial localization region.
Obviously the size of the localization region needed to obtain a certain accuracy depends on the decay properties of the density matrix as well as on the selected accuracy threshold. It also depends on the quantity one wants to study. Generally, the total energy as well as derived quantities such as the geometric equilibrium configurations are surprisingly insensitive to finite localization regions, because these quantities are not strongly influenced by the exponentially small tails which are cut off by the introduction of a localization region. This insensitivity also holds true, even though to a much lesser extent, for metals. As we have seen above the introduction of a finite temperature leads to an exponential decay of the density matrix which in turn justifies truncation. In a metal, the difference between the finite and the zero temperature total energy $\Delta E$ is proportional to the square of the temperature, $\Delta E \propto T^2$, (Ashcroft and Mermin 1976) and thus rather small. There are however quantities which are very sensitive to finite localization regions. In the modern theory of polarization in solids (King-Smith and Vanderbilt 1989), the polarization can be expressed in terms of the centers of the Wannier functions $\int W({\bf r }) {\bf r } W({\bf r }) d{\bf r }$. Using this formula (Fernandez [*et al.,*]{} 1997) one has a strong influence of the tails of the Wannier functions because they get strongly weighted by the factor of ${\bf r }$ in the integral. Since the tails are much more influenced by the boundary of the localization region than the central part, this quantity is more sensitive to the size of the localization region.
There are even quantities which are not at all directly accessible by a solution which is given in terms of density matrices or Wannier functions. The Fermi surface in a metal which can be calculated via the eigenvalues of the band structure $\epsilon_n({\bf k})$ is such an example.
It is also clear that one can gain significant computational efficiency only if the size of the system is larger than the size of the localization region. When this criterion is fulfilled depends not only on the decay properties of the density matrix of the system but also on its dimensionality. In the case of a linear chain molecule with a large band gap, it might be enough to have a localization region containing just two neighboring atoms on each side. So the localization region would just contain 5 atoms and for systems larger than 5 atoms one might potentially gain computational efficiency by using an O(N) method. If one has a 3 dimensional system with a comparable gap, then a spherical localization region extending out to the second neighbors would contain some 60 atoms and the crossover point would already be much larger. For a system with a small gap such as silicon or for metallic systems the crossover point is even larger.
There are essentially six basic approaches to achieve linear scaling.
- The Fermi Operator Expansion (FOE) is based on Equation (\[foe\]). In this approach one finds a computable functional form of $F$ as a function of $H$ to build up the density matrix. Two possible representations based on a Chebychev expansion and a rational expansion will be discussed.
- The Fermi Operator Projection (FOP) is closely related to the FOE method. The computable form of $F$ is however not used to construct the entire density matrix but to find the space spanned by the occupied states, i.e. the space corresponding to the eigenfunctions associated with the unit eigenvalues of the Density matrix at zero temperature. These eigenfunctions can be considered as Wannier functions in the generalized sense defined before.
- In the Divide and Conquer (DC) method for the density matrix the relevant parts of the density matrix are patched together from pieces that were calculated for smaller subsystems.
- In the Density Matrix Minimization (DMM) approach, one finds the density matrix by a minimization of an energy expression based on the density matrix.
- In the Orbital Minimization approach (OM), one finds a set of Wannier functions by minimization of an energy expression.
- The Optimal Basis Density Matrix Minimization scheme (OBDMM) contains aspects of both the OM and DMM methods. In addition to finding a density matrix with respect to the basis, one also finds an optimal basis by additional minimization steps. The number of basis functions has to be at least equal to the number of electrons in the system, but can be bigger as well.
A major difference between these methods is whether they calculate the full density matrix or only its representation in terms of Wannier functions. The later approach applies only to insulators while the former is in also applicable to systems with fractional occupation numbers (i.e. $f(\epsilon_n)$ is not either 1 or 0) such as metals or systems at finite electronic temperature.
In the following each of these six approaches will be presented in detail.
The Fermi Operator Expansion
----------------------------
The FOE (Goedecker and Colombo 1994, Goedecker and Teter 1995) is the most straightforward approach for the calculation of the density matrix. The basic idea in this approach is to find a representation of the matrix function (\[foe\]) which can be evaluated on a computer. Several such representations are possible. We will discuss a Chebychev and a rational representation.
### The Chebychev Fermi Operator Expansion
One of the most basic operations a computer can do are matrix times vector multiplications. The simplest representation of the density matrix would therefore be a polynomial representation $$F \approx p(H) = c_0 I + c_1 H + c_2 H^2 + \: ... \: + c_{n_{pl}} H^{n_{pl}} \: .$$ where $I$ is the identity matrix. Unfortunately polynomials of high degree become numerically unstable. This instability can however be avoided by introducing a Chebychev polynomial representation, which is a widely used numerical method (Press [*et al.,*]{} 1986) $$p(H) = \frac{c_0}{2} I + \sum_{j=1}^{n_{pl}} c_j T_j(H) \: .$$ Since the Chebychev polynomials are defined only within the interval \[-1:1\], we will assume in the following that the eigenvalue spectrum of $H$ falls within this interval. This can always be easily achieved by scaling and shifting of the original Hamiltonian. The Chebyshev matrix polynomials $T_j(H)$ satisfy the recursion relations $$\begin{aligned}
T_0(H) & = & I \\
T_1(H) & = & H \\
T_{j+1}(H) & = & 2\: H \: T_j(H) - T_{j-1}(H) \: .\end{aligned}$$ The expansion coefficients of the Chebychev expansion can easily be determined. The eigenfunction representation (Equation (\[fevcs\])) of $F$ is, $$< \Psi_n|F|\Psi_m> = f(\epsilon_n) \: \delta_{n,m} \: .
\label{eq:fdia1}$$ Evaluating the polynomial expansion in the same eigenfunction representation we obtain $$< \Psi_n|p(H)|\Psi_m> = p(\epsilon_n) \: \delta_{n,m} \: ,
\label{eq:fdia2}$$ where $$p(\epsilon) = \frac{c_0}{2} + \sum_{j=1}^{n_{pl}} c_j T_j(\epsilon) \: .$$ Comparing Equation (\[eq:fdia1\]) and Equation (\[eq:fdia2\]) we see that the polynomial $p(\epsilon)$ has to approximate the Fermi distribution in the energy interval \[-1:1\] where the scaled and shifted Hamiltonian has its eigenvalues. How to find the Chebychev expansion coefficients for a scalar function is described in standard textbooks on numerical analysis (Press [*et al.,*]{} 1986). Actually it is not necessary to take the exact Fermi distribution. In practically all situations one is interested in the limit of zero temperature. Hence any function which approaches a step functions in the limit of zero temperature can be used. In the case of simulations of insulators for instance it is advantageous to take the function $f(\epsilon) = \frac{1}{2} (1-{\rm erf}(\frac{\epsilon-\mu}{\Delta \epsilon}))$ shown in Figure \[foedist\] since it decays faster to 0 respectively 1 away from the chemical potential. The term Fermi distribution will in the following always be used in this broader sense. The energy resolution $\Delta \epsilon$ is chosen to be a certain fraction of the size of the gap (Goedecker and Teter 1995). In the case of metals, $\Delta \epsilon$ is chosen by considerations of numerical convenience. Large values of $\Delta \epsilon$ will give lower accuracy results. However as pointed out before, the convergence of the total energy with respect to $\Delta \epsilon$ is quadratic and so highly accurate total energies can be obtained with rather high values of $\Delta \epsilon$ (Goedecker and Teter 1995). Small values of $\Delta \epsilon$ make the calculation numerically expensive. The detailed scaling behavior of the numerical effort in the limit of vanishing gaps is analyzed in section \[compare\], where it is found that actually the increase in the size of the localization region is the limiting factor in all methods.
( 8.,5.0) (-1.5,-0.5)
Even if one wants to study electronic properties in the limit of zero electronic temperature it is important that one nevertheless uses a finite temperature Fermi distribution for the Chebychev fit. Using the zero temperature step function introduces so-called Gibbs oscillations in the fit and spoils the Chebychev fit over the whole interval.
How to eliminate these Gibbs oscillations in the zero temperature case by the so called kernel polynomial method (Voter [*et al.,*]{} 1996, Silver [*et al.,*]{}1996) can be used as a starting point for an alternative derivation of the FOE method. The basic idea is to expand a delta function as a polynomial using damping factors to suppress large oscillations. This representation of an approximate delta function can then be integrated to obtain a smooth representation of the Fermi distribution. Used this way the kernel polynomial method is thus just another way to derive the expansion coefficients for the Chebychev expansion (Kress [*et al.,*]{} 1998). In addition the kernel polynomial method can also be used to smear out the density of states rather than the zero temperature Fermi distribution resulting in a method with practically identical computational requirements but some slightly different properties. One useful property is that the smeared density of states energy is an approximate lower bound to the energy, whereas the smeared Fermi energy is an approximate upper bound (Voter [*et al.,*]{} 1996).
Coming back to the original motivation for a polynomial representation, let us now show how the density matrix can be constructed using only matrix times vector multiplications. Let us denote by $t^j_l$ the l-th column of the Chebychev matrix $T_j$. Now each column of these Chebychev matrices satisfies the same recursion relations $$\begin{aligned}
|t^0_l> & = & |e_l> \label{recurs} \\
|t^1_l> & = & H|e_l> \nonumber \\
|t^{j+1}_l> & = & 2\: H \: |t^j_l> - |t^{j-1}_l> \nonumber \: .\end{aligned}$$ where $e_l$ is a unit vector that has zeroes everywhere except at the $l$-th entry. So Equation (\[recurs\]) demonstrates that we indeed need only matrix vector multiplications. Once we have generated the $l$-th columns of all the Chebychev matrices, we can obtain the $l$-th column $f_l$ of the density matrix just by forming linear combinations $$|f_l> = \frac{c_0}{2} |t^0_l> + \sum_{j=1}^{n_{pl}} c_j |t^j_l> \: .$$ As we have described the method so far it has a quadratic scaling instead of the linear scaling we finally want to achieve. If we have $M_b$ basis functions, the density matrix is a $M_b \times M_b$ matrix and we have to calculate $M_b$ full columns. For the calculation of each column, we have to do $n_{pl}$ matrix times vector multiplications, each of which costs $M_b n_H$ operations assuming the matrix $H$ is a sparse matrix with $n_H$ off-diagonal elements per row/column. So the total computational cost is $M_b^2 \: n_{pl} \: n_H$. The degree of the polynomial $n_{pl}$ and the width $n_H$ of the Hamiltonian are independent of the size of the system, whereas $M_b$ is proportional to the number of atoms in the system. The overall scaling with respect to the number of atoms is therefore quadratic.
In order to do the correct shifting and scaling of the original Hamilton to map its eigenvalue spectrum on the interval \[-1:1\] we have to know its lowest and highest eigenvalues $\epsilon_{min}$ and $\epsilon_{max}$. In addition we have to know the chemical potential $\mu$. There are auxiliary matrix functions of $H$ that can help us to determine these quantities. These functions of $H$ can be build up in the same way as the density matrix. Since the recursive build up of the Chebychev matrices is the most costly part, the additional cost for evaluating other functions is negligible. To determine whether we have a vanishing density of states beyond an energy $\epsilon_{up}$ we can for instance construct a Chebychev fit $p_{up}(\epsilon)$ to a function which is zero (to within a certain tolerance) for energies below $\epsilon_{up}$, but blows up for energies larger than $\epsilon_{up}$. If $Tr[p_{up}(H)]$ does not vanish we have a non-vanishing density of states beyond $\epsilon_{up}$. A similar procedure can be applied to determine a lower bound for the density of states. The determination of the chemical potential in an insulator can be done along the same lines as well (Bates and Scuseria 1998). Without any significant extra cost one can build up several Fermi distributions with different chemical potentials until one finds the correct chemical potential leading to charge neutrality. In a metallic system the search for the chemical potential can be accelerated since it is possible to predict with high accuracy how the number of electrons changes in response to a change in the chemical potential. From Equation (\[nel\]) it follows $$\label{dnel}
\frac{\partial N_{el}}{\partial \mu} = - Tr [p'(H)] \: ,$$ where $p'$ is the derivative of the Chebychev polynomial $p$ that approximates the Fermi distribution. The Chebychev expansion coefficients of $p'$ can be calculated from the coefficients for $p$ (Press [*et al.,*]{} 1986). Using the finite difference approximation of Equation (\[dnel\]), $$\label{fddnel}
\Delta \mu = \frac{ \Delta N_{el}}{Tr [p'(H)]} \: ,$$ it is possible to find the correction $\Delta \mu$ to the chemical potential which will nearly exactly eliminate an excess of $\Delta N_{el}$ electrons due to an incorrect initial chemical potential. The correct chemical potential in a metallic system can thus be found with very high accuracy with a few iterations.
The desired linear scaling can be obtained by introducing a localization region for each column, outside of which the elements are negligibly small. For the $k$-th column, this localization region will be centered on the $k$-th basis function. If we use atom centered basis functions, then the localization region will consequently be centered on the atom to which this $k$-th basis function belongs. We have then to calculate only that part of each column which corresponds to this localization region. This means that we can use a truncated Hamiltonian $H(k)$ which retains only the matrix elements corresponding to the basis functions contained within the localization region $k$. Denoting the number of basis functions in this region by $M_{loc}$ (which might actually depend on the localization region $k$ being considered), the overall computational cost is then $M_b M_{loc} \: n_{pl} \: n_H$ and thus scales linearly. Let us stress, that the size of the localization region is independent of the degree of the polynomial. If one uses for instance a polynomial of degree $n_{pl}=50$, the recursion in Equation (\[recurs\]) will extend over the 50 nearest neighbor shells without localization constraint for a Hamiltonian coupling only nearest neighbors. The localization region however is typically much smaller comprising just a few nearest neighbor shells. Imposing a localization region introduces some subtleties. For instance the eigenvalues of the truncated density matrix are not anymore exactly given by $p(\epsilon_n)$ and $F$ is not any more strictly symmetric. More importantly, strictly speaking we can no longer use the Trace notation, since we use different local Hamiltonians $H(k)$ to build up the different columns of the density matrix. The band-structure energy $E_{BS}$ has now to be written as $$\label{etrun}
E_{BS} = \sum_k \sum_j [p(H(k))]_{k,j} [H(k)]_{j,k} \: .$$
Another important quantity are the forces. The force acting on the $\alpha$-th atom at position $R_{\alpha}$ is obtained by differentiating the total energy with respect to these positions. The total energy consists of the band structure part and possibly other contributions. We will only discuss the non-trivial part of the force arising from the differentiation of the band structure energy $E_{BS}$. For simplicity let us assume that we have a simple polynomial expansion and not a Chebychev expansion. Let us also assume that we calculate the full density matrix, i.e. that we do not truncate $H$ by introducing a localization region. We then obtain $$\frac{d E_{BS}}{d R_{\alpha}} =
\frac{ d}{d R_{\alpha}} Tr \left[ H \sum_{\nu} c_{\nu} H^{\nu} \right] =
\sum_{\nu} c_{\nu} Tr \left[ \frac{\partial H^{\nu+1}}{\partial R_{\alpha}} \right] \: .$$ Let us consider for instance the term for which $\nu = 2$ $$\label{term3full}
\frac{d Tr[H^{3}]}{d R_{\alpha}} =
Tr \left[ H H \frac{\partial H}{\partial R_{\alpha}} \right] +
Tr \left[ H \frac{\partial H}{\partial R_{\alpha}} H \right] +
Tr \left[ \frac{\partial H}{\partial R_{\alpha}} H H \right]
= 3 Tr \left[ H H \frac{\partial H}{\partial R_{\alpha}} \right] \: ,$$ where we used that $Tr[A B] = Tr[B A]$. The final result for the force, which also holds in the case of a Chebychev expansion, is thus $$\label{foeforce}
\frac{d E_{BS}}{d R_{\alpha}} =
Tr \left[ \left( p(H) + H p'(H) \right) \frac{\partial H}{\partial R_{\alpha}} \right] \: .$$ In the case of an insulator, the second term in the brackets $H p'(H)$ is very small compared to the first term $p(H)$ at small but finite temperatures and it vanishes in the limit of zero temperature. The reason for this is that the eigenvalues of the matrix $p'(H)$ are $p'(\epsilon_n)$. Since at zero temperature $p'(\epsilon)$ is nonzero only at the chemical potential which is in the middle of the gap, all eigenvalues are zero and the matrix is identically zero. Nevertheless it is recommendable to retain this term in numerical calculations because it leads to forces consistent with the total energy.
In the case where we calculate only part of the density matrix, i.e. where we have a truncated Hamiltonian $H(k)$ going with the energy expression (\[etrun\]) we cannot use the properties of the trace to simplify the force expression as we did in Equation (\[term3full\]). The equation corresponding to Equation (\[term3full\]) therefore reads
$$\begin{aligned}
\label{term3trun}
\sum_{k,j1,j2,k}
& & [H(k)]_{k,j1} [H(k)]_{j1,j2}
\left( \frac{\partial H(k)}{\partial R_{\alpha}} \right)_{j2,k} + \\
& & [H(k)] _{k,j1}
\left( \frac{\partial H(k)}{\partial R_{\alpha}}\right)_{j1,j2} [H(k)]_{j2,k} + \nonumber \\
& & \left( \frac{\partial H(k)}{\partial R_{\alpha}} \right)_{k,j1} [H(k)]_{j1,j2} [H(k)]_{j2,k} \nonumber \: .\end{aligned}$$
Similar results hold for all the other terms with different values of $\nu$. In the case of a Chebychev expansion the situation is completely analogous, just the formulas are more complicated. The force formula has been worked out in this case by Voter [*et al.*]{} (1996) and is given by $$\label{forcevoter}
\frac{d T_j(H)}{d R_{\alpha}} =
\frac{d T_{j-2}(H)}{d R_{\alpha}} +
\sum_{i=0}^{j-1} (1+k_i)(1+k_{j-1-i}) T_i(H)
\frac{\partial H}{\partial R_{\alpha}}
T_{j-1-i}(H) \: ,$$ where $k_j = 0$ if $j \leq 0$ and $k_j = 1$ otherwise. In the typical Tight Binding context $\frac{\partial H}{\partial R_{\alpha}}$ is a very sparse matrix. If it contains $n_{D}$ non-zero elements, we need of the order of $n_{pl}^2 \: n_{D} M_b$ operations to evaluate all the forces according to Equation (\[forcevoter\]). The error incurred by using the approximate formula (\[foeforce\]) based on the trace is usually negligible if the localization region is large enough. Since the approximate formula can be evaluated with order $n_{pl} \: n_{D} M_b$ operations, it might actually be preferable to do so. In a molecular dynamics simulation, the largest deviations in the conservation of the total energy come from events where atoms enter or leave localization regions and this kind of error is not taken into account by either force formula.
All the above force formulas were derived for the case where we have a constant chemical potential and where the polynomial representing the Fermi distribution does thus not change. Frequently one wants however to do simulations for a fixed number of electrons rather than for a fixed chemical potential. In this case one has to readjust the chemical potential for each new atomic configuration. The chemical potential is thus a function of all the atomic positions $\mu = \mu(R_{\alpha})$, but the explicit functional form of this dependence is not known. The force formula can however also be adapted to this case (B. Roberts and P. Clancy 1998). Ignoring the above warnings and using again trace notation for simplicity we have $$\begin{aligned}
E_{BS} & = & Tr[H \: p(H-\mu I)] \\
N_{el} & = & Tr[p(H-\mu I)] \end{aligned}$$ and consequently $$\begin{aligned}
\frac{d E_{BS}}{d R_{\alpha}} & = &
Tr \left[ ( H \: p' + p) \frac{\partial H}{\partial R_{\alpha}} \right] -
Tr [ ( H \: p' ) ] \frac{\partial \mu}{\partial R_{\alpha}} \label{foededr} \\
\frac{d N_{el}}{d R_{\alpha}} & = &
Tr \left[ p' \frac{\partial H}{\partial R_{\alpha}} \right] -
Tr [ p' ] \frac{\partial \mu}{\partial R_{\alpha}}
\label{foedndr} \: .\end{aligned}$$ Since $\frac{d N_{el}}{d R_{\alpha}}$ has to be equal to zero, we can solve Equation (\[foedndr\]) for $\frac{\partial \mu}{\partial R_{\alpha}}$ and insert it into equation (\[foededr\]) to obtain the force under the constraint of a constant number of electrons.
Let us finally derive a force formula for the case where a local charge neutrality condition is enforced (Kress [*et al.,*]{} 1998). The motivation for this approach is that in non-self-consistent Tight Binding calculations one frequently finds an unphysically large transfer of charge between atoms. In a self-consistent calculation the electrostatic potential, built up by a charge transfer, is counteracting a further charge flow and thus limits charge transfer to reasonably small values. Some Tight Binding schemes (Horsfield [*et al.,*]{} 1996a) enforce a so-called local charge neutrality condition requiring that the total charge associated with an atom in a molecule or solid be equal to the charge of the isolated atom. This is done by determining a potential offset $u_{\alpha}$ for each atom $\alpha$ in the system which will ensure this neutrality. The total Hamiltonian $H$ of the system is then given by $H_0+U$ where $H_0$ is the Hamiltonian without any potential bias and $U$ a diagonal matrix containing the atomic potential offsets $u_{\alpha}$. The band structure energy is given by $$\label{foe273}
E_{BS} = Tr[(H_0 + U) \: p(H_0 + U)] - \sum_{\alpha} Q_{\alpha} u_{\alpha} \: ,$$ where the term containing the atomic valence charges $Q_{\alpha}$ has been subtracted to make the expression invariant under the application of a uniform potential bias to all atoms in the system. Expressed in terms of the density matrix the local charge neutrality condition becomes $$\label{foe499}
\sum_l p(H)_{\alpha,l;\alpha,l} = Q_{\alpha} \: .$$ In Equation (\[foe499\]) we have labeled the basis functions by a composite index where $\alpha$ indicates on which atom the basis function is centered and where $l$ describes the character of the atom centered basis function. If we have carbon atoms, for which $Q_{\alpha}=4$, $l$ would for instance denote the 4 orbitals $2s$, $2px$, $2py$, $2pz$. Using Equation (\[foe499\]), Equation (\[foe273\]) then simplifies to $$\label{foe473}
E_{BS} = Tr[H_0 \: p(H_0 + U)] \: .$$ Taking the derivative we get $$\label{foe573}
\frac{d E_{BS}}{d R_{\alpha}} =
\sum_{\beta} \frac{\partial E_{BS}}{\partial u_{\beta}}
\frac{\partial u_{\beta}}{\partial R_{\alpha}} +
\frac{\partial E_{BS}}{\partial R_{\alpha}} \: ,$$ where $$\label{foe673}
\frac{\partial E_{BS}}{\partial u_{\beta}} =
Tr \left[ H_0 \: p'(H) \frac{\partial H}{\partial u_{\beta}} \right] \: .$$ As discussed above, the matrix $p'(H)$ is close to zero in an insulator at sufficiently low temperature and can often be neglected. The forces are therefore approximately given by $$\label{foe773}
\frac{d E_{BS} }{d R_{\alpha} } =
Tr \left[ H_0 \: p(H) \frac{\partial H}{\partial R_{\alpha}} \right] \: .$$ It has to be pointed out that to get sufficiently high accuracy the degree of the polynomial has to be higher than in the case of Tight Binding case without local charge neutrality.
The degree $n_{pl}$ of the polynomial needed to represent the Fermi distribution is proportional to $$\label{npldegree}
\frac{\epsilon_{max}-\epsilon_{min}}{\Delta \epsilon} \: .$$ This follows from the fact, that the $n$th order Chebychev polynomial has $n$ roots and so a resolution that is roughly proportional to $1/n$. For the usual Tight Binding Hamiltonians the ratio in Equation (\[npldegree\]) is not very large and for silicon and carbon systems without gap states polynomials of degree 50 are sufficient. In contexts other than Tight Binding this ratio can however be fairly large and polynomial representation would become very inefficient.
### The Rational Fermi Operator Expansion
A rational representation of the density matrix (Goedecker 1995) is in this case more efficient $$\label{foepade}
F = \sum_{\nu} \frac{w_{\nu}}{H-z_{\nu}} \: .$$ As is well known, the function $f(\epsilon)$ given by $$\label{zercont}
f(\epsilon) = \frac{1}{2 \pi i} \oint \frac{dz}{\epsilon-z}$$ is equal to 1 if $\epsilon$ is within the volume encircled by the contour integration path and zero otherwise. If the integration path contains the occupied states as shown in Figure (\[figcontint\]) it can therefore be used as a zero temperature Fermi distribution. The exact finite temperature Fermi distribution $f(\epsilon_n({\bf k}))$ can be obtained by replacing the path in this contour integral by a sequence of paths which do not intersect the real axis (Goedecker 1993, Gagel 1998, Nicholson and Zhang 1997). Actually, as already mentioned above, it is usually not necessary to have the exact Fermi distribution. The electronic temperature is just determined by the slope (and possibly some higher derivatives) of the distribution at the Fermi energy. We will also refer to such generalized distributions as Fermi distributions. A distribution of this type can be obtained by discretizing the zero temperature contour integral from Equation (\[zercont\]) as shown in Figure \[figcontint\].
( 8.,3.5) (-.5,-.5)
In principle any other set of $z_{\nu}$’s and $w_{\nu}$’s can be used as long as it satisfies $$f(\epsilon) \approx \sum_{\nu=1}^{n_{pd}} \frac{w_{\nu}}{\epsilon-z_{\nu}} \: ,$$ where $n_{pd}$ is the degree of the rational approximation. How can we now evaluate Equation (\[foepade\]) on a computer? Denoting $\frac{I}{H-z_{\nu}}$ by $F_{\nu}$ we have $$\begin{aligned}
(H-z_{\nu}) F_{\nu} & = & I \label{padeeq} \\
F & = & \sum_{\nu} w_{\nu} F_{\nu} \label{padesm} \: .\end{aligned}$$ So we have first to invert all the matrices $H-z_{\nu}$ and then to form linear combinations of them. The inversion is equivalent to the solution of $M_b$ linear systems of equations. This can be effectuated using iterative techniques so that in the end everything can again be done by matrix times vector multiplications. A rational approximation can represent the sharp variation near the chemical potential of a low temperature Fermi distribution in a more efficient way than a Chebychev approximation. Whereas in the Chebychev case the degree of the polynomial is given by Equation (\[npldegree\]) the degree of the rational approximation $n_{pd}$ is given by $$\label{npddegree}
n_{pd} = \frac{\mu-\epsilon_{min}}{\Delta \epsilon} \: .$$ This $n_{pd}$ in contrast to $n_{pl}$ does not depend on the largest eigenvalue $\epsilon_{max}$. Once $n_{pd}$ is of the order of magnitude given by Equation (\[npddegree\]) one has exponential convergence to the zero temperature Fermi distribution. In the case where the integration points and weights are obtained by discretizing the contour integral of Figure \[figcontint\] this exponential behavior is immediately comprehensible since an equally spaced integration scheme gives exponential convergence for periodic functions (Sloan and Joe, 1994). Since $n_{pd}$ is usually reasonably small, the success of the method will hinge upon whether it is possible to solve the linear system of equations associated with each integration point with a small number of iterations. The number of iterations in an iterative method such as a conjugate gradient scheme (Press [*et al.,*]{} 1986) is related to whether it is possible to find a good preconditioning scheme. In the case of plane wave calculations a good preconditioner can be obtained from the diagonal elements and of the order of 10 iterations are required. In other schemes using Gaussians for instance it is not quite clear whether good preconditioners can be found. When the Hamiltonian depends on the atomic positions $R_{\alpha}$, Equations (\[padeeq\]) and (\[padesm\]) can be differentiated to obtain the derivative $\frac{d F}{d R_{\alpha}}$, which is needed for the calculation of the forces.
The Fermi Operator Projection Method
------------------------------------
The FOE method is used to calculate the full density matrix. This can be inefficient if the number of basis functions per atom is very large. As was mentioned before, the density matrix at zero temperature does not have full rank. In the case of an insulator it can be constructed from $N_{el}$ Wannier functions (\[denst0\]). If one has a numerical representation of the zero temperature density operator, which is actually a projection operator, that eliminates all components belonging to eigenvalues above the Fermi level, one can apply it to a set of trial Wannier functions $\tilde{V}_n$, $n=1,...,N_{el}$ to generate a set of orbitals which span the space of the Wannier functions. The numerical representation of the density operator can again either be a Chebychev or rational one. We will first discuss the rational case (Goedecker 1995).
To do the projection with a rational representation, a system of equations analogous to (\[padeeq\]) and (\[padesm\]) has to be solved for each trial Wannier function $\tilde{V}_n$ and at each integration point $\nu$ $$\begin{aligned}
(H-z_{\nu}) \tilde{W}_{n,\nu} & = & \tilde{V}_n \label{wanneq} \\
\tilde{W}_n & = & \sum_{\nu} w_{\nu} \tilde{W}_{n,\nu} \label{wannsm} \: .\end{aligned}$$ Thus the saving comes from the fact that one has to solve this system of equations (\[wanneq\]) just for $N_{el}$ right hand sides, whereas one has $M_b$ right hand sides in Equation (\[padeeq\]). Obviously the solution of the Equation (\[wanneq\]) has to be done not within the whole computational volume but only within the localization region to obtain linear scaling. The functions $\tilde{W}_n$ will now span our subspace unless one of our trial functions $\tilde{V}_n$ was chosen in such a way that it has zero overlap with the space of the occupied orbitals, which is highly unlikely. To obtain a set of valid Wannier functions $W_n$ one has still to orthogonalize the orbitals $\tilde{W}_n$. Since the $W_n$’s are localized the overlap matrix is a sparse matrix and can be calculated with linear scaling. In the typical Density Functional context, the inversion of this matrix is a rather small part, even if it is done with cubic scaling. In a Tight Binding context it is much more important and a linear scaling method has been devised by Stephan [*et al.*]{} (1998) for the inversion. The construction of the Wannier functions by projection according to Equations (\[wanneq\]) and (\[wannsm\]) is illustrated in Figure \[project\] in the case of a silicon crystal. In this case one knows that the Wannier functions are bond centered and it is therefore natural to choose a set of bond centered functions as an initial guess. In this example we took simple Gaussians. As shown in Figure \[project\], the projection modifies the details of the Gaussian but does not significantly change its localization properties.
( 8.,4.5) (-4.,-2.0)
Chebychev based projection methods have been introduced in connection with other techniques by Sankey [*et al.*]{} (1994) and by Stephan [*et al.*]{} (1998).
The Divide and Conquer method
-----------------------------
The original formulation of the DC method (W. Yang 1991a, W. Yang 1991b, Zhao and Yang 1995) was based on a subdivision of the electronic density. To calculate the density at a certain point an ordinary electronic structure calculation is done for a sub-volume containing this point. Since the electronic density is only influenced by features in a rather small neighborhood, the density obtained in this way is a good approximation to the density one would obtain by doing a calculation in the whole volume occupied by the molecule or solid under consideration. The more recent formulation of this theory (W. Yang and T-S. Lee 1995) is however also based on the density matrix and we will discuss this version in more detail. The idea is to calculate certain regions of the density matrix by considering sub-volumes and then to generate the full density matrix by adding up these parts with the appropriate weights. How to calculate the density matrix for the sub-volumes is in principle unspecified but is usually done using ordinary electronic structure calculations based on exact diagonalization. Let us illustrate the principle for synthesizing the density matrix from its subparts in a pictorial way by considering a one-dimensional chain molecule. For simplicity let us also assume that we have atom centered basis functions.
First one divides the whole computational volume into sub-volumes which we will call central regions. Three such central regions are displayed in Figure (\[dcregs\]) by dark shading. Around each central region one puts a buffer region denoted by light shading. The sum of these two regions corresponds to the localization region of the other O(N) methods.
( 10.,4.7) (-2.,-0.0)
Once one has set up this subdivision, one does an electronic structure calculation within each localization region to obtain the density matrix belonging to this region. These different calculations are only coupled by the requirement that the Fermi level is the same in all the localization regions. Typically a very small temperature is used to stabilize the search for the overall Fermi level. From the density matrices obtained in this way, one cuts off all the corners, i.e. the regions where both matrix indices belong to basis functions in the buffer region. This transformation is shown for one localization region in Figure (\[dctrans\]).
( 10.,4.0) (1.,-0.3)
Using these cross-shaped blocks one then finally synthesizes the density matrix as shown in Figure (\[dcmat\]) by adding up the different contributions. The regions shown in dark shading which do not overlap with other regions have thereby weight one, whereas the overlapping regions indicted by light shading have each weight one half such that the sum of the weights is one as well in the overlap regions.
( 10.,5.5) (0.,-.3)
The achievement of linear scaling in the DC and OM methods is conceptually related. In both methods certain parts of the density matrix are calculated independently. The main difference is that in the FOE method no calculated parts of the density matrix are discarded in the way done in the DC method as depicted in Figure \[dctrans\]. The FOE method can thus be considered as some DC method where only the central part of the density matrix which is not contaminated by the boundary of the localization region is calculated. The fact that in the DC method only a small part of the density matrix obtained by costly diagonalizations is used, while the largest part associated with the buffer region is thrown away results in a large prefactor (Equation (\[cpu1\])) for this method. This is evidently a particularly serious disadvantage if large localization regions are needed as will be discussed in more detail in section \[compare\].
The calculation of the forces acting on the atoms within the DC method is also described by Yang and Lee (1995). Their force formula is based on the Hellmann-Feynman theorem (Feynman 1939) as well as some other terms such as Pulay forces (Pulay 1977) which arise from the use of atom centered basis sets and auxiliary charge densities. As has been discussed in the case of the FOE method the Hellman Feynman expression for the force (\[foeforce\]) is not exactly consistent with the total energy expression in a non-variational method, since it is based on the assumption that one is allowed to take traces. Even though the density matrix in the DC method is not calculated via a polynomial expansion, the analysis given for the FOE method also applies to the DC method since conceptually one can represent any matrix functional of $H$ as a polynomial Taylor expansion. The total energy will consequently not have its minimum exactly at the same place where the Hellman Feynman forces vanish if both quantities are calculated with the DC method. In the case of the FOE method there is a simple analytic expression for the calculation of the total energy, even in the case where localization constraints are imposed (Equation (\[etrun\])). One can therefore differentiate it without using the simplifications arising from the use of traces to obtain consistent forces. No such simple prescription can be written down for the DC method which would allow the calculation of consistent forces. Evidently this compatibility problem becomes negligible for large localization regions and there are certainly practical applications where small inconsistencies of forces and energies are tolerable.
The Density Matrix Minimization approach
----------------------------------------
The DMM approach of Li, Nunes and Vanderbilt (1993) is another approach where the full density matrix is constructed. In contrast to the FOE method one obtains the density matrix $F$ in the limit of zero temperature, so no adjustable temperature parameter enters the calculation. The density matrix is obtained by minimizing the following functional for the grand potential $\Omega$ with respect to $F$ $$\label{dmmfunc}
\Omega = Tr[ (3 F^2 - 2 F^3) (H - \mu I)] \: .$$ There is no constraint imposed during the minimization so all the matrix elements of $F$ are independent degrees of freedom. Nevertheless the final density matrix will obey the correct constraint of being a projector if no localization constraints are imposed. This is related to the fact that the matrix $3 F^2 - 2 F^3$ is a purified version of $F$ as can be seen from Figure \[weeny\]. If $F$ has eigenvalues close to zero or one then the purified matrix will have eigenvalues that are even closer to the same values. It is also clear from Figure \[weeny\] that the eigenvalues of the purified matrix are contained in the interval \[0;1\] as long as the eigenvalues of $F$ are in the interval \[ -1/2 ; 3/2 \].
( 8.,5.0) (-2.,-1.0)
The gradient of $\Omega$ as given by Equation (\[dmmfunc\]) with respect to $F$ is itself a matrix and it is given by $$\label{dmmgrad}
\frac{\partial \Omega}{\partial F} = 3 ( F \: H' + H' \: F) -
2 (F^2 \: H' + F \: H' \: F + H' \: F^2) \: ,$$ where $H' = (H - \mu I)$. In order to verify that Equation (\[dmmfunc\]) defines a valid functional we have to show two things. First, that the grand potential expression (\[dmmfunc\]) gives the correct result if we insert the exact density matrix $F$, and second, that the gradient (\[dmmgrad\]) vanishes in this case. From Equation (\[denst0proj\]) we see that the exact $F$ is a projection operator, i.e that $F^2 = F$. Therefore $(3 F^2 - 2 F^3) = F$ and the grand potential expression (\[dmmfunc\]) agrees indeed with the correct result (\[omegatrace\]). Using in addition the fact that $H'$ and the exact $F$ commute (as follows from Equations (\[hevcs\]), (\[fevcs\]) ) it is also evident that the gradient in Equation (\[dmmgrad\]) vanishes. The gradient vanishes however not only for the ground state density matrix $F$ but also for any excited state density matrix. In order to exclude the possibility of local minima, we have to verify, that these stationary points are no minima. This can easily be done (D. Vanderbilt, private communication) using the fact that the functional is a cubic polynomial with respect to all its degrees of freedom. Let us suppose that there are two minima. Inspecting the functional along the line connecting these two minima we would obviously again find these two minima, which is a contradiction because a cubic polynomial cannot have two minima. Thus we have proved by contradiction that the DMM functional has only one single minimum.
There is a second thing which is worrying at first sight with this functional. If the density matrix for an insulator at zero temperature is of the correct form (i.e. if the occupation numbers $n_l$ are integers) the gradient (\[dmmgrad\]) will vanish independently of the value of the chemical potential. This ambiguity however disappears as soon as one has fractional occupation numbers. Let us consider an approximate density matrix of the form $$\label{fap}
F = \sum_l n_l |\Psi_l> <\Psi_l| \: .$$ Then it is easy to see that $$\begin{aligned}
\Omega & = &
\sum_l (\epsilon_l -\mu) ( 3 n_l^2 - 2 n_l^3) \label{dmmmmx} \\
\frac{\partial \Omega}{\partial F} & = &
\sum_l 6 (\epsilon_l -\mu) n_l (1- n_l) |\Psi_l> <\Psi_l| \label{dmmmix} \: .\end{aligned}$$ The polynomial of Equation (\[dmmmmx\]) is the same as the one shown in Figure \[weeny\] and we see that components corresponding to eigenvalues larger than the chemical potential are damped until they vanish in the minimization process, whereas components corresponding to smaller eigenvalues are amplified until they reach the value one. Thus the chemical potential will determine the number of electrons to be found in the system as it should. The above statements are actually only correct if all the $n_l$’s are contained in the interval \[-1/2: 3/2\]. If this is not the case then one can see from Figure \[weeny\], that there can be runaway solutions, where some $n_l$ tend to $\pm \infty$. When we implemented the scheme we however never encountered in practice such a runaway case.
Having convinced ourselves, that the functional defined in Equation (\[dmmfunc\]) is well behaved, let us now estimate the number of iterations which are necessary to minimize it. As is well known, the error reduction per iteration step depends on the condition number $\kappa$ which is the ratio of the largest curvature $a_{max}$ to the smallest curvature $a_{min}$ at the minimum. These curvatures could be determined exactly by calculating the Hessian matrix at the minimum. Let us instead only derive an estimate of these curvatures by calculating the curvature along some representative directions. To do this let us now consider a ground state density matrix where some fraction $x$ of an excited state is mixed in
$$F(x) = \sum_{n=1}^{N_{el}} \Psi^*_n({\bf r}) \Psi_n({\bf r})
- x \Psi^*_I({\bf r}) \Psi_I({\bf r}) + x \Psi^*_J({\bf r}) \Psi_J({\bf r}) \: .$$
The index $I$ is a member of the $N_{el}$ eigenstates below $\mu$ and the index $J$ refers to a state above $\mu$. The expectation value of the OM functional for this density matrix is given by $$\begin{aligned}
\Omega(x) & = & Tr[ (3 F(x)^2 - 2 F(x)^3) (H - \mu I)] \\
& = & \sum_{n=1}^{N_{el}} \epsilon_n + \left( 3x^2-2x^3 \right) (\epsilon_J-\epsilon_I)
\nonumber\end{aligned}$$ and its second derivative by $$\label{dmmgscurve}
\left. \frac{\partial^2 \Omega(x)}{\partial x^2} \right|_{x=0} =
6 ( \epsilon_J - \epsilon_I ) \: .$$ The largest curvature will roughly be $\epsilon_{max} - \epsilon_{min}$ and the smallest curvature of the order of the HOMO-LUMO separation $\epsilon_{gap} = \epsilon_{N_{el}+1} - \epsilon_{N_{el}}$ The condition number is thus given by $$\label{dmmkappa}
\kappa = \frac{a_{max}}{a_{min}} \approx \frac{\epsilon_{max} - \epsilon_{min}}{\epsilon_{gap}} \: .$$
In the conjugate gradient method, which is the most efficient method to minimize the DMM functional, the error $e_k$ decreases as follows (Saad) $$e_k \propto \left( \frac{ \sqrt{\kappa}-1 }
{ \sqrt{\kappa}+1 } \right)^k \: .$$ The error $e_k$ is defined in this context as the length of the vector which is the difference between the exact and approximate solution at the k-th iteration step. Under realistic conditions $\kappa$ is large and the number of iterations $n_{it}$ to achieve a certain accuracy is therefore proportional to $$\label{cgnit}
n_{it} \propto \sqrt{ \kappa }
= \sqrt{ \frac{\epsilon_{max} - \epsilon_{min}}{\epsilon_{gap}} } \: .$$ This is an important result since it indicates that in an insulator the number of iterations is independent of system size. This result is also confirmed by numerical tests.
The use of a conjugate gradient scheme requires line minimizations along these conjugate directions. For arbitrary functional forms this has to be done by numerical techniques such as bisection (Press [*et al.,*]{} 1986). In the case of the DMM functional we have however a cubic form along each direction. The four coefficients determining the cubic form can be calculated with four evaluations of the functional. Once these 4 coefficients are known the minimum along this direction can easily be found.
Doing a series of minimization steps as outlined above will in general result in a density matrix which does not lead to the correct number of electrons. Thus one has to do some outer loops where one searches for the correct value of the chemical potential. For better efficiency, these two iterations loops can however be merged into one loop where one alternatingly minimizes the energy and adjusts the chemical potential (S-Y. Qui [*et al.,*]{} 1994).
The forces on the atoms are given by $$\label{dmmfrc}
\frac{d \Omega}{d R_{\alpha}} =
\frac{\partial \Omega}{\partial F} \frac{\partial F}{\partial R_{\alpha}} +
\frac{\partial \Omega}{\partial H} \frac{\partial H}{\partial R_{\alpha}} \: .$$ Since the method is variational, $\frac{\partial \Omega}{\partial F}$ vanishes at the solution and the force formula simplifies to $$\label{dmmforce}
\frac{d \Omega}{d R_{\alpha}} =
\frac{\partial \Omega}{\partial H} \frac{\partial H}{\partial R_{\alpha}} =
Tr \left[ (3 F^2 - 2 F^3) \frac{\partial H}{\partial R_{\alpha}} \right]$$ which can easily be evaluated.
The introduction of a localization region leads again to some subtleties. Whereas in the unconstrained case the eigenvalues of the final density matrix $F$ will all be either zero or one, this is not any more the case when a localization region is introduced. So the truncated $F$ is not any more a projection matrix but it is given by $$F = \sum_{m=1}^{M_b} n_m \Psi^*_m({\bf r}) \Psi_m({\bf r}) \: ,$$ where now $\Psi_m$ are the eigenfunctions of the truncated $F$ and the occupation numbers $n_m$ their eigenvalues. In a certain sense the localization constraint introduces a finite electronic temperature. This is actually not surprising after the discussion of the relation between the temperature and the localization properties in section \[general\]. Figure \[dmmfig\] shows the energy expectation values of the eigenvectors of $F$ versus the occupation numbers, for the case of a crystalline Si cell of 64 atoms, where the localization region extends up to the second nearest neighbors. As one sees, the energy expectation values $<\Psi_m|H|\Psi_m>$ of the eigenvectors of $F$ are very close to the exact eigenvalues of $H$.
( 8.,8.0) (-4.,-1.5)
This close correspondence of the eigenvectors of $F$ to the eigenvectors of $H$ explains why the number of iterations needed to find the minimum does not increase as one introduces localization constraints. Equation (\[dmmgscurve\]) remains approximately valid if the occupation numbers for the occupied states are close to 1 and if the occupation numbers for the unoccupied states are very small as well as if the energy expectation values $<\Psi_m|H|\Psi_m>$ are close to to the exact eigenvalues of the Hamiltonian. These conditions are fulfilled as discussed above. Hence the condition number for the minimization process does not change appreciably in the truncated case.
All the arguments used to prove the absence of local minima remain valid in the truncated case as well. The force formula Equation (\[dmmforce\]) remains equally valid.
An alternative derivation of this algorithm has been given by Daw (1993). He considers a differential equation which describes the evolution of a density matrix when the electronic temperature is cooled down from infinity to zero. The change of the density matrix during this process is equal to the gradient of Equation (\[dmmgrad\]).
The Orbital Minimization approach {#ommethod}
---------------------------------
The OM method (Mauri [*et al.,*]{} 1993, Ordejon [*et al.,*]{} 1993, Mauri and Galli 1994, Ordejon [*et al.,*]{} 1995, Kim [*et al.,*]{} 1995) also calculates the grand potential in the limit of zero temperature. In contrast to the two previous methods, it does not calculate the density matrix directly but expresses it via the Wannier functions according to Equation (\[denst0\]). These Wannier functions are obtained by minimizing the following unconstrained functional.
$$\label{omfunc}
\Omega = 2 \sum_n \sum_{i,j} c^n_i H'_{i,j} c^n_j
- \sum_{n,m} \sum_{i,j} c^n_i H'_{i,j} c^m_j \sum_l c^n_l c^m_l \: ,$$
where $c^n_i$ is the expansion coefficient of the $n$-th Wannier orbital with respect to the $i$-th basis function and $H'_{i,j}$ are the matrix elements of the shifted Hamiltonian $H-\mu I$ with respect to the basis functions. In the original formulation (Mauri [*et al.,*]{} 1993, Ordejon [*et al.,*]{} 1993, Mauri and Galli 1994, Ordejon [*et al.,*]{} 1995) only $N_{el}$ orbitals were included in the orbital sums in Equation (\[omfunc\]) (i.e. $n = 1 ... N_{el}$ , $m = 1 ... N_{el}$). In the formulation of Kim (1995) more than $N_{el}$ orbitals are included in the sums. The functional of Equation (\[omfunc\]) can be derived by considering the ordinary band structure energy expression $$\label{convfunc}
E_{BS} = \sum_n \sum_{i,j} c^n_i H'_{i,j} c^n_j$$ and by incorporating the orthogonality constraint by a Taylor expansion of the inverse of the overlap matrix $O$ between the occupied orbitals $$\label{omover}
O_{n,m} = \sum_l c^n_l c^m_l$$ up to first order. A family of related functionals can be obtained by Taylor expansions to higher order (Mauri and Galli 1994, Galli 1996). Since these functionals do not offer any significant advantage and are not used in calculations we will not discuss them. The gradient of the functional of Equation (\[omfunc\]) is given by $$\label{omgrad}
\frac{ \partial \Omega}{\partial c^n_k} = 4 \sum_j H'_{k,j} c^n_j
- 2 \sum_m \sum_j H'_{k,j} c^m_j \sum_l c^n_l c^m_l
- 2 \sum_m c^m_k \sum_{i,j} c^n_i H'_{i,j} c^m_j \: .$$
Let us first discuss this functional in the case where no localization constraint is imposed on the orbitals. It is easy to see that the functional (\[omfunc\]) is invariant under unitary transformations of the occupied (i.e. $N_{el}$ lowest) orbitals. So we can derive our results in terms of eigenorbitals rather than Wannier orbitals. The coefficients $c^n_i$ are then the expansion coefficients of the eigenorbitals. Using the fact that in this case $ \sum_l c^n_l c^m_l = \delta_{n,m}$ and that $\sum_{i,j} c^n_i H'_{i,j} c^m_j = \delta_{n,m} (\epsilon_n -\mu) $ we obtain $$\begin{aligned}
\Omega & = & 2 \sum_n \sum_{i,j} c^n_i H'_{i,j} c^n_j
- \sum_{n,m} \sum_{i,j} c^n_i H'_{i,j} c^m_j \delta_{n,m} \nonumber \\
& = & \sum_n \sum_{i,j} c^n_i H'_{i,j} c^n_j \nonumber \\
& = & \sum_n \epsilon_n - \mu N_{el} \nonumber \end{aligned}$$ which is the standard expression (\[omegatrace\]) for the grand potential. Similarly the gradient expression can be simplified obtaining $$\begin{aligned}
\frac{ \partial \Omega}{\partial c^n_k} & = & 4 \sum_j H'_{k,j} c^n_j
- 2 \sum_m \sum_j H'_{k,j} c^m_j \delta_{n,m}
- 2 c^n_k \sum_m \delta_{n,m} (\epsilon_m -\mu) \\
& = & 2 \sum_j H'_{k,j} c^n_j - 2 c^n_k (\epsilon_n -\mu) = 0 \nonumber \: .\end{aligned}$$ So the functional has indeed a vanishing gradient at the ground state and it gives the correct ground state energy. As was the case for the DMM functional the gradient vanishes not only for the set of ground state orbitals but also for any set of excited states. So we have to verify that these stationary points are not local minima but saddle points. We do this by picking a certain direction along which the curvature is negative. In the OM case an excited state is described by a set of $N_{el}$ orbitals $\Psi_n$ where at least one index $n=I$ corresponding to an occupied orbital $I$ is replaced by an unoccupied orbital $J$. Let us now consider the variation of the grand potential $\Omega(x)$ under a transformations of the form $\Psi_J \rightarrow \cos (x) \Psi_J + \sin (x) \Psi_I$. One can show that the curvatures at these stationary points is given by $$\label{dmmcurve}
\left. \frac{\partial^2 \Omega(x)}{\partial x^2} \right|_{x=0} =
- 4 ( \epsilon_J - \epsilon_I ) \: .$$ Since the unoccupied eigenvalue $\epsilon_J$ is higher in energy than the occupied one $\epsilon_I$, the curvature is negative and we have indeed a saddle point. In the same way we can also again show that the condition number is given by Equation (\[dmmkappa\]).
Also in analogy to the DMM functional one can show (Kim [*et al.,*]{} 1995) that in the formulation of Kim, the chemical potential $\mu$ determines the number of electrons by amplifying components below $\mu$ and annihilating components above it. Considering a state consisting of a set of eigenfunctions $\Psi_n$ of $H$ where each eigenfunction is multiplied by a scaling factor $a_n$, the expectation value for the grand potential becomes $$\label{ompl}
\Omega = \sum_n (2-a_n^2) a_n^2 (\epsilon_n-\mu) \: .$$ The relevant function $(2-x^2) x^2 $ is shown in Figure (\[ompoly\]). One can see that the minimum of Equation (\[ompl\]) is attained by $a_n=0$ if $\epsilon_n > \mu$ and by $a_n= \pm 1$ if $\epsilon_n < \mu$. Again this is only true if $a_n$ is within a certain safety interval. Otherwise there can be runaway solutions. Infinitesimally close to the solution $\mu$ becomes ill defined in an insulator as it should.
( 8.,5.5) (-2.,-1.5)
Whereas the DMM functional keeps all its good properties when one introduces a localization constraint, the OM functional looses most of them. The localization constraint is introduced in the OM functional by allowing each Wannier orbital to deviate from zero only within its own localization region. These localization regions are usually atom centered and contain a few shells of neighboring atoms. The basic idea of the OM functional, namely of describing an electronic system by a set of Wannier functions with finite support is already problematic. Orthogonality and finite support are mutually exclusive properties and so the orbitals which one obtains in the minimization process are necessarily non-orthogonal. The true Wannier functions are however orthogonal. In addition, as we have seen in the DMM case, a density matrix which is truncated has full rank, i.e. none of its eigenvalues is exactly zero. Thus $N_{el}$ Wannier orbitals are not sufficient to represent the density matrix in this case. The generalized formulation of Kim (1995) where more than $N_{el}$ orbitals are used alleviates this problem, but does not completely fix it unless the number of orbitals is equal to the number of basis functions $M_b$.
When implemented with localization constraints the OM functional exhibits the following problems:
- The functional has multiple minima (Ordejon [*et al.,*]{} 1995). Depending on the initial guess one obtains thus different answers, some of which are physically meaningless (Kim 1995). As we have shown above in Equation (\[dmmcurve\]) the functional has no multiple minima in the non-truncated case. The analysis we used to show this was based on the eigenfunctions. Since the Wannier functions have no resemblance to the eigenfunctions this analysis cannot be carried over into the localized regime. In the case of the DMM method the absence of multiple minima could be proven using the fact that the DMM functional is cubic. The OM functional (\[omfunc\]) is however quartic with respect to its degrees of freedom and will thus in general have multiple minima. The problem of the multiple minima is attenuated by the formulation of Kim (1995) but it is not completely eliminated since the functional still has quartic character. As a byproduct of the multiple minimum problem, the total energy cannot be conserved in molecular dynamics simulations, which is an important requirement. Here again, energy conservation is better in the Kim formulation but still far from perfect (Kim 1995).
In practical applications of the OM method for electronic structure methods great care is usually taken to construct input guesses which correspond to the physical bonding properties of the molecule under consideration (Itoh [*et al.,*]{} 1996). If the minima that is closest in distance is always selected during the subsequent line minimizations then one will most likely end up in a physically reasonable minima that reflects the bonding properties of the input guess (Stephan private communication). This is especially true if the localization regions are large and if the topology of the total energy surface within a reasonably large region around the physical minimum is not too different from the one of the non-truncated case. Such a procedure is of course not applicable in systems where the exact bonding properties are unknown.
- The number of iterations is very large whenever any localization constraint is imposed together with tight convergence criteria. This is due to the deterioration of the condition number, a phenomenon which is easy to understand (Ordejon 1995). Introducing a localization region destroys the strict invariance of the band structure energy under unitary transformations among the occupied orbitals. When the localization region is large, this invariance will still approximately exist and one can find certain directions around the minimum where the energy varies extremely slowly and where the curvature is therefore much smaller than the smallest curvature $\epsilon_{gap}$ in the unconstrained case. Whereas directions where the curvature is strictly zero do not affect the condition number, these very small curvatures will have a negative effect on the condition number (Equation (\[dmmkappa\])) and the required number of iterations is consequently much larger in the constrained case than in the unconstrained case. Even though the condition number deteriorates with increasing localization region, the detrimental effect on the number of iterations will disappear at a certain point where the gradient due to these small curvatures becomes smaller than the numerical threshold determining the convergence criterion of the minimization procedure.
- The optimal localization regions would be centered on the centers of the Wannier functions. Since these centers are not known a priori, atom centered localization regions are usually chosen. In this case the Wannier functions do not generally exhibit the correct symmetry (Ordejon 1995). As a consequence molecular geometries obtained from this functional can have broken symmetry as well. In a $C_{60}$ molecule for instance there are only two equivalent sites. When treated with the OM functional they are however all slightly different (Kim [*et al.,*]{} 1995).
- As follows from Equation (\[ompl\]) there can be runaway solutions. We have encountered this problem in test cases with random numbers as input guess. If one would construct a more sophisticated input guess, based on the bonding properties of the system, this would however probably not occur. In the DMM method the possibility of runaway solutions also exists but is never found in practice even with the most trivial input guess.
- If the method is used in the context of self-consistent calculations, where the electronic charge density is used to calculate the Hartree and exchange correlation potential, problems arise, since the total charge is not conserved during the minimization iteration (Mauri and Galli 1994).
To overcome the competing requirements of orthogonality and localization, a related approach has recently been proposed by Yang (1997) where the orbitals are allowed to be non-orthogonal. This approach of Yang has up to now not been applied in connection with a localization constraint.
The Optimal Basis Density Matrix Minimization method
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Despite its many advantages in the Tight Binding context, the DMM method has the big disadvantage that it is very inefficient if one needs very large basis sets (i.e. many basis functions per atom). Large basis sets are typically required in grid based Density Functional calculations. In this case it just becomes impossible to calculate and store the full density matrix in the DMM method even though it is a sparse matrix. From this point of view the Wannier function based methods are advantageous since they do not require the full density matrix. The basic idea of the OBDMM method (Hierse and Stechel 1994, Hernandez and Gillan 1995) is now to contract first the fundamental basis functions into a small number of new basis functions and then to set up the Hamiltonian and overlap matrix in this new small basis. A generalized version of the DMM method which can be applied to the non-orthogonal context (a subject which will be discussed later in the article) is then used to solve the electronic structure problem in this basis. The essential point is that one tries to do the contraction in an optimal way. This is done by minimizing the total energy also with respect to the degrees of freedom determining the contracted basis functions $\Psi_n$. Formulated mathematically the density matrix is given by $$\label{gillan}
F({\bf r},{\bf r}') = \sum_{i,j} \Psi^*_i({\bf r}) K_{i,j} \Psi_j({\bf r}') \: .$$ The matrix $K$ is a purified version of the the density matrix within the contracted basis $L$ and it is given by $$\label{gillan2}
K = 3 LOL -2 LOLOL \: ,$$ where $O$ is the overlap matrix among the contracted orbitals. The main difference between the formulation of Hierse and Stechel and of Hernandez and Gillan is that in the first formulation the number of contracted basis functions $\Psi_i$ is equal to the number of electrons, whereas in the second approach it can be larger. In the formulation of Hernandez and Gillan the basis set can for instance be chosen to have the size of a minimal basis set. The difference to standard minimal basis sets from quantum chemistry is that it is optimally adapted to its chemical environment since the contraction coefficients are not predetermined but found variationally. In practice the full density matrix is found by a double loop minimization procedure. In the inner loop one has the ordinary DMM procedure to find the density matrix for a given contracted basis set. In the outer loop one searches for the optimally contracted basis functions $\Psi_i$ for fixed $L$.
Unfortunately the minimization of the contracted basis functions $\Psi_i$ is ill conditioned (Gillan [*et al.,*]{} 1998) and the number of iterations is therefore at present very large. As already explained before ill-conditioning occurs if the curvatures in the minimum along different directions are widely different. Three causes for the ill-conditioning are reported by Gillan (1998).
- Length scale ill-conditioning: This problem is actually not related to the OBDMM algorithm itself but to the (uncontracted) basis functions which are taken to be so-called ”Blip“ functions in the present implementation. This kind of problem can be found in all iterative electronic structure algorithms if grid based basis functions such as finite elements are used. Its origin is easy to understand. Let us imagine that we are searching for the lowest state of jellium using a localized basis set associated with an equally spaced grid. By symmetry the solution is a constant vector, i.e. all basis functions have the same amplitude in the solution vector. Let us now assume that we explore the energy surface around the minimum along several directions. Let us first ”go” into a direction where we add components in such a way that the sign of the amplitude of each neighboring basis function changes. This corresponds to a high frequency plane wave and since the kinetic energy of such a plane wave is big, the total energy will rapidly increase if we add a such a contribution to our solution vector. If on the other hand we add contributions that correspond to low frequency plane waves the energy will increase much more slowly. Since in grid based methods the basis functions are usually narrow and since one can thus construct high frequency functions the condition number can be very bad. As one can suspect from the above explanation the different curvatures can be estimated by doing a Fourier analysis. With this information one can then use preconditiong techniques to cure the length scale ill-conditioning problem. Such a scheme has been proposed by Bowler and Gillan (1998).
- Superposition ill-conditioning: This ill-conditioning problem is essentially identical to the ill-conditioning problem of the OM functional. If we have $N_{el}$ contracted basis functions and no localization constraints the total energy is invariant with respect to unitary transformations of these functions. The introduction of a localization constraint destroys this invariance but there is an approximate invariance left with manifests itself in very small curvatures in the minimum along certain directions.
- Redundancy ill-conditioning: This problem can only be found in the formulation of Hernandez and Gillan, where the number of contracted basis functions is larger than the number of electrons. In this case one spans a space that contains not only the occupied orbitals but also some unoccupied. As was shown before in the context of the DMM functional introducing a localization constraint will not assign zero occupation numbers, but only very small occupation numbers to components corresponding to the unoccupied states in the unconstrained case. Since these components corresponding to the unoccupied states have now very little weight they have little influence on the total energy and one has again certain directions where the total energy changes very slowly resulting in very small curvatures.
Another open question is whether the OBDMM has local minima. The functional is a 6-th order polynomial with respect to the expansion coefficients of the contracted basis functions as can be seen from Equation (\[gillan\]) and (\[gillan2\]). The two overlap matrices in Equation (\[gillan2\]) give each a quadratic term, the two contracted orbitals in Equation (\[gillan\]) a linear term. Minimization with respect to the contracted basis functions should therefore exhibit local minima. Local minima have however not been reported with this method so far. Perhaps the following DMM minimization step which is free of local minima saves the method from overall local minima.
Comparison of the basic methods {#compare}
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It is certainly not possible to claim that a specific method is the best for all applications. Nevertheless the methods presented so far differ in many respects and one can therefore clearly judge under which limiting circumstances certain methods will fail or perform well. In the following the methods presented so far will therefore be compared under several important aspects. The comparison will be done in two categories. The first category applies to electronic structure methods requiring a small number of degrees of freedom per atom. The Tight Binding method belongs to the first category requiring a few basis functions per atom (or just a few degrees of freedom in the case of semiempirical Tight Binding). But we will also include the standard quantum chemistry methods into this first category, where one typically needs from a few up to a few dozen Gaussian type basis functions per atom. The second category contains methods which are grid based such as finite difference schemes (Chelikowsky [*et al.,*]{} 1994), or where the basis functions can be associated with grid points such as in finite element basis functions (White [*et al.,*]{} 1989) or blip (Hernandez [*et al.,*]{} 1997) basis functions. In these methods one has typically many hundred degrees of freedom per atom. Even though the density matrix is a sparse matrix, O(N) methods which calculate the full density matrix can not be applied to the second category of electronic structure methods. The memory requirements alone are already prohibitive. As pointed out before we can expect that the localization region in a 3-dimensional structure comprises on the order of 100 atoms. The density matrix will exhibit significant sparseness only for larger system. Assuming that we just have 100 basis functions per atom the storage of the density matrix would require about 1 Gigabyte of memory which is the upper limit of current workstations. The comparison in the large basis set class will therefore comprise only the methods which are Wannier function based namely FOP, OM and OBDMM. The comparison in the small basis set class will comprise FOE, DC, DMM and OM, excluding two methods which are explicitly targeted at large basis sets, namely FOP and OBDMM.
Small basis sets
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The comparison of the methods applicable to small basis sets is based on the following criteria:
- Scaling with respect to the size of the localization region: The size of the localization region is taken as the number of atoms contained within it. Only the FOE method has a linear scaling with respect to the size of the localization region. As one increases the size of the localization region the nonzero part of each column of the Chebychev matrices increases linearly implying also a linear increase in the basic matrix time vector multiplication part. In the DMM method the CPU time increases quadratically since the numerical effort for the basic matrix times matrix multiplications grows quadratically with respect to the number of off-diagonal elements of the matrix. Neglecting ill-conditioning problems, the OM method exhibits quadratic scaling, since the numerical effort for the calculation of the overlap matrix (Equation (\[omover\])) between the Wannier orbitals increases quadratically. As the localization region grows there are more matrix elements and the calculation of each matrix element is more expensive since each orbital is described by a longer vector. Because the ill-conditioning problem becomes more severe for large localization regions the number of iterations increases in reality in a way which is difficult to model resulting in an effective scaling that is stronger than linear. The DC method has a cubic scaling with respect to the size of the localization region if each sub-volume is treated with diagonalization schemes. From the comparison of the scaling behavior of all these methods one can thus conclude, that the FOE method will clearly perform best if large localization regions are needed. The FOE method is thus also the only method which can be faster than traditional cubically scaling algorithms if no localization constraints are imposed. In this case its overall scaling behavior is quadratic whereas all other methods have a cubic scaling with a prefactor which is significantly larger than the one for exact diagonalization.
- Scaling with respect to the accuracy: A detailed comparison of the polynomial FOE method and the DMM method under this aspect has recently been given by Baer and Head-Gordon (1997) for systems of different dimensionality. Their analysis is based on the assumption that the decay constant $\gamma$ is given by the tight binding limit of Equation (\[insulator\]). They conclude, that in the one dimensional case the DMM has the best asymptotic behavior, but its prefactor is much larger than the one of the FOE method, so that the FOE method is more efficient in the relevant accuracy regime. In the two dimensional case they have the same asymptotic behavior, but the FOE method has again a much smaller prefactor. In the most relevant three dimensional case the FOE method has both the best asymptotic behavior and prefactor. These results are plausible after the preceeding discussion of the scaling with respect to increasing localization region size. When one wants to improve the accuracy the most important factor is the enlargement of the localization region. It is also clear that in higher dimensions the number of atoms within the localization region grows faster than in lower dimensions and that the scaling with respect to the number of atoms will thus become the decisive factor in 3 dimensions. In lower dimensions the number of iterations has higher relative importance, favoring the DMM method. A comparison of the FOE and DMM method applied to quasi two-dimensional huge tight binding fullerenes by Bates and Scuseria (1998) is also in agreement with the above statements. They found that the FOE and DMM methods give nearly the same performance with a small advantage for the FOE method.
As discussed before, the scaling of the OM and DC methods is stronger than quadratic with respect to the size of the localization region. It is therefore clear that the required numerical effort for increased accuracy will grow even faster than in the DMM method.
- Scaling with respect to the size of the gap: In the FOE method the degree $n_{pl}$ of the Chebychev polynomial increases linearly with decreasing gap (Equation (\[npldegree\])). At the same time the density matrix decays more slowly. It follows from Equation (\[insulator\]) that the linear extension of the localization region grows as $\epsilon_{gap}^{-1}$ in the applicable weak binding limit. The volume of the localization region and the number of atoms contained in it grow consequently as $\epsilon_{gap}^{-3}$. Taking into account the number of iterations (Equation (\[npldegree\])), the total numerical effort increases thus as $\epsilon_{gap}^{-4}$. In the DMM method the number of iterations also increases with decreasing gap but more slowly namely like $\epsilon_{gap}^{-1/2}$ as follows from Equation (\[cgnit\]). Taking into account the above discussion of the scaling properties of the DMM method with increasing localization region we obtain the overall scaling of $\epsilon_{gap}^{-13/2}$, which is higher than the scaling behavior of the FOE method. Obviously the scaling behavior of the OM and DC methods is worse. So in contrast to what one might first think the FOE methods performs best in this limit. In three-dimensional metallic systems, the FOE method is thus to be expected to be the only method which will work efficiently at good accuracies.
- Finding a first initial guess: No initial guess is required in the FOE and DC methods (except perhaps for the potential in a selfconsistent calculation). In the DMM method an extremely simple and efficient input guess for the density matrix is just a diagonal matrix that sums up to the correct number of electrons. In the OM method this point is somewhat problematic. As mentioned above a Wannier function represents typically a bond or lone electron pair. So if one can draw the standard Lewis structure of a molecule, where bonds are denoted by lines and a lone electron pair by a pair of dots, one knows where the Wannier functions should be centered and the Lewis formula can be the basis for the initial guess. This procedure can not be used any more if the molecule is characterized by two or more Lewis structures that are resonating. Especially if the two Lewis structures correspond to an electron transfer over a distance which is larger than the range of the localization region, serious problems are to be expected with Wannier function based methods. In such a case it might actually not only be impossible to find an initial guess, but it might also be impossible at all to describe such a molecule by $N_{el}$ localized Wannier functions.
- Number of iterations in electronic structure calculations: In the variational methods (OM, DMM) the number of iterations depends on the condition number of the energy expression. As pointed out the OM energy expression is ill conditioned under localization constraints and therefore the required number of iterations is very large. Even for modest accuracy several hundred iterations are required ( Mauri and Galli 1994, Ordejon [*et al.,*]{} 1995). In the DMM method on the other hand the number of iterations is equal to the number of iterations one needs in ordinary electronic structure calculations (non O(N)), namely 20 to 30 (Nunes private communication). The quantity that corresponds to the number of iterations in the FOE method is the degree $n_{pl}$ of the polynomial. While it is difficult to compare the cost of one Chebychev recursion step with the cost of a DMM minimization step, such a comparison can be done in the case of the OM method. In each OM line minimization step one has to calculate the minimum of a quartic polynomial which requires at least 3 applications of $H$ to the wavefunction. One Chebychev recursion step requires one application of $H$.
- Number of iterations in molecular dynamics simulations: In molecular dynamics simulations as well as structural relaxations steps and selfconsistent mixing schemes the density matrix or respectively the Wannier functions from the previous step are a good input guess for the next step. Good initial input guesses are beneficial in all methods except in the polynomial FOE method and the DC method. It is difficult to quantify the possible savings of such a reuse. To preserve the quality of the solution of the preceeding step as an input guess in a Molecular Dynamics simulation, it can be necessary to make the time step smaller than the integration scheme would allow. How large the maximum time step can be depends of course also on the order and properties of the time integration scheme used to propagate the Molecular Dynamics simulation. Similar remarks apply to the case of structural relaxations. The decisive factor determining the number of iterations per molecular dynamics step is in this context again the condition number of the functional. With the DMM methods of the order of 2 to 3 steps are needed both for accurate molecular dynamics simulations (Qiu 1994) and for structural relaxations (Nunes private communication). The smallest number of iterations that was used in molecular dynamics simulations with the OM method was 10, but at the price of a very poor energy conservation.
- Cross over point for standard Tight Binding systems: The FOE method has the lowest reported cross over point for the standard carbon test-system in the crystalline diamond structure. For the FOE method it is around 20 atoms (Goedecker 1995), and for the DMM it is estimated (Li [*et al.,*]{} 1993) to be around 90 atoms. No crossover points were ever given for the OM and DC method and presumably they are much higher. All quoted cross over points for electronic structure calculations are for an accuracy of roughly 1 percent in the cohesive energies, but in the relevant publications not all computational details are listed to ensure that these numbers are really comparable in all respects. The low crossover point of the FOE method can be understood in terms of its scaling behavior with respect to the size of the localization region discussed above. For small systems the size of the localization region equals the size of the whole system. The FOE method therefore starts off with a quadratic scaling behavior, whereas all other methods start off with a cubic behavior. Consequently the cross over point for all other methods can only be for system sizes larger than the localization region, whereas the crossover point in the FOE method can already be at smaller system sizes if it is implemented efficiently.
In the context of molecular dynamics simulations the cross over points are different because some of the variational methods can benefit from good input guesses. For the FOE method the cross over point remains at 20 atoms, for the OM method Mauri and Galli quote 40 atoms, and for the DMM method Qui [*et al.*]{} (1994) quote 60 atoms. Again no crossover point is given for the DC method. It has to be stressed however that the number quoted by Qui [*et al.,*]{} (1994) was for highly accurate molecular dynamics runs where the total energy was conserved, while in the benchmarks of Mauri and Galli no satisfactory energy conservation was obtained.
- Influence of the range of a sparse Hamiltonian matrix on the performance: In the FOE method the numerical effort increases strictly linearly with respect to the number of nonzero elements per column which depends cubically on the range of the Hamiltonian matrix. In the case of the DMM method it can be shown (Li [*et al.,*]{} 1993) that one has to calculate intermediate product matrices only up to a range which is the sum of the range of the density matrix and the Hamiltonian matrix. As long as the range of the Hamiltonian is small compared to the range of the density matrix the number of operation increases therefore only very weakly with respect to an increasing Hamiltonian range. The DMM method therefore outperforms the FOE method under such circumstances (Daniels and Scuseria 1998). Hamiltonian matrices of relatively large range are found in the context of Density Functional calculations using Gaussian basis sets. For Tight Binding calculations, in contrast, the range of the Hamiltonian is usually small. The OM method shows the same behavior as the FOE method. The numerical effort increases linearly with respect to the number of nonzero elements per column of the Hamiltonian. In the DC method the numerical effort is independent of the bandwidth, but it is not expected that even in this case the DC method might outperform the FOE or DMM method.
- Scaling with respect to the size of the basis set: Let us now consider the case, where the number of atoms as well as all other relevant quantities, such as the size of the localization region, are fixed and where we only increase the number of basis functions per atom $m_b$. Both the size and the number of off-diagonal elements per column of the density matrix will then increase. We will also assume that the Hamiltonian is a sparse matrix with $m_b$ off diagonal elements per column. In the DMM method the numerical effort will consequently grow cubically with respect to $m_b$, since the number of operations needed for the multiplication of two sparse matrices of size $n$ with $m$ off-diagonal elements per column is proportional to $n \: m^2$. The DC method scales cubically as well since it is based on diagonalization. The FOE method equally scales cubically, since three factors are increasing. The number of columns in the density matrix, the number of coefficients in each column and the number of off-diagonal elements of the Hamiltonian matrix. In addition to the arguments showing the unrealistically large memory requirements of these methods when used with large basis sets, we thus also find a cubic scaling which prohibits the use of these algorithms in this context.
In the OM method both the application of the Hamiltonian to the orbitals as well as the calculation of the overlap between the orbitals scale quadratically with respect to $m_b$.
- Memory requirements: The DMM method requires the storage of the whole sparse density matrix. If one takes advantage of the fact that the matrix is symmetric storage can actually be cut into half. The OM method requires only the storage of the truncated Wannier orbitals and so the memory requirements are reduced by about 50 percent in the typical Tight Binding context compared to the case where one stores the all the nonzero elements of the density matrix without taking advantage of its symmetry. If the Kim formulation is used the gain can come down to less. In both the DC and FOE method only the subparts respectively the columns of the density matrix which are consecutively calculated need to be stored. The storage requirements are therefore greatly reduced compared to the DMM and OM methods, namely by a factor of roughly $N_{el}$.
- Parallel implementation: Parallel computers and clusters of workstations are standard tools in the high performance computing environment. The suitability of an algorithm for parallelization is therefore also an important aspect. It is of course always possible to parallelize any program, the question is just whether this can be done in a coarse or fine grained way, i.e. with a small or large communication to computation ratio. Only a coarsed grained parallel program will run efficiently on clusters of workstations with relatively slow communications as well as on a very large number of processors of a massively parallel computer. Both the FOE and the DC algorithms are intrinsically parallel algorithms, meaning that the big computational problem is subdivided into smaller subproblems which can be solved practically independently. In the case of the FOE method (Goedecker and Hoisie 1997) the calculations of the different columns of the density matrix are practically independent (Equation (\[recurs\])). In the case of the DC method the calculations of the different patches of the density matrix are practically independent as well. Both methods can therefore be implemented in a coarse grained way. A parallel program based on the FOE method won the 1993 Gordon Bell prize in parallel computing for its outstanding performance on a cluster of 8 workstations, obtaining half of the peak speed of the whole configuration (Goedecker and Colombo 1994). Impressive speedups of up to 400 have been obtained with the FOE method on a 800 processor parallel machine (Kress [*et al.,*]{} 1998). Even though it is more difficult to implement the OM method in parallel two such implementations have been reported. In the OM method two parallelization schemes are conceivable. In the first scheme (Canning [*et al.,*]{} 1996) one associates to each processor a certain number of localized orbitals. This data structure is optimal for the application of the Hamiltonian to the orbitals, but requires communication for the calculation of the overlap between the orbitals. The second scheme (Itoh [*et al.,*]{} 1995) associates the coefficients of all the orbitals whose localization region has an intersection with a certain region of space to a certain processor. This data layout is optimal for the calculation of the overlap matrix, but requires communication for the application of the Hamiltonian. The OBDMM method has also been parallelized (Goringe [*et al.,*]{} 1997b). Since the OBDMM is more complex than the other methods that have been implemented on parallel machines three different parallelization paradigms are required.
- Quality of forces: In the case of the variational (DMM and OM) methods the force formula is particularly simple (Equation \[dmmforce\]) since only the Hellman Feynman term survives. It has to be stressed that this formula is however only exact if one has succeeded in reducing the gradient with respect to all variational quantities really to zero. If, in a simulation, the gradient is not reduced to zero within the required precision because too many iterations would be required, errors will creep into the calculated forces, making them inconsistent with the total energy. From this point of view the situation is easier in the FOE method. Since the FOE method is not an iterative method (in the sense that one iterates until a certain accuracy criterion is met), the force formula of Voter (Equation (\[forcevoter\])) will always give forces consistent with the total energy. As has already been discussed no consistent force formula exists for the DC method.
Consistent forces are a prerequisite for the conservation of the total energy in Molecular Dynamics simulations. Even with consistent forces there are however other factors which can cause deviations from perfect total energy conservation in Molecular Dynamics simulations such as finite time steps and events where atoms enter or leave the localization region.
- Cases where the methods become inefficient: Cases where different methods become inefficient have already implicitly been pointed out when discussing the previous criteria. Let us finally mention a special case where the FOE method is inefficient. As discussed above a small gap implies usually highly extended density matrices and the FOE method is highly competitive. There can however be exceptions to this rule. If by symmetry restrictions there is practically no coupling between two well localized states, which are close together, their energy levels can be split by a very small amount. If the Fermi level falls in between these two levels a very high degree polynomial is needed to separate these two levels in an occupied and an unoccupied one. This scenario can be found in the case of a vacancy in the silicon crystal. A Jahn Teller distortion leads to a very small splitting between an occupied and an unoccupied gap level. Using a high electronic temperature and a low degree polynomial in the FOE method does not reproduce this Jahn Teller distortion. A detailed study of this effect is given by Voter [*et al.,*]{} (1996) showing that a polynomial of degree 200 is needed instead of the typical polynomial of degree 50 needed for bulk silicon in the Tight Binding context. From an energetic point of view it is frequently not necessary to track such Jahn Teller distortions, since they lead to a rather small energy lowering. In molecular dynamics simulations of metallic systems this suppression of the opening of a gap can even be beneficial (Goedecker and Teter 1995) since it leads to a smoother density of states around the Fermi level.
In summary, we see that the performance depends critically on many parameters which can change from one application to another. Performance superiority claims based on test runs of a particular system have therefore to be taken with caution.
Large basis sets
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Whereas the methods which are mainly applicable in the context of small basis sets showed important differences under the various comparison criteria, the behavior of the FOP, OM and OBDMM methods are quite similar under most of these criteria. The comparison of the methods which are applicable to large basis sets will therefore be based only on a smaller set of important criteria.
- Scaling with respect to the size of the basis set: As pointed out before the methods compared in this sections all have a reasonable scaling with respect to the size of the basis set allowing thus their use in the context of very large basis sets. In contrast to the discussion of the same point within the small basis set framework, the number of nonzero matrix elements of the Hamiltonian is typically independent of the resolution of the grid, i.e. of the number of basis functions. The most important part of the FOP, OM and OBDMM algorithms, the application of the Hamiltonian matrix to a wave vector scales therefore linearly. At the same time all these algorithms require at some stage the calculation of an overlap matrix among the occupied orbitals. This part scales quadratically as discussed before.
- Finding a first initial guess: As discussed in the comparison part dealing with small basis sets, it can be difficult to find an initial guess for Wannier function based methods. This problem does not exist in the OBDMM method if the number of orbitals is larger than the number of electrons. In this case the orbitals are just basis functions and by analogy with the usual tight binding or LCAO basis sets it should always be possible to generate a physically motivated initial guess for these orbitals.
- Required number of iterations: As mentioned both the OM and OBDMM methods suffer from ill-conditioning problems and require therfore a frequently excessive number of iterations for the iterative minimization. No such ill-conditioning exists for the FOP method.
- Cases where the methods become highly inefficient: None of the 3 methods have ever been applied to metallic systems, and they are all expected to fail in this case.
Other O(N) methods
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The recursion method is a well established method which also exhibits O(N) scaling. It is principally a method to calculate the density of states $D(\epsilon)$, but once the density of states is known, the band structure energy can be calculated by integrating $\epsilon D(\epsilon)$ up to the Fermi level. The recursion method has been described extensively (Haydock 1980, Gibson [*et al.,*]{} 1993) and we will therefore not review it. Let us just point out that within the original formulation of the recursion method only the diagonal elements of the density matrix could be calculated. For the calculation of the forces one would however also need the off-diagonal elements. So the applicability of the recursion method is significantly reduced compared to the O(N) method described in section \[four\], which all gave access to the forces. There have been several attempts to overcome this limitation (Aoki 1993, Horsfield [*et al.,*]{} 1996b, Horsfield [*et al.,*]{} 1996c, Bowler [*et al.,*]{} 1997). In contrast to the methods of section \[four\] these Bond Order Potential (BOP) methods are fairly complex and difficult to implement. The basic idea in the BOP method is to calculate the off diagonal elements of the density matrix as the derivative of diagonal elements of a density matrix defined with respect to a transformed basis. Even though it is now possible to calculate forces within the BOP method, they are unfortunately not consistent with the total energy. In another version of the recursion method the GDOS method (Horsfield [*et al.,*]{} 1996b, Horsfield [*et al.,*]{} 1996c) the exact forces can be calculated. It is however necessary to calculate some generalized moments $H^k$ called interference terms from the recursion coefficients. This inversion is ill conditioned and becomes unstable if one tries to calculate more than 20 moments (Bowler [*et al.,*]{} 1997). With such a low number of moments it is however not possible to describe many realistic systems (Kress and Voter 1995) and the error one reaches when the instability sets in is frequently still much too large to be acceptable (Bowler [*et al.,*]{} 1997). So recursion based methods seem not to be a general purpose tool for electronic structure calculations where accurate energies and forces are required. BOP methods can however give insight into basic bonding properties of crystalline solids (Pettifor). Since in these methods related to the recursion algorithm all the diagonal elements of the density matrix have to be calculated, they are obviously not very efficient if a very large number of basis functions per atom is used and they were indeed primarily proposed for Tight Binding or other schemes with a small number of basis function. The only exception is a version proposed by Baroni [*et al.*]{} (Baroni and Giannozzi 1995) who suggest to use delta functions as a basis in a Density Functional type calculation. With his basis set the diagonal elements of the density matrix are enough to determine the charge density, whereas for more general basis functions the off diagonal elements are needed as well (Equation (\[rhotrace\])). Because the number of delta functions has to be very large even in the most favorable case of silicon, the crossover point was estimated to be around 1000 silicon atoms. This method is therefore clearly not competitive with most other methods where the cross over point is much lower.
Another approach to improve the scaling properties is based on so called pseudo-diagonalization (Stewart [*et al.,*]{} 1982). The method is closely related to the well known Jacobi method for matrix diagonalization. Whereas in the original Jacobi method rotation transformations are applied until all off-diagonal elements vanish, only the entries which couple occupied and unoccupied states are annihilated in the pseudo-diagonalization method. One obtains thus not the occupied eigenvectors of the Hamiltonian but an arbitrary set of vectors which span the same occupied space. In its original formulation (Stewart and Pulay 1982) this method still had cubic scaling however with a smaller prefactor than complete matrix diagonalization. Nearly linear scaling has been reported with this method (Stewart 1996) if the Hamiltonian matrix is constructed with respect to a set of well localized orbitals. In this way most of the elements in the block coupling occupied and unoccupied states are zero at the start of the transformations. The annihilation of certain matrix elements during the rotation steps causes only a controlled spread of other rows and columns in the matrix. So at the end each column and row extends over a region which is comparable to the localization region in other O(N) methods.
A method proposed by Galli and Parrinello (1992) can be considered as a predecessor of the OM method. The total energy is minimized with respect to a set of localized Wannier functions. In contrast to the OM method one has however the old conventional functional (Equation (\[convfunc\])) which has to be minimized under the constraint of orthogonality, whereas in the OM method the orthogonalization constraint is contained in the modified functional (Equation (\[omfunc\])). In this scheme it is necessary to calculate the inverse of the overlap matrix between the Wannier functions. From timing considerations this is not a serious drawback since this part is only a small fraction of the total workload even for large systems and even if it is done with a scheme which scales cubically. There are, however, problems of numerical stability. As pointed out by Pandey [*et al.*]{} (1995) the overlap matrix becomes close to singular and the introduction of localization constraints can under these circumstances falsify the results.
Also vaguely related to the OM functional are methods where certain constrains are included by a penalty function. Wang and Teter (1992) included the orthogonality constraint in this way, Kohn (1996) the idempotency condition of the zero temperature density matrix.
O(N) implementations of electronic structure methods based on the multiple scattering theory have also been reported. In the simplest version (Wang [*et al.,*]{} 1995) it is essentially a DC method with the difference that within each localization region the calculation is done with the multiple scattering method. A further development was to replace the buffer region by an effective medium (Abrikosov [*et al.,*]{} 1995, Abrikosov [*et al.,*]{} 1996). This can considerably reduce the prefactor of the calculation, especially in metallic systems where large localization regions are needed. For this class of methods no force formulas have however been reported, a deficiency restricting their applicability.
A scheme which possibly leads to a reduced scaling behavior has also be proposed by Alavi [*et al.,*]{} (1994) It is based on a direct representation under the form of a sparse matrix of the Mermin finite temperature functional (Mermin 1965). So it allows for a finite temperature as does the FOE method.
As was already mentioned the polynomial FOE method becomes inefficient in cases where one has a large basis set causing the highest eigenvalue to grow very large. This would necessitate a Chebychev polynomial of very high degree. A elegant method to overcome this bottleneck within a polynomial method has been proposed by Baer and Head-Gordon (1998). They write the density matrix at a low temperature $T$ as a telescopic expansion of differences of density matrices at higher temperatures $Tq^j$. $$F_{T} = F_{Tq^n} + \left( F_{Tq^{n-1}} - F_{Tq^n} \right)
+ \left( F_{Tq^{n-2}} - F_{Tq^{n-1}} \right)
+ ...
+ \left( F_{Tq^{0}} - F_{Tq^{1}} \right)$$ The factor $q$ determining the geometric sequence of temperatures is chosen from considerations of numerical convenience. As the temperature is lowered the numerical rank of each difference term becomes smaller and smaller since the difference of two Fermi distributions of similar temperature is vanishing to within numerical precision over most of the spectrum. Hence it is necessary to find Chebychev expansions only over successively smaller regions of the spectrum and it is also possible to calculate the traces (which in turn determine all physically relevant quantities) within spaces of smaller and smaller dimension.
Ordejon [*et al.*]{} (1995) proposed a method based on the OM method to calculate phonons from the electronic structure with linear scaling.
This article concentrated on O(N) methods which are able to calculate the total energy of a system as well as its derived quantities such as the forces acting on the atoms. Let us finally still point out that there are also several O(N) methods which primarily calculate the density of states and thus give information about the eigenvalue spectrum of a system. These methods (Drabold and Sankey 1993, Wang 1994, Silver and Roeder 1997) are not described in this article. In principle, it would also be possible to derive band structure and total energies from the spectrum by an integration up to the chemical potential. Attempts in this direction have however not been very successful up to now with the exception of the smeared density of states Kernel Polynomial method (Voter [*et al.,*]{} 1996) which is closely related to the FOE method.
Some further aspects of O(N) methods
====================================
Hierse and Stechel (1994, 1996) examined whether non-orthogonal Wannier like orbitals are transferable from one chemical environment to another similar one. If this was the case one could reuse precalculated Wannier orbitals as a very good initial starting guess. Such a property would thus reduce the prefactor of any method regardless of its scaling behavior. Unfortunately, they could find reasonable transferability only under rather restrictive conditions. When they added $C H_2$ units to a $C_n H_{2n+2}$ polymer they found good transferability of the orbitals associated with this building block within a Density Functional scheme (Hierse and Stechel 1994). As soon as they started bending the polymer (Hierse and Stechel 1996), the transferability was however lost in the Density Functional scheme. Only in a non-self-consistent Harris functional scheme some efficiency gains were still possible.
Hernandez [*et al.*]{} (1997) suggest a solution to the basis set problem in O(N) methods. As was mentioned in section \[four\] O(N) techniques are difficult to reconcile with extended basis sets such as plane waves. Plane waves have however several important advantages and are therefore widely used in conventional (i.e. not O(N)) electronic structure calculations. One of their main advantages is that the accuracy can easily be improved by increasing one single parameter, namely the minimal wavelength which corresponds to the resolution in real space. Hernandez [*et al.*]{} proposed now a basis set of “blip" functions which combines both advantages. It is localized and its resolution can systematically be improved. As an alternative to the “blip" functions one could also use finite difference techniques (Chelikowsky [*et al.,*]{} 1994) or wavelets (Lippert [*et al.,*]{} 1998, Goedecker and Ivanov 1998b, Arias 1998, Goedecker 1998b) since they share the same advantages. As shown by Goedecker and Ivanov (1998c) wavelets allow for highly compact representation of both the density matrix and the Wannier functions, since they are localized both in real and Fourier space.
Much of the work of Ordejon (1996) is also based on a new set of basis functions proposed by Sankey and Niklewski (1989). This basis set consists of atomic orbitals which are modified in such a way that they are strictly zero outside a certain spherical support region. This then gives rise to a Hamiltonian matrix which is strictly sparse. By tabulating these matrix elements it is possible to do Density Functional calculations whose computational requirements are in between the requirements of traditional Density Functional calculations and Tight Binding calculations. Obviously one has to find a compromise between accuracy and speed. If the basis functions extend about a larger support region, one has a more accurate basis, but the the numerical effort increases because the Hamiltonian is less sparse.
Horsfield and Bratkovsky (1996d) have incorporated entropy terms in O(N) methods within the FOE algorithm. As soon as one has systems at nonzero temperature, these terms should in principle be added, however in most systems their effects are very small at room temperature. For computational convenience temperatures much larger than room temperature can however be used. Wentzcovitch [*et al.*]{} (1992) and Weinert and Davenport (1992) showed that the inclusion of entropy gives simplified force formulas, since only the Hellmann-Feynman term survives. The free energy $A$ is given by $$A = Tr [F \: H - k_B T \: S(F)] \: ,$$ where the entropy $S$ is a matrix function of $F$ $$S = - \left( F \: \ln (F) + (1-F) \: \ln (1-F) \right) \: .$$ Writing $S$ as a Chebychev polynomial in $H$ and analyzing everything in terms of the eigenfunctions of $H$ they find that one has to do a Chebychev fit to a distribution very similar to the Fermi distribution just with some additional features close to the chemical potential. Using a formalism by Gillan (1998) they then extrapolate their results to zero temperature obtaining faster convergence to the zero temperature limit in this way. Let us stress again, that with the FOE method it is possible to build up density matrices corresponding to several temperatures without significant extra cost. A set of generalized Fermi distributions which allow an efficient extrapolation to the zero temperature limit by eliminating arbitrarily high powers of $T$ has also been proposed by Methfessel and Paxton (1989)
As was mentioned in the introduction the fundamental cubic term in an electronic structure calculation based on orbitals comes from the orthogonalization requirement. In traditional pseudopotential calculations based on a Fourier space formulation there is however a second very large cubic term, arising from the nonlocal part of the pseudopotential. This second cubic term can be eliminated by using pseudopotentials which are separable in real space (King-Smith [*et al.,*]{} 1991, Goedecker [*et al.,*]{} 1996, Hartwigsen [*et al.,*]{} 1998)
Non-orthogonal basis sets {#nonorthog}
=========================
Up to now we have always implicitly assumed that we are dealing with orthogonal basis sets, i.e. that $$\label{ortho}
\int \phi^*_i({\bf r}) \phi_j({\bf r}) d{\bf r} = \delta_{i,j} \: .$$ Non-orthogonal basis sets give rise to an overlap matrix $S$, $$S_{i,j} = \int \phi^*_i({\bf r}) \phi_j({\bf r}) d{\bf r} \: .$$ An orthogonality relation similar to Equation (\[ortho\]) can be obtained by introducing the dual basis functions $\tilde{\phi}_i({\bf r})$ given by $$\tilde{\phi}_i({\bf r}) = \sum_j S^{-1}_{i,j} \phi_j({\bf r}) \: ,$$ where $S^{-1}$ is the inverse of the overlap matrix $S$. It is then easy to verify that $$\int \tilde{\phi}^*_i({\bf r}) \phi_j({\bf r}) d{\bf r} = \delta_{i,j} \: .$$ As has been mentioned in section \[four\] all realistic atom centered localized basis sets are non-orthogonal. Within the Tight Binding context, there are also many non-orthogonal schemes on the market. There is thus certainly a need to apply O(N) techniques also for these schemes. Most of the basic O(N) algorithms presented previously have therefore been generalized to the non-orthogonal case and we will present these generalizations in the following. In the context of a non-orthogonal scheme one has to distinguish carefully between the eigenfunctions $\Psi_n$ and the associated eigenvector ${\bf c}^n$ which contain the expansion coefficients $c_i^n$ such that $\Psi_n({\bf r}) = \sum_i c_i^n \phi_i({\bf r})$. The eigenvector ${\bf c}^n$ satisfies the generalized eigenvalue equation $$H {\bf c}^n = \epsilon_n S {\bf c}^n \label{gevcs} \: .$$ In the same way one has to distinguish carefully between the density matrix operator $F({\bf r},{\bf r}')$ and the density matrix $F_{i,j}$ itself. While the expression (\[fdens\]) for the density matrix operator remains the same, $$\begin{aligned}
\label{frrno}
F({\bf r},{\bf r}') = \sum_n f(\epsilon_n) \Psi^*_n({\bf r}) \Psi_n({\bf r}')
= \sum_n f(\epsilon_n) \sum_{i,j} c^{n *}_i c_j^n \phi^*_i({\bf r}) \phi_j({\bf r}') \end{aligned}$$ the expression for the number of electrons (Equation (\[nel\])) is modified to $$N_{el} = Tr[F \: S] \: .$$ The expression for the band structure energy (\[ebstrace\]) however remains valid. In the definition of the density matrix $F_{i,j}$ (Equation (\[fijdef\])) one has to use now the dual basis functions instead of the ordinary, $$\label{fijdefn}
F_{i,j} = \int \int \tilde{\phi}^*_i({\bf r}) F({\bf r},{\bf r}') \tilde{\phi}_j({\bf r}') d{\bf r} d{\bf r}' = \sum_n c^{n *}_i c_j^n f(\epsilon_n)$$ This replacement can have important consequences for the locality of the density matrix $F_{i,j}$. If we have a set of localized orthogonal basis functions (the only known set of basis functions with this property are the Daubechies scaling functions), whose extension is less than the oscillation period of the density matrix operator then Equation (\[fijdef\]) ensures that $F_{i,j}$ will have the same decay properties as $F({\bf r},{\bf r}')$. This does not necessarily hold true for Equation (\[fijdefn\]). Even if the basis functions $\phi_i({\bf r})$ are well localized this is in general not true for their duals $\tilde{\phi}_i({\bf r})$. If the duals have a very slow decay then this slow decay will be inherited by $F_{i,j}$ and it might not be possible to use O(N) algorithms for the calculation of $F_{i,j}$. Problems might for instance arise if high quality Gaussian basis sets containing diffuse functions are used. Preliminary experience indicates that for small basis sets of moderate quality the duals are not so delocalized as to destroy the localization of $F_{i,j}$.
In the case of the DC method the generalization to the non-orthogonal case is trivial. Since it is based on diagonalization within each subvolume one has to solve just the generalized eigenvalue problem (Equation (\[gevcs\])) instead of an ordinary one. In the Density Functional context, the DC method has actually only been used with non-orthogonal orbitals.
The FOE method using a Chebychev representation of the density matrix has been generalized by Stephan [*et al.*]{}(1998). It is easy to see that all the central equations of the FOE method remain correct if the Hamiltonian $H$ is replaced by a modified Hamiltonian $\bar{H}$ (that is not any more hermitian) given by $$\label{modham}
\bar{H} = S^{-1} H \: .$$ In particular, it remains true that the density matrix is given within arbitrary precision by $$F \approx \frac{c_0}{2} I + \sum_{j=1}^{n_{pl}} c_j T_j(\bar{H})$$ if a sufficiently large $n_{pl}$ is used. The problem is how to find $\bar{H}$ efficiently. Even if $S$ is a sparse matrix the inverse $S^{-1}$ is not sparse in general and $\bar{H}$ would be a full matrix as well, destroying immediately the linear scaling. However, it turns out that the matrix elements of $\bar{H}$ decay faster than the corresponding matrix elements of $H$ (Stephan 1998, Gibson 1993). One can therefore cut off the Tight Binding Hamiltonian $\bar{H}$ at the same distance where one would usually cut off $H$. In this way all the matrices involved are reduced to sparse matrices and $\bar{H}$ can be constructed by solving the set of linear systems $$S \bar{H} = H \: .$$ Since both the right hand sides in $H$ and the solution vectors making up $\bar{H}$ are sparse different systems of equations are only coupled by subblocks of $S$. So the big matrix inversion problem is decoupled into many small systems of equations and the scaling is therefore strictly linear.
If the FOP method is used in connection with a rational approximation the generalization to the non-orthogonal case can be done straightforwardly and without any approximation (Goedecker 1995) $$F = \sum_{\nu} \frac{w_{\nu} }{H-z_{\nu} S } \: .$$ One has then to solve a systems of linear equations which is a generalization of Equation (\[padeeq\]) $$\begin{aligned}
(H-z_{\nu} S ) F_{\nu} & = & I \label{green} \\
F & = & \sum_{\nu} w_{\nu} F_{\nu} \: .\end{aligned}$$ A similar approach was adopted by Jayanthis [*et al.*]{} (1998). They formulated their method in terms of the Green function. However, $F_{\nu}$ in Equation (\[green\]) is a Green function for a complex energy $z_{\nu}$ and the methods are essentially equivalent.
If the FOE method is used to calculate Wannier functions the required projection operator $F_p$ is slightly different from the density matrix and it is given by $$F_p = \sum_{\nu} \frac{w_{\nu} S }{H-z_{\nu} S } \: .$$
The DMM method has also been generalized to the non-orthogonal case (Nunes and Vanderbilt 1994). Introducing a modified density matrix $\bar{F}$ defined as $$\bar{F} = S^{-1} F S^{-1}$$ the DMM functional (\[dmmfunc\]) becomes $$\Omega = Tr[ (3 \bar{F} S \bar{F} - 2 \bar{F} S \bar{F} S \bar{F}) (H - \mu I)] \: .$$ This has the advantage that one does not have to invert $S$ if one minimizes directly with respect to $\bar{F}$. The calculation of the gradient of the DMM functional in the non-orthogonal case is a tricky point. The definition of the gradient is not as absolute as one might think. It is the direction of steepest descent per unit change of the variables, and one must therefore define a norm for the multidimensional space of variables before defining the gradient (D. Vanderbilt private communication). Two different gradient expression have been proposed by Nunes and Vanderbilt (1994) and by White [*et al.*]{} (1997) which correspond to two different choices of metric. The gradient of White [*et al.*]{} (1997) requires less minimization steps (Gillan [*et al.,*]{} 1998), but each minimization step is more expensive since it requires the calculation of the inverse of the overlap matrix. From the point of view of overall numerical efficiency it is therefore not clear which gradient expression is more efficient.
Another generalization (Millam and Scuseria 1997, Daniels [*et al.,*]{} 1997) of the DMM method which is similar in spirit to Stephan’s generalization of the FOE method consists in first performing a transformation to an orthogonal basis set by finding the LU decomposition of the overlap matrix $$\label{ludec}
S = U^T U \: ,$$ where U is an upper triangular matrix. If in addition $U$ is approximated as a sparse matrix with $m$ off-diagonal elements then Equation (\[ludec\]) can be solved with a scaling proportional to $n \: m^2$ where $n$ is the dimension of the matrices involved. Thus the scaling with respect to the size of the system is linear as it should be.
The OM method can easily be generalized to the non-orthogonal case (Ordejon [*et al.,*]{} 1996). The orbital overlap $\sum_l c^n_l c^m_l$ in the OM functional (Equation (\[omfunc\])) has to be generalized to $\sum_{l,k} c^n_l S_{l,k} c^m_k$.
The calculation of the selfconsistent potential
===============================================
We will now discuss an issue which is relevant only in selfconsistent electronic structure calculations, namely the calculation of the potential arising from the electronic charge. This potential consists essentially of two parts, the electrostatic or Hartree potential and the exchange correlation potential.
The electrostatic potential {#coulomb}
---------------------------
The solution of Poisson’s equation to find the electrostatic potential arising from a charge distribution $\rho$ is a basic problem found in many branches of physics. Solution techniques are described in a wide variety of books and articles. We will therefore only point out a few aspects which are important in the special context of O(N) electronic structure calculations.
If one is dealing with an electronic charge density which has only one length scale Poisson’s equation can be solved efficiently and with a scaling which is close to linear. Charge densities of this type are encountered in the context of pseudopotential calculations where one has eliminated the core electrons and where the characteristic length scale is the typical extension of an atomic valence electron. One can, for instance, use plane wave or multigrid techniques (Briggs) which both have a scaling proportional to $n \: \log (n)$ where $n$ is the number of grid points.
The situation becomes problematic when core electrons are included. In this case one could in principle still use the above mentioned techniques with a increased resolution. One would still have linear scaling, but the prefactor would be so large to make it completely impractical both from timing and memory considerations. Exactly the same arguments apply to the representation of the wave functions themselves and for this reason ordinary plane waves are not used for all electron calculations.
A widely used basis set for all electron calculations are Gaussian type basis sets (Boys 1960). By varying the width of the Gaussians one can describe both core and valence electrons. The popularity of Gaussian type basis functions comes from the fact that many important operations can be done analytically (S. Obara and Saika, 1986) One property which we will use is that the product of two Gaussians is again a Gaussian centered in between the two original Gaussian type functions. The matrix elements of the electrostatic potential part of the Hamiltonian with respect to a set of Gaussian orbitals $g_i({\bf r}), i = 1, ... , M_b$ are given by $$\label{gtohij}
H_{i,j} = \int d{\bf r} g_i({\bf r})
\left( \int d{\bf r}' \sum_{k,l} \frac{ F_{k,l} \: g_k({\bf r}') \: g_l({\bf r}') }{|{\bf r}-{\bf r}'|} \right) g_j({\bf r}) \: .$$ The elementary integral $$\int d{\bf r} \int d{\bf r}' \frac{ g_i({\bf r}) g_j({\bf r}) g_k({\bf r}') g_l({\bf r}') }{|{\bf r}-{\bf r}'|}$$ can also be calculated analytically (S. Obara and Saika, 1986). A straightforward evaluation of Equation (\[gtohij\]) would then result in a quartic scaling. There are, however, many well known techniques (Challacombe [*et al.,*]{} 1995) to reduce this scaling. The most obvious trick starts from the observation that there is a negligible contribution to the charge density $\rho$ if the Gaussians $g_l$ and $g_k$ are centered far apart. Consequently the charge density is not a sum over $M_b^2$ product Gaussians $G_m$, ($G_m = g_i g_j$), but only over $M_a$ such Gaussians $$\label{rhoprod}
\rho({\bf r}') = \sum_{k=1}^{M_b} \sum_{k=1}^{M_b} F_{k,l} \: g_k({\bf r}') \: g_l({\bf r}')
\approx \sum_{m=1}^{M_a} c_m G_m({\bf r}') \: .$$ The size of the auxiliary basis set $M_a$ is proportional to $M_b$ with a large prefactor which depends on the ratio of the largest to the smallest extension of the Gaussians as well as on the accuracy target. In a similar way matrix elements $H_{i,j}$ become negligible if the basis functions $g_i$ and $g_j$ are centered very far apart. Using these two approximations one obtains a quadratic scheme with a very big prefactor. An approximate quadratic scaling is also found in numerical tests (Strout and Scuseria 1995).
Another widely used method consists in fitting the charge density by a set of $M_a$ auxiliary Gaussians $G_i$. Even though we use the same symbols ($M_a$, $G_i$) as above they denote now somewhat different quantities. The allowed number of auxiliary Gaussians $M_a$ is now much smaller, namely a few times $M_b$. The auxiliary functions themselves are therefore not anymore taken to be all the possible product functions, but determined by empirical rules in such a way as to give the best possible fit in Equation (\[rhoprod\]). The fitting involves the solution of an ill conditioned system of linear equations and has therefore cubic scaling, however with a small prefactor. The evaluation of the matrix elements using this representation of the charge density has then quadratic scaling if one again neglects small elements.
To obtain an even better scaling behavior requires the introduction of completely new concepts. One possibility is to build upon algorithms which solve the classical Coulomb problem for point particles. The classical Coulomb problem requires the calculation of the electrostatic potential arising from all the $N$ particles with charge $Z_j$ at all the positions ${\bf r}_i$ $$\label{fmm}
U( {\bf r}_i ) = \sum_j \frac{Z_j}{|{\bf r}_i-{\bf r}_j|} \: .$$ By grouping particles into hierarchical groups and by describing their potential far away from such groups in an controlled approximate way by multipoles these fast algorithms allow to evaluate Equation (\[fmm\]) with linear scaling instead of the expected quadratic one. There are now several proposals (Strain [*et al.,*]{} 1996, White [*et al.,*]{} 1994, Perez-Jorda 1997), how one can modify one of these fast algorithms, the Fast Multipole Method (Greengard 1994) in such a way that it can handle also the continuous charge distributions arising in the context of electronic structure calculations. The basic idea is fairly straightforward. As we saw the charge distributions is always given as a weighted sum of auxiliary Gaussians (Equation (\[rhoprod\])). Now the electrostatic potential of such a Gaussian particle looks the same from a distance as the potential of a point particle. Concerning the far field, one can thus essentially take over the existing algorithms. To account for the non point like nature of the Gaussian particles one has however to correct the near field. Since the calculation of these local corrections have linear scaling the whole procedure has linear scaling as well. There are two problems with this kind of approach. First, one has only linear scaling with respect to the size of the basis set if the volume covered by the basis set grows at least as fast as the size of the basis set. If one adds for example basis functions for a molecule of fixed size, to improve the accuracy of the basis set one does not any more have linear scaling. This is related to the fact that one can apply the fast far field treatment now to a smaller number of Gaussian particle interactions. This behavior can be easily understood by considering the extreme case where all Gaussian particles are centered very close together within a radius which is smaller than their width. In this case one evidently cannot use any more any far field techniques. The second problem is closely akin to the first. If one adds extended Gaussians to the system the efficiency deteriorates.
A method which scales strictly linear with respect to the size of the basis set, independently of whether the volume is increased at the same time or not and which can be applied within the context of any basis set, is based on wavelets (Goedecker and Ivanov 1998a). As the input to this method the charge density is needed on a real space grid which can have varying resolution. THus near the core region of the atoms in a molecule the resolution can be arbitrarily increased. Using interpolating wavelets this charge density can uniquely be mapped to a wavelet expansion, since a wavelet expansion can compactly describe nonuniform functions. In the wavelet basis one can then iteratively solve Poissons equation $$\label{poiss}
\nabla^2 V = -4 \pi \rho \: .$$ The matrix representing the Laplace operator $\nabla^2$ is sparse and the matrix times vector multiplications needed for the iterative solution of Equation (\[poiss\]) scale linearly. Using a preconditioning scheme in a basis of lifted wavelets the condition number is independent of the size and of the maximal resolution of the wavelet expansion and the number of iterations is therefore constant. Thus one obtains an overall linear scaling.
The exchange correlation potential
----------------------------------
Within the most popular versions of Density Functional theory the exchange correlation potential is a purely local function. In the case of the Local Density Approximation (Parr and Yang) the exchange correlation potential at a certain point depends only on the density at that point, in the case of Generalized Gradient Approximations (Perdew [*et al.,*]{} 1996, Becke 1988, Lee [*et al.,*]{} 1988) it depends in addition still on the gradient of the density at that point. If one uses real space methods such as finite differences or finite elements as well as plane wave methods where the calculation of the exchange correlation potential is done on a real space grid as well, it is obvious that the numerical effort is linear with respect to the system size. If one uses more extended basis functions such as Gaussian type orbitals it becomes more difficult to achieve linear scaling (Stratmann [*et al.,*]{} 1996).
In the case of Hartree Fock calculations the exchange energy $$\label{hfex}
\sum_{i,j} \int \int d{\bf r} d{\bf r}'
\frac{ \Psi_i({\bf r}) \Psi_j({\bf r}) \Psi_i({\bf r}') \Psi_j({\bf r}') }
{|{\bf r}-{\bf r}'|}$$ seems to be as nonlocal as the Coulomb potential. Using Equation (\[denst0\]) one can however rewrite the expression (\[hfex\]) to obtain $$\int \int d{\bf r} d{\bf r}'
\frac{ F({\bf r},{\bf r}') F({\bf r},{\bf r}')} {|{\bf r}-{\bf r}'|}$$ showing that the exchange energy in an insulator is indeed a local quantity whose locality is determined by the decay properties of the density matrix. A linear method to evaluate exchange terms within Gaussian type electronic structure calculations based on the aforementioned locality properties has been devised by Schwegler and Challacombe (1996). An alternative method based on the Fast Multipole Method has been developed by Burant [*et al.,*]{} (1996).
Obtaining self-consistency
==========================
To do a self-consistent electronic structure calculation, two ingredients have to be blended. The first is the calculation of the density matrix in a fixed external potential, a topic which is the main focus of this article. The second is the calculation of the potential from a given electronic charge density which was discussed in the preceeding section (\[coulomb\]). Even if both of these basic parts exhibit linear scaling, it is however not yet granted, that one has overall linear scaling. It might happen that the number of times one has to repeat these two basic parts increases with the size of the system.
The easiest scheme to combine the calculation of the density matrix and the calculation of the potential is the so called scalar mixing scheme. Given an input charge density $\rho_{in}$ which determines the potential one obtains after the calculation of the density matrix for this potential via Equation (\[rhotrace\]) a new output density $\rho_{out}$. The new input density $\rho_{in}^{new}$ is now not the output density $\rho_{out}$, but rather a linear combination of the old input density and the output density $$\rho_{in}^{new}({\bf r}) =
\rho_{in}({\bf r}) + \alpha ( \rho_{out}({\bf r}) - \rho_{in}({\bf r})) \: .$$ Here $\alpha$ is the mixing parameter. Overall linear scaling is endangered if one has to decrease $\alpha$ for reasons of numerical stability as the system becomes bigger and if one consequently needs a larger number of iterations.
The standard theory of mixing (Dederichs and Zeller 1983, Ho [*et al.,*]{} 1982) is based on the dielectric response function in ${\bf k}$ space. Within this theory numerical instabilities arise if the the dielectric response functions tends to infinity as $k$ tends to zero. This happens in a metal but not in an insulator where the dielectric response function always remains finite. Following the general philosophy of this paper to remain within a real space formalism, we will elucidate mixing under this perspective. The final conclusions are of course the same as the one based on the Fourier space theory.
Let us first consider a metal. We assume that we are doing a calculation for a one-dimensional metallic structure of length $L$. Let us also assume that due to deviations from the converged self-consistent charge density we transfer an incremental charge $\Delta Q_{in}$ from one end of the sample to the other. Using the basic formula for the potential in a capacitor we get a constant electric field in the sample giving rise to a potential difference of $\Delta U = L \: \Delta Q_{in}$ between the two ends. In a metal this potential difference will most likely be larger than the HOMO LUMO separation (which vanishes for large systems) and we get a large charge transfer $\Delta Q_{out}$. This charge transfer is related to the density of states at the Fermi level, $D(\mu)$, which in our one dimensional case is the number of states per length unit and per energy unit. So the total charge transfer $\Delta Q_{out}$ is given by $$\Delta Q_{out} \approx D(\mu) \: L \: \Delta U = D(\mu) \: L^2 \: \Delta Q_{in} \: .$$ If this induced charge transfer $\Delta Q_{out}$ is larger than the initial transfer $\Delta Q_{in}$ then the charge transfer will exponentially increase in subsequent iterations and we have the numerical instability called “charge sloshing”. To avoid it the mixing factor $\alpha$ has to be proportional to $\frac{1}{D(\mu) \: L^2}$. Doing the same analysis in a three-dimensional structure all the lateral dimensions cancel and we get the same result concerning $\alpha$. Denoting the volume of our sample by $v$ we find that $\alpha$ is proportional to $v^{-2/3}$. So $\alpha$ has to be decreased with increasing volume and the number of iterations in the mixing schemes increases with increasing system size. Fortunately and contrary to the implications of Annett (1995), this charge sloshing can be eliminated by state-of-the-art techniques (Kresse, 1996). One possibility (Kerker 1981) is just to do the mixing in Fourier space and to have a $k$ dependent mixing parameter $\alpha(k)=\alpha_0 \frac{k^2}{k^2+k_0^2}$ $$\rho_{in}^{new}({\bf k}) = \rho_{in}({\bf k}) + \alpha_0 \frac{k^2}{k^2+k_0^2} ( \rho_{out}({\bf k}) - \rho_{in}({\bf k})) \: .$$ As we see long wavelength components (corresponding to small $k$ values) are now strongly damped by $$\alpha_0 k^2 = \alpha_0 \left( \frac{2 \pi}{\lambda} \right)^2$$ and the dampening has the correct dimensional behavior with respect to the wavelength $\lambda$ which corresponds to the length $L$ in our above dimensional analysis. Short wavelength contributions are just damped by $\alpha_0$ and this constant dampening sets in when $k$ becomes comparable in magnitude to $k_0$. We know, that for wavelengths of the order of the interatomic spacing a mixing parameter somewhat smaller than 1 works well and so we can determine by these conditions the values of $\alpha_0$ and $k_0$.
Let us next examine whether we can have charge sloshing in an insulator. We will assume that the potential difference across the sample is not larger than the gap, in which case the discussion for the metallic case would rather apply. Again we consider a sample of length $L$. According to the modern theory of polarization in solids (King-Smith and Vanderbilt 1989) a polarization arises because the centers of the Wannier functions are displaced under the action of an electric field. Since the Wannier functions are exponentially localized, the charge which will build up at the two surfaces of our sample is mainly due to the displacements of the Wannier function in the elementary cells of the crystal right on the surface and the charge $\Delta Q_{out}$ is thus practically independent of the length of the sample. So the optimal mixing constant $\alpha$ is nearly independent of the size of the system and the number of iterations as well.
In conclusion, we see that linear scaling can also be obtained in the selfconsistent case and that even in a metal charge sloshing problems can be overcome.
Mixing is the natural choice if the DC or the FOE methods are used in a self-consistent calculation. If methods based on minimization (DMM and OM) are used one can alternatively also obtain the ground state by a single minimization loop (R. Car and M. Parrinello 1985, I. Štich [*et al.,*]{} 1989, M. P. Teter [*et al.,*]{} 1989, M. Payne, [*et al.,*]{} 1992 ) without distinguishing between density matrix optimization cycles and potential mixing cycles. As is not surprising after our discussion of mixing one finds (Annett 1995) that in an insulator the number of iterations does not depend upon whether one has a self-consistent type of calculation where the potential is varying during each minimization step or whether one has a fixed potential. In other words there is no charge sloshing. In metallic systems it is essential to have finite electronic temperature (Wentzcovitch [*et al.,*]{} 1992, Weinert and Davenport 1992, Kresse 1996) and therefore the minimization schemes cannot be applied straightforwardly in any case. Insofar Annett’s analysis (Annett 1995) showing that in this case the scaling is at least proportional to $N_{at}^{4/3}$ is irrelevant.
A completely different approach to the mixing problem has recently been proposed by Gonze (1996). He calculates the gradient of the total energy with respect to the potential. His gradient expression does not depend on the wavefunctions and could thus well be combined with O(N) schemes.
Applications of O(N) methods
============================
This chapter is not intended to be a comprehensive or even complete review of all the applications which were done using O(N) methods. It is rather intended as an illustration of the wide range of areas where O(N) methods allowed to study systems which were too big to be studied with conventional methods. In general one can say that most large-scale Tight Binding studies are nowadays done in the connection with O(N) methods. In those cases systems comprising from a few hundred up to many thousand atoms are typically studied. Treating such a large number of atoms with O(N) Density Functional methods is much more difficult. In the case of Density Functional calculations the benchmarking and verification aspect is usually dominating whereas in tight binding calculations the focus was in most cases on how to solve challenging physical problems.
Questions concerning extended defects in crystalline materials were one of the main focus of these Tight Binding studies. Because several good Tight Binding parameters are available for silicon, most studies were done for this material. The $90^0$ partial dislocation in silicon was at the focus of interest of a series of tight binding studies. The three structures that were examined are shown in Figure \[nunes\].
( 8.,9.5) (-0.,-1.0)
The energy difference between the structure (a) and (b) of Figure \[nunes\] was studied both by Nunes [*et al.*]{} (1996) and by Hansen [*et al.*]{} (1995). Even though they used different TB parameters and different O(N) algorithms (DMM and FOE) the both obtained exactly the energy difference of .18eV/$\AA$ in favor of structure (b). Later Benetto and al. (1997) discovered a new structure (c) that is even lower in energy. To validate their tight binding results they did conventional density functional calculations for smaller subsystems finding perfect agreement with the O(N) tight binding results. This new structure is experimentally difficult to distinguish form structure (b) and so this result is a convincing illustration of the power of these new O(N) algorithms in materials science. All these tight binding calculations necessitated electronic structure calculations involving a few hundred atoms and would therefore have been unfeasible with standard algorithms.
Extended {311} defects in silicon systems containing more than 1000 atoms and their relation to point defects were studied by Kim [*et al.*]{} (1997) using the OM method in the improved version of Kim [*et al.*]{} (1995). An understanding of these processes is important for the fabrication of semiconductor devices, since defects have a strong influence of the diffusion properties of semiconductors. Unfortunately, the more realistic questions involving boron dopant atoms in addition to the bulk silicon atoms can not be treated with current tight binding models.
Ismail-Beigi and Arias (1998) examined the surface reconstruction properties of silicon nanobars, finding that the influence of edges in these small structures is strong enough to lead to surface reconstructions that are different form the ones in bulk silicon. They also both employed traditional density functional calculations and O(N) FOE tight binding calculations and also found good agreement between both for small subsystem which are accessible to both approaches.
Roberts and Clancy (1998) simulated vacancy and interstitial diffusion processes in silicon using the FOE tight binding method. The diffusion constants they obtain are in good agreement with similar calculations based on classical force fields and density functional calculations. Compared to the density functional calculations they could also significantly enlarge both the number of atoms (216) and the simulation times. The diffusion constant predicted by all these simulations is however orders of magnitude larger than the experimental one, a fact for which no explanation is known up to now.
Besides silicon there is another material for which several good Tight Binding schemes are available, namely carbon. Fullerene systems are therefore another focus of Tight Binding studies.
Galli and Mauri (1994) did molecular crash test of $C_{60}$ fullerenes colliding with a diamond surface using the OM method. They found three different impact energy regimes where the impinging fullerenes either survive the collision undamaged, slightly damaged or get completely destroyed. Even though the interaction region between the impinging fullerene and the surface does not comprise a very large number of atoms, their computational box contained more than 1000 atoms. The reason why the box has to be so large is that the phonons emitted during the collision may not be reflected back from the walls of the box during the time scale of the collision. This reflection of phonons is also a serious problem in classical force field simulations of crack propagation where for this reason systems comprising several million atoms are sometimes necessary (Zhou [*et al.,*]{} 1997). In the case of this molecular crash test most of the carbon atoms are propagating the phonons. Phonons are well described by classical force fields and one could use this scheme for the majority of the atoms, while it would be necessary to use the more expensive tight binding scheme only for the atoms in the collision region. Unfortunately such schemes combining molecular methods of different speed and accuracy have not yet been developed and thus it is therfore a feature of many O(N) calculations that one is doing an overkill in a certain sense, treating a large number of essentially inactive atoms with highly accurate methods. Canning [*et al.*]{} (1997) examined thin films of $C_{28}$ fullerenes with the same method, finding that thin superatom films can be formed.
The equilibrium geometries of large fullerenes such as $C_{240}$ was also studied by several groups with O(N) techniques. The central question here is whether such large fullerenes have a spherical form or a polyhedrally faceted shape, where nearly flat polyhedral regions are alternating with edges where the curvature is concentrated. York [*et al.*]{} (1994) used the original formulation of the DC method in terms of densities to do density functional calculations of $C_{240}$ and found spherical shapes. Itoh and al. using both empirical and ab initio tight binding in the context of the OM method found however polyhedral shapes. This result is also supported by Xu and Scuseria (1996) who investigated fullerenes up to $C_{8640}$ using the DM method. The optimized geometries they found for various large fullerenes are shown in Figure \[scuseria\].
( 8.,9.5) (0.,0.7)
Ajayan [*et al.*]{} (1998) used tight binding FOE molecular dynamics to simulate irradiation mediated knock-out of carbon atoms out of carbon nanotubes. In agreement with experimental observations they found that this atom removal leads to a shrinking of the diameter of the nanotube, but leaves the nanotube essentially intact until the diameter is practically zero. They could identify in their virtual 400 atom sample processes on the atomic level that are responsible for the rapid healing of the defects created by the removal of atoms.
Recently developed Tight Binding parameters (Horsfield 1996a) made it possible to study also a composite system, namely hydrocarbons. This set of tight binding parameters includes some kind of self-consistency by imposing a local charge neutrality requirement. Local charge neutrality is essential if different phases are studied because it prevents any unphysically large charge transfer between different phases arising from different chemical potentials in different phases. Using this new tight binding scheme, Kress [*et al.*]{} (1998) studied the dissociation of $CH_4$ under high pressure and at high pressure using the FOE molecular dynamics. Previous Density functional based molecular dynamics studies by Ancilotto [*et al.*]{} (1997) were limited to very small system sizes of 16 $CH_4$ molecules and short simulation times of 2 ps. At variance with experimental findings these density functional simulations could not find a phase separation of methane into hydrogen and carbon. FOE molecular dynamics allowed to treat much larger systems of 128 molecules and also much longer simulation times of 8 ps. After 4 ps a phase separation was indeed observed.
Sanchez-Portal [*et al.*]{} (1997) compared the experimental X-ray structure of a large DNA molecule comprising 650 atoms with the geometric structure obtained from a density functional based OM relaxation. They obtained a root mean square deviation from the experimental geometry of 0.23 $\AA$. Their method relies however on fairly drastic approximations resulting in errors that are by far larger than the error one generally expects from a density functional calculation.
A similar study of a large biomolecule is reported by Lewis [*et al.*]{} (1997).
York [*et al.*]{} used the DC method in the context of the semi-empirical AM1 method (Dewar [*et al.,*]{} 1985) to calculate heats of formation, solvation free energies and densities of states for protein and DNA systems containing up to 2700 atoms.
Daniels and Scuseria reported AM1 semi-empirical benchmark calculations for up to 20000 atoms using DMM, FOE and pseudo-diagonalization methods.
Applications of O(N) methods within Density Functional theory, that use basis sets large enough such the basis set errors are not dominating the Density Functional error at present do not exist. If calculations of this type were done, such as the calculations of a cell containing 6000 silicon atoms by Goringe [*et al.*]{} (1997), then the performance evaluation aspect was always the dominating one. With the advance of faster computers and improved algorithms this situation will, however, certainly soon change. It is also interesting to note in this context that the 1998 version of the very popular Gaussian software package will contain O(N) algorithms.
Let us finally come back to a point briefly mentioned in the introduction. The development of O(N) methods has also deepened our understanding of locality in quantum mechanical systems and has thereby also fostered the development of theories based on a local picture rather than the conventional nonlocal Bloch function picture. An example is the theory of polarization in crystalline materials by King-Smith and Vanderbilt (1989). Their theory is based on a local picture in terms of Wannier functions and allows for an intuitive understanding of these phenomena that were difficult to understand before.
Conclusions
===========
O(N) methods have become an essential part of most large scale atomistic simulations based either on Tight Binding or semiempirical methods. The physical foundations of O(N) methods are well understood. They are related to the decay properties of the density matrix. The use of O(N) methods within Density Functional methods is still waiting for their widespread appearance. All the algorithms that would allow us to treat very large basis sets within density functional theory have certain shortcomings. The OM and OBDMM method suffer from ill-conditioning problems, and in both the OM and FOP method detailed knowledge about the bonding properties is required to form the input guess. Thus there is probably still some algorithmic progress necessary before these obstacles can be overcome. It is also not quite clear what the localization properties of very large complicated molecules are and whether perhaps a quadratic scaling rather than a linear scaling is the optimum one can obtain in certain cases. It is clear that the elimination of the cubic scaling bottleneck is a very important achievement and that it will pave the way for calculations of unprecedented size in the future. Such calculations will not only be beneficial to physics, but they will also nourish progress in many related fields such as chemistry, materials science and biology. Even with O(N) algorithms it will not be possible in the foreseeable future to treat systems containing millions of atoms at a highly accurate Density functional level using large basis sets, as would be necessary for certain materials science applications. Such problems can only be approached if one succeeds in combining methods of different accuracy such as density functional methods with classical force fields, applying the high accuracy method only to regions where the low accuracy method is expected to fail. Hybrid methods of this type will certainly be based on the same notions of locality and similar techniques as O(N) methods.
Acknowledgments
===============
I would like to express my deep gratitude to all the people who helped me to discover errors in the manuscript and who made valuable suggestions how to improve the paper. They are in alphabetical order David Bowler, David Drabold, Olle Gunnarson, Eduardo Hernandez, Andrew Horsfield, Jurg Hutter, Ove Jepsen, Joel Kress, Klaus Maschke, Richard Martin, Chris Mundy Ricardo Nunes, Michele Parrinello, Anna Puttrino, Gustavo Scuseria, Uwe Stephan, and David Vanderbilt .
Appendix: Decay properties of Fourier transforms
================================================
The density matrix is defined in terms of a Fourier transform given by Equation (\[ftrans\]). The decay properties of the density matrix are thereby closely related to the decay properties of Fourier transforms. All the properties described in this paragraph are well known, for completeness we will briefly outline them.
For simplicity let us just consider a one-dimensional Fourier transform $$\label{frdef}
g(r) = \int_{-\infty}^{\infty} e^{i k r} g(k) dk \: ,$$ where $g(k)$ is an integrable piecewise continuous function tending rapidly to $0$ for $k$ tending to $\pm \infty$.
For any function $g(k)$ of this type, $g(r)$ will obviously tend to zero when $r$ tends to infinity. In this case $e^{i k r}$ is a very rapidly oscillating function and the product $ e^{i k r} g(k)$ will therefore change sign very rapidly and thus the integral will tend to zero. The exact decay properties depend on how many derivatives are continuous. Let us consider first a function which is piecewise constant and has only a finite number of discontinuities. A function which falls into this class is the function $g(k)$ which is $1$ in the interval \[-1:1\] and zero everywhere else. Calculating the Fourier transform one finds $g(r) = 2 \frac{\sin (r)}{r}$. Since any piecewise function $g(k)$ can be written as a linear combination of the above prototype function, its transform $g(r)$ will always decay like $1/r$. Using integration by parts we see that $$r^l g(r) = \int_{-\infty}^{\infty} g(k) \: i^{-l} \left( \frac{\partial}{\partial k} \right)^l e^{ikr} dk
= (-i)^{-l} \int_{-\infty}^{\infty} e^{ikr} \left( \frac{\partial}{\partial k} \right)^l g(k) dk \: .$$ If the $l$-th derivative is integrable then the integral will vanish for the reasons discussed above. So if we can do $l$ integrations by parts each transformation will accelerate the decay by one inverse power of $r$ and we can do such a transformation whenever our function has at least continuous first derivatives. Hence we arrive at the rule that if $l$ derivatives of $g(k)$ are continuous, $g(r)$ will decay like $r^{-(l+1)}$.
If we have a function $g(k)$ which is analytic, i.e. one for which an infinite number of derivatives exists, then the transform will decay faster than any power of $r$. One then says that it decays exponentially instead of algebraically. This notion of exponential decay does not necessarily mean that it decays strictly like an exponential function. As an example we could for instance take $g(k)= \exp(-k^2)$, where we know that the transform is again a Gaussian and decays thus faster than an ordinary exponential function. The rate of decay will be related to the smallest length scale of $g(k)$. If the smallest length scale of $g(k)$ is $k_{min}$ then $g(r)$ will roughly decay like $\exp (- c |r| k_{min} )$, where $c$ is a constant of order of 1. This follows from the fact that one will have an important cancellation of terms of opposite sign in the integral in Equation (\[frdef\]) only if several oscillations occur within the interval $k_{min}$.
Another qualitative feature of the Fourier transform is that it will have oscillations whenever $g(k)$ is shifted off center. The oscillation period is determined by this shift. As an example let us look at the Fourier transform of a shifted Gaussian $g(k)=\exp (-\frac{1}{2}(k-a)^2)$. The result is $g(r)=\sqrt{2 \pi} \: \exp (i a r) \: \exp (-\frac{1}{2} r^2)$, which is the transform of the unshifted Gaussian times an oscillatory term.
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---
abstract: 'Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology.'
author:
- Achmad Choiruddin
- 'Francisco Cuevas-Pacheco'
- 'Jean-François Coeurjolly'
- Rasmus Waagepetersen
bibliography:
- 'masterbib.bib'
title: Regularized estimation for highly multivariate log Gaussian Cox processes
---
[**Key words:**]{} cross pair correlation, elastic net, LASSO, log Gaussian Cox process, multivariate point process, proximal Newton method.
Introduction
============
Highly multivariate point pattern data are becoming increasingly common. Tropical rain forest ecologists, for example, collect data on locations of thousands of trees belonging to hundreds of species. Likewise, huge space-time data sets regarding scene, time and type of crimes are recorded and made publicly available for many major cities across the world. Research on statistical methodology for multivariate point patterns has mainly considered bivariate or trivariate point patterns. Some exceptions are [@diggle:etal:05] and [@baddeley:jammalamadaka:nair:14] who considered four- and six-variate multivariate Poisson processes and more recently [@jalilian:etal:15] and [@waagepetersen:etal:16] who considered five- and nine-variate multivariate Cox processes. A truly high-dimensional analysis was conducted by [@rajala:murrell:olhede:17] who introduced a multivariate Gibbs point process and applied it to a point pattern data set containing locations of 83 species of rain forest trees.
A particular challenge regarding modeling of highly multivariate point patterns is that models easily become very complex with large numbers of parameters. To enhance interpretability of fitted models and numerical stability of estimation, [@rajala:murrell:olhede:17] used regularization methods such as the group lasso. The possibility of using regularization was also mentioned in the discussion of [@waagepetersen:etal:16] in the context of multivariate log Gaussian Cox processes.
The type of multivariate log Gaussian Cox process considered by [@waagepetersen:etal:16] and reviewed in Section \[sec:mlgcp\] has a simple and natural interpretation and e.g. enables the user to decompose variation according to different sources and to group different types of point patterns according to similarities in their spatial distributions, see [@waagepetersen:etal:16] for details. However, the fitting of these models is very challenging in the highly multivariate case due to model complexity. In Section \[sec:rlse\] of this paper, we develop a numerically stable and efficient parameter estimation methodology by introducing regularization and using efficient convex optimization algorithms. We test the methodology in a simulation study in Section \[sec:sim\] and apply it to a tropical rain forest data in Section \[sec:app\]. Section \[sec:disc\] contains some concluding remarks.
Multivariate log Gaussian Cox processes {#sec:mlgcp}
=======================================
A multivariate log Gaussian Cox point process [see @moeller:syversveen:waagepetersen:98] is a multivariate point process ${\mathbf{X}}=(X_1,\ldots,X_p)$, $p>1$, where each component $X_i$, $i=1,\ldots,p$, is a Cox process driven by a log Gaussian random intensity function ${\Lambda}_i$. Conditionally on the $\Lambda_i$, the $X_i$ are independent Poisson point processes each with intensity function ${\Lambda}_i$. As in [@waagepetersen:etal:16], we assume that the random intensity functions are of the form ${\Lambda}_i({\mathbf{u}})=\exp[Z_i({\mathbf{u}})]$ with $$\begin{aligned}
Z_i({\mathbf{u}})=\mu_i({\mathbf{u}})+Y_i({\mathbf{u}})+U_i({\mathbf{u}}), \; {\mathbf{u}}\in \R^2. \label{eq:Z}
\end{aligned}$$ The terms $\mu_i$ are deterministic and typically given in terms of regressions on observed covariates. The terms $Y_i$ and $U_i$ are zero-mean Gaussian fields. The $Y_i$ can be mutually correlated while the $U_i$ are assumed to be independent. The $U_i$ are assumed to be stationary with variances $\sigma_i^2 > 0$ and correlation functions $c_i$, $i=1,\ldots,p$. For the $Y_i$ we assume that $$Y_i(u)= \sum_{l=1}^q {\alpha}_{il}E_l(u)$$ where $q \ge 1$, ${\boldsymbol{\alpha}}=[{\alpha}_{ij}]_{ij}$ is a $p\times q$ real valued coefficient matrix, and the $E_l$, $l=1,\ldots,q$, are independent zero-mean stationary Gaussian fields with variance one. In our applications we also consider the case $q=0$ meaning that the $Y_i$ are omitted in . The $Y_i$ can be interpreted as effects of unobserved spatial covariates while the $U_i$ represent sources of clustering which are specific to each type of points. We denote by $r_l$ the correlation function of $E_l$. For the correlation functions $r_l$ and $c_i$ we introduce isotropic parametric models $r_l(\cdot;\phi_l)=r(\|\cdot\|/\phi_l)$ and $c_i(\cdot;\psi_i)=c(\|\cdot\|/\psi_i)$, where $\phi_l$ and $\psi_i$ are correlation scale parameters. Specifically, we consider in this paper exponential correlation functions $r(t)=c(t)=\exp(-t)$, $t \ge 0$, although many other choices are available [@chiles:delfiner:99].
Intensity function and pair correlation function {#sec:moments}
------------------------------------------------
Let ${\boldsymbol{\alpha}}_{i \cdot}$ denote the $i$th row of ${\boldsymbol{\alpha}}$. Following [@moeller:syversveen:waagepetersen:98], the intensity function of $X_i$ is $\rho_i({\mathbf{u}}) = \exp\big [\mu_i({\mathbf{u}}) + {\boldsymbol{\alpha}}_{i\cdot} {\boldsymbol{\alpha}}^{{{\mbox{\scriptsize \sf T}}}}_{i\cdot}/2 +\sigma^2_i/2\big ]$ while the cross pair correlation function for the pair $X_i$ and $X_j$ is $$\label{eq:crosspair}
g_{ij}({{t}}) = \exp\big[ \sum_{l=1}^{q}{\alpha}_{il}{\alpha}_{jl}r_{l}({{t}};\phi_l) + 1(i=j)\sigma^2_ic_i({{t}};\psi_i) \big ]$$ for ${{t}}\ge 0$. Consider two spatial locations ${\mathbf{u}}$ and ${\mathbf{v}}$. Then $\rho_j({\mathbf{v}})g_{ij}(\|{\mathbf{v}}-{\mathbf{u}}\|)$ represents the cross-Palm intensity function [@coeurjolly:moeller:waagepetersen:17] and can be interpreted as the intensity function of $X_j$ conditional on that ${\mathbf{u}}\in X_i$. Hence $g_{ij}(\|{\mathbf{v}}-{\mathbf{u}}\|)>1$ ($<1$) implies that presence of a point from $X_i$ at ${\mathbf{u}}$ increases (decreases) the intensity of $X_j$ at ${\mathbf{v}}$. Thus $\sum_{l=1}^{q}{\alpha}_{il}{\alpha}_{jl}r_{l}({{t}}) < 0$ ($>0$) implies repulsion (attraction) between points of $X_i$ and $X_j$ at lag ${{t}}$. Similarly, a large value of $\sum_{l=1}^{q} {\alpha}_{il}^2 r_{l}({{t}})+\sigma^2_ic_i({{t}})$ leads to strong attraction among points of $X_i$ separated by a lag ${{t}}$.
Non-parametric kernel estimates of the $g_{ij}$ are given by $$\label{eq:ghat}
\hat{g}_{ij}(t) = \frac{1}{2\pi t} \sum_{\substack{{\mathbf{u}}\in X_{i} \cap W,\\ {\mathbf{v}}\in X_{j} \cap W,\\ {\mathbf{u}}\neq {\mathbf{v}}}} \frac{k_{b}(t-\|{\mathbf{u}}-{\mathbf{v}}\|)}{\hat{\rho}_i({\mathbf{u}})\hat{\rho}_j({\mathbf{v}}) |W \cap W_{{\mathbf{u}}- {\mathbf{v}}}|}, \quad t>0,$$ where $W$ is the observation window, $k_{b}$ is a kernel function depending on a smoothing parameter $b>0$, $|\cdot|$ denotes area and $W_{{\mathbf{h}}}$ denotes the translate of $W$ by the vector ${\mathbf{h}}\in \R^{2}$ [@moeller:waagepetersen:03]. The quantities $\hat{\rho}_i$ and $\hat{\rho}_j$ are estimates of the intensity functions of $X_i$ and $X_j$, typically obtained from regression models depending on observed covariates through maximizing the composite likelihood [see e.g. @waagepetersen:07; @moeller:waagepetersen:07] or its regularized versions [e.g. @thurman2015regularized; @choiruddin2018convex].
Least squares estimation {#sec:lse}
------------------------
Let ${\boldsymbol{\theta}}$ be the parameter vector consisting of the components of ${\boldsymbol{\alpha}}$, $\boldsymbol{\sigma}^2 = (\sigma_{1}^{2},\ldots,\sigma_{p}^{2})^{{{\mbox{\scriptsize \sf T}}}}$, $\boldsymbol{\phi} = (\phi_{1},\ldots,\phi_{q})^{{{\mbox{\scriptsize \sf T}}}},$ and $\boldsymbol{\psi} = (\psi_{1},\ldots,\psi_{p})^{{{\mbox{\scriptsize \sf T}}}}$. Let further $$\begin{aligned}
{\boldsymbol{\beta}}_{ij}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2)& =
({\alpha}_{i1}{\alpha}_{j1},\ldots,{\alpha}_{iq}{\alpha}_{jq})^{{\mbox{\scriptsize \sf T}}},
i\neq j, \nonumber \\
{\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2) & =
({\alpha}^2_{i1},\ldots,{\alpha}^2_{iq},\sigma_i^2)^{{\mbox{\scriptsize \sf T}}}. \label{eq:betaii}\end{aligned}$$ The objective function used by [@waagepetersen:etal:16] for parameter estimation is of the form $$\begin{aligned}
\label{eq:lso}
Q({\boldsymbol{\theta}}) = \sum_{i,j=1}^{p} \| Y_{ij}- X_{ij}({\boldsymbol{\phi}},{\boldsymbol{\psi}}) {\boldsymbol{\beta}}_{ij}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2) \|^2,
\end{aligned}$$ where $$Y_{ij}= (\sqrt{w_{ij1}}\log \hat{g}_{ij}(t_1),\ldots, \sqrt{w_{ijL}}\log \hat g_{ij}(t_L))^{{\mbox{\scriptsize \sf T}}},$$ $\hat g_{ij}(t_k)$, $k=1,\ldots,L$, are obtained using for lags $0<t_1 < t_2 < \ldots <t_L$ and the $w_{ij} \ge 0$ are non-negative weights. The matrix $X_{ij}({\boldsymbol{\phi}},{\boldsymbol{\psi}})$ is $L \times q$ ($i\neq j$) or $L \times (q+1)$ ($i=j$) with rows $\sqrt{w_{ijk}}\mathbf r(t_k;{\boldsymbol{\phi}})$ ($i \neq j$) or $\sqrt{w_{iik}} [ \mathbf r(t_k;{\boldsymbol{\phi}}),c_i(t_k;\psi_i)]$ ($i=j$), $k=1,\ldots,L$, where $$\begin{aligned}
\mathbf r(t_k;{\boldsymbol{\phi}})= (r_1(t_k;\phi_1),\ldots,r_q(t_k;\phi_q)).
\end{aligned}$$ [@waagepetersen:etal:16] minimized $Q({\boldsymbol{\theta}})$ using a standard quasi-Newton method.
Inference regarding multivariate dependence structure {#sec:multvarinference}
-----------------------------------------------------
The model enables us to decompose the covariances of the latent Gaussian fields $Z_i$ into contributions from the common fields $E_l$ and the type-specific fields $U_i$. Specifically, [@waagepetersen:etal:16] considered for each type $i$ and lag $t$ the proportion of variance (PV) due to the common fields: $$\begin{aligned}
\mathrm{PV}_i(t) &=\frac{\mathrm{cov}\{Y_i({\mathbf{u}}), Y_i({\mathbf{u}}+ {\mathbf{h}})\}}{\mathrm{cov}\{Z_i({\mathbf{u}}), Z_i({\mathbf{u}}+ {\mathbf{h}})\}} \\ &=\frac{\sum_{l=1}^{q}{\alpha}_{il}^2r_{l}({{t}};\phi_l)}{\sum_{l=1}^{q}{\alpha}_{il}^2r_{l}({{t}};\phi_l) + \sigma^2_ic_i({{t}};\psi_i)} , \quad \|{\mathbf{h}}\|=t.
\end{aligned}$$ These are useful e.g. for grouping species based on how much of the variation is due to common factors respectively type-specific factors. Furthermore, from ${\boldsymbol{\alpha}}$ and ${\boldsymbol{\sigma}}^2$ we can compute the matrix of lag zero inter-type covariances ${\boldsymbol{\alpha}}{\boldsymbol{\alpha}}^{{\mbox{\scriptsize \sf T}}}$ due to the common latent fields with $ij$th entry $$\mathrm{cov}\{Y_i({\mathbf{u}}), Y_j({\mathbf{u}})\}={\boldsymbol{\alpha}}_{i.}{\boldsymbol{\alpha}}_{j.}^{{\mbox{\scriptsize \sf T}}}$$ as well as the lag zero covariances between the fields including both common and type-specific effects, $$\label{eq:covZ} \mathrm{cov}\{Z_i({\mathbf{u}}), Z_j({\mathbf{u}})\}={\boldsymbol{\alpha}}_{i.}{\boldsymbol{\alpha}}_{j.}^{{\mbox{\scriptsize \sf T}}}+1[i=j]\sigma^2_i. $$ A row ${\boldsymbol{\alpha}}_{i\cdot}$ informs on the dependence of $X_i$ on the common latent fields. Considering the norms of differences $\|{\boldsymbol{\alpha}}_{i.}- {\boldsymbol{\alpha}}_{j.} \|$, we are able to group the different types of point patterns according to their dependence on the latent factors $E_l$.
As discussed in [@waagepetersen:etal:16], the distribution of our multivariate log Gaussian Cox process is invariant to 1) simultaneous permutation of columns in ${\boldsymbol{\alpha}}$ and corresponding $\phi_i$’s and 2) multiplication of a column in ${\boldsymbol{\alpha}}$ by $-1$. Thus we can not identify individual parameters ${\alpha}_{il}$ and $\phi_l$ without imposing constraints on the parameter space.
In our simulation studies in Section \[sec:sim\], we therefore follow [@waagepetersen:etal:16] by restricting attention to identifiable functions of ${\boldsymbol{\alpha}}$ and $\boldsymbol{\psi}$ such as the aforementioned proportions of variances and covariances and norms of differences between rows of ${\boldsymbol{\alpha}}$. In the application, we also consider the percentage of zero entries when ${\boldsymbol{\alpha}}$ is estimated using elastic net regularization with $\xi>0$, see next section. The more zeros, the less complex is the dependence structure of the multivariate log Gaussian Cox process.
Regularized least squares estimation {#sec:rlse}
====================================
The parameter vector ${\boldsymbol{\theta}}$ is of potentially very high dimension, especially due to the many components of the $p\times q$ parameter matrix ${\boldsymbol{\alpha}}$. To enhance interpretability and numerical stability of estimation we suggest to introduce regularization and thus consider the regularized least squares criterion $$\begin{aligned}
\label{eq:reglso}
Q_{\lambda}({\boldsymbol{\theta}}) = \; Q({\boldsymbol{\theta}})+ {\lambda}\sum_{i=1}^p\sum_{l=1}^q p({\alpha}_{il})
\end{aligned}$$ where $Q({\boldsymbol{\theta}})$ is given by , ${\lambda}$ is a nonnegative tuning parameter and $p(\cdot)$ is a convex penalty function. We consider in the following the elastic net penalization [@zou:hastie:05] $p({\alpha}_{il})=(1-\xi) {\alpha}_{il}^2/2+\xi |{\alpha}_{il}|$, $0 \le \xi \le 1$, which embraces LASSO [@tibshirani:96] and ridge regression [@hoerl1988ridge] techniques by setting $\xi=1$ or $\xi=0$ respectively.
Using regularization in a related factor analysis was previously suggested by [@choi2010penalized]. Their simpler setting corresponds to directly observing vectors $(Z_i(u_k))_{i=1}^p$, $k=1,\ldots,n$, where $Z_i(u_k)$ is modeled as in but with zero spatial correlation. In contrast, our $Z_i$ are unobserved with spatial correlation modeled via the correlation functions $r_l$ and $c_i$. Thus the computational methodology suggested by [@choi2010penalized] is not applicable in our situation.
To minimize with respect to ${\boldsymbol{\theta}}$, we employ a cyclical block descent algorithm where ${\boldsymbol{\sigma}}^2$, ${\boldsymbol{\alpha}}$, ${\boldsymbol{\phi}}$ and ${\boldsymbol{\psi}}$ are updated in turn. The updating is iterated until relative function convergence of the criterion . The details of the block updates are given in the following two sections and Appendices \[prox\]-\[sec:CDA\]. Pseudo-code for the full algorithm is given in Appendix \[algorithm\].
Update for ${\boldsymbol{\sigma}}^2$ and ${\boldsymbol{\alpha}}$ {#sec:afsigma}
----------------------------------------------------------------
Our strategy for updating ${\boldsymbol{\sigma}}^2$ and ${\boldsymbol{\alpha}}$ is to use for $i=1,\ldots,p$, a least squares update of $\sigma^2_i$ followed by an update of ${\boldsymbol{\alpha}}_{i\cdot}$ using a cyclical coordinate descent algorithm. The motivation for updating rows ${\boldsymbol{\alpha}}_{i\cdot}$ instead of other subsets of ${\boldsymbol{\alpha}}$ is that the update of ${\boldsymbol{\alpha}}_{i \cdot}$, keeping all other parameters fixed, is quite close to a standard least squares problem, as will be evident in the following.
The relevant part of the objective function for the updates of $\sigma_i^2$ and ${\boldsymbol{\alpha}}_{i \cdot}$ given all other parameters is $$\begin{aligned}
\label{eq:blockdescent}
Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i) = \; 2 \sum_{\substack{j=1\\j \neq i}}^p \| Y_{ij}-
\tilde X_{ij} {\boldsymbol{\alpha}}_{i \cdot} \|^2 + \|Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2) \|^2+ {\lambda}\sum_{l=1}^q p({\alpha}_{il})
\end{aligned}$$ where the $l$th column of $\tilde X_{ij}$ is the $l$th column of $X_{ij}$ multiplied by ${\alpha}_{jl}$. In other words, for $i \neq j$, $\tilde X_{ij}= X_{ij} {\text{Diag}}({\alpha}_{j1},\ldots,{\alpha}_{jq} )$ where ${\text{Diag}}({\alpha}_{j1},\ldots,{\alpha}_{jq} )$ is the diagonal matrix with diagonal entries ${\alpha}_{j1},\ldots,{\alpha}_{jq}$. For ease of notation we here omit the dependence of $\tilde
X_{ij}$ and $X_{ii}$ on the fixed parameters ${\boldsymbol{\psi}}$ and ${\boldsymbol{\phi}}$. Note that is equivalent to a standard least squares objective function for ${\boldsymbol{\alpha}}_{i \cdot}$ except for the middle term that depends on ${\alpha}_{il}^2$, $l=1,\ldots,q$, cf. .
The minimization of $Q_{{\lambda},i}$ with respect to $\sigma^2_i$ only involves the middle term in . This is a standard least squares problem except that we require $\sigma^2_i$ to be non-negative. Thus, $$\begin{aligned}
\hat \sigma^2_i & = \max \{ 0,\arg\min_{\sigma^2_i} Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i) \}.
\end{aligned}$$ An explicit formula for this update is given in Appendix \[sigma\].
To update ${\boldsymbol{\alpha}}_{i \cdot}$ (given $\sigma_i^2$ and all other parameters), we use a so-called proximal Newton update [@lee:sun:saunders:14 and Appendix \[prox\]] where the middle term in is replaced by a quadratic approximation around the current value ${\boldsymbol{\alpha}}_{i \cdot}^{(k)}$. We denote by $\hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i|{\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ the resulting approximate objective function (to be detailed in the next paragraph). Since $\hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i|{\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ is a regularized linear least squares objective function, minimization can be performed using a standard coordinate descent algorithm [see e.g. @hastie:tibshirani:wainwright:15].
A very simple quadratic approximation of the middle term of is $$\|Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2) \|^2 \approx \|Y_{ii}- \tilde X_{ii}^k [{\boldsymbol{\alpha}}_{i \cdot}^{{\mbox{\scriptsize \sf T}}},\sigma_i^2]^{{\mbox{\scriptsize \sf T}}}\|^2,$$ where $\tilde X_{ii}^k = X_{ii}{\text{Diag}}\big\{{\alpha}_{i1}^{(k)},\ldots,{\alpha}_{iq}^{(k)},1\big \}$. Nevertheless, the curvature of this quadratic approximation does not match the curvature of the original term at ${\boldsymbol{\alpha}}_{i \cdot}^{(k)}$. Instead we use a second-order Taylor approximation as detailed in the Appendix \[taylor\] which results in the explicit expression for $\hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i|{\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ given by $$\begin{aligned}
Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i) & \approx \hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot}|{\boldsymbol{\alpha}}_{i \cdot}^{(k)}) \nonumber \\
& = \sum_{\substack{j=1}}^p \| Y^*_{ij}-
X^*_{ij}{\boldsymbol{\alpha}}_{i \cdot} \|^2 + {\lambda}\sum_{l=1}^q p({\alpha}_{il}) \label{finalobj},
\end{aligned}$$ where $$\begin{aligned}
Y^*_{ij} & = \sqrt{2}Y_{ij}, \mbox{ for } i \neq j, \nonumber \\
X^*_{ij} & = \sqrt{2}X_{ij}D({\alpha}_{j \cdot}^{(k)}), \mbox{ for } i \neq j, \nonumber \\
Y^*_{ii} & = Y_{ii}+ X_{ii,\cdot (1:q)}{\boldsymbol{\alpha}}_{i \cdot}^{2,(k)} - X_{ii,\cdot (q+1)}\sigma^{2}_i, \nonumber \\
X^*_{ii} & = 2X_{ii,\cdot (1:q)} D({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) \label{eq:YstarXstar}
\end{aligned}$$ and $X_{ii,\cdot (1:q)}$ denotes the first $q$ columns in $X_{ii}$.
We obtain $$\begin{aligned}
\hat {\boldsymbol{\alpha}}_{i \cdot} & = \arg\min_{{\boldsymbol{\alpha}}_{i \cdot}} \hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot}|{\boldsymbol{\alpha}}_{i \cdot}^{(k)})
\end{aligned}$$ using coordinate descent with an explicit formula for the updates given in Appendix \[alpha\]. Further, define for some $t>0$, $$\begin{aligned}
\label{direction}
{\boldsymbol{\alpha}}_{i \cdot}^{(k+1)} = {\boldsymbol{\alpha}}_{i \cdot}^{(k)}+ t (\hat {\boldsymbol{\alpha}}_{i \cdot}-{\boldsymbol{\alpha}}_{i \cdot}^{(k)} ).
\end{aligned}$$ Thus, ${\boldsymbol{\alpha}}_{i \cdot}^{(k+1)}$ is obtained using $(\hat {\boldsymbol{\alpha}}_{i
\cdot}-{\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ as a search direction with step size controlled by $t$. Following @lee:sun:saunders:14 [Proposition 2.3], one can show (see Appendix \[theory\]) that $Q_{i,{\lambda}}({\boldsymbol{\alpha}}_{i \cdot}^{(k+1)})< Q_{i,{\lambda}}({\boldsymbol{\alpha}}_{i
\cdot}^{(k)})$ if $t$ is small enough. That is, if the minimization of $ \hat Q_{i,{\lambda}}$ is combined with a line search the resulting update is guaranteed to decrease the objective function $Q_{i,{\lambda}}$ written in .
Update for ${\boldsymbol{\psi}}$ and ${\boldsymbol{\phi}}$ {#profap}
----------------------------------------------------------
To update ${\boldsymbol{\phi}}$ and ${\boldsymbol{\psi}}$ given all other parameters, we first reparameterize the objective function in terms of ${\bf f}=(\log \phi_1,\ldots,\log \phi_q )^{{\mbox{\scriptsize \sf T}}}$ and ${\bf s}=(\log \psi_1,\ldots,\log \psi_p)^{{\mbox{\scriptsize \sf T}}}$. We then update ${\bf f}$ and ${\bf s}$ in turn using a standard quasi-Newton update as implemented in the `optim` routine in the `R` language with method `bfgs` (Broyden-Fletcher-Goldfarb-Shanno update). Finally, we transform back using the exponential to get updates of ${\boldsymbol{\phi}}$ and ${\boldsymbol{\psi}}$.
We also tried other options: joint update of $({\boldsymbol{\phi}},{\boldsymbol{\psi}})$ without log-transformation but introducing box constraints to avoid negative values and joint quasi-Newton update of the log-transformed parameters $(\bf f, \bf s)$. For simulated data examples, the option with separate updates of $\bf f$ and $\bf s$ performed best.
Initialization
--------------
We initialize the components ${\boldsymbol{\alpha}}$ by a sample of independent random normals with mean zero and standard deviation 0.05 while we choose 1 for the initial values of the components in ${\boldsymbol{\sigma}}^2$. For ${\boldsymbol{\phi}}$ and ${\boldsymbol{\psi}}$ we choose initial values that depend on the scale of the observation window to avoid that the corresponding covariance functions become essentially constant equal to zero (too small initial values) or to one (too large initial values). For the unit square observation window, for example, the initial values for ${\boldsymbol{\phi}}$ and ${\boldsymbol{\psi}}$ were chosen randomly from the uniform distribution on $[0.01,0.05]$. Regarding the choice of weights $w_{ijk}$ introduced in Section \[sec:lse\], we follow arguments by [@waagepetersen:etal:16] and fix, for $i,j=1,\ldots,p$ and $k=1,\ldots,L$, $w_{ijk}=\hat g_{ij}(t_k)/2$ for $i \neq j$ and $w_{iik}=\hat g_{ii}(t_k)$.
Strategy to determine $q$ and regularization parameters $\lambda$ and $\xi$ {#sec:cv}
---------------------------------------------------------------------------
In our applications we consider just a few values $\xi=0$ (ridge), $\xi=0.5$ (mix of ridge and LASSO, i.e. elastic net) and $\xi=1$ (LASSO). For each of the values of $\xi$ we use a two-dimensional $K$-fold cross validation approach to select optimal values ${\lambda}_{\text{opt}}$ and $q_{\text{opt}}$ among prespecified values ${\lambda}_1,\ldots,{\lambda}_M$ and $q_1,\ldots,q_N$ [e.g. @hastie:tibshirani:friedman:13 Chapter 7]. The procedure is as follows.
1. We split indices $ijk$ ($i,j=1,\ldots,p$ and $k=1,\ldots,L$) into $K$ sets $S_1,\ldots,S_K$ (see details below). \[step1\]
2. For each ${\lambda}\in \{{\lambda}_1,\ldots,{\lambda}_M\}$ and $q
\in \{q_1,\ldots,q_N\}$, we obtain an estimate $\boldsymbol{\hat \theta}_c$ by minimizing equation with $w_{ijk}$ replaced by 0 for $ijk \in S_c,c=1,\ldots,K$. The cross validation score for $\lambda$ and $q$ is then obtained by $$\begin{aligned}
\mathrm{CV}({\lambda},q)=\frac{1}{K} \sum_{c=1}^{K} \mathrm{CV}_c, \label{eq:CV}
\end{aligned}$$ where $\mathrm{CV}_c= \sum_{ijk \in S_c} (Y_{ijk}- \hat{Y}_{ijk}(\boldsymbol{\hat \theta}_c))^2$ and $\hat{Y}_{ij}(\boldsymbol{\hat \theta}_c)=X_{ij}(\hat {\boldsymbol{\phi}}_c,,\hat {\boldsymbol{\psi}}_c) {\boldsymbol{\beta}}_{ij}(\hat {\boldsymbol{\alpha}}_c,\hat {\boldsymbol{\sigma}}^2_c)$. \[step2\]
3. To obtain ${\lambda}_{\text{opt}}$ and $q_{\text{opt}}$, we minimize $\mathrm{CV}({\lambda},q)$ w.r.t ${\lambda}$ and $q$, i.e., $$\begin{aligned}
({\lambda}_{\text{opt}}, q_{\text{opt}})=\operatorname*{arg\,min}_{m=1,\ldots,M,n=1,\ldots,N} \mathrm{CV}({\lambda}_m,q_n). \label{eq:qopt}
\end{aligned}$$
The sets $S_c$ in Step \[step1\] need to be chosen carefully. First, since $\log(\hat g_{ijk})$ and $\log(\hat
g_{ijk'})$ are strongly correlated when $k$ and $k'$ are close, we leave out blocks of consecutive indices. Second, we do not include diagonal indices $iik$ in the sets $S_c$ since values $Y_{iik}$ include contributions from the type-specific random fields. The diagonal values thus do not provide so much information about $q$ and omission of these values further makes the estimation procedure less stable regarding ${\boldsymbol{\sigma}}^2$ and ${\boldsymbol{\psi}}$. So, to determine each subset $S_c$, we arrange the $ijk$ with $i<j$ lexicographically in a vector $(121,122,\ldots)$ and split this vector into consecutive blocks of length $b$. These blocks are then assigned to the different $S_c$ at random.
The one standard error (1-SE) rule is an alternative way to select $\lambda$ and $q$ based on the CV scores obtained from [e.g. @hastie:tibshirani:friedman:13]. In case of $q$ fixed, the 1-SE rule chooses the largest $\lambda$ for which the CV score is less than the smallest CV score plus one standard deviation. In the case where both $\lambda$ and $q$ is to be selected, we adapt the 1-SE rule by starting with $({\lambda}_{\text{opt}}, q_{\text{opt}})$ given by and then choosing $({\lambda},q)$ to be the smallest $q$ and largest $\lambda$ possible such that the following condition holds: $$\begin{aligned}
\mathrm{CV}({\lambda},q) \leq \mathrm{CV}({\lambda}_{\text{opt}},q_{\text{opt}}) + \mathrm{SE}({\lambda}_{\text{opt}},q_{\text{opt}}),
\end{aligned}$$ where $$\begin{aligned}
\mathrm{SE}({\lambda}_{\text{opt}},q_{\text{opt}})=\sqrt{\frac{\sum_{c=1}^{K}(\mathrm{CV}_c-\mathrm{CV}({\lambda},q))^2}{(K-1)K}}.
\end{aligned}$$ Hence, the 1-SE rule attempts to select the most simple model whose CV score is within one standard error of the minimal CV score.
Finally, note that when $\xi=0.5$ or $\xi=1$ and $\lambda>0$ is chosen, the resulting estimate of ${\boldsymbol{\alpha}}$ may contain columns that consist entirely of zeros. The effective number $q_{\text{eff}}$ of columns in ${\boldsymbol{\alpha}}$ then becomes smaller than $q_{\text{opt}}$.
Simulation study {#sec:sim}
================
We conduct two simulation studies to evaluate the regularized least squares technique for parameter estimation and the cross-validation (CV) method to select $q$ and $\lambda$. The setting of the first study corresponds to the simulation study in [@waagepetersen:etal:16]. We first compare the estimates obtained using the new cyclical block descent (CBD) algorithm developed in Section \[sec:rlse\] with the method proposed by [@waagepetersen:etal:16]. In this regard, we consider values of $q=1,\ldots,5$ and for comparison purposes, we fix $\lambda=0$ since regularization was not used in [@waagepetersen:etal:16]. Next we consider only the new algorithm with the objective of comparing different CV options for selecting $q$ and $\lambda$, cf. Section \[sec:cv\], and to study the effect of regularization. The second study has the same objective but with a more complex setting for the simulations. In both simulation studies we use $K=8$ for the CV and we only consider the LASSO option ($\xi=1$) for regularization.
To asses the parameter estimates, we consider the root mean squared errors (RMSEs) of the estimates. For a real parameter $\omega$ and estimate $\hat \omega$, the RMSE is $$\begin{aligned}
\mathrm{RMSE}(\hat \omega)=\sqrt{\mathbb{E} \big( (\hat \omega - \omega)^2\big)}.
\end{aligned}$$ For each of the parameter matrices/vectors ${\boldsymbol{\alpha}}{\boldsymbol{\alpha}}^{{{\mbox{\scriptsize \sf T}}}}$, ${\boldsymbol{\sigma}}^2$, ${\boldsymbol{\psi}}$, or the vector of proportions of variances at lag 0 (PV), we evaluate the average of RMSEs for the components in these quantities. For example, we compute the average of RMSEs for each entry in the $p \times p$ matrix ${\boldsymbol{\alpha}}{\boldsymbol{\alpha}}^{{\mbox{\scriptsize \sf T}}}$.
Comparison of methods for least squares estimation {#sec:simp5:study}
--------------------------------------------------
The first study follows the one in [@waagepetersen:etal:16] for which 200 point patterns in $W=[0,1]^2$ are generated from multivariate log Gaussian Cox processes as defined in Section \[sec:mlgcp\], with $p=5$ and $q=2$. The true parameters are: $\boldsymbol{\sigma}^2=(1,1,1,1,1), \; \boldsymbol{\psi}=(0.01,0.02,0.02,0.03,0.04), \; \boldsymbol{\phi}=(0.02,0.1)$ and $$\begin{aligned}
\boldsymbol{\alpha}^{{\mbox{\scriptsize \sf T}}}& =
\begin{bmatrix}
\sqrt{0.5} & 1& -1& 0& 0 \\
0& 0& 1& -1& 0.5 \\
\end{bmatrix}.
\end{aligned}$$ The trend models $\mu_i(u)=m_i$ are set such that the expected number of points is 1000 for each $i=1,\ldots,5$. A uniform kernel with bandwidth 0.005 is used for the non-parametric estimation of the cross pair correlation function at $L=25$ equispaced lags between 0.025 and 0.25.
For each simulation we compare two methods for minimizing with $\lambda=0$ and $q \in \{1,\cdots,5\}$:
1. The standard quasi-newton (SQN) optimization algorithm considered by [@waagepetersen:etal:16] and implemented in the `R` package `optimx`. This algorithm updates all parameters jointly.
2. The new CBD algorithm described in Section \[sec:rlse\].
The comparison is in terms of minimization of the objective function, computing time and RMSEs.
----- ------ ------ ------ ------ ------
1 2 3 4 5
SQN 6.61 4.76 5.39 6.32 4.51
CBD 3.55 1.96 1.73 1.62 1.57
SQN 0.96 1.98 3.97 6.45 8.99
CBD 1.99 3.11 4.26 5.30 5.92
----- ------ ------ ------ ------ ------
: Averages of the minimized objective function $Q({\boldsymbol{\theta}})$ given by and the computing time (in seconds) based on 200 simulations from a multivariate log Gaussian Cox process ($p=5, q=2$), modeled with $q \in \{1,2,3,4,5\}$, for two optimization methods.
\[tab:QT\]
[crrrrrr]{} & &\
& 1 & 2 & 3 & 4 & 5 &\
\
SQN & 0.41 & 0.93 & 1.10 & 1.17 & 1.09 & 10.3\
CBD & 0.41 & 0.25 & 0.29 & 0.32 & 0.39 & 0\
\
SQN & 0.58 & 0.54 & 0.44 & 0.89 & 0.98 & 1.1\
CBD & 0.34 & 0.18 & 0.28 & 0.39 & 0.50 & 0\
\
SQN & 0.0791 & 0.1752 & 0.1337 & 0.4091 & 0.4566 & 11.5\
CBD & 0.0050 & 0.0091 & 0.0110 & 0.0005 & 0.0004 & 0\
\[tab:RMSE:no\_out\]
Table \[tab:QT\] reports the averages of the values of the minimized objective functions and the computational times over the 200 simulations. All timings are carried out on a Dell R740 2 x 14 cores (Intel(R) Xeon(R) Gold 6132 CPU @ 2.60GHz) 768 GB RAM 2x200gb SSD 960 GB NVME. CBD performs considerably better in terms of minimizing the objective function than SQN. SQN is somewhat faster than CBD for small $q$ but slower for larger $q$. The computing times for SQN grow quite quickly with increasing $q$ while the computing times seems more stable for CBD.
The RMSE results are shown in Table \[tab:RMSE:no\_out\]. For the calculation of the RMSEs, we exclude small percentages of very extreme parameter estimates. These percentages are reported in the last column of Table \[tab:RMSE:no\_out\]. CBD performs better than SQN since smaller RMSEs are obtained and there are no outlying parameter estimates. For SQN quite large percentages of extreme parameter estimates are observed.
Assessment of cross-validation and regularization methods with $p=5$ {#sec:simp5:res}
--------------------------------------------------------------------
In this section we continue with the simulations from the previous setting but restrict attention to CV selection of $q$ and $\lambda$ using CBD for optimization with the LASSO regularization ($\xi=1$). We select values of $q$ in $\boldsymbol{q}=\{1,2,3,4,5\}$ and values of $\lambda$ in $\boldsymbol{\lambda}=\{0,10^{-3},\ldots,5\}$ which has 20 elements and where the non-zero values of $ \boldsymbol \lambda$ grow log-linearly from $\log 10^{-3}$ to $\log 5$. We consider three situations: (1) we select $q$ from $\boldsymbol{q}$ with $\lambda=0$ fixed, thus least squares estimation (LSE) is performed; (2) we search for the jointly optimal $(q,\lambda)$; (3) we fix $q=5$ and select $\lambda$ from $\boldsymbol{\lambda}$. Recall that the selection of a relatively big $\lambda$ may lead to zero columns in the ${\boldsymbol{\alpha}}$ estimate. We therefore consider the effective $q_{\text{eff}}$ as defined in Section \[sec:cv\]. Thereby we can also evaluate the selection of $q$ in situation (3). In case of (2) we both consider the minimum CV (Min) and the 1-standard error (1-SE) rules to select $q$ and $\lambda$.
Table \[tab:distp5:final\] shows the distribution of absolute distance between $q_{\text{eff}}$ and the true $q=2$. For LSE, using the Min rule, $q_{\text{eff}}$ coincides with the true $q$ for 47% of the simulations and differs at most by 1 from the true $q$ in 75% of the simulations. The results with the 1-SE rule are similar with percentages $46$ and $78$. LASSO with Min rule for joint selection of $(q,\lambda)$ performs similarly to LSE with the corresponding percentages 42 and 74 %. With fixed $q=5$ the percentages are reduced to 16% and 53 %. Using 1-SE rule, the LASSO forces many columns to be zero leading to quite small percentages where $|q_{\text{eff}}-2| \le 1$.
----------------------- ---- ---- ---- ---- ---- ---- ---- --- ---- ---- ---- ----
$ |q_{\text{eff}}-2|$ 0 1 2 3 0 1 2 3 0 1 2 3
Min 47 28 13 12 42 32 21 5 16 37 30 17
1-SE 46 32 22 0 15 20 65 0 10 22 65 3
----------------------- ---- ---- ---- ---- ---- ---- ---- --- ---- ---- ---- ----
: Distribution of $ |q_{\text{eff}}-2|$ (in %) over 200 simulations from a multivariate log Gaussian Cox process ($p=5,q=2$) using CBD for minimization. []{data-label="tab:distp5:final"}
------------------------------------------------------------------------------------- ------------- -------------------------------------- ------ ------ ------ ------ ------ ------
$\lambda=0$ $\lambda \in {\boldsymbol{\lambda}}$
Min Min Min 1-SE Min 1-SE Min 1-SE
$\hat {\boldsymbol{\alpha}}\hat {\boldsymbol{\alpha}}^{{\mbox{\scriptsize \sf T}}}$ 0.26 0.33 0.33 0.40 0.36 0.54 0.40 0.54
$\hat {\boldsymbol{\sigma}}^2$ 0.42 0.54 0.54 0.58 0.56 0.75 0.63 0.76
$\hat {\boldsymbol{\psi}}$ 0.04 0.05 0.05 0.02 0.03 0.01 0.04 0.01
$\hat {\mathrm{PV}}$ 0.28 0.31 0.32 0.35 0.33 0.41 0.37 0.42
------------------------------------------------------------------------------------- ------------- -------------------------------------- ------ ------ ------ ------ ------ ------
: Average RMSEs obtained from 200 simulations from a multivariate log Gaussian Cox process ($p=5, q=2$) for different methods of selecting $q$ and $\lambda$.[]{data-label="tab:RMSEp5:final"}
RMSEs are reported in Table \[tab:RMSEp5:final\] for all three situations. In addition, in the first columns, we consider the case fixed $q=2$ assuming the true $q$ is known. We first note that LASSO gives worse results than LSE when $q=2$ is fixed. In general, for unknown $q$, LSE and LASSO perform quite similarly when the Min rule is used. The results are worse when 1-SE is used and in particular for LASSO. When $q$ is fixed to $5$ and only $\lambda$ is selected the results are worse than for LASSO with $q$ selected by the Min rule while the results with $q=5$ are similar to LASSO with $q$ selected by the 1-SE rule.
The overall impression is that LSE performs slightly better than LASSO, especially in estimating ${\boldsymbol{\alpha}}{\boldsymbol{\alpha}}^{{\mbox{\scriptsize \sf T}}}$. This may indicate that when $p$ is relatively small, selection of $q$ with $\lambda=0$ (LSE) already gives sparse results. Another reason that LASSO does not improve RMSE may be that the true ${\boldsymbol{\alpha}}$ is not that sparse having only 40% zero components. Thus the bias introduced by regularization is not counterbalanced by a reduction in variance. Also, the 1-SE rule does not seem preferable in this situation. In the next section we consider a more complex setting with $p=10$.
Assessment of cross-validation and regularization methods with $p=10$ {#sec:simp10}
---------------------------------------------------------------------
In this experiment, we study a more complex situation with a higher $p$ and more variation in the parameters. We simulate 200 point patterns from a multivariate log Gaussian Cox process with $p=10$, $q=4$, $W=[0,1]^2$, and parameters $$\begin{aligned}
\boldsymbol{\phi} & = (0.02,0.03,0.03,0.05)^{{\mbox{\scriptsize \sf T}}}, \\
\boldsymbol{\sigma}^2 & = (1,1,1.5,1,0.2,0.2,1,1.5,1.5,1.5)^{{\mbox{\scriptsize \sf T}}},
\end{aligned}$$
$$\begin{aligned}
\boldsymbol{\alpha}& =
\begin{bmatrix}
& \sqrt{0.5} & 0.10 & -1& 0 \\
& 0 & 0 & -0.70 & 1 \\
& 0 & -0.15 & \sqrt{0.5} & 0.10 \\
& -1& 0 & 0 & 0 \\
& -0.70 & 1& 0 & -0.15 \\
& \sqrt{0.5} & 0.10 & -1& 0 \\
& 0 & 0 & -0.70 & 1 \\
& 0 & -0.15 & \sqrt{0.5} & 0.10 \\
& -1& 0 & 0 & 0 \\
& -0.70 & 1& 0 & -0.15 \\
\end{bmatrix},
\end{aligned}$$
and ${\boldsymbol{\psi}}$ equal to $$(0.01,0.02,0.02,0.03,0.04,0.04,0.05,0.06,0.06,0.07)^{{\mbox{\scriptsize \sf T}}}.$$ The settings for the trend models, the kernel estimation and the cross validation are as in the previous simulation study except that ${\boldsymbol{q}}=\{0,\ldots,8\}$. In ${\boldsymbol{\alpha}}$, 40% of the components are zeros and 20% are of absolute value less than 0.15. The remaining components have absolute value greater than 0.7.
Table \[tab:distp10:final\] shows the distribution of the absolute distance $|q_{\text{eff}}-4|$ between $q_{\text{eff}}$ and the true $q=4$. Considering first the Min rule, with LSE, $q_{\text{eff}}$ concurs with the true $q$ in 19% of the simulations and differs at most by 2 from the true $q$ in 58% of the simulations. The corresponding percentages are 14% and 65 % for LASSO, and 6% and 41 % for LASSO with $q=8$ fixed. In this situation, the 1-SE rule seems advantageous for selecting $q$. For example, the percentage of $q_{\text{eff}}$’s which differ from the true $q$ by at most 2 improves from 58% to 83 % for LSE, from 65% to 80 % for LASSO, and from 41% to 68 % for LASSO with fixed $q=8$.
----------------------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
$ |q_{\text{eff}}-4|$ 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
Min 19 21 18 19 23 14 31 20 19 16 6 15 20 21 38
1-SE 27 36 20 12 5 22 37 21 8 12 21 22 25 11 21
----------------------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
Table \[tab:RMSEp10:final\] details the RMSE results. The superiority of the 1-SE rule when selecting $q$ does not translate into better results in terms of RMSE except for LASSO with fixed $q=8$ where better results are obtained with 1-SE than with Min. The best results are obtained with LASSO using the Min rule for selecting $q$ and $\lambda$. This indicates that regularization is indeed helpful in complex settings with relatively large $p$.
[l | cc | ccH |cc]{} & & &\
& Min & 1-SE & Min & 1-SE & SE ($\lambda$ & $q$) & Min & 1-SE\
$\hat {\boldsymbol{\alpha}}\hat {\boldsymbol{\alpha}}^{{\mbox{\scriptsize \sf T}}}$ &0.50&0.67&0.44&0.48&&0.78&0.51\
$\hat {\boldsymbol{\sigma}}^2$ &0.58&0.89&0.54&0.70&&0.88&0.76\
$\hat {\boldsymbol{\psi}}$ &0.02&0.02&0.01&0.02&&0.02&0.02\
$\hat{\mathrm{PV}}$ &0.35&0.35&0.34&0.39&&0.35&0.40\
Based on the simulation studies, for analyzing highly multivariate point pattern data, we recommend to use regularization with the Min rule for selecting $q$ and $\lambda$.
Application {#sec:app}
===========
In a 50-hectare $1,000 \; \mathrm{m} \times 500 \; \mathrm{m}$ region of the tropical moist forest of Barro Colorado Island (BCI) in central Panama, censuses have been carried out where all free-standing woody stems with at least 10 mm diameter at breast height were identified, tagged, and mapped, resulting in maps of over 350,000 individual trees with around 300 species [see e.g. @hubbell:foster:83; @condit:hubbell:foster:96; @condit:98]. In addition, 13 spatial covariates are also available containing topological attributes and soil nutrients (see Figure \[fig:cov\]). Our main objective is to study the impact of regularization and the computational feasibility of our method. We first consider 9 tree species, [*Psychotria, Protium t., Capparis, Protium p., Swartzia, Hirtella, Tetragastris, Garcinia, Mourmiri*]{}, with intermediate abundances ranging from 2500 to 7500 and previously analyzed by [@waagepetersen:etal:16]. The plots of locations of each species are shown in Figure \[fig:spec9\]. The main aim of this analysis is to compare the results with our new algorithm to those obtained by [@waagepetersen:etal:16]. Secondly, to test our algorithm in a more challenging situation, we analyze a highly multivariate point pattern involving species of trees with at least 400 individuals, resulting in 86 species. For each species, we use maximum composite likelihood to fit log-linear regression models involving the spatial covariates for the $\mu_i$-terms in . We then estimate the cross pair correlation function using . Therefore, the variation due to observed covariates are filtered out and the non-parametric estimates of cross pair correlation function hence capture the residual correlation due to unobserved covariates, species-specific factors, and any other sources.
Application with 9 species
--------------------------
For each value of $\xi=0,0.5,1$ we apply $8$-fold CV to select $q$ and ${\lambda}$ where ${\lambda}\in {\boldsymbol{\lambda}}= \{0,10^{-3},\ldots,5\}$ as in the simulation studies and $q \in {\boldsymbol{q}}=\{0,\ldots,9\}$. The upper left plot in Figure \[fig:CV9\] shows for each $\xi$, $\min_{{\lambda}\in {\boldsymbol{\lambda}}} \mathrm{CV}(q,{\lambda})$ as a function of $q$. For comparison with [@waagepetersen:etal:16] we also show in this plot $\mathrm{CV}(q,0)$ against $q$ (LSE). A general pattern for ridge, elastic net and LASSO is that the cross validation scores decrease quite quickly as a function of $q$ until around $q=4$ and after that the CV scores stabilize or decrease slowly. The CV scores for ridge ($\xi=0$) are consistently smaller than those for elastic net ($\xi=0.5$) and LASSO ($\xi=1$). Hence we select $\xi=0$. The minimal CV score for $\xi=0$ is obtained with $q=9$. However, in the interest of model simplicity, we choose $q=4$ and $\lambda=0.29$ since the decrease in CV score is rather minor from $q=4$ to $q=9$.
For comparison, the minimal CV score with LASSO is obtained with $q=8$ and ${\lambda}=0.11$. However, in this case, the resulting effectively selected $q_{\text{eff}}$ is three since the resulting estimate of ${\boldsymbol{\alpha}}$ has 5 zero columns. In case of LSE ($\lambda=0$), the CV procedure chooses $q=1$. The second-smallest CV with LSE is obtained with $q=4$ which was the value chosen in [@waagepetersen:etal:16]. The difference in cross validation results for LSE compared with [@waagepetersen:etal:16] is due to our new more efficient optimization algorithm, cf. the comparison in Section \[sec:simp5:study\].
The middle plot in Figure \[fig:CV9\] is an image plot of the CV scores for ridge ($\xi=0$) where darker color corresponds to smaller CV score. The development of the CV scores across values of $q$ for fixed ${\lambda}$ appears quite erratic with several local minima. In contrast, for each $q$ there appears to be a well-defined minimum for $\lambda$. As an example, the right plot in Figure \[fig:CV9\] shows $\mathrm{CV}(4,\lambda)$ plotted against $\log \lambda$ (where we replace the undefined $\log 0$ by $\log 5e-4$). The computing time required to run the CV method with $\xi=0$ is $2.4$ hours with the same processor as used in the simulation study. Approximately $16$ seconds is required to estimate the parameters for the 9-species application using ridge with $q=4$ and ${\lambda}=0.29$.
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{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
------------------------------------------- ------------------------------------------------- -------------------------------------------------
------------------------------------------------- -------------------------------------------------
{width="50.00000%"} {width="50.00000%"}
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The results regarding the multivariate dependence structure of the 9 species are qualitatively similar to those obtained by [@waagepetersen:etal:16]. The estimated inter-species correlations $\mathrm{corr}\{Z_i(u),Z_j(u)\}$, cf. , are shown in the left plot of Figure \[fig:mult9\]. Most of the pairs of species have a positive correlation. However, the correlations between [*Psychotria*]{} and the other species are mainly close to zero. The right plot in Figure \[fig:mult9\] shows a hierarchical clustering of the species based on the estimated coefficient rows ${\boldsymbol{\alpha}}_{i \cdot}$, where [*Psychotria*]{} appears to form its own cluster in agreement with the estimated inter-species correlations. This clustering may have some relation to the families of species as shown by the cluster of [*Protium p.*]{}, [*Protium t.*]{} and [*Tetragastris*]{} which come from the same family (see Table \[supp-tab:86spec\] in the supplementary material).
Application with 86 tree species
--------------------------------
For the 86-species application, we apply the 8-fold CV procedure with $\xi=0,0.5,1$ and ${\lambda}\in \{0,10^{-3},\ldots,5\}$ as in the previous section and $q \in \{0, \ldots, 10\}$. Figure \[fig:CV86\] is similar to Figure \[fig:CV9\]. The left plot shows that consistently smaller CV scores are obtained with elastic net ($\xi=0.5$) and the smallest CV score is obtained with $q=4$. The remaining plots are obtained with $\xi=0.5$. The image plot of cross validation scores in the middle plot looks much smoother than in the 9 species case. The right plot shows a well defined minimum for $\lambda=1.94$ given $q=4$.
-------------------------------------------- ------------------------------------------------- ----------------------------------------------
{width="33.30000%"} {width="33.30000%"} {width="33.30000%"}
-------------------------------------------- ------------------------------------------------- ----------------------------------------------
-------------------------------- ------ ------ ------ ----- ----- -----
Lower -1 -0.5 -0.2 0 0.2 0.5
Upper -0.5 -0.2 0 0.2 0.5 1
$\mathrm{corr}[Y_i(u),Y_j(u)]$ 2 6 9 13 22 48
$\mathrm{corr}[Z_i(u),Z_j(u)]$ 0 2 15 60 19 4
-------------------------------- ------ ------ ------ ----- ----- -----
: Distribution (in %) of estimated inter-species correlations $\mathrm{corr}[Y_i(u),Y_j(u)]$ and $\mathrm{corr}[Z_i(u),Z_j(u)]$, $i \neq j$, over different intervals $[\text{Lower},\text{Upper}]$ for the 86 species application using elastic net ($\xi=0.5$) with $q=4$ and ${\lambda}=1.94$. []{data-label="tab:correlation"}
Interval 0-0.25 0.25-0.5 0.5-0.75 0.75-1
------------------- -------- ---------- ---------- --------
Number of species 46 20 10 10
Species (%) 53 23 12 12
: Distribution of estimated $\mathrm{PV}_i(0)$ for 86 species application using elastic net ($\xi=0.5$) with $q=4$ and ${\lambda}=1.94$.[]{data-label="tab:PV"}
The computing time for the CV is 7.6 hours for $\xi=0.5$ and the computing time to estimate the parameters for the chosen $q=4$ and $\lambda=1.94$ is 3.2 minutes. Out of $4\times 86$ parameters in the estimated ${\boldsymbol{\alpha}}$, 13 were set to zero by the elastic net regularization. We thereby model $86\times 87/2=3741$ distinct pair and cross pair correlation functions using only $6\times 86-13+4=507$ parameters. Thus we have indeed obtained a sparse model for the given data.
The distribution of estimated PVs is shown in Table \[tab:PV\]. Most species ($53\%$) have estimated proportions of variances due to common factors less than 0.25.
Table \[tab:correlation\] shows the distribution of estimated inter-species correlations due to common latent fields and the combination of common and species-specific fields (see Section \[sec:multvarinference\]) across 6 intervals. Most estimated correlations are positive. However, the correlations decrease a lot in absolute value when the species-specific fields are included (last row of Table \[tab:correlation\]).
Figure \[fig:clus86\] shows a clustering of species based on estimated ${\boldsymbol{\alpha}}_{i\cdot}$, $i=1,\ldots,86$. The leaves are marked with species life form. There may be some indication that species of life form “Tree” (life form number 4) tend to cluster together. However, one should be careful with this interpretation since apparent patterns like this could be due to sampling variation.
{width="100.00000%" height="80.00000%"}
Conclusion {#sec:disc}
==========
We developed in this study a regularized estimation method for highly multivariate point patterns modeled by multivariate log Gaussian Cox processes. The procedure is numerically stable and performs well both in the considered simulations and applications. In our truly highly multivariate second application, we were able to fit a sparse model for a multivariate point pattern with 86 types of points.
An interesting application of obtained estimates is to group types of points according to their estimated dependence on common latent fields as expressed by the rows ${\boldsymbol{\alpha}}_{i\cdot}$. Hence a further development could be to consider an extension of the so-called fused LASSO [@Tibshirani05sparsityand] by introducing regularization for differences ${\boldsymbol{\alpha}}_{i \cdot}-{\boldsymbol{\alpha}}_{j\cdot}$. A further possibility would be to consider a sparse group LASSO [@simon2013sparse] to obtain estimates of ${\boldsymbol{\alpha}}$ with some zeros of $\alpha_{il}$ as developed in this paper and, in addition, with entire rows of zeros implying independence of corresponding types of points and all other types of points.
The research by A. Choiruddin, F. Cuevas-Pacheco, and R. Waagepetersen is supported by The Danish Council for Independent Research | Natural Sciences, grant DFF – 7014-00074 “Statistics for point processes in space and beyond”, and by the “Centre for Stochastic Geometry and Advanced Bioimaging”, funded by grant 8721 from the Villum Foundation.
The BCI forest dynamics research project was made possible by National Science Foundation grants to Stephen P. Hubbell: DEB-0640386, DEB-0425651, DEB-0346488, DEB-0129874, DEB-00753102, DEB-9909347, DEB-9615226, DEB-9615226, DEB-9405933, DEB-9221033, DEB-9100058, DEB-8906869, DEB-8605042, DEB-8206-992, DEB-7922197, support from the Center for Tropical Forest Science, the Smithsonian Tropical Research k+1 Institute, the John D. and Catherine T. MacArthur Foundation, the Mellon Foundation, the Celera Foundation, and numerous private individuals, and through the hard work of over 100 people from 10 countries over the past two decades. The plot project is part of the Center for Tropical Forest Science, a global network of large-scale demographic tree plots.
The BCI soils data set were collected and analyzed by J. Dalling, R. John, K. Harms, R. Stallard and J. Yavitt with support from NSF DEB021104, 021115, 0212284, 0212818 and OISE 0314581, STRI and CTFS. Paolo Segre and Juan Di Trani provided assistance in the field. The covariates `dem`, `grad`, `mrvbf`, `solar` and `twi` were computed in SAGA GIS by Tomislav Hengl (`http://spatial-analyst.net/`). We thank Dr. Joseph Wright for sharing data on dispersal modes and life forms for the BCI tree species.
Proximal Newton Method {#prox}
======================
Suppose we want to find the solution of $$\begin{aligned}
\min_{\boldsymbol{\theta} \in \mathbb{R}^n} f(\boldsymbol{\theta}):= a(\boldsymbol{\theta}) + c(\boldsymbol{\theta}),
\end{aligned}$$ where the function $f(\cdot)$ can be separated into two parts: the function $a(\cdot)$ which is a convex and twice continuously differentiable loss function and the function $c(\cdot)$ which is a convex but not necessarily differentiable penalty function. The proximal-Newton method is an iterative optimization algorithm that uses a quadratic approximation of the differentiable part $a(\cdot)$: $$\begin{aligned}
f(\boldsymbol{\theta}) \approx & \; \hat f(\boldsymbol{\theta}) \nonumber \\
= & \; \hat a(\boldsymbol{\theta}) + c(\boldsymbol{\theta}) \nonumber \\
= & \; a(\boldsymbol{{\theta}^{(k)}}) + \nabla a(\boldsymbol{{\theta}^{(k)}})^{{\mbox{\scriptsize \sf T}}}(\boldsymbol{\theta} - \boldsymbol{{\theta}^{(k)}}) + (\boldsymbol{\theta} - \boldsymbol{{\theta}^{(k)}})^{{\mbox{\scriptsize \sf T}}}H(\boldsymbol{{\theta}^{(k)}}) (\boldsymbol{\theta} - \boldsymbol{{\theta}^{(k)}}) + c(\boldsymbol{\theta}) \label{proxmethod},
\end{aligned}$$ where ${\boldsymbol{\theta}}^{(k)}$ is the current value of ${\boldsymbol{\theta}}$, $\nabla a(\cdot)$ is the first derivative of $a(\cdot)$ and $H(\cdot)$ is an approximation to the Hessian matrix $\nabla^2 a(\cdot)$. Letting $\tilde {\boldsymbol{\theta}}= \operatorname*{arg\,min}_{{\boldsymbol{\theta}}} \hat f({\boldsymbol{\theta}})$, the next value of ${\boldsymbol{\theta}}$ is obtained as $${\boldsymbol{\theta}}^{(k+1)}={\boldsymbol{\theta}}^{(k)}+ t (\tilde {\boldsymbol{\theta}}- {\boldsymbol{\theta}}^{(k)})$$ for some $t>0$. That is, $\tilde {\boldsymbol{\theta}}$ is used to construct a search direction for the $k+1$th value of ${\boldsymbol{\theta}}$. Theoretical results in [@lee:sun:saunders:14] show that $t$ can be chosen so that $f({\boldsymbol{\theta}}^{(k+1)})<f({\boldsymbol{\theta}}^{(k)})$. The matrix $H(\cdot)$ can be chosen in various ways, see [@lee:sun:saunders:14] and [@hastie:tibshirani:wainwright:15] for more details.
In the following sections, we adapt the proximal Newton method to minimization of our objective function.
Quadratic approximation for updating ${\boldsymbol{\alpha}}_{i\cdot}$ {#taylor}
---------------------------------------------------------------------
Let us first regard as a function of ${\boldsymbol{\alpha}}_{i \cdot}$, $$\begin{aligned}
Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma^2_i) = & \; 2 \sum_{\substack{j=1\\j \neq i}}^p \| Y_{ij}-
\tilde X_{ij}{\boldsymbol{\alpha}}_{i \cdot} \|^2 + \|Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2) \|^2 + {\lambda}\sum_{l=1}^q p({\alpha}_{il}) \nonumber \\
= & \; a({\boldsymbol{\alpha}}_{i \cdot}) + b({\boldsymbol{\alpha}}_{i \cdot}) + c({\boldsymbol{\alpha}}_{i \cdot}).\label{eq:abc}
\end{aligned}$$
To minimize , we consider the proximal Newton method stated in . In particular, we approximate $b({\boldsymbol{\alpha}}_{i \cdot})$ by a quadratic approximation around the current value ${\boldsymbol{\alpha}}_{i \cdot}^{(k)}$: $$\begin{aligned}
b({\boldsymbol{\alpha}}_{i \cdot}) \approx & \; \hat b({\boldsymbol{\alpha}}_{i \cdot}) \nonumber\\
= & \; b({\boldsymbol{\alpha}}_{i \cdot}^{(k)})+ \nabla b({\boldsymbol{\alpha}}_{i \cdot}^{(k)})^{{\mbox{\scriptsize \sf T}}}({\boldsymbol{\alpha}}_{i\cdot} -{\boldsymbol{\alpha}}_{i \cdot}^{(k)}) + \frac{1}{2} ({\boldsymbol{\alpha}}_{i\cdot}-{\boldsymbol{\alpha}}_{i \cdot}^{(k)})^{{\mbox{\scriptsize \sf T}}}H({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) ({\boldsymbol{\alpha}}_{i\cdot} -{\boldsymbol{\alpha}}_{i \cdot}^{(k)}) \label{approxb}.
\end{aligned}$$ Here, the first derivative is $$\nabla b({\boldsymbol{\alpha}}_{i \cdot}^{(k)})= -4 D({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) X_{ii,\cdot(1:q)}^{{\mbox{\scriptsize \sf T}}}\Big(Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}}^{(k)},{\boldsymbol{\sigma}}^2)\Big)$$ while $H({\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ is an approximation of the second derivative, $$\nabla^2 b({\boldsymbol{\alpha}}_{i \cdot}^{(k)})= 8 D({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) X_{ii,\cdot(1:q)}^{{\mbox{\scriptsize \sf T}}}X_{ii,\cdot(1:q)} D({\boldsymbol{\alpha}}^{(k)}_{i \cdot}) - C({\boldsymbol{\alpha}}^{(k)}_{i \cdot}),$$ where $D({\boldsymbol{\alpha}}_{i \cdot}^{(k)})={\text{Diag}}({\alpha}_{i1}^{(k)},\ldots,{\alpha}_{iq}^{(k)})$, $X_{ii,\cdot (1:q)}$ denotes the first $q$ columns in $X_{ii}$, and $C({\boldsymbol{\alpha}}^{(k)}_{i \cdot})=4 {\text{Diag}}\bigg(X_{ii,\cdot (1:q)}^{{\mbox{\scriptsize \sf T}}}\Big(Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}}^{(k)},{\boldsymbol{\sigma}}^2)\Big)\bigg)$. Specifically, $$\begin{aligned}
H({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) & = 8 D({\alpha}^{(k)}_{i \cdot}) X_{ii,\cdot(1:q)}^{{\mbox{\scriptsize \sf T}}}X_{ii,\cdot(1:q)} D({\alpha}^{(k)}_{i \cdot}) \\
& \approx \nabla^2 b({\boldsymbol{\alpha}}_{i \cdot}^{(k)}).
\end{aligned}$$
To ease the presentation and computation, we write $\hat b({\boldsymbol{\alpha}}_{i \cdot})$ from in the form of a least squares problem $$\begin{aligned}
\hat b({\boldsymbol{\alpha}}_{i \cdot}) = & \; \|Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}}^{(k)},{\boldsymbol{\sigma}}^2) \|^2 \\ &- 2 \Big(Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}}^{(k)},{\boldsymbol{\sigma}}^2) )\Big)^{{\mbox{\scriptsize \sf T}}}[2X_{ii,\cdot (1:q)} D({\boldsymbol{\alpha}}_{i \cdot}^{(k)})] ({\boldsymbol{\alpha}}_{i\cdot} -{\boldsymbol{\alpha}}_{i \cdot}^{(k)}) \\
& + \frac{1}{2} (2) ({\boldsymbol{\alpha}}_{i\cdot} -{\boldsymbol{\alpha}}_{i \cdot}^{(k)})^{{\mbox{\scriptsize \sf T}}}[2 D({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) X_{ii,\cdot (1:q)}^{{\mbox{\scriptsize \sf T}}}] [2X_{ii,\cdot (1:q)} D({\boldsymbol{\alpha}}_{i \cdot}^{(k)})] ({\boldsymbol{\alpha}}_{i\cdot} -{\boldsymbol{\alpha}}_{i \cdot}^{(k)}) \\
= & \; \mathbf{v}^{{\mbox{\scriptsize \sf T}}}\mathbf{v} - 2\mathbf{v}^{{\mbox{\scriptsize \sf T}}}X^*_{ii}\boldsymbol{\gamma} + \boldsymbol{\gamma}^{{\mbox{\scriptsize \sf T}}}(X^*_{ii})^{{\mbox{\scriptsize \sf T}}}X^*_{ii}\boldsymbol{\gamma}\\
= & \; \|\mathbf{v} - X^*_{ii}\boldsymbol{\gamma}\|^2 \\
= & \; \|Y^*_{ii} - X^*_{ii} {\boldsymbol{\alpha}}_{i \cdot}\|^2
\end{aligned}$$ where $$\begin{aligned}
\mathbf{v} & = Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}}^{(k)},{\boldsymbol{\sigma}}^2) ,\\
X^*_{ii} & = 2X_{ii,\cdot (1:q)} D({\boldsymbol{\alpha}}_{i \cdot}^{(k)}), \\
\boldsymbol{\gamma} & = {\boldsymbol{\alpha}}_{i\cdot} -{\boldsymbol{\alpha}}_{i \cdot}^{(k)}, \\
Y^*_{ii} & = Y_{ii}+ X_{ii,\cdot (1:q)}{\boldsymbol{\alpha}}_{i \cdot}^{2,(k)} - X_{ii,\cdot (q+1)}\sigma^{2}_i.
\end{aligned}$$ Replacing $b$ in with $\hat b$ we obtain the approximate objective function $\hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot}|{\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ given in . Since is a standard regularized least squares problem, we minimize using a coordinate descent algorithm to obtain $\hat {\boldsymbol{\alpha}}_{i \cdot}$ as detailed in Section \[alpha\].
Theoretical result regarding proximal Newton update {#theory}
---------------------------------------------------
Let $\Delta({\boldsymbol{\alpha}}_{i \cdot}^{(k)})=\hat {\boldsymbol{\alpha}}_{i \cdot}-{\boldsymbol{\alpha}}_{i \cdot}^{(k)}$ where $\hat {\boldsymbol{\alpha}}_{i \cdot}$ is the minimizer of and according to a line search strategy let $${\boldsymbol{\alpha}}_{i \cdot}^{(k+1)}= {\boldsymbol{\alpha}}_{i \cdot}^{(k)}+ t\Delta({\boldsymbol{\alpha}}_{i \cdot}^{(k)})$$ for some $t>0$. Following the proof of Proposition 2.3 in [@lee:sun:saunders:14], we can verify the following theorem.
\[descentdirection\] Let $ H({\boldsymbol{\alpha}}_{i \cdot}^{(k)})=8 D({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) X_{ii}^{{\mbox{\scriptsize \sf T}}}X_{ii} D({\boldsymbol{\alpha}}_{i \cdot}^{(k)})$. Then $$\begin{aligned}
Q_{i,{\lambda}}({\boldsymbol{\alpha}}_{i \cdot}^{(k+1)},\sigma^2_i) \leq & \; Q_{i,{\lambda}} ({\boldsymbol{\alpha}}_{i \cdot}^{(k)},\sigma^2_i) - t \Delta ({\boldsymbol{\alpha}}_{i \cdot}^{(k)})^{{\mbox{\scriptsize \sf T}}}H({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) \Delta ({\boldsymbol{\alpha}}_{i \cdot}^{(k)}) + O(t^2).
\end{aligned}$$
Thus, by Theorem \[descentdirection\], if $H({\boldsymbol{\alpha}}_{i \cdot}^{(k)})$ is positive definite, we can choose $t>0$ so that $Q_{i,{\lambda}}({\boldsymbol{\alpha}}_{i \cdot}^{(k+1)},\sigma^2_i)< Q_{i,{\lambda}}({\boldsymbol{\alpha}}_{i \cdot}^{(k)},\sigma^2_i)$. That is, the update of ${\boldsymbol{\alpha}}_{i \cdot}$ results in a decrease of the objective function .
Algorithm {#sec:CDA}
=========
In our block descent algorithm, we minimize with respect to $ {\boldsymbol{\sigma}}^2, {\boldsymbol{\alpha}}, {\boldsymbol{\phi}}$, and ${\boldsymbol{\psi}}$ in turn. For $i=1,\ldots,p$, we first update $\sigma^2_i$ by minimizing using least squares estimation followed by an update of ${\boldsymbol{\alpha}}_{i\cdot}$ by minimizing using a coordinate descent method. We denote by $X_{ij,\cdot k}$ the $k$th column of $X_{ij}$ for $k=1,\ldots,q$ ($i \neq j$) or $k=1,\ldots,q+1$ ($i=j$). We detail, respectively in Appendices \[sigma\] and \[alpha\], the updates of $\sigma^2_i$ and the coordinate descent updates of ${\alpha}_{il}$ for $l=1,\ldots,q$. A summary of the final algorithm is given by Appendix \[algorithm\].
Update of $\sigma_i^2$ {#sigma}
----------------------
The parameter $\hat \sigma_i^2$ is updated using least squares methods. More precisely, the gradient of with respect to $\sigma_i^2$ is $$\begin{aligned}
\frac{\partial Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma_i^2)}{\partial \sigma_i^2} = & \; -2 X_{ii,\cdot (q+1)}^{{\mbox{\scriptsize \sf T}}}(Y_{ii}- X_{ii} {\boldsymbol{\beta}}_{ii}({\boldsymbol{\alpha}},{\boldsymbol{\sigma}}^2) ).
\end{aligned}$$ By solving $\frac{\partial Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i \cdot},\sigma_i^2)}{\partial \sigma_i^2} = 0$, we obtain the update $$\begin{aligned}
\label{updatesigma}
\sigma_i^2 \leftarrow \max \left\{ \frac{ X_{ii,\cdot (q+1)}^{{\mbox{\scriptsize \sf T}}}\left (Y_{ii}- \sum_{l=1}^q X_{ii,\cdot l} {\alpha}^2_{i l}
\right )
}{X_{ii,\cdot (q+1)}^{{{\mbox{\scriptsize \sf T}}}} X_{ii,\cdot (q+1)}} ,0 \right \}
\end{aligned}$$ where $\max\{a,0\}$ is used to avoid negative results of the update.
Update of ${\alpha}_{il}$ {#alpha}
-------------------------
Let $ r_{ij}= Y^*_{ij} - \sum_{\substack{k=1\\k \neq l}}^q X^*_{ij,\cdot k} {\alpha}_{ik}$, where $Y^*_{ij}$ and $X^*_{ij}$ are specified in . Then we rewrite as $$\begin{aligned}
\hat Q_{{\lambda},i}({\boldsymbol{\alpha}}_{i\cdot}) = & \; \sum_{\substack{j=1}}^p \|r_{ij}- X^*_{ij,\cdot l} {\alpha}_{il}\|^2 + {\lambda}\sum_{\substack{k=1\\k \neq l}}^q \Big( (1-\xi)\frac{1}{2}{\alpha}_{ik}^2 + \xi | {\alpha}_{ik}|\Big) \\ & + {\lambda}\Big((1-\xi)\frac{1}{2}{\alpha}_{il}^2 + \xi | {\alpha}_{il}| \Big).
\end{aligned}$$ The gradient with respect to ${\alpha}_{il}$ is $$\begin{aligned}
\frac{\partial \hat Q_{{\lambda},i}({\alpha}_{il})}{\partial {\alpha}_{il}} & = \; -2 \sum_{\substack{j=1}}^p (X^*_{ij,\cdot l})^{{\mbox{\scriptsize \sf T}}}(r_{ij}- X^*_{ij,\cdot l} {\alpha}_{il}) + {\lambda}\Big((1-\xi){\alpha}_{il} + \xi \operatorname{sign}({\alpha}_{il}) \Big).
\end{aligned}$$ Following the main argument by [@friedman:hastie:tibshirani:10], the coordinate-wise update for ${\alpha}_{il}$ is of the form $$\begin{aligned}
\label{updatealpha}
{\alpha}_{il} \leftarrow
\frac{S \left (2 \sum_{\substack{j=1}}^p (X^*_{ij,\cdot l})^{{\mbox{\scriptsize \sf T}}}r_{ij},{\lambda}\xi \right)}{2 \sum_{\substack{j=1}}^p (X^*_{ij,\cdot l})^{{\mbox{\scriptsize \sf T}}}X^*_{ij,\cdot l} + {\lambda}(1-\xi)},
\end{aligned}$$ where $ S(A,{\lambda}\xi) =\text{sign}(A)(|A|-{\lambda}\xi)_+$.
Algorithm to update ${\boldsymbol{\alpha}}, {\boldsymbol{\sigma}}^2, {\boldsymbol{\phi}}, {\boldsymbol{\psi}}$ {#algorithm}
--------------------------------------------------------------------------------------------------------------
For a given $q$ and sequence of $\lambda$ values $0 \le {\lambda}_1,\ldots,{\lambda}_M$, the overall procedure to estimate the parameters: ${\boldsymbol{\alpha}}, {\boldsymbol{\sigma}}^2, {\boldsymbol{\phi}}, {\boldsymbol{\psi}}$ is described by Algorithm \[Algorithm:BDA\]. Note that estimates obtained with ${\lambda}_{s-1}$ are used as initial values for the estimation with ${\lambda}_{s}$, $s=2,\ldots,M$.
Set initial values $\hat {\boldsymbol{\alpha}}^{(0)}, \hat {\boldsymbol{\sigma}}^{2,(0)},\hat {\boldsymbol{\phi}}^{(0)}$ and $\hat {\boldsymbol{\psi}}^{(0)}$ ${\boldsymbol{\sigma}}^2 :=\hat {\boldsymbol{\sigma}}^{2,(s-1)}$ ${\boldsymbol{\alpha}}:=\hat {\boldsymbol{\alpha}}^{(s-1)}$ ${\boldsymbol{\phi}}:=\hat {\boldsymbol{\phi}}^{(s-1)}$ ${\boldsymbol{\psi}}:=\hat {\boldsymbol{\psi}}^{(s-1)}$
Update $\sigma_i^2$ using Update ${\boldsymbol{\alpha}}_{i\cdot}$ using cyclical descent over ${\alpha}_{il}$, $l=1,\ldots,q$ using Apply line search for ${\boldsymbol{\alpha}}_{i\cdot}$ update ${\boldsymbol{\phi}}$ using quasi-Newton update ${\boldsymbol{\psi}}$ using quasi-Newton $\hat {\boldsymbol{\sigma}}^{2,(s)}:={\boldsymbol{\sigma}}^2$ $\hat {\boldsymbol{\alpha}}^{(s)}:={\boldsymbol{\alpha}}$ $\hat {\boldsymbol{\phi}}^{(s)}:={\boldsymbol{\phi}}$ $\hat {\boldsymbol{\psi}}^{(s)}:={\boldsymbol{\psi}}^2$
Plots and detail information of BCI data used in the analysis {#app:bci}
=============================================================
Plots of 13 spatial covariates used for analysis are depicted in Figure \[fig:cov\]. Figure \[fig:spec9\] shows locations of the 9 selected tree species.
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|
---
author:
- 'P.O. Kazinski'
title: 'Comment on “Finite size corrections to the radiation reaction force in classical electrodynamics”'
---
1\. The authors of Letter [@FSCRRFCE] claim that they “prove that leading order effect due to the finite radius $R$ of a spherically symmetric charge is order $R^2$ rather than order $R$ in any physical model, as widely claimed in the literature” since “symmetries prohibit linear corrections”. I shall show that this is an incorrect statement.
Indeed, according to the effective field theory approach exploited in [@FSCRRFCE], we should augment the initial classical action of a point charge by every possible local combination of fields, which does not spoil any symmetry of the initial model. In the case considered in [@FSCRRFCE], these terms should have a mass dimension one or two. The authors assert that there is only one term Eq. (17) complying with these requirements. Its mass dimension is two, whence the main statement of the Letter follows. However, it is not difficult to find two more terms: (in the proper time parameterization) $$\label{rigid term}
A=\int d\tau\ddot{x}_\mu \ddot{x}^\mu,\qquad B=\int d\tau\partial_\mu F^{\mu\nu} \dot{x}_\nu$$ with dimensions one and two, respectively. The term $B$ is the first low energy correction to the form factor of a charged particle due to its finite size [@Foldy]. The term $A$ provides a counterexample to the main claim of Letter [@FSCRRFCE].
The “rigid” relativistic terms like $A$ are well-known [@Pisar]. They appear naturally in studying higher dimensional generalizations of the Lorentz-Dirac (LD) equation [@higher]. Also, if one smears the current of a point charge in a Lorentz-invariant manner [@CarBatUz] $$\label{regulariz}
j^\mu(x)\rightarrow j^\mu_{\varepsilon}(x)=\Box_x \int d^4yG_{\varepsilon}(x-y)j^\mu(y),$$ where $G_{\varepsilon}(x)=\theta(x^0)\delta(x^2-{\varepsilon}^2)/2\pi$, then the first correction to the LD equation is of order ${\varepsilon}$ and is obtained by a variation of the term $A$ (see Eq. (26), [@rrmm]). It is that term which is “prohibited” as the authors of the Letter claim.
2\. In order to get rid of divergences, the authors of [@FSCRRFCE] use an improper regularization: “one should regularize the divergences... by using... dimensional regularization which... sets all power-divergent integrals... to zero”. This assertion contradicts the standard renormalization procedure [@Collins]. We can take the power-like divergences to be equal to zero only if: i) we know that such terms can be canceled by appropriate counterterms added to the initial Lagrangian; ii) experiments or symmetries require that the coefficients at these terms vanish. Using their regularization scheme, the authors missed one possible divergent structure (the term $A$), which is not prohibited by symmetries and cannot be set to zero at will. In [@brane] it was proven that if one uses the regularization , which has a clear physical interpretation and does not spoil any symmetry, or some equivalent to it then all the arising divergences are Lagrangian and can be renormalized.
3\. Another flaw is concerned with an ignorance of the relation between the regularization parameter $\Lambda$ and the characteristic size $R$ of a charged body. As it is given in [@FSCRRFCE], the particle creates the electromagnetic field as a point object (the current is localized on a world-line). This field is substituted to the Lorentz-like force taken on the trajectory of the particle, i.e., taken on scales much lesser than $R$, where the particle can not be considered as a point. To make this procedure consistent, the regularization parameter characterizing the wave-length cutoff must be of the same order or even larger than $R$ so that the particle “looks” like a point object for the electromagnetic field. Then the expansion of the radiation reaction force in terms of $R$ rearranges (see Eq. (25) with $\Lambda\propto1/R$) and requires a more careful examination. The contributions of higher multipoles, which were discarded in [@FSCRRFCE], can be greater than the terms (25), (26).
[999]{}
Ch.R. Galley, A.K. Leibovich, and I.Z. Rothstein, Phys. Rev. Lett. **105**, 094802 (2010).
L.L. Foldy, Rev. Mod. Phys. **30**, 471 (1958).
R.D. Pisarski, Phys. Rev. D **34**, 670 (1986).
B.P. Kosyakov, Theor. Math. Phys. **119**, 493 (1999); P.O. Kazinski, S.L. Lyakhovich, and A.A. Sharapov, Phys. Rev. D **66**, 025017 (2002).
B. Carter, R.A. Battye, and J.-Ph. Uzan, Commun. Math. Phys. **235**, 289 (2003); K. Lechner, P.A. Marchetti, Ann. Phys. (NY) **322**, 1162 (2007).
P.O. Kazinski, J. Exp. Theor. Phys. **105**, 327 (2007).
J.C. Collins, Renormalization (CUP, Cambridge, 1984).
P.O. Kazinski, A.A. Sharapov, Theor. Math. Phys. **143**, 798 (2005).
|
---
abstract: '13MW of electron cyclotron current drive (ECCD) power deposited inside the $q=1$ surface is likely to reduce the sawtooth period in ITER baseline scenario below the level empirically predicted to trigger neo-classical tearing modes (NTMs). However, since the ECCD control scheme is solely predicated upon changing the local magnetic shear, it is prudent to plan to use a complementary scheme which directly decreases the potential energy of the kink mode in order to reduce the sawtooth period. In the event that the natural sawtooth period is longer than expected, due to enhanced $\alpha$ particle stabilisation for instance, this ancillary sawtooth control can be provided from $>10$MW of ion cyclotron resonance heating (ICRH) power with a resonance just inside the $q=1$ surface. Both ECCD and ICRH control schemes would benefit greatly from active feedback of the deposition with respect to the rational surface. If the $q=1$ surface can be maintained closer to the magnetic axis, the efficacy of ECCD and ICRH schemes significantly increases, the negative effect on the fusion gain is reduced, and off-axis negative-ion neutral beam injection (NNBI) can also be considered for sawtooth control. Consequently, schemes to reduce the $q=1$ radius are highly desirable, such as early heating to delay the current penetration and, of course, active sawtooth destabilisation to mediate small frequent sawteeth and retain a small $q=1$ radius. Finally, there remains a residual risk that the ECCD+ICRH control actuators cannot keep the sawtooth period below the threshold for triggering NTMs (since this is derived only from empirical scaling and the control modelling has numerous caveats). If this is the case, a secondary control scheme of sawtooth stabilisation via ECCD+ICRH+NNBI, interspersed with deliberate triggering of a crash through auxilliary power reduction and simultaneous pre-emptive NTM control by off-axis ECCD has been considered, permitting long transient periods with high fusion gain. The power requirements for the necessary degree of sawtooth control using either destabilisation or stabilisation schemes are expected to be within the specification of anticipated ICRH and ECRH heating in ITER, provided the requisite power can be dedicated to sawtooth control.'
address: |
$^{1}$EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK\
$^{2}$CRPP, Association EURATOM/Confédération Suisse, EPFL, 1015 Lausanne, Switzerland\
$^{3}$Association EURATOM-Tekes, Aalto University, Department of Applied Physics, P.O.Box 14100 FI-00076 AALTO, Finland\
$^{4}$General Atomics, PO Box 85608, San Diego, CA 92186, USA\
$^{5}$MPI fur Plasmaphysik, EURATOM-Ass D-85748 Garching, Germany\
$^{6}$EURATOM-VR Association, EES, KTH, Stockholm, Sweden\
$^{7}$JET-EFDA, Culham Science Centre, Abingdon, UK OX14 3DB\
$^{*}$See the Appendix of F. Romanelli *et al.*, Fusion Energy 2012 (Proc. 24th Int. Conf. San Diego, 2012) IAEA, (2012)
author:
- 'IT Chapman$^{1}$, JP Graves$^{2}$, O Sauter$^{2}$, C Zucca$^{2}$, O Asunta$^{3}$, RJ Buttery$^{4}$, S Coda$^{2}$, T Goodman$^{2}$, V Igochine$^{5}$, T Johnson$^{6}$, M Jucker$^{2}$ RJ La Haye$^{4}$, M Lennholm$^{7}$ and JET-EFDA Contributors$^*$'
title: Power requirements for electron cyclotron current drive and ion cyclotron resonance heating for sawtooth control in ITER
---
Introduction and Background
===========================
Sawtooth control remains an important unresolved issue for baseline scenario [@Casper] operation of ITER. Since the monotonic $q$-profile of such baseline ELMy H-mode plasmas have a large $q=1$ radius, $r_{1}$, with low magnetic shear at the $q=1$ surface, $s_{1}=r_{1}\textrm{d}q/\textrm{d}r$, these plasmas are expected to be unstable to the internal kink mode. Furthermore, the energetic trapped fusion-born $\alpha$-particles are predicted to lead to significant stabilisation of the internal kink mode [@Porcelli1996; @ChapmanEPS], resulting in very long sawtooth periods. However, such long sawtooth periods have been observed to result in triggering NTMs at lower plasma $\beta$ [@Sauter; @Buttery; @Chapman2010] (where $\beta$ is the pressure normalised to the magnetic pressure) which in turn can significantly degrade plasma confinement. Consequently, there is an urgent need to assess whether sawtooth control will be achievable in ITER and how much power is required from the actuators at our disposal to attain an acceptable sawtooth period. Our understanding of internal kink mode stability that underlies sawteeth has improved significantly recently through a combination of analytic understanding, experimental verification and detailed modelling, as reviewed in [@ChapmanRev]. This enhanced understanding now provides a platform from which to make an improved assessment of sawtooth control requirements in ITER.
The two approaches to sawtooth control are to (i) either eliminate or delay the sawtooth crash for as long as possible (stabilisation) or (ii) decrease the sawtooth period to reduce the likelihood of triggering other MHD instabilities (destabilisation). In ITER, it is foreseen that destabilisation will be employed to keep the sawteeth small and frequent to help flush He ash from the plasma core and to avoid triggering NTMs. Sawtooth control can be achieved by tailoring the distribution of energetic ions; by changing the radial profiles of the plasma current density and pressure, notably their local gradients near the $q=1$ surface; by rotating the plasma, or changing the rotation shear local to the $q=1$ surface; by shaping the plasma; or by heating the electrons inside the $q=1$ surface. The primary actuators to achieve these perturbations are electron cyclotron current drive (ECCD), ion cyclotron resonance heating (ICRH) and neutral beam injection (NBI). The highly localised perturbations to the current density profile achievable with ECCD have been employed to significantly alter sawtooth behaviour on a number of devices. By driving current just inside the $q=1$ surface, the magnetic shear at $q=1$ can be increased, and thus result in more frequent sawtooth crashes. ECCD is foreseen as the primary sawtooth control actuator in the ITER design [@IPB2] due to both the highly localised current density that can be achieved when compared to ion cyclotron current drive for instance, and because of the ability to provide real time control of the current drive location by changing the launcher angle of the injected EC beam by using steerable mirrors. However, complementary control schemes which work via kinetic effects, such as ICRH or NBI, are also useful for sawtooth control in the presence of a population of core energetic particles.
An open question that predicates the assessment of required actuator power level is what an acceptable sawtooth period will be in ITER. In order to provide some empirical basis for an acceptable sawtooth period in ITER, a multi-machine database has been established and an empirical scaling law derived [@Chapman2010], as described in section \[sec:empirical\].
For many years it has been known that trapped energetic particles result in strong stabilisation of sawteeth. However, passing fast ions can also significantly influence sawtooth behaviour. For highly energetic ions, the radial drift motion becomes comparable to the radial extent of the kink mode. In this regime, the kinetic contribution to the mode’s potential energy (together with a non-convective contribution to the fluid part of $\delta W$) becomes increasingly important. When the passing fast ion population is asymmetric in velocity space, there is an important finite orbit contribution to the mode stability. The effect of passing ions is enhanced for large effective orbit widths, which is to say, for highly energetic ions (like ICRH or negative ion NBI (N-NBI) in ITER) or for a population with a large fraction of barely passing ions (like ICRH). Passing fast ions can destabilise the internal kink mode when they are co-passing and the fast ion distribution has a positive gradient from inside to outside $q=1$, or when they are counter-passing, but the deposition is peaked outside the $q=1$ surface. This mechanism is described in detail in references [@Graves2004] and [@Graves2009]. The effect of passing fast ions has been confirmed in NBI experiments in JET [@Chapman2007; @Chapman2008] and ASDEX Upgrade [@Chapman2009] and using He$^{3}$ minority ICRH in JET [@GravesAPS]. By employing He$^{3}$ minority heating schemes (which are envisaged for ITER ICRF heating), the resultant current drive is negligible [@Graves2009; @Laxaback]. Nonetheless, the ICRH can still strongly influence the sawtooth stability, demonstrating that sawtooth control via ICRH can be achieved via a kinetic destabilisation mechanism rather than through local modification of the magnetic shear at $q=1$ [@Graves2012; @Graves2010]. Experimental evidence that both ECCD and ICRH control is effective in plasmas with a significant fraction of core energetic particles is given in section \[sec:exp\].
Sawtooth stability is strongly influenced by the energetic particles arising from neutral beam injection, ion cyclotron resonance heating and fusion alpha particles. Previous assessments [@ChapmanEPS; @Chapman2008] have shown that the N-NBI ions, like the fusion-born alphas, will be strongly stabilising if the resultant distribution function is peaked inside the $q=1$ surface, as is envisioned for baseline scenario operation. The tools used to model the fast ion distribution functions and their effect on stability are outlined in section \[sec:tools\], and the energetic particle distributions are detailed in section \[sec:fastions\].
The effect on the sawtooth period from electron cyclotron current drive (the main actuator planned for ITER) is described in section \[sec:ECCD\] and the effect of ion cyclotron resonance heating is outlined in section \[sec:ICRH\]. Finally, an alternative approach to sawtooth control through deliberate stabilisation is discussed in section \[sec:stabilisation\]. The conclusions of the study in terms of required power levels are summarised in section \[sec:conclusions\].
An Acceptable Sawtooth Period in ITER {#sec:empirical}
=====================================
The neoclassical tearing mode (NTM) is one of the most critical performance-limiting instabilities for baseline scenarios in ITER. The NTM is a metastable mode which requires a ‘seed’ perturbation in order to be driven unstable and grow [@Carrera], except at very high plasma pressure [@Brennan]. Various effects have been proposed to prevent NTM growth for small island widths, namely (i) incomplete pressure flattening which occurs when the connection length is long compared to the island width [@Fitzpatrick], (ii) ion polarisation currents arising due to finite orbit width $E \times B$ drifts occurring for ions and electrons across the island region [@Wilson; @Smolyakov], which act to replace the missing bootstrap current, and (iii) curvature effects [@Kotschenreuther; @Lutjens]. Consequently, NTM growth is generally prohibited in the absence of a sufficiently large seed island in the plasma [@LaHaye2000]. Whilst this seed may be caused by edge localised modes (ELMs) [@Buttery2008; @Gerhardt] or fast particle-driven fishbones [@Gude], the trigger of most concern is the sawtooth oscillation which typically triggers the NTMs at lower plasma pressures [@Gude]. Many theories have been proposed to explain how the sawtooth crash triggers the NTM, including magnetic coupling [@Hegna], nonlinear ‘three-wave’ coupling [@Nave], changes in the classical tearing stability due to current redistribution inside $q=1$ [@Reimerdes; @Maget; @Koslowski] or changes in the rotation profile resulting in a reversal of the ion polarisation current [@Buttery2003] in the modified Rutherford equation governing NTM stability [@Sauter1997]. These models predict that the salient features of the sawtooth crash that should determine the onset of the NTM are the amplitude of the magnetic perturbation, the coupling to the NTM rational surface and any shielding effects such as rotational screening or diamagnetic effects. However, empirical observation and neural network analysis have determined that the sawtooth period shows far stronger correlation to the triggering of the NTM than the sawtooth amplitude [@Buttery2003; @Buttery2004; @Sauter; @Belo; @Coda].
Although the seeding of the NTM by the sawtooth crash remains poorly understood, the empirical observation that deliberately increasing the sawtooth frequency helps to avoid triggering NTMs is now universally accepted and routinely used as a method for NTM mitigation. The issue of whether a sawtooth period in ITER in the range of 20-50s, as predicted by transport simulations [@Jardin; @Bateman1998; @Waltz; @Onjun; @Budny2008; @Bateman] will avoid triggering NTMs is currently poorly understood, and so a multi-machine empirical scaling is presented here in order to provide some basis for extrapolation and specification of sawtooth control actuators in ITER. A database of plasma parameters has been established for discharges which exhibit sawteeth, including both crashes which trigger NTMs and those which do not. This dataset contains details for over 200 shots from nine tokamaks; namely ASDEX Upgrade, DIII-D, HL-2A, JET, JT-60U, MAST, NSTX, TCV and Tore Supra [@Chapman2010]. Naturally, comparing discharges between a large range of tokamaks means that the database contains a wide range of plasma shapes, $q$-profiles, fast ion pressures and fast ion distribution functions, all of which will influence the sawtooth behaviour. Similarly, the different $q$-profiles, and thus different magnetic shear between rational surfaces, as well as the different rotation profiles will undoubtedly influence the coupling between the sawtooth oscillations at $q=1$ and NTMs at higher rational surfaces. The database also incorporates triggered NTMs at three different rational surfaces, $q=4/3, 3/2, 2/1$. However, retaining such a wide range of plasma parameters means that a “safe” operating space, where sawteeth are less likely to trigger NTMs, can be inferred.
The dynamics which determine when the sawtooth crash will occur (in the absence of any deliberate sawtooth control actuators) are predominantly determined by the evolution of the $q$-profile, particularly of the radial position of the $q=1$ surface and the local magnetic shear at $q=1$ [@Porcelli1996]. Since these quantities evolve on the timescale of the resistive diffusion in the plasma core, the sawtooth period has been normalised accordingly.
Notwithstanding the individual constraints of each tokamak, there is a significant scatter in the database meaning that it is difficult to draw any conclusion about the permissible sawtooth period in ITER that avoids triggering NTMs. Consequently, in order to make a more reliable extrapolation to ITER, a subset of the data has been considered which retains only discharges with ITER like shape ($\delta \in [0.3,0.4]$ and $\kappa \in [1.65,1.85]$), a broad flat $q$-profile with a wide $q=1$ surface ($r_{1}/a \in [0.33,0.45]$) and with auxilliary heating power only slightly above the L-H threshold given by in reference [@Martin] ($P_{aux}/P_{LH} \in [1.3,1.7]$) as expected in the ELMy H-mode baseline scenario in ITER [@Doyle]. This reduced database of “ITER-like” sawtoothing discharges is illustrated in figure \[fig:trimmed\]. It is clear that this subset exhibits a general trend that NTMs are triggered at lower $\beta_{N}$ for longer sawtooth periods with respect to the resistive diffusion time.
Also shown in figure \[fig:trimmed\] is the range of sawtooth periods that could be expected in ITER. A period of 20-50s predicted by transport modelling [@Onjun; @Budny2008] would lie in the range $\tau_{st}/\tau_{r} \in [0.0178,0.0446]$ which approaches the period at which this empirical extrapolation suggests NTMs would be triggered by the sawtooth crashes at the target plasma pressure of $\beta_{N}=1.8$ in ITER baseline scenario. However, if the natural sawtooth period is approximately the same as the critical period for triggering NTMs, there is the opportunity to apply control actuators to sufficiently reduce $\tau_{st}$ and avoid NTMs, which would not be the case if the natural period was significantly longer than the critical period.
An empirical scaling law developed from the entire database suggests that the critical $\beta_{N}$ for triggering an NTM by a sawtooth crash in ITER is 2.09 for a sawtooth period of 50s. Figure \[fig:trimmed\_fit\] shows the critical $\beta_{N}$ for triggering NTMs for the ITER-like subset, compared to the predictions of the derived scaling law, showing good agreement between the two. Also overlaid is the critical achievable pressure predicted for a range of sawtooth periods in ITER. At the target operating pressure for ITER ELMy H-mode scenario – $\beta_{N}=1.8$ – this scaling law suggests that a sawtooth period of around 70s will be permissible. It is evident that a sawtooth period in the range of 20-50s predicted by transport simulations is predicted to avoid triggering NTMs at the scenario target operating pressure. It is also clear that the critical $\beta_{N}$ for NTM onset increases as the sawtooth period is reduced, highlighting the need for provision of sawtooth control actuators. This scaling law is, of course, only an empirical fitting and not based on any physics model, so its application to future devices should only be for guidance, and certainly not quantitative. It should be noted that the empirical scaling law derived from the experimental database is primarily for unidirectional NBI-heated plasmas. Supplementing this database with extra plasmas run at more ITER-relevant low torques would help to clarify whether the rotation plays an important role in mediating the coupling between the sawtooth crash and the NTM onset and is likely to lead to an additional parameter in the scaling law.
Experimental evidence of sawtooth control in the presence of core energetic particles {#sec:exp}
=====================================================================================
When a sawtooth crash occurs in the presence of stabilising fast ions it is often more violent and more likely to trigger NTMs, leading to a degradation in pressure and thus in fusion performance. Therefore, it is important to demonstrate that both ECCD and ICRH can be used to control sawteeth in the presence of a population of core fast ions.
Whilst ECCD has been shown to control sawteeth effectively for decades, only recently have such demonstrations been replicated in the presence of core energetic particles. Sawtooth destabilisation of long period sawteeth induced by ICRH generated core fast ions with energies $\geq 0.5$MeV was achieved in Tore Supra, even with modest levels of ECCD power [@Lennholm; @Lennholm2009]. Similarly, ECCD destabilisation has also been achieved in the presence of ICRH accelerated neutral beam injection (NBI) ions in ASDEX Upgrade [@Igochine] as well as with normal NBI fast ions in ASDEX Upgrade [@Muck] and JT-60U [@Isayama]. More recently sawtooth control using ECCD has even been demonstrated in ITER-like plasmas with a large fast ion fraction, wide $q=1$ radius and long uncontrolled sawtooth periods in DIII-D [@ChapmanDIIID]. As expected from simulation, the sawtooth period is minimised when the ECCD resonance is just inside the $q=1$ surface. Active sawtooth control using driven current inside $q=1$ allows the avoidance of sawtooth-triggered NTMs, even at much higher pressure than required in the ITER baseline scenario. Operation at $\beta_{N}=3$ without 3/2 or 2/1 neoclassical tearing modes has been achieved in ITER demonstration plasmas when sawtooth control is applied using only modest ECCD power [@ChapmanDIIID]. Such avoidance of NTMs permitting operation at higher pressure than otherwise achievable by application of core ECCD sawtooth control has also recently been demonstrated in ASDEX Upgrade [@ChapmanAUG].
A major advantage of current drive schemes is that ECCD provides a simple external actuator in a feedback-control loop through the angle of inclination of the launcher mirrors. Consequently, there has been considerable effort to develop real-time control of the deposition location in order to obtain requested sawtooth periods. TCV has demonstrated feedback control of the sawtooth period by actuating on the EC launcher injection angle in order to obtain the sawtooth period at a pre-determined value [@Paley]. Recently, fine control over the sawtooth period has been demonstrated on TCV using either ‘sawtooth pacing’ via modulated ECCD with real-time crash detection [@Goodman], or ‘sawtooth locking’, where the sawtooth period is controlled even in the absence of crash detection in a reduced region of duty-cycle versus pulse-period parameter space [@Lauret; @Witvoet]. Meanwhile, Tore Supra have implemented a ‘search and maintain’ control algorithm to vary the ECCD absorption location in search of a location at which the sawteeth are minimised; having achieved this, the controller maintains the distance between the ECCD deposition location and the measured inversion radius despite perturbations to the plasma [@Lennholm2009].
Control of sawteeth by ICRH in the presence of core energetic particles has been widely exploited on JET [@Sauter; @Eriksson2006; @Eriksson2004; @Westerhof2002; @Mayoral]. Furthermore, ICRH control has also been demonstrated in plasmas with significant heating power on-axis from neutral beam injection and high $\beta_{p}$, well above the critical threshold for triggering of 3/2 NTMs in the absence of sawtooth control [@Coda]. Subsequently it was noted that the sensitivity of sawtooth destabilisation required accuracy of the resonance position with respect to the $q=1$ surface of less than 0.5% (ie within 1cm of the $q=1$ surface in JET) [@Coda], far more sensitive than expected from a control mechanism involving a modification of the magnetic shear. Graves *et al* showed that the sawtooth control mechanism from localised off-axis toroidally propagating waves is due to the radial drift excursion of the energetic ions distributed asymmetrically in the velocity parallel to the magnetic field [@GravesAPS]. This kinetic mechanism results in a deep and narrow minimum in the change of the potential energy when the peak of the passing fast ion distribution is just inside the $q=1$ surface, helping to explain the extreme sensitivity of the sawtooth behaviour to the deposition location of the ICRH waves. Recent JET experiments using $^{3}$He minority heating (so that the driven current was negligible) on the high-field side just outside the $q=1$ surface lead to a strong destabilisation for counter-propagating waves (-90$^{\circ}$) and a strong stabilisation for co-propagating waves (+90$^{\circ}$) [@Graves2010]. This sawtooth control scheme via affecting the kink mode potential energy has subsequently been demonstrated in H-mode plasmas with significant core heating too [@Graves2012], adding credence to its applicability in ITER. Finally, real-time control through variation of the ICRH frequency has been attempted on JET [@Lennholm2011], though the frequency variation is much slower than anticipated in ITER [@Lamalle].
Energetic Particle Modelling Tools Used {#sec:tools}
=======================================
Modelling the energetic particle distributions
----------------------------------------------
In order to model the neutral beam fast ion distribution, the <span style="font-variant:small-caps;">Transp</span> [@Budny1992] and <span style="font-variant:small-caps;">Ascot</span> [@Heikennen; @Kurki] codes have been used. <span style="font-variant:small-caps;">Ascot</span> has been used to model the alpha particle population whilst <span style="font-variant:small-caps;">Selfo</span> [@Hedin] and <span style="font-variant:small-caps;">Scenic</span> [@Jucker; @Jucker2] codes have been used to simulate the ICRH distribution. Finally, the <span style="font-variant:small-caps;">Hagis</span> drift kinetic code [@Pinches] has been employed to study the effect of the various fast ion populations on internal kink stability. The plasma equilibrium for the ITER baseline scenario is taken from integrated transport modelling using the <span style="font-variant:small-caps;">Corsica</span> code as reported in [@Casper].
The <span style="font-variant:small-caps;">Transp</span> code was used to simulate the NBI fast ion population since it enables the use of the beam module <span style="font-variant:small-caps;">Nubeam</span> in a convenient, integrated plasma simulation environment. The <span style="font-variant:small-caps;">Nubeam</span> module is a Monte Carlo package for time dependent modelling of fast ion species using classical physics. Multiple fast ion species can be present, due to either beam injection of energetic neutral particles or as a product of nuclear fusion reactions. The model self consistently handles guiding center drift orbiting, collisional and atomic physics effects during the slowing down of the fast ions. In order to reduce the risk of a result dependent entirely upon the prediction of one code, the <span style="font-variant:small-caps;">Ascot</span> code has also been used to simulate the NNBI distribution. <span style="font-variant:small-caps;">Ascot</span> [@Kurki] is a guiding-centre orbit following Monte Carlo code which integrates the particles’ equation of motion in time over a five-dimensional space. Collisions with the background plasma are modelled using Monte Carlo operators allowing an acceleration of collisional time scales and reduced computational time. The alpha particle markers are initialised by the local $\langle \sigma v \rangle_{DT}$ whereas the beam ions are followed starting from the injector taking into account the beamlet position, direction, beam species, energy, total power, and its bi-Gaussian dispersion. The ionization cross-section is calculated at each step using the local temperature and density, and analytic fits from [@Suzuki]. In addition to thermal fusion reactions, also fusion reactions between the fast NBI particles and thermal plasma particles are included in the <span style="font-variant:small-caps;">Ascot</span> code. The ICRH fast ion populations are simulated using <span style="font-variant:small-caps;">Selfo</span> and <span style="font-variant:small-caps;">Scenic</span>. The <span style="font-variant:small-caps;">Selfo</span> code [@Hedin] determines self-consistently the power absorption and the fast ion acceleration by coupling the global wave solver <span style="font-variant:small-caps;">Lion</span> [@Villard] and the Monte-Carlo code <span style="font-variant:small-caps;">Fido</span> [@Carlsson]. <span style="font-variant:small-caps;">Fido</span> solves the 3D orbit averaged kinetic equations, including quasilinear ICRF acceleration from the <span style="font-variant:small-caps;">Lion</span> wave field. <span style="font-variant:small-caps;">Fido</span> accounts for guiding centre orbits, including all possible shapes of banana and potato orbits. It should be noted that <span style="font-variant:small-caps;">Lion</span> does not include the upshift of the parallel wave number, and therefore <span style="font-variant:small-caps;">Selfo</span> can be used to treat harmonic heating schemes, but not mode conversion. A limitation in the <span style="font-variant:small-caps;">Fido</span> code is the assumption of circular flux surfaces. To minimise the error caused by this assumption, the ITER equilibrium has been mapped so that the poloidal flux function in the outboard midplane $\{\psi(R,Z =
Z_{axis}) | R > R_{axis} \}$ is the same in <span style="font-variant:small-caps;">Selfo</span> as in the non-circular ITER equilibrium. Furthermore, the ICRH power is normalised so that the power absorbed per resonant ion is the same.
To reduce the uncertainty in simulating the ICRH distribution with just one code, the <span style="font-variant:small-caps;">Scenic</span> code has also been used. The <span style="font-variant:small-caps;">Scenic</span> integrated code package [@Jucker; @Jucker2] takes an equilibrium from <span style="font-variant:small-caps;">Animec</span> [@Cooper], the wave fields and wave numbers from LEMan [@Popovich] and iterates with the distribution function evolved by <span style="font-variant:small-caps;">Venus</span> [@Fischer; @Cooper2007]. These codes are iterated to form a self-consistent solution which can incorporate anisotropic equilibria in full 3D geometry. For the equilibrium and wave field computations, a bi-Maxwellian distribution is used for the hot minority, allowing for pressure anisotropy and stronger poloidal dependence of the pressure and dielectric tensor. Whereas LEMan is limited to leading order FLR effects, and thus to fundamental harmonic without mode conversion, it computes the wave vectors with the help of an iterative scheme, and can therefore treat correctly upshifted wave numbers without the use of a local dispersion relation.
Internal kink mode linear stability criteria
--------------------------------------------
The fundamental trigger of the sawtooth crash is thought to be the onset of an $m=n=1$ internal kink mode. However, the dynamics of this mode are constrained by many factors, including not only the macroscopic drive from ideal MHD, but collisionless kinetic effects related to high energy particles [@Porcelli1991] and thermal particles [@Kruskal] as well as non-ideal effects localised in the narrow layer around $q=1$. A heuristic model predicts that a sawtooth crash will occur when one of three criteria is met [@Porcelli1996; @SauterVarenna]. In the presence of fast ions, two conditions are unlikely to be satisfied since the magnetic drift frequency of the hot ions, $\omega_{dh}$, will be large and $\hat{\delta W}$ may have a large positive contribution from $\hat{\delta W}_{h}$. The change in the kink mode potential energy is defined such that $\delta \hat{W} = \delta \hat{W}_{core} + \delta \hat{W}_{h}$ where $\delta \hat{W}_{core}=\delta \hat{W}_{MHD} + \delta \hat{W}_{KO}$ and $\delta \hat{W}_{KO}$ is the change in the mode energy due to the collisionless thermal ions [@Kruskal], $\delta \hat{W}_{h}$ is the change in energy due to the fast ions and $\delta \hat{W}_{MHD}$ is the ideal fluid mode drive [@Bussac]. The potential energy is normalised such that $\delta \hat{W} \equiv 4 \delta W/(s_{1} \xi_{0}^{2} \epsilon_{1}^{2} R B^{2})$ and $\xi_{0}$ is the plasma displacement at the axis, $\epsilon_{1}=r_{1}/R$, $R$ is the major radius and $B$ is the magnetic field. When $\hat{\delta W}$ is large and positive, the mode takes the structure of a tearing mode, which is resistive and can be weakly unstable. It is assumed that these drift-tearing modes are stabilised by diamagnetic effects, so do not drive sawtooth crashes. When the potential energy is sufficient to weakly drive a resistive kink mode, the sawtooth crash is determined by the domain in which the resistive mode can be destabilised, that is to say when $$-c_{\rho} \hat{\rho} < -\hat{\delta W} < \frac{1}{2} \omega_{*i}\tau_{A}
\label{eq:crashresistive}$$ where $\tau_{A}= \sqrt{3}R/v_{A}$ is the Alfvén time, $c_{\rho}$ is a normalisation coefficient of the order of unity that determines the threshold at which the mode is considered to result in a sawtooth crash, $\omega_{*i}$ is the ion diamagnetic frequency, $\hat{\rho}=\rho_{\theta i}/r_{1}$ and the poloidal ion Larmor radius is $\rho_{\theta i}=v_{thi}m_{i}/eB_{\theta}$ where $B_{\theta}=\mu_{0}I_{p}/2\pi a$ and $v_{thi}=(kT_{i}/m_{i})^{1/2}$. In the presence of fast ions, the sawtooth crash is typically triggered by a resistive kink mode when inequality \[eq:crashresistive\] is satisfied. However, it should be noted that the crash can still be triggered by an ideal internal kink if the magnetic shear is sufficiently large that the normalisation of $\delta W$ results in a crash. In ITER, $\rho_{\theta i}$ will be small and $r_{1}$ is expected to be large, meaning that satisfying \[eq:crashresistive\] by increasing $s_{1}$ alone may not be possible if $\delta W$ is large and positive, so it is prudent to find ways to directly reduce the fast ion stabilisation arising from core energetic particles.
Stability modelling {#sec:hagis}
-------------------
The effect of the fast ions on the kink mode stability is tested using the Monte-Carlo guiding centre drift kinetic code <span style="font-variant:small-caps;">Hagis</span> [@Pinches]. The equilibrium is calculated with the static fixed-boundary 2D Grad-Shafranov solver <span style="font-variant:small-caps;">Helena</span> [@Huysmans]. The stability of this equilibrium is then tested using the linear MHD code <span style="font-variant:small-caps;">Mishka</span> [@ChapmanMishka]. The perturbation and equilibrium are then fed into <span style="font-variant:small-caps;">Hagis</span> together with the distribution functions of fast ions from the modelling described above. This coupling of numerical codes is illustrated schematically in figure \[fig:code\_overview\].
<span style="font-variant:small-caps;">Hagis</span> solves the non-linear drift guiding centre equations of motion. It allows the evolution of a fast ion population to be studied in the presence of electromagnetic perturbations in a toroidal plasma. The <span style="font-variant:small-caps;">Hagis</span> code has been used extensively for studying the stability of the internal kink mode, successfully replicating experimental signatures of sawtooth behaviour on JET [@Chapman2007; @ChapmanEPS; @Graves2010], TEXTOR [@Chapman2008] and ASDEX Upgrade [@Chapman2009; @ChapmanPoP].
Energetic Particle Distributions in ITER {#sec:fastions}
========================================
Core energetic particles
------------------------
The distribution of alpha particles has been tested with <span style="font-variant:small-caps;">Ascot</span> in the case when there is a 3D equilibrium field due to the presence of ferritic inserts. The alphas are well confined within $\rho \sim 0.6$ and are approximately isotropic, as seen in figure \[fig:alpha\_ascot\], where the velocity distribution is averaged across the minor radius to q=1 in order to represent all fast ions which affect the behaviour of the internal kink mode.
In order to penetrate the hot, dense plasmas in ITER, neutral deuterium beam energies of the order of 0.5-1.0MeV are necessary. In this study, the N-NBI is assumed to consist of 1MeV (D) neutrals from a negative ion-beam system injected in the co-current direction, at a tangency radius of 6m. This generates a broad beam-driven current profile with a total driven current of 1.2 MA [@Budny2002]. The beam can be aimed at two extreme (on-axis and off-axis) positions by tilting the beam source around a horizontal axis on its support flange, resulting in N-NBI injection in the range of $Z$ = -0.25 to -0.95 m [@ITER].
<span style="font-variant:small-caps;">Transp</span> and <span style="font-variant:small-caps;">Ascot</span> simulations have been carried out to predict the fast ion distribution function due to the N-NBI when it is aimed either on- or off-axis [@Budny2002]. The off-axis fast ion population is peaked at approximately $r/a=0.22$, as seen in figure \[fig:nnbi\_radial\]. This fast ion population is strongly passing. The current driven by the neutral beams results in the $q=1$ surface being slightly closer to the magnetic axis than when on-axis NBI is applied.
Ion cyclotron resonance heating
-------------------------------
The application of $^{3}$He minority heating in baseline scenario with the resonance on-axis, slightly off-axis and near mid-radius have been simulated for a range of different minority concentrations. The phasing of the antenna has also been investigated, and it is found that the inward pinch with +90$^{\circ}$ phasing enhances the on axis fast ion pressure. For the case with 20MW injected on-axis and minority concentration of $n_{^{3}He}/n=0.01$ simulated with <span style="font-variant:small-caps;">Selfo</span>, around 70% of the power absorbed goes into heating the $^{3}$He ions (around 7MW). The off-axis resonance has also been simulated in order to generate a strong radial gradient in the asymmetry of the passing fast ion distribution near the $q=1$ surface necessary for sawtooth destabilisation. Whilst the far-off-axis heating gives rise to a low power per particle and no highly energetic tails in the distribution, it does nonetheless incur fast ion distributions capable of affecting internal kink stability.
The orbit width effects upon which the internal kink destabilisation mechanism are predicated [@Graves2009] are much smaller in ITER than in JET. In ITER, with $^{3}$He minority heating at 52MHz (ie resonance 0.16m from the magnetic axis) and toroidal field $B_{T}=5.3$T, 1MeV ions have an orbit width $\Delta_{r}/a = 0.06$, whereas for comparison, 1MeV ions in JET with $^{3}$He minority at $B_{T}=2.75$T (as per experiments in reference [@Graves2010]) have an orbit width of $\Delta_{r}/a = 0.25$. The fast ion effects are the only way in which the ICRH can contribute to internal kink stability since the strong electron drag means that the change in the magnnetic shear due to ICCD will be negligible [@Laxaback].
![The fast ion currents arising from ion cyclotron resonance heating of $^{3}$He minority near the $q=1$ surface on the low-field side for different minority concentrations predicted by <span style="font-variant:small-caps;">Selfo</span>. The position of the $q=1$ surface is shown by a vertical line.[]{data-label="fig:fastioncurrents"}](figure6a.eps){width="\textwidth"}
![The fast ion currents arising from ion cyclotron resonance heating of $^{3}$He minority near the $q=1$ surface on the low-field side for different minority concentrations predicted by <span style="font-variant:small-caps;">Selfo</span>. The position of the $q=1$ surface is shown by a vertical line.[]{data-label="fig:fastioncurrents"}](figure6b.eps){width="\textwidth"}
![The fast ion currents arising from ion cyclotron resonance heating of $^{3}$He minority near the $q=1$ surface on the low-field side for different minority concentrations predicted by <span style="font-variant:small-caps;">Selfo</span>. The position of the $q=1$ surface is shown by a vertical line.[]{data-label="fig:fastioncurrents"}](figure6c.eps){width="\textwidth"}
Figure \[fig:fastioncurrents\] shows the passing and trapped contributions of the deduced flux-averaged fast ion current density predicted by <span style="font-variant:small-caps;">Selfo</span> as a function of minor radius for ITER at full magnetic field when the ICRH frequency is chosen so that the resonance layer is on the low-field side near the $q=1$ surface. The current drive predicted by <span style="font-variant:small-caps;">Selfo</span> is smaller than in <span style="font-variant:small-caps;">Scenic</span> and the passing current in <span style="font-variant:small-caps;">Selfo</span> and <span style="font-variant:small-caps;">Scenic</span> is different. This may be due to differences in the trapped-passing boundary (the boundary between deeply trapped orbits and passing orbits near low-field side stagnation) in <span style="font-variant:small-caps;">Selfo</span> and <span style="font-variant:small-caps;">Scenic</span>, such that some passing orbits in <span style="font-variant:small-caps;">Scenic</span> are counted as trapped in <span style="font-variant:small-caps;">Selfo</span>. Also, the precession of deeply trapped ions and stagnation passing ions is different. A similar benchmark between <span style="font-variant:small-caps;">Selfo</span> and <span style="font-variant:small-caps;">scenic</span> and the iterative coupling between the <span style="font-variant:small-caps;">Aorsa</span> wave field code [@Jaeger] and the Fokker-Planck CQL3D code [@Harvey] (which neglects orbit widths) was reported in reference [@Chapman2011].
Sawtooth control with ECCD in ITER {#sec:ECCD}
==================================
The ITER electron cyclotron heating and current drive system consists of up to 26 gyrotrons operating at 170GHz and delivering 1-2 MW each, for a nominal injected power into the plasma of up to 24 MW [@Omori]. The system has two types of antennas to inject the power into the plasma: the equatorial launcher (EL), which occupies one port in the equatorial plane, and the upper launcher (UL), occupying four ports in the upper plane. The EL is designed to access the inner half of the plasma, encompassing all physics applications other than NTM stabilization, including current profile tailoring for steady-state operation, central heating to assist the transition from L- to H-mode and control of the sawtooth instability. The UL provides a more focussed, peaked driven current density profile, ideal for control of instabilities, including sawteeth after a design modification allowed the UL to access towards the plasma core [@Ramponi]. In order to access the region from $0.4 < \rho < 0.5$ with the EC power, the access range of the Upper Steering Mirror (USM) and Lower Steering Mirror (LSM) is spread out. This forms essentially three access zones from the UL: an inner zone accessible with the USM (13MW, 16 beams), an overlap zone accessible with both the USM and the LSM (therefore up to 20MW, 24 beams) and an outer zone accessible with the LSM (13MW, 16 beams). Using this method, the overall access region from the UL is increased from about $0.51 < \rho < 0.87$ to about $0.3 < \rho < 0.86$ [@Omori; @Ramponi].
Modelling the change in magnetic shear
--------------------------------------
The effect of local EC heating on the $q$-profile has been modelled with the <span style="font-variant:small-caps;">Astra</span> transport code [@Pereverzev], which solves a reduced set of 1-D equations for the evolution of the electron and ion temperatures, the helium density and the poloidal magnetic flux. The equilibrium is self-consistently calculated with a 2D fixed-boundary solver. The electron density is kept fixed and the impurity densities are assumed to be known fractions of it. The deuterium and tritium densities are determined from quasi-neutrality assuming they are equal, and the effective charge profile is uniform. The electron and ion heat diffusion coefficients are normalized to achieve a thermal confinement improvement $H_{(y2,98)} \sim 1$ for the standard ELMy H-mode. The neoclassical conductivity and the bootstrap coefficients are evaluated by formulas obtained by solving the Fokker-Planck equation with the full collision operator [@Sauter1999; @Sauter1999b]. The Neutral Beam (NB) components are self-consistently evaluated with a Fokker-Planck subroutine which calculates the separate NB contributions to the electrons and ions. The EC power density and current driven profiles are evaluated by the beam tracing <span style="font-variant:small-caps;">Gray</span> code [@Gray].
The ECCD components provided as an input to <span style="font-variant:small-caps;">Astra</span> are Gaussian profiles with amplitude, width and total EC current derived from averaged values output from <span style="font-variant:small-caps;">Gray</span>. Figure \[fig:currentprofile\] shows the time evolution of the electron cyclotron driven current during the sawtooth ramp phase together with the Ohmic current, the beam current and the bootstrap current predicted by <span style="font-variant:small-caps;">Astra</span> when the equatorial launchers are used to provide 13.3MW of ECCD just inside the $q=1$ surface.
Figure \[fig:change\_in\_q\] shows the $q$-profile and the magnetic shear profiles as they evolve in time when the ECCD is applied inside $q=1$. It is evident that the shear at $q=1$ (here assumed at a radius $\rho_{1} = 0.48$) increases in time after the sawtooth crash (which happens every 26.6s in this simulation) due to the electron cyclotron driven current. Without ECCD, the value of the magnetic shear at $q = 1$ is 0.15, which is a typical value expected at sawtooth crashes in present experiments in the absence of fast particle stabilization. By depositing co-ECCD inside or outside the $q = 1$ radius, the shear at $q = 1$ spans a rather large range, as it changes from just above 0 to 0.4.
The change in the magnetic shear generated by the application of ECRH has been tested for a range of different launcher configurations [@GravesFST]. When the deposition location is far outside the $q = 1$ surface, there is no significant effect on the shear at the $q = 1$ location: the $s_1$ value stays approximately constant around 0.15. With the deposition just outside $q=1$, the $s_{1}$ value drops close to zero, and then as the resonance moves inward, the shear rapidly increases and stays constant at approximately $s_{1}=0.4$, even for very on-axis heating. It should also be noted that the $q = 1$ radius changes rapidly as the ECCD deposition moves across its initial position, meaning that the deposition needs to be adjusted in real time in order to follow the $q = 1$ radius and allow optimum sawtooth destabilization. That said, the significant increase in $s_{1}$ achievable with core ECCD means that good shear control can be achieved irrespective of the exact resonance provided the EC deposition is inside $q=1$, relaxing requirements on the real-time control system.
Figure \[fig:shear\_comp\] shows the difference of $s_1$ from the original value without ECCD as a function of $\rho_{dep}$ minus the $q = 1$ radius of the case without ECCD; the light blue shaded region corresponds to deposition inside $q = 1$ and the yellow region to deposition outside $q = 1$. With deposition inside $q = 1$, $s_1^{CD} - s_1^0$ increases, indicating that a sawtooth crash is likely to be triggered more rapidly and $\tau_{ST}$ decreases (and vice-versa with deposition outside $q = 1$). In fact, the shear increases when $\rho_{dep} - \rho_{1} \leq -0.02$, rather than strictly anywhere inside $q=1$, though this does not affect the real-time control scheme. The equilibrium modelling shows that the sawtooth destabilization should be somewhat easier to obtain than stabilization, because the radial extent inside $q = 1$ at which one can deposit co-ECCD and still obtain a significant shear increase is large, whereas one has to be very well localized around a specific region outside $q = 1$ to obtain a significant decrease in $s_1$ and thus have a chance to stabilize sawteeth.
Modelling the effect on the sawtooth period
-------------------------------------------
In order to model the nonlinear sawtooth period, the <span style="font-variant:small-caps;">Astra</span> transport code includes a heuristic model for when a sawtooth crash will occur, as described in reference [@SauterVarenna]. The sawtooth period in ITER is predicted to be considerable, due to the influence of $\alpha$-particle stabilization [@Chapman2011]. Since the sawtooth period is related to the free parameter of the model, $c_{\rho}$, this has been chosen to provide $\tau_{ST} = 40s$ for the reference case without any additional EC power, which is a lower bound of the sawtooth period for triggering NTMs predicted from empirical evidence in section \[sec:empirical\]. The corresponding value for $c_{\rho}$ is 4.3. If the free parameter in the Porcelli model is taken to be $c_{\rho}=1$ (as originally in the model [@Porcelli1996]), the sawtooth period approaches 200s and the safety factor on axis drops very low, making these predictions unreliable.
The results obtained with 13.3MW of co-ECCD driven from the upper launcher are shown in Figure \[fig:shear\_time\] [@Zucca; @Zucca2]. The plot shows the time evolution of the $s_1$ value in the case with no EC injection (top plot), co-ECCD deposited just inside $q = 1$ (middle plot) and just outside $q = 1$ (bottom plot). The sawtooth period can be easily estimated from this plot as $\tau_{ST} = 40$s(top plot), $\tau_{ST} = 30$s (middle plot) and $\tau_{ST} = 70$s (bottom plot).
Different launcher variations have been tested for their efficiency in destabilising the sawteeth. Figure \[fig:tau\_ECCD\] shows the sawtooth period as a function of the radial deposition location, $\rho_{dep}$, of the injected co-ECCD for different mixtures of EC power from the equatorial launcher or the upper launcher. The most efficient design uses 20MW of EC power from the equatorial launcher. In this case, the sawtooth period can be reduced from the reference case of 40s to 23-24s with $\rho_{dep}=0.35$ meaning that the ECCD also leads to efficient heating of the core and minimal impact on the fusion gain, $Q=P_{fusion}/P_{aux}$ [@Kirneva]. A combination of co-ECCD driven by 2 rows of the equatorial launcher (13.3MW) and the remaining power driven by the upper launcher, at a fixed location, can also decrease the sawtooth period down to less than 30s.
The degraded control in both destabilizing and stabilizing the sawteeth by using only the upper launcher can be ameliorated with a real-time control (RTC) algorithm, through which the deposition location is recalculated every time step by the simple formula: $\rho_{dep} = \rho_{1} + \eta w_{CD}$, where $\eta$ is a real-time control parameter that was scanned between -2 and +2, and $w_{CD}$ is the width of the Gaussian ECCD profile. With real-time feedback controlling the $\rho_{dep}$, the sawtooth period can be increased up to 70s, i.e. more than 50% increase on the fixed deposition case. Note that the time evolution of the value of the magnetic shear at the $q = 1$ surface, shown in Figure \[fig:shear\_time\](b) and (c), result from the RTC simulations and correspond to a RTC parameter $\eta = 0.75 (\tau_{ST} = 30s)$ and $\eta = 0.25 (\tau_{ST} = 75s)$ respectively.
The <span style="font-variant:small-caps;">Astra</span> modelling suggests that the natural sawtooth period can be reduced by around 30% by using 13.3MW from the equatorial launchers. Real-time control further enhances the ability of the ECCD to destabilise the kink mode and increase the sawtooth frequency. 13.3MW was assumed for control in order to leave more than 5MW available for NTM control if required (references [@Sauter2010; @LaHaye2009] suggests that relatively low ECCD power is likely to be sufficient for NTM island suppression in ITER). This assumption also allows for some margin if the ECCD effect is not as efficient as simulated. Assuming that the natural sawtooth period is approximately 50s as predicted by various transport simulations [@Jardin; @Bateman; @Waltz; @Onjun; @Bateman1998], then a reduction of $\sim 30\%$ to 35s is likely to avoid triggering NTMs according to the empirical scaling presented in section \[sec:empirical\]. However, the largest uncertainty is what the natural sawtooth period will be. The reference of 40s can be justified by scaling by resistive diffusion time from the 1s monster sawtooth crashes in JET, and further lengthening to account for the stabilising effect of the alphas by scaling the period in proportion to $\delta W_{\alpha}$ in ITER with respect to that in JET. Furthermore, the value of $c_{\rho}$ results in a crash when the magnetic shear at $q=1$ is in the range 0.5-0.6, which is in line with typical empirical evidence on a number of devices when active control is applied in plasmas with a large fast ion fraction of the total pressure.
Whilst the requirements of 13.3MW from the equatorial launcher, preferably in real-time control, may be sufficient, the next two sections detail a further ancillary control scheme using ICRH to aid the ECCD destabilisation (section \[sec:ICRH\]) or an alternative scheme of sawtooth stabilisation (section \[sec:stabilisation\]).
Sawtooth control with ICRH {#sec:ICRH}
==========================
The largest risk to controlling sawteeth only with ECCD is that this control scheme works through modification of the magnetic shear profile alone and does not directly influence the free energy to drive the kink mode. Consequently, it is recommended that a prudent approach will be to consider a complementary scheme which directly affects $\delta W$ and can ideally compete with the stabilisation afforded by the presence of the alpha population. In order to assess the effect of the ICRH born energetic particles on the stability of the internal kink, the distribution of fast ions simulated by <span style="font-variant:small-caps;">Selfo</span> and <span style="font-variant:small-caps;">Scenic</span> have been fed as a Monte Carlo set of markers into the <span style="font-variant:small-caps;">Hagis</span> code, as described in section \[sec:hagis\], and the $\delta W_{ICRH}$ calculated. This is then compared to the potential energy contributions from the NNBI, the fusion-born alphas and the thermal ions and fluid drive to assess the linear stability of the kink mode. Whilst such a linear assessment cannot be used to infer the sawtooth period, it qualitatively provides insight into the applicability of ICRH as a control tool, and the ratio of these contributions can be considered as a guide to its efficacy. Applying the ICRH off-axis means deposition at lower temperature at mid-radius and therefore shorter slowing down, which makes it more difficult to generate as many fast particles. This means that there are not very energetic tails to the distribution, and the absence of very fast ($>$10MeV) particles means that the finite orbit width effects are diminished. That said, however, the ICRH ions do still have a relatively strong impact on the internal kink stability.
Figure \[fig:ICRH\_stability\] shows the change in the potential energy of the mode arising due to the ICRH energetic ions as a function of the difference between the resonance radial location and the radius of the $q=1$ surface. There is a clear narrow well in the potential energy when the RF resonance is just inside the rational surface, that is to say when the gradient of the distribution of energetic passing ions is strong and positive. This narrow region ($\sim 2$cm) in which the sawteeth will be sensitive to the destabilising influence of the ICRH energetic ions implies that real-time control will be required in order that the resonance location be held in the right location with respect to the $q=1$ surface, though this is expected to be available between 40-55MHz in ITER with requisite latency [@Lamalle]. Despite the fact that the power absorbed by the minority species increases with the concentration [@Chapman2011], the strongest effect on mode stability is for a $^{3}$He concentration of only 1%. When there is too much $^{3}$He, the energy of the particles in the tail of the distribution becomes too low to have a strong effect on the kink mode whereas too little $^{3}$He means that the absorbed power is low and the broader distribution function leads to increased fast ion losses. This strong sensitivity to minority concentration introduces a risk in using ICRH as a sawtooth control tool since accurate control of the minority concentration and radial profile is difficult to achieve.
A comparison of $\delta W_{ICRH}$ when the distribution of markers is taken from both <span style="font-variant:small-caps;">Selfo</span> or <span style="font-variant:small-caps;">Scenic</span> is shown in figure \[fig:selfoscenic\] for the case with 0.5% $^{3}$He, -90$^{\circ}$ antenna phasing and 20MW injected power with the resonance at $r=0.32$m (f=48.58MHz in <span style="font-variant:small-caps;">Selfo</span> and f=48.9MHz in <span style="font-variant:small-caps;">Scenic</span>). It is clear that slightly stronger destabilisation is observed using the <span style="font-variant:small-caps;">Scenic</span> distribution, though in general the agreement is good. This could be due to the inclusion of the shaping effects. The main purpose of this comparison, though, is to mitigate the risk in the modelling uncertainty by taking the predictions from the least favourable result, in this case, the <span style="font-variant:small-caps;">Selfo</span> distribution. The influence of the ICRH fast ions compared to the stabilising effect of the alpha particles and NNBI distributions is shown in figure \[fig:fastion\_stability\] for the case when the ICRH resonance is at $r=0.32$m ($f_{ICRH}=48.9\textrm{MHz}$) and when it is at $r=0.43m$ ($f_{ICRH}=50\textrm{MHz}$). In these simulations the $q=1$ surface is moved by changing the equilibrium rather than re-simulating the fast ion distribution for different resonance locations. It is evident that the mid-radius ICRH fast ions, despite the poor power absorption and low energy tails, retain a strongly destabilising influence, comparable to the magnitude of stabilisation afforded by the alphas or the NBI heating. Whilst the power absorption is better when the resonance layer is nearer to the axis, resulting in improved core heating, the passing fast ions are only destabilising when the radial location of the $q=1$ surface is inside $\rho=0.2$. These simulations are for 1% $^{3}$He concentration and +90$^{\circ}$ phasing of the antenna, though the $-90^{\circ}$ phasing gives similar results, with a slightly diminished destabilisation. The fact that the ICRH is able to completely negate the stabilising term from the presence of the $\alpha$ population is significant and important, and makes the ICRH an essential part of the portfolio of control tools in ITER.
Having reduced the risk in uncertainty of the ICRH fast ion distribution by utilising independent RF wave field codes, the largest residual uncertainty in this modelling is the location of the $q=1$ surface. The ITER baseline scenario designed using <span style="font-variant:small-caps;">Astra</span> transport simulations suggests that the $q=1$ surface will approach mid-radius. However, the $q$-profile has a wide region of very low shear in the core, meaning that a small change in $q_{0}$ can significantly affect the radial location of the rational surface. Figure \[fig:fastion\_stability\] also shows that if the $q=1$ surface could be maintained closer to the magnetic axis, sawtooth control would be significantly easier to achieve, since the alphas would be less stabilising, the NNBI would be less stabilising, and could even be used as a destabilising control tool in the most off-axis orientation, and the control and flexibility afforded by the ICRH would be increased. Furthermore, the ECCD used to control the sawteeth would be closer to the plasma core, and so have the dual benefit of heating the plasma, hence affording a potential reduction in other auxiliary heating power and subsequent increase in $Q$. This may be possible with early heating to delay the current penetration into the core, as regularly employed on JET, and then deliberate sawtooth destabilisation to mediate the $q$-profile once the $q=1$ surface enters.
Since 20MW of ICRH power is unlikely to be dedicated for sawtooth control, lower ICRH powers have also been considered. <span style="font-variant:small-caps;">Selfo</span> simulations of the ICRH fast ion population from 10MW of ICRH with 1% $^{3}$He minority concentration, $\pm$90$^{\circ}$ antenna phasing and $f_{ICRH}=48.6$MHz have been performed and used as input for <span style="font-variant:small-caps;">Hagis</span>. The change in the potential energy of the mode at the optimal relative position of $q=1$ with respect to the ICRH resonance location scales more favourably than linearly at reduced power. The $\delta W_{ICRH}$ when $\rho_{1}-\rho_{res}=0.02$ (the optimal resonance position as seen in figure \[fig:ICRH\_stability\]) is shown in figure \[fig:powerscan\] for both $\pm 90^{\circ}$ antenna phasings for 20MW and 10MW injected power. Whilst it is not possible to infer the sawtooth period resultant from the ICRH application from this linear modelling, it is clear that 10MW of RF heating does significantly destabilise the kink mode, meaning that it is likely to be useful as an ancillary control actuator, even with half the available power. This is supported by empirical evidence from recent JET experiments demonstrating sawtooth control with $^{3}$He minority schemes [@Graves2010]. Figure \[fig:icrh\_power\_jet\] shows the sawtooth period with respect to the ICRH power normalised to the total auxiliary power (NBI+RF) for the series of pulses described in reference [@Graves2010]. Each curve is for different injected NBI power. This can be considered congruent to including alphas where the fixed quantity is $P_{NBI}+P_{\alpha}=P_{fast}-P_{ICRH}$. Whilst there is initially a strong monotonic reduction of the sawtooth period as the ICRH power is increased for all different background fast ion stabilisation levels, as the ICRH power continues to increase, the destabilisation does not increase. This happens as the saturated sawtooth period is close to that of Ohmic sawteeth, or at least Ohmic sawteeth with modified energy and resistive diffusion time due to the heating effect of the ICRH. Whatever level of ICRH power available in ITER for sawtooth control, it is likely to have a significant beneficial contribution to make in reducing the sawtooth period.
Sawtooth Stabilisation {#sec:stabilisation}
======================
If it proves that a combination of $>10$MW of ICRH power on top of the primary actuator of 13.3MW of ECCD from the equatorial launcher is insufficient for successful sawtooth control, then it is important that an alternative solution is provided within the design of the heating and current drive facilities available on ITER. It is worth noting that in D-T plasmas in JET the best performance was achieved in the transient phases during which sawteeth were avoided [@Nave2002]. Consequently, a sawtooth stabilisation scenario has been envisioned, whereby the natural sawtooth period is deliberately lengthened, and the (very probable) NTM that ensues at the crash is pre-emptively stabilised before it reaches its saturated width. This was considered the most desirable route to sawtooth amelioration in the original ITER Physics Basis [@IPB2], and was only superseded by destabilising control tools as anxiety grew about the ramifications of triggering performance-degrading NTMs and due to the need for frequent expulsion of the on-axis accumulation of higher-$Z$ impurities that would otherwise cause degradation of energy confinement due to impurity radiation.
Long sawtooth periods are naturally achieved by applying early heating during the current ramp-up phase to increase the conductivity and so slow down the current penetration. Combining this with achieving early ignition will further stabilise the sawteeth due to the $\alpha$ particle stabilisation. ICRH could then be used as an ancillary control tool, with core heating providing a further population of strongly stabilising fast ions. Furthermore, in order to meet the $Q=10$ goal of ITER baseline scenario, it is desirable to turn off the ECRH power whenever it is not being actively used for mode control, or to use it for core heating and reduce the NBI/ICRH power, and so maximise $Q$. Thus, rather than being constantly required to modify the shear at $q=1$, an alternative can be foreseen whereby fast ions are used to deliberately stabilise the sawteeth, and before each crash the ECCD is pre-emptively applied near the $q=2$ surface to stabilise the ensuing NTM [@Sauter2010; @LaHaye2009]. Provided the seed island is sufficiently small ($\sim 7$cm), all models for ECCD island suppression indicate that 13.3MW from the upper launchers can fully suppress the mode within 6s [@Sauter2010]. It is important to apply the ECCD pre-emptively in order to tackle the NTM when the island width is relatively small, thus requiring much less time of applied ECCD in order to fully suppress the mode, and therefore having a less deleterious effect on fusion gain. However, it is worth noting that the seed island for NTM growth can be created at large island size at the sawtooth crash as sometimes observed in JET [@Sauter; @Graves2010], meaning that the pre-emptive ECCD is only useful if it modifies the dynamics of the island seeding during the sawtooth crash.
It is anticipated that by using either counter-CD in the core, off-axis co-ECCD or on-axis ICRH to maximise the sawtooth period, transient (though this could be $>100$s) periods of very high $Q$ could be attained. It could be that if the $\alpha$ stabilisation proves to be stronger than anticipated, and the natural sawtooth period is $>100$s, then control by deliberate stabilisation, followed by provocation of a crash through dropping auxilliary heating coupled with simultaneous pre-emptive application of ECCD near the $q=2$ surface to suppress the subsequent NTM growth, could be the best route to long-periods of excellent performance. This has been demonstrated in TCV where sawtooth pacing using modulated core ECCD coupled with pre-emptive NTM avoidance by ECCD at $q=3/2$ has avoided the triggering of NTMs [@Goodman; @Felici]. It should be noted that deliberate stabilisation of the sawteeth using either ICRH or core counter-ECCD/off-axis co-ECCD will require real-time control, as discussed in section \[sec:ECCD\]. Whilst sawtooth stabilisation may even seem advantageous for maximising $Q$ due to the reduced application time of the control actuators, the removal of frequent sawteeth from the baseline scenario means an alternative strategy is required to reverse the on-axis accumulation of higher-$Z$ impurities that would otherwise cause degradation of energy confinement due to impurity radiation. Although this has been demonstrated using core ECH in ASDEX Upgrade [@Dux; @Neu], its application for $He$-ash removal is untested, and the avoidance of potentially disruptive NTMs (even with pre-emptive ECCD) is preferable, so sawtooth destabilisation remains the optimal solution for ITER.
Conclusions {#sec:conclusions}
===========
An empirical scaling of the sawtooth period that will trigger an NTM in ITER suggests that the “natural” sawtooth period predicted by transport modelling is approximately at the threshold for NTM seeding. Whilst this means that active sawtooth control is essential, it suggests that sufficient control can be achieved through a relatively small reduction in the sawtooth period. Transport modelling coupled to ray-tracing predictions and using the linear stability thresholds for sawtooth onset suggests that 13MW of ECCD from the equatorial launcher could be sufficient to reduce the sawtooth period by $\sim$30%, and this being the case, dropping it below the NTM triggering threshold. This modelling is predicated upon choosing a natural sawtooth period of 40s; should the stabilising contribution from the fusion-born alpha particles and on-axis NBI injection prove to give rise to a significantly longer natural sawtooth period, the ability of the ECCD to control sawteeth will be diminished. There are naturally large uncertainties associated with this modelling, and it is prudent to plan to use more than one control actuator in order to reduce this risk. Consequently, it is recommended that $>10$MW of ICRH at $\sim 47$MHz (with real-time feedback) just inside $q=1$ is also reserved for sawtooth control. The largest uncertainty in the modelling of the effect of the fast ions is the position of the $q=1$ surface. If the $q=1$ surface could be maintained closer to the magnetic axis, sawtooth control would be significantly easier to achieve, since both the alphas and the beam-induced fast ions would be less stabilising. Furthermore, the ICRH and ECCD used to control the sawteeth would be further towards the plasma core, thus heating in the good confinement region and so affording a potential reduction in other auxiliary heating power and subsequent increase in $Q$. Finally, should active sawtooth destabilisation prove to be unattainable due to unexpectedly large stabilising contribution from the $\alpha$ particles, plant availability or inefficiency in power absorption or current drive, then there is a viable alternative strategy relying upon sawtooth stabilisation coupled with pre-emptive NTM suppression, which would provide long periods of good performance. The power requirements for the necessary degree of sawtooth control using either destabilisation and stabilisation schemes are expected to be within the specification of anticipated ICRH and ECRH heating in ITER, provided the requisite power can be dedicated to sawtooth control. It is worth reiterating that the results presented in sections \[sec:empirical\] and \[sec:exp\] are derived from present day experiments in the absence of alpha particles. The alpha particles are found to significantly stabilise the internal kink mode, as shown in section \[sec:ICRH\], which means there is a relatively high uncertainty in the modelling predictions at high alpha fraction, that is to say at high $Q$.
**Acknowledgements**
This work was conducted under the auspices of the ITPA MHD Stability Topical Group. This work was partly funded by the United Kingdom Engineering and Physical Sciences Research Council under grant EP/I501045, the European Communities under the contract of Association between EURATOM and CCFE and supported in part by the Swiss National Science Foundation and US Department of Energy under DE-FC02-04ER54698 and DE-AC52-07NA27344. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
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---
abstract: 'Multiple quantum coherences are typically characterised by their coherence number and the number of spins that make up the state, though only the coherence number is normally measured. We present a simple set of measurements that extend our knowledge of the multiple quantum state by recording the coherences in both the $x$ basis and the usual $z$ basis. The coherences in the two bases are related by a similarity transformation. We characterize the growth of the multiple quantum coherences via measurements in the two bases, and show that the rate varies with the coefficient of the driving term in the Hamiltonian. Such measurements in non-commuting bases provides additional information over the 1D method about the state of the spin system. In particular the measurement of coherences in a basis other than the usual $z$ basis allows us to study the dynamics of the spin system under Hamiltonians, such as the secular dipolar Hamiltonian, that conserve $z$ basis coherence number.'
address: 'Department of Nuclear Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139'
author:
- 'C. Ramanathan, H. Cho, P. Cappellaro, G. S. Boutis, D. G. Cory'
bibliography:
- 'Bibliography.bib'
title: 'Encoding multiple quantum coherences in non-commuting bases'
---
Introduction {#introduction .unnumbered}
============
The many-body behaviour of nuclear spins in a solid was described, for a long time, in the language of spin thermodynamics [@Jeener-1968; @Goldman-1970; @Wolf-1979]. This description is, however, essentially static, and the dynamical behaviour of the system was typically ignored, or addressed in terms of memory functions [@Demco-1975; @Mehring-1983]. With the advent of multiple quantum NMR techniques [@Hatanaka-1975a; @Pines-1976; @Aue-1976; @Vega-1976], multi-spin processes could be described by multiple spin correlations and multiple quantum (MQ) coherences. The selective excitation and transformation [@Warren-1980] of MQ coherences led to a new picture of many-body spin dynamics in a dipolar solid [@Baum-1985; @Munowitz-1987b; @Levy-1992; @Lacelle-1993]. The focus in these experiments was on transitions of coherence number, which were observable, rather than the number of spins involved in the MQ coherence states. However, the object of the experiment continued to be “spin counting” as these MQ experiments were called. These techniques have been used to probe the spatial relationships between spins in large macromolecules, polymers and crystalline systems, including determining the dimensionality and size of localized spin clusters (see [@Weitekamp-1983; @Munowitz-1987a; @Lacelle-1991; @Emsley-1994] for reviews).
In principle, all the spins in a solid are coupled together through their dipolar fields, and the Hilbert space of the spin system is determined by the total number of spins in the sample. However, at high field and for temperatures above a few degrees Kelvin, it is sufficient to consider a much smaller spin system to predict the NMR spectrum of a solid [@Slichter-1990]. The sample resembles an ensemble of weakly coupled subsystems, within each of which an effective number of coupled spins is postulated to exist. In equilibrium, at high field, the spin number is one, as $\rho \approx \sum_{i} I_{z}^{i}$.
In a strong magnetic field ($B_{0}\hat{z}$), an N spin-1/2 system has $2^{N}$ stationary states. These can be classified according to the magnetic quantum number, $M_{z} = \sum_{i} m_{zi} = (n_{|+1/2>} - n_{|-1/2>})/2 $, where $m_{zi} = \pm 1/2$ is the eigenvalue of the $i$th spin in the system, and the energy eigenvalue corresponding to $M_{z}$ is $E_{z} = -\gamma\hbar B_{0}M_{z}$. For non-degenerate stationary states there are on the order of $2^{2N-1}$ possible transitions between any two levels. The difference in $M_{z}$ values between the two levels is referred to as the [*coherence number*]{}. If the density operator is expanded in the basis of irreducible tensor operators $T_{lm}$, $$\rho = \sum_{lm} a_{lm} T_{lm}$$ the rank $l$ of the tensor element defines the spin number, while the order $m$ characterizes the coherence number.
While these coherences refer to transitions between levels, it is useful to discuss multiple quantum coherences for states of a system. When the state is expressed in the eigenbasis of the system, the presence of a non-zero matrix element $<z_{i}|\rho|z_{j}>$, indicates the presence of an $n$ quantum coherence, where $n = M_{z}(z_{j}) - M_{z}(z_{i})$. Since $M_{z} = \sum_{i}m_{zi}$ is a good quantum number, we use a collective rotation about the axis of quantization, $\sum_{i} I_{z}^{i}$, to characterize it $$<z_{i}|\exp(-i\phi\sum_{i}I_{z}^{i})\rho\exp(+i\phi\sum_{i}I_{z}^{i})|z_{j}> = \exp(in\phi) <z_{i}|\rho|z_{j}> \: \: .$$
In the usual MQNMR experiment, the system (initially in Zeeman equilibrium) is allowed to evolve under the action of a Hamiltonian that generates single quantum (SQ) [@Suter-1987], or double quantum (DQ) [@Warren-1980; @Yen-1983] transitions. This progressively increases the coherence numbers of the state of the system, as well as causing its spin number to increase.
While the coherences have a physical meaning in the eigenbasis of the system, a generalized coherence number reports on the response of the system to any collective rotation of the spins (about the $x$ axis for example). This is equivalent to expressing the state of the spins in a basis where the apparent axis of quantization is given by the axis of rotation, and can be obtained from the eigenbasis via a similarity transformation. For example, the similarity transform $\mathcal{P}$ connects the density matrices of the system in the two (the $z$ or eigen-basis, and the $x$ basis) representations. $$\left[\rho^{x}\right] = \mathcal{P}^{-1} \left[\rho^{z}\right] \mathcal{P}$$ where the elements of the matrices are $\left[\rho^{x}\right]_{ij} = <x_{i}|\rho|x_{j}>$, $\left[\rho^{z}\right]_{ij} = <z_{i}|\rho|z_{j}>$ and $\{x_{i}\}$ and $\{z_{i}\}$ are complete sets of basis operators. Under a collective rotation about the $x$ axis, we obtain, $$<x_{i}|\exp(-i\phi\sum_{i}I_{x}^{i})\rho\exp(+i\phi\sum_{i}I_{x}^{i})|x_{j}> = \exp(in\phi) <x_{i}|\rho|x_{j}>$$ where $n$ is the $x$ basis coherence number. Similarity transformations do not change the eigen-energies or the physics of the system [@Cornwell-1997]. Suter and Pearson [@Suter-1988] previously used a combination of phase shifts and a variable flip angle pulse to encode for coherences in the $y$ basis as well as the $z$ basis. Their technique was recently used to study the dynamics of polarization and coherence echoes [@Tomaselli-1996]. Requantization in an alternative basis has also been applied to analyzing RF gradient NMR spectroscopy [@Zhang-1995].
In this letter we demonstrate an improved technique for the encoding of coherences in the $x$ basis as well as for encoding coherences simultaneously in the $x$ and $z$ bases. While the measurement of coherence number in an orthogonal basis does provide more information about the state, it does not yield a direct measure of the spin number. Under a collective rotation of the spins, the different orders within a given rank are mixed, but there is no mixing between terms of different rank. Thus contributions to a given coherence order from the different ranked tensors cannot be separated out, without some measure of the distribution of tensor ranks in the system. In order to unambiguosly determine the spin number, N independent measurements are required.
Measurements in non-commuting basis are central to the task of quantum state tomography. Eigenbasis measurements provide information on the amplitudes of the terms in the density matrix (in the eigenbasis), but not on the associated phase factors. Changing the basis and repeating the measurements allows reconstruction of the exact state of the system, a familiar process used to measure the Wigner function in optics experiments [@Leonhardt-1996]. Measuring multiple quantum coherences in a basis other than the usual $z$ basis is particularly important if we wish to study the dynamics of the spin system under a Hamiltonian that conserves $z$ basis coherence number, such as the secular dipolar Hamiltonian.
Methods {#methods .unnumbered}
=======
Table 1 shows the initial state, Hamiltonian, and selection rules for the standard MQ experiment (using a DQ Hamiltonian), in both the standard $z$ basis and the $x$ basis. Reference [@Zhang-1995] tabulates the transformations between quantization in the different Cartesian bases. Thus, starting from an initial Zeeman state, we see that under the DQ Hamiltonian we should get only even order coherences in the $z$ basis and only odd order coherences in the $x$ basis.
$z$ basis $x$ basis
--------------------------------- --------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------
initial state $I_{z}$ $-\frac{\imath}{2}\left\{ I_{x}^{+} - I_{x}^{-}\right\}$
initial coherence number $0$ $\pm 1$
MQ Hamiltonian $\displaystyle{\sum_{i<j}} d_{ij} \left\{ I_{i}^{+}I_{j}^{+} + I_{i}^{-}I_{j}^{-} \right]$ $\displaystyle{\sum_{i<j}} d_{ij} \left[ \{2I_{xi}I_{xj} - \frac{1}{2}(I_{xi}^{+}I_{xj}^{-}+I_{xi}^{-}I_{xj}^{+})\} - \right.$
$\left. \frac{1}{2}\{ I_{xi}^{+}I_{xj}^{+} + I_{xi}^{-}I_{xj}^{-}\} \right\}$
coherence number selection rule $\pm2$ $0,\pm2$
spin number selection rule $\pm1$ $\pm1$
coherences even odd
: Description of the MQ experiment in the $z$ and $x$ bases.
The experimental methods presented here improve on those of Suter and Pearson, as their variable flip angle pulse is replaced by a sequence of phase shifted pulses, whose duration is fixed. The dipolar evolution during the variable angle pulse, as it is sampled out to multiples of $2\pi$, can significantly attenuate the signal and compromise the resolution of the coherences in the $x$ or $y$ basis, especially in a strongly dipolar coupled system. In our experiment, the dipolar evolution is refocused, and the $\phi I_{x}$ rotation achieved purely with phase shifts.
The pulse sequences shown in Figures 1(a) and 1(b) allow us to encode coherences in the two bases under essentially identical conditions. Figure 1(a) is a $z$ basis encoding experiment while Figure 1(b) is an $x$ basis encoding experiment. The only difference between the two is that the first $\pi/2$ pulse is phase shifted along with $U_{\phi}$ in the $x$ basis experiment. Figure 1(c) shows the 16 pulse DQ selective sequence used. It consists of two cycles of the standard 8 pulse sequence, phase shifted by $\pi$ with respect to each other. The sequence compensates for pulse imperfections and resonance offsets. $U_{\phi}$ is created by phase shifting all the pulses in the 16 pulse experiment by $\phi$. The two ($\pi/2$) pulses and the Cory 48-pulse sequence are not required for the $z$ basis experiment. However, they are included in order to perform the two experiments under identical conditions. In the $x$ basis experiment, the two ($\pi/2$) pulses perform the basis transformation, and the phase encoding of the coherences. Placing them back-to-back leads to unwanted switching transients, so they are separated by the Cory 48-pulse sequence which prevents evolution of the spin system under the secular dipolar Hamiltonian between the two ($\pi/2$) pulses, and has been described previously [@Cory-1990b].
The operator corresponding to the observable signal is $I_{z}$. The measured signal for the experiment in Figure 1(a) corresponds to $<I_{z}>_{\phi} = Tr \left[\rho_{f} I_{z} \right]$, where the final density matrix is given by $$\begin{aligned}
\rho_{f} & = & U^{\dag}R_{y}(-\pi/2)R_{y}(\pi/2)U_{\phi} \rho_{i} U_{\phi}^{\dag}R_{y}(-\pi/2)R_{y}(\pi/2)U \nonumber \\ \nonumber \\
% & = & U^{\dag}U_{\phi} I_{z} U_{\phi}^{\dag}U \nonumber \\ \nonumber \\
% & = & U^{\dag}R_{z}(-\phi)UR_{z}(\phi) I_{z} R_{z}(-\phi)U^{\dag}R_{z}(\phi)U \nonumber \\ \nonumber \\
& = & U^{\dag}R_{z}(-\phi)U I_{z} U^{\dag}R_{z}(\phi)U\end{aligned}$$ where $R_{\alpha}(\phi) = \exp(i\phi I_{\alpha})$, and we have used the fact that the initial state $I_{z}$ is invariant to $z$-rotations. Defining $\rho_{s} = U \rho_{i} U^{\dag} = U I_{z} U^{\dag}$, the state of the system after evolution under the DQ Hamiltonian, we obtain the measured signal in the $z$ basis experiment $$<I_{z}>_{\phi} = Tr \left[ R_{z}(-\phi) \rho_{s} R_{z}(\phi) \rho_{s} \right] \label{eq:C} \: \: .$$
For the experiment in Figure 1(b), the final density matrix is given by $$\begin{aligned}
\rho_{f} & = & U^{\dag}R_{y}(-\pi/2)R_{\phi}(\pi/2)U_{\phi} \rho_{i} U_{\phi}^{\dag}R_{\phi}(-\pi/2)R_{y}(\pi/2)U \nonumber \\ \nonumber \\
%& = & U^{\dag}R_{y}(-\pi/2)R_{\phi}(\pi/2)R_{z}(-\phi)UR_{z}(\phi) I_{z} R_{z}(-\phi)U^{\dag}R_{z}(\phi)R_{\phi}(-\pi/2)R_{y}(\pi/2)U \nonumber \\ \nonumber \\
%& = & U^{\dag}R_{y}(-\pi/2)R_{z}(-\phi)R_{y}(\pi/2)R_{z}(\phi)R_{z}(-\phi)UI_{z}U^{\dag}R_{z}(\phi)R_{z}(-\phi)R_{y}(-\pi/2)R_{z}(\phi)R_{y}(\pi/2)U \nonumber \\ \nonumber \\
& = & U^{\dag}R_{y}(-\pi/2)R_{z}(-\phi)R_{y}(\pi/2)U I_{z} U^{\dag}R_{y}(-\pi/2)R_{z}(\phi)R_{y}(\pi/2)U \nonumber \\ \nonumber \\
& = & U^{\dag}R_{x}(-\phi)UI_{z}U^{\dag}R_{x}(\phi)U\end{aligned}$$ and the observed magnetization in the $x$ basis experiment is $$<I_{z}>_{\phi} = Tr \left[ R_{x}(-\phi) \rho_{s} R_{x}(\phi) \rho_{s} \right] \label{eq:D} \: \: .$$ In both cases, the experiment is repeated multiple times as $\phi$ is uniformly sampled out to a multiple of $2\pi$, and the resulting data Fourier transformed with respect to $\phi$ to obtain the distribution of coherence numbers.
Note that if Equations (\[eq:C\]) and (\[eq:D\]) could be written in terms of $<I_{z}>_{\phi} = Tr \left[ A(\phi) \rho_{s} \right] $, where the set of operators $A(\phi)$ form a complete basis for the Hilbert space of the spin system, it would be possible to perform quantum state tomography on the spins [@Leonhardt-1996]. However, the experiments described here cannot completely characterize the state, or even just its collective properties.
Results {#results .unnumbered}
=======
The experiments were performed at 2.35 T with a Bruker Avance spectrometer and a home-built RF probe. The 90 degree pulse time was 0.51 $\mu$s. The pulse spacing $\Delta$ used in the DQ sequence was 1.3 $\mu$s, and the cycle time for the 16 pulse cycle was 43.4 $\mu$s. The pulse spacing in the 48 pulse time suspension sequence was 1.5 $\mu$s. The T$_{1}$ of the single crystal calcium fluoride sample used was 7 s, and the recycle delay used in the experiment was 10 s.
Figure 2 shows the results obtained in the $z$ and $x$ basis encoding experiments. The maximum coherence encoded was $\pm32$, with $\Delta\phi = 2\pi/64$. The phase incrementation was carried out to $8 \pi$. It is seen that the $z$ and $x$ basis measurements give only even and odd coherences respectively as expected from Table 1. The data shown correspond to 1, 3 and 5 loops of the 16 pulse cycle. The higher order coherences are seen to grow in both bases, as the system evolves under the DQ Hamiltonian.
In Figure 3 we plot the effective spin number, obtained by a Gaussian fit to the coherence number distributions in the $x$ and $z$ bases ($N(\tau) = 2\sigma^{2}$) as a function of $\tau$ [@Baum-1985]. The fits were performed on the 1D data. The variance of the $x$ basis measurements is consistently smaller than that of the $z$ basis measurements. The points appear to lie on a straight line, and a best linear fit has a slope of 0.54, which is very close to the value of 0.5 expected from the ratio between the double quantum selective terms in the DQ Hamiltonian expressed in the two bases, as shown in Table 1. The linear fit was performed on the mean spin numbers $N_x$ and $N_z$ without considering the standard deviations. The error in the fit is negligible. In the $z$ basis, evolution under the DQ Hamiltonian forces the system to change coherence number, while in the $x$ basis the presence of zero quantum terms permits mixing without changing the coherence number. Thus the growth of the coherence numbers is slowed relative to the $z$ basis, leading to a narrower distribution. It should be emphasised that the basis change does not change the spin number, only the experimentally observable coherences. The change in spin number with basis representation demonstrates the limitations of the Gaussian statistical model as an accurate predictor of spin number in strongly coupled spin systems. Lacelle has also discussed the limitations of the model [@Lacelle-1991].
As first shown by Suter and Pearson [@Suter-1988], a two dimensional experiment illustrates the correlation between $x$ and $z$ basis coherences. The two-dimensional experiment is obtained from the $x$ basis experiment in Figure 1(b) by phase cycling the refocussing sequence $U^{\dag}$ by $\beta$ independently of $\phi$. The phases $\phi$ and $\beta$ are incremented independently to sample a rectangular grid and a 2D Fourier transform is performed to yield the coherences. The measured data in a single experiment is $$<I_{z}>_{\beta\phi} = Tr \left[R_{z}(-\beta)R_{x}(-\phi) \rho_{s} R_{x}(\phi)R_{z}(\beta) \rho_{s} \right] \label{eq:E} \: .$$ It is straightforward to show that the order of the $x$ and $z$ phase shifts does not matter when both of them are sampled over a $2\pi$ range. The $(\phi_{x})(\beta_{z})$ experiment is equivalent to the $(-\beta_{z})(-\phi_{x})$ experiment. The two dimensional experiment separates out the different terms that contribute to a particular $z$ basis coherence as can be seen in Figure 4. We used $\tau$ = 130.3 $\mu$s, corresponding to 3 loops of the MQ cycle. The maximum coherence encoded in each direction was $\pm12$, with $\Delta\phi = 2\pi/24$. The phase incrementation was carried out to $8 \pi$ along each axis, resulting in a $96\times96$ data grid, which was Fourier transformed to yield the coherences shown.
This two dimensional technique can be used to examine the evolution of multiple quantum coherences under the dipolar Hamiltonian. Figure 5 shows the attenuation of the $z$ basis zero quantum signal as it evolves under the dipolar Hamiltonian. Also shown on the figure are the various $x$ basis contributions to the $z$ basis zero quantum signal obtained from the 2D data. The decay is clearly non-exponential. It can be seen that the different $x$ basis terms attenuate at different rates, and that the measured decay of the 1D $z$ basis data represents some sort of average attenuation of all these terms. Thus this technique allows us to probe the details of spin dynamics beyond the ability of existing techniques. We have also recently used this technique to study the evolution of the spin system following a Jeener-Broekaert pulse pair, and observed the evolution of the system to a dipolar ordered state [@Cho-2002].
Conclusions {#conclusions .unnumbered}
===========
We have shown that by encoding MQ coherences in different bases ($x$ and $z$) additional information about the state of the spin system may be obtained. In particular, $x$ basis encoding could be useful in determining the size of multiple spin correlations under the action of a Hamiltonian that preserves $z$ basis coherence number, but changes the number of spins in the state, such as the secular dipolar Hamiltonian.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Dr. Joseph Emerson for stimulating discussions, and the NSF and DARPA DSO for financial support.
Figure Captions {#figure-captions .unnumbered}
===============
1. \(a) The $z$ basis encoding experiment, $U_{\phi} = R_{z}(-\phi)UR_{z}(\phi)$, and $U = \exp(iH_{DQ}\tau)$. The propagator for the 48 pulse time-suspension sequence is the Identity Operator I. (b) The $x$ basis encoding experiment. where the first ($\pi/2$) pulse is phase shifted, $(\pi/2)_{\phi} = R_{z}(-\phi)(\pi/2)_{y}R_{z}(\phi)$. (c) The 16 pulse sequence used to generate the effective DQ Hamiltonian; $\Delta = 1.3$ $\mu$s, $t_{\pi/2} = 0.51$ $\mu$s, $t_{c} = 43.4$ $\mu$s.
2. Comparison of $z$ basis and $x$ basis coherences at preparations times $\tau =$ 43.4, 130.3 and 217.2 $\mu$s, corresponding to 1, 3 and 5 loops of the 16 pulse cycle in Figure 1(c), showing the presence of purely even and odd coherences in the two bases respectively.
3. Plot of the effective spin number ($N(\tau) = 2\sigma^{2}$) obtained by fitting the coherence distributions obtained in the 1D $x$ and $z$ basis measurements to a Gaussian. Also shown is the best linear fit to the data, whose slope is 0.54. The slope expected from the coefficient of the DQ selective terms in the two bases (see Table 1) is 0.5.
4. Results of the 2D experiment - showing correlations between encoding in the $x$ and $z$ bases. The preparation time used was $\tau$ = 130.3 $\mu$s, corresponding to 3 loops of the 16 pulse cycle in Figure 1(c).
5. Decay of the $z$ basis zero quantum signal under the dipolar Hamiltonian. The decay of the different $x$ basis contributions to the zero quantum signal in the $z$ basis, obtained from the 2D experiment are also shown. The data were normalized to the observed intensity at 2 $\mu$s. Note that the data are plotted on a log scale. The preparation time used was $\tau$ = 130.3 $\mu$s, corresponding to 3 loops of the 16 pulse cycle in Figure 1(c).
|
---
abstract: |
The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This work is a geometric fulfillment of Fock’s characterization of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [@Ca14]. This framework leads to a true geometric understanding of conventional choices for boundary conditions. For example, the boundary condition that the pull-back of the 1-form on the cotangent space of fields at the boundary vanish, i.e. $\Pi^*\alpha=0$ , is shown to be a consequence of our finding that the boundary fields of the theory lie in the 0-level set of the moment map of the gauge group of the theory.
It is also shown that the natural way to interpret Euler-Lagrange equations as an evolution system near the boundary is as a presymplectic system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions at the boundary are analyzed and the reduced phase space of the system is determined to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations. This setting makes it possible to define well-posed boundary conditions, and provides the adequate setting for the canonical quantization of the system.
The notions of the theory will be tested against three significant examples: scalar fields, Poisson $\sigma$-model and Yang-Mills theories.
address:
- 'ICMAT and Depto. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain.'
- 'Dept. of Mathematics, Univ. of California at Berkeley, 903 Evans Hall, 94720 Berkeley CA, USA'
author:
- 'A. Ibort'
- 'A. Spivak'
title: 'Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories'
---
Introduction {#sec:introduction}
============
Multisymplectic geometry provides a convenient framework for describing first order covariant field theories both in the Lagrangian and Hamiltonian formalism [@Ca91]. See also the GimMsy papers [@Go98] and [@Go04]. However, the role of boundaries in the multisymplectic formalism, relevant as it is in the construction of the corresponding quantum field theories, has not been incorporated in a unified geometrical picture of the theory.
In this paper we will extend the multisymplectic formalism to deal with first-order covariant Hamiltonian field theories on manifolds with boundary, providing a consistent geometrical framework, for instance, for the perturbative quantization program recently set up by A. Cattaneo, P. Mnev and N. Reshetikhin [@Ca14] for theories on manifolds with boundary. The restriction to the boundary will provide the canonical Hamiltonian formalism needed for the canonical quantization picture that will be developed in detail elsewhere. We will concentrate on the classical setting and we will prepare the ground to introduce a graded setting that will become useful when dealing with the quantization of gauge theories.
The history of the construction of a geometrical picture for field theories is extensive with many relevant contributions. We refer the reader to the comprehensive texts [@Gi09] and [@Bi11] and to references therein. The first author’s earlier work on the subject, [@Ca91], benefited from [@Ga72], [@Go73], [@Ki76], [@Ki79] and many others. The ambitious GimMsy papers [@Go98] and [@Go04] were the first parts of a project that aimed to reconcile a multisymplectic geometrical formalism for field theories with the canonical picture needed for quantization. For recent work on the geometry of of first and higher order classical field theories see [@Gr12] and [@Gr15]. For recent work extending the multisymplectic formalism to higher order Hamiltonian theories see [@Le05], [@Ec07], [@Ro09], [@Vi10], [@Pr14] and references therein.
In [@Ca11] the authors lay out Fock’s unpublished account of the general structure of Lagrangian field theories on manifolds with boundary. Adjusting this account in the obvious way, one arrives at the general structure of Hamiltonian field theories on manifolds with boundary $\mathit{a}$ $\mathit{la}$ Fock. The multisymplectic formalism we develop in this work to describe first-order Hamiltonian field theories on manifolds with boundary, puts meat and ligaments onto the bones, so to speak, of the Fock-inspired account of a Hamiltonian field theory. The immediate fruits of our formalism include a formula and a proof for the differential of the action functional, valid for any classical theory. In [@Ca11] and [@Ca14] and in many other works, for each classical field they consider, the authors have to come up with a different action functional and a different expression for the differential of the action functional. To do so they need to decide what the momenta of the theory need to be. In the multisymplectic formalism we develop, having one expression for the action functional and one expression for its differential that works for all classical theories, means in particular that we do not have to choose what the momenta fields should be for each physical theory we want to examine, our multisymplectic formalism identifies them for us. From there we are able to prove once and for all, that for all classical theories $\Pi(EL)$, the boundary values of the solutions the Euler-Lagrange equations, is an isotropic submanifold of $T^*\mathcal{F}_{\partial M}$ (see Sect. \[sec:general\]). Our formalism is then shown to yield geometric insight into conventional choices for boundary conditions.
This paper is devoted to systematically describing the classical ingredients in the proposed Hamiltonian framework. The description of the theory in the bulk, while following along the lines already established in the literature, also includes analysis of the role of boundary terms in the computation of the critical points of the action functional. Thus a natural relation emerges between the action functional and the canonical symplectic structure on the space of fields at the boundary. It is precisely this relations that allows a better understanding of the role of boundary conditions. The canonical 1-form on the space of fields at the boundary can be directly related to the charges of the gauge symmetries of the theory, allowing us in this way to explain why admissible boundary conditions are determined by Lagrangian submanifolds on the space of fields at the boundary.
Once the geometrical analysis of the theory has been performed, the space of quantum states of the theory would be obtained, in the best possible situations, by canonical or geometrical quantization of a reduced symplectic manifold of fields at the boundary that would describe its “true” degrees of freedom. The propagator of the theory would be obtained by quantizing a Lagrangian submanifold of the reduced phase space of the theory provided by the specification of admissible boundary conditions. The latter should preserve the fundamental symmetries of the theory, in the sense that the charges associated to them should be preserved. The resulting overall picture as descrbed in the case of Chern-Simons theory [@At90], is that the functor defining a quantum field theory is obtained by geometric quantization of the quasi-category of Lagrangian submanifolds associated to admissible boundary conditions at the boundaries of space-times and their corresponding fields. (This picture is being currently extended to the Poisson $\sigma$-model [@Co13], [@CC14].)
The level of rigor of this work is that of standard differential geometry: When dealing with finite-dimensional objects, they will be smooth differentiable manifolds, locally trivial bundles, etc., however when dealing with infinite-dimensional spaces, we will assume, as customary, that the rules of global differential calculus apply and we will use them freely without providing constructions that will lead to bona fide Banach manifolds of maps and sections. Also, the notation of variational differentials and derivatives will be used for clarity without attempting to discuss the classes of spaces of generalised functions needed to justify their use.
The paper is organized as follows: Section \[sec:general\] is devoted to summarizing the basic geometrical notions underlying the theory. The multisymplectic formalism is briefly reviewed, the action principle and a fundamental formula exhibiting the differential of the action functional of the theory is presented and proved. The role of symmetries, moment maps at the boundary and boundary conditions are elucidated. Section \[sec:presymplectic\] presents the evolution formulation of the theory near the boundary. The presymplectic picture of the system will be established and the subsequent constraints analysis is laid out. Its relation with reduction with respect to the moment map at the boundary is pointed out. Real scalar fields and the Poisson $\sigma$-model are analyzed to illustrate the theory. Finally, Section \[sec:Yang-Mills\] concentrates on the study of Yang-Mills theories on manifolds with boundary as first-order Hamiltonian field theories in the multisymplectic framework and the Hamiltonian reduced phase space of the theory is described.
The multisymplectic formalism for first order covariant Hamiltonian field theories on manifolds with boundary {#sec:general}
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The setting: the multisymplectic formalism {#sec:multisymplectic}
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The geometry of Lagrangian and Hamiltonian field theories has been examined in the literature from varying perspectives. For our purposes here we single out for summary the Hamiltonian multisymplectic description of field theories on manifolds without boundary found in [@Ca91]. Everything in this section will apply also to manifolds possessing boundaries. In the next section we will consider only manifolds having boundaries and we will extend the multisymplectic formalism to deal with Hamiltonian field theories over such manifolds.
A manifold $M$ will model the space or spacetime at each point of which the classical field under discussion assumes a value. We will therefore take $M$ to be an oriented $m = 1+d$ dimensional smooth manifold. In most situations $M$ is either Riemannian or Lorentzian and time-oriented. We will denote the metric on $M$ by $\eta$. In either case we will denote by $\mathrm{vol}_M$ the volume form defined by the metric $\eta$ on $M$. In an arbitrary local chart $x^{\mu}$ this volume form takes the form $\mathrm{vol}_M = \sqrt{|\eta|} {\mathrm{d}}x^0 \wedge {\mathrm{d}}x^1 \cdots \wedge {\mathrm{d}}x^d $. Notice however that the only structure on $M$ required to provide the kinematical setting of the theory will be a volume form $\mathrm{vol}_M$ and, unless specified otherwise, local coordinates will be chosen such that $\mathrm{vol}_M = {\mathrm{d}}x^0 \wedge {\mathrm{d}}x^1 \cdots \wedge {\mathrm{d}}x^d$.
The fundamental geometrical structure of a given theory will be provided by a fiber bundle over $M$, $\pi \colon E \to M$. Local coordinates adapted to the fibration will be denoted as $(x^\mu, u^a)$, $a= 1, \ldots, r$, where $r$ is the dimension of the standard fiber.
Let $J^1E$ denote the first jet bundle of the bundle $E$, i.e., at each point $(x,u) \in E$, the fiber of $J^1E$ consists of the set of equivalence classes of germs of sections of $\pi\colon E \rightarrow M$. If we let $\pi_1^0$ be the projection map, $\pi_1^0 \colon J^1E \rightarrow E$, then $(J^1E,\pi_1^0,E)$ is an affine bundle over $E$ modelled on the linear bundle $VE \otimes \pi^*(T^*M)$ over $E$. (See [@Sa89], [@Ca91] and [@Gr15] for details on affine geometry and the construction of the various affine bundles naturally associated to $E \to M$.)
If $(x^{\mu};u^a)$, is a bundle chart for the bundle $\pi \colon E \to M$ then we will denote by $(x^{\mu},u^a;u_{\mu}^a)$ a local chart for $J^1E$.
The affine dual of $J^1E$ is the vector bundle over $E$ whose fiber at $\xi = (x,u)$ is the linear space of affine maps $\mathrm{Aff}(J^1E_\xi, \mathbb{R})$. The vector bundle $\mathrm{Aff}(J^1E, \mathbb{R})$ possesses a natural subbundle defined by constant functions along the fibers of $J^1E \to E$, that we will denote again, abusing notation, as $\mathbb{R}$. The quotient bundle $\mathrm{Aff}(J^1E, \mathbb{R})/\mathbb{R}$ will be called the covariant phase space bundle of the theory, or the phase space for short. Notice that such bundle, denoted in what follows by $P(E)$ is the vector bundle with fiber at $\xi = (x,u) \in E$ given by $(V_uE\otimes T_x^*M)^* \cong T_xM \otimes(V_uE)^*\cong \mathrm{Lin}(V_uE,T_xM)$ and projection $\tau_1^0 \colon P(E) \to E$.
Local coordinates on $P(E)$ can be introduced as follows: Affine maps on the fibers of $J^1E$ have the form $u_{\mu}^a \mapsto \rho_0 + \rho_a^{\mu}u_{\mu}^a$ where $u_{\mu}^a$ are natural coordinates on the fiber over the point $\xi$ in $E$ with coordinates $(x^{\mu},u^a)$. Thus an affine map on each fiber over $E$ has coordinates $\rho_0, \rho^{\mu}_a$, with $\rho^\mu_a$ denoting linear coordinates on $TM \otimes VE^*$ associated to bundle coordinates $(x^\mu, u^a)$. Functions constant along the fibers are described by the numbers $\rho_0$, hence elements in the fiber of $P(E)$ have coordinates $\rho_a^{\mu}$. Thus a bundle chart for the bundle $\tau_1^0\colon P(E) \to E$ is given by $(x^\mu, u^a; \rho^\mu_a)$.
The choice of a distinguished volume form $\mathrm{vol}_M$ in $M$ allows us to identify the fibers of $P(E)$ with a subspace of $m$-forms on $E$ as follows ([@Ca91]): The map $u_{\mu}^a \mapsto \rho_a^{\mu}u_{\mu}^a$ corresponds to the $m$-form $\, \, \rho_a^\mu \, {\mathrm{d}}u^a\wedge \mathrm{vol}_{\mu}$ where vol$_{\mu}$ stands for $i_{{\partial}/{\partial x^{\mu}}}$vol$_M.$ Let ${\bigwedge}^m (E)$ denote the bundle of $m$-forms on $E$. Let ${\bigwedge}_k^m(E)$ be the subbundle of ${\bigwedge}^m (E)$ consisting of those $m$-forms which vanish when $k$ of their arguments are vertical. So in our local coordinates, elements of ${\bigwedge}_1^m(E)$, i.e., $m$-forms on $E$ that vanish when one of their arguments is vertical, commonly called semi-basic 1-forms, have the form $\rho_a^{\mu} \, {\mathrm{d}}u^a\wedge \mathrm{vol}_\mu + \rho_0 \mathrm{vol}_M$, and elements of ${\bigwedge}_0^m(E)$, i.e., basic $m$-forms, have the form $\rho_0\mathrm{vol}_M$. These bundles form a short exact sequence: $$0\rightarrow\textstyle{\bigwedge}^m_0E\hookrightarrow
\textstyle{\bigwedge}^m_1E\rightarrow P(E)\rightarrow0 \, .$$ Hence ${\bigwedge}_1^m E$ is a real line bundle over $P(E)$ and, for each point $\zeta = (x,u,\rho)\in P(E)$, the fiber is the quotient $\bigwedge_1^m (E) _\zeta/ \bigwedge_0^m (E)_\zeta$.
The bundle $\textstyle{\bigwedge}_1^m(E)$ carries a canonical $m$–form which may be defined by a generalization of the definition of the canonical 1-form on the cotangent bundle of a manifold. Let $\sigma \colon \textstyle{\bigwedge}_1^m(E) \to E$ be the canonical projection, then the canonical $m$-form $\Theta$ is defined by $$\Theta_\varpi(U_1,U_2,\ldots,U_m) = \varpi(\sigma_*U_1, \ldots, \sigma_*U_m)$$ where $\varpi\in\bigwedge^m_1(E)$ and $U_i\in T_\varpi(\bigwedge^m_1(E))$. As described above, given bundle coordinates $(x^\mu,u^a)$ for $E$ we have coordinates $(x^\mu,u^a,\rho, \rho^\mu_a)$ on $\bigwedge^m_1(E)$ adapted to them and the point $\varpi\in\bigwedge^m_1(E)$ with coordinates $(x^\mu,u^a;\rho, \rho^\mu_a )$ is the $m$-covector $\varpi = \rho^\mu_a\, {\mathrm{d}}u^a\wedge \mathrm{vol}_\mu + \rho \, \mathrm{vol}_M$. With respect to these same coordinates we have the local expression $$\Theta = \rho^\mu_a \, {\mathrm{d}}u^a \wedge \mathrm{vol}_\mu + \rho\, \mathrm{vol}_M \, ,$$ for $\Theta$, where $\rho$ and $\rho^\mu_a$ are now to be interpreted as coordinate functions.
The $(m+1)$-form $\Omega = {\mathrm{d}}\Theta $ defines a multisymplectic structure on the manifold $\bigwedge_1^m(E)$, i.e.$(\bigwedge^m_1(E),\Omega)$ is a multisymplectic manifold. There is some variation in the literature on the definition of multisymplectic manifold. For us, following [@Ca91], [@Go98] and [@Ca99], a multisymplectic manifold is a pair $(X,\Omega)$ where $X$ is a manifold of some dimension $m$ and $\Omega$ is a $d$-form on $X$, $d \geq 2$, and $\Omega$ is closed and nondegenerate. By nondegenerate we mean that if $i_v{\Omega} = 0$ then $v=0$.
We will refer to $\bigwedge^m_1E$, by $M(E)$ to emphasize that it is a multisymplectic manifold. We will denote the projection $M(E) \to E$ by $\nu$, while the projection $M(E) \to P(E)$ will be denoted by $\mu$. Thus $\nu = \tau^0_1\circ\mu$, with $\tau_1^0 \colon P(E) \to E$ the canonical projection.(See figure 1.)
A Hamiltonian $H$ on $P(E)$ is a section of $\mu$. Thus in local coordinates $$H(\rho_a^{\mu}\, {\mathrm{d}}u^a\wedge\mathrm{vol}_{\mu}) = \rho_a^{\mu}{\mathrm{d}}u^a\wedge \mathrm{vol}_{\mu}-\mathbf{H}(x^{\mu},u^a, \rho_a^{\mu}) \mathrm{vol}_M \, ,$$ where $\mathbf{H}$ is here a real-valued function also called the Hamiltonian function of the theory.
We can use the Hamiltonian section $H$ to define an $m$-form on $P(E)$ by pulling back the canonical $m$-form $\Theta$ from $M(E)$. We call the form so obtained the Hamiltonian $m$-form associated with $H$ and denote it by $\Theta_H$. Thus if we write the section defined in local coordinates $(x^\mu, u^a;\rho, \rho_a^\nu )$ as $$\label{rhoH}
\rho = - \mathbf{H}(x^{\mu}, u^a, \rho_a^\mu ) \, ,$$ then $$\label{ThetaH}
\Theta_H = \rho_a^\mu\, {\mathrm{d}}u^a \wedge \mathrm{vol}_\mu - \mathbf{H}(x^\mu,u^a, \rho_a^\mu) \, \mathrm{vol}_M \, .$$ In Eqs. and , the minus sign in front of the Hamiltonian is chosen to be in keeping with the traditional conventions in mechanics for the integrand of the action over the manifold. When the form $\Theta_H$ is pulled back to the manifold $M$, as described in Section \[sec:sections\], the integrand of the action over $M$ will have a form reminiscent of that of mechanics, with a minus sign in front of the Hamiltonian. See equation . In what follows, unless there is risk of confussion, we will use the same notation $H$ both for the section and the real-valued function $\mathbf{H}$ defined by a Hamiltonian.
The action and the variational principle {#sec:variational}
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### Sections and fields over manifolds with boundary {#sec:sections}
From here on, in addition to being an oriented smooth manifold with either a Riemannian or a Lorentzian metric, $M$ has a boundary $\partial M$. The orientation chosen on $\partial M$ is consistent with the orientation on $M$. Everything in the last section applies. The presence of boundaries, apart from being a natural ingredient in any attempt of constructing a field theory, will enable us to enlarge the use to which the multisymplectic formalism can be applied, starting with the statement and proof of Lemma 2.1.
The fields $\chi$ of the theory in the Hamiltonian formalism constitute a class of sections of the bundle $\tau_1 : P(E) \to M$. $P(E)$ is a bundle over $E$ with projection $\tau_1^0$ and it is a bundle over $M$ with projection $\tau_1 = \pi\circ \tau_1^0$. The sections that will be used to describe the classical fields in the Hamiltonian formalism are those sections $\chi\colon M \to P(E)$, i.e. $\tau_1\circ \chi = \mathrm{id}_M$, such that $\chi = P \circ \Phi$ where $\Phi \colon M \rightarrow E$ is a section of $\pi: E\rightarrow M$, i.e. $\pi \circ \Phi = \mathrm{id}_M$, and $P \colon E \to P(E)$ is a section of $\tau_1^0 \colon P(E) \rightarrow E$ i.e. $\tau_1^0 \circ P = \mathrm{id}_P$. (See Figure \[sections\]). The sections $\Phi$ will be called the configuration fields or just the configurations, and the sections $P$ the momenta fields of the theory. In other words $u^a = \Phi^a(x)$ and $\rho_a^\mu = P_a^\mu (\Phi(x))$ will provide local expression for the section $\chi = P \circ \Phi$. We will denote such a section $\chi$ by $(\Phi, P)$ to stress the iterated bundle structure of $P(E)$ and we will refer to $\chi$ as a double section.
![Bundles, sections and fields: configurations and momenta[]{data-label="sections"}](sections.pdf){width="8cm"}
We will denote by $\mathcal{F}_M$ the space of sections $\Phi$ of the bundle $\pi \colon E \rightarrow M $, that is $\Phi \in \mathcal{F}_M$, and we will denote by $\mathcal{F}_{P(E)}$ the space of double sections $\chi = (\Phi, P)$. Thus $\mathcal{F}_{P(E)}$ represents the space of fields of the theory, configurations and momenta, in the first order covariant Hamiltonian formalism.
The equations of motion of the theory will be defined by means of a variational principle, i.e., they will be related to the critical points of an action functional $S$ on $\mathcal{F}_{P(E)}$. Such action will be given simply by $$\label{action}
S(\chi ) = \int_M \chi^*\Theta_H \, ,$$ or in a more explicit notation, $$\label{action_phip}
S(\Phi,P) = \int_M \left( P_a^\mu (x) \partial_\mu \Phi^a (x) - H(x,\Phi(x), P(x)) \right) \mathrm{vol}_M ,$$ where $P_a^\mu (x)$ is shorthand for $P_a^\mu (\Phi (x))$. Of course, as is usual in the derivations of equations of motion via variational principles, we assume that the integral in Eq. is well defined. It is also assumed that the ‘differential’ symbol in equation $(2.5)$ below, defined in terms of directional derivatives, is well defined and that the same is true for any other similar integrals that will appear in this work.
\[dS \] With the above notations we obtain, $$\label{dSfirst}
\mathrm{d} S (\chi) (U) = \int_M \chi^* \left(i_{\widetilde U} {\mathrm{d}}\Theta _H \right) + \int_{\partial M} (\chi\circ i)^* \left(i_{\widetilde U} \Theta_H \right) \, ,$$ where $U$ is a vector field on $P(E)$ along the section $\chi$, $\widetilde{U}$ is any extension of $U$ to a tubular neighborhood of the range of $\chi$, and $i\colon \partial M \to M$ is the canonical embedding.
If $\chi$ is a section of $P(E)$, then we denote by $T_\chi \mathcal{F}_{P(E)}$ the tangent bundle to the space of fields at $\chi$. Tangent vectors $U$ to $\mathcal{F}_{P(E)}$ at $\chi$, i.e., $U \in T_\chi \mathcal{F}_{P(E)}$, are just vector fields $U$ on $P(E)$ along the map $\chi$ or, in other words, maps $U \colon M \to TP(E)$ such that $\tau_{P(E)}\circ U = \chi$, where $\tau_{P(E)} \colon TP(E) \to P(E)$ denotes the canonical tangent bundle projection.
Thus if $U\in T_\chi \mathcal{F}_{P(E)}$, with $U(x) \in T_{\chi (x)}P(E)$, then consider a curve $\chi_\lambda (x) = \chi (\lambda, x) \colon (-\epsilon, \epsilon) \times M \to P(E)$, such that $\chi (0, x) = \chi (x)$, and $$U(\chi(x)) =\left. \frac{\partial}{\partial \lambda}\right|_{\lambda = 0} \chi(\lambda, x) \, .$$ We can extend the vector field $U$ to a tubular neighborhood $T_\chi$ of the image of $\chi$ in $P(E)$ and we will denote it by $\tilde{U}$. Consider the local flow $\varphi_\lambda$ of $\tilde{U}$, $$\frac{{\mathrm{d}}}{{\mathrm{d}}\lambda} \varphi_\lambda = \tilde{U} \circ \varphi_\lambda \, ,$$ or in other words, let us denote the integral curves of $\tilde{U}$ by $\varphi_\lambda (\xi)$, $\xi \in T_\chi \subset P(E)$. Then if $\xi = \chi (x)$ we have, $\varphi_\lambda(\xi) = \varphi_\lambda (\chi (x))= \chi (\lambda, x) = (\chi\circ \chi_\lambda)(x)$, i.e., $\varphi_\lambda \circ \chi = \chi_\lambda$. We thus obtain, $$\begin{aligned}
{\mathrm{d}}S(\chi) (U) &=& \left.\frac{{\mathrm{d}}}{{\mathrm{d}}\lambda}\right|_{\lambda = 0} S(\chi_\lambda) = \left. \frac{{\mathrm{d}}}{{\mathrm{d}}\lambda}\right|_{\lambda = 0} \int_M \chi_\lambda^* \Theta_H = \nonumber \\
&=& \int_M \left. \chi^*\frac{\partial}{\partial\lambda}\right|_{\lambda = 0} \varphi_\lambda^*\Theta_H = \int_M \chi^* (\mathcal{L}_{\tilde{U}} \Theta_H) = \nonumber \\
&=& \int_M \chi^* d(i_{\tilde{U}}\Theta_H) + \int_M \chi^* i_{\tilde{U}} {\mathrm{d}}\Theta _H \, . \label{3rdline}\end{aligned}$$
Applying Stokes’ theorem to the first term in eq. then yields eq. .
### The cotangent bundle of fields at the boundary {#sec:cotangent_boundary}
The boundary term contribution to $\mathrm{d} S$ in eq. , that is, $\int_{\partial M} (\chi\circ i)^* \left(i_{\tilde U} \Theta_H\right)$, suggests that there is a family of fields at the boundary that play a special role. Actually, we notice that the field $\tilde{U}$ being vertical with respect to the projection $\tau_1\colon P(E) \to M$ has the local form $\tilde{U} = A^a \, \partial/\partial u^a + B_\mu^a \, \partial/\partial \rho_\mu^a$. Hence we obtain for the boundary term, $$\label{boundaryfirst}
\int_{\partial M} (\chi\circ i)^* \left(i_{\widetilde U} \Theta_H \right) = \int_{\partial M} (\chi\circ i)^* \rho_a^\mu \, A^a \, \mathrm{vol}_\mu = \int_{\partial M} i^*(P_a^\mu\, A^a \, \mathrm{vol}_\mu)$$ for $\chi = (\Phi, P)$.
We will assume now and in what follows, that there exists a collar around the boundary $U_\epsilon \cong (-\epsilon, 0] \times \partial M$. We choose local coordinates $(x^0,x^k)$, on the collar such that $x^0 = t \in (-\epsilon, 0]$ , and $x^k$, $k = 1, \ldots, d$, define local coordinates for $\partial M$. In these coordinates $\mathrm{vol}_{U_\epsilon} = {\mathrm{d}}t \wedge \mathrm{vol}_{\partial M}$ with $\mathrm{vol}_{\partial M}$ a volume form on $\partial M$. The r.h.s. of eq. becomes, $$\label{boundary_final}
\int_{\partial M} i^*(P_a^\mu\, A^a \, \mathrm{vol}_\mu) = \int_{\partial M} p_a \, A^a \, \mathrm{vol}_{\partial M} \, ,$$ where $p_a = P_a^0\circ i$ is the restriction to $\partial M$ of the zeroth component of the momenta field $P_a^\mu$ in a local coordinate chart of the previous form.
Consider the space of fields at the boundary obtained by restricting the zeroth component of sections $\chi$ to $\partial M$, that is the fields of the form (see Figure \[sections\]) $$\varphi^a = \Phi^a \circ i \, , \qquad p_a = P_a^0 \circ i \, .$$ Notice that the fields $\varphi^a$ are nothing but sections of the bundle $i^*E$, the pull-back along $i$ of the bundle $E$, while the space of fields $p_a$ can be thought of as 1-semibasic $d$-forms on $i^*E \to \partial M$. This statement is made precise in the following:
\[decomposition\] Given a collar around $\partial M$, $U_\epsilon \cong (-\epsilon, 0] \times \partial M$, and a volume form $\mathrm{vol}_{\partial M}$ on $\partial M$ such that $\mathrm{vol}_{U_{\epsilon}} = {\mathrm{d}}t \wedge \mathrm{vol}_{\partial M}$ with $t$ the normal coordinate in $U_\epsilon$, then the pull-back bundle $i^*(P(E))$ is a bundle over the pull-back bundle $i^*E$ and decomposes naturally as $i^*P(E) \cong \bigwedge_1^m(i^*E) \oplus \bigwedge_1^{m-1}(i^*E)$. If $i^*\zeta \in i^*P(E)$, we will denote by $p$ and $\beta$ the components of the previous decomposition, that is, $i^*\zeta = p + \beta$.
By definition of pull-back, the fiber over a point $x \in \partial M$ of the bundle $i^*E$, consists of all vectors in $E_x$. The pull-back bundle $i^*P(E)$ is a bundle over $i^*E$, the fiber over $(x,u) \in i^*E$ is $T_xM\otimes VE_u^*$. Using the volume form $\mathrm{vol}_M$, we identify this fiber with $\bigwedge^{m-1}(T_xM)\otimes VE_u^*$ by contracting elements $\varpi = v \otimes \alpha \in T_xM\otimes VE_u^*$ with $\mathrm{vol}_M(x)$. The collar neighborhood $U_\epsilon$ introduces a normal coordinate $t\in (-\epsilon, 0]$ such that $\mathrm{vol}_{U_{\epsilon}}
= {\mathrm{d}}t \wedge \mathrm{vol}_{\partial M}$. Notice that such decomposition depends on the choice of the collar. We obtain $\varpi = \rho_a^0 {\mathrm{d}}u^a \wedge \mathrm{vol}_{\partial M} + \rho_a^k {\mathrm{d}}u^a \wedge {\mathrm{d}}t \wedge i_{\partial /\partial x^k}\mathrm{vol}_{\partial M}$. Finally, the assignment $\varpi \mapsto (\rho_a^0 {\mathrm{d}}u^a \wedge \mathrm{vol}_{\partial M} , \rho_a^k {\mathrm{d}}u^a \wedge \wedge i_{\partial /\partial x^k}\mathrm{vol}_{\partial M} )$ provides the decomposition we are after and $p_a = \rho_a^0$, $\beta_a^k = \rho_a^k$.
If we denote by $\mathcal{F}_{\partial M}$ the space of configurations of the theory, $\varphi^a$, i.e., $\mathcal{F}_{\partial M} = \Gamma(i^*E)$, then the space of momenta of the theory $p_a$ can be identified with the space of sections of the bundle $\bigwedge_1^m(i^*E) \to i^*E$, according to Lemma \[decomposition\]. Therefore the space of fields $(\varphi^a, p_a)$ can be identified with the contangent bundle $T^*\mathcal{F}_{\partial M}$ over $\mathcal{F}_{\partial M}$ in a natural way, i.e., each field $p_a$ can be considered as the covector at $\varphi^a$ that maps the tangent vector $\delta\varphi$ to $\mathcal{F}_{\partial M}$ at $\varphi$ into the number $\langle p, \delta \varphi \rangle$ given by, $$\label{pairing_cotangent}
\langle p, \delta \varphi \rangle = \int_{\partial M} p_a(x)\delta\varphi^a (x) \, \mathrm{vol}_{\partial M} \, .$$
Notice that the tangent vector $\delta \varphi$ at $\varphi$ is a vertical vector field on $i^*E$ along $\varphi$, and the section $p$ is a 1-semibasic $m$-form on $i^*E$ (Lemma \[decomposition\]). Hence the contraction of $p$ with $\delta\varphi$ is an $(m-1)$-form along $\varphi$, and its pull-back $\varphi^*\langle p, \delta\varphi \rangle$ along $\varphi$ is an $(m-1)$-form on $\partial M$ whose integral defines the pairing above, Eq. .
Viewing the cotangent bundle $T^*\mathcal{F}_{\partial M}$ as double sections $(\varphi, p)$ of the bundle $\bigwedge_1^m(i^*E) \to i^*E \to \partial M$ described by Lemma \[decomposition\], the canonical 1-form $\alpha$ on $T^*\mathcal{F}_{\partial M}$ can be expressed as, $$\label{alpha}
\alpha_{(\varphi, p)} (U) = \int_{\partial M} p_a (x) \delta\varphi^a (x) \, \mathrm{vol}_{\partial M}$$ where $U$ is a tangent vector to $T^*\mathcal{F}_{\partial M}$ at $(\varphi, p)$, that is, a vector field on the space of 1-semibasic forms on $i^*E$ along the section $(\varphi^a, p_a)$, and therefore of the form $U = \delta\varphi^a \, \partial /\partial u^a + \delta p_a \, \partial /\partial \rho_a$.
Finally, notice that the pull-back to the boundary map $i^*$, defines a natural map from the space of fields in the bulk, $\mathcal{F}_{P(E)}$, into the phase space of fields at the boundary $T^*\mathcal{F}_{\partial M}$. Such map will be denoted by $\Pi$ in what follows, that is, $$\Pi \colon \mathcal{F}_{P(E)}\to T^*\mathcal{F}_{\partial M} \, , \qquad \Pi(\Phi, P) = (\varphi, p) , \, \quad \varphi = \Phi\circ i, \, p_a = P_a^0\circ i \, .$$
With the notations above, by comparing the expression for the boundary term given by eq. , and the expression for the canonical 1-form $\alpha$, eq. , we obtain, $$\int_{\partial M} (\chi\circ i)^* \left(i_{\tilde U} \Theta_H\right) = (\Pi^*\alpha)_\chi (U) \, .$$ In words, the boundary term in eq. is just the pull-back of the canonical 1-form $\alpha$ at the boundary along the projection map $\Pi$.
In what follows it will be customary to use the variational derivative notation when dealing with spaces of fields. For instance, if $F(\varphi,p)$ is a differentiable function defined on $T^*\mathcal{F}_{\partial M}$ we will denote by $\delta F / \delta \varphi^a$ and $\delta F / \delta p_a$ functions (provided that they exist) such that $$\label{dF}
{\mathrm{d}}F_{(\varphi,p)}(\delta \varphi^a, \delta p_a) = \int_{\partial M} \left( \frac{\delta F}{\delta \varphi^a} \delta \varphi^a + \frac{\delta F}{\delta p_a} \delta p_a \right) \mathrm{vol}_{\partial M} \, ,$$ with $U = (\delta \varphi^a, \delta p_a)$ a tangent vector at $(\varphi,p)$. We also use an extended Einstein’s summation convention such that integral signs will be omitted when dealing with variational differentials. For instance, $$\delta F = \frac{\delta F}{\delta \varphi^a} \delta \varphi^a + \frac{\delta F}{\delta p_a} \delta p_a \, ,$$ will be the notation that will replace ${\mathrm{d}}F$ in Eq. . Also in this vein we will write, $$\alpha = p_a \, \delta \varphi^a \, ,$$ and the canonical symplectic structure $\omega_{\partial M} = -{\mathrm{d}}\alpha$ on $T^*\mathcal{F}_{\partial M}$ will be written as, $$\omega_{\partial M} = \delta \varphi^a \wedge \delta p_a \, ,$$ by which we mean $$\omega_{\partial M} ((\delta_1\varphi^a, \delta_1p_a), (\delta_2\varphi^a, \delta_2p_a)) = \int_{\partial M} \left(
\delta_1\varphi^a(x) \delta_2 p_a(x) - \delta_2\varphi^a (x) \delta_1p_a(x) \right) \mathrm{vol}_{\partial M} \, ,$$ where $(\delta_1\varphi^a, \delta_1p_a), (\delta_2\varphi^a, \delta_2p_a)$ are two tangent vectors at $(\varphi, p)$.
### Euler-Lagrange’s equations and Hamilton’s equations
We now examine the contribution from the first term in ${\mathrm{d}}S$, eq. . Notice that such a term can be thought of as a 1-form on the space of fields on the bulk, $\mathcal{F}_{P(E)}$. We will call it the Euler-Lagrange 1-form and denote it by $\mathrm{EL}$, thus with the notation of Lemma \[dS \], $$\mathrm{EL}_\chi (U) = \int_M \chi^* \left(i_{\tilde U} {\mathrm{d}}\Theta _H \right) \, .$$ A double section $\chi = (\Phi, P)$ of $P(E) \to E \to M$ will be said to satisfy the Euler-Lagrange equations determined by the first-order Hamiltonian field theory defined by $H$, if $\mathrm{EL}_\chi = 0$, that is, if $\chi$ is a zero of the Euler-Lagrange 1-form $\mathrm{EL}$ on $\mathcal{F}_{P(E)}$. Notice that this is equivalent to $$\label{formEL}
\chi^*(i_{\tilde{U}} \\d \Theta _H)=0 \, ,$$ for all vector fields $\tilde{U}$ on a tubular neighborhood of the image of $\chi$ in $P(E)$. The set of all such solutions of Euler-Lagrange equations will be denoted by $\mathcal{EL}_M$ or just $\mathcal{EL}$ for short. In local coordinates $x^\mu$ such that the volume element takes the form $\mathrm{vol_M}= dx^0 \wedge\cdots \wedge dx^d$, and for natural local coordinates $(x^\mu,u^a,\rho^\mu_a)$ on $P(E)$, using eqs. , , we have, $$\begin{aligned}
i_{\partial/\partial \rho^\mu_a}{\mathrm{d}}\Theta _H &=& -\frac{\partial
H}{\partial \rho^\mu_a} d^mx + {\mathrm{d}}u^a\wedge {\mathrm{d}}^{m-1}x_\mu \\
i_{\partial/\partial u^a} {\mathrm{d}}\Theta _H &=& -\frac{\partial
H}{\partial u^a} d^mx - d\rho^\mu_a\wedge {\mathrm{d}}^{m-1}x_\mu.\end{aligned}$$ Applying Eq. to these last two equations we obtain the Hamilton equations for the field in the bulk:
$$\label{hamilton_equations}
\frac{\partial u^a}{\partial x^\mu} = \frac{\partial H}{\partial \rho^\mu_a}\, ; \qquad
\frac{\partial \rho^\mu_a}{\partial x^\mu} = -\frac{\partial H}{\partial u^a} \, ,$$
where a summation on $\mu$ is understood in the last equation. Note that had we not changed to normal coordinates on $M$, the volume form would not have the above simple form and therefore there would be related extra terms in the previous expressions and in Eqs. .
These Hamilton equations are often described as being covariant. This term must be treated with caution in this context. Clearly, by writing the equations in the invariant form $\chi^*(i_{\tilde{U}}{\mathrm{d}}\Theta _H)=0$ we have shown that they are in a sense covariant. However, it is important to remember that the function $H$ is, in general, only locally defined; in other words, there is in general no true ‘Hamiltonian function’, and the local representative $H$ transforms in a non-trivial way under coordinate transformations. When $M(E)$ is a trivial bundle over $P(E)$, so that there is a predetermined global section, then the Hamiltonian section may be represented by a global function and no problem arises. This occurs for instance when $E$ is trivial over $M$. In general, however, there is no preferred section of $M(E)$ over $P(E)$ to relate the Hamiltonian section to, and in order to write the Hamilton equations in manifestly covariant form one must introduce a connection. (See [@Ca91] for a more detailed discussion and [@Gr12] for a general treatment of these issues.)
The fundamental formula {#sec:fundamental}
-----------------------
Thus we have obtained the formula that relates the differential of the action with a 1-form on a space of fields on the bulk manifold and a 1-form on a space of fields at the boundary. $$\label{fundamental}
\mathrm{d} S_\chi = \mathrm{EL}_\chi + \Pi^* \alpha_\chi \, , \qquad \chi \in \mathcal{F}_{P(E)} \, .$$ In the previous equation $\mathrm{EL}_\chi$ denotes the Euler-Lagrange 1-form on the space of fields $\chi = (\Phi, P)$ with local expression (using variational derivatives): $$\label{ELform}
\mathrm{EL}_\chi = \left( \frac{\partial \Phi^a}{\partial x^\mu} - \frac{\partial H}{\partial P^\mu_a} \right) \delta P_a^\mu - \left( \frac{\partial P^\mu_a}{\partial x^\mu} + \frac{\partial H}{\partial \Phi^a} \right) \delta \Phi^a \, ,$$ or, more explicitly: $$\mathrm{EL}_\chi (\delta \Phi, \delta P) = \int_M \left[ \left( \frac{\partial \Phi^a}{\partial x^\mu} - \frac{\partial H}{\partial P^\mu_a} \right) \delta P_a^\mu - \left( \frac{\partial P^\mu_a}{\partial x^\mu} + \frac{\partial H}{\partial \Phi^a} \right) \delta \Phi^a \right] \, \mathrm{vol}_M \, .$$
In what follows we will denote by $(P(E), \Theta_H)$ the covariant Hamiltonian field theory with bundle structure $\pi \colon E \to M$ defined over the $m$-dimensional manifold with boundary $M$, Hamiltonian function $H$ and canonical $m$-form $\Theta_H$.
We will say that the action $S$ is regular if the set of solutions of Euler-Lagrange equations $\mathcal{EL}_M$ is a submanifold of $\mathcal{F}_{P(E)}$. Thus we will also assume when needed that the action $S$ is regular (even though this must be proved case by case) and that the projection $\Pi(\mathcal{EL})$ to the space of fields at the boundary $T^*\mathcal{F}_{\partial M}$ is a smooth manifold too.
This has the immediate implication that the projection of $\mathcal{EL}$ to the boundary $\partial M$ is an isotropic submanifold:
\[isotropicEL\] Let $(P(E), \Theta_H)$ be a first order Hamiltonian field theory on the manifold $M$ with boundary, with regular action $S$ and such that $\Pi(\mathcal{EL})\subset T^*\mathcal{F}_{\partial M}$ is a smooth submanifold. Then $\Pi (\mathcal{EL}) \subset T^*\mathcal{F}_{\partial M}$ is an isotropic submanifold.
Along the submanifold $\mathcal{EL} \subset T^*\mathcal{F}_{\partial M}$ we have, $${\mathrm{d}}S \mid_{\mathcal{EL}} = \Pi^*\alpha\mid_{\mathcal{EL}} .$$ Therefore $\mathrm{d} (\Pi^*\alpha) = \mathrm{d}^2 S = 0$ along $\mathcal{EL}$, and $\mathrm{d} (\Pi^*\alpha) = \Pi^* \mathrm{d} \alpha$ along $\mathcal{EL}.$ But $\Pi$ being a submersion then implies that $\mathrm{d}\alpha = 0$ along $\Pi(\mathcal{EL})$.
In many cases $\Pi(\mathcal{EL})$ is not only isotropic but Lagrangian. We will come back to the analysis of this in later sections.
Symmetries and the algebra of currents
--------------------------------------
Without attempting a comprehensive description of the theory of symmetry for covariant Hamiltonian field theories, we will describe some basic elements needed in what follows (see details in [@Ca91]). Recall from Sect. \[sec:multisymplectic\], $(M(E),\Omega)$ is a multisymplectic manifold with $(m+1)$-dimensional multisymplectic form $\Omega = d \Theta $, where dim $M = m$. Canonical transformations in the multisymplectic framework for Hamiltonian field theories are diffeomorphisms $\Psi \colon M(E) \to M(E)$ such that $\Psi^*\Omega = \Omega$. Notice that if $\Psi$ is a diffeomorphism such that $\Psi^*\Theta = \Theta$, then $\Psi$ is a canonical transformation.
A distinguished class of canonical transformations is provided by those transformations $\Psi$ induced by diffeomorphisms $\psi_E \colon E \to E$, i.e., $\Psi (\varpi) = (\psi_E^{-1})^*\varpi$, $\varpi \in M(E)$. If the diffeomorphism $\psi_E$ is a bundle isomorphism, there will exist another diffeomorphism $\psi_M \colon M \to M$ such that $\pi\circ \psi_E = \psi_M \circ \pi$. Under such circumstances it is clear that the induced map $ (\psi_E^{-1})^* \colon \bigwedge^m(E) \to \bigwedge^m(E)$ preserves both $\bigwedge_1^m(E)$ and $\bigwedge_0^1(E)$, thus the map $\Psi = (\psi_E^{-1})^*\colon M(E) \to M(E)$ induces a natural map $\psi_* \colon P(E) \to P(E)$ such that $\mu \circ (\psi_E^{-1})^* = \psi_* \circ \mu$. Canonical transformations induced from bundle isomorphisms will be called covariant canonical maps.
Given a one-parameter group of canonical transformations $\Psi_t$, its infinitesimal generator $U$ satisfies $$\mathcal{L}_U \Omega = 0 \, .$$ Vector fields $U$ on $M(E)$ satisfying the previous condition will be called (locally) Hamiltonian vector fields. Locally Hamiltonian vector fields $U$ for which there exists a $(m-1)$-form $f$ on $M(E)$ (we are assuming that $\Omega$ is a $(m+1)$-form) such that $$i_U \Omega = {\mathrm{d}}f \ ,$$ will be called, in analogy with mechanical systems, (globally) Hamiltonian vector fields. The class $\mathbf{f} = \{ f + \beta \mid {\mathrm{d}}\beta = 0, \, \beta \in \Omega^{m-1}(M(E)) \}$ determined by the $(m-1)$-form $f$ is called the Hamiltonian form of the vector field $U$ and such a vector field will be denoted as $U_{\mathbf{f}}$.
The Lie bracket of vector fields induces a Lie algebra structure on the space of Hamiltonian vector fields that we denote as $\mathrm{Ham}(M(E),\Omega)$. Notice that Hamiltonian vector fields whose flows $\Psi_t$ are defined by covariant canonical transformations are globally Hamiltonian because $\mathcal{L}_U \Theta = 0$, and therefore $i_U {\mathrm{d}}\Theta = - {\mathrm{d}}i_U\Theta$. The Hamiltonian form associated to $U$ is the class containing the $(m-1)$-form $f = i_U\Theta$.
The space of Hamiltonian forms, denoted in what follows by $\mathcal{H}(M(E))$, carries a canonical bracket defined by $$\{ \mathbf{f}, \mathbf{f}' \} = i_{U_f} i_{U_{f'}} \Omega + Z^{m-1}(M(E)) \, ,$$ where $Z^{m-1}(M(E))$ denotes the space of closed $(m-1)$-forms on $M(E)$. The various spaces introduced so far are related by the short exact sequence [@Ca91]: $$0 \to H^{m-1} (M(E)) \to \mathcal{H}(M(E)) \to \mathrm{Ham}(M(E), \Omega) \to 0 \, .$$
Let $G$ be a Lie group acting on $E$ by bundle isomorphisms and $\psi_g \colon E \to E$, the diffeomorphism defined by the group element $g \in G$. This action induces an action on the multisymplectic manifold $(M(E), \Omega)$ by canonical transformations. Given an element $\xi \in \mathfrak{g}$, where now and in what follows $\mathfrak{g}$ denotes the Lie algebra of the Lie group $G$, we will denote by $\xi_{M(E)}$ and $\xi_E$ the corresponding vector fields defined by the previous actions on $M(E)$ and $E$, respectively. The vector fields $\xi_{M(E)}$ are Hamiltonian with Hamiltonian forms $\mathbf{J}_\xi$, that is, $$\label{ixiJxi}
i_{\xi_{M(E)}} \Omega = {\mathrm{d}}J_\xi \, ,$$ with $J_\xi = i_{\xi_{M(E)}}\Theta$. It is easy to check that $$\{Ê\mathbf{J}_\xi, \mathbf{J}_\zeta \} = \mathbf{J}_{[\xi , \zeta]} + c(\xi, \zeta) \,$$ where $c(\xi, \zeta)$ is a cohomology class of order $m-1$. The bilinear map $c(\cdot, \cdot)$ defines an element in $H^2(\mathfrak{g}, H^{m-1}(M(E)))$ (see [@Ca91]). In what follows we will assume that the group action is such that the cohomology class $c$ vanishes. Such actions are called strongly Hamiltonian (or just Hamiltonian, for short).
So far our discussion has not involved a particular theory, that is, a Hamiltonian $H$. Let $(P(E), \Theta_H)$ be a covariant Hamiltonian field theory and $G$ a Lie group acting on $\mathcal{F}_{P(E)}$. Among all possible actions of groups on the space of double sections $\mathcal{F}_{P(E)}$ those that arise from an action on $P(E)$ by covariant canonical transformations are of particular significance. Let $G$ be a group acting on $E$ by bundle isomorphisms. Let $\psi_*(g)$ denote the covariant diffeomorphism on $P(E)$ defined by the group element $g$. Then the transformed section $\chi^g$ is given by $\chi^g (x) = \psi_*(g)(\chi(\psi_M(g^{-1}) x))$ where $\psi_M(g)$ is the diffeomorphism on $M$ defined by the action of the group. We will often consider only bundle automorphisms over the identity, in which case $\chi^g (x) = \psi_*(g)(\chi (x))$. Such bundle isomorphisms will be called gauge transformations and the corresponding group of all gauge transformations will be called the gauge group of the theory and denoted by $\mathcal{G}(E)$, or just $\mathcal{G}$ for short, in what follows.
The group $G$ will be said to be a symmetry of the theory if $S(\chi^g) = S(\chi)$ for all $\chi \in \mathcal{F}_{P(E)}$, $g\in G$. Notice that, in general, an action of $G$ on $M(E)$ by bundle isomorphisms will leave $\Theta$ invariant and will pass to the quotient space $P(E)$, however it doesn’t have to preserve $\Theta_H$. Hence, it is obvious that a group $G$ acting on $P(E)$ by covariant transformations will be a symmetry group of the Hamiltonian field theory defined by $H$ iff $g^*\Theta_H = \Theta_H + \beta_g$, where now, for the ease of notation, we indicate the diffeomorphism $\psi_*(g)$ simply by $g$, and $\beta_g$ is a closed $m$-form on $M$. In what follows we will assume that the group $G$ acts on $E$ and its induced action on $P(E)$ preserves the $m$-form $\Theta_H$, that is $\beta_g = 0$ for all $g$.
Because the action of the group $G$ preserves the $m$-form $\Theta_H$, the group acts by canonical transformation on the manifold $(P(E), d \Theta_H)$ with Hamiltonian forms $\mathbf{J}_\xi$ given by (the equivalence class determined by the $m$-forms): $$J_\xi = i_{\xi_{P(E)}}\Theta_H \, .$$
\[Noether\] Let $G$ be a Lie group acting on $E$ which is a symmetry group of the Hamiltonian field theory $(P(E), \Theta_H)$ and such that it preserves the $m$-form $\Theta_H$. If $\chi \in \mathcal{EL}$ is a solution of the Euler-Lagrange equations of the theory, then the $(m-1)$-form $\chi^*J_{\xi}$ on $M$ is closed.
Because $\chi$ is a solution of Euler-Lagrange equations, recalling eq. we have $$0 = \chi^*(i_{\xi_{P(E)}}\Omega_H) = \chi^*{\mathrm{d}}J_\xi = {\mathrm{d}}(\chi^*J_\xi) \, .$$
The de Rham cohomology classes determined by the closed $(m-1)$-forms $\chi^*J_{\xi}$ on $M$ will be called currents and denoted by $\mathbf{J}_\xi[\chi]$. Using the Poisson bracket $\{ \cdot, \cdot \}$ defined on the space of Hamiltonian forms $\mathcal{H}(P(E))$ we define a Lie bracket in the space of currents $\mathbf{J}_\xi[\chi] \in H^{m-1}(M)$ by $$\{ \mathbf{J}_\xi[\chi], \mathbf{J}_\zeta[\chi] \} = \chi^* \{ \mathbf{J}_\xi, \mathbf{J}_\zeta \} = \mathbf{J}_{[\xi, \zeta]}[\chi] \, .$$ By Stokes’ theorem, the $(m-1)$-forms $i^*(\mathbf{J}_\xi[\chi])$ on $\partial M$ satisfy $$\label{boundary_conservation}
\int_{\partial M} i^*\mathbf{J}_\xi[\chi] = 0 \, .$$ We will refer to the quantity $Q \colon \mathcal{F}_{P(E)} \to \mathfrak{g}^*$, where $\mathfrak{g}^*$ denotes the dual of the Lie algebra $\mathfrak{g}$, defined by $$\label{chargeQ}
\langle Q(\chi), \xi \rangle = \int_{\partial M} i^* \mathbf{J}_\xi[\chi] \, , \qquad \forall \xi \in \mathfrak{g} \, ,$$ as the charge defined by the symmetry group. Notice that the pairing $\langle \cdot, \cdot \rangle$ on the left hand side of Eq. is the natural pairing between $\mathfrak{g}$ and $\mathfrak{g}^*$. As a consequence of Noether’s theorem we get $Q\mid_{\mathcal{EL}} = 0$.
### The moment map at the boundary
Suppose that there is an action of a Lie group $G$ on the bundle $E$ that leaves invariant the restriction of the bundle $E$ to the boundary, that is, the transformations $\Psi_g$ defined by the elements of the group $g\in G$ restrict to the bundles $i^*(P(E))$ and $E_{\partial M} := i^*E$ (see Figure \[sections\]). We will denote such restriction as $\Psi_g\mid_{\partial M} = g_{\partial M}$.
Two elements $g, g' \in G$ will induce the same transformation on the bundle $E_{\partial M}$ if there exists an element $h$ such that $g' = gh$ and $h_{\partial M} = \mathrm{id}_{E_{\partial M}}$. If we consider now the group $\mathcal{G}$ of all gauge transformations, then the set of group elements that restrict to the identity at the boundary is a normal subgroup of $\mathcal{G}$ which we will denote by $\mathcal{G}_0$. The induced action of $\mathcal{G}$ at the boundary is the action of the group $\mathcal {G}_{\partial M} = \mathcal{G}/\mathcal{G}_0$ which is the group of gauge transformations of the bundle $E_{\partial M} = i^*E$.
In particular the group $\mathcal{G}$ induces an action on $\mathcal{F}_{\partial M}$ by $$g\cdot \varphi (x) = \psi_E(g)(\Phi (g^{-1}x)) = g_{\partial M} (\varphi (g^{-1} x)) \, , \qquad \forall x \in \partial M,\, g \in \mathcal{G}\, ,$$ and similarly for the momenta field $p$.
\[boundary\_moment\] Let $\mathcal{G}_{\partial M}$ denote the gauge group at the boundary, that is, the group whose elements are the transformations induced at the boundary by gauge transformations of $E$. Then the action of $\mathcal{G}_{\partial M}$ in the space of fields at the boundary is strongly Hamiltonian with moment map $$\mathcal{J} \colon T^*\mathcal{F}_{\partial M} \to \mathfrak{g}_{\partial M}^*$$ given by, $$\langle \mathcal{J}(\varphi, p), \xi \rangle = \langle Q(\chi), \xi \rangle \, \qquad \forall \xi \in \mathfrak{g}_{\partial M} \, ,$$ where $\Pi(\chi) = (\varphi,p)$, and $\mathfrak{g}_{\partial M}$, $\mathfrak{g}^*_{\partial M}$ denote respectively the Lie algebra and the dual of the Lie algebra of the group $\mathcal{G}_{\partial M}$. In other words, the projection map $\Pi$ composed with the moment map at the boundary $\mathcal{J}$ is the charge $Q$ of the symmetry group.
The action of the group $\mathcal{G}_{\partial M}$ on $T^*\mathcal{F}_{\partial M}$ is by cotangent liftings, thus its moment map $\mathcal{J}$ takes the particularly simple form, $$\langle \mathcal{J}(\varphi,p) , \xi \rangle = \langle p, \xi_{\mathcal{F}_{\partial M}}(\varphi) \rangle \, ,$$ where $\xi_{\mathcal{F}_{\partial M}}$ denotes, consistently, the infinitesimal generator defined by the action of $\mathcal{G}_{\partial M}$ on $\mathcal{F}_{\partial M}$. Such generator, because the action is by gauge transformations, i.e., bundle isomorphisms over the identity, has the explicit expression: $$\xi_{\mathcal{F}_{\partial M}} = \xi \circ \varphi \frac{\delta}{\delta \varphi} \, ,$$ where $\xi$ is the infinitesimal generator of the action of $\mathcal{G}_{\partial M}$ on $E_{\partial M}$. Notice that the Lie algebra ·$\mathfrak{g}$ of the group of gauge transformations is precisely the algebra of vertical vector fields on $E$ (similarly for $\mathfrak{g}_{\partial M}$ and $E_{\partial M}$). Hence $\xi \in \mathfrak{g}_{\partial M}$ is just a vertical vector field and the infinitesimal generator $\xi_{\mathcal{F}_{\partial M}}$ at $\varphi$, which is just a tangent vector to $\mathcal{F}_{\partial M}$ at $\varphi$, is the vector field along $\varphi$ given by the composition $\xi \circ \varphi$.
However because the action of $\mathcal{G}_{\partial M}$ on $E_{\partial M}$ is exactly the action of $\mathcal{G}$ on $E\mid_{\partial M}$, $\xi$ can be considered as an element on $\mathfrak{g}$ and (recalling the definition of the pairing in $T^*\mathcal{F}_{\partial M}$, eq. , and the discussion at the end of Sect. \[sec:cotangent\_boundary\] on the conventions with variational derivatives) we get, $$\begin{aligned}
\langle p, \xi_{\mathcal{F}_{\partial M}}(\varphi) \rangle &=& \int_{\partial M} p_a\, \xi^a (\varphi (x))\, \mathrm{vol}_{\partial M} \label{momentum_Q1} \\ &=& \int_{\partial M} i^*\left(\chi^* \left(i_{\xi_{P(E)}}\Theta_H \right) \right) = \int_{\partial M} i^*\mathbf{J}_\xi [\chi] = \langle Q(\chi), \xi \rangle \, , \label{momentum_Q2}\end{aligned}$$ with $\Pi (\chi) = (\varphi, p)$. In the previous computations we have used that $i_{\xi_{P(E)}}\Theta_H = \rho_a^\mu \,\xi^a (x,\rho^a) \, \mathrm{vol}_\mu$, therefore $\chi^*\left(i_{\xi_{P(E)}}\Theta_H \right)= P_a(\Phi(x))^\mu \, \xi^a (\Phi(x)) \, \mathrm{vol}_\mu$ and thus $$i^*\left(\chi^*\left(i_{\xi_{P(E)}}\Theta_H \right) \right)= p^a(\varphi(x)) \, \xi^a(\varphi(x)) \, \mathrm{vol}_{\partial M} \, .$$ Notice the particularly simple form that the currents take in this situation $J_\xi[\chi] = p_a \xi^a(\varphi)$.
Thus Noether’s Theorem (which implies that $Q|_{\mathcal{EL}} = 0$) together with Prop. \[boundary\_moment\], imply that for any $\chi = (\Phi,P) \in \mathcal{EL}$, then $(\varphi,p) \in \mathcal{J}^{-1}(\mathbf{0})$, with $(\varphi, p) = \Pi (\chi)$.\
The main, and arguably the most significant, example of symmetries is provided by theories such that the symmetry group is the full group of automorphisms of the bundle $\pi \colon E \to M$, and in particular its normal subgroup of bundle automorphisms over the identity map, i.e., diffeomorphisms $\psi_E\colon E \to E$ preserving the structure of the bundle and such that $\pi \circ \psi_E = \psi_E$. As indicated before, such group will be called the gauge group of the theory (or, better the group of gauge transformations of the theory) and we will denote it by $\mathcal{G}(E)$ (or just $\mathcal{G}$ if there is no risk of confusion). In such case, eqs. -, imply the following:
\[Qalpha\] With the above notations, for the group of gauge transformations we obtain, $$Q = \Pi^*\alpha \, ,$$ where $\alpha$ is the canonical 1-form on $T^*\mathcal{F}_{\partial M}$, in the sense that for any $\xi\in \mathfrak{g}$ and $\chi\in J^1\mathcal{F}^*$, $$\langle Q(\chi), \xi \rangle = \alpha_{\Pi(\chi)}(\xi_{\mathcal{F}_{\partial M}}) = \Pi^*\alpha_{\chi}(\xi_{P(E)}) \, .$$
Boundary conditions
-------------------
Because of the boundary term $\Pi^*\alpha$ arising in the computation of the critical points of the action $S$, the propagator of the corresponding quantum theory will be affected by such contributions and the theory could fail to be unitary [@As15]. One way to avoid this problem is by selecting a subspace of the space of fields $\mathcal{F}_{P(E)}$ such that $\Pi^*\alpha$ will vanish identically when restricted to it (see for instance an analysis of this situation in Quantum Mechanics in [@As05].)
Moreover, we would like to choose a maximal subspace with this property. Then these two requirements will amount to choosing boundary conditions determined by a maximal submanifold $\mathcal{L} \subset T^*\mathcal{F}_{\partial M}$ such that $\alpha\mid_{\mathcal{L}} = 0$, that is, $\mathcal{L}$ is a special Lagrangian submanifold of the cotangent bundle $T^*\mathcal{F}_{\partial M}$.
In general we will consider not just a single boundary condition but a family of them defining a Lagrangian fibration of $T^*\mathcal{F}_{\partial M}$. An example of such a choice is the Lagrangian fibration $\mathcal{L}$ corresponding to the vertical fibration of $T^*\mathcal{F}_{\partial M}$. For $\varphi \in \mathcal{F}_{\partial M}$, the subspace of fields defined by the leaf $\mathcal{L}_{\varphi}$, $\varphi \in \mathcal{F}_{\partial M}$ is just the subspace of fields $\chi = (\Phi, P)$ such that $\Phi\mid_{\partial M} = \varphi$.
Another argument justifying the use of special Lagrangian submanifolds of $T^*\mathcal{F}_{\partial M}$ as boundary conditions, relies just on the structure of the classical theory and its symmetries and not on its eventual quantization. Recall from Cor. \[Qalpha\], that if a theory $(P(E), \Theta_H)$ has the group of gauge transformations $\mathcal{G}$ of the bundle $E$ as a symmetry group, then $Q = \Pi^*\alpha$. Therefore the admissible fields of the theory - not necessarily solutions of Euler-Lagrange equations - are those such that the charge $Q$ is preserved along the boundary, that is, those that lie on a special Lagrangian submanifold $\mathcal{L} \subset T^*\mathcal{F}_{\partial M}$.
We will say that a (classical field) theory is Dirichlet if, for any $\varphi \in \mathcal{F}_{\partial M}$, there exists a unique solution $\chi = (\Phi, P)$ of the Euler-Lagrange equations, i.e., $\chi \in \mathcal{EL}$ such that $\Phi |_{\partial M} = \varphi$.
\[Lagrangian\_sub\] Let $S$ be a regular action defined by a first order Hamiltonian field theory $(P(E), \Theta_H)$ then, if the theory is Dirichlet, $\Pi(\mathcal{EL})$ is a Lagrangian submanifold of $T^*\mathcal{F}_{\partial M}$.
Recall the discussion in Sec. \[sec:fundamental\]. If the action is regular, i.e. if the solutions of the Euler-Lagrange equations $\mathcal{EL}$ define a submanifold of $\mathcal{F}_{P(E)}$, then from the fundamental relation eq. , we get Prop. \[isotropicEL\], that $\Pi(\mathcal{EL})$ is an isotropic submanifold of $T^*\mathcal{F}_{\partial M}$.
Let the functional $W$ denote the composition $W = S\circ D$ where $S : \mathcal{F}_{P(E)} \rightarrow \mathbb{R}$ is the action of our theory defined by $H$, i.e. $S(\chi)= \int_M \chi^*\Theta_H \,$ and $D : \mathcal{F}_{\partial M} \rightarrow \mathcal{F}_{P(E)}$ is the map that assigns to any boundary data $\varphi$ the unique solution $(\Phi, P)$ of the Euler-Lagrange equations such that $\Phi |_{\partial M} =\varphi$. Thus $$W(\varphi)= \int_M \chi^*\Theta_H = S (\chi) \, ,$$ for any $\varphi \in \mathcal{F}_{\partial M}$. By Eq. and since $D(\phi)= (\Phi,P)=\chi \in \mathcal{EL}$ it follows that $\mathrm{EL}_{\chi}=0$. A simple computation then shows that $$\mathrm{d}W (\varphi) (\delta \varphi) = p_a\delta \varphi^a \, ,$$ where $p_a = P_a^0\mid_{\partial M}$. Thus the graph of the 1-form $\mathrm{d} W =\Pi(\mathcal{EL})$, i.e., $W$ is a generating function for $\Pi(\mathcal{EL})$ and therefore $\Pi(\mathcal{EL})$ is a Lagrangian submanifold of $\mathcal{T^*F}_{\partial M}$.
The Dirichlet condition can be weakened and a corresponding proof for a natural extension of Theorem \[Lagrangian\_sub\] can be provided. See our work on Hamiltonian dynamics [@Ib16].
The presymplectic formalism at the boundary {#sec:presymplectic}
===========================================
The evolution picture near the boundary {#sec:dynamical_eqs}
---------------------------------------
We discuss in what follows the evolution picture of the system near the boundary. As discussed in Section \[sec:cotangent\_boundary\], we assume that there exists a collar $U_\epsilon \cong (-\epsilon , 0]\times\partial M$ of the boundary $\partial M$ with adapted coordinates $(t; x^1,\ldots, x^d)$, where $t = x^0$ and where $x^i$, $i = 1,\ldots, x^d$ define a local chart in $\partial M$. The normal coordinate $t$ can be used as an evolution parameter in the collar. We assume again that the volume form in the collar is of the form $\mathrm{vol}_{U_{\epsilon}} = dt\wedge \mathrm{vol}_{\partial M}$.
If $M$ happens to be a globally hyperbolic space-time $M \cong [t_0,t_1]\times \Sigma$ where $\Sigma$ is a Cauchy surface, $[t_0,t_1] \subset \mathbb{R}$ denotes a finite interval in the real line, and the metric has the form $-dt^2 + g_{\partial M}$ where $g_{\partial M}$ is a fixed Riemannian metric on $\partial M$, then $t$ represents a time evolution parameter throughout the manifold and the volume element has the form $\mathrm{vol}_M = dt \wedge \mathrm{vol}_{\partial M}$. Here, however, all we need to assume is that our manifold has a collar at the boundary as described above.
Restricting the action $S$ of the theory to fields defined on $U_\epsilon$, i.e., sections of the pull-back of the bundles $E$ and $P(E)$ to $U_\epsilon$, we obtain, $$\label{Sepsilon}
S_\epsilon (\chi ) = \int_{U_\epsilon} \chi^*\Theta_H = \int_{-\epsilon}^0 {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M} \left[ P_a^0 \partial_0 \Phi^a + P_a^k\partial_k \Phi^a - H(\Phi^a, P_a^0, P_a^k) \right]\, .$$ Defining the fields at the boundary as discussed in Lemma \[decomposition\], $$\varphi^a = \Phi^a|_{\partial M} \, , \qquad p_a = P_a^0|_{\partial M} \, , \qquad \beta_a^{k} = P_a^{k}|_{\partial M} \, ,$$ we can rewrite as $$S_\epsilon (\chi) = \int_{-\epsilon}^0 {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M}[ p_a \dot{\varphi}^a + \beta_a^k\partial_k\varphi^a -H(\varphi^a,p_a,\beta_a^k)] \, .$$ Letting $\langle p, \dot{\varphi} \rangle = \int_{\partial M} p_a \dot{\varphi}^a \,\, \mathrm{vol}_{\partial M}$ denote, as in , the natural pairing and, similarly, $$\langle \beta, \mathrm{d}_{\partial M}\varphi \rangle = \int_{\partial M} \beta_a^k \partial_k \varphi^a \, \mathrm{vol}_{\partial M},$$ we can define a density function $\mathcal{L}$ as, $$\label{densityL}
\mathcal{L}(\varphi,\dot{\varphi},p,\dot{p},\beta,\dot{\beta})=\langle p,\dot{\varphi}\rangle + \langle\beta, \mathrm{d}_{\partial M}\varphi \rangle - \int_{\partial M} H(\varphi^a,p_a,\beta_a^k) \, \mathrm{vol}_{\partial M} \, ,$$ and then $$S_\epsilon (\chi) = \int_{-\epsilon}^0 {\mathrm{d}}t \, \, {\mathcal{L}}(\varphi,\dot{\varphi},p,\dot{p},\beta,\dot{\beta}) \, .$$
Notice again that because of the existence of the collar $U_\epsilon$ near the boundary and the assumed form of $\mathrm{vol}_ {U_\epsilon}$, the elements in the bundle $i^*P(E)$ have the form $\rho_a^0 {\mathrm{d}}u^a \wedge \mathrm{vol}_{\partial M} + \rho_a^k {\mathrm{d}}u^a \wedge {\mathrm{d}}t \wedge i_{\partial /\partial x^k}\mathrm{vol}_{\partial M}$ and, as discussed in Lemma \[decomposition\], the bundle $i^*P(E)$ over $i^*E$ is isomorphic to the product $\bigwedge_1^m(i^*E) \times B$, where $B = \bigwedge_1^{m-1}(i^*E)$. The space of double sections $(\varphi,p)$ of the bundle $\bigwedge_1^m(i^*E) \to i^*E \to \partial M$ correspond to the cotangent bundle $T^*\mathcal{F}_{\partial M}$ and the double sections $(\varphi, \beta)$ of the bundle $B \to i^*E \to \partial M$ correspond to a new space of fields at the boundary denoted by $\mathcal{B}$.
We will introduce now the total space of fields at the boundary $\mathcal{M}$ which is the space of double sections of the iterated bundle $i^*P(E) \to i^*E \to \partial M$. Following the previous remarks it is obvious that $\mathcal{M}$ has the form, $$\mathcal{M}= \mathcal{T}^*\mathcal{F}_{\partial M} \times_{\mathcal{F}_{\partial M}} \mathcal{B} = \{(\varphi, p, \beta)\} \, .$$
Thus the density function $\mathcal{L}$, Eq. , is defined on the tangent space $T\mathcal{M}$ to the total space of fields at the boundary and could be called accordingly the boundary Lagrangian of the theory.
Consider the action $ A = \int_{-\epsilon}^0 \mathcal{L} \,\, {\mathrm{d}}t$ defined on the space of curves $\sigma\colon ( -\epsilon, 0] \to \mathcal{M}$. If we compute $\mathrm{d}A$ we obtain a bulk term, that is, an integral on $(-\epsilon, 0]$, and a term evaluted at $\partial [-\epsilon,0] = \{-\epsilon, 0\}$. Setting the bulk term equal to zero, we obtain the Euler-Lagrange equations of this system considered as a Lagrangian system on the space $\mathcal{M}$ with Lagrangian function $\mathcal{L}$, $$\label{Euler-Lagrange Equations 1}
\frac{{\mathrm{d}}}{{\mathrm{d}}t} \frac{{\delta}{\mathcal{L}}}{{\delta}{\dot{{\varphi}^a}}}= \frac{{\delta}{\mathcal{L}}}{{\delta}{{\varphi}^a}} \, ,$$ which becomes, $$\label{pidot}
\dot{p_a}= -{\partial}_k{\beta}_a^k - \frac{{\partial}H}{{\partial}{\varphi}^a} \, .$$ Similarly, we get for the fields $p$ and $\beta$: $$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\frac{{\delta}{\mathcal{L}}}{{\delta}{\dot{p}_a}}=\frac{{\delta}{\mathcal{L}}}{{\delta}{p_a}} \, , \quad
\frac{{\mathrm{d}}}{{\mathrm{d}}t}\frac{{\delta}{\mathcal{L}}}{{\delta}{\dot{{\beta}}_a^k}}=\frac{{\delta}{\mathcal{L}}}{{\delta}{{\beta}_a^k}}$$ that become respectively, $$\label{phidot}
\dot{\varphi}_a = \frac{\partial H}{\partial p_a} \, ,$$ and, the constraint equation: $$\label{dconstraint}
{\mathrm{d}}_{\partial M}{\varphi}-\frac{\partial H}{{\partial}{{\beta}_a^k}}=0 \, .$$ Thus, Euler-Lagrange equations in a collar $U_\epsilon$ near the boundary, can be understood as a system of evolution equations on $T^*\mathcal{F}_{\partial M}$ depending on the variables $\beta_a^k$, together with a constraint condition on the extended space $\mathcal{M}$. The analysis of these equations, Eqs. , and , is best understood in a presymplectic framework.
The presymplectic picture at the boundary and constraints analysis
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We will introduce now a presymplectic framework on $\mathcal{M}$ that will be helpful in the study of Eqs. -.
Let $\varrho :\mathcal{M} \longrightarrow \mathcal{T}^*\mathcal{F}_{\partial M}$ denote the canonical projection $\varrho(\varphi,p,\beta)=(\varphi,p)$. (See Figure \[diagram\].) Let $\Omega$ denote the pull-back of the canonical symplectic form ${\omega}_{\partial M}$ on $\mathcal{T}^*\mathcal{F}_{\partial M}$ to $\mathcal{M}$, i.e., let $\Omega=\varrho^*\omega_{\partial M}$. Note that the form $\Omega$ is closed but degenerate, that is, it defines a presymplectic structure on $\mathcal{M}$. An easy computation shows that the characteristic distribution $\mathcal{K}$ of $\Omega$, is given by $$\mathcal{K} = \ker\Omega= \mathrm{span} \left\{ \frac{\delta}{{\delta}{\beta}_a^k} \right\} \, .$$
Let us consider the function defined on $\mathcal{M}$, $$\mathcal{H}(\varphi,p,\beta)= -\langle \beta, d_{\partial M}\varphi \rangle + \int_{\partial M} H({\varphi}^a, p_a, {\beta}_a^k)\, \mathrm{vol}_{\partial M} \, .$$
![The space of fields at the boundary $\mathcal{M}$ and its relevant structures.[]{data-label="diagram"}](diagram.pdf){width="10cm"}
We will refer to $\mathcal{H}$ as the boundary Hamiltonian of the theory. Thus $\mathcal{L}$ can be rewritten as $$\mathcal{L}(\varphi, \dot{\varphi},p,\dot{p},\beta, \dot{\beta})=\langle p,\dot{\varphi} \rangle - \mathcal{H}(\varphi,p,\beta)$$ and $$\label{Lagrangian with Boundary Hamiltonian}
S_{\epsilon}(\varphi, p, \beta) =\int_{-\epsilon}^0 [\langle p,\dot{\varphi} \rangle - \mathcal{H}(\varphi,p,\beta)] {\mathrm{d}}t \, ,$$ and therefore the Euler-Lagrange equations , and can be written as $$\label{Hamilton's Evolution Equations}
\dot{\varphi}^a = \frac{\delta \mathcal{H}}{\delta p_a}\, ,\qquad
\dot{p}_a = -\frac{\delta \mathcal{H}}{\delta{\varphi}^a} \, ,$$ and $$\label{Hamilton's Constraint Equation}
0= \frac{\delta\mathcal{H}}{\delta{\beta}_a^k} \, .$$
Now it is easy to prove the following:
\[presymplectic\_equation\] The solutions to the equations of motion defined by the Lagrangian $\mathcal{L}$ over a collar $U_\epsilon$ at the boundary, $\epsilon$ small enough, are in one-to-one correspondence with the integral curves of the presymplectic system $(\mathcal{M},\Omega,\mathcal{H})$, i.e., with the integral curves of the vector field $\Gamma$ on $\mathcal{M}$ satisfying $$\label{presymplectic_equation1}
i_\Gamma \Omega = {\mathrm{d}}\mathcal{H} \, .$$
Let $\Gamma = A^a\frac{\delta}{\delta{\varphi}^a} + B^a\frac{\delta}{{\delta}p^a} + C^a\frac{\delta}{{\delta}{\beta}_a^k}$ be a vector field on $\mathcal{M}$ (notice that we are using an extension of the functional derivative notation introduced in Section \[sec:cotangent\_boundary\] on the space of fields $\mathcal{M}$). Then because $\Omega = \delta{\varphi}^a \wedge \delta p_a$, we get from $i_{\Gamma}\Omega= {\mathrm{d}}\mathcal{H}$ that, $$A^a = \frac{{\delta}{\mathcal{H}}}{{\delta} p_a},\qquad
B^a = -\frac{{\delta}{\mathcal{H}}}{{\delta}{\varphi}^a} \, ,\qquad 0 = \frac{{\delta}{\mathcal{H}}}{{\delta}{\beta}_a^k} \, .$$
Thus, $\Gamma$ satisfies Eq. iff $$\dot{\varphi}^a =\frac{{\delta}{\mathcal{H}}}{{\delta}p_a}, \qquad
\dot{p}_a = -\frac{{\delta}{\mathcal{H}}}{{\delta}{\varphi}^a} \, , \quad
\mathrm{and} \quad
0= \frac{{\delta}{\mathcal{H}}}{{\delta}{\beta}_a^k} \, .$$
Let us denote by $\mathcal{C}$ the submanifold of the space of fields $\mathcal{M} =T^*\mathcal{F}_{\partial M} \times \mathcal{B}$ defined by eq. $(3.9)$. It is clear that the restriction of the solutions of the Euler-Lagrange equations on $M$ to the boundary ${\partial}M$, are contained in $\mathcal{C}$; i.e., $ \Pi (\mathcal{EL})\subset\mathcal{C}.$
Given initial data $\varphi, p$ and fixing $\beta$, existence and uniqueness theorems for initial value problems when applied to the initial value problem above, would show the existence of solutions for small intervals of time, i.e., in a collar near the boundary.
However, the constraint condition given by eq. , satisfied automatically by critical points of $S_\epsilon$ on $U_\epsilon$, must be satisfied along the integral curves of the system, that is, for all $t$ in the neighborhood $U_\epsilon$ of $\partial M$. This implies that consistency conditions on the evolution must be imposed. Such consistency conditions are just that the constraint condition eq. , is preserved under the evolution defined by eqs. . This is the typical situation that we will find in the analysis of dynamical problems with constraints and that we are going to summarily analyze in what follows.
### The Presymplectic Constraints Algorithm (PCA)
Let $i$ denote the canonical immersion $\mathcal{C}=\{(\varphi,p,\beta)| \frac{\delta {\mathcal{H}}}{\delta{\beta}}=0\}\to \mathcal{M}$ and consider the pull-back of $\Omega$ to $\mathcal{C}$, i.e., $\Omega_1 = i^*\Omega$. Clearly then, $\ker \Omega_1 = \ker \varrho_* \cap T\mathcal{C}$. But $\mathcal{C}$ is defined as the zeros of the function $\delta \mathcal{H}/\delta \beta$. Therefore if $\delta^2 \mathcal{H}/\delta^2\beta$ is nondegenerate (notice that the operator $\delta^2 \mathcal{H}/\delta
{\beta}_a^i{\delta}{\beta}_b^j$ becomes the matrix $\partial^2 H /\partial \beta_a^i \partial\beta_b^j$), by an appropriate extension of the Implicit Function Theorem, we could solve $\beta$ as a function of $\varphi$ and $p$. In such case, locally, $\mathcal{C}$ would be the graph of a function $F\colon T^*\mathcal{F}_{\partial M} \to \mathcal{B}$, say $\beta = F(\varphi, p)$. This is precisely the situation we will see in the simple example of scalar fields in the next section. Collecting the above yields:
The submanifold $(\mathcal{C}, \Omega_1)$ of $(\mathcal{M},\Omega,\mathcal{H})$ is symplectic iff $H$ is regular, i.e., $\partial^2 H /\partial \beta_a^i \partial\beta_b^j$ is non-degenerate. In such case the projection $\varrho$ restricted to $\mathcal{C}$, which we denote by $\varrho_C$, is a local symplectic diffeomorphism and therefore $\varrho_C^*\omega_{\partial M} = \Omega_1$.
When the situation is not as described above, and $\beta$ is not a function of $\varphi$ and $p$, then $(\mathcal{C},\Omega_1)$ is indeed a presymplectic submanifold of $\mathcal{M}$ and $i_{\Gamma}{\Omega}=d\mathcal{H}$ will not hold necessarily at every point in $\mathcal{C}$. In this case we would apply Gotay’s Presymplectic Constraints Algorithm \[Go78\], to obtain the maximal submanifold of $\mathcal{C}$ for which $i_{\Gamma}{\Omega}=d\mathcal{H}$ is consistent and that can be summarized as follows.
Consider a presymplectic system $(\mathcal{M}, \Omega, \mathcal{H})$ where $\mathcal{M}= T^*\mathcal{F}_{\partial M}\times\mathcal{B}$ and, $\Omega$ and $\mathcal{H}$ are as defined above. Let $\mathcal{M}_0 = \mathcal{M}$, $\Omega_0 = \Omega$, $\mathcal{K}_0 = \ker \Omega_0$, and $\mathcal{H}_0 = \mathcal{H}$. We define the primary constraint submanifold $\mathcal{M}_1$ as the submanifold defined by the consistency condition for the equation $i_\Gamma \Omega_0 = {\mathrm{d}}\mathcal{H}_0$, i.e., $$\mathcal{M}_1 = \{ \chi \in \mathcal{M}_0 \mid \langle Z_0(\chi) , {\mathrm{d}}\mathcal{H}_0(\chi) \rangle = 0, \, \, \forall Z_0 \in \mathcal{K}_0 \} \, .$$ Thus $\mathcal{M}_1= \mathcal{C}$. Denote by $i_1 \colon \mathcal{M}_1 \to \mathcal{M}_0$ the canonical immersion. Let $\Omega_1 = i_1^*\Omega_0$, $\mathcal{K}_1 = \ker \Omega_1$, and $\mathcal{H}_1 = i_1^*\mathcal{H}_0$. We now define recursively the $(k+1)$-th constraint submanifold as the consistency condition for the equation $i_\Gamma \Omega_k = {\mathrm{d}}\mathcal{H}_k$, that is, $$\mathcal{M}_{k+1} = \{ \chi \in \mathcal{M}_k \mid \langle Z_k(\chi) , {\mathrm{d}}\mathcal{H}_k(\chi) \rangle = 0, \, \, \forall Z_k \in \mathcal{K}_k \} \, \qquad k \geq 1 \, ,$$ and $i_{k+1}\colon \mathcal{M}_{k+1} \to \mathcal{M}_k$ is the canonical embbeding (assuming that $\mathcal{M}_{l+1}$ is a regular submanifold of $\mathcal{M}_k$), and $\Omega_{k+1} = i_{k+1}^*\Omega_k$, $\mathcal{K}_{k+1} = \ker \Omega_{k+1}$ and $\mathcal{H}_{k+1} = i_{k+1}^*\mathcal{H}_k$.
The algorithm stabilizes if there is an integer $r> 0$ such that $\mathcal{M}_{r} = \mathcal{M}_{r+1}$. We refer to this $\mathcal{M}_r$ as the final constraints submanifold and we denote it by $\mathcal{M}_\infty$. Letting $i_\infty\colon \mathcal{M}_\infty \to \mathcal{M}_0$ denote the canonical immersion, we define, $$\Omega_\infty = i_\infty^*\Omega_0, \qquad \mathcal{K}_\infty = \ker \Omega_\infty\, , \qquad \mathcal{H}_\infty = i_\infty^*\mathcal{H}_0 \, .$$ Notice that the presymplectic system $(\mathcal{M}_\infty, \Omega_\infty, \mathcal{H}_\infty )$ is always consistent, that is, the dynamical equations defined by $i_\Gamma \Omega_\infty = d\mathcal{H}_\infty$ will always have solutions on $\mathcal{M}_\infty$. The solutions will not be unique if $\mathcal{K}_\infty \neq 0$, hence the integrable distribution $\mathcal{K}_\infty$ will be called the “gauge” distribution of the system, and its sections (that will necessarily close a Lie algebra), the “gauge” algebra of the system.\
In the particular theories considered in this work we found that $\mathcal{M}_{\infty}=\mathcal{M}_1=\mathcal{C}$ and we do not needed to go beyond the first step of the algorithm to obtain the final constraints submanifold.\
The quotient space $\mathcal{R} = \mathcal{M}_\infty / \mathcal{K}_\infty$, provided it is a smooth manifold, inherits a canonical symplectic structure $\omega_\infty$ such that $\pi_\infty^*\omega_\infty = \Omega_\infty$, where $\pi_\infty \colon \mathcal{M}_\infty \to \mathcal{R}$ is the canonical projection. We will refer to it as the reduced phase space of the theory. Notice that the Hamiltonian $\mathcal{H}_\infty$ also passes to the quotient and we will denote its projection by $h_\infty$ i.e., $\pi_\infty^* h_\infty = \mathcal{H}_\infty$.
Thus the Hamiltonian system $(\mathcal{R}, \omega_\infty, h_\infty)$ will provide the canonical picture of the theory at the boundary and its quantization will describe the states and dynamics of the theory with respect to observers sitting at the boundary $\partial M$.
Of course all the previous constructions depend on the boundary $\partial M$ of the manifold $M$. For instance, if we assume that $M$ is a globally hyperbolic space-time of the form $M \cong [t_0,t_1] \times \Sigma$, then $\partial M = \{t_0\}\times \Sigma \cup \{t_1\}\times \Sigma$. But if we use a different Cauchy surface $\Sigma'$, the boundary of our space-time will vary and we will get a new reduced phase space $(\mathcal{R}', \omega', h')$ for the theory. However in this case it is easy to show that there is a canonical symplectic diffeomorphism $S\colon \mathcal{R} \to \mathcal{R}'$ such that $h = S^*h'$. (Recall that in such case there will exist a canonical diffeomorphism $\Sigma \to \Sigma'$ that will eventually induce the map $S$ above.)
Recall that $\Pi (\mathcal{EL}) \subset \mathcal{C}$. We easily show then that after the reduction to $\mathcal{R}$, the reduced submanifold of boundary values of Euler-Lagrange solutions of the theory, $\widetilde{\Pi}(\mathcal{E}\mathcal{L})$, is an isotropic submanifold, now of the reduced phase space.
\[reduction\_theorem\] The reduction $\widetilde{\Pi}(\mathcal{EL})$ of the submanifold of Euler-Lagrange fields of the theory is an isotropic submanifold of the reduced phase space $\mathcal{R}$ of the theory.
It is clear that $\Pi(\mathcal{EL}) \subset \Pi(\mathcal{EL}_\epsilon) \subset \mathcal{M}_{\infty}$ where $\mathcal{EL}_\epsilon = \mathcal{EL}_{U_\epsilon}$ are the critical points of the action $S_\epsilon$, i.e., solutions of the Euler-Lagrange equations of the theory on $U_\epsilon$.
The reduction $\widetilde{\Pi}(\mathcal{EL}) = \Pi(\mathcal{EL})/ (\mathcal{K}_\infty\cap T\, \Pi (\mathcal{EL}))$ of the isotropic submanifold $ \Pi(\mathcal{EL})$ to the reduced phase space $\mathcal{R} = \mathcal{M}_\infty / \mathcal{K}_\infty$ is isotropic because $\pi_\infty^*\omega_\infty = \Omega_\infty$, hence $\pi_\infty^* (\omega_\infty\mid_{\widetilde{\Pi}(\mathcal{EL})}) = (\pi_\infty^* \omega_\infty)\mid_{\Pi(\mathcal{EL})} = \varrho^*\mathrm{d} \alpha \mid_{\Pi(\mathcal{EL})} = 0$.
Reduction at the boundary and gauge symmetries {#reduction_gauge}
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If our theory $(P(E), \Theta_H)$ has $\mathcal{G}$ as a covariant symmetry group, then because of Noether’s theorem, Thm. \[Noether\], and Eq. , we have that $ J_\xi[\chi]$, with $\Pi (\chi) = (\varphi, p)$ a closed $(m-1)$-form. Hence $\int_{\partial M} i^*J_\xi[\chi] = 0$, and so $$\langle \mathcal{J}(\varphi, p), \xi \rangle = \int_{\partial M} i^*J_\xi[\chi] = 0 \, .$$ Then $\mathcal{J}(\Pi (\chi )) = 0$, and therefore, $$\Pi (\mathcal{EL}) \subset \mathcal{J}^{-1} (\mathbf{0}) \, .$$
There is a natural reduction of the theory at the boundary defined by the covariant symmetry $\mathcal{G}$ for the following reason: Provided that the value $\mathbf{0}$ of the moment map $\mathcal{J}$ is weakly regular, the submanifold $\mathcal{J}^{-1}(\mathbf{0}) \subset T^*\mathcal{F}_{\partial M}$ is a coisotropic submanifold and the characteristic distribution $\ker i_0^*\omega_{\partial M}$ of the pull-back of the canonical symplectic form on $T^*\mathcal{F}_{\partial M}$ to it, is the distribution defined by the orbits of the group $\mathcal{G}_{\partial M}$. From Prop. \[boundary\_moment\], $\mathcal{J}$ is the moment map of the canonical lifting of the action of the group $\mathcal{G}_{\partial M}$ on $\mathcal{F}_{\partial M}$.
From the above and by Thm \[presymplectic\_equation\], Lemma $3.4$ follows easily.
$$\mathcal{J}^{-1}(\mathbf{0}) \subset \varrho (\mathcal{M}_\infty)\, .$$
If $\mathcal{G}$ is a symmetry group of the Hamiltonian $H$ of the theory, then it is clear that $\mathcal{G}_{\partial M}$ is a symmetry group of the function $\mathcal{H}$, with the canonical action of $\mathcal{G}_{\partial M}$ on the total space of fields at the boundary $\mathcal{M}$.
Then if $\zeta \in \mathcal{M}_\infty$, there exists $\Gamma$ at $\zeta$ such that $i_\Gamma \Omega_\infty = {\mathrm{d}}\mathcal{H}_\infty$ and the integral curve $\gamma$ of $\Gamma$ passing through $\zeta$ lies in $\mathcal{M}_\infty$. But $\varrho(\gamma) \subset \mathcal{J}^{-1}(\mathbf{0})$, because it is the projection of an integral curve of a solution of Euler-Lagrange equations in $U_\epsilon$. But because the Hamiltonian $\mathcal{H}$ is invariant, the trajectory must lie in a level surface of the moment map $\mathcal{J}$. Hence $\mathcal{J}^{-1}(\mathbf{0}) \subset \varrho (\mathcal{M}_\infty)$.
Because, $\mathcal{R} = \mathcal{M}_\infty /\mathcal{K}_\infty$ and $\ker \varrho_* \cap T\mathcal{M}_\infty \subset \mathcal{K}_\infty$, we get that $\mathcal{M}_\infty /\mathcal{K}_\infty \cong \varrho (\mathcal{M}_\infty) / \varrho_*(\mathcal{K}_\infty)$. Now if we are in the situation where $\varrho(\mathcal{M}_\infty) = \mathcal{J}^{-1}(\mathbf{0})$, then $\mathcal{R} \cong \varrho (\mathcal{M}_\infty) / \varrho_*(\mathcal{K}_\infty) = \mathcal{J}^{-1}(\mathbf{0}) / \ker \omega_{\partial M}\mid_{\mathcal{J}^{-1}(\mathbf{0})}$. Hence because of the standard Marsden-Weinstein reduction theorem the reduced phase space of the theory is obtained simply as, $$\label{MWR}
\mathcal{R} \cong \mathcal{J}^{-1}(\mathbf{0})/ \mathcal{G}_{\partial M} \, .$$
A simple example: the scalar field {#scalar}
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We will consider the simple example of a real scalar field on a globally hyperbolic space-time $(M, \eta)$ of dimension $m = 1 +d$ with boundary $\partial M$ a Cauchy surface and hence $M \cong (-\infty, a] \times \partial M$. The configuration fields of the system are sections of the (real) line bundle $\pi \colon E \to M$, where $\pi$ is projection onto the first factor. Bundle coordinates will have the form $(x^\mu,u)$, $\mu=0,1,...,d$.
If the bundle $E \to M$ were trivial, $E \cong M \times \mathbb{R}$, the first jet bundle $J^1E$ would be the affine bundle $J^1E \cong T^*M \times \mathbb{R} \rightarrow E$ with bundle coordinates $(x^\mu,u; u_\mu)$, $\mu = 0,1,...,d$. The covariant phase space $P(E)$, in such case, would be isomorphic to $TM\times \mathbb{R}$ with bundle coordinates $(x^{\mu},u; \rho^\mu)$.
As explained in Section \[sec:multisymplectic\], by using the volume form $\mathrm{vol}_M = \sqrt{| \eta |} \, {\mathrm{d}}^mx$ defined by the metric $\eta$ (in arbitrary local coordinates $x^\mu$), elements in $P(E)$ can be identified with semi-basic $m$-forms on $E$, $w \in\bigwedge^m_1(E)$, $w = \rho^\mu {\mathrm{d}}u \wedge \mathrm{vol}_\mu^d + \rho_0 \mathrm{vol}_M$, $\mathrm{vol}_\mu^d = i_{\partial/\partial x^\mu} \mathrm{vol}_M$, after we mod out basic $m$-forms, $\rho_0 \mathrm{vol}_M$.
The space of fields in the bulk, $\mathcal{F}_{P(E)}= \{ \chi = ( \Phi , P)\}$ consists of double sections of the iterated bundle $P(E) \to E \to M$, $\Phi \colon M \to E$, $u = \Phi (x)$, and $P\colon E \to P(E)$, $\rho = P(u)$ that, in the instance of a trivial bundle $E$, can be described as maps $\Phi \colon M \to \mathbb{R}$, the configuration fields, and $(m-1)$-forms, $P = P^{\mu}(x)\mathrm{vol}^d_\mu$, the momenta fields.
The Hamiltonian $H$ of the theory determines a section of the projection $M(E) \to P(E)$ by fixing the variable $\rho_0$ above, i.e., $\rho_0 = -H(x^\mu,u, \rho^\mu)$. One standard choice for $H$ in such case is: $$H(x^\mu,u; \rho^\mu) = \frac{1}{2}\eta_{\mu\nu}\rho^\mu \rho^\nu + V(u) \, ,$$ with $V(u)$ a smooth function on $\mathbb{R}$. The particular instance of $V(u) = m^2u^2$ gives us the Klein-Gordon system.
The canonical $m$-form $\Theta$ in $\bigwedge^m_1(E)$ can be pulled back to $P(E)$ along $H$ and takes the form, $$\Theta_H = \rho^{\mu} du \wedge \mathrm{vol}^d_{\mu} - H (u) \mathrm{vol}_M \, .$$
With the above choice for $H$, the action functional of the theory becomes: $$S(\Phi,P) = \int_M \left[ P^\mu(x) \partial_\mu\Phi (x) - \frac{1}{2} \eta_{\mu \nu} P^\mu P^\nu - V(\Phi) \right] \sqrt{|\eta |}\, {\mathrm{d}}^mx \, .$$
The space of boundary fields $T^*\mathcal{F}_{\partial M} = \{ (\varphi , p) \}$ is given by $\varphi = \Phi\mid_{\partial M}$, $p = P^0\mid_{\partial M}$. Computing the differential of the action we get, $$\begin{aligned}
\mathrm{d} S_{(\Phi, P)} ( \delta \Phi, \delta P ) &=& \int_M [ \delta P^\mu ( \partial_\mu\Phi -
\eta_{\mu \nu} P^\nu) + \delta \Phi (-\frac{1}{\sqrt{|\eta|}} \partial_\mu (P^\mu\sqrt{|\eta |}) \\ &-& V'(\Phi))] \sqrt{|\eta |}\, {\mathrm{d}}^mx + \int_{\partial M}p \delta \varphi \, \mathrm{vol}_{\partial M} \, ,\end{aligned}$$ and the Euler-Lagrange equations of the theory are given by, $$\label{scalar_EL}
\frac{1}{\sqrt{|\eta |}}{\partial}_{\mu}(P^{\mu}\sqrt{|\eta |})+ V'(\Phi) =0 \, , \qquad {\partial}_\mu \Phi - \eta_{\mu\nu}P^\nu = 0 \,
.$$
From the second of the Euler equations we get, $P^\nu = \eta^{\mu\nu} \partial_\mu \Phi$, and substituting into the first we get $$\label{Laplace}
\frac{1}{\sqrt{|\eta |}} \partial_\mu(\sqrt{|\eta|} \eta^{\mu\nu} \partial_\nu\Phi)= -V'(\Phi) \, .$$ The first term is the Laplace-Beltrami operator of the metric $\eta$, i.e., the d’Alembertian in the case of the Minkowski metric.\
Note that had we instead chosen normal local coordinates on $M$, the volume element in such charts would take the form $$\mathrm{vol}_M = {\mathrm{d}}x^0 \wedge {\mathrm{d}}x^1 \wedge \cdots \wedge {\mathrm{d}}x^d$$ and then equations $(3.13)$ would just be Hamilton’s equations:
$$\partial_\mu P^\mu = -\frac{\partial H}{ \partial \Phi } \, \qquad \partial_\mu\Phi = \frac{\partial H}{\partial P^\mu} \, .$$
### The evolution picture near the boundary {#sec:scalar_boundary .unnumbered}
We consider a collar around the boundary $U_\epsilon = (- \epsilon,0] \times \partial M$ with coordinates $t=x^0$ and $x^i$, $i = 1, \ldots, d$. We assume that $\eta=-{\mathrm{d}}t^2 + \eta_{0i}(x){\mathrm{d}}t \otimes {\mathrm{d}}x^i + g_{ij}(x){\mathrm{d}}x^i \otimes {\mathrm{d}}x^j$ and $g = g_{ij}(x) {\mathrm{d}}x^i\otimes {\mathrm{d}}x^j$ defines a Riemannian metric on $\partial M$. Writing again the action functional S restricted to fields $\Phi , P$ defined on $U_{\epsilon}$, we have, $$S_\epsilon (\Phi , P) = \int_{-\epsilon}^0 {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M} \sqrt{|\eta |} (P^0 \partial_0 \Phi + P^i \partial_i \Phi - \frac{1}{2} \eta_{\mu\nu} P^\mu P^\nu - V(\Phi)) \, .$$ Consider the fields at the boundary $\varphi$ and $p$ defined before and $\beta^i = P^i\mid_{\partial M}$. Also, let $\Delta = \sqrt{|\eta|}/\sqrt{|g|}$. Then $\sqrt{|\eta |} d^mx = \Delta \, {\mathrm{d}}t \wedge \mathrm{vol}_{\partial M}$.
Therefore we can write, $$\begin{aligned}
S_{\epsilon}({\Phi},P) &=& \int_{-\epsilon}^0 {\mathrm{d}}t \int_{{\partial}M}\, \mathrm{vol}_{\partial M} \, \Delta\, [ p \dot{\varphi} + \beta^i \partial_i \phi + \frac{1}{2} p^2 - \eta_{0i} p \beta^i - \frac{1}{2} g_{ij} \beta^i \beta^j - V(\phi)]
\\ &=& \int_{-\epsilon}^0 {\mathrm{d}}t\, \, \left[ \langle p, \dot{\varphi} \rangle - \mathcal{H}(\varphi,p,\beta ) \right]\, \end{aligned}$$ where $$\label{Deltaproduct}
\langle p, \dot{\varphi} \rangle = \int_{\partial M} \, p(x) \dot{\varphi}(x) \Delta\, \mathrm{vol}_{\partial M} \, ,$$ denotes the scalar product on functions on $\partial M$ defined by the volume $\Delta\, \mathrm{vol}_{\partial M}$, and $\mathcal{H} \colon \mathcal{M} \to \mathbb{R}$ denotes the Hamiltonian function induced from the Hamiltonian $H$ of the theory, $$\mathcal{H} (\varphi,p,\beta) = -\langle \beta, d_{\partial M} \varphi \rangle - \frac{1}{2} \langle p, p \rangle + \langle p, \tilde{\beta} \rangle + \frac{1}{2} \langle \beta,\beta \rangle + \int_{\partial M} V(\varphi) \Delta\, \mathrm{vol}_{\partial M} \, ,$$ with $\tilde{\beta} = \eta(d/{\mathrm{d}}t,\beta) = \eta_{i0}\beta^i$, $\langle p, p \rangle$ and $\langle p, \tilde{\beta} \rangle$ defined as in eq. . The product $\langle \beta,\beta \rangle$ denotes the scalar product of vector fields defined by the metric $g$, i.e., $$\langle \beta,\beta \rangle = \int_{\partial M} g_{ij} \beta^i(x) \beta^j(x) \Delta\, \mathrm{vol}_{\partial M} \, ,$$ and $\langle \beta, d_{\partial M} \varphi \rangle$ is the natural pairing between vector fields and 1-forms on $\partial M$, that is $$\langle \beta,d_{\partial M} \rangle = \int_{\partial M} \beta^i(x) \partial_i \varphi(x) \Delta\, \mathrm{vol}_{\partial M} \, .$$
As in Section \[sec:dynamical\_eqs\] we denote the space of all fields at the boundary, the dynamical fields $\varphi$, $p$ and the fields $\beta^i$, as $\mathcal{M} = T^*\mathcal{F}_{\partial M} \times \mathcal{B} = \{ (\varphi, p; \beta ) \}$ and Hamilton’s equations for $\mathcal{H}$ are given by, $$\dot{\varphi} = \frac{\delta \mathcal{H}}{\delta p} = -p + \tilde{\beta} \, , \qquad
\dot{p} = - \frac{\delta \mathcal{H}}{\delta \varphi} = -V'(\varphi)-\mathrm{div\,}\beta \, ,$$ together with the constraint equation obtained from the variation of $S_\epsilon$ with respect to $\beta$, $$0 = \frac{{\delta}\mathcal{H}}{{\delta}{\beta}^i} = -\partial_i \phi + \eta_{0i}p + g_{ij} \beta^j \, .$$ Thus we get, $$\begin{aligned}
\dot{p} &=& - \mathrm{div\,}\beta - V'(\varphi) \\
\dot{\varphi} &=& - p + \tilde{\beta} \end{aligned}$$ and the constraints equations, $$\label{scalar_constraint}
- {\mathrm{d}}_{\partial M} \varphi + p^\flat + \beta^\flat = 0 \, ,$$ where $\beta^\flat = g(\beta, \cdot)$ is the 1-form associated to the vector $\beta$ by the metric $g$, and $p^\flat$ is the 1-form associated to the vector $p\, \partial / \partial t$.
Let $\mathcal{C}= \{ (\varphi,p,\beta)\in\mathcal{M} \mid \delta \mathcal{H}/ \delta \beta = 0 \}$, the submanifold of $\mathcal{M}$ defined by the constraints , and let $\varrho \colon \mathcal{M} \to T^*\mathcal{F}_{\partial M}$ denote the canonical projection. We can solve for $\beta^i$ as a function of $\varphi$ and $p$ in the constraint equation , obtaining $\beta^j = g^{ij}(\partial_i \varphi - g_{0i}p)$ or, more intrisically, $$\beta = {\mathrm{d}}_{\partial M}\varphi^\sharp - p \frac{\partial}{\partial t} \, ,$$ where $d_{\partial M}\varphi^\sharp$ is the vector field associated to the 1-form $d_{\partial M}\varphi$ by means of the metric $g$. Thus the restriction of $\varrho$ to $\mathcal{C}$ is a diffeomorphism onto $T^*\mathcal{F}_{\partial M}$. If we denote by $\Omega$ the pull-back $\varrho^*\omega_{\partial M}$ to $\mathcal{M}$ of the canonical symplectic form on $T^*\mathcal{F}_{\partial M}$ and by $\Omega_{\mathcal{C}}$ its restriction to the submanifold $\mathcal{C}$, the restriction of the canonical projection $\varrho \colon \mathcal{M} \to T^*\mathcal{F}_{\partial M}$ to $\mathcal{C}$ provides a symplectic diffeomorphism $(\mathcal{C}, \Omega_\mathcal{C}) \cong (T^*\mathcal{F}_{\partial M}, \omega_{\partial M})$.
Moreover, the projection $\Pi (\mathcal{EL})$ of the space of solutions to the Euler-Lagrange equations to the boundary, defines, wherever it is a smooth submanifold, an isotropic submanifold of $T^*\mathcal{F}_{{\partial}M}$, as shown in Thm. \[Lagrangian\_sub\]. $\Pi (\mathcal{EL})$ is not necessarily a Lagrangian submanifold because in general the Dirichlet problem defined by boundary conditions $(\varphi, p)$ for Eq. doesn’t have a solution. The situation is different in the Euclidean case, i.e., if $(M, \eta)$ is a Riemannian manifold, the Laplace-Beltrame operator would be elliptic and the Dirichlet problem would always have a unique solution. In such case the space $\Pi (\mathcal{EL})$ would certainly be a Lagrangian submanifold of $T^*\mathcal{F}_{\partial M}$.
Another example: The Poisson $\sigma$-model
-------------------------------------------
We will illustrate the previous ideas as they apply to the case of the Poisson $\sigma$-model. We note that the Poisson $\sigma$-model (P$\sigma$M for short) was analyzed in depth by A. Cattaneo *et al* [@Ca00] and provides a quantum field theory interpretation of Konsevitch’s quantization of Poisson structures. We will just concentrate on its first order covariant Hamiltonian formalism along the lines described earlier in this paper.
We will consider a Riemann surface $\Sigma$ with smooth boundary $\partial \Sigma \neq \emptyset$. We may assume that $\Sigma$ also carries a Lorentzian metric. This will not play a significant role in the discussion and we can stick to a Euclidean picture by selecting a Riemannian metric on $\Sigma$. Local coordinates on $\Sigma$ will be denoted as always by $x^\mu$, $\mu = 0,1$.
Let $(P, \Lambda)$ be a Poisson manifold with local coordinates $u^a$, $a = 1, \ldots, r$. The Poisson tensor $\Lambda$ will be expressed in local coordinates as $$\Lambda = \Lambda^{ab}(y) \frac{\partial}{\partial u^a} \wedge \frac{\partial}{\partial u^b} \, ,$$ and it defines a Poisson bracket on functions $f,g$ on $P$, $$\{ f, g \} = \Lambda ({\mathrm{d}}f, {\mathrm{d}}g) \, .$$ The bundle $E$ of the theory, will be the trivial bundle $E = \Sigma \times P$ with projection $\pi$, the canonical projection on the first factor. The first jet bundle $J^1E$ is the affine bundle over $E$ modeled on $VE \otimes T^*\Sigma$, however in this case, because of the triviality of $E$, we have that $VE \cong TP$ and the affine bundle is trivial. Now the dual bundle $ P(E)$ will be naturally identified with the vector bundle over $E$ modeled on $T^*P \otimes T\Sigma$, that is, its sections will be vector fields on $\Sigma$ with values on 1-forms on $P$. However as shown in the general case, we may use a volume form $\mathrm{vol}_\Sigma$ on $\Sigma$ (for instance that provided by a Riemannian metric) to identify elements on $P(E)$ with 1-semibasic forms on $E$, i.e. $$P = P_a^\mu \, {\mathrm{d}}u^a \wedge i_{\partial / \partial x^\mu} \mathrm{vol}_\Sigma \, ,$$ and the corresponding double sections $\chi = (\Phi, P)$ of $P(E) \to E \to \Sigma$, with 1-forms $\eta$ on $\Sigma$ with values on 1-forms on $P$ along the map $\Phi \colon \Sigma \to P$, that is, $$P \colon T\Sigma \to T^*P \, , \qquad \tau_P \circ P = \Phi \, .$$ The covariant Hamiltonian of the theory will be given by, $$H ( x, u ; P) = \frac{1}{2} \Lambda^{ab}(u) (P_a^\mu, P_b^\nu) \epsilon_{\mu\nu} \,$$ with $\mathrm{vol}_{\Sigma} = \epsilon_{\mu\nu} {\mathrm{d}}x^\mu\wedge {\mathrm{d}}x^\nu$. The action of the theory is thus $$\label{action_PsM}
S_P (\chi ) = \int_\Sigma \chi^*\Theta_H = \int_{\Sigma} \left[ P_a^\mu \partial_\mu \Phi^a - H \right] \mathrm{vol}_{\Sigma}.$$ Notice that $P_a = P_a^\mu {\mathrm{d}}x_\mu$ and that ${\mathrm{d}}x_\mu = i_{\partial / \partial x^\mu} \mathrm{vol}_\Sigma$ is a 1-form on $\Sigma$. $H \mathrm{vol}_{\Sigma}$ can be expressed as $$H ( x, u ; P) \mathrm{vol}_{\Sigma}= \frac{1}{2} \Lambda^{ab}(u) (P_a \wedge P_b)$$ and the first term in the action becomes simply $P_a\wedge d\Phi^a$. Thus the action of the theory is simply given as $$S_P (\Phi, P) = \int_{\Sigma} P_a (x) {\mathrm{d}}\Phi^a (x) - \frac{1}{2} \Lambda^{ab}(\Phi (x)) (P_a(x) \wedge P_b(x)) \, ,$$ or more succinctly, $$S_P (\Phi, P) = \int_{\Sigma} \langle P \wedge {\mathrm{d}}\Phi \rangle - \frac{1}{2} (\Lambda \circ \Phi) (P \wedge P ) \, ,$$ where $\langle \cdot, \cdot \rangle$ now denotes the natural pairing between $T^*P$ and $TP$.
To get the evolution picture of the theory near the boundary, we choose a collar $U_\epsilon \cong (-\epsilon, 0 ]\times \partial \Sigma$ around the boundary $\partial \Sigma$ and we expand the action $S_P$ of the theory, eq. restricted to fields defined on $U_\epsilon$. We obtain, $$S_{P, U_\epsilon} = \int_{-\epsilon} {\mathrm{d}}t \int_{\partial \Sigma} {\mathrm{d}}u \left[ p_a \dot{\varphi}^a + \beta_a \acute{\varphi}^a - \Lambda^{ab} p_a \beta_b \right] \, ,$$ where the boundary fields $p_a$ and $\beta_a$ are defined as before, $$p_a = P_a^0 \mid_{\partial \Sigma} \, , \qquad \beta_a = P_a^1\mid_{\partial \Sigma} \, .$$ The volume form and the coordinate $u$ along the boundary $\partial \Sigma$ have been chosen so that $\mathrm{vol_\Sigma} = {\mathrm{d}}t \wedge {\mathrm{d}}u$, and $\acute{\varphi}^a$ denotes $\partial \varphi^a /\partial u$.
As before, the cotangent bundle of boundary fields is $T^*\mathcal{F}_{\partial \Sigma}$ with the canonical form $\alpha = p_a \delta \varphi^a$. In order to analyze the consistency of the Hamiltonian theory at the boundary, we introduce the extended phase space $\mathcal{M} = T^*\mathcal{F}_{\partial \Sigma} \times \mathcal{B}$, with its presymplectic structure $\Omega = \delta \varphi^a \wedge \delta p_a$ and the boundary Hamiltonian $$\mathcal{H} (\varphi, p, \beta) = - \beta_a \acute{\varphi}^a + \Lambda^{ab}(\varphi)p_a\beta_b \, .$$ Solving for the Euler-Lagrange equations we obtain two evolution equations,
$$\dot{\varphi}^aÊ= \frac{\delta \mathcal{H}}{\delta p_a} = \Lambda^{ab} \beta_b \, , \qquad \dot{p_a} = -\frac{\delta \mathcal{H}}{\delta \varphi^a} = - \acute{\beta}_a - \frac{\partial \Lambda^{bc}}{\partial \xi^a} p_b\beta_c \, ,$$ and one constraint equation equation, $$\label{constraint_PsM}
0 = \frac{\delta \mathcal{H}}{\delta \beta_a} = - \acute{\varphi}^a - \Lambda^{ab}(\varphi)p_b \, .$$ Thus the first constraints submanifold $\mathcal{M}_1$ will be defined by eq. . Notice the constraint defining $\mathcal{M}_1$ does not depend on the fields $\beta^a$, thus $\mathcal{M}_1$ is a cylinder along the projection $\varrho$ over its projection $W = \varrho (\mathcal{M}_1) \subset T^* \mathcal{F}_{\partial M}$.
Notice that $\Omega = \varrho^*\omega_{\partial M}$ is such that $\ker \Omega = \mathcal{K} = \{ \delta/ \delta \beta^a \} $. Thus, $\mathcal{K} \subset \ker \Omega_1$, where $\Omega_1$ is the restriction of $\Omega$ to $\mathcal{M}_1$. It is easy to check that $\ker \Omega_1 = \mathcal{K} \oplus \ker \Omega_{\mathcal{C}}$, where $\Omega_{\mathcal{C}}$ is the pull-back of $\omega_{\partial M}$ to $\mathcal{C}$.
The submanifold $ W \subset T^* \mathcal{F}_{\partial M}$ is defined by the constraint $$\Psi^a(\varphi, p) = - \acute{\varphi}^a - \Lambda^{ab}(\varphi ) p_b \, ,$$ whose Hamiltonian vector field $X_a$, i.e. $X_a$ such that $$i_{X_a} \omega_{\partial M} = {\mathrm{d}}\Psi^a \, ,$$ is given by $$X_a (\varphi, p) = \Lambda^{ab}(\varphi) \frac{\delta}{\delta \varphi^b} - \left( \partial_u \delta_a^c- p_b \frac{\partial \Lambda^{ab}}{\partial \varphi^c} \right) \frac{\delta}{\delta p_c} \, .$$ A simple computation shows that $$X_a (\Psi^b)\mid_{\mathcal{C}} = 0 \, .$$ Hence $TW^\perp \subset TW$ and consequently, not only $W$, but also $\mathcal{M}_1$ are coisotropic submanifolds.( In describing $\mathcal{M}_1$ as a coisotropic submanifold of the presymplectic manifold $\mathcal{M}$ we mean simply that $T\mathcal{M}_1^\perp \subset T\mathcal{M}_1$.)
The stability of the constraints shows that the PCA algorithm stops at $\mathcal{M}_1$. Then the reduced (or physical) phase space of the theory is $$\mathcal{R} = \mathcal{M}_1/\ker \Omega_1 \cong \mathcal{C}/ \mathrm{span}\{X_a \} \, .$$ The reduced phase space is a symplectic manifold, that in this case happens to be finite-dimensional.
In some particular cases it can be computed explicitly (for instance $\Sigma = [0,1]\times [0,1]$ with appropriate boundary conditions). In some instances it happens to inherit a groupoid structure that becomes the symplectic groupoid integrating the Poisson manifold $P$ [@Ca01].
Yang-Mills theories on manifolds with boundary as a covariant Hamiltonian field theory {#sec:Yang-Mills}
======================================================================================
The multisymplectic setting for Yang-Mills theories
---------------------------------------------------
Recall from the introduction, $(M, \eta)$ is an oriented smooth manifold of dimension $m = 1 + d$ with boundary $\partial M \neq \emptyset$. It carries either a Riemannian or a Lorentzian metric $\eta$, in the later case of signature $(-+\cdots +)$ and such that the connected components of $\partial M$ are space-like submanifolds, that is, the restriction $\eta_{\partial M}$ of the Lorentzian metric to them is a Riemannian metric.
Yang-Mills fields are principal connections $A$ on some principal fiber bundle $\rho \colon P \to M$ with structural group $G$. For clarity in the exposition we are going to make the assumption that $P$ is trivial (which is always true locally), i.e., $P \cong M \times G \to M$ where (again, for simplicity) $G$ is a compact semi-simple Lie group with Lie algebra $\mathfrak{g}$.
Under these assumptions, principal connections on $P$ can be identified with $\mathfrak{g}$-valued 1-forms on $M$, i.e., with sections of the bundle $E=T^*M \otimes \mathfrak{g} \longrightarrow M$. Local bundle coordinates in the bundle $E \to M$ will be written as $(x^\mu, A_\mu^a)$, $\mu = 1, \ldots, m$, $a= 1, \ldots, \dim\mathfrak{g}$, where $A = A_\mu^a \xi_a \in \mathfrak{g}$ with ${\xi}_a$ a basis of the Lie algebra $\mathfrak{g}$. Thus, a section of the bundle can be written as $$\label{connectionA}
A(x) = A^a_{\mu}(x)\, {\mathrm{d}}x^{\mu}{\otimes}{\xi}_a \, .$$
The covariant Hamiltonian formalism will be formulated by considering the bundle $P(E)$, the affine dual of the first jet bundle $J^1E$. Let us recall from the general discussion on Sect. \[sec:general\], that $J^1E$ is an affine bundle modeled on the vector bundle $T^*M\otimes VE \cong T^*M\otimes T^*M \otimes \mathfrak{g}$. The affine dual of $J^1E$ can thus be modeled on the vector bundle $TM\otimes TM \otimes \mathfrak{g}^*$.
The multisymplectic formalism is described in the manifold $P(E)$ whose elements can be identified with 1-semibasic $m$-forms $$P = P^{\mu \nu}_a{{\mathrm{d}}A}^a_{\mu} \wedge {\mathrm{d}}^{m-1}x_{\nu} \, ,$$ where $\\d^{m-1}x_\nu = i_{\partial/\partial x^\nu} \mathrm{vol}_\eta$ and $\mathrm{vol}_M$ is the canonical volume form on $M$ defined by the metric $\eta$. Thus the fields of the theory in the multisymplectic picture are provided by sections $(A,P)$ of the double bundle $P(E)\to E \to M$.
We will formulate our theory directly in terms of the natural fields $A, P$, and we will write the action functional following the general principle, eq. : $$\label{ymap}
S_{\mathrm{YM}}(A,P) = \int_M P_a^{\mu\nu} {\mathrm{d}}A_\mu^a \wedge {\mathrm{d}}x^{m-1}_\nu - H(A,P) \mathrm{vol}_M \, .$$ The Hamiltonian function is defined as, $$\label{hamiltonian}
H(A,P) = \frac{1}{2} \epsilon_{bc}^aP^{\mu\nu}_a A_\mu^bA_\nu^c + \frac{1}{4}P^{\mu\nu}_a P_{\mu\nu}^a \, ,$$ where the indexes $\mu\nu$ ($a$) in $P_a^{\mu\nu}$ have been lowered (raised) with the aid of the Lorentzian metric $\eta$ (the Killing-Cartan form on $\mathfrak{g}$, respect.). Expanding the right hand side of eq. , we get[^1], $$\label{ymP}
S_{\mathrm{YM}}(A,P) = -\int_M \frac{1}{2} \left[ P^{\mu\nu}_a (\partial_\mu A_\nu^a - \partial_\nu A_\mu^a + \epsilon_{bc}^a A_\mu^b A_\nu^c) + \frac{1}{2} P^{\mu\nu}_aP^a_{\mu\nu} \right] \, \mathrm{vol}_M \, .$$ Notice that if $A$ is given by eq. , then, its curvature is given by, $$\begin{aligned}
\label{Fmunu}
F_A &=& {\mathrm{d}}_A A = {\mathrm{d}}A + \frac{1}{2}[A\wedge A] = F_{\mu\nu} {\mathrm{d}}x^\mu \wedge {\mathrm{d}}x^\nu \\
&=& \frac{1}{2}\left( \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + \epsilon_{bc}^a A_\mu^b A_\nu^c\right) {\mathrm{d}}x^\mu \wedge {\mathrm{d}}x^\nu \otimes \xi_a\, . \nonumber\end{aligned}$$ Thus the previous expression for the Yang-Mills action becomes, $$S_{\mathrm{YM}}(A,P) = -\int_M \left[ P_a^{\mu\nu} F_{\mu\nu}^a + \frac{1}{4}P^{\mu\nu}_aP^a_{\mu\nu} \right] \, \mathrm{vol}_M \,.$$ The Euler-Lagrange equations of the theory are very easy to obtain from the previous expression, they are, $$\label{ymfo}
\frac{1}{2}P_{\mu\nu}^a = - F_{\mu\nu}^a \, , \qquad \partial_\mu P_a^{\mu\nu} + \epsilon_{ab}^c A_\mu^b P_c^{\mu\nu}= 0 \, .$$
The canonical formalism near the boundary {#sec:canonical}
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In order to obtain an evolution description for Yang-Mills and to prepare the ground for the discussion of its canonical quantization, we need to introduce a local time parameter.
In the case that $M$ is a Lorenztian manifold it is customary to assume that $M$ is globally hyperbolic (even if far less strict causality assumptions on $M$ would suffice), therefore the time parameter can be chosen globally. Actually we will only assume that a collar $U_{\epsilon} = (-\epsilon, 0]\times \partial M$ around the boundary can be chosen and so that a choice of a time parameter $t = x^0$ can be made near the boundary that would be used to describe the evolution of the system. The fields of the theory would then be considered as fields defined on a given spatial frame that evolve in time for $t \in (-\epsilon, 0]$.
The dynamics of such fields would be determined by the restriction of the Yang-Mills action to the space of fields on $U_{\epsilon}$, $$\label{ymPepsilon}
S_{\mathrm{YM},U_{\epsilon}}(A,P)= - \int_{-\epsilon}^0 {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M} \left[ P^{\mu\nu}_a F_{\mu\nu}^a + \frac{1}{4} P^{\mu\nu}_aP^a_{\mu\nu} \right] \, ,$$ where now we are assuming that the collar $U_\epsilon$ is strongly hyperbolic and $\mathrm{vol}_{U_\epsilon} = {\mathrm{d}}t \wedge \mathrm{vol}_{\partial M}$ where $\mathrm{vol}_{\partial M}$ is the canonical volume defined by the restriction of the metric $\eta$ to the boundary.
Expanding we obtain, $$\begin{aligned}
S_{\mathrm{YM},U_\epsilon} (A,P) & = & -\frac{1}{2} \int^0_{- \epsilon} {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M}
\left[ P^{\mu\nu}_a \left( \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + \epsilon_{bc}^a A_\mu^b A_\nu^c \right) + \frac{1}{2}P^{\mu\nu}_a P_{\mu\nu}^a \right] \\
& = & - \frac{1}{2} \int^0_{- \epsilon} {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M}
\left[ P^{k0}_a \left( \partial_k A_0^a - \partial_0 A^a_k + \epsilon^a_{bc} A^b_k A^c_0 \right) \right. + \\
&+& P^{0k}_a \left(\partial_0 A^a_k - \partial_k A_0^a + \epsilon^a_{bc} A^b_0 A^c_k \right) + \\
&+& \left. P^{kj}_a \left( \partial_k A^a_j - \partial_j A^a_k + \epsilon^a_{bc}A^b_kA^c_j \right) + \frac{1}{2} P^{k0}_a P^a_{k0} + \frac{1}{2} P^{0k}_a P^a_{0k} + \frac{1}{2}P^{kj}_aP_{kj}^a \right] \\
&=& \int^0_{- \epsilon} {\mathrm{d}}t \int_{\partial M} \mathrm{vol}_{\partial M}
\left[ P^{k0}_a \left( \partial_0 A^a_k - \partial_k A_0^a - \epsilon^a_{bc} A^b_k A^c_0 \right) \right. + \\
&- & \left. \frac{1}{2} P^{kj}_a \left( \partial_k A^a_j - \partial_j A^a_k + \epsilon^a_{bc}A^b_kA^c_j \right) - \frac{1}{2}P^{k0}_a P^a_{k0} - \frac{1}{4}P^{kj}_aP_{kj}^a \right] \, .\end{aligned}$$ In the previous expressions $\epsilon_{bc}^a$ denote the structure constants of the Lie algebra $\mathfrak{g}$ with respect to the basis $\xi_a$, that is $[\xi_b, \xi_c] = \epsilon_{bc}^a \xi_a$. Notice that $\epsilon^a_{bc}A^b_0A^c_0=0$ because for fixed a, ${\epsilon}^a_{bc}$ is skew-symmetric. Moreover the indexes $\mu$ and $a$ have been pushed down and up by using the metric $\eta$ and the Killing-Cartan form $\langle\cdot, \cdot \rangle$ respectively.
In equation we introduced the assumption that $P$ is a bivector, i.e., $P_a^{\mu\nu}$ is skew symmetric in $\mu$ and $\nu$. Therefore $P_a^{00} = 0$, and also $P^{k0}_a P^a_{k0} = P^{0i}_a P^a_{0i}$, because $P^{k0} = - P^{0k}$, etc. This assumption will be justified later on (see Sect. \[sect:Legendre\])
The previous expression acquires a clearer structure by introducing the appropriate notations for the fields restricted at the boundary and assuming that they evolve in time $t$. Thus the pull-backs of the components of the fields $A$ and $P$ to the boundary will be denoted respectively as, $$\begin{aligned}
a^a_k &:=& A^a_k\mid_{\partial M} ; \qquad a = (a^a_k) \, , \qquad a^a_0 := A^a_0\mid_{\partial M} ; \qquad a_0 = (a^k_0) \, , \\
p^k_a &:=& P^{k0}_a \mid_{\partial M} ; \qquad p = (p^k_a)\, , \qquad p^0_a := P^{00}_a\mid_{\partial M}= 0 ; \qquad p_0 = (p^0_a)=0 \, , \\
\beta^{ki}_a &:=& P^{ki}_a\mid_{\partial M} ; \qquad \beta =( \beta^{ki}_a) \, .\end{aligned}$$ Given two fields at the boundary, for instance $p$ and $a$, we will denote as usual by $\langle p, a\rangle$ the following expression: $$\langle p, a\rangle = \int_{\partial M} p_a^\mu a_\mu^a \, \mathrm{vol}_{\partial M}\, ,$$ and the contraction of the inner (Lie algebra) indices by using the Killing-Cartan form and the integration over the boundary is understood.
Introducing the notations and observations above in the expression for $S_{\mathrm{YM}, U_\epsilon}$ we obtain, $$\begin{aligned}
S_{\mathrm{YM},U_\epsilon}(A,P) &=& \int^0_{- \epsilon}{\mathrm{d}}t \int_{ \partial M} \mathrm{vol}_{\partial M}
\left[ p^k_a \left( \dot{a}^a_k - \partial_ka_0^a -\epsilon^a_{bc} a^b_k a^c_0 \right) \right. + \nonumber
\\
&-& \frac{1}{2} \beta^{ki}_a \left( \partial_k a^a_i - \partial_i a^a_k + \epsilon^a_{bc}a^b_ka^c_i \right) -
\frac{1}{4} \beta^{ki}_a \beta^a_{ki} - \frac{1}{2}p_a^kp^a_k = \nonumber \\
&=& \int^0_{- \epsilon} {\mathrm{d}}t \, \mathcal{L}(a,\dot{a},a_0,\dot{a}_0,p,\dot{p}, \beta,\dot{\beta}) \, \label{ym_boundary}\end{aligned}$$ where now $\mathcal{L}$ denotes the boundary Lagrangian, Eq. , and depends on the restrictions to the boundary of the fields of the theory. Collecting terms and simplifying we can then write $\mathcal{L}$ as, $$\mathcal{L}(a,\dot{a},a_0,\dot{a}_0,p,\dot{p},\beta,\dot{\beta})
= Ê\langle p,\dot{a} - {\mathrm{d}}_a a_0 \rangle - \langle \beta , F_a \rangle - \frac{1}{2} \langle p,p \rangle - \frac{1}{4} \langle \beta, \beta \rangle \, .$$
Now we can find the Euler-Lagrange equations corresponding to the Lagrangian function $\mathcal{L}$ as an infinite-dimensional mechanical system defined on the configuration space $P(E) = \{ a, a_0, p, \beta \}$. Notice that the fields $a$, $p$ are 1-forms on $\partial M$ with values in the Lie algebra $\mathfrak{g}$, while the field $a_0$ is a function on $\partial M$ with values in $\mathfrak{g}$, and the field $\beta$ is a 2-form on $\partial M$ with values in $\mathfrak{g}$ too. Thus the configuration space is the space of sections of the bundle $(T^*M\oplus T^*M\oplus \Lambda^2(T^*M)\oplus \mathbb{R} )\otimes \mathfrak{g}$.
Euler-Lagrange equations will have the form: $$\frac{d}{{\mathrm{d}}t} \frac{\delta \mathcal{L}}{\delta \dot{\chi}} = \frac{\delta \mathcal{L}}{\delta \chi} \, ,$$ where $\chi \in P(E)$ and $\delta /\delta \chi$ denotes the variational derivative of the functional $\mathcal{L}$.
Thus for $\chi = p$ we obtain, $$\frac{\delta \mathcal{L}}{\delta \dot{p}}=0, \quad \mathrm{hence} \quad 0 = \frac{\delta \mathcal{L}}{\delta p} = -p + \dot{a} - {\mathrm{d}}_a a_0 \, ,$$ and thus, $$\label{control}
\dot{a} = p + {\mathrm{d}}_a a_0 \, .$$ This equation corresponds to the Legendre transformation of the velocity and agrees with the standard minimal coupling definition of the momenta $p = \dot{a} - \mathrm{d}_a a_0$.
For $\chi = \beta$ we obtain, $$\frac{\delta \mathcal{L}}{\delta \dot{\beta}} = 0, \quad \mathrm{thus} \quad 0=\frac{\delta \mathcal{L}}{\delta \beta } = - F_a - \frac{1}{2}{\beta}$$ and consequently, $$\label{betafa}
\beta = -2 F_a \, .$$
For $\chi = a$ we obtain, $$\frac{\delta \mathcal{L}}{\delta \dot{a}} = p , \quad \mathrm{hence} \quad \dot{p} = \frac{d}{{\mathrm{d}}t}\frac{\delta \mathcal{L}}{\delta \dot{a}} = \frac{\delta \mathcal{L}}{\delta a} = \mathrm{d}^*_a\beta + [p,a_0].$$ Thus we get the equation determining the evolution of the momenta field (the Yang-Mills electric field) $p$: $$\label{pdot}
\dot{p} = \mathrm{d}^*_a\beta + [p,a_0] \, .$$
Finally for $\chi = a_0$ we obtain, $$\frac{\delta \mathcal{L}}{\delta \dot{a_0}} = 0, \quad \mathrm{and \,\,\, therefore,} \quad \frac{\delta \mathcal{L}}{\delta a_0} = \mathrm{d}_a^*p \, .$$ Thus we obtain, $$\label{gauss}
\mathrm{d}^*_ap = 0$$ that must be interpreted as Yang-Mills Gauss law (in the absence of charges). Thus we have two evolution equations, and , and two constraint equations and .
Notice that the field $a_0$ is undetermined. This fact, clearly a consequence of the gauge invariance of the theory, will be interpreted in the next section.
We will study the consistency of the previous equations in the following section.
The Legendre transform {#sect:Legendre}
----------------------
### The Legendre transform in the bulk
So far we have presented a covariant Hamiltonian theory, equation following (4.5), whose Euler-Lagrange equations are equivalent to Yang-Mills equations. However it is not automatically true that such theory is equivalent to the standard Yang-Mills theory. The standard Yang-Mills theory is a Lagrangian theory determined by a Lagrangian density which is nothing but the square norm of the curvature $F_A$ of the connection 1-form $A$, and its action the $L^2$ norm of $F_A$, i.e. $$\label{yms}
S = - \frac{1}{4}\int_{M} \mathrm{Tr\,} (F_A \wedge \star F_A) = \int_M L_{\mathrm{YM}}(A) \mathrm{vol}_M \, .$$ Standard quantum field theories describing gauge interactions use exactly this Lagrangian description (and provide accurate results). Thus if we will assume that the correct Yang-Mills theory is provided by the action above, eq. , then we would like to relate the covariant Hamiltonian picture above to this Lagrangian picture.
For this task we have to introduce the natural extension of Legendre transform to the setting of covariant first order Lagrangian field theories. The Legendre transform is defined [@Ca91] as the bundle map $\mathcal{F}L_{YM} \colon J^1E \to P(E)$, as $\mathcal{F}L_{YM}(x^\mu, A_\mu^a; A_{\mu\nu}^a) = (x^\mu, A_\mu^a; P_a^{\mu\nu})$, where $$P_a^{\mu\nu} = \frac{\partial L_{\mathrm{YM}}}{\partial A_{\mu\nu}^a} \,$$ and $L_{\mathrm{YM}} = - \frac{1}{4} \mathrm{Tr\,} (F_A \wedge \star F_A)$. Now recall that $\alpha \wedge \star \beta = (\alpha, \beta )_\eta \mathrm{vol}_M$, $\alpha, \beta$, $k$-forms, where $(\cdot, \cdot )_\eta$ denotes the inner product on $k$-forms. Thus we will write $\alpha \wedge \star \beta = \alpha_{\mu_1\cdots \mu_k} \beta^{\mu_1 \cdots \mu_k} \mathrm{vol}_M$ where we have raised the indexes by using the $\eta^{\mu\nu}$. Hence, $$\label{LYM}
L_{YM} = \frac{1}{2} F_{\mu\nu} F^{\mu\nu} \, .$$ Hence in bundle coordinates $(x^\mu, A_\mu^a; A_{\mu\nu}^a)$, we have, $$F_{\mu\nu} = \frac{1}{2}\left( A_{\nu\mu}^a - A_{\mu\nu}^a + \epsilon_{bc}^a A_\mu^b A_\nu^c\right) \, .$$ Thus $$P_a^{\mu\nu} = F^{\mu\nu}_a \, .$$ Notice that on the graph of the Legendre map, the Yang-Mills action in the Hamiltonian first order formalism, eq. , is just, up to a coefficient, the previous action eq. .
It was mentioned at the end of Section \[sec:canonical\] that the momenta fields $P^{\mu\nu}$ are skew-symmetric in the indices $\mu$ and $\nu$. Notice that from the definition of the momenta fields as sections of the bundle $P(E)$ there is no restriction on them. However because Yang-Mills theories are Lagrangian theories, the Legendre transform selects a subspace on the space of momenta that corresponds to fields $P$ which are skew-symmetric on the indices $\mu$, $\nu$.
The presymplectic formalism: Yang-Mills at the boundary and reduction
---------------------------------------------------------------------
As discussed in general in section $3.2$, we define the extended Hamiltonian, $\mathcal{H}$, so that $\mathcal{L} = \langle p, \dot{a}\rangle - \mathcal{H}$. Thus $$\label{PH}
\mathcal{H} (a,\beta) = \langle p, {\mathrm{d}}_a a_0 \rangle + \frac12 \langle p, p \rangle + \langle \beta, F_a + \frac{1}{2} \beta \rangle \, .$$
Thus the Euler-Lagrange equations can be rewritten as $$\label{PMP1}
\dot{a} = \frac{\delta \mathcal{H}}{\delta p}; \quad \dot{p} = - \frac{\delta \mathcal{H}}{\delta a} \, ,$$ $$\label{PMP0}
\frac{\delta \mathcal{H}}{\delta a_0} = 0 \,$$
$$\label{PMP2}
\frac{\delta \mathcal{H}}{\delta \beta} = 0 \, .$$
We denote again by $\varrho \colon \mathcal{M} \to T^*\mathcal{F}_{\partial M}$ the canonical projection $\varrho(a,a_0,p,\beta)=(a,a_0,p)$. Let $\omega_{\partial M}$ denote the form on the cotangent bundle $T^*\mathcal{F}_{\partial M}$, $$\omega_{\partial M} = \delta a \wedge \delta p .$$ We will denote again by $\Omega$ the pull-back of this form to $\mathcal{M}$ along $\varrho$, i.e., $\Omega = \varrho^*\omega_{\partial M}$. Clearly, $\ker {\Omega} = \mathrm{span} \{ \delta /\delta \beta, \delta/\delta a_0 \}$, and we have the particular form that Thm. \[presymplectic\_equation\] takes here.
The solution to the equation of motion defined by the Lagrangian $L_{\mathrm{YM}}$, i.e. the Yang-Mills equations, are in one-to-one correspondence with the integral curves of the presymplectic system $(\mathcal{M},\Omega,\mathcal{H})$, i.e. with the integral curves of the vector field $\Gamma$ on $\mathcal{M}$ such that $i_\Gamma\Omega= \mathrm{d} \mathcal{H}$.
The primary constraint submanifold $\mathcal{M}_1$ is defined by the two constraint equations, $$\mathcal{M}_1= \{(a,a_0,p,\beta)|\mathcal{F}_a + \beta = 0, d_a^*p = 0 \} \, .$$
Since $\beta$ is just a function of $a$, we have that $\mathcal {M}_1\cong \{(a,a_0,p)| d_a^*p = 0\}$ and $\ker \Omega|_{\mathcal{M}_1} = \mathrm{span} \{\frac{\partial}{\partial a_0}\}$. Thus $\mathcal{M}_2 = \mathcal{M}_1/(\ker \Omega|\mathcal{M}_1) \cong \{(a,p)| \mathrm{d}_a^*p = 0\}.$
Gauge transformations: symmetry and reduction
---------------------------------------------
The group of gauge transformations $\mathcal{G}$, i.e, the group of automorphisms of the principal bundle $P$ over the identity, is a fundamental symmetry of the theory. Notice that the action $S_{\mathrm{YM}}$ is invariant under the action of $\mathcal{G}$ (however it is not true that $H$ is $\mathcal{G}$-invariant).
The quotient of the group of gauge transformations by the normal subgroup of identity gauge transformations at the boundary defines the group of gauge transformations at the boundary $\mathcal{G}_{\partial M}$, and it constitutes a symmetry group of the theory at the boundary, i.e. it is a symmetry group both of the boundary Lagrangian $\mathcal{L}$ and of the presymplectic system $(\mathcal{M}, \Omega, \mathcal{H})$. We may take advantage of this symmetry to provide an alternative description of the constraints found in the previous section.
With the notations above, $\mathcal{J}(a,p) = {\mathrm{d}}_a^*p$.
The moment map $\mathcal{J} \colon T^*\mathcal{F}_{\partial M} \to \mathfrak{g}_{\partial M}^*$ is given by, $$\langle \mathcal{J} (a,p), \xi \rangle = \langle p, \xi_{\mathcal{F}_{\partial M}}\rangle = \langle
p, {\mathrm{d}}_a\xi \rangle \, ,$$ because the gauge transformation $g_s = \exp s \xi$ acts in $a$ as $a \mapsto g_s\cdot a = g_s^{-1} a g_s + g_s^{-1}{\mathrm{d}}g_s$ and the induced tangent vector is given by, $$\xi_{\mathcal{A}_{\partial M}} (a)= \frac{{\mathrm{d}}}{{\mathrm{d}}s} g_s \cdot a \mid_{s = 0} = {\mathrm{d}}_a\xi \, .$$
Let $\mathcal{A}_{\partial M}$ denote the space of connections $a$ defined on the boundary $\partial M $. The constraint submanifold $\mathcal{M}_1$ projected to the space $T^*\mathcal{A}_{\partial M}$, by means of the projection map $(a,a_0,p) \mapsto (a,p)$, is such that $\mathcal{C} = \mathcal{J}^{-1}(\mathbf{0})$. This is exactly the situation depicted in Sect.\[reduction\_gauge\]. Hence the standard Marsden-Weinstein reduction, eq. , will give the reduced phase space, $$\mathcal{R}_{\mathrm{YM}} = \mathcal{J}^{-1}(\mathbf{0})/ \mathcal{G}_{\partial M} \, .$$ and its Hamiltonian, $$h ([a], [p] )= \frac{1}{2}\langle p, p \rangle - \frac{1}{2}\langle F_a, F_a\rangle \, ,$$ where $[a]$ and $[p]$ denote equivalence classes of connections and momenta with respect to the action of the gauge group $\mathcal{G}_{\partial M}$. Notice that both terms in the Hamiltonian function $h$ are $\mathcal{G}_{\partial M}$-invariant, and the Hamiltonian system $h$ defined on the reduced phase space $\mathcal{R}_{\mathrm{YM}}$ has the structure of an infinite-dimensional mechanical system with potential function $V([a]) = \frac{1}{2}|| F_a ||^2$.
The reduction of the boundary values of solutions of Yang-Mills equation in the bulk is of course, an isotropic submanifold of the reduced space. In the case where $M$ is Riemannian, an existence and uniqueness theorem for solutions of Yang-Mills equations on manifolds with boundary can be proved and hence this submanifold, following the proof of Theorem 2.7, is a Lagrangian submanifold.
Conclusions and discussion
==========================
It has been shown that the multisymplectic geometry of the covariant phase space $P(E)$ provides a convenient framework to study first order covariant Hamiltonian field theories on manifolds with boundaries. In particular it induces a natural presymplectic structure on the total space of fields at the boundary whose reduction provides the symplectic phase space of the theory. The solution of the Euler-Lagrange equations on the bulk induce an isotropic submanifold in the reduced symplectic phase space at the boundary. Provided that the boundary conditions are well-posed, this submanifold is in fact Lagrangian.
The gauge symmetries of the theory fit nicely into the picture and the symplectic reduction of the theory at the boundary induced by the moment map, i.e., by the conserved charges of the theory, is in perfect agreement with the presymplectic analysis of the theory. Various instances are discussed illustrating the main features of the theoretical framework: the real scalar field, the Poisson $\sigma$-model and Yang-Mills theories. Each of them allows as to stress different aspects of the theory. The regular situation for the scalar field, the coisotropic structure at the boundary in the case of the Poisson $\sigma$-model and the reduction using the moment map at the boundary in the case of Yang-Mills theories.
The theory presented in this work is particularly well suited for describing Palatini’s gravity. C. Rovelli’s [@Ro04], [@Ro06], can be read in part as seeking and arguing for precisely such a theory. We interpret Rovelli’s canonical form ${\Theta}_H$ as alluding to a multisymplectic structure in the bulk. Such aspects will be discussed in a subsequent paper where the reduction of Topological Field Theories at the boundary and Palatini’s gravity will be discussed from a common perspective.
Acknowledgements {#sec:acknowledgements .unnumbered}
================
A.I. was partially supported by the Community of Madrid project QUITEMAD+, S2013/ICE-2801, and MINECO grant MTM2014-54692-P. Part of this work was completed while A.S. was a guest of the Mathematics department at University Carlos III de Madrid, and supported by Spain’s Ministry of Science and Innovation Grant MTM2010-21186-C02-02. She thanks the Mathematics department for their warm hospitality and for their financial support. A.S. also thanks Nicolai Reshetikhin for suggesting to her a problem that motivated this work. The solution to that other problem will appear in another publication. Finally, the authors would like to thank the referee for carefully reading over the manuscript and for drawing their attention to important points that were in need of clarification.
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[^1]: The minus sign in front comes form the expansion of $P_a^{\mu\nu} dA_\mu^a \wedge {\mathrm{d}}x^{m-1}_\nu$, that gives $P^{\mu\nu}_a (\partial_\nu A_\mu^a - \partial_\mu A_\nu^a) \mathrm{vol}_M$.
|
harvmac ł Ø § Ł
P.A. Grassi$^{~a,b,}$, R. Ricci$^{~a,}$, and D. Robles-Llana$^{~a,}$
$^{(a)}$ [*C.N. Yang Institute for Theoretical Physics,*]{}
*State University of New York at Stony Brook, NY 11794-3840, USA*
$^{(b)}$ [*Dipartimento di Scienze, Università del Piemonte Orientale,*]{}
*C.so Borsalino 54, Alessandria, 15100, ITALY*
We study (anti-) instantons in super Yang-Mills theories defined on a non anticommutative superspace. The instanton solution that we consider is the same as in ordinary $SU(2)$ $N=1$ super Yang-Mills, but the anti-instanton receives corrections to the $U(1)$ part of the connection which depend quadratically on fermionic coordinates, and linearly on the deformation parameter $C$. By substituting the exact solution into the classical Lagrangian the topological charge density receives a new contribution which is quadratic in $C$ and quartic in the fermionic zero-modes. The topological charge turns out to be zero. We perform an expansion around the exact classical solution in presence of a fermionic background and calculate the full superdeterminant contributing to the one-loop partition function. We find that the one-loop partition function is not modified with respect to the usual $N=1$ super Yang-Mills.
It is a common belief that in the standard formulation of superspace, only the bosonic subspace may have a non-trivial topology. A superspace ${\cal M}^{(n|m)}$, where $n$ is the dimension of its bosonic subspace ${\cal M}_{b}$ and $m$ is the dimension of spinor representation, is a Grassmanian vector bundle with no topology in the fibers. However, despite some attempts to construct models with non trivial superspace topology (see for example , and ) and interesting arguments suggesting that only the topology of the bosonic subspace really matters (see for example ), a new superspace formulation based on a construction in superstring theory , reopened the debate. For related considerations on deformed superspaces see also , .
In this approach, the fermionic coordinates $\t^{\a},
\bar\t^{\dot \a}$ are no longer Grassmann variables, but they are promoted to elements of a Clifford algebra where $C^{\dot\a \dot\b}$ is the constant self-dual RR field strengh of the closed string theory background. As a consequence the $N=1$ supersymmetry algebra is deformed and broken down to $N=1/2$ .
From a more physical perspective, and after several perturbative studies of $N=1/2$ supersymmetric quantum field theories , one is tempted to ask about their non-perturbative aspects. The issue is not unrelated to the problem addressed in the previous paragraph: it is by now well established that the main sources of non-perturbative physics are objects which have also a special topological significance. One would then hope that knowing more about the non-perturbative regime of $N=1/2$ supersymmetric theories might in addition shed some light on a possible non-trivial topology of superspace. In particular, in this paper we study instantons (anti-instantons), [*i.e.*]{} finite-action anti-selfdual (self-dual) solutions to the Euclidean equations of motion of (super) Yang-Mills theories, which have proven to be one of the main sources of insights in both the non-perturbative regime of quantum field theories, and the topology of four-manifolds (for a physics review, see for example ; for a mathematical introduction see ). As is well-known, the instanton charge is topological and completely computable in terms of the bosonic solution to the self-dual Yang-Mills equations. Moreover, instantons are degenerate solutions, and it is crucial to study their moduli space, which is parametrized by a set of variables which are referred to as collective coordinates. In some special instances a complete parametrization of this space can be obtained through the generators of the symmetries of the equations of motion which are broken by the classical solution (an example is $N=1$ SYM with $SU(2)$ gauge group). More generally, however, one has to find the most general solution to the equations of motion through the ADHM construction (see for example , and references therein).
Supersymmetry adds many interesting features to the study of instantons. The most salient is perhaps the fact that instantons in supersymmetric theories break half of the supersymmetries of the original action, in addition to translations, dilatations, and half of the Lorentz symmetry (this is possible only in Euclidean space). To be more specific, instantons ($F^{+}_{\mu\nu} =0$) break the supersymmetries generated by $\bar
Q^{\dot\a},S^{\a}$, while anti-instantons ($F^{-}_{\mu\nu} =0$) break $Q^{\a},\bar S^{\dot\a}$ . These broken supercharges give rise to fermionic collective coordinates, which can be thought of as the fermionic superpartners of the bosonic coordinates introduced above. Again, finding the complete set of fermionic collective coordinates requires solving the full equations of motion. In geometrical terms, the fermionic collective coordinates can be seen to parametrize the symplectic tangent space to the moduli space.
When considered from the quantum field theoretical perspective, instantons characterize topological vacua of the Euclidean theory around which one must expand in the computation of the path integral. Going back to Minkowski space they give the main contribution to tunneling processes which go as the square of the inverse of the coupling constant, and thus can never be seen in ordinary perturbation theory. In the semi-classical approximation, one must expand the classical action up to quadratic terms in quantum fluctuations around the instanton. The measure in the path integral is then modified by the degeneracy of solutions, which translates into the presence of zero-modes in the functional determinants. The correct normalization is determined by the jacobian obtained by trading the integration over the zero-modes for the collective coordiantes. As we saw, in super Yang-Mills theories, instantons have fermionic counterparts which depend on fermionic collective coordinates. These must be accounted for by including the Pfaffian associated to their inner products in the measure, and the Grassmann variables are to be integrated over using Berezin integration. This leads to new genuinely non-perturbative effects, such as the well-known non-vanishing result for the gluino condensate.
In this paper we would like to see how the characteristics of instanton calculations in ordinary supersymmetric theories are modified when one considers quantum field theories defined in deformed superspace. We will concentrate on pure $U(2)$ $N=1/2$ SYM, as an already non-trivial illustration of how these differences arise.
First and foremost, in $N=1/2$ theories the commutation relations for half of the supercharges, say $Q^{\a}$, are deformed to The Lagrangian for pure $N=1/2$ SYM contains two additional operators of dimensions 5 and 6 with respect to the $N=1$ case , but it is nonetheless renormalizable (see later).
The fields obeying the anti-self-duality equations of conventional SYM ($F^{+}_{\m\n}=0$) furnish a complete solution to the equations of motion even in the presence of the extra couplings. Moreover, the supersymmetries broken by these solutions are $\bar Q_{\dot \a}$,$S_{\a}$, which are not broken by the C-deformation. Consequently, the subalgebra generated by these supercharges should lead to the complete set of collective coordinates as usual, and the path integral measure can be constructed in the conventional way, at least at the classical level. The classical instanton action is the same as in $N=1$ SYM. The fermionic zero-modes of the Dirac operator in the instanton background could however be lifted in perturbation theory due to the presence of extra non-supersymmetric couplings in the action, thereby giving corrections to the effective action at one-loop.
On the other hand, $N=1$ anti-instantons ($F^{-}_{\mu\nu}=0$) do not provide a full solution to the classical equations of motion of the deformed theory. As noticed in the extra couplings contribute fermionic source terms for the Dirac equation in the anti-instanton background. In this paper we revisit the procedure of and solve the equations of motion exactly. Our perspective, however, differs from that of , and is more in line with the overall philosophy of . We derive the complete solution through an iterative procedure which consists in systematically expanding the equations of motion in powers of the fermionic quasi-collective coordinates. In the end we arrive at the conclusion that the ordinary $SU(2)$ supersymmetric anti-instanton is supplemented with a non-trivial $U(1)$ connection which depends quadratically on fermionic variables (the Grassmann collective coordinates of the $N=1$ solution). When we substitute this solution into the lagrangian we find the density charge
The new feature of is the fact that it depends on the fermionic parameters $\xi_{\a}$ and $\bar\eta^{\dot\a}$. The appearance of the second term in the exponent of can be traced back to the fact that the anti-instanton breaks the supersymmetries which are already broken by the C-deformation. Consequently the fermionic parameters entering the anti-selfdual solution cannot be viewed as collective coordinates in the usual sense. Notice however that performing the integration over the bosonic variables $x$, the total charge, namely the instanton action, reproduce the usual undeformed $N=1$ answer (in the text, an argument based on the Atiyah Singer index Theorem will be used) .
A similar situation occurs in $N=4$ SYM. In that case, not only the charge density, but also the total charge depends upon the fermionic “quasi-collective coordinates”, but this relies on the approximate nature of the solutions of the equations of motion , whereas in the present case the appearance of zero-modes in the Lagrangian is a direct consequence of the broken supersymmetry of the classical action . Note that, as the C-deformation also breaks dilatational symmetry, it is perfectly reasonable that the parameter $\r$, which labels the size of the anti-instanton, appears in .
The second step in path integral computations around a classical solution is the evaluation of radiative corrections. In the conventional case, the unbroken supersymmetries pair up the bosonic and fermionic fluctuations in the one-loop determinant, with the net effect of only a prefactor in the path integral depending on the regulator mass $\m$. Together with the measure coming from the zero modes, it encodes the one-loop renormalization of the coupling constant. In section 6 we will discuss the one loop corrections to the effective action for N=1/2 SYM in anti-instanton background. The novelty is the treatment of the fermionic modes. As discussed in one can either treat the fermions perturbatively, expanding around a purely bosonic configuration, or [*exactly*]{} including them already in the classical background. Here we follow the second route. This has the following important consequence for the one-loop calculation: Performing an expansion around the full background of the C-deformed action induces new bilinear couplings between bosonic and fermionic fluctuations. One is then prompted to calculate a superdeterminant in the space of all fields (unlike the conventional situation in which bosonic and fermionic determinants decouple). This computation, to the best of our knowledge, has never before been performed in this context (see however ). The one loop effective action turns out to be zero also. The physical reason behind the cancellation has to be found in the fact that the deformation in the sector considered breaks exactly the same supersymmetry generators broken already by the anti-instanton solution. Therefore we are left with the same supersymmetry generators as in N=1 with anti-instanton backgorund. The pairing mechanism which is responsible for the cancellation of the one loop contribution in the familiar N=1 case turns out to be effective also in the present situation.
From this result we can derive interesting consequences. As usual in super-instanton computations one can obtain the the renormalization group $\b$-function by combining the measure for zero-modes with the one-loop contribution to the path-integral. We find that the part independent of the Grassmann variables in the effective action gives the usual running of the coupling constant for $U(2)$ super Yang-Mills.
The gluino condensate is not deformed by $C$ since the anti-instanton moduli space measure is unchanged with respect to the N=1 case.
Finally one word of caution is in order as to the physical interpretation of our results. This model makes sense only in Euclidean spacetime and therefore the usual physical interpretation of (anti)-instanton solutions as tunneling processes between topologically different bosonic Minkowski vacua of the theory is elusive.
The paper is organized as follows: in Sec. 2, we recall the basic facts about deformed superspace and super Yang-Mills theory. In Sec. 3, we discuss the role of collective coordinates for instanton and anti-instanton in deformed superspace. In Sec. 4, we solve the equations of motion iteratively for both instanton and (anti)-instanton. In Sec. 5, we compute the classical Lagrangian. In Sec. 6 we derive the (quasi) zero-mode measure, and the one-loop contributions to the effective action. Conclusions and future directions are given in Sec. 7.
The lagrangian in deformed superspace for general $U(N)$ gauge group is Note that our conventions are opposite to the ones in . We have chosen anti-hermitian generators [*i.e.*]{} for $U(2)$ we take $T^{a}=i{\s^{a}\over 2}$ for the $SU(2)$ subgroup and $T^{4}={i\over 2}$ for the $U(1)$ part. In this way ${\rm tr}
\{T^{a}T^{b}\}=-{1\over 2} \d^{ab}$. $C^{\m\n}=C_{\dot\a \dot\b}
\e^{\dot\b \dot\g} {{\bar\s}^{\m\n \dot\a}}_{\dot\g}$ is anti-symmetric and anti-selfdual. In these conventions the covariant derivative for any field in the adjoint is $D_{\m}=\del_{\m} + g [A_{\m},\cdot]$.
For a generic group, we have that $\bar\l_{\a} \bar\l_{\b}
\e^{\a\b} = {1\over 2}\bar\l^{a}_{\a} \bar\l^{b}_{\b} \e^{\a\b}
\{T^{a}, T^{b} \}$ and in the case of $G= SU(N)$, this is equal to ${1\over 2}\bar\l^{a}_{\a} \bar\l^{b}_{\b} d^{a b c} \e^{\a\b}
T^{c}$ . This is clearly zero for $SU(2)$. The action is not hermitian and it does not preserve $R$ symmetry. Notice that the operators in the second line on have dimensions 5 and 6. Nevertheless it turns out that the action is renormalizable . The two operators break conformal invariance of the classical theory since the constant $C^{\a\b}$ has mass dimension -1. It has been shown in that unusual mass dimensions can be assigned to $\l$, $\bar\l$ and $C$. This is the key to the renormalizability by power counting.
We work in chiral superspace with $y^{\m}=x^{\m}-i\t\s^{\m}\bar{\t}$ and supercharges In the supersymmetry algebra only the anticommutator of the $Q_{\a}$’s gets modified when we turn on $C^{\dot\a \dot\b}$ The explicit presence of the $C$ deformation in the algebra breaks the amount of supersymmetry from N=1 to N=1/2. The only preserved charges are the $\bar{Q}^{\dot{\a}}$’s.
The symmetry of the action for $C=0$ is the complete superconformal group generated by $D, \Pi,
Q_{\a}, \bar Q_{\dot\a}, S_{\a}, \bar S_{\dot \a}, P_{\a\dot \a},
M_{\a\b}, \bar M_{\dot \a\dot\b}$ where $\Pi$ is the generator of $U(1)$ R-symmetry and $D$ is the generator of the dilatations. Turning on the parameter $C^{\dot\a \dot\b}$, the symmetry group is broken to the group generated by $Q_{\a},
P_{\a\dot\a}, M_{\dot \a\dot \b}, \bar S_{\dot \a}$ which form a subalgebra.
We now describe the collective coordinates which parametrize the coset space $G/H$ where $G$ and $H$ are the symmetry groups of the action and of the solution, respectively. Therefore $G/H$ represents the symmetries of the action which are broken by an explicit instanton solution. In $N=1$ $SU(2)$ super Yang-Mills, given a bosonic solution to the equations of motion one can reconstruct the most general solution depending on the set of bosonic and fermionic collective coordinates $\{b_{i}\}, \{ f_{i}\}$ applying the generalized shift operator $V(\{b_{i}\},\{f_{i}\})= \prod_{i,j}
e^{Q^{bos}_{i}b_{i}} e^{Q^{ferm}_{j}f_{j}}$ where $Q^{bos}_{i}$ and $Q^{ferm}_{j}$ are the fermionic and bosonic generators of broken symmetries . Let us begin considering the deformed anti-instanton solution . The usual solution is modified by the presence of Grassmann variables. The novelty here is that the bosonic background breaks the chiral supersymmetries which are already broken by the C deformation from the outset. The axial R symmetry generated by $\Pi=\t^{\a}\del/\del{\t}^{\a}-\bar{\t}^{\dot{\a}}\del/\del\bar{{\t}}^{\dot{\a}}$ is also explicitly broken by the C deformation $\{\t^{\dot{\a}},\t^{\dot{\b}}\}=C^{\dot{\a}\dot{\b}}$. The R symmetry can be restored introducing the term $-2C^{\a\b}\del/\del
C^{\a\b}$ in $\Pi$ assigning R symmetry number -2 to $C^{\a\b}$ As already remarked the dilatations are broken: the classical action is not conformally invariant. It is also easy to check that only the chiral superconformal generators $S_{\a}$’s are symmetries. The Lorentz invariance of the action is also broken down to the Lorentz generators $M_{\a\b}$. The symmetry group of the Lagrangian is therefore $G=(\bar{Q}_{\dot{\a}},S_{\a},P_{\m},M_{\a\b})$.
The symmetry group preserved by the anti-instanton solution is $H=(\bar{Q}_{\dot{\a}},S_{\a},M_{\a\b})$ and $G/H=\{P_{\m}\}$. The moduli space of the instanton solution is therefore parameterized by the collective coordinate corresponding to the translations. In particular the Grassmann parameters $\xi_{\a}$ and $\bar{\eta}_{\dot{\a}}$ which, in the usual $N=1$ superspace, parameterize the moduli space do no longer index exact zero modes of the action. We therefore expect them to appear in the classical action once we substitute the exact solution. Still they must be integrated over in the path integral, with the modified measure given by their potential in the classical action.
In the instanton background the group of symmetries preserved by the solution is $H=(Q_{\a},\bar{S}_{\dot{\a}},\bar{M}_{\dot{\a}\dot{\b}})$. Therefore we have the usual coset $G/H=(\bar{Q}_{\dot{\a}},S_{\a},P_{\mu},M_{\a\b})$.
In this section we solve the classical equations of motion (we restrict our attention to $U(2)$ gauge group) order by order in fermionic quasi-collective coordinates. We will see that the usual anti-instanton ($F_{\m\n}^{-}=0$) receives corrections of order $g^{0}$ quadratic in fermionic coordinates, whereas the instanton ($F_{\m\n}^{+}=0$) remains an exact solution. The equations of motion for the lagrangian are $$i \s^{\mu} D_{\mu} \bar\l = \l \Big(i g C^{\m\n} F_{\mu\nu}^{-} +g^{2} {{|C|^{2}}\over 2} \l \l \Big)\,,~~~~~
\bar\s^{\mu} D_{\mu} \l = 0\,.$$
We want to solve in an anti-instantonic background. Anticipating the final result, and as a kind of synopsis, we can already say that the fields admit an expansion
We start by setting the fermions to zero, $\l = \bar\l =0$. The equations become As customary, this equation admits self-dual solutions for which The gauge field configuration which solves is the usual anti-instanton where the index $a$ belongs to $SU(2)$, and ${\eta}_{\m\n}^{a}$ are the ’t Hooft symbols. We have written in regular gauge. Note that at this stage the $U(1)$ part of the connection is zero, as usual $U(1)$ (anti-) instantons are necessarily flat. However, we must substitute the background into the Dirac operator. The equations for the fermions become where $D_{\m}^{(0)}$ is the Dirac operator with respect to the connection . Now, the second equation in has non-trivial zero-modes given by where $F_{\m\n}^{(0)}= F_{\m\n}^{(0),a}T^a$. As an aside, note that the modes are not generated by any symmetry of the equations of motion, as superconformal transformations are explicitely broken in this sector. Nevertheless their presence is required by the index theorem.
Once $\l^{(0)}$ has been switched on the equations of motion are modified to In order to satisfy both equations simultaneously, notice that $C_{\m\n}$ is anti-selfdual, and thus $C_{\m\n}F_{\m\n}=C_{\m\n}F_{\m\n}^{-}$. Using this and the Bianchi identity $D^{\m}F_{\m\n}^{+}=D^{\m}F_{\m\n}^{-}$ we see that one can set $\bar\l=0$ and impose The analysis of shows that the correction to $F^{-}_{\m\n}$ to second order in Grassmann coordinates affects only the $U(1)$ part of the curvature, and takes the form F\_\^[(1), 4-]{} = - [14g ]{} C\_ [ł]{}\^[(0), a]{} [ł]{}\^[(0),a]{} with $\l^{4}$=0. Using some spinor algebra, we can calculate the square on the right-hand side. We find We see that the $U(1)$ part of the connection is of order $g^{0}$ and depends quadratically on the Grassmann coordinates.
For later use we write down explicit expressions for the different Grassmann components of $A^{(2),4}_{\m}$. The strategy to follow is based on the fact that if $F_{\m\n}^{-4}$ can be written as $BC_{\m\n}
\del^{2}K(x)$ then $A_{\m}^{4}=-BC_{\m\n}{\del}_{\n}K(x)$, where $B$ is a constant. A simple way to prove this is using spinor notation: assuming $A_{\a\dot{\a}}=C_{\a}^{\g}\del_{\g\dot{\a}}K$. Then where we used $\del_{\a}^{\dot{\a}}\del_{\g\dot{\a}}={1\over 2}
\epsilon_{\g\a}\del^2$.
In our case we have $$F_{\m\n}^{({\bar\eta}^{2}), 4-} = 2 \r^{2} {\bar\eta}^{2} C_{\m\n} \del^{2} K_{2}(x,x_{0},\r)$$ $$F_{\m\n}^{(\bar\eta \xi), 4-} = 4 \r^{2} C_{\m\n} \del^{2} K_{3}(x,x_{0},\r)$$ where It is easy to see that the iteration procedure stops at order ${C\over g}$: the covariant derivative in $SU(2)$ does not get corrections in Grassmann variables, and the covariant derivative in $U(1)$ is the normal derivative for fields in the adjoint, and is thus insensitive to the Grassmann-corrected part of the $U(1)$ connection. Therefore there are no further $U(1)$ normalizable zero modes and $\l^{4}$ stays zero at further orders in the coupling constant . The full solution can then be written
The instanton solution is the same as in the undeformed case. Starting now with ${F_{\m\n}^{(0),+}}=0$ and $\bar\l = \l = 0$, we note that in this case the Dirac operator $\sigma^{\m}D_{\m}$ has non-trivial zeromodes for $\bar\l$ whereas $\bar\sigma^{\m}D_{\m}$ has none, and thus $\l=0$. Once again $\bar\l^{(1)}$ are given by the usual fermions required by the index theorem. The difference with the previous case is that now, as $\l$ has to be zero at this order in Grassmann variables the equations of motion do no acquire a fermion source term, and the initial bosonic solution remains a solution. We can then write the full solution as
In order to do a semi-classical calculation around the (anti-)instanton background, we would like to substitute the solutions found in the previous section into the classical action given in . We consider anti-instantons and instantons in turn.
To find the classical action for the anti-instanton solution we perform the Bogomol’ny trick of and write the action as Ł[ S\_[bos]{}= - d\^[4]{}x [Tr]{} (F\_\^[-]{} - [i2]{} g C\_ łł)\^[2]{} - [14]{} \^ F\_ F\_ .]{} The first term is saturated for the soution found above, and thus the contribution to the classical action comes entirely from the Chern-Simons-like term. However one has to be aware of the fact that the field strengths now depend on the Grassmann variables, and thus the lagrangian will itself depend on fermionic coordinates.
To calculate the topological term we find it convenient to rewrite it as We have to recall that for $a$ in $SU(2)$ $F_{\m\n}^{a}$ is the usual anti-instanton field strength such that $F_{\m\n}^{a,-}=0$ , and $F_{\m\n}^{4,-} = {\xi}^2 {F_{\m\n}^{(\xi)^{2}4,-}} +
{\bar\eta^2} F_{\m\n}^{(\bar\eta^{2})4,-} + F_{\m\n}^{(\bar\eta \xi)4,-}$. The total anti-instanton action will have contributions $$\hskip2cm
= {1\over 4} \Big[F_{\m\n}^{a}F_{\m\n}^{a} + \left(F_{\m\n}^{(\bar\eta \xi),4} F_{\m\n}^{(\bar\eta
\xi),4} - 2 F_{\m\n}^{(\bar\eta \xi),4} F_{\m\n}^{(\bar\eta
\xi),4-}\right) +$$ $$+ (\bar\eta^{2} \xi^{2})\left(2
F_{\m\n}^{(\bar\eta^2),4}F_{\m\n}^{(\xi^2),4}
-2\left(F^{(\bar\eta^2),4-}_{\m\n}F_{\m\n}^{(\xi^2),4}+F^{(\bar\eta^2),4}_{\m\n}F_{\m\n}^{(\xi^2),4-}\right)\right)\Big]\,.$$ From the expressions for $K_1$, $K_2$, $K_3$ we can calculate this quantity. Recalling Ø[ \^2 F\_\^[(\^2),4]{} = 2 \^2 (C\_ \_ \_ - C\_ \_ \_)K\_[1]{} ]{} $$\bar\eta^2 F_{\m\n}^{(\bar\eta^2),4} = -2 \r^{2} \bar\eta^2 (C_{\n\r} \del_{\m} \del_{\r} - C_{\m\s} \del_{\n} \del_{\s})K_{2}$$ $$F_{\m\n}^{(\bar\eta \xi),4} = -4 \r^{2} (C_{\n\r} \del_{\m} \del_{\r} - C_{\m\s} \del_{\n} \del_{\s})K_{3}$$ we find Putting everything together the classical action for the anti-instanton is We then see that, although the $U(1)$ anti-intanton charge density is non vanishing, only the $SU(2)$ part contributes to the total topological charge. This is in accordance with the index theorem.
Because in the instanton solution to the equations of motion one has $\l=0$ the classical instanton action does not suffer any modification from the usual $N=1$ super Yang-Mills. We then have
Before we proceed to set up a semi-classical calculation around the (anti-) instanton solutions found in previous sections, we review the conventional approach to better contrast the new elements which arise in our situation.
The general procedure consists in splitting the fields in the path integral into a classical part which satisfies the classical equations of motion and a quantum part which describes the fluctuations around the classical solution. After fixing a gauge, one plugs the field expansion into the gauge-fixed action and keeps terms up to second order in quantum fluctuations. This yields a product of determinants corresponding to the different fields in the path integral. A characteristic feature of instanton calculations is that the quadratic operators corresponding to these determinants have zero modes which must be treated separately in order for the path integral to give a sensible result. The presence of these zero modes is due to a degeneracy of lowest-energy configurations, which is parameterized by a set of collective coordinates. One must choose a gauge to fix this degeneracy and trade in the integration over zero-modes in the path integral for an integration over the collective coordinates. In the process one picks up a jacobian factor which determines the measure of integration over the collective coordinates.
To see this in a little more detail consider the generic field expansion where $\phi(x;X)$ is a classical background with degeneracy parameterized by collective coordinates denoted generically by $X^{A}$ (these encompass both fermionic and bosonic collective coordinates), and $\delta\phi(x;X)$ is the quantum fluctuation. The zero modes are then $\delta_{A}\phi_{n}(x)$, where $\delta_{A}$ denotes differentiation with respect to $X^{A}$. The action is then expanded to quadratic order in the fluctuations, $S=\delta\phi_{n}^{N} {\cal O}_{nm}^{NM}\delta\phi_{m}^{M}$. Expanding the fluctuations in terms of a complete set of eigenvectors of ${\cal O}$ where $\delta_{A} \phi_{n}^{M}$ are the zero-modes and $\tilde{\phi}_{n}^{M}$ are the non-zero eigenvectors, the path integral measure becomes where $g_{AB}= k \int d^{4}x {\rm} Tr \delta_{A}\phi_{r}^{M} \delta_{B}\phi_{r}^{M}$ is the suitably normalized metric of inner products of the zero-modes.
The next step is to fix the gauge for both non-zero and zero-modes, and trade $d\xi^A$ for $dX^a$ in the integration over the zero-modes. To do that one performs a BRST quantization inserting unity into the path integral in the form where $G(\tilde\phi)=D_{m}\tilde\phi_{m}$ (background gauge) and $f_{A}= k \int d^{4}x {\rm} Tr \delta\phi_{m}^{M} \delta_{A}\phi_{m}^{M}= \sum_{A} \xi^{B}g_{AB}$. $\delta(G)$ fixes the gauge for the non-zero modes, and ${\rm det} \Big| {\delta G(\phi^{\tilde\Omega})\over \delta \Omega} \Big|$ gives the usual Faddeev-Popov determinant. For the zero-modes $\delta(f_{A}(X))$ enforces $\xi^{A}=0$, as $g_{AB}$ is invertible. Moreover to leading order in g, ${\delta f_{A}\over \delta X^{B}}=g_{AB}$ so that The measure for the gauge-fixed action becomes Note that one can introduce additional anticommuting ghosts $\bar{c}^{(0)a}$ and $c^{(0)b}$ for the bosonic zero-modes, and commuting ghosts $\bar{\gamma}^{(0)\a}$ and $\gamma^{(0)\b}$ for the fermionic zero-modes, and write assuming orthogonality between bosonic and fermionic zero-modes. We now see how these considerations apply to our situation.
The splitting into background plus fluctuation of the fields in the path integral takes the form with the expressions for the different fields given in . We now must compute the quadratic part in the fluctuations of the classical lagrangian. However, we should stress that the background is non-conventional, insofar as it includes fermions. Usually fermion zero-modes are not included in the classical background, and are treated in perturbation theory . The main difference with the conventional treatment boils down to the fact that the background fermions induce additional fermion-boson and fermion-fermion couplings in the quadratic expansion of the classical action. In the one-loop calculation one is then forced to compute the [*superdeterminant*]{} of these fields.
To perform the computation we choose a $U(2)$ background gauge fixing $$D_{\m}^{(0)}Q_{\m} = 0$$ which adds to the action. The gauge-fixed action action, up to quadratic fluctuations, can be written as where and we have suppressed gauge indices for clarity. The different elements of this matrix can be read from $$\hskip-1.8cm
+ \Big[ - {1 \over 2} (\del_{\m} Q^{4}_{\n})^{2} + {1 \over 2}(\del_{\m} Q^{4}_{\m})^{2} \Big]$$ $$\hskip -2cm
2 {\rm tr} \l \not\!\!D \bar\l |_{quad}= -\bar{q}^{4}
\bar\s^{\m}{\del}_{\m}q^{4} - q^4 \s^\m \del_\m \bar{q}^4 -
\bar{q}^{a}(\bar\s^{\m}D_{\m}^{(0)}q)^{a} - q^a (\s^\m
D_\m^{(0)}\bar q)^a$$ $$- \left( \bar{q}^{a} \bar{\s}^{\m}[Q_{\m},\l^{(0)}]^{a}+\l^{(0),a} \s^{\m} [Q_\m , \bar q]^a\right)$$ and $$\hskip+2.5cm
+ {1\over 2} \Big[C_{\m\n} \left( \Big( \del_{[\m}
Q_{\n]}^{a}+ [A_\m^{(0)},Q_\n ]^{a} \Big)q^{4} \l^{(0),a} +
\del_{[\m}Q^{4}_{\n]} q^{a} \l^{(0),a}\right) \Big]$$ $$%\hskip-1.2cm
%+ 2 (\l^{(0),a}q^{a})(\l^{(0),b}q^{b}) \Big]$$ The expansion of the superdeterminant can be done systematically using In our case $X_{bb}={\cal A}_{\n\m}$ is the usual bosonic quadratic operator for $U(2)$ (super) Yang-Mills , $Z_{ff}= \left( \matrix{{{\cal E}_{\b}}^{\a} & {\cal F}_{\b \dot{\a}} \cr {\cal H}^{\dot{\b}\a}
& {{\cal I}^{\dot{\b}}}_{\dot{\a}}} \right) = \Delta_{D}+\Delta_{E}$. $\Delta_{D}$ is the usual operator for adjoint Dirac fermions in $U(2)$ (super) Yang-Mills and $\Delta_{E}$ encodes the additional fermion-fermion couplings arising from the first line in and from . We see that ${{\cal I}^{\dot{\b}}}_{\dot{\a}}=0$. Finally $Y_{bf}=\left(\matrix{{{\cal B}_{\n}}^{\a} & {\cal C}_{\n \dot{\a}}} \right)$ and $W$ its fermionic transpose. We can expand the logarithm of the superdeterminant as $$\hskip1.5cm
- {\rm Tr}\,{\rm log} \left(1+\Delta_{D}^{-1}\Delta_{E}-\Delta_{D}^{-1} \left( \matrix{{\cal D}_{\b \m} \cr {{\cal G}^{\dot{\b}}}_{\m}} \right) ({\cal A}_{\m\n})^{-1} \left( \matrix{{{\cal B}_{\n}}^{\a} & {\cal C}_{\n \dot{\a}}} \right) \right)$$ The first two terms in give the same contributions as in $N=1$ $SU(2)$ super Yang-Mills, as the $U(1)$ part in these terms yields only an infinite constant which is canceled once we normalize with respect to the vacuum. The calculation is standard, but we reproduce it here for completeness. Integration over $A_{\m}^{a}$ yields where ${\cal A}_{\m\n} = - (D^{(0)})^{2} \delta_{\m\n} - 2 F^{(0)}_{\m\n}$ , and the prime indicates that the determinant has to be amputated because, as usual, it has zero-modes. Integration over the $SU(2)$ fermions gives Here $\Delta_{+} = - \bar\sigma^{\m} D_{\m} \sigma^{\n}D_{\n} = -D^{2}$ is the hermitian operator for the $SU(2)$ $\bar\l$ fluctuations (with the same spectrum as $\sigma^{\m}D_{\m}$). As there are no $\bar\l$ zeromodes the determinant is the full one. On the other hand $\Delta_{-} = - \sigma^{\m} D_{\m} \bar\sigma^{\n}D_{\n} = -D^{2} - \sigma_{\m\n}F_{\m\n}^{(0)}$ has zeromodes, and the determinant has to be amputated. The spectrum of non-zero eigenvalues of $\Delta_{+}$ and $\Delta_{-}$ is the same. For the ghosts one gets ${\rm det} \Delta_{ghosts}$, where $\Delta_{ghosts} = - D^{2}$.
It is a standard fact that Using this the total product of determinants is given by which is formally one, since as mentioned before both determinants have the same spectrum of non-zero eigenvalues. One has to introduce a regularization scheme to make sense of these determinants, however. The usual procedure is to use a Pauli-Villars regulator mass. With this the total product of determinants picks up a factor of $\m^{n_{b}-{1\over 2}n_{f}}=\m^{6}$ where $n_{b(f)}$ is the number of bosonic (fermionic) zero-modes.
The new part of the calculation comes from the second line in . One has to expand the logarithm in powers of the background fields and keep gauge invariant combinations. The possible one-loop diagrams contributing have been analyzed in . The non vanishing one-loop Feynman diagrams come in two different topologies (see figs. 3 and 6 in ). We now show how these diagrams appear from the super-determinant expansion. The first diagram contributes to the renormalization of the $A\l\l$ vertex and contains a loop made of two fermionic and one bosonic propagator, in which the bosonic propagator runs between two external $\l$, and the fermionic between external $A$ and $\l$. The modified C dependent vertex can be any vertex of the diagram. Using the notation of the diagrams $(1_c,2,3),(1,2_c,3),(1,2,3_c)$ all contribute (the index “c” specifies the position of the modified vertex). As usual in background field formalism only a reduced set of diagrams contribute. For this topology it is easy to see, using the background Feynman rules, that only the diagram $(1,2,3_c)$ contribute with a background $U(1)$ photon entering vertex 3, 2 $SU(2)$ fluctuation fermions and one $SU(2)$ fluctuation photon circulating in the loop. Putting $x=\Delta_{D}^{-1}\Delta_{E}$, $y=-\Delta_{D}^{-1} \left( \matrix{{\cal D}_{\b \m} \cr {{\cal
G}^{\dot{\b}}}_{\m}} \right) ({\cal A}_{\m\n})^{-1} \left(
\matrix{{{\cal B}_{\n}}^{\a} & {\cal C}_{\n \dot{\a}}} \right)$ and expanding the logarithm we see that in our approach the three vertex diagram arises from the term $xy$. Indeed in $xy$ one finds the term with two background fermions $\l^a\l^a$ and one background $A_{\m}^{4}$ (note that terms with an $A_{\m}^{a}$ background vanish due to the anti-selduality of $C$). All the other terms in $xy$ do not give diagrams consistent with the Feynman rules and therefore do not contribute. We can also draw a new diagram which is identical to $(1,2,3_c)$ apart from substituting the modified vertex with a new vertex originating from the term $-|C|^2/16
(q^aq^a)(\l^{(0)b}\l^{(0),b})$ present in $-|C|^2/4(\l\l)^2_{quad.}$. The new diagram cancels exactly the previous one. Indeed the modified vertex in $(1,2,3_c)$ comes from $-1/4
C_{\m\n}\partial_{[\m}A_{\n]}^4q^aq^a=1/16|C|^2\l^{(0),a}\l^{(0),a}q^bq^b
$ where we used the deformed anti-instanton equation $F_{\m\n}^{(-),4}=-1/4C_{\m\n}\l^{(0),a}\l^{(0),a}$.
By power counting at the vertices (and accounting for the $g$-scalings of the different background fields) this diagram goes as $|C|^2$. Moreover, the external field structure implies that it is proportional to $\bar\eta^2 \xi^2$. The other non-vanishing diagram is the one that renormalizes the $\l\l\l\l$ coupling and has four external fermions, a loop with two fermionic propagators and two bosonic propagators. Using the background Feynman rules we find that, in agreement with , the only consistent diagrams are $(1_c,2_c,3,4)$ and $(1,2,3_c,4_c)$ with one U(1) photon connecting the C-modified vertices and SU(2) fermions circulating in the loop. This diagrams come from the $y^2$ term of the super-determinant expansion. For instance the diagram $(1_c,2_c,3,4)$ arises from The contribution of this diagram is also zero because we have four fermions in the background and $(\l^{(0),a}\l^{(0),a})^2=0$. Once again this goes as $|C|^2 \bar\eta^2 \xi^2$. It is also possible to check that all the other terms coming from the super-determinant expansion do not generate consistent diagrams. Therefore the one-loop effective action is zero and the zero modes remain unlifted to this order in perturbation theory. Much as in the $N=1$ case, the supersymmetries left unbroken by both the anti-instanton and the C-deformation are still effective in compensating the bosonic and fermionic fluctuations in the one-loop (super)determinant.
Finally, the last part of the calculation corresponds to the integration the modes that were amputated in the determinants of the first line in , namely the zero-modes of $A_{\m\n}$ and $\Delta_{-}$. The calculation reduces to the computation of the superdeterminant for ordinary $N=1$ SYM. $$\hskip-3cm
= 2^{10} \pi^{6} g^{-8} \r^{3} \Big({g^{2}\over 32\pi^{2}} \Big) \Big({g^{2}\over 64\pi^{2}\r^{2}} \Big)$$ In the formula above $b_{a}$ stands for bosonic coordinates, namely $x_{0},\r$ and gauge orientations, and $f_{\alpha}$ stands for $\bar\eta, \xi$. The normalization for the inner products is Boson and fermion zero-modes are orthogonal. Putting everything together the total semi-classical partition function for the anti-instanton is $$\int d^{4}x_{0} \int \r^{3} d\r \int d^{2}\xi \Big({g^{2}\over 32 \pi^{2}} \Big) \int d^{2} \bar\eta \Big({g^{2}\over 64 \pi^{2} \r^{2}} \Big)
%{\rm exp} \left[ \left(- {26\over 21} \pi^{2} {|C|_r^{2}(\r)\over \r^{2}} \right) %\bar\eta^{2} \xi^{2}\right]$$
As a final remark we notice that the gluino condensate is unchanged with respect to the familiar N=1 result. This is due to the fact that volume form in the anti-instanton moduli space does not get C-corrections. In the case of the instanton the g-expansion of the different fields is as follows The expansion, up to quadratic terms in fluctuations, of the C-deformed part of the action is then The undeformed part of the action also acquires a new coupling with respect to usual $U(2)$ $N=1$ SYM when we substitute the background fermion $\bar\l^{(0)}$. It is given by however, in this case it is not strictly necessary to incorporate the $\bar\l$ fermions into the classical background, as the classical instanton action does not depend on the background fermions. We can then follow the usual prescription and treat them in perturbation theory. We should however take into account the extra bilinear fermion coupling in . Writing the fermions in Dirac form it can be seen to contribute a (gauge off-diagonal) mass term, which can be treated as a perturbation to the usual Dirac operator $\Delta_{D}$. However, because of the chirality of the two fermions of this new vertex, it does not contribute to the determinant. The analysis can be reduced to standard techniques and we will not further pursue it here.
The results in this paper should be seen as a preliminary step to an instanton calculation with matter included. Indeed, for the weak coupling semi-classical approximation to be self-consistent, in the moduli space integration one should introduce an infrared cutoff to prevent the coupling constant from becoming strong. The one-loop behavior of $g$ for $N=1/2$ supersymmetric theories is the same as in $N=1$ SYM, so one inevitably runs into strongly coupled regions. It is hoped that the Higgs field $VEV$ will provide the infrared cutoff, just as in the ordinary case. (It would be interesting to explore higher-loop renormalizations, using explicit supergraph techniques for example, as well as genuinely non-perturbative consequences our results could have for $N=1/2$ supersymmetric theories.)
A different direction would be to explore whether more general (anti-) instanton solutions to the C-deformed equations of motion exist, and what relevance these could have both at the mathematical and the physical level. The case of multi-instantons immediately comes to mind.
Another very interesting application of the present analysis would be the extension to N=2 SYM (see for example ) where the complete many instanton computation can be performed along the lines of using the methods of equivariant cohomology.
Finally, a dual picture can be constructed. Using the analysis of Ooguri and Vafa one can see that the supersymmetry can be restored by changing the commutation properties of the gluino $\bar\l$ However, this prescription has to be handled at the quantum level in the process of quantizing the gluinos. This implies that the system is constrained and therefore has be carefully discussed. We notice however that there is a similarity in the eqs. of Imaapur and the constraints on the gluinos. We see that the eqs. coincides with if $F^{a,+}_{\mu\nu} \propto C_{\mu\nu}$.
We thank Martin Roček and Peter van Nieuwenhuizen for useful discussions. We also wish to express our gratitude to the organizers of the Simons Workshop in Mathematics and Theoretical Physics at Stony Brook for providing a stimulating atmosphere which encouraged this project. This work is partially supported by NSF grant PHY-0098527.
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[**Addendum**]{} After completion of the first version of this paper, but before submission to the archives, appeared, there is a partial overlap with our results. Also, in the first version, it was erroneously claimed that the $U(1)$ topological charge was not vanishing, as implicitly pointed out in . We believe, however, that the C independence of gluino condensates, and of the volume of the deformed $U(2)$ anti-instanton moduli space, pointed out in that previous version, are independent of that flaw. Moreover, the general iterative method of solving the equations of motion in fermionic quasi-collective coordinates, and the dependence of the topological charge density on these quasi-collective coordinates and C, are of course unaffected.
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abstract: 'We derive a local, gauge invariant action for the $SU(N)$ non-linear $\sigma$-model in $2+1$ dimensions. In this setting, the model is defined in terms of a self-interacting pseudo-vector field $\theta_\mu$, with values in the Lie algebra of the group $SU(N)$. Thanks to a non-trivially realized gauge invariance, the model has the correct number of physical degrees of freedom: only one polarization of $\theta_\mu$, like in the case of the familiar Yang-Mills theory in $2+1$ dimensions. Moreover, since $\theta_\mu$ is a pseudo-vector, the physical content corresponds to one massless [*pseudo-scalar*]{} field in the Lie algebra of $SU(N)$, as in the standard representation of the model. We show that the dynamics of the physical polarization corresponds to that of the $SU(N)$ non-linear $\sigma$-model in the standard representation, and also construct the corresponding BRST invariant gauge-fixed action.'
author:
- |
C. D. Fosco$^a$[^1]\
and\
C. P. Constantinidis$^b$[^2]\
[*$^a$Department of Physics, Theoretical Physics,*]{}\
[*1 Keble Rd., Oxford OX1 3NP, United Kingdom*]{}\
[*$^b$Universidade Federal do Espírito Santo,*]{}\
[*29060-900 Vitória - ES, Brasil*]{}
title: 'A gauge invariant formulation for the $SU(N)$ non-linear $\sigma$-model in $2+1$ dimensions'
---
Introduction {#sec:intro}
============
The non-linear $\sigma$-model [@weinberg1] is a very important tool for the description of the effective, low-energy dynamics of systems with a broken continuous (global) symmetry [@zinn]. Many of its interesting and distinctive features stem from the fact that the symmetry group is realized in a non-linear way, as this endows the theory with a rich structure of interactions. Indeed, it has an infinite number of interaction vertices, when defined in terms of field variables which are themselves group coordinates. Nonetheless, this holds true in spite of the model having a ‘universality’: its properties are completely determined when the symmetry group and the spacetime dimension are known.
Of course, the same non-linearity is also responsible for the fact that, except for the $1+1$ dimensional case, the theory becomes non-renormalizable from the point of view of the usual loop expansion [@zinn]. However, even in more than two spacetime dimensions, the model still has a reasonable predictive power, if properly understood as an effective theory [@weinberg2]. This approach has been successfully applied to chiral perturbation theory [@cpt], as a convenient effective model for $QCD$. Note, however, that in $2+1$ dimensions, the non-linear $\sigma$-model is renormalizable if a large-$N$ expansion is used [@largen], instead of the standard loopwise perturbation theory.
The non-linearity may usually be tackled by resorting to an auxiliary, ‘Lagrange multiplier’ field, which enforces a constraint on the (otherwise free) field variables. The typical example of this is, perhaps, the $O(N)$ non-linear $\sigma$-model, where an auxiliary field imposes a constant-modulus constraint on an $N$-component scalar field ${\vec \phi}\,=\,(\phi_1,\ldots,\phi_N)$, which is a vector field in internal space. An important by-product of this construction is that the auxiliary field is a $O(N)$ singlet, hence, the large-$N$ expansion is easier to formulate after one ‘integrates out’ the $\phi$ field, leaving an action for the Lagrange multiplier.
Indeed, the procedure of ‘linearizing’ an action, by the introduction of auxiliary fields, and afterwards integrating the original fields out to obtain an effective theory for the auxiliary fields, has frequently proven to be very useful. This is particularly true when the auxiliary field has some convenient symmetry or transformation properties [@rivers]. In particular, it allows one to obtain an effective theory where the symmetry properties are inherited from the ones of the Lagrange multiplier in the linearized theory.
In this paper, we introduce a gauge invariant, non-trivially realized Abelian quantum field theory model in $2+1$ dimensions, which is derived by the procedure of integrating out the original variables, in order to obtain an effective theory for the auxiliary field. Since our starting point shall be a representation of the non linear $\sigma$-model where the Lagrange multiplier has a local gauge symmetry, that feature will be preserved in the resulting action. The realization of the Abelian gauge symmetry is non trivial, because the commutator of two gauge transformations is zero only on-shell, i.e., on the configurations that satisfy the equations of motion. Equivalently, the commutator between two ‘true’ gauge transformations yields a trivial, ‘equation of motion’ gauge transformation [@deWit:1978cd; @teit].
The structure of this paper is as follows: in section \[sec:themodel\] we derive the action for model, showing that it is indeed defined by a gauge invariant action. Then we consider the realization and structure of the gauge and global symmetries in section \[sec:gauge\], leaving for section \[sec:quantum\] the quantum treatment of the model. Section \[sec:conc\] contains our conclusions.
The model {#sec:themodel}
=========
We shall begin by reviewing the main features of the polynomial representation for the $SU(N)$ non-linear $\sigma$-model in $2+1$ dimensions, as presented in [@fm1; @fm2]. This formulation may be defined in terms of a gauge invariant Euclidean action $S_{inv}$, which determines the dynamics of two fields $L_\mu$ (vector) and $\theta_\mu$ (pseudo-vector) in the Lie algebra of $SU(N)$: $$\label{eq:defseuc}
S_{inv}[L,\theta] \;=\; \int d^3x \,{\mathcal L}_{inv} (L,\theta)$$ with $$\label{eq:defleuc}
{\mathcal L}_{inv}(L,\theta)\;=\; \frac{1}{2} g^2 L_{\mu}\cdot L_\mu
+ i g \,\theta_\mu \cdot {\tilde F}_\mu(L)$$ where $g$ is a constant with the dimensions of a mass (it is in fact the exact analog of $f_\pi$ in the $3+1$ dimensional case), and ${\tilde F}_\mu (L)$ denotes the dual of the non Abelian field strength tensor for the vector field $L_\mu$, namely, $$\label{eq:defFt}
{\tilde F}_\mu (L) \;=\; \frac{1}{2}\epsilon_{\mu\nu\lambda} F_{\nu\lambda}(L)
\;\;\;,\;\;\;
F_{\mu\nu}(L) \;=\; \partial_\mu L_\nu - \partial_\nu L_\mu
\,+\, g^{\frac{1}{2}} [L_\mu , L_\nu ] \;.$$ Being $L_\mu$ an element in the Lie algebra, with the convention that $L_\mu = - L_\mu^\dagger$, it can be written as $$L_\mu(x) \;=\; L_\mu^a (x) \lambda_a \;\;,\;\; \lambda_a^\dagger
\;=\; - \lambda_a \;,$$ $$\label{eq:conv}
{\rm tr} (\lambda_a \lambda_b) \;=\; - \delta_{a b}\;\;,\;\;
[\lambda_a , \lambda_b ] \;=\; f_{a b c} \, \lambda_c$$ where $f_{a b c}$ is real and completely antisymmetric. Group indices will be indistinctly written as subscripts or superscripts; no meaning should be assigned to the difference. In (\[eq:defleuc\]), we also used the notation: $U \cdot V \equiv U_a V_a$, and $(U \times V)_a =
f_{a b c} U_b V_c$ for any two elements $U$, $V$ in the algebra. Also, both $L$ and $\theta$ have the mass dimensions of $g^{1/2}$.
The ‘inv’ subscript in the action has been introduced in order to emphasize the fact that it is, indeed, invariant under the (local) gauge transformations: $$\label{eq:pgtrns}
\delta_\omega L_\mu \;=\; 0 \;\;\;\;\;\;\;
\delta_\omega \theta_\mu \;=\; D_\mu \omega \;,$$ where the covariant derivative is compatible with the parallel transport defined by $L$, namely, $$D_\mu \omega = \partial_\mu \omega + g^{\frac{1}{2}} [L_\mu , \omega]
\;,$$ or in components: $$(D_\mu \omega)^a = \partial_\mu \omega^a + g^{\frac{1}{2}}
\, f_{a b c} \, L_\mu^b \, \omega_c \;.$$
It must be noted that this gauge symmetry is valid [*of-shell*]{}, namely, it holds true regardless of whether the fields verify the equations of motion or not. Besides, equation (\[eq:pgtrns\]) tells us that $L$ is a gauge-invariant object, and this implies that the commutator of two gauge transformations vanishes: $$\label{eq:polcomm}
\left[\delta_\eta \,,\, \delta_\omega \right]\;=\; 0 \;.$$ Here $\delta_\omega$ and $\delta_\eta$ denote the operators that perform a gauge transformation on a given functional (eventually a function) of the fields. Namely, if $I$ is a functional of $L$ and $\theta$, $$\delta_\omega I[L,\theta] \,=\, \int d^3x \, \delta_\omega
\theta_\mu^a (x) \, \frac{\delta I[L,\theta]}{\delta \theta_\mu^a (x)}
\;,$$ where $\delta_\omega \theta_\mu^a$ is defined as in (\[eq:pgtrns\]). This of course means that the gauge group is [*Abelian*]{}, in spite of the non-Abelian looking transformation rule for $\theta$.
Had we wanted to work with this representation, we should have considered fixing the gauge as the next step. Rather than doing that, we shall move on to derive an ‘effective theory’ for $\theta_\mu$, an auxiliary field which transforms as a vector field in the adjoint representation. To that end, we define the effective action $S_{inv}[\theta]$ by the following expression: $$\label{eq:defst}
\int [{\mathcal D} \theta ] \; e^{ - S_{inv}[\theta] }
\;=\; \int {\mathcal D}\theta \, {\mathcal D}L \; e^{-S_{inv}[L,\theta]}$$ where $[{\mathcal D}\theta]$ denotes the integration measure for $\theta$ in the effective theory (the brackets denote possible group factors). Of course, the integration over $\theta_\mu$ is ill-defined, since the theory is gauge invariant. There is, however, no obstruction to the integration of the $L$-field, since $\theta_\mu$ is, in that case, regarded as a background field. We shall, of course, have to deal with the gauge-fixing for $S_{inv}[\theta]$ afterwards.
The integral over $L_\mu$ in (\[eq:defst\]) is a Gaussian, and its evaluation yields the result: $$\label{eq:st}
S_{inv}[\theta] \;=\; \int d^3x \, {\mathcal L}_{inv}(\theta)\;\;,\;\;
{\mathcal L}_{inv}(\theta) \;=\; \frac{1}{2} {\tilde f}_\mu^a
G_{\mu\nu}^{ab}(\theta){\tilde f}_\nu^b$$ where $\tilde f$ is the dual of the [*Abelian*]{} field [^3] strength: , and $$\label{eq:defg}
G_{\mu\nu}^{ab} \,=\, [ M^{-1} ]_{\mu\nu}^{ab}
\;\;,\;\;
M_{\mu\nu}^{ab}\,=\, \delta_{\mu\nu}\delta^{ab} + i g^{-\frac{1}{2}}
\epsilon_{\mu\lambda\nu} f^{acb} \theta_\lambda^c\;.$$ The fact that $G$ is the inverse of $M$ must be understood in the sense that the relations: $$\label{eq:definv}
G_{\mu\lambda}^{ac} M_{\lambda\nu}^{cb} \;=\; \delta_{\mu\nu} \delta^{ab}$$ are valid. Fortunately, the explicit for of $G$ is not required for most of our presentation. Note, however, that one may easily obtain an approximate expression for it by performing an expansion in powers of the (dimensionless) object $\theta g^{-\frac{1}{2}}$. There arises also from the Gaussian integral a factor which modifies the $\theta$-field integration measure, $$\label{eq:thmeas}
[{\mathcal D}\theta]\;=\; {\mathcal D}\theta \, [\det(M)]^{-\frac{1}{2}}$$
A question that immediately presents itself at this point is what has happened to the gauge invariance; indeed, the gauge invariance in the polynomial representation, equation (\[eq:pgtrns\]), involves $L_\mu$ in its definition, and $L_\mu$ is precisely the field that has been eliminated from the action.
Of course, a standard Maxwell-like gauge transformation will not do, since, although ${\tilde f}_\mu$ is invariant under the Abelian gauge transformations of the Maxwell theory, $G$, that depends on $\theta_\mu$, is not. Indeed, looking for example at the explicit form of the action (\[eq:st\]), with $G$ expanded up to terms of order $\frac{\theta^2}{g}$, we see that: $$S_{inv}[\theta] \;=\; \int d^3x \, \left[ \frac{1}{2} {\tilde
f}_\mu(\theta) \cdot {\tilde f}_\mu(\theta) \,-\,\frac{i}{2}
g^{-\frac{1}{2}} \epsilon_{\mu\nu\lambda} \, \theta_\mu \cdot
{\tilde f}_\nu (\theta) \times {\tilde f}_\lambda(\theta) \right.$$ $$\left. - \frac{1}{2 g} ( \theta_\mu\cdot{\tilde f}_\mu
\theta_\nu\cdot{\tilde f}_\nu \,-\,
\theta_\mu\cdot{\tilde f}_\nu \, \theta_\mu\cdot{\tilde f}_\nu \,
+\, {\tilde f}_\mu\cdot{\tilde f}_\mu \, \theta_\nu\cdot \theta_\nu
-\, {\tilde f}_\mu\cdot{\tilde f}_\nu \, \theta_\mu \cdot
\theta_\nu )
\right] \;,$$ where only the term in the first line is invariant under Abelian gauge transformations. In spite of this, we do expect a gauge invariance to exist for $S_{inv}[\theta]$, since we know there are two unphysical components (for each value of $a$) in $\theta_\mu$, which do appear in the free propagator. This propagator will of course be determined by the free action $$\label{eq:mxwll}
S^{(0)}_{inv} [\theta]\;=\; \int d^3x\, \frac{1}{2} {\tilde f}_\mu^a(\theta)
{\tilde f}_\mu^a(\theta) \;=\; \int d^3x \;\frac{1}{4} f^a_{\mu\nu}(\theta) f^a_{\mu\nu}(\theta)$$ after adding a gauge fixing term.
It is then reasonable to assume that the gauge transformations for $\theta$ should be of the form $$\label{eq:egtrns}
\delta_\omega \theta_\mu \;=\; \partial_\mu \omega + g^{\frac{1}{2}} [L_\mu(\theta) , \omega]$$ where $L_\mu(\theta)$ is a [*dependent*]{} field which plays the role of a connection, and should of course be defined in terms of $\theta$.
A possible hint to find the explicit form of $L_\mu(\theta)$ comes from the fact that performing the Gaussian integration is tantamount to ‘replacing the integrated field by their values at the extreme of the exponent’. Denoting by ${\hat L}_\mu(\theta)$ the expression that maximizes the exponent, we see that it is given by: $$\label{eq:ltheta}
{\hat L}^a_\mu \;=\; - i g^{-1} \, G_{\mu\nu}^{ab}(\theta) {\tilde f}_\nu^b\;.$$ Thus we shall adopt the ansatz $L_\mu(\theta) \equiv {\hat L}_\mu
(\theta)$, the consistency of which we will verify now: to see whether the transformation (\[eq:egtrns\]) is a (gauge) symmetry of the action (\[eq:st\]) or not, we first evaluate the first variation of $S_{inv}[\theta]$ under a general, not necessarily gauge, infinitesimal variation of $\theta$.
After some elementary algebra, we obtain: $$\delta S_{inv}[\theta] \;=\; \int d^3x \,\delta\theta_\mu^a \,
\left\{ \epsilon_{\mu\nu\lambda}\,\partial_\nu
[ G_{\lambda\rho}^{ab}(\theta) {\tilde f}_\rho^b(\theta) ] \right.$$ $$\label{eq:fvar}
\left. -\frac{i}{2} g^{-\frac{1}{2}} \, \epsilon_{\mu\nu\lambda}
\, f_{abc}\, G_{\nu\alpha}^{bd}(\theta)\,{\tilde f}_\alpha]^d
\, G_{\lambda\beta}^{ce}(\theta) \, {\tilde f}_\beta^e \right\}$$ where we used the symmetry property $G_{\mu\nu}^{ab} = G_{\nu\mu}^{ba}$, and the relation $$\delta G_{\mu\nu}^{ab} \;=\; - i g^{-\frac{1}{2}}\,
G_{\mu\lambda}^{ac}(\theta)\epsilon_{\lambda\rho\sigma} \, f^{cde} \,
\delta\theta_{\rho}^d \, G_{\sigma\nu}^{eb} (\theta)\;,$$ both of them consequences of the fact that $G = M^{-1}$. Recalling the definition of $L_\mu(\theta)$, we may also write (\[eq:fvar\]) as: $$\label{eq:fvar1}
\delta S_{inv}[\theta] \;=\; i g \int d^3x \, \delta\theta_\mu^a \,
{\tilde F}_\mu^a (L(\theta))$$ where $${\tilde F}_\mu^a (L(\theta)) \;=\; \frac{1}{2}
\epsilon_{\mu\nu\lambda} F^a_{\nu\lambda}(L(\theta)) \;,$$ $$\label{eq:defF}
F_{\mu\nu}^a (L(\theta)) \;=\; \partial_\mu L_\nu^a (\theta) - \partial_\nu
L_\mu^a(\theta) + g^{\frac{1}{2}} f^{abc}L_\mu^b(\theta) L_\nu^c
(\theta)\;.$$ Using now the explicit form for $\delta\theta_\mu$ that corresponds to a gauge variation, equation (\[eq:egtrns\]), we see that $$\delta S_{inv}[\theta] \;=\; - i g \int d^3x \; \omega^a (x)
[D_\mu {\tilde F}_\mu ]^a(L) \;=\;0 \;,$$ as a consequence of the Bianchi identity, which is of course true regardless of $L$ being an independent field or not. We shall henceforth omit writing the dependence of $L$ on $\theta$ explicitly, since $L$ shall always be assumed to be a dependent field. A small technical point (absent in the real time formulation) is that the relation (\[eq:ltheta\]) includes complex factors: an $i$ multiplying $G$, but also $G$ itself has both real an imaginary parts. That should be hardly surprising, since the action itself is not purely real, as it happens with Euclidean actions including Chern-Simons terms (and with other topological objects in different numbers of dimensions). Thus the relation (\[eq:ltheta\]), to have non-trivial solutions, require the continuation of the fields to complex values. Of course, the gauge invariant action in Minkowski spacetime, $S_{inv}^M$, is real, $$\label{eq:mink}
S_{inv}^M \,=\, \int d^3x \,\frac{1}{2} {\tilde f}^\mu_a
G_{\mu\nu}^{ab}(\theta){\tilde f}^\nu_b$$ where ${\tilde f}^\mu_a = \epsilon^{\mu\nu\lambda}\partial_\nu
\theta_\lambda^a$ and $G_{\mu\nu}^{ab}(\theta)$ is determined by the equations: $$\label{eq:defgm}
G_{\mu\rho}^{ac}(\theta)\, M^{\rho\nu}_{cb}(\theta) \;=\;
\delta_\mu^\nu \delta^a_b
\;\;,\;\;
M^{\mu\nu}_{ab}\,=\, g^{\mu\nu}\delta_{ab} + g^{-\frac{1}{2}}
\epsilon^{\mu\lambda\nu} f^{acb} \theta_\lambda^c\;.$$
Thus we have verified the consistency of the definition of the covariant derivative with the gauge invariance of the action. Note, however, that there is an important difference with the polynomial formulation, in that the gauge transformations for $\theta$ involve $L$, which is itself a function on $\theta$. Thus $L$ will, in general, change under a gauge transformation in this formulation. In particular, this implies that finite gauge transformations will be different to infinitesimal ones. This is a consequence in fact of the algebra of gauge transformations being open, as it will be discussed in the next section.
Also, expression (\[eq:fvar1\]) tells us that the classical equations of motion deriving from $S_{inv}[\theta]$ are: $$\label{eq:mc}
F_{\mu\nu}(L) \;=\; 0 \;.$$ i.e., the Maurer-Cartan equations for $L$, which obviously have a gauge invariant set of solutions.
Regarding the integration measure $[{\mathcal D}\theta]$, it is straightforward to verify that the gauge variation of $[{\mathcal
D}\theta]$ is zero. We conclude that the action (\[eq:st\]) is indeed gauge invariant. The gauge invariance is not of the Yang-Mills type, but rather involves as a connection a vector field $L_\mu$ which is a composite field, defined in terms of $\theta_\mu$ and its derivatives. As we shall see in the next section, the gauge group is indeed Abelian, but the albegra of gauge transformations is not closed off-shell.
It may seem surprising at first sight that the only ‘content’ of the classical equations of motion is that the Maurer-Cartan equations for a field are satisfied, since we still need the dynamics for the true degrees of freedom. Of course, such a dynamics is also present in this description: $L$ is a pure gauge field, i.e., $L_\mu = U^\dagger
\partial_\mu U$ with $U(x) \in SU(N)$, and besides (see (\[eq:global\]) below) $\partial_\mu \cdot L_\mu = 0$. These two equations are the equivalent to the classical equations of motion for the non-linear $\sigma$-model.
Symmetries {#sec:gauge}
==========
The actual form of the gauge transformations, as acting on the field $\theta_\mu$, has been obtained by the procedure of borrowing the (known) form of the corresponding transformations from the polynomial version, and afterwards replacing the field $L_\mu$ by its value at the extreme (a function of $\theta$). This yields, for a transformation parametrized by the function $\omega(x)$, the variation: $$\label{eq:vartheta}
\delta_\omega \theta_\mu(x) \;=\; D^L_\mu \omega(x)$$ where $$\label{eq:defdl}
D^L_\mu \omega \;=\; \partial_\mu \omega + g^{\frac{1}{2}} [L_\mu,\omega] \;,$$ with: $$\label{eq:loft}
L_\mu^a \,=\, -i g^{-1} \, G_{\mu\nu}^{ab}(\theta)\, {\tilde f}_\nu^b (\theta)\;.$$ In spite of the presence of a covariant derivative, the transformations do not correspond to a non-Abelian Yang-Mills theory. Indeed, it should be noted that the transformations (\[eq:vartheta\]) involve the covariant derivative, defined in terms of a composite field which plays the role of a connection. However, they are not of the strictly Abelian type either, since the transformation law for $\theta$ does not correspond to that case.
We shall now see that what happens is that the transformations are, indeed, Abelian, but only [*on-shell*]{}, i.e., on the equations of motion. To be specific, consider the commutator of two gauge transformations, corresponding to the gauge functions $\omega$ and $\eta$. We find that the result may be written, after some algebraic manipulations, as follows: $$\label{eq:gaugecomm}
[\delta_\eta, \delta_\omega ] \theta_\mu^a \;=\;
\Sigma_{\mu\nu}^{ab} (\theta) \, \frac{\delta S_{inv}[\theta]}{\delta \theta_\nu^b}$$ where we introduced the object: $$\label{eq:defsigma}
\Sigma_{\mu\nu}^{ab}(\theta) \;=\; - \frac{1}{g} \,
\eta^h \, \omega^c \, ( f^{aec} f^{dbh} - f^{aeh} f^{dbc})
G_{\mu\nu}^{ed}(\theta) \;.$$ It is important to realize that $\Sigma_{\mu\nu}^{ab}$ is antisymmetric, namely, $$\label{eq:asymm}
\Sigma_{\mu\nu}^{ab} \;=\; -\Sigma_{\nu\mu}^{ba} \;,$$ since this means that the right hand side of (\[eq:gaugecomm\]) is a trivial gauge transformation [@teit]. Indeed, for a given action $S[\theta]$, a transformation of the kind $$\delta \theta_\mu^a \;=\; \Lambda_{\mu\nu}^{ab}(\theta) \frac{\delta
S[\theta]}{\delta \theta_\nu^b}$$ with an arbitrary antisymmetric function $\Lambda_{\mu\nu}^{ab}=-
\Lambda_{\nu\mu}^{ba}$, is a symmetry of $S[\theta]$, regardless of the form of $S[\theta]$. It can also be shown [@teit], that the commutator between a non-trivial gauge transformation and a trivial one yields a trivial gauge transformation. Thus, we see that the physically relevant gauge group is Abelian, and isomorphic to $U(1)^{(N^2 -1)}$ (for $SU(N)$), although realized in a non-trivial way, since the ‘trivial’ part of the gauge transformations cannot be easily eliminated within the present formulation of the model.
A related property is that the composite field $L_\mu$, which is gauge invariant in the polynomial transformation, is now also gauge-invariant but only on-shell: $$\label{eq:delf}
\delta_\omega L_\mu^a \;=\; - i g^{-\frac{1}{2}} \, G_{\mu\nu}^{ab}(\theta)
f^{bcd} {\tilde F}_\nu^c (L) \omega_d \;,$$ i.e., it vanishes when ${\tilde F}_\mu (L)= 0$.
The question that immediately presents itself is what are the conditions a gauge invariant functional must verify. This is of course important, since gauge invariant functionals are naturally associated to physical observables. Besides, in the functional integral approach to a quantum gauge field theory, the condition a gauge invariant functional must satisfy is an important part of the formulation.
So, assuming $I[\theta]$ to be a gauge invariant functional of $\theta$, it must verify the condition: $$\label{eq:ginvi}
\delta_\omega \, I[\theta] \;=\; 0 \;,$$ where $$\delta_\omega \;=\; \int d^3x \, \delta_\omega \theta_\mu^a(x) \,
\frac{\delta}{\delta \theta_\mu^a(x)} \;.$$ However, if such a gauge invariant functional exists, one immediately gets a consistency condition by applying two successive gauge transformations on $I$ and subtracting them, namely: $$\label{eq:ccond}
\delta_\omega I[\theta]=0 \;\;\Rightarrow \;\;
\left[ \delta_\eta \,,\, \delta_\omega \right] I[\theta] \,=\, 0 \;.$$ On the other hand, we may of course evaluate the commutator of two gauge transformations; after some algebra, we find: $$\label{eq:comm}
[ \delta_\eta \,,\, \delta_\omega ] \;=\; \int d^3x \,
\Sigma_{\mu\nu}^{ab}(\theta) \, \frac{\delta S_{inv}}{\delta
\theta_\mu^a (x)} \frac{\delta }{\delta\theta_\nu^b (x)} \;.$$ Thus, for non-trivial gauge invariant functional $I$ to exist, since $\Sigma$ depends on the arbitrary functions $\eta$ and $\omega$, we have to impose the additional condition: $$\label{eq:extra}
F_{\mu\nu}(L) \;=\; 0\;.$$ This is nothing new from the classical point of view, but it makes a difference for the quantum theory, where all the configurations matter, and not just the extrema of the action. This seems to lead us to the inclusion of (\[eq:extra\]) as a constraint, what is not what we want. Fortunately, there are ways out of this [@teit], that does not require the introduction of extra constraints (wich might even reduce the number of degrees of freedom.
Regarding the global symmetries, we know that $L_\mu$ is a conserved current, associated to a global symmetry of the non-linear $\sigma$-model. To see that $L_\mu$ is conserved in this formalism is a bit tricky. One possible way to prove that is to use the property that the composite field $L_\mu$ as given by (\[eq:loft\]) may also be written, after some algebra, as: $$\label{eq:eqlt}
L_\mu \;=\; -i g^{-1} \, \epsilon_{\mu\nu\lambda} D_\nu \theta_\lambda \;,$$ where we used the property: $$G_{\mu\nu}^{ab} (\theta) \;=\; \delta_{\mu\nu}^{ab} \,-\, i g^{-\frac{1}{2}}
\epsilon_{\mu\lambda\sigma} f^{acd} \theta_\lambda^c G_{\sigma\nu}^{db}(\theta)
\;.$$ Then it follows that $$\label{eq:global}
\partial_\mu L_\mu \;=\; D_\mu L_\mu \;=\; -i g^{-1} \, \epsilon_{\mu\nu\lambda}
D_\mu D_\nu \theta_\lambda \;=\; -i g^{-\frac{1}{2}} [ {\tilde F}_\mu (L),
\theta_\mu ]$$ which vanishes on shell, and implies the conservation of $L_\mu$. The conserved charge is of course given by the space integral of $L_0$. It is instructive to consider the particular case of a static point-like static charge of color $a$ and strength $q$ located at ${\mathbf x} = {\mathbf x}_0$. This corresponds to a charge density $$L^a_0 (x) \;=\; - i q \delta ({\mathbf x} - {\mathbf x}_0) ,\;\;\;
L_j (x) \;=\; 0\;.$$ Inserting this into the relation (\[eq:loft\]) yields $${\tilde f}_\mu^a \;=\; q \, \delta_{\mu 0} \,
\delta({\mathbf x} - {\mathbf x}_0)$$ i.e., it corresponds to a point like magnetic flux sitting on the same point. The conserved charge is then equal to the total magnetic flux (for that color).
Quantum theory {#sec:quantum}
==============
We shall consider here the quantum theory corresponding to this gauge invariant model, from the path integral approach. The natural object to consider is then of course the generating functional for $\theta$-field correlation functions. The ill-defined (gauge invariant) partition function shall be given by the expression: $$\label{eq:defzinv}
{\mathcal Z}_{inv}[J] \;=\; \int [{\mathcal D}\theta] \,
\,\exp \left\{- S_{inv}[\theta] + \int d^3x J_\mu \cdot \theta_\mu \right\} \;.$$
The generating functional (\[eq:defzinv\]), being gauge invariant, requires the introduction of a gauge-fixing term and its companion ghost action to be well-defined. However, a standard Faddeev-Popov approach to the definition of the gauge-fixed action will not do, since the resulting action is neither BRST invariant, nor the transformation becomes nilpotent. The difficulty lies, of course, in the fact that the algebra of the gauge transformations is ‘open’, namely, it closes only when the equations of motion are satisfied. However, a modified action, which generally involves quartic ghost terms may be constructed, such that the action is invariant under an extended BRST transformation [@deWit:1978cd; @teit]. By an application of such method to this case, we obtain the gauge-fixed action $S$: $$\label{eq:defsfx}
S[\theta_\mu;b,{\bar c}, c] \;=\; S_{inv}[\theta] \,+\,
S_{gf}[b,\theta]\,+\, S_{gh}[{\bar c},c;\theta]$$ where We shall adopt the covariant gauge-fixing term: $$\label{eq:defsgf}
S_{gf}[\theta] \;=\; \int d^3x \,( - \frac{1}{2\lambda} b^2
\,+\, b \cdot \partial_\mu \theta_\mu )$$ and the corresponding ghost action becomes $$\label{eq:defsgh}
S_{gh}[{\bar c},c;\theta]\,=\, \int d^3x \left[
\partial_\mu {\bar c} \cdot D_\mu^L c \,+\,
\frac{1}{2g} (\partial_\mu {\bar c} \times c)^a
G_{\mu\nu}^{ab}(\theta) (\partial_\nu{\bar c}\times c)^b \right]\;.$$ The existence of a quartic term in the ghosts makes it evident that the BRST transformations are not of the standard form. Indeed, we find that the precise form for the transformations is: $$\delta \theta_\mu^a \;=\; \xi \, (D_\mu c)^a \,+\, \xi \frac{i}{g} f^{abe}
G_{\mu\nu}^{bd} (\partial_\nu {\bar c} \times c)^d c^e$$ $$\label{eq:defbrst}
\delta c \,=\, 0 \;,\;\;\;
\delta {\bar c} \,=\, i \xi b \;,\;\;\;
\delta b \,=\, 0 \;.$$ They leave the action $S$ invariant, and the transformation is besides nilpotent.
The generating functional for the gauge-fixed action is then defined as follows: $${\mathcal Z}[J;j, {\bar\eta}, \eta] \;=\; \int [{\mathcal D}\theta]
\,{\mathcal D}b \, {\mathcal D}{\bar c} \, {\mathcal D}c \;$$ $$\label{eq:defztheta}
\times \,\exp \left\{- S[\theta;b, {\bar c}, c] +
\int d^3x ( J_\mu \cdot \theta_\mu + j \cdot b \,+\,
{\bar\eta}\cdot c + {\bar c} \cdot \eta ) \right\} \;.$$
It should be noted that, in all of the above equations, the covariant derivative is defined in terms of the dependent field $L$, which is a function of $\theta$.
This may be thought of as the main result of this letter, namely, there exists a gauge invariant description for the non-linear $\sigma$-model in $2+1$ dimensions; that description is built in terms of $\theta$, a pseudo-vector field in the algebra of the group. The gauge algebra is however open, what makes the BRST quantization less immediate than for the Yang-Mills case (although the algebra is Abelian on-shell). The resulting gauge fixed action contains terms quartic in the ghosts, and is invariant under a global BRST symmetry. This BRST symmetry may be applied to, for example, the derivation of Ward identities that will restrict the form of the counterterms.
Regarding the quantum corrections, it should be noted that there is another (equivalent) possibility to tackle the problem of open gauge algebras, through the introduction of auxiliary field. Their function is to render the on-shell symmetry into an off-shell one, where the Faddeev-Popov trick may be applied. The upshot of this procedure here, leads one to the ‘polynomial formulation’ Lagrangian of (\[eq:defleuc\]), whose renormalization properties have been considered in [@fm1].
Conclusions {#sec:conc}
===========
We have shown that the $SU(N)$ non-linear $\sigma$-model in $2+1$ dimensions may indeed be described by a gauge invariant action $S_{inv}[\theta]$, for a single pseudo-vector field $\theta$. That action has a gauge invariance which involves a composite field $L$ (a function of $\theta$) that plays a role similar to a connection. This, however, is so only when one considers infinitesimal gauge transformations. Finite gauge transformations, and the composition of two gauge transformations show that the gauge algebra is open. The resulting classical theory shows no difference with the standard formulation of the non-linear $\sigma$-model, since the classical trajectories are the only important part of the action, and there the algebra closes.
For the quantum theory, however, the situation is more complicated, as the BRST quantization requires the introduction of a term which is quartic in the ghosts. However, the corresponding global BRST symmetry exists, and may indeed be used as a starting point in the construction of the quantum effective action. We also note that this open algebra formulation is also equivalent to the polynomial formulation, where the algebra is closed and Abelian.
Acknowledgments: {#acknowledgments .unnumbered}
================
The authors wish to thank The Abdous Salam ICTP, where this work was initiated, for the warm hospitality. C. D. F. was supported by Fundación Antorchas, Argentina. C. P. C. thanks Olivier Piguet for useful discussions and a careful reading of the manuscript.
[99]{} S. Weinberg, Phys. Rev. [**166**]{}, 1568 (1968). See, for example:\
section [**14**]{} of: J. Zinn-Justin, [*Quantum Field Theory and Critical Phenomena*]{}, Clarendon Press, Oxford (2002). S. Weinberg, Phys. Lett. B[**91**]{}, 51 (1980). See for example:\
A. Pich, Rept. Prog. Phys. [**58**]{}, 563 (1995), for a modern review. I. Y. Arefeva, Annals Phys. [**117**]{}, 393 (1979). See, for example, Chapter [**17**]{} of:\
R. J. Rivers, [*Path Integral Methods in Quantum Field Theory*]{}, Cambridge University Press, Cambridge (1988). For an early reference, see for example:\
B. de Wit and J. W. van Holten, Phys. Lett. B [**79**]{}, 389 (1978). For a review, see for example:\
M. Henneaux and C. Teitelboim, [*Quantization of Gauge Systems*]{}, Princeton University Press, Princeton, NJ (1992). C. D. Fosco and T. Matsuyama, Int. J. Mod. Phys. A [**10**]{} (1995) 1655. C. D. Fosco and T. Matsuyama, Prog. Theor. Phys. [**93**]{} (1995) 441.
[^1]: CONICET
[^2]: CNPq
[^3]: We adopt the convention that a lowercase $f_\mu$ refers to the dual of the [*Abelian*]{}field strength, while the uppercase one is reserved for the dual non Abelian one.
|
---
abstract: 'The search for habitable exoplanets and life beyond the Solar System is one of the most compelling scientific opportunities of our time. Nevertheless, the high cost of building facilities that can address this topic and the keen public interest in the results of such research requires the rigorous development of experiments that can deliver a definitive advance in our understanding. Most work to date in this area has focused on a “systems science” approach of obtaining and interpreting comprehensive data for individual planets to make statements about their habitability and the possibility that they harbor life. This strategy is challenging because of the diversity of exoplanets, both observed and expected, and the limited information that can be obtained with astronomical instruments. Here we propose a complementary approach that is based on performing surveys of key planetary characteristics and using statistical marginalization to answer broader questions than can be addressed with a small sample of objects. The fundamental principle of this comparative planetology approach is maximizing what can be learned from each type of measurement by applying it widely rather than requiring that multiple kinds of observations be brought to bear on a single object. As a proof of concept, we outline a survey of terrestrial exoplanet atmospheric water and carbon dioxide abundances that would test the habitable zone hypothesis and lead to a deeper understanding of the frequency of habitable planets. We also discuss ideas for additional surveys that could be developed to test other foundational hypotheses is this area.'
author:
- 'Jacob L. Bean'
- 'Dorian S. Abbot'
- 'Eliza M.-R. Kempton'
bibliography:
- 'ms.bib'
title: A Statistical Comparative Planetology Approach to the Hunt for Habitable Exoplanets and Life Beyond the Solar System
---
Introduction {#sec:intro}
============
Through the study of exoplanets, humanity stands on the threshold of making significant progress towards answering the age-old question of whether there is life elsewhere in the Universe. Exoplanet surveys, and in particular NASA’s *Kepler* mission, have revealed that small planets are common in circumstellar habitable zones in our Galaxy [@petigura13; @burke15; @dressing15]. The search for exoplanets has recently culminated in the discovery of the first Earth-size planets in the habitable zones of nearby stars [@angladaescude16; @gillon17; @dittmann17]. By virtue of their orbiting nearby stars, and with the pending advent of powerful new instruments and facilities, these newly discovered planets are the first bona fide targets for future efforts using the techniques of astronomical remote sensing to determine if they are truly habitable and possibly even inhabited [@kreidberg16; @meadows16; @barstow16; @turbet16; @lovis17].
The exciting possibility of finding other Earth-like worlds and life beyond our Solar System has motivated the development of new instruments for existing telescopes and is a key justification for the construction of the next generation of extremely large ground-based telescopes (“ELTs”). The characterization of potentially habitable planets is also expected to be a major part of the science program for the *James Webb Space Telesscope* (*JWST*), which is planned for launch in 2018 [@deming09; @beichman14; @cowan15]. Furthermore, two of the four flagship mission concepts currently being developed by NASA in preparation for the next Astronomy and Astrophysics Decadal Survey have characterization of terrestrial exoplanets as a main driver[^1], while a third is being designed with this science case as an option[^2].
With the fast-approaching opportunity to make a search for habitable environments and life on exoplanets comes the very real challenge of actually designing an experiment that will deliver clear results. The scientific challenges of designing experiments on this topic are formidable because terrestrial exoplanets are expected to be diverse in the structures and compositions of their interiors and atmospheres. This expectation is based on our knowledge of the current and past states of the terrestrial planets of the Solar System, the diversity of bulk compositions inferred for the known low-mass exoplanets, the different properties of the stars in the solar neighborhood, and the random nature of the planet formation and evolution processes.
The challenges of designing a robust experiment are compounded by the fact that even the best telescopes and instruments currently under construction or in design will only reveal a small piece of a planet’s puzzle on their own. This is due to the practical limitations of certain approaches [e.g., the challenge of determining planet masses with direct imaging, @brown15] and the finite grasp of instruments as set by technological constraints or fundamental physics (e.g., not having sufficient spectral coverage to detect all the chemical species of interest in an exoplanet’s atmosphere using a single instrument). Therefore, the questions of how many planets need to be characterized, which planetary properties need to be determined, and what level of precision is needed in the measurements are not trivial to answer, yet they have profound implications for the cost, risk, and timescale of the program [@stark15; @stark16].
To answer this experimental design challenge, we suggest that a statistical comparative planetology approach should be a key element in efforts to address the topics of habitable worlds and life beyond the Solar System. Statistical comparative planetology requires a broad survey that will necessarily be less detailed than what could be obtained with an approach focused on a small number of planets. However, the fundamental principle of this approach is maximizing what can be learned from each type of measurement by applying it widely as an alternative to bringing multiple detailed measurements to bear on a single object. The advantages of this approach are that a broad survey can give context to aid the understanding of individual planets, and it enables conclusions to be reached statistically despite ambiguous results for individual objects. The statistical comparative planetology approach also uses the diversity of exoplanets as an advantage to be exploited rather than a challenge to be overcome. Additionally, this approach can be built around simple physical models instead of the more complex models needed to make accurate statements about an individual planet.
The statistical comparative planetology approach we advocate is an extension of the hugely successful efforts to determine planetary frequency, and it is informed by lessons learned in the atmospheric characterization of close-in transiting exoplanets. However, our proposal is somewhat at odds with the prevailing “planetary systems science” approach to the problem. We therefore begin in §\[sec:systems\] with a review of the planetary systems science approach to addressing the topics of habitable worlds and life beyond the Solar System. In §\[sec:lessons\] we place the future characterization of Earth-like exoplanets in the context of recent work on the frequencies and atmospheres of transiting exoplanets. We describe example experiments to test the concept of the habitable zone in §\[sec:hz\]. We conclude with some suggestions for future work to expand on these ideas in §\[sec:conclusion\].
The Systems Science Approach Reviewed {#sec:systems}
=====================================
Most work to date has been focused on the development of a planetary systems science framework (“systems science” hereafter for brevity) for designing and interpreting observations of potentially habitable exoplanets [@seager2005vegetation; @meadows08; @seager2010exoplanet; @kaltenegger2012rocky; @rugheimer2015effect; @meadows16; @robinson17]. The systems science framework aims to reveal the nature of individual planets based on a combination of empirical data and theoretical modeling. The empirical data should be as complete as possible to minimize model dependencies, while the theoretical models are benchmarked on the Solar System planets to maximize their accuracy [e.g., @robinson11; @robinson14]. The ultimate goal of the systems science approach is to identify particular exoplanets that are habitable and to make statements about the possibility that they harbor life.
One strength of the systems science approach to the question of life on other planets is that a sequential roadmap can be written down that guides the construction of new facilities and the interpretation of the data they will obtain. However, the focus on individual planets in the systems science approach necessitates extensive characterization using multiple techniques and instruments, which increases the cost of the program and results in a long lead time for getting robust answers.
The systems science approach can also be interpreted to suggest that we should first characterize a small number of promising planets since that would require a smaller telescope, or that efforts to obtain more data for a small sample of planets should take precedence over a broad survey when allocating time on a larger telescope. Both of these approaches runs the risk of delivering detailed characterization of a small number of planets, but not finding any definitive indications of biosignatures and with the end result being no clearer understanding of the prevalence of life. Furthermore, the dependency on complex theoretical models will never be fully relaxed when the aim is to understand individual planets due to the limitations of astronomical remote sensing. The expected diversity of exoplanets also suggests that the models trained on the Solar System planets will be stretched in ways that may undermine their accuracy. It is telling that a range of false positive and false negative scenarios for habitability and life have already been identified [e.g., @wordsworth14; @domogal-goldman14; @reinhard17].
We propose here to reframe the question from “Are there other habitable or inhabited planets?” to “What are the frequencies of habitable and inhabited planets?”. This new question requires a larger, statistical sample and an altered observational strategy compared to what would be motivated from a simple interpretation of the systems science approach. However, we think that this question can be more robustly answered, and ultimately this information may be required to interpret the characterization of individual planets anyway.
Lessons From Recent Transiting Exoplanet Studies {#sec:lessons}
================================================
The ongoing characterization of close-in exoplanets using transit techniques offers compelling lessons on the power of statistical comparative planetology. The *Kepler* mission in particular has been transformative for not just what we know about exoplanets, but also how we go about obtaining the information. A key breakthrough in the analysis of *Kepler* data was the calculation of the false positive rate for transiting planet candidates [@torres11; @morton11]. This opened a shortcut past confirmation of individual targets, which requires a host of other observations (e.g., high resolution imaging, host star characterization, and high precision radial velocity measurements), and yielded assessments of planet frequency directly from the *Kepler* data alone [e.g., @fressin13; @morton14; @dressing15; @burke15]. Besides initially requiring no additional data, another strength of the statistical approach is that it could also be extended as results from other observations became available [e.g., precise host star characterization, @fulton17].
The study of exoplanet atmospheres has also benefited from applying a statistical comparative planetology approach. For example, @sing16 were able to show statistically that high altitude aerosols were the cause of the muted spectral features in the transmission spectra of hot Jupiters rather than low water abundances by performing a comparative study of ten planets. The water abundances determined from the individual spectra in the @sing16 study had very large uncertainties, but by using the diversity of the sample the authors were able to show that low water abundances couldn’t explain the observed continuum of spectral features. Another strength of the @sing16 result is that it only depended on simple and generic physical models for how water abundances and aerosols affect transmission spectra.
Beyond broad statistical conclusions, comparative studies also enable the identification of outliers, which are useful for homing in on model inadequacies. For example, a study of heat transport for highly irradiated planets found that WASP-43b is a unique exception to the expectations of theoretical models and an empirical trend of heat transport vs. irradiation temperature [@schwartz15]. This finding has drawn attention to the possibility of clouds on the nightsides of tidally-locked planets [@kataria15]. Another example is the unusually high dayside albedo of Kepler-7b, which was discovered in a survey of optical secondary eclipse measurements [@demory11]. The properties of this planet have also motivated the development of a more comprehensive model for clouds in hot Jupiter atmospheres [@heng13; @parmentier16].
The statistical approach also has the benefit of enabling the combination of results from different types of observations even if they aren’t targeted on the same objects. For example, @schwartz15 were able to compare the geometric albedos of planets that had been studied in the optical with *Kepler* to the Bond albedos of more nearby planets that had been studied in the thermal infrared with *Spitzer* and *Hubble*. This kind of approach has also been useful for constraining planet frequency over a broad range of parameter space by combining results from surveys that were performed with different techniques [e.g., @clanton17].
Worked Examples: Empirical Tests of the Habitable Zone Concept {#sec:hz}
==============================================================
One of the guiding principles in the search for other Earth-like planets is the concept of the liquid water habitable zone [@Kasting93; @kasting14]. However, the link between the canonical habitable zone (in terms of orbital distance) and the existence of surface liquid water has not yet been shown observationally, and therefore it does not currently provide a rigorous framework for interpreting the characterization of individual planets. Testing the habitable zone hypothesis with a statistical comparative planetology approach would thus be an important step towards determining the frequency of habitable planets.
We outline here two applications of the statistical comparative planetology approach that could be used to test for the inner and outer boundaries of the habitable zone and the prevalence of planets with temperate climates regulated by a carbonate-silicate cycle within it. We focus on how measurements of atmospheric water (H$_{2}$O) and carbon dioxide (CO$_{2}$) abundances can be used because these are likely to be the first chemical species that can be detected for terrestrial exoplanets. Water and carbon dioxide have numerous strong absorption bands throughout the optical and infrared, and thus will likely be accessible to both the transit and direct imaging approaches to studying exoplanet atmospheres. For now we remain agnostic about which technique is used. In §\[sec:conclusion\] we discuss how the technique being used could matter.
Water abundances
----------------
One possible test of the habitable zone concept is to survey the atmospheric H$_{2}$O abundances of exoplanets with a range of orbital separations. The hypothesis is that the presence of H$_{2}$O as a function of irradiation will be correlated with the boundaries of the habitable zone as predicted by models. Inside the inner edge of the habitable zone H$_{2}$O should not be abundant in the atmospheres of mature planets because it is expected to be lost to space following a runaway greenhouse process like is thought to have occurred on Venus. Similarly, beyond the outer edge of the habitable zone H$_{2}$O would be less abundant because it should freeze out.
The strengths of the statistical approach in the case of measuring atmospheric H$_{2}$O abundances are that it can likely be performed with relatively low precision per planet and that the random aspects of the planet formation process can be marginalized over. Precisely and accurately measuring the atmospheric abundance of any chemical species for an exoplanet puts stringent requirements on spectral resolution and signal-to-noise, as well as knowledge of the planet’s physical and orbital properties [@konopacky13; @kreidberg14b; @lupu16; @nayak17; @batalha17]. Therefore, detecting the presence of H$_{2}$O on many planets may be easier to accomplish than robustly determining the atmospheric H$_{2}$O abundances for individual planets. Furthermore, determining whether an individual exoplanet has surface liquid water requires knowing not just the atmospheric H$_{2}$O abundance but also the full chemical inventory of the atmosphere so that accurate calculations of the surface temperature can be performed. Observations of a small number of planets also run the risk of encountering planets that are dry due to the stochastic nature of planet formation [e.g., @raymond07].
Carbon dioxide abundances
-------------------------
The habitable zone concept rests on the assumption of a functioning silicate-weathering feedback [@Walker-Hays-Kasting-1981:negative]. This feedback should increase the atmospheric CO$_2$ to maintain surface temperatures that allow liquid water as a planet receives less stellar radiation. There is some evidence that the silicate-weathering feedback has operated in Earth history, but it is not definitive [@zeebe08]. A statistical approach to exoplanet astronomy would allow us to test the silicate-weathering feedback directly, and therefore the habitable zone concept, by making a number of low-precision CO$_2$ measurements on Earth-size and -mass planets inside the traditional habitable zone. Specifically, we can use a radiative-transfer model to calculate the amount of CO$_2$ needed to maintain habitable conditions for a given stellar irradiation while marginalizing over surface temperatures and pressures consistent with liquid water as well as uncertainties such as clouds and other greenhouse gases. By comparing these estimates of the amount of CO$_2$ necessary to maintain surface liquid water with low-precision estimates of the CO$_2$ made on many planets, we can perform a statistical test of the habitable zone concept.
Here we make a simple calculation to illustrate this methodology. A full evaluation of this idea is beyond the scope of the current work. In Figure \[fig:olr\_co2\] we plot in blue the theoretical prediction for the CO$_2$ that would lead to radiative balance based on a fit to planetary albedo and infrared emission to space from a one-dimensional radiative-convective model [@williams1997habitable]. We assume a saturated atmosphere with 1 bar of N$_2$, Earth-like clouds, a cosine of the solar zenith angle of 0.5, and a surface temperature of 290 K. We then create five bins of artificial data with CO$_2$ perturbed off this curve assuming Gaussian noise with a 1-$\sigma$ log error of 0.5 decades. This assumes a measurement log error of 0.5 decades; a model log error of 0.5 decades, which includes uncertainty in clouds, pressure broadening and scattering by other background gasses, and the presence of other greenhouse gases such as CH$_4$; and four planets per bin. Using these assumptions it would be possible to detect the downward trend in CO$_2$ as the stellar irradiation increases by measuring 20 planets. Such an inference is only possible if enough planets are measured to marginalize over the many factors other than stellar irradiation that could determine CO$_2$ even with a functioning silicate-weathering feedback. This calculation also shows that measurements on planets near Earth’s irradiation (toward the inner edge of the habitable zone) will be critical for evaluating the theory because the model predicts much lower CO$_2$ values for them.
Suggested Future Research {#sec:conclusion}
=========================
If we can step away from the idea that the only way to address the topic of habitable planets is through detailed characterization of individual objects then a comparative planetology approach could provide a useful basis for designing future experiments. We have presented some first ideas for statistical surveys that could be carried out based on a re-framing of the problem as determining the frequency of habitable worlds. Other ideas include measuring reflected stellar radiation or emitted planetary radiation to determine the planetary albedo. A statistical transition from low to high planetary albedo would represent a detection of the outer edge of the habitable zone. Another idea would be to measure the surface temperature using gap regions of the infrared spectrum. The null hypothesis would be that surface temperature would scale as irradiation to the ${\frac{1}{4}}$ power when measured for many planets. In contrast, if planetary surface temperature is regulated in the habitable zone, then it would show little dependence on stellar irradiation.
More work is needed to study the details of these ideas and those presented above. For example, beyond the outer edge of the habitable zone planets may have water in their atmospheres due to sublimation. If the water abundances are being measured through transmission spectroscopy then it may be difficult to distinguish a low stratospheric water abundance due to cold trapping from an atmosphere with a low bulk water abundance. It would also be valuable to explore whether statements could be made about the frequency of planetary inhabitance through a survey of biosignature gases such as O$_{2}$ or O$_{3}$.
A key assumption of our proposed statistical experiments is that the observations can be concentrated on terrestrial planets. This is possible in principle for transiting exoplanets, which will necessarily be orbiting M dwarfs due to their uniquely close-in habitable zones, but may be more difficult for true Earth analogues orbiting Sun-like stars, the atmospheres of which can only be probed with direct imaging and for which we will likely always lack density measurements.
The required assumption for our proposed statistical tests is an example of the more general problem that too many unknowns may make it impossible to discern the underlying trends even in a large statistical sample. Therefore it is likely that multiple types of data will need to be combined even for the statistical approach. It may also be necessary to characterize a few planets in greater detail to identify the key diagnostics for statistical investigations. Future work to flesh out the details of a statistical approach will focus on quantifying these aspects of the observing strategy. Ultimately, there is also a chance that some questions in the topic of planetary habitability and life are too complex for the statistical approach. However, we still suspect that the statistical approach will be crucial for progress in this area because getting data for more planets can help get around the unknowns, whereas it may be impossible to determine the habitability of individual exoplanets from astronomical remote sensing at high confidence.
We thank Edwin Kite, Jean-Michel Désert, and the anonymous referee for helpful comments on an early draft of this paper. J.L.B. thanks the members of the *LUVOIR* Science and Technology Definition Team for stimulating discussions that motivated this work, and acknowledges support from the David and Lucile Packard Foundation and NASA through STScI grants GO-13021, 13467, 14792, and 14793. D.S.A. acknowledges partial support from NASA grant number NNX16AR85G, which is part of the “Habitable Worlds” program and from the NASA Astrobiology Institute’s Virtual Planetary Laboratory, which is supported by NASA under cooperative agreement NNH05ZDA001C. E.M.-R.K. acknowledges support from the Research Corporation for Science Advancement through the Cottrell Scholar program and from Grinnell College’s Harris Faculty Fellowship.
[^1]: The *Large UV/Optical/IR Surveyor* (*LUVOIR*: <https://asd.gsfc.nasa.gov/luvoir/>) and the *Habitable Exoplanet Imaging Mission* (*HabEx*: <http://www.jpl.nasa.gov/habex/>).
[^2]: The *Origins Space Telescope* (*OST*: <https://asd.gsfc.nasa.gov/firs/>).
|
---
abstract: 'Big data has become a critically enabling component of emerging mathematical methods aimed at the automated discovery of dynamical systems, where first principles modeling may be intractable. However, in many engineering systems, abrupt changes must be rapidly characterized based on limited, incomplete, and noisy data. Many leading automated learning techniques rely on unrealistically large data sets and it is unclear how to leverage prior knowledge effectively to re-identify a model after an abrupt change. In this work, we propose a conceptual framework to recover parsimonious models of a system in response to abrupt changes in the low-data limit. First, the abrupt change is detected by comparing the estimated Lyapunov time of the data with the model prediction. Next, we apply the sparse identification of nonlinear dynamics (SINDy) regression to update a previously identified model with the fewest changes, either by addition, deletion, or modification of existing model terms. We demonstrate this sparse model recovery on several examples for abrupt system change detection in periodic and chaotic dynamical systems. Our examples show that sparse updates to a previously identified model perform better with less data, have lower runtime complexity, and are less sensitive to noise than identifying an entirely new model. The proposed abrupt-SINDy architecture provides a new paradigm for the rapid and efficient recovery of a system model after abrupt changes.'
author:
- Markus Quade
- Markus Abel
- 'J. Nathan Kutz'
- 'Steven L. Brunton'
bibliography:
- 'main.bib'
title: Sparse Identification of Nonlinear Dynamics for Rapid Model Recovery
---
**Dynamical systems modeling is a cornerstone of modern mathematical physics and engineering. The dynamics of many complex systems (e.g., neuroscience, climate, epidemiology, etc.) may not have first-principles derivations, and researchers are increasingly using data-driven methods for system identification and the discovery of dynamics. Related to discovery of dynamical systems models from data is the *recovery* of these models following abrupt changes to the system dynamics. In many domains, such as aviation, model recovery is mission critical, and must be achieved rapidly and with limited noisy data. This paper leverages recent advances in sparse optimization to identify the fewest terms required to recover a model, introducing the concept of *parsimony of change*. In other words, many abrupt system changes, even catastrophic bifurcations, may be characterized with relatively few changes to the terms in the underlying model. In this work, we show that sparse optimization enables rapid model recovery that is faster, requires less data, is more accurate, and has higher noise robustness than the alternative approach of re-characterizing a model from scratch.**
Introduction
============
The data-driven discovery of physical laws and dynamical systems is poised to revolutionize how we model, predict, and control physical systems. Advances are driven by the confluence of big data, machine learning, and modern perspectives on dynamics and control. However, many modern techniques in machine learning (e.g., neural networks) often rely on access to massive data sets, have limited ability to generalize beyond the attractor where data is collected, and do not readily incorporate known physical constraints. These various limitations are framing many state-of-the-art research efforts around learning algorithms [@goodfellow2016deep], especially as it pertains to generalizability, limited data and [*one-shot learning*]{} [@fei2006one; @vinyals2016matching; @delahunt2018putting]. Such limitations also frame the primary challenges and limitations associated with data-driven discovery for real-time control of strongly nonlinear, high-dimensional, multi-scale systems with abrupt changes in the dynamics. Whereas traditional methods often require unrealistic amounts of training data to produce a viable model, this work focuses on methods that take advantage of prior experience and knowledge of the physics to dramatically reduce the data and time required to characterize dynamics. Our methodology is similar in philosophy to the machine learning technique of [*transfer learning*]{} [@pan2010survey], which allows networks trained on one task to be efficiently adapted to another task. Our architecture is designed around the goal of rapidly extracting parsimonious, nonlinear dynamical models that identify only the fewest important interaction terms so as to avoid overfitting.
There are many important open challenges associated with data-driven discovery of dynamical systems for real-time tracking and control. When abrupt changes occur in the system dynamics, an effective controller must rapidly characterize and compensate for the new dynamics, leaving little time for recovery based on limited data [@Brunton2015amr]. The primary challenge in real-time model discovery is the reliance on large quantities of training data. A secondary challenge is the ability of models to generalize beyond the training data, which is related to the ability to incorporate new information and quickly modify the model. Machine learning algorithms often suffer from overfitting and a lack of interpretability, although the application of these algorithms to physical systems offers a unique opportunity to enforce known symmetries and physical constraints (e.g. conservation of mass). Inspired by biological systems, which are capable of extremely fast adaptation and learning based on very few trials of new information [@rankin2004invertebrate; @whitlock2006learning; @johansen2011molecular], we propose model discovery techniques that leverage an *experiential framework*, where known physics, symmetries, and conservation laws are used to rapidly infer model changes with limited data.
Previous work in system identification
--------------------------------------
There are a wealth of regression techniques for the characterization of system dynamics from data, with varying degrees of generality, accuracy, data requirements, and computational complexity. Classical linear model identification algorithms include Kalman filters [@Kalman1960jfe; @Kalman1965AAC; @gershenfeld1999nature], the eigensystem realization algorithm (ERA) [@Juang1985jgcd], dynamic mode decomposition (DMD) [@Schmid2010jfm; @Rowley2009jfm; @Tu2014jcd; @Kutz2016book], and autoregressive moving average (ARMA) models [@Akaike1969annals; @Brockwell2017], to name only a few. The resulting linear models are ideal for control design, but are unable to capture the underlying nonlinear dynamics or structural changes. Increasingly, machine learning is being used for nonlinear model discovery. Neural networks have been used for decades to identify nonlinear systems [@gonzalez1998identification], and are experiencing renewed interest because of the ability to train deeper networks with more data [@goodfellow2016deep; @Yeung2017arxiv; @Takeishi2017nips; @Wehmeyer2017arxiv; @Mardt2017arxiv; @Lusch2017arxiv] and the promise of transformations that linearize dynamics via the Koopman operator [@Koopman1931pnas; @Mezic2005nd]. Neural networks show good capacity to recover the dynamics in a so-called “model-free” way [@Lukosevicius2009; @Lu2017]. These methods are also known as “reservoir computers”, “liquid state machines”, or “echo state networks”, depending on the context. However, a real-time application is unrealistic, and the output is generally not analytically interpretable. In another significant vein of research, genetic programming [@dantzig1985mathematical; @koza1992genetic] is a powerful bio-inspired method that has successfully been applied to system identification [@Bongard2007pnas; @Schmidt2009science; @schmidt2011automated; @LaCava2016b], time-series prediction [@LaCava2016a; @Quade2016] and control [@Gout2018; @Duriez2017]. However, evolutionary methods in their pure form, including genetic programming, are computationally complex and thus are not suitable for real-time tracking.
Recently, *interpretability* and *parsimony* have become important themes in nonlinear system identification [@Bongard2007pnas; @Schmidt2009science]. A common goal now is to identify the fewest terms required to represent the nonlinear structure of a dynamical system model while avoiding overfitting [@Brunton2016pnas]. Symbolic regression methods [@voss1998; @Schmidt2009science; @McConaghy2011; @Brunton2016pnas] are generally appealing for system identification of structural changes, although they may need to be adapted to the low-data limit and for faster processing time. Nonparametric additive regression models [@abel2005additive; @voss1999; @abel2004] require a backfitting loop which allows general transformations, but may be prohibitively slow for real-time applications. Generalized linear regression methods are slightly less general but can be brought to a fast evaluation and sparse representation [@McConaghy2011; @Brunton2016pnas]. These leading approaches to identify dynamical equations from data usually rely on past data and aim at reliable reproduction of a stationary system, i.e. when the underlying equations do not change in the course of time [@Schmidt2009science; @abel2004; @Brunton2016pnas].
Contributions of this work
--------------------------
In this work, we develop an adaptive modification of the sparse identification of nonlinear dynamics (SINDy) algorithm [@Brunton2016pnas] for real-time recovery of a model following abrupt changes to the system dynamics. We refer to this modeling framework as *abrupt-SINDy*. Although this is not the only approach for real-time change detection and recovery, parsimony and sparsity are natural concepts to track abrupt changes, focusing on the fewest modifications to an existing model. SINDy already requires relatively small amounts of data [@Kaiser2017arxivB], is based on fast regression techniques, and has been extended to identify PDEs [@Rudy2017sciadv; @Schaeffer2017prsa], to include known constraints and symmetries [@Loiseau2016arxiv], to work with limited measurements [@Brunton2017natcomm] and highly corrupted and noisy data [@Tran2016arxiv; @Schaeffer2017pre], to include control inputs [@Brunton2016nolcos; @Kaiser2017arxivB], and to incorporate information criteria to assess the model quality [@Mangan2017prsa], which will be useful in abrupt model recovery.
Here, we demonstrate that the abrupt-SINDy architecture is capable of rapidly identifying sparse changes to an existing model to recover the new dynamics following an abrupt change to the system. The first step in the adaptive identification process is to detect a system change using divergence of the prediction from measurements. Next, an existing model is updated with sparse corrections, including parameter variations, deletions, and additions of terms. We show that identifying sparse model changes from an existing model requires less data, less computation, and is more robust to noise than identifying a new model from scratch. Further, we attempt to maintain a critical attitude and caveat limitations of the proposed approach, highlighting when it can break down and suggesting further investigation. The overarching framework is illustrated in Fig. \[Fig:Overview\].
{width="\textwidth"}
State of the art
================
Recently, sparse regression in a library of candidate nonlinear functions has been used for sparse identification of nonlinear dynamics (SINDy) to efficiently identify a sparse model structure from data [@Brunton2016pnas]. The SINDy architecture bypasses an intractable brute-force search through all possible models, leveraging the fact that many dynamical systems of the form $$\frac{d}{dt}{\bf {x}} = {\bf f}({\bf x})$$ have dynamics ${\bf f}$ that are sparse in the state variable ${\bf x}\in\mathbb{R}^n$. Such models may be identified using a sparsity-promoting regression [@Tibshirani1996lasso; @Hastie2009book; @James2013book] that penalizes the number of nonzero terms $\xi_{ij}$ in a generalized linear model: $$\hat{f_i}({\bf x}) = \sum_{j=1}^p\xi_{ij} \theta_{j}({\bf x}),
\label{eq:glm}$$ where $\theta_j({\bf x})$ form a set of nonlinear candidate functions. The candidate functions may be chosen to be polynomials, trigonometric functions, or a more general set of functions [@Brunton2016pnas; @McConaghy2011]. With poor choice of the candidate functions $\theta_j$, i.e. if library functions are non-orthogonal and/or overdetermined, the SINDy approach may fail to identify the correct model.
Sparse models may be identified from time-series data, which are collected and formed into the data matrix $${\bf X} = \begin{bmatrix} {\bf x}_1 & {\bf x}_2 & \cdots {\bf x}_m\end{bmatrix}^T.$$ We estimate the time derivatives using a simple forward Euler finite-difference scheme, i.e. the difference of two consecutive data, divided by the time difference: $${{\bf \dot{X}}} = \begin{bmatrix} {{\bf \dot{x}}}_1 & {{\bf \dot{x}}}_2 & \cdots {{\bf \dot{x}}}_m\end{bmatrix}^T.$$ This estimation procedure is numerically ill-conditioned if data are noisy, although there are many methods to handle noise which work very well if used correctly [@Ahnert-Abel-2007; @Chartrand2011isrnam]. Noise-robust derivatives were investigated in the original SINDy algorithm [@Brunton2016pnas]. Next, we consider a library of candidate nonlinear functions $\boldsymbol{\Theta}({\bf X})$, of the form $$\boldsymbol{\Theta}({\bf X}) = \begin{bmatrix} \mathbf{1} & {\bf X} & {\bf X}^2 & \cdots & {\bf X}^d & \cdots & \sin({\bf X}) & \cdots \end{bmatrix}.$$ Here, the matrix ${\bf X}^d$ denotes a matrix with column vectors given by all possible time-series of $d$-th degree polynomials in the state ${\bf x}$. The terms in $\boldsymbol{\Theta}$ can be functional forms motivated by knowledge of the physics. Within the proposed work, they may parameterize a piecewise-affine dynamical model. Following best practices of statistical learning [@Hastie2009book], to preprocess, we mean-subtract and normalize each column of $\boldsymbol{\Theta}$ to have unit variance. The dynamical system can now be represented in terms of the data matrices as $${{\bf \dot{X}}} \approx \boldsymbol{\Theta}({\bf X})\boldsymbol{\Xi}.$$ The coefficients in the column $\boldsymbol{\Xi}_k$ of $\boldsymbol{\Xi}$ determine the active terms in the $k$-th row of Eq. . A parsimonious model has the fewest terms in $\boldsymbol{\Xi}$ required to explain the data. One option to obtain a sparse model is via convex $\ell_1$-regularized regression: $$\boldsymbol{\Xi} = \text{argmin}_{\boldsymbol{\Xi}'}\|{\mathbf{\dot{X}}} - \boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}'\|_2+\gamma \|\boldsymbol{\Xi}'\|_1.$$ The hyper parameter $\gamma$ balances complexity and sparsity of the solution. Sparse regression, such as LASSO [@Tibshirani1996lasso] and sequential thresholded least-squares [@Brunton2016pnas], improves the robustness of identification for noisy overdetermined data, in contrast to earlier methods [@Wang2011prl] using compressed sensing [@Donoho2006ieeetit; @Candes2006picm]. Other regularization schemes may be used to improve performance, such as the elastic net regression [@Li2016].
In this paper we use the sequentially thresholded ridge regression [@Rudy2017sciadv], which iteratively solves the ridge regression $$\boldsymbol{\Xi} = \text{argmin}_{\boldsymbol{\Xi}'}\|{\mathbf{\dot{X}}} - \boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}'\|_2+\alpha \|\boldsymbol{\Xi}'\|_2.$$ and then thresholds any coefficient that is smaller than $\gamma$. The procedure is repeated on the non-zero entries of $\boldsymbol{\Xi}$ until the model converges. The convergence of the SINDy architecture has been discussed in [@Zhang2018]. After a sparse model structure has been identified in normalized coordinates, it is necessary to regress onto this sparse structure in the original unnormalized coordinates. Otherwise, non-physical constant terms appear when transforming back from normalized coordinates due to the mean-subtraction.
In La Cava et al. [@LaCava2016b] the authors pursue a complementary although more computationally intensive idea of adaptive modeling in the context of generalized linear models. Starting from an initial guess for the model, a brute force search is conducted to scan a larger set of candidate functions $\theta \rightarrow \theta \theta'^{\gamma}$, where $\theta'$ are multiplicative extensions to the initial set of candidate functions and $\gamma$ are real valued exponents. The intended use of this method is the refinement of first-principle based models by discovery of coupling terms. It is possible to combine this refinement with our proposed scheme for dealing with abrupt changes. In addition, sparse sensors [@Manohar2017csm] and randomized algorithms [@Erichson2016arxivA] may improve speed.
Methods
=======
The viewpoint of sparsity extends beyond model discovery, and we propose to extend SINDy to identify systems undergoing abrupt changes. It may be the case that abrupt model changes will only involve the addition, deletion, or modification of a few terms in the model. This is a statement of the *parsimony of change*, and indicates that we can use sparse regression to efficiently identify the new or missing terms with considerably less data than required to identify a new model from scratch. In general, each additional term that must be identified requires additional training data to distinguish between joint effects. Thus, having only a few changes reduces the amount of data required, making the model recovery more rapid. This section will describe a procedure that extends SINDy to handle three basic types of model changes: If the structure of the model is unchanged and only the parameters vary, we will perform least-squares regression on the known structure to identify the new parameters. This is computationally fast, and it is easy to check if the model explains the new dynamics, or if it is necessary to explore possible additions or deletions of terms. If the model changes by the removal of a few terms, then SINDy regression can be applied on the sparse coefficients in order to identify which terms have dropped out. If a term is added, then SINDy regression will find the sparsest combination of inactive terms that explain the model error. Since least squares regression scales asymptotically $\mathcal{O}(p^3)$, with $p$ the number of columns in the library, this is computationally less expensive than regression in the entire library.
Combinations of these changes, such as a simultaneous addition and deletion, are more challenging and will also be explored. This approach is known as *abrupt-SINDy*, and it is depicted schematically in Fig. \[fig:flow-chart\].
![Adaptive SINDy flow chart. For an initial model and hyper parameter selection, a gridsearch is conducted. Next, we apply a predictor corrector scheme checking every $t_{\text{error}}$ for model divergence using estimated Lyapunov time, and eventually update the model in a two step fashion.[]{data-label="fig:flow-chart"}](figures_flowchart){width="\columnwidth"}
Baseline model
--------------
First, we must identify a baseline SINDy model, and we use a gridsearch to determine the optimal hyper parameter selection. In gridsearch, all combinations of hyper parameters are tested and the best performing set is selected. This search is only performed once, locking in hyper parameters for future updates. The baseline model is characterized by the sparse coefficients in $\boldsymbol{\Xi}_0$.
Detecting model divergence {#sec:divergence}
--------------------------
It is essential to rapidly detect any change in the model, and we employ a classical predictor-corrector scheme [@gershenfeld1999nature]. The predictor step is performed over a time $\tau_{\text{pred}}$ in the interval $t,t+\tau_{\text{pred}}$ using the model valid at time $t$. The divergence of the predicted and measured state is computed at $t+\tau$ as $\|\Delta {\bf x}\| = \| \hat{{\bf x}}(t+\tau) - {\bf x}(t+\tau) \|$, where $\hat{{\bf x}}$ is the prediction and ${\bf x}$ is the measurement. The idea is to identify when the model and the measurement diverge faster than predicted by the dynamics of the system. For a chaotic system, the divergence of a trajectory is measured by the largest Lyapunov exponent of the system [@ott2002chaos], although a wealth of similar measures have been suggested [@scholl2008handbook]. The Lyapunov exponent is defined as $$\lambda = \lim_{\tau \to \infty} \lim_{\Delta {\bf x}(t_0) \to 0} \frac{\left\langle \log \left( \frac{ \Delta {\bf x}(t_0+\tau) }{ \Delta {\bf x}({t_0}) }\right) \right\rangle}{\tau},$$ and its inverse sets the fastest time scale. Here, the analogy of ensemble and time average is used, more precisely the local, finite-time equivalent [@lai2011transient; @ding2007nonlinear]. An improvement can be achieved by exploiting an ensemble, e.g. by adding noise to the state ${\bf x}$ that corresponds to the given measurement accuracy. Since we know the dynamical system for the prediction step, the Lyapunov exponent is determined by evolving the tangent space with the system [@kantz1994robust; @pikovsky1998dynamic].
In our detection algorithm, we fix a fluctuation tolerance $\Delta {\bf x}$ and measure if the divergence time we find deviates from the expectation. If data are noisy, this tolerance must be significantly larger than the the typical fluctuation scale of the noise. Formally, the model and measurements diverge if the time-scale given by the local Lyapunov exponent and prediction horizon disagree. The local Lyapunov exponent is computed directly from the eigenvalues of the dynamical system [@kantz1994robust; @pikovsky1998dynamic]. The prediction horizon $T(t)$ is the first passage time where prediction $\hat{{\bf x}}(t + \Delta t)$ with initial condition ${\bf x}(t)$ and and measurement ${\bf x}(t + \Delta t)$ differ by more than $\Delta {\bf x}$: $$T(t) = \operatorname*{arg\rm{}max}_{\Delta t} {\left\lVert\hat{{\bf x}}(t + \Delta t) - {\bf x}(t + \Delta t)\right\rVert} < {\left\lVert\Delta {\bf x}\right\rVert}.
\label{eq:horizon}$$
Analogous to the local Lyapunov exponent, we compute the ratio $\log \left\| \Delta {\bf x}(t_0+\tau)/\Delta {\bf x}(t_0) \right \|$ as a measure for the divergence based on the measurement. For the model, we compute the local Lyapunov exponent as the average maximum eigenvalue $\bar{\lambda}(t) = \langle \lambda (t') \rangle_{t' \in [t, t+T]}$ with $\lambda (t) = \max(\lambda_i(t))$ and $\lambda_i v_i(t) = \left. {\partial f_j}/{\partial {\bf x}_k} \right| _{{\bf x}(t)} v_i(t)$. Thus we compare the expected and observed trajectory divergence. Model and measurement have diverged at time $t$ if the model time scale and the measured one differ: $$\bar{\lambda}(t) > \alpha \dfrac{\log(\Delta {\bf x}) - \log(\bar{\Delta}(t))}{T(t)}.
\label{eq:estlyap}$$ If the model is not chaotic, but the measurement is chaotic, one must invert the inequality, as in Fig. \[fig:divergence\]. The empirical factor $\alpha$ accounts for finite-time statistics.
![Sketch of the prediction horizon estimation. We use the observation ${\bf x}(t)$ as initial condition for the current model. Integration gives $\hat{{\bf x}}(t)$. The prediction horizon $T(t)$ is calculated according to Eq. . The prediction horizon is a function of time and the current model. It indicates divergence of model and observation. For details see text.[]{data-label="fig:divergence"}](figures_divergence){width="\columnwidth"}
This method depends heavily on the particular system under investigation, including the dynamics, time scales, and sampling rate. In a practical implementation, these considerations must be handled carefully and automatically. It is important to note that we are able to formulate the divergence in terms of dynamical systems theory, because our model *is* a dynamical system, in other cases, such as artificial neural networks, this is not possible due to the limited mathematical framework.
Adaptive model fitting
----------------------
After a change is detected, the following procedure is implemented to rapidly recover the model:
1. First, the new data is regressed onto the existing sparse structure $\boldsymbol{\Xi}_0$ to identify varying parameters.
2. Next, we identify deletions of terms by performing the sparse regression on the sparse columns of $\boldsymbol{\Theta}$ that correspond to nonzero rows in $\boldsymbol{\Xi}_0$. This is more efficient than identifying a new model, as we only seek to delete existing terms from the model.
3. Finally, if there is still a residual error, then a sparse model is fit for this error in the inactive columns of $\boldsymbol{\Theta}$ that correspond to zero rows in $\boldsymbol{\Xi}_0$. In this way, new terms may be added to the model.
If the residual is sufficiently small after any step, the procedure ends. Alternatively, the procedure may be iterated until convergence. We are solving smaller regression problems by restricting our attention to subsets of the columns of $\boldsymbol{\Theta}$. These smaller regressions require less data and are less computationally expensive [@Li2016], compared to fitting a new model. The deletion-addition procedure is performed after a model divergence is detected, using new transient data collected in an interval of size $t_{\text{update}}$.
Results {#sec:gridsearch}
=======
In this section, we describe the results of the abrupt-SINDy framework on dynamical systems with abrupt changes, including parameter variation, deletion of terms, and addition of terms. The proposed algorithm is compared against the original SINDy algorithm, which is used to identify a new model from scratch, in terms of data required, computational time, and model accuracy.
In each case, we begin by running a gridsearch algorithm[@pedregosa2011scikit][^1] to identify the main parameters: $\alpha$, the ridge regression regularization parameter; $\gamma$, the thresholding parameter; $n_{\text{degree}}$, the maximum degree of the polynomial feature transformation; and $n_{\text{fold}}$, the number of cross-validation runs. For scoring we use the explained variance score and conduct a five-fold cross validation for each point in the $(\alpha, \gamma, n_{\text{degree}})$ parameter grid.
\[tab:parameters\_gridsearch\]
Parameter Value
--------------------- --------------------------
$\alpha$ $0,0.2,0.4,0.6,0.8,0.95$
$\gamma $ $0.1,0.2,0.4$
$n_{\text{degree}}$ $2,3$
$n_{\text{fold}}$ $5$
Seed $42$
CV k-fold
Score explained variance score
: Parameters for the grid search.
Lorenz system {#Sec:Lorenz}
-------------
The Lorenz system is a well-studied, and highly simplified, conceptual model for atmospheric circulation [@Lorenz1963jas]: $$\begin{aligned}
\dot{x} &= \sigma(y -x) \\
\dot{y} &= \rho x - xz - y \\
\dot{z} &= xy -\beta z
\end{aligned}$$ where the parameter $\rho$ represents the heating of the atmosphere, corresponding to the Rayleigh number, $\sigma$ corresponds to Prandtl number, and $\beta$ to the aspect ratio [@strogatz2014nonlinear]. The parameters are set to $\rho=28$, $\beta=8/3$, $\sigma=10$.
In the following we integrate the system numerically to produce a reference data set. We deliberately change the parameter $\rho$ at $t=40$ to $\rho=15$ and at $t=80$ back to $\rho=28$, as shown in Figs. \[fig:lorenz\_y\] and \[fig:lorenz\_3d\]. These parametric changes lead to a bifurcation in the dynamics, and they are detected quickly. The subsequent adapted parameters are accurately detected up to two digits, as shown in Table \[tab:lorenz\]. Because we are identifying the sparse model structure on a normalized library $\boldsymbol{\Theta}$, with zero mean and unit variance, we must de-bias the parameter estimates by computing a least-squares regression onto the sparse model structure in the original unnormalized variables. Otherwise, computing the least-squares regression in the normalized library, as is typically recommended in machine learning, would result in non-physical constant terms in the original unnormalized coordinates.
![Time-series of the $y$ coordinate of the Lorenz system. The blue and green segments correspond to the system parameters $\sigma=10, \rho=28, \beta=\frac{8}{3}$. The orange segment from $t=40$ and $t=80$ corresponds to the modified parameter $\rho=15$. The initial condition is ${\mathbf{x}}_0 = (1, 1, 1)$.[]{data-label="fig:lorenz_y"}](figures_lorenz_trajectory_y){width="\columnwidth"}
![Lorenz system: Colors and parameters as in Fig. \[fig:lorenz\_y\]. In **A**, **B**, and **C** we show the first, second, and third segments of the trajectory in color with the concatenated trajectory in grey. The system changes from a butterfly attractor to a stable fixed point and back to a butterfly attractor.[]{data-label="fig:lorenz_3d"}](figures_lorenz_trajectory_3d){width="\columnwidth"}
Abrupt changes to the system parameters are detected using the prediction horizon from Eq. . When the system changes, the prediction horizon of the system should decrease, with smaller horizon corresponding to a more serious change. Conversely, the inverse time, corresponding to the Lyapunov exponent, should diverge. Figure \[fig:lorenz\_horizon\] exhibits this expected behavior. After a change is detected the model is rapidly recovered as shown in Table \[tab:lorenz\]. It is important to confirm that the updated model accurately represents the structure of the true dynamics. Figure \[fig:lorenz\_horizon\] shows the norm of the model coefficients, $\|{\boldsymbol{\xi}} - \hat{{\boldsymbol{\xi}}}\|$, which is a measure of the distance between the estimated and true systems. Except for a short time ($t_{\text{update}}=1$) after the abrupt change, the identified model closely agrees with the true model.
![Lorenz system: **A** We show the model accuracy over time. For coefficients, see Table \[tab:lorenz\]. For $t\leq 10$ no model is available and $\|{\boldsymbol{\xi}} - \hat{{\boldsymbol{\xi}}}\| = -1$. At both switch points, $t=40$ and $t=80$, $t_{\text{update}} = 1$ is needed to update the model. During this interval, a fallback solution, e.g. DMD could be implemented. Note that the accuracy metric requires knowledge about the ground truth and thus is only available in a hindcast scenario. **B** Evaluation of Eq. . At both switch points, $t=40$ and $t=80$, we quickly detect the divergence of model and measurement. The parameters are $t_{\text{model}}= 10, t_{\text{update}}= 1, t_{\text{error}}=0.5,$ and $\Delta {\bf x}= 1.0$.[]{data-label="fig:lorenz_horizon"}](figures_lorenz_error_horizon){width="\columnwidth"}
$t_{\text{detected}}$ $t_{\text{update}}$ Equations
----------------------- --------------------- ------------------------------------------------------------------------------------------------------------------
0.00 10.0 $\begin{aligned}\dot{x} & = -10.0x+10.0y\\\dot{y} & = 27.96x-0.99y-1.0xz\\\dot{z} & = -2.67z+1.0xy\end{aligned}$
40.01 41.0 $\begin{aligned}\dot{x} & = -10.0x+10.0y\\\dot{y} & = 15.0x-1.0y-1.0xz\\\dot{z} & = -2.67z+1.0xy\end{aligned}$
80.02 81.0 $\begin{aligned}\dot{x} & = -10.0x+10.0y\\\dot{y} & = 27.98x-1.0y-1.0xz\\\dot{z} & = -2.67z+1.0xy\end{aligned}$
: Lorenz system: detection and update times, along with identified equations. The detection time coincides up to the second digit with the true switching time. The rapidly identified model agrees well with the true model structure and parameters. Coefficients are rounded to the second digit.
\[tab:lorenz\]
### Effects of noise and data volume
An important set of practical considerations include how noise and the amount of data influence the speed of change detection and the accuracy of subsequent model recovery. Both the noise robustness and the amount of data required will change for a new problem, and here we report trends for this specific case. In addition, the amount of data required is also related to the sampling rate, which is the subject of ongoing investigation; in some cases, higher sampling time may even degrade model performance due to numerical effects [@Ahnert-Abel-2007].
Figure \[fig:lorenz\_update\_scaling\] shows the model fit following the abrupt change, comparing both the abrupt-SINDy method, which uses information about the existing model structure, and the standard SINDy method, which re-identifies the model from scratch following a detected change. In this figure, the model quality is shown as a function of the amount of data collected after the change.
The abrupt-SINDy model is able to identify more accurate models in a very short amount of time, given by $t_{\text{update}}\approx 0.1$. At this point, the standard SINDy method shows comparable error, however for even smaller times, the data are no longer sufficient for the conventional method. Since the adaptive method starts near the optimal solution, larger data sets do not degrade the model, which was an unexpected additional advantage.
Figure \[fig:lorenz\_noise\] explores the effect of additive noise on the derivative on the abrupt-SINDy and standard SINDy algorithms. Note that in practice noise will typically be added to the measurement of ${\mathbf{x}}$, as in the original SINDy algorithm [@Brunton2016pnas], requiring a denoising derivative [@Ahnert-Abel-2007; @Chartrand2011isrnam]; however, simple additive noise on the derivative is useful to investigate the robustness of the regression procedure. Abrupt-SINDy has considerably higher noise tolerance than the standard algorithm, as it must identify fewer unknown coefficients. In fact, it is able to handle approximately an order of magnitude more noise before failing to identify a model. Generally, increasing the volume of data collection improves the model. The critical point in the abrupt-SINDy curves corresponds to when small but dynamically important terms are mis-identified as a result of insufficient signal-to-noise. Although the noise and chaotic signal cannot be easily distinguished for small signal-to-noise, it may be possible to distinguish between them using a spectral analysis, since chaos yields red noise in contrast to the white additive noise.
![Lorenz system: We show the model accuracy versus the amount of data used to update (blue $\times$) or re-fit (orange dot) respectively. Data is collected from in the interval $[40, 40 + t_{\text{update}}]$ just after the first change of the system dynamics. The number of data points for $t_{\text{update}} =0.1$ are $N=25$, for $t_{\text{update}} =10$ we have 2500 points. At $t_{\text{update}} \simeq 1$, updating and re-fitting methods become comparable. However, for smaller update times, or less data, respectively, the fraction of transient data becomes too small for identifying the exact model from scratch. Updating the model needs less data for the same accuracy or achieves higher accuracy with the same amount of data.[]{data-label="fig:lorenz_update_scaling"}](figures_noise_no_noise_comparison){width="\columnwidth"}
![Lorenz system: We show the noise robustness of model accuracy. In A) we use the previous knowledge and update the model; in B) we make a new fit only re-using the previously discovered hyper-parameters. The curves are parametrized by $t_{\text{update}}$, c.f. Fig. \[fig:lorenz\_update\_scaling\]. The accuracy measure is very noise sensitive, as distinction between library functions gets lost. At a signal to noise ration of approximately $1$, no accurate model can be obtained with either model. At lower noise ratios, updating the model achieves higher accuracy (the library is smaller). In both cases, accuracy scales approximately logarithmically with $t_{\text{update}}$.[]{data-label="fig:lorenz_noise"}](figures_noise_sigma_vs_error){width="\columnwidth"}
Van der Pol
-----------
As a second example, we consider the famous nonlinear Van der Pol oscillator [@van1920theory]. We include additional quadratic nonlinearities $\alpha x^2$ and $\alpha y^2$ to study the ability of our method to capture structural changes when these terms are added and removed abruptly. This example focuses on the important class of periodic phenomena, in contrast to the chaotic Lorenz dynamics. The modified Van der Pol oscillator is described by the following equations: $$\begin{aligned}
\dot{x} &= y - \alpha y^2 \\
\dot{y} &= \mu (1 - x^2)y - x + \alpha x^2\;,\\
\end{aligned}$$ where $\mu > 0$ is a parameter controlling the nonlinear damping, and $\alpha$ parameterizes the additional quadratic nonlinearity. The reference data set is shown in Fig. \[fig:vdp\], with $\mu=7.5$ and $\alpha=0$ for $t \in [0, 100]$, which results in a canonical periodic orbit. At $t=100$ we introduce a structural change, switching on the quadratic nonlinearity ($\alpha = -0.25$), and driving the system to a stable fixed point. We also modify the parameter $\mu$, setting it to $\mu=6.0$. Finally, at $t=200$, we switch off the additional nonlinearity ($\alpha = 0$) and keep $\mu = 6$.
![Van der Pol system with parameters $\mu=5, \alpha=0$ (blue), $\mu=7.5, \alpha=-0.25$ (orange), $\mu=6.0, \alpha=0$ (green). **A**: time evolution of the $y$-coordinate. **B** phase-space-trajectory $x, y$.[]{data-label="fig:vdp"}](figures_vanderpol_traj_phase){width="\columnwidth"}
Table \[tab:vdp\] shows the corresponding models recovered using the abrupt-SINDy method. The change is detected using the Lyapunov time defined in Eq. , as shown in Fig. \[fig:vdp\_error\_horizon\]. Again, the estimated Lyapunov time (Fig. \[fig:vdp\_error\_horizon\] **B**) captures the the changes in the model, which correspond to peaks in structural model error (Fig. \[fig:vdp\_error\_horizon\] **A**). While the first and third stage are indeed identified correctly, the term $-1.25 x$ is preferred over $-x - 0.25x^2$ in the sparse estimate for $\dot{y}$ in the orange trajectory. However, since both terms look similar near the fixed point at $x \sim 1$, this describes the dynamics well. This type of mis-identification often occurs in data mining when features are highly correlated [@Li2016] and is more related to sparse regression in general than the proposed abrupt-SINDy. For dynamic system identification, the correct nonlinearity could be resolved by obtaining more transient data, i.e. by perturbing the system through actuation. However, this model may be sufficient for control while a more accurate model is identified.
![Van der Pol system: Evaluation of Eq. . Parameters: $t_{\text{model}}= 20, t_{\text{update}}=10, t_{\text{error}}=1, \Delta x= 1.5$.[]{data-label="fig:vdp_error_horizon"}](figures_vanderpol_error_horizon){width="\columnwidth"}
$t_{\text{detected}}$ $t_{\text{update}}$ Equations
----------------------- --------------------- --------------------------------------------------------------------------------------------
0.00 20.01 $\begin{aligned}\dot{x} & = 1.0y\\\dot{y} & = -1.0x+4.99y-4.99x^2y\end{aligned}$
106.39 116.00 $\begin{aligned}\dot{x} & = 0.99y+0.25y^2\\\dot{y} & = -1.26x+7.46y-7.46x^2y\end{aligned}$
200.12 210.00 $\begin{aligned}\dot{x} & = 1.0y\\\dot{y} & = -1.0x+5.98y-5.98x^2y\end{aligned}$
: Van der Pol system: Summary of the discovered equations. Coefficients are rounded to the second digit.[]{data-label="tab:vdp"}
Conclusions
===========
In this work, we develop an adaptive nonlinear system identification strategy designed to rapidly recover nonlinear models from limited data following an abrupt change to the system dynamics. The sparse identification of nonlinear dynamics (SINDy) framework is ideal for change detection and model recovery, as it relies on parsimony to select the fewest active terms required to model the dynamics. In our adaptive abrupt-SINDy method, we rely on previously identified models to identify the fewest *changes* required to recover the model. This modified algorithm is shown to be highly effective at model recovery following an abrupt change, requiring less data, less computation time, and having improved noise robustness over identifying a new model from scratch. The abrupt-SINDy method is demonstrated on several numerical examples exhibiting chaotic dynamics and periodic dynamics, as well as parametric and structural changes, enabling real-time model recovery.
There are limitations of the method which can be addressed by several promising directions that may be pursued to improve the abrupt-SINDy method:
1. **Fallback models:** In the current implementation, after a change has been detected, the old model will be used until enough data is collected to identify a new model. The dynamic mode decomposition [@Kutz2016book] provides an alternative fallback model, that may be identified rapidly with even less data. Additionally, instead of relying on a sparse update to the current model, it is sensible to also maintain a library of past models for rapid characterization [@Brunton2014siads].
2. **Hyperparameterization:** In the initial prototype, the hyper-parameters $\Delta_x$ and $t_{\text{update}}$ are fixed. Over time, an improved algorithm may learn and adapt optimal hyper-parameters.
3. **Comprehensive Lyapunov time estimation:** According to Eq., the Lyapunov time $T(t | \Delta {\bf x})$ is estimated for a fixed $\Delta {\bf x}$. Estimating the time for a range of values, i.e. $\Delta {\bf x} \in (0, \Delta {\bf x}_{\max}]$, will be more robust and may provide a richer analysis without requiring additional data. Further investigation must be made into the case of chaotic systems, where the numerical calculation of the Lyapunov exponent may fail to reveal divergence due to the fact of simple averaging over time. Because of the importance of the detection of model divergence, this is a particularly important area of future research.
4. **Advanced optimization and objectives:** Looking forward, advanced optimization techniques may be used to further improve the adaptation to system changes. Depending on the system, other objectives may be optimized, either by including regularization or in a multi-objective optimization.
The proposed abrupt-SINDy framework is promising for the real-time recovery of nonlinear models following abrupt changes. It will be interesting to compare with other recent algorithms that learn local dynamics for control in response to abrupt changes [@Ornik2017arxiv]. Future work will be required to demonstrate this method on more sophisticated engineering problems and to incorporate it in controllers. To understand the limitations for practical use, many further studies are needed, it will be particularly useful to test this method on a real experiment. The abrupt-SINDy modeling framework may also help inform current rapid learning strategies in neural network architectures [@fei2006one; @vinyals2016matching; @delahunt2018putting], potentially allowing dynamical systems methods to inform rapid training paradigms in deep learning.
Acknowledgements {#acknowledgements .unnumbered}
================
MQ was supported by a fellowship within the FITweltweit program of the German Academic Exchange Service (DAAD). MQ and MA acknowledge support by the European Erasmus SME/HPC project (588372-EPP-1-2017-1-IE-EPPKA2-KA). SLB acknowledges support from the ARO and AFOSR Young Investigator Programs (ARO W911NF-17-1-0422 and AFOSR FA9550-18-1-0200). SLB and JNK acknowledge support from DARPA (HR0011-16-C-0016). We would like to thank the anonymous reviewers for their comments which helped to improve this manuscript. We also acknowledge valuable discussions related to abrupt model recovery and programming wisdom with Dennis Bernstein, Karthik Duraisamy, Thomas Isele, Eurika Kaiser, and Hod Lipson.
References {#references .unnumbered}
==========
[^1]: The user manual is located at <http://scikit-learn.org/stable/modules/generated/sklearn.model_selection.GridSearchCV.html>.
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abstract: 'We present the results of near-infrared imaging and spectroscopic observations of the young, core-collapse supernova remnant (SNR) G11.2–0.3. In the 164 image, we first discover long, clumpy filaments within the radio shell of the SNR, together with some faint, knotty features in the interior of the remnant. The filaments are thick and roughly symmetric with respect to the NE-SW elongation axis of the central pulsar wind nebula. We have detected several lines and line toward the peak position of the bright southeastern filament. The derived extinction is large ($A_V=13$ mag) and it is the brightest 164 filament detected toward SNRs to date. By analyzing two 164 images obtained in 2.2 yrs apart, we detect a proper motion corresponding to an expansion rate of $0.''''035\pm 0.''''013$ yr$^{-1}$ ($830\pm 310$ ). In addition to the features, we also discover two small 212 filaments. One is bright and along the SE boundary of the radio shell, while the other is faint and just [*outside*]{} of its NE boundary. We have detected (2-1) S(3) line toward the former filament and derive an excitation temperature of 2,100 K. We suggest that the filaments are dense clumps in a presupernova circumstellar wind swept up by the SNR shock while the filaments are probably composed of both shocked wind material and shocked supernova (SN) ejecta. The distribution of filaments may indicate that the SN explosion in 11p2 was asymmetric as in Cassiopeia A. Our results support the suggestion that 11p2 is a remnant of a SN IIL/b interacting with a dense red supergiant wind.'
author:
- 'Bon-Chul Koo'
- 'Dae-Sik Moon'
- 'Ho-Gyu Lee and Jae-Joon Lee'
- Keith Matthews
title: 'and filaments in the Supernova Remnant G11.2$-$0.3: Supernova Ejecta and Presupernova Circumstellar Wind'
---
Introduction
============
G11.2$-$0.3 is a composite-type SNR with a central pulsar wind nebula (PWN) surrounded by a circular shell. The shell is bright both in radio and X-rays, and has an outer diameter of $4'$ and a thickness of $0.'5$ [@gre88; @rob03]. The shell is clumpy with several clumps protruding its outer boundary. The bright radio shell with high circular symmetry indicates that the remnant is young and it is thought to be the best candidate for the possible historical supernova of AD 386 [@ste02]. The PWN with an associated pulsar was discovered at the very center of the remnant in X-rays with [*ASCA*]{}, and later its detailed structure was studied with the Chandra X-ray Observatory [@vas96; @kas01; @rob03 and references therein]. The PWN is elongated along the NE-SW direction with a total extent of $1'$, and appears to be surrounded by a radio synchrotron nebula with similar extent and shape. The distance to G11.2$-$0.3 determined from absorption is 5 kpc [@gre88]. The overall morphology of G11.2$-$0.3 resembles Cassiopeia A (Cas A). Both have a thick, bright, and clumpy shell, although the shell of Cas A is much brighter than that of 11p2, i.e., 2720 Jy vs 22 Jy at 1 GHz [@gre04] [^1]. At a distance of 3.4 kpc, the outer radius of Cas A shell is 2.0 pc and its expansion velocity is 4,000–6,500 [e.g., @fes96], while they are 2.9 pc and $\sim 1,000$ for G11.2$-$0.3. Cas A has a faint $5'$ (or 2.5 pc)-diameter plateau extending beyond the bright shell [e.g., @hwa04]. The plateau represents swept-up circumstellar (or ambient) medium while the bright shell is thought to be mainly the ejecta swept up by a reverse shock. Such plateau has not been detected in 11p2. [@che05] classified both Cas A and G11.2$-$0.3 into SN IIL/b category which has a red supergiant (RSG) progenitor star with some H envelope but most lost [cf. @hwa03; @you06]. Detailed observations have revealed that the explosion of Cas A was turbulent and asymmetric and that the ejecta is now interacting with a clumpy circumstellar wind [see @hwa04; @che03 and references therein]. However, very little is known about the explosion and the interaction of the 1620(?) yr-old 11p2 despite its close similarity to Cas A.
In this paper, we report the discovery and detailed studies of and filaments in the SNR 11p2 using near-infrared (IR) imaging and spectroscopic observations. Although the recent mid-IR data obtained with the Spitzer Space Telescope show the presence of very faint wispy emission close to its SE boundary [@lee05; @rea06], our near-IR observations reveal much more prominent and extended features both at the boundary and interior of the remnant, which provide important clues on the origin and evolution of 11p2.
Observations
============
We carried out near-IR imaging observations of the SNR G11.2-0.3 with Wide-field Infrared Camera (WIRC) on the Palomar 5-m Hale telescope using several narrow- and broad-band filters in 2003 June and 2005 August (Table 1). WIRC is equipped with a Rockwell Science Hawaii II HgCdTe 2K infrared focal plane array, covering $\sim 8.'5\times 8.'5$ field of view with $0.''25$ pixels scale. For the basic data reduction, we subtracted dark and sky background from each individual dithered image and then normalized it by a flat image. We finally combined the individual images to make a final image. The seeing was typically $0.''8$–$1''$ over the observations. We obtained the flux calibration of our narrow-band filter (i.e., and ) images using the H (for 164) and Ks (for 212) band magnitudes of $\ge 20$ nearby isolated, unsaturated 2MASS stars. For this, we first converted the 2MASS magnitudes to fluxes [@coh03], and then obtained the fluxes of the and emission after we deconvolved the responsivities of both WIRC (and ) and 2MASS ($H$ and $K_s$) filters. The overall uncertainty in the flux calibration is less than $10 \%$. We attribute the major source of uncertainty to the different band response of each filter as the uncertainty in photometry itself is typically a few percent. For the astrometric solutions of our images, we used all the cataloged 2MASS stars in the field, and found that they are consistent with that of 2MASS with rms uncertainty of $0.''15$.
After identifying several emission-line features in the aforementioned imaging observations, we have carried out follow-up spectroscopic observations of them using Long-slit Near-IR Spectrograph of the Palomar 5-m Hale Telescope [@lar96] in 2005 August. The spectrograph has a $256\times 256$ pixel HgCdTe NICMOS 3 array with a fixed slit of $38''$ length. We placed the slit along the bright and filaments crossing their peak positions. Toward the filament, four spectra around 1.25, 1.52, 1.63 $\mu$m (for emission), and 2.16 $\mu$m (for ) were obtained, while, toward the filament, one spectrum around 2.11 $\mu$m was obtained. Over the observations, the slit width was fixed to be $1''$, resulting in spectroscopic resolution of 650–850 with 0.06–0.12 $\mu$m usable wavelength coverage. For all the obtained spectra, the individual exposure time was 300 s with the same amount of exposure of nearby sky for sky background subtraction. For the lines, we performed the exposure twice (with different sky positions, but the same source position), and combined them, while, for the line, we performed the exposure only once.
Just after the source observations, we obtained the spectra of the G3V star HR 8545, which was at the similar airmass of the source by uniformly illuminating the slit using the f/70 chopping secondary of the telescope. We then divided the source spectra by those of HR 8545 and multiplied by a blackbody radiation curve of the G3V star temperature, which is equivalent to simultaneous flat fielding and atmospheric opacity correction. G stars, however, have numerous intrinsic (absorption) feautres, so that this procedure could inflate the intensities of emission lines if they fall on these stellar features. We have estimated the errors using the G2V solar spectrum [@liv91][^2] as a template of our reference star HR 8545 [cf. @mai96; @vac03]. The estimated errors in the observed line fluxes are $\le 5$% for all lines except line for which it is $10$%. The resulting errors in the line ratios, which are used for the derivation of physical parameters, are $\le 2$% except for the /164 ratio for which it is 7%. These calibration errors are all less than their statistical ($1\sigma$) errors (see Table 2). Another source of error is different atmospheric condition. All spectra of the source and HR 8545 were obtained at air mass of 1.6–1.8 except the H$_2$ spectra of the source which was obtained at an air mass of 2.27. There is a strong atmospheric CO$_2$ absorption line between 2.05 and 2.08 $\mu$m, and the different airmasses can give an error in the intensity of H$_2$ (2–1) S(3) line at 2.0735 $\mu$m. According to [@han96], the error is about $2$% when the air masses differ by 0.3, so that it would be $\simlt 5$% for our H$_2$ (2–1) S(3) line. Again, this is less than the ($1\sigma$) statistical error. We therefore consider that the uncertainty due to the calibration errors is less than the statistical errors quoted in this paper (Table 2). For the wavelength solutions of the spectra, we used the OH sky lines [@rou00].
Results
=======
Fig. 1 (right) is our three-color image representing the near-IR 164 (B), 212 (G), and 2.166 $\mu$m (R) emission of the SNR 11p2. We also show an 1.4 GHz VLA image for comparison which was obtained by [@gre88] in 1984–85 with $3''$ resolution. Note that the expansion rate of 11p2 at 1.4 GHz is $0.''057\pm 0.''012$ yr$^{-1}$ [@tam03] which amounts to $\sim 1''$ over the last 20 years (see also §3.3). The near-IR emission features in Fig. 1 can be summarized as follows: (1) an extended ($\sim 2.'5$), bright (blue) filament along the SE radio shell; (2) some faint, knotty emission features along the NW radio shell as well as in the interior of the source; (3) a small ($30''$), bright (green) filament along the outer boundary of the source in the SE; (4) another small, faint filament outside the NE bounday of the source. Overall the filaments are located either within the radio shell or inside of the source, while the filaments are along the radio boundary or even outside of it. We have not found any apparent filament in our rather shallow imaging observation, although we have detected faint line emission toward the peak position in our spectroscopic observation. In the following, we summarize the results on the and emission features.
164 emission
------------
### Photometry
In order to see the emission features more clearly, we have produced an ‘star-subtracted’ image (Fig. 2). We first performed PSF photometry of H-cont and 164 images, and removed stars in the 164 image if they had corresponding ones in the H-cont image. This PSF photometric subtraction left residuals around bright stars which we masked out. The faint stars, which were not removed by the PSF subtraction because the H-cont image is not as deep as that of , were then removed by subtraction of median value of $15 \times 15$ nearby pixels. Fig. 2 is the final star-subtracted image where we can see the detailed features of emission more clearly.
As in Fig. 1 (right), the extended filament within the southeastern SNR shell, -SE filament hereafter, is most prominent. The filament is composed of two bright, $30''$-long, elongated segments in the middle and two clumpy segments at the ends. The one at the southern end is a little bit apart from the other three. The total extent of the filament is $\sim 2.'5$. The filament is not very thin but has a width of $\simlt 10''$. Fig. 3 shows a detailed structure of the filament, where we have just masked out stars using the K-cont image in order to avoid any possible artifacts associated with the PSF photometric subtraction. We can see that the filament has a very good correlation with the radio shell both in morphology and brightness. The peak 164 surface brightness of the filament is $1.9 \pm 0.2\times 10^{-3}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$, which is larger than any previously reported brightness of 164 filaments in other remnants, e.g., 1.1–3$\times 10^{-4}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ in IC 443 and Crab [@gra87; @gra90] or $1.5\times 10^{-3}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$ in RCW103 [@oli89].
On the opposite side of the SE filament lies another long ($\sim 2.'5$) filament within the northwestern SNR shell (Fig. 4). This filament (-NW filament) is relatively faint and appears to be clumpy. It has little correlation with the radio emission. We note that -SE and NW filaments lie roughly symmetric with respect to the line of position angle $\approx 60^\circ$, which is close to the inclination of the central PWN of 11p2 in X-ray [@rob03]. In addition to these two extended filaments, some faint, knotty emission features are also seen in the interior of the remnant, particularly in the southern area (Fig. 5). These features spread over an area of $\sim 2'$ extent and filametary, with some of them having a partial ring-like structure. There are also several bright clumps of $\sim 5''$ size. Most of the clumps appear to be connected to the filaments, although some are rather isolated. The brightnesses of these central emission features and NW filament are $\simlt 3\times 10^{-4}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$. The observed total 164 flux is estimated to be $1.1\pm 0.2 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$, $76\pm 12$% of which is from the SE filament.
### Spectroscopy
We have detected several lines toward the peak position of the -SE filament (-pk1). Table 2 summarizes the detected lines and their relative strengths, and Fig. 6 shows the spectra. 1.257 $\mu$m and 164 lines originate from the same upper level, so that their unreddened flux ratio is fixed by relative Einstein $A$ coefficients which is 1.04 according to [@qui96]. Toward -pk1, the ratio is 0.31, which implies $A_V=13$ mag ($A_{1.644\mu {\rm m}}=2.43$ mag) or H-nuclei column density of $2.49\pm 0.07 \times 10^{22}$ cm$^{-2}$ using the extinction cross section of the carbonaceous-silicate model for interstellar dust with $R_V=3.1$ of [@dra03][^3]. This is a little larger than the column density to the remnant derived from X-ray observations $(1.7-2.4)\times 10^{22}$ cm$^{-2}$ [@rob03]. We note that the numerical values of the Einstein $A$ coefficients for near-IR lines in the literature differ as much as 50%: using the values of [@nus88], the expected 1.257 $\mu$m to 164 line-intensity ratio is 1.36, while [@smi06] empirically derived 1.49 from their spectroscopy of P Cyg. If the intrinsic ratio is 1.36 or 1.49, we obtain a little (20–30%) higher column density. We adopt the $A$-values of [@qui96] in this paper which yield a column density closer to the X-ray one. According to [@har04], they also yield extinction more consistent with optical spectroscopic result for a protostellar jet.
The ratios of the other three lines, e.g., 1.534 , 1.600 , and 1.664 , to 1.644 are good indicators of electron density [e.g., @oli90]. We solved the rate equation using the atomic parameters assembled by CLOUDY [version C05.05, @fer98] which adopts the Einstein A coefficients of [@qui96] and collision strengths of [@pad93] and [@zha95]. We have included 16 levels which is enough at temperatures of our interest ($\simlt 10^4$ K). We consider only the collisions with electrons, neglecting those with atomic hydrogen, even if the degree of ionization of the emitting region could be low (see § 4.2). This should be acceptable since the rate coefficients for atomic hydrogen collisions are more than two orders of magnitude smaller than those for electron collisions [@hol89]. The ratios of 1.534 and 1.664 lines yield consistent results, e.g., $6,000\pm 400$ cm$^{-3}$ and $5,900\pm 400$ cm$^{-3}$, while 1.600 line ratio yields a little higher density ($7,800\pm 400$ cm$^{-3}$) at $T=5,000$ K which is the mean temperature estimated for line-emitting regions in other SNRs [@gra87; @oli89; see also § 4.2]. The result is not sensitive to temperature, e.g., a factor of 2 variation in temperature causes 10–20% in density. We adopt the average value $6,600\pm 900$ cm$^{-3}$ at $T=5,000$ K as the characteristic electron density of the filaments.
We also detected line toward -pk1. The dereddened ratio of 164 to line is $77^{+14}_{-10}$, which is much greater than that ($\simlt 0.1$) of HII regions but comparable to the ratios observed in other SNRs (see § 4.2).
### Proper Motion during 2003–2005
We have two 164 images taken in 2.2 years apart, i.e., in 2003 June and 2005 August. The time interval is not long enough to notice the proper motion of the filaments in the difference image obtained by subtracting one from the other. We instead inspect one-dimensional intensity profiles of the bright -SE filament to search for its proper motion associated with an expansion.
Fig. 7 shows the intensity profiles across the two bright segments of the -SE filament along the cuts (dashed lines) in Fig. 3. The cuts are made to point to the central pulsar which is very close to the geometrical center of the SNR shell [@kas01]. The distance in the abscissa is measured from the upper right end of the cuts, so that it increases outward from the remnant center. Note that the profiles of the filament in 2005 (solid lines) are slightly shifted outward from those in 2003 (dashed lines). We fit the profiles along the cuts A and B with a Gaussian and obtain shifts of $0.''063\pm 0.''032$ and $0.''095\pm 0.''064$ in their central positions, respectively. For comparison, the profiles of nearby stars, e.g., the strong peak at $24''$ in Fig. 7 (left), do not show any appreciable shift. The mean shift in stellar positions from the same 1-dimensional Gaussian analysis of nearby seven stars is found to be $-0.''0067 \pm 0.''0029$. Therefore, the mean proper motion of the SE filament with respect to the nearby stars during 2.2 years amounts to $0.''076 \pm 0.''029$, which corresponds to a rate of $0.''035\pm 0.''013$ yr$^{-1}$.
212 emission: Photometry and Spectroscopy
-----------------------------------------
Fig. 8 is a star-subtracted and median-filtered 212 image. The image has been made in the same way as Fig. 2. Two small ($\sim 30''$) filaments, one at the southern SNR radio boundary and another fainter one outside the NE boundary are now clearly seen. The one in the southeast (-SE filament) is bright and elongated along the radio boundary. Its peak surface brightness is $3.0\pm 0.3 \times 10^{-4}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ and its flux is $4.3 \pm 0.4 \times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$. The NE filament (H$_2$-NE filament) is just outside of the SNR boundary and is located where the radio continuum boundary is distorted. Its surface brightness is $\simlt 40$% of the SE filament peak brightness, while its flux density is $\sim 50$% of the SE filament. There is no 164 emission associated with either filament. A long ($\sim 2'$) filamentary feature seems to be present well outside the southeastern SNR boundary, but it is too faint to be confirmed.
Fig. 9 shows a detailed structure of the -SE filament. It is composed of two bright segments surrounded by a diffuse envelope. It is just outside of the bright -SE filament, but there is no apparent correlation between the two (cf. Fig. 3). We have detected two lines, (1,0) S(1) and (2,1) S(3), toward the peak position of the filament, -pk1 (Fig. 10). Their dereddened ratio, using the column density derived from line ratios ($A_{2.12 \mu {\rm m}}=1.59$ mag), is $0.14\pm 0.01$ (Table 2), which gives $\tex\approx 2,100$ K using the transition probabilities of [@wol98].
Discussion
==========
G11.2$-$0.3 has been proposed to be a young remnant of an SN IIL/b interacting with a dense RSG wind based on its PWN and the small size of the SNR shell [@che05]. The thick, bright shell is thought to be shocked SN ejecta in contact with shocked wind material. The outer edge of the shell is not sharp and it was suggested that the ambient shock propagating into wind material could be at a larger distance [@gre88; @che05]. In the following, we first discuss the physical properties of the filaments that we have discovered in this paper, and show that our results support the SN IIL/b scenario. Then we discuss the physical properties of the filaments which are thought to be composed of both shocked wind material and shocked SN ejecta.
Filaments and Presupernova Circumstellar Wind
---------------------------------------------
### Excitation of filaments
The -SE filament is located at the rim of the bright SNR shell and elongated along the rim, which suggests that it is excited by the SNR shock. The derived $v=2$–1 excitation temperature ($\approx 2,100$ K) is also typical for shocked molecular gas [@bur89]. The dereddened peak 212 surface brightness is $1.3\pm 0.1 \times 10^{-3}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$, and the dereddened total flux of the SE filament is $1.9\pm 0.2 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$.
The interstellar ultraviolet (UV) photons in principle could excite and heat the gas to produce similar excitation temperature if the gas is dense enough for collisions to dominate deexcitation [@ste89; @bur90]. However, the expected 212 surface brightness by UV photon excitation is low unless the density is high and the radiation field is very strong, e.g., $\nh\ge 10^5$ cm$^{-3}$ and $G_0\ge 10^4$ for $\ge 1\times 10^{-4}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ where $\nh$ is the number density of H nuclei and $G_0$ is far UV (FUV) intensity relative to the interstellar radiation field in the solar neighborhood [@bur90]. Note that $G_0=10^4$ corresponds to an O4-type star at a distance of $\sim 1$ pc [@tie05]. No such strong FUV source exists around the filament.
X-ray emission from the remnant is another source that could possibly excite and heat the filament. We may consider a molecular clump situated at some distance from an SN explosion. As the SN explodes and the SNR evolves, the X-ray flux increases and, in principle, an ionization-dissociation front may develop and propagate into the clump. If the density is sufficiently high, the lines from heated molecular gas could show ‘thermal’ line ratios [@gre95]. The line intensities from such clump depend on details, and no model calculations that may be directly applicable to our case are available [cf. @dra90; @dra91; @mal96]. In the following, we instead simply consider the energy budget. If the 212 line is emitted by reprocessing the X-ray photons from the SNR falling onto the molecular clump, its luminosity may be written as $L_{2.122} \sim \epsilon L_X (\Omega_{\rm cl}/4\pi)$ where $\epsilon$ is an efficiency of converting the incident X-ray energy flux into 212 line emission, $L_X$ is the X-ray luminosity of the remnant, and $\Omega_{\rm cl}$ is the solid angle of the clump seen from the SNR center. The above formula is accurate if the clump is small and if the X-ray source is spherically symmetric. Although 11p2 is not a spherically symmetric source in X-rays, we may use the formula to make a rough estimate of the expected 212 line luminosity. The conversion efficiency for SNRs embedded in molecular clouds was calculated to be $\simlt 1 \times 10^{-3}$ [@lep83; @dra90; @dra91]. The efficiency is a function of X-ray energy absorbed per H-nucleon and the above inequality might be valid for X-ray irradiated small clumps too. Now if we assume that the clump has the line-of-sight extent similar to the extent on the sky ($\sim 0.'5$), then $ \Omega_{\rm cl}/4\pi \sim 4 \times 10^{-3}$. Since the X-ray luminosity of 11p2 is $L_X\sim 10^{36}$ erg s$^{-1}$ in 0.6–10 keV band [@vas96], we have $L_{2.122}\simlt 4 \times 10^{30} $ erg s$^{-1}$. This is much less than the observed 212 luminosity of the SE filament, which is $\sim 6 \times 10^{33}$ erg s$^{-1}$. Therefore, the X-ray excitation/heating does not appear to be important for the -SE filament.
The above consideration leads us to conclude that the -SE filament is excited by the SNR shock associated with 11p2. The absence of associated 164 or emission suggests that the emission from the -SE filament might be from warm molecules swept-up by a slow, non-dissociative $C$ shock not from reformed molecules behind a fast, dissociative $J$ shock. The critical velocity for a shock to be a non-dissociative $C$ shock is $\simlt 50$ [@dra83; @mck84]. The dereddened mean surface brightness of the -SE filament is $\sim 8\times 10^{-4}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$. This is comparable to the (normal) brightness of a $\sim 30$ shock propagating into molecular gas of $n_{\rm H}=10^4$ cm$^{-3}$ according to the $C$-shock model of [@dra83]. We were unable to find model calculations for lower densities. But, since the intensity will be proportional to the preshock density, provided that the density in the emitting gas is less than the critical density [$\simgt 10^5$ cm$^{-3}$; @bur89], the results of [@dra83] indicates that a 40–50 shock propagating into molecular gas of $n_{\rm H}=10^3$ cm$^{-3}$ might have similar (normal) surface brightness. A slower shock with a lower preshock density would be possible if the shock propagating into the filament is tangential along the line of sight, so that the brightness normal to the shock front is lower.
The situation is not so clear for the -NE filament for which we lack spectroscopic information. Its flux density, however, is comparable to that of the SE filament and we may rule out the excitation by X-rays from 11p2. We checked 2MASS colors of nearby ($\le 2'$) stars, but found no OB stars that would be responsible for the UV excitation. This leaves again the shock excitation for the origin of the emission. A difficulty with the shock excitation is that the filament is located outside the radio SNR boundary. But as have been pointed out in previous studies [e.g., @gre88], the radio continuum boundary is not sharp and the ambient shock is thought to have propagated beyond the apparent radio boundary. It therefore seems to be reasonable to consider that the -NE filament is excited by the SNR shock too, although we need spectroscopic observations to understand the nature of the -NE filament.
### Circumstellar Origin of filaments
The filaments are more likely of circumstellar origin than interstellar. If interstellar, they must be dense clumps originally in an ambient or parental molecular cloud. We do not expect to observe molecular material around small, young core-collapse SNe in general because massive stars clear out the surrounding medium with their strong UV radiation and strong stellar winds during their lifetime. Some molecular material may survive if the progenitor star is an early B-type (B1–B3) star, which does not have strong UV radiation nor strong stellar winds [@mck84b; @che99]. A difficulty with this scenario, however, is that then the swept up mass at the current radius (3 pc) is likely to be much greater than the ejecta mass, so that the remnant should have been already in Sedov stage where it would appear as a thin, limb-brightened shell. The thick-shell morphology of 11p2, however, indicates that it is not yet in Sedov stage. We therefore consider that the filaments are of circumstellar origin which fits well into the SN IIL/b scenario.
It is plausible that the progenitor of 11p2 had a strong wind which contains dense clumps. Numerous such clumps have been observed in Cas A, e.g., “Quasi-stationary flocculi (QSF)", which are slowly moving, dense optical clumps immersed within a smoother wind [@van71; @van85]. In the 320-yr old Cas A, the shock propagating into the clump is fast [100–200 ; @che03] while in the 1620-yr old 11p2 it is slow (30–50 ). Their velocity ratio is comparable to the ratio ($\sim 1/5$) of SNR expansion velocities, which suggests that the winds in 11p2 and Cas A have similar properties. We may estimate the density contrast between the clump and the smoother wind from the ratio of the shock speed into the clump ($v_c=30-50$ ) to the SNR forward shock speed $\vexp$. If we adopt the result of the radio (20 cm) expansion studies by [@tam03], $\vexp=1350\pm280$ so that the density contrast would be $(\vexp/v_c)^2=700-3,000$. For comparison, [@che03] estimated a density contrast of $3,000$ for Cas A.
Filaments and SN Ejecta
-----------------------
### Shock Parameters of the -SE filament
The filaments are located within the bright SNR shell in contrast to the filaments. The -SE filament has a remarkable correlation with the radio shell in both morphology and brightness. The knotty emission features inside the remnant might be within the shell too, but projected on the sky. The location of the filaments suggests that the emission is almost certainly from the shocked gas. The shock must be radiative and the emission should originate from the cooling layer behind the shock.
The -SE filament is very bright with the dereddened 164 peak surface brightness of $1.80 \pm 0.18\times 10^{-2}$ ergs cm$^{-2}$ s$^{-1}$ sr$^{-1}$. It is in fact the brightest among the known 164 filaments associated with SNRs. The total dereddened 164 flux is $1.0\pm 0.1 \times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$. The ratio of 164 to line ($\sim 80$) toward the peak position of the -SE filament is larger or comparable to the ratios observed in other SNRs, e.g., 27 to $\ge 71$ in IC 443 [@gra87] or 34 in RCW 103 [@oli89]. It was pointed out in previous studies that the high ratio can result from SNR shocks [*interacting with the ISM*]{} by the combined effects of ‘shock excitation’ and the enhanced gas-phase iron abundance. First, since the ionization potential of iron atom is only 7.9 eV, FUV photons from the hot shocked gas can penetrate far downstream to maintain the ionization state of Fe$^+$ where H atoms are primarily neutral [@mck84; @hol89b; @oli89]. Therefore, lines are emitted mainly in gas with a low degree of ionization at $T=10^3$–$10^4$ K. This partly explains the observed high ratio of 164 to lines, but not all. Shock model calculations showed that the ratio is $\sim 1$ if the gas-phase iron abundance is depleted as in normal ISM. According to [@hol89b], the ratio is $\sim 1.5$ for shocks at velocities 80–150 propagating into a molecular gas of $\nh=10^3$ cm$^{-3}$ with iron depletion $\deltafe\equiv{\rm [Fe/H]/[Fe/H]_\odot}=0.03$ where \[Fe/H\]$_\odot$=$3.5\times 10^{-5}$. [@mck84] presented the results on atomic shock calculations including grain destruction: for a 100 shock propagating into atomic gas of $\nh=10$ and 100 cm$^{-3}$, \[Fe II\] 1.2567 $\mu$m/=2.7 and 3.7 with $\deltafe=0.53-0.58$ in the far downstream. If we use 0.033 as the ratio of to line intensities which corresponds to a Case B nebula at 5,000 K [@ost89], the ratio corresponds to 164/=80 and 110, comparable to the observed ratio. Therefore, gas-phase iron abundance close to the solar is required to explain the observed 164 to ratio toward -pk1.
The preshock density may be estimated from the 164 brightness. The 164 surface brightness toward the -SE filament varies $\sim 1-10\times 10^{-3}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$. Its morphology in Fig. 4 suggests that the shock front might be tangential along the line of sight to enhance the surface brightness of the filament. The normal surface brightness of the 100 shock propagating into atomic gas of $\nh=100$ cm$^{-3}$ is $2.5\times 10^{-4}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ [@mck84]. It is $0.3-2\times 10^{-3}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ for 80–150 shocks propagating into a molecular gas of $\nh=10^3$ cm$^{-3}$ if the gas-phase abundance of iron was solar [@hol89b]. Therefore, the preshock density needs to be $\simgt 1,000$ cm$^{-3}$. This appears to be roughly consistent with the electron density derived from lines ratios. As we pointed out above, the ionization fraction of the -emitting region is expected to be low. [@oli89] estimated a mean ionization fraction of 0.11, in which case $\nh\approx n_e/0.11\approx 6\times 10^4$ cm$^{-3}$. For a 100 shock, the final compression factor would be $\sim 80$ [@hol89], so that the above postshock density implies a preshock density of $\sim 800$ cm$^{-3}$. This is close to the density required to explain the surface brightness considering the uncertainties in various parameters. Therefore, a 100 shock propagating into a gas of $\nh\simgt 1,000$ cm$^{-3}$ and destroying dust grains seems to explain the observed parameters of the -SE filament.
### Origin of filaments
The filaments could be either shocked circumstellar medium (CSM) or shocked ejecta, or both under the context of the Type IIL/b scenario. In the SE filament, line is detected at the peak position and its ratio to 164 line is consistent with a $100$ [*interstellar*]{} shock (§ 4.2.1), which implies that the emission is not from metal-rich ejecta but from shocked CSM. For example, when the clumps in the previous section are swept up by shocked dense ejecta, a stronger shock will propagate into the clumps to dissociate and ionize the gas to produce emission. Radio observation also suggests that the remnant is more heavily affected by the ambient medium toward this direction: [@kot01] showed that the magnetic field structure of the bright radio shell is radial in general except the bright SE shell where the degree of polarization is significantly low compared to the other parts of the shell. The non-radial magnetic field and the low degree of polarization suggest that the synchrotron emission is dominated by shocked ambient gas not by shocked ejecta. On the other hand, the SE filament is located in the middle of the radio shell and has a large radial proper motion. If the proper motion is due to expansion of the SNR shell, which is very likely, it implies an expansion velocity of $\ge 830\pm 310$ (see next). This suggests that the filament is associated with ejecta. It is possible that some emission originates from dense, Fe-rich ejecta recently swept-up and excited by a reverse shock. We suppose that the -SE filament consists of both the shocked CSM and the shocked ejecta, although it is not obvious how the two interact to develop the observed properties.
The derived proper motion of the -SE filament ($0.''035\pm 0.''013$ yr$^{-1}$) may be compared to the expansion rate of the radio shell. [@tam03] obtained a mean expansion rate of $0.''057\pm 0.''012$ yr$^{-1}$ at 1.465 GHz and $0.''040\pm 0.''013$ yr$^{-1}$ at 4.860 GHz by comparing radio images separated by 17 years. Our proper motion is comparable to the 4.860-GHz expansion rate but is smaller than the 1.465-GHz expansion rate which was considered to be more reliable by the authors. It is possible that the -SE filament is not moving perpendicularly to the sight line, so that the true space motion is greater. But, considering that the filament is located near the boundary of the remnant, the projection effect is probably not large. Instead the difference may be because the proper motion that we have derived in this paper represents the velocity of the brightest portion of the filament while the radio expansion rate might be close to the pattern speed, e.g., the SNR shock speed. Since the velocities of shocked ambient gas and shocked ejecta in the shell might be less than the SNR shock velocity, it is plausible that our ‘expansion rate’ is less than the radio one. We will explore the dynamical properties of 11p2 in our forthcoming paper.
The -NW filament and the knotty emission features are considered to be mostly, if not all, dense SN ejecta. The radial magnetic field supports this interpretation [@kot01]. Their filamentary and ring-like structure may be a consequence of bubbly Fe ejecta [e.g., @blo01]. It is worth to note that the emission is distributed mainly along the NW-SE direction (Fig. 2), the direction perpendicular to the PWN axis. The long and symmetric morphology of the -SE and -NW filaments resembles the main optical shell of Cas A. Cas A, in optical forbidden lines of O, S ions, shows a complex northern shell composed of several bright, clumpy filamentary structures at varying distances from the center and a relatively simple-structured southern shell [e.g., @fes01]. These northern and southern portions of the main optical shell are opposite across the jet-axis along the NE-SW axis. The optical shell is generally believed to be dense clumps in ejecta recently swept up by reverse shock, although it contains QSFs too. The similarity to Cas A suggests that the explosion in 11p2 was asymmetric as in Cas A.
The total 164 luminosity is $\sim 75 L_\odot$. This is two orders of magnitude greater than Kepler or Crab, but comparable to RCW 103 or IC 443 [@oli89; @kel95]. In collisional equilibrium at $T=5,000$ K with $n_e\approx 6,600$ cm$^{-3}$, this converts to Fe mass of $\sim 5.3\times 10^{-4}$ . Both the shocked ejecta and the shocked CSM constitue this. The $^{56}$Fe mass that would have been formed from the radioactive decay of $^{56}$Ni in 15–25 SN explosion is 0.05–0.13 [@woo95; @thi96]. Therefore, the Fe ejecta detected in 164 emission is less than one percent of the total Fe ejecta. On the other hand, the observed Fe mass corresponds to H (+He) mass of 0.27 $M_\odot$ for the solar abundance, which implies that the mass of the shocked CSM comprising the Fe filaments should be a tiny fraction of the swept-up CSM too.
Conclusion
==========
G11.2$-$0.3 has been known as an evolved version of Cas A, both being SN IIL/b with a significant mass loss before explosion. Our results confirm that 11p2 is indeed interacting with a clumpy circumstellar wind as in Cas A. Clumps with a density contrast of $\sim 3,000$ may be common in presupernova circumstellar wind of SN IIL/b. As far as we are aware, 11p2 is the first source where the presupernova wind clumps are observed in emission. The filament in the northeast is of particular interest because it could provide a strong evidence for an ambient shock beyond the bright radio shell. Future spectroscopic studies will reveal the nature of this filament.
The filaments in 11p2 are probably composed of both shocked CSM and shocked ejecta. The one in the southeast is the brightest among the known 164 filaments associated with SNRs and is thought to be where the ejecta is heavily interacting with dense CSM. We note that RCW 103, which is another young remnant of SN IIL/b [@che05], has a very bright filament too. The source is similar to 11p2 in the sense that emission is detected beyond the apparent SNR boundary, although the emission in RCW 103 extends along the entire bright SNR shell [@oli90]. It is possible that the filaments in the two remnants are of the same origin. The other faint -emitting features of 11p2 are thought to be mostly SN ejecta. The distribution of filaments suggests that the explosion produced 11p2 was asymmetric as in Cas A. In Cas A, however, Fe ejecta have been observed mainly in X-rays although faint 164 lines have been detected toward several fast-moving ejecta knots from spectroscopic observations by [@ger01]. Future detailed spectroscopic studies will help us to understand the nature of the filaments and knots in 11p2 as well as the SN explosion itself.
We thank Dave Green for providing his VLA images of 11p2. We also wish to thank Chris McKee and Roger Chevalier for their helpful comments. D-SM acknowledges a Millikan fellowship from California Institute of Technology. This work was supported by the Korea Science and Engineering Foundation (ABRL 3345-20031017).
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[lccccl]{}
$^d$ & 1.644 & 0.0252 & 60 & 12 & 2003. 06. 17, 2005. 08. 27\
H$_2$ & 2.120 & 0.0329 & 20 & 36 & 2005. 08. 14\
Br$\gamma$ & 2.166 & 0.0327 & 30 & 20 & 2005. 08. 28\
$K_s$ & 2.150 & 0.312 & 15 & 90 & 2003. 06. 16\
$H$-cont & 1.570 & 0.0236 & 30 & 10 & 2005. 08. 14\
$K$-cont & 2.270 & 0.0330 & 60 & 12 & 2005. 08. 27\
[lllll]{}
-pk1$^c$ & 1.2567 & a $^4D_{7/2}\rightarrow a ^6D_{9/2}$ & 0.314 (0.010) & 1.04\
&1.5335 & a $^4D_{5/2}\rightarrow a ^4F_{9/2}$ & 0.116 (0.004) & 0.151 (0.005)\
&1.5995 & a $^4D_{3/2}\rightarrow a ^4F_{7/2}$ & 0.102 (0.003) & 0.113 (0.003)\
&1.6436 & a $^4D_{7/2}\rightarrow a ^4F_{9/2}$ & 1.0 & 1.0\
&1.6638 & a $^4D_{1/2}\rightarrow a ^4F_{5/2}$ & 0.052 (0.002) & 0.050 (0.002)\
&2.1661 & H 4-7 Br$\gamma$ & 0.030 (0.004) & 0.013 (0.002)\
-pk1$^d$ & 2.0735 & (2-1) S(3) & 0.13 (0.01) & 0.14 (0.01)\
&2.1218 & (1-0) S(1) & 1.0 & 1.0\
[^1]: Also available at http://www.mrao.cam.ac.uk/surveys/snrs/.
[^2]: Also available at http://diglib.nso.edu/contents.html.
[^3]: Data available at http://www.astro.princeton.edu/ draine/dust/dustmix.html.
|
---
abstract: 'In recent years a two-scale expansion was established to study reactions of the type $NN\to NN\pi$ within chiral perturbation theory. Then the diagrams of some subclasses that are invariant under the choice of the pion field no longer appear at the same chiral order. In this letter we show that the proposed expansion still leads to well defined results. We also discuss the appropriate choice of the heavy baryon propagator.'
address: |
Institut für Kernphysik,\
Forschungszentrum Jülich GmbH,\
D–52425 Jülich, Germany
author:
- 'C. Hanhart'
- 'A. Wirzba'
title: 'Remarks on $NN\to NN\pi$ beyond leading order'
---
and
Chiral Lagrangians,Effective interactions,Pion production 21.30.Fe,12.39.Fe,25.10.+s,25.40.Ve
Introduction
============
Pion production in nucleon-nucleon ($NN$) collisions is subject of theoretical and experimental investigations already since the 1960s — for a review of the history of the field see Ref. [@garmiz]. However, when new high precision data became available due to advanced accelerator technology in the beginning of the 1990s it became clear that all phenomenological studies performed so far were not capable of describing the data. Several mechanisms were proposed to cure the problem; however, no clear picture emerged [@report].
There was the hope that chiral perturbation theory (ChPT) could resolve the issue. As the effective field theory for the standard model at low energies it should provide a framework to investigate the reactions $NN\to NN\pi$ in a field-theoretically consistent way. In a first attempt a scheme proposed by Weinberg to study elastic and inelastic pion reactions on nuclei [@wein92] was applied to investigate also pion production in $NN$ collisions. However, in doing so up to next–to–leading order (NLO) the discrepancy between the calculations and data became even worse [@firsts]. In addition, loop contributions, formally of order NNLO, gave even larger effects [@loop_dmit; @loop_ando] shedding doubts on an applicability of chiral perturbation theory to $NN\to NN\pi$.
In parallel, already in Refs. [@bira] it was stressed that the large momentum transfer, typical for meson production in $NN$ collisions, needs to be taken care of in the power counting. This idea was further developed in Refs. [@pwaves; @mitnorbert]. The appropriate expansion parameter for $NN\to NN\pi$ therefore is $$\chi_{\rm prod}=p_{\rm thr}/M=\sqrt{{m_\pi}/{M}} \ ,
\label{chiprod}$$ where $p_{\rm thr}=\sqrt{Mm_\pi}$ denotes the threshold momentum for pion production in $NN$ collisions. $M$ and $m_\pi$ are the masses of the nucleon and pion, respectively. Here the leading-order (LO) scales as ${\mathcal O}(\chi_{\rm prod}^1)$ and subleading orders N$^n$LO scale as ${\mathcal O}(\chi_{\rm prod}^{n+1})$. For the most recent developments for the reaction $NN\to NN\pi$ within chiral perturbation theory we refer to Refs. [@pp2dpi].
Thus in the reactions $NN\to NN\pi$ one is faced with a two-scale expansion, since both $m_\pi$ as well as $p_{\rm thr}$ appear explicitly in the expressions. For tree-level diagrams this does not cause any problem. To perform the power counting for loop integrals, however, a rule has to be given what scale to assign to the components of the loop momentum. After subtraction of the nucleon mass $M$, the residual energy of each external nucleon at threshold is $m_\pi/2$, whereas the corresponding momentum is of order $p_{\rm thr}$. One therefore would be tempted to take over this scaling also for the loop momentum. On the other hand the new power counting is based on two scales, $p_{\rm thr}\gg m_\pi$, and the pions in loops are off-shell. Therefore there is no reason why the scaling of the pion energies in loops should be different from the scaling of the pertinent three-momenta. In Appendix E of Ref. [@report] it is shown that for all diagrams that do not have a two-nucleon cut, each component of the loop momentum should be counted of order of the largest external momentum in the loop. The argument there is based on the observation that in time ordered perturbation theory (TOPT) there is no ambiguity for the order assignment of energies since it is a 3 dimensional theory in the first place. On the other hand the leading order of a given TOPT amplitude should agree to that of the corresponding Feynman amplitude. This allows to identify the proper scale for the energy of the loop momentum. The assignment was checked by explicit calculations in Refs. [@loop_dmit; @mitnorbert]. As a result, all components of the loop momentum in diagram (a) and (b) of Fig. \[loops\] scale as $\chi_{\rm prod}M$, but those of diagram (c) and (d) scale as $m_\pi\sim\chi_{\rm prod}^2 M$ and are therefore suppressed[^1]. One further consequence of the presence of two scales in the problem is that the individual loops no longer contribute to only a single order, but each loop contributes to infinitely many orders, since $m_\pi/p_{\rm thr}= \chi_{\rm prod}$ appears as the argument of non–analytic functions. The power counting only identifies the lowest order where the particular loop starts to contribute [@mitnorbert].
In Ref. [@novel] it was shown that the sum of all diagrams of Fig. \[loops\] is independent of the choice of the pion field. However, based on the scheme developed in Refs. [@pwaves; @mitnorbert] only diagram $(a)$ and $(b)$ contribute at NLO whereas diagrams $(c)$ and $(d)$ start to contribute not until order N$^4$LO (see Table 11 of Ref. [@report]). The main purpose of this letter is to investigate the consistency of these two statements.
As we go along we also need to discuss the appropriate choice of the nucleon propagator in the heavy baryon formulation. This is done in Section \[subleading\]. Section \[conclusion\] contains our conclusions. Moreover, for clarification two appendices are added, one is about reparameterizations of the chiral matrix $U$, the other is about the $1/M$ expansion of the nucleon propagator.
Dependence on the pion field to NLO
===================================
The Lagrangian relevant for our study may be written as [@GSS] $${\mathcal L}=\frac{f_\pi^2}{4}\left<u^\mu u_\mu\right>
+\frac{f_\pi^2}{4}\left<\chi_+\right>+
\bar \Psi \left(i\gamma^\mu D_\mu -M+\frac{g_A}{2}\gamma^\mu
u_\mu\gamma_5\right)\Psi \ .
\label{LGSS}$$ Here $\left< \dots \right>$ denotes a trace in the isospin-space, $\Psi$ is the relativistic spinor of the nucleon, $D_\mu$ is its covariant derivative containing the Weinberg-Tomozawa term [@WeinbergTomozawa] and other $\pi^{2n}NN$ terms, $g_A$ is the axial-vector constant, $f_\pi$ the pion decay constant. Furthermore, $$u_\mu = i\left( u^\dagger \partial_\mu u- u \partial_\mu
u^\dagger\right)\quad\mbox{and}\quad
\chi_+ = u^\dagger \chi u^\dagger + u \chi^\dagger u$$ are the chiral vielbein and the mass term, respectively, with $\chi=2B{\mathcal M}$, where ${\mathcal M}$ is the quark mass matrix and $B$ is proportional to the $SU(2)$ quark condensate in chiral limit. In the isospin symmetric case one may write $
\chi = m_\pi^2 \, \mathbf{1}
$.
In order to investigate the dependence of our results on the choice made for the pion field $\pi$ (= $\vec\tau\cdot\vec\pi$ in terms of the Pauli matrices for isospin), we start from the following general expression for the chiral matrix $U=u^2$ (see Appendix \[app:chiral\_matrix\]) $$U = \exp \left(\frac{i}{f_\pi}(\vec \tau \cdot \vec \pi) g(\pi ^2/f_\pi^2)
\right) \ .
\label{Umat}$$ Here $g(\pi^2/f_\pi^2)$ is an arbitrary regular function with $g(0)=1$. For our purposes it is sufficient to expand $g$ up to second order in the pion field. We may write $$g (\pi^2/f_\pi^2)
= 1+\left(\alpha+\frac{1}{6}\right)\frac{{\vec \pi}^2}{f_\pi^2}+\cdots \ .$$ Obviously, for $\alpha=-1/6$ we work with $U$ in the so-called [*exponential gauge*]{}. In the $\sigma$–gauge one uses $$U=\sqrt{1-\frac{{\vec \pi}^2}{f_\pi^2}}+\frac{i}{f_\pi}\vec \tau \cdot
\vec\pi
=1+\frac{i}{f_\pi}\vec\tau \cdot \vec\pi -
\frac{1}{2f_\pi^2}{\vec\pi}^2-\frac{1}{8f_\pi^4}{\vec\pi}^4-\cdots \ .$$ By explicit evaluation one finds, that $\alpha=0$ reproduces this expression up to and including terms of order $(\pi / f_\pi)^4$. This is sufficient for our purposes. For more details including a justification of the notion “gauge choice” for the pion parameterizations see Appendix \[app:chiral\_matrix\].
All building blocks of the chiral Lagrangian may now be expressed in terms of the field $u$ defined above. One finds for the operators relevant in this work $$u_\mu =
-\frac1{f_\pi}\vec \tau \cdot \partial_\mu \vec \pi
-\frac1{2f_\pi^3}\left(2\alpha{\vec \pi}^2 \left(\vec \tau \cdot
\partial_\mu \vec \pi\right)+\left(1+4\alpha\right)
\left(\vec \pi\cdot \partial_\mu \vec \pi\right)\left(\vec \tau\cdot
\vec \pi\right)\right)
+ \cdots$$ for the chiral vielbein and $$\chi_+=m_\pi^2\left(u^\dagger u^\dagger +
uu\right)=m_\pi^2\left(2-\frac{{\vec \pi}^2}{f_\pi^2}-
\frac{{\vec \pi}^4}{4f_\pi^4}\left(1+8\alpha\right) - \cdots \right)$$ for the mass term. In both cases terms of higher order in the pion field were not displayed, since they are not relevant for the present work.
We use the framework of heavy-baryon chiral perturbation theory (HBChPT) [@JenMano; @BKKM]. It is then straightforward to find the Feynman rules for the relevant building blocks of the diagrams (see Fig. \[blocks\], in all cases the pion momenta $q_i=(q^0_i,{\vec q}_i)$, $i=1,2,3,4$ are chosen as outgoing):
$$\begin{aligned}
\nonumber
i V_{4\pi} &=& \frac{i}{f_\pi^2}\Bigl\{\Bigl[
(q_1+q_2)^2-m_\pi^2+2\alpha\sum_{i=1}^4
(q_i^2-m_\pi^2)\Bigr]\delta^{ab}\delta^{cd}
\\
& & \qquad +
\Bigl[ {\mbox{\scriptsize$
\left(\begin{array}{c} a b;c d \\[-1mm] 12;34 \end{array}\right)$}} \rightarrow
{\mbox{\scriptsize$
\left(\begin{array}{c} ac;bd \\[-1mm] 13;24 \end{array}\right)$}}\Bigr]
+ \Bigl[ {\mbox{\scriptsize$
\left(\begin{array}{c} a b;c d \\[-1mm] 12;34 \end{array}\right)$}} \rightarrow
{\mbox{\scriptsize$
\left(\begin{array}{c} ad;cb \\[-1mm] 14;32 \end{array}\right)$}}\Bigr]
\Bigr\}\ ,
\label{V4pi} \\
i V_{NN\pi} &=& -\frac{g_A}{2f_\pi}\tau^a (\vec \sigma\cdot {\vec q}_i) \ ,
\label{VNNpi}\\
i V_{NN3\pi} &=& -\frac{g_A}{4f_\pi^3}\left\{\delta^{ab}\tau^c \vec
\sigma\cdot \left[\vec q_1+\vec q_2
+4\alpha(\vec q_1+\vec q_2+\vec q_3)\right]+ \mbox{cyclic}\right\} \ ,
\label{VNN3pi}\end{aligned}$$
where $\vec\sigma$ is the Pauli-matrix vector for spin and $a,b,c,d\in\{1,2,3\}$ are the isospin indices.[^2] In addition we need both the pion propagator and the nucleon propagator. The former is given by the standard expression $i D_\pi(q)^{ab}= i\delta^{ab}(q^2-m_\pi^2+i\epsilon)^{-1}$. To leading order we use for the latter $$i S_N(p-q)=\frac{i}{-q_0+i\epsilon} \ .
\label{nprop1}$$ We chose the momenta such that the initial, on-shell, nucleon with momentum $P^\mu=Mv^\nu+p^\mu$ is pushed off its mass shell by the emission of a virtual pion with momentum $q$, where $v^\mu$ is a four-vector with the properties $v^2=1$ and $v^0\geq1$. The standard choice of HBChPT, also used here, is $v^\mu=(1,0,0,0)$. Throughout the paper we follow the convention that uppercase nucleon momenta contain $Mv^\mu$, whereas this term is subtracted out from their lowercase counterparts. Note that for loop momenta temporal ($q_0$) as well as spacial ($q_i$) components are assumed of order $p_{\rm thr}$, if not stated otherwise, as outlined in the introduction. The residual energy $p_0$ of the incoming [*on-shell*]{} proton, however, is of order $\chi_{\rm prod}^2 M\sim m_\pi$ because of the on-shell condition. Our rule for the nucleon propagator is different to the one applied in Refs. [@loop_dmit; @fred], where $i(p_0-q_0+i\epsilon)^{-1}$ is used for the propagator. It is justified in the next section and in Appendix \[app:propagator\]. For a very explicit derivation of the rules of the heavy baryon formalism we refer to Ref. [@scherer1] — see chapter 5.5.6 and Eq.(5.112) for another justification that, to leading order in the $1/M$ expansion, $v\cdot p$ has to vanish.
With the building blocks at hand we can now evaluate diagram $(a)$ of Fig. \[loops\]. Especially let us focus on those terms that are proportional to $\alpha$. These read $$\begin{aligned}
\nonumber
i\tilde
A_{(a)}^{\rm
NLO}&=&-2\frac{i\alpha}{f_\pi^2}\left(\frac{g_A}{2f_\pi}\right)^3\frac{i^4}
{k^2-m_\pi^2}(\vec\sigma_2\cdot \vec k)
\tau_2^c \\
& & \ \times
\int\frac{d^4l}{(2\pi)^4}\frac{\vec \sigma_1\cdot ({\vec p}'-\vec l-\vec p)
\tau_1^a(\vec \sigma_1\cdot \vec
l)\tau_1^b}{(l^2-m_\pi^2)((p'-l-p)^2-m_\pi^2)(l_0+i\epsilon)}i\tilde
V_{4\pi}^{ab cd} \ ,\end{aligned}$$ where $$i\tilde
V_{4\pi}^{ab cd}
=(\delta^{ab}\delta^{cd}+\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc})
\left[(l^2-m_\pi^2)+((p'\mbox{$-$}l\mbox{$-$}p)^2-m_\pi^2)+(k^2-m_\pi^2)\right]
.
\label{v4pi}$$ Here the indices $j$ = 1, 2 of the Pauli matrices $\tau_j$ and $\sigma_j$ refer to the left and right nucleon lines in Fig. \[loops\], respectively; $l\sim p_{\rm thr}$ denotes the momentum of the pion loop, $p$ and $p'$ are the momenta of the incoming and outgoing leg of the left nucleon line ($j$=1), respectively, whereas $k$ is the difference of the incoming momentum minus the outgoing one of the right nucleon line ($j$=2). Note that the temporal components $p_0$, $p_0'$ and $k_0$ of the nucleon momenta or, respectively, nucleon-momentum difference scale all as $\chi_{\rm prod}^2 M~\sim m_\pi$, whereas their spatial counterparts $p_i$, $p_i'$ and $k_i$ scale as $p_{\rm thr}$. The uncontracted index $d$ refers to the isospin of the produced pion. Its momentum is equal to $k+p-p'$ and is of course on-shell and scales as $m_\pi$. The corresponding term for diagram $(b)$ gives $$\begin{aligned}
\nonumber
i\tilde
A_{(b)}^{\rm NLO}&=&-2i^3\frac{\alpha}
{f_\pi^3}\left(\frac{g_A}{2f_\pi}\right)^3(\vec\sigma_2\cdot \vec k)\,
(\delta^{ab}\tau_2^d+\delta^{ad}\tau_2^b+\delta^{bd}\tau_2^a)
\\
& & \qquad \times
\int\frac{d^4l}{(2\pi)^4}\frac{\vec \sigma_1\cdot ({\vec p}'-\vec l-\vec p)
\tau_1^a(\vec \sigma_1\cdot \vec l)
\tau_1^b}{(l^2-m_\pi^2)((p'-l-p)^2-m_\pi^2)(l_0+i\epsilon)} \ .\end{aligned}$$ Note that the particular combination of momenta as it appears in the $\alpha$-dependent terms of the three-pion vertex is independent of the integration variable $l$ and was therefore pulled out of the integral. As a consequence the integral $\tilde A_{(b)}^{\rm NLO}$ exactly cancels that part of $\tilde A_{(a)}^{\rm NLO}$ that corresponds to the last term of Eq. (\[v4pi\]). What remains to be studied are the other two terms. Each of them cancels one of the pion propagators inside the integral. We get $$\begin{aligned}
\nonumber
&&i\left(\tilde A_{(b)}^{\rm NLO}+\tilde A_{(a)}^{\rm NLO}\right)
=-10\frac{i\alpha}{f_\pi^2}\left(\frac{g_A}{2f_\pi}\right)^3
\frac{i^4}{k^2-m_\pi^2}(\vec \sigma_2\cdot \vec k)\tau_2^d \hfill \\
& &
{\times}
\int\frac{d^4l}{(2\pi)^4}\frac{\vec \sigma_1\cdot
({\vec p}'\mbox{$-$}\vec l\mbox{$-$}\vec p)
(\vec \sigma_1\cdot \vec l)}{(l_0+i\epsilon)}
\left\{\frac1{l^2-m_\pi^2}+\frac1{(p'\mbox{$-$}l\mbox{$-$}p)^2-m_\pi^2}\right\}
\ .
\label{firststep}\end{aligned}$$ Using the variable transformation $l\to l'=p'\!-\!l\!-\!p$ in the second term we find $$\begin{aligned}
\nonumber
i\left(\tilde A_{(b)}^{\rm NLO}+\tilde A_{(a)}^{\rm NLO}\right)
&=&-10\frac{i\alpha}{f_\pi^2}\left(\frac{g_A}{2f_\pi}\right)^3
\frac{i^4}{k^2-m_\pi^2}(\vec \sigma_2\cdot \vec k)\tau_2^d \hfill \\
& &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{\times}
\int\frac{d^4l}{(2\pi)^4}\frac{\vec \sigma_1\cdot (\vec w-\vec l)
(\vec \sigma_1\cdot \vec l)}{l^2-m_\pi^2}
\left\{\frac1{l_0+i\epsilon}+\frac1{w_0-l_0+i\epsilon}\right\}
\ ,
\label{intermediate}\end{aligned}$$ where we defined $w\equiv p'-p$ and renamed $l'$ back to $l$. The integrand now does not contain the large scale $|\vec p|$ anymore in the denominator. Consequently, $l$ will now be of order $m_\pi$ and no longer of order $p_{\rm thr}$. This is why $w_0$, which is also of order $m_\pi$ (while $|\vec w|\sim p_{\rm thr}$), is to be kept in the denominator of the last term. The angular integration leads to $$\vec \sigma_1\cdot ({\vec p}'-\vec l-\vec p)
(\vec \sigma_1\cdot \vec l) \ \to \
- {\vec l}^2 = (l^2-m_\pi^2)-(l_0^2-m_\pi^2) \ .$$ Inserting the first bracket into the above integral leads to a vanishing result, since the spatial integration is free of scales. We may therefore write $$\begin{aligned}
\nonumber
i\left(\tilde A_{(b)}^{\rm NLO}+\tilde A_{(a)}^{\rm NLO}\right)
&=&10\frac{i\alpha}{f_\pi^2}\left(\frac{g_A}{2f_\pi}\right)^3
\frac{i^4}{k^2-m_\pi^2}(\vec \sigma_2\cdot \vec k)\tau_2^d \hfill \\
& &
{\times} w_0
\int\frac{d^4l}{(2\pi)^4}\frac{l_0^2-m_\pi^2}{l^2-m_\pi^2}
\left\{\frac{1}{(l_0+i\epsilon)(w_0-l_0+i\epsilon)}\right\}
\ .
\label{nloadep}\end{aligned}$$ The only external scales in the integral are $m_\pi$ and $w_0\sim m_\pi$. In addition, the integral is quadratically divergent. Therefore, when being evaluated in dimensional regularization, the resulting expression will scale as $|\vec k|^{-1}
\times w_0\times m_\pi^2\sim 1/p_{\rm thr}\times m_\pi^3$.
As was explained in Ref. [@mitnorbert], the leading loops (including pion-field independent terms) can be well estimated by identifying all momentum/energy scales by $p_{\rm thr}$. Thus the $\alpha$-dependent terms of the sum of diagram $(a)$ and $(b)$ are suppressed by a power of $(m_\pi/p_{\rm
thr})^3=\chi_{\rm prod}^3$ compared to the leading loops that start to contribute at order NLO. This implies that the pion-field dependent terms start to contribute only at order N$^4$LO.[^3] At this order the sum of Eq. (\[nloadep\]) cancels against the sum of the $\alpha$-dependent contributions of diagrams $(c)$ and $(d)$ of Fig. \[loops\] which are separately of order N$^4$LO, when the same Feynman rules are applied in the calculation. This, however, does not exclude the possibility that there may exist other $\alpha$-dependent terms of order N$^4$LO that result from subleading Feynman rules. This we will investigate in the next section.
Beyond leading order {#subleading}
====================
We found that to NLO the sum of diagram $(a)$ and $(b)$ (and – trivially – the sum of diagram $(c)$ and $(d)$) of Fig. \[loops\] is invariant under the choice of the pion field. All terms that depend on the pion field vanish to this order. In this section we investigate the pion-field dependent terms of the diagrams shown in Fig. \[loops\] to NNLO. Please note that there are several additional diagrams contributing to this order that are potentially pion-field dependent — one example being the so–called football diagrams shown in Fig. \[loops2\]. The proof that to order lower than N$^4$LO, [i.e.]{} to ${\mathcal O}(\chi_{\rm prod}^n)$ with $n\leq 4$, there are no $\alpha$-dependent terms resulting from the additional diagrams is analogous to the one given here for the diagrams of Fig. \[loops\] and thus we do not present it in detail.
To order NNLO the only thing that needs to be considered is the subleading contribution to the nucleon propagator, which is suppressed by one power in $\chi_{\rm prod}$ compared to (\[nprop1\]).[^4] The subleading $\pi NN$ vertices on the other hand are already down by $m_\pi/M\sim \chi_{\rm prod}^2$. There are two pieces to the subleading propagator: one from treating $p_0$ as subleading in Eq. (\[nprop1\]) $$\nonumber
i\Delta_1
S_N(p-q)\equiv\frac{i}{p_0-q_0+i\epsilon}-\frac{i}{-q_0+i\epsilon}
=-i\frac{p_0}{(q_0-i\epsilon)^2}
\left[1+{\mathcal O}\left({\frac{p_0}{q_0}}\right)\right] \ ,$$ and one coming from the $1/M$ corrections [@ulfbible] given by $$\nonumber
i\Delta_2
S_N(p-q)=\frac{i}{2M}\left(1-\frac{(p-q)^2}{(p_0-q_0+i\epsilon)^2}\right)
=i\frac{(\vec p-\vec q)^2}{2M(q_0-i\epsilon)^2}
\left[1+{\mathcal O}\left({\frac{p_0}{q_0}}\right)\right]
\ .$$ Putting both pieces together we get the following next-to-leading contribution of the nucleon propagator in HBChPT, using the on-shell condition $p_0={\vec p}^2/2M+\mathcal{O}({\vec p}^4/M^3)$ and neglecting higher-order terms: $$\begin{aligned}
i\Delta S_N(p-q)=i\Delta_1 S_N(p-q)+i\Delta_2 S_N(p-q)
=i\frac{{\vec q}^2-2\vec p\cdot \vec q}{2M(q_0-i\epsilon)^2} \ .
\label{subleadingprop}\end{aligned}$$ This illustrates nicely why $p_0$ should be treated as order $\chi_{\rm prod}^2M$: if the nucleon leg attached to the propagator is on-shell, the $p_0$ term gets canceled by the ${\vec p}^2/(2 M)$ term of the $1/M$ corrections, as soon as both contributions are treated on equal footing. Note that each of the two steps of the derivation was based on $p_0/q_0\sim \chi_{\rm prod}$, whereas the total result holds in general. Therefore we present in Appendix \[app:propagator\] a straight forward derivation of this result – based on the covariant propagator – that still is valid even when $q_0\sim |\vec q|\sim p_0 \sim m_\pi$.
Using Eq. (\[subleadingprop\]) for the nucleon propagator we get for the NNLO contribution of the $\alpha$-dependent terms of diagram $(a)$ of Fig. \[loops\] with $\tilde V_{4\pi}^{ab cd}$ as in Eq. (\[v4pi\]): $$\begin{aligned}
\nonumber
i\tilde
A_{(a)}^{\rm NNLO}&=&
-2\frac{i\alpha}{f_\pi^2}\left(\frac{g_A}{2f_\pi}\right)^3\frac{i^4}
{k^2-m_\pi^2}(\vec\sigma_2\cdot \vec k)
\tau_2^c\int\frac{d^4l}{(2\pi)^4}
\left(\frac{2\vec l\cdot \vec p+{\vec l}^2}{2M}\right) \\
& & \qquad \times
\frac{\vec \sigma_1\cdot ({\vec p}'-\vec l-\vec p)
\tau_1^a(\vec \sigma_1\cdot \vec l)\tau_1^b}
{(l^2-m_\pi^2)((p'-l-p)^2-m_\pi^2)(l_0+i\epsilon)^2}i\tilde
V_{4\pi}^{ab cd} \ .
\label{tildeannlo}\end{aligned}$$ As before, the integral that emerges when introducing the last term of Eq. (\[v4pi\]) into Eq. (\[tildeannlo\]) gets canceled by the corresponding term for diagram $(b)$ and we refrain from showing the expression explicitly. After the same variable transformation ($l\to l'=p'-l-p$) as above, the remainder reads $$\begin{aligned}
\nonumber
&& i\left(\tilde A_{(b)}^{\rm NNLO}+\tilde A_{(a)}^{\rm NNLO}\right)
=-10\frac{i\alpha}{f_\pi^2}\left(\frac{g_A}{2f_\pi}\right)^3
\frac{i^4}{k^2-m_\pi^2}(\vec \sigma_2\cdot \vec k)\tau_2^d
\frac1{2M}
\\
& &
\times \int\frac{d^4l}{(2\pi)^4}
\frac{\vec \sigma_1\cdot (\vec w\mbox{$-$}\vec l)
(\vec \sigma_1\cdot \vec l)}
{l^2-m_\pi^2}\left\{\frac{2\vec l\cdot \vec p+{\vec l}^2}
{(l_0+i\epsilon)^2}
+\frac{2(\vec w\mbox{$-$}\vec l)\cdot \vec p+
(\vec w\mbox{$-$}\vec l)^2}{(w_0-l_0+i\epsilon)^2}\right\}
,
\label{tildeapbnnlo}\end{aligned}$$ again using $w=p'- p$. As before, this integral diverges (at least) quadratically with the only scale in the denominator given by $m_\pi\sim w_0$. In addition there is now an overall scale of order ${\vec p}^2/M$ present, which is also of order $m_\pi$. Therefore the integral given in Eq. (\[tildeapbnnlo\]) also starts to contribute only at order N$^4$LO. Again this sum cancels against the summed $\alpha$-dependent contributions of diagram $(c)$ and $(d)$ of Fig. \[loops\], if the subleading nucleon propagator (\[subleadingprop\]) is inserted into the latter diagrams.
Finally note that the next-to-subleading correction to the nucleon propagator and vertices would necessarily involve an additional factor $\chi_{\rm prod}$ relative to the above presented N$^4$LO result. The summed $\alpha$-dependent contributions of diagram $(a)$ and $(b)$ (and of $(c)$ and $(d)$) of Fig. \[loops\] resulting from this next-to-subleading order should therefore contribute only at order N$^5$LO. In other words, the proof that the $\alpha$-dependent terms in the sum of all diagrams of Fig. \[loops\] cancels to order N$^4$LO is now complete.
Conclusions {#conclusion}
===========
Of course, the fact that there is a cancellation of the summed $\alpha$-dependent terms of the four diagrams of Fig. \[loops\] does not come as a surprise, see Ref. [@novel], since these terms would cancel also in a relativistic calculation of the type [@GSS] where the nucleon propagator (\[nprop1\]) is replaced by the non-expanded covariant form (\[covprop\]) and where the terms $-\vec \sigma \cdot {\vec q}_i$ appearing in the vertices (\[VNNpi\]) and (\[VNN3pi\]) are replaced by their covariant Dirac-analogs $\gamma_\mu\gamma_5 (q_i)^\mu$. In fact, this cancellation between the $\alpha$-dependent contributions of diagram $(a)$ on the one hand and the ones of diagrams $(b)$, $(c)$ and $(d)$ on the other hand is solely based on the cancellations of the inverse pion-propagators appearing in Eq. (\[v4pi\]) and the various pion propagators appearing in diagram $(a)$ which all are of covariant nature – even in HBChPT. The point, however, is that now it is clear that this cancellation is also consistent with the two-scale expansion scheme of Refs.[@pwaves; @mitnorbert]: (i) we have explicitly shown that the pion-field dependent contributions of the diagrams $(a)$ and $(b)$ of Fig.\[loops\], although both are NLO diagrams, cancel at NLO and at N$^2$LO when calculated with leading and next-to-leading input, respectively, for the nucleon-propagators and vertices. (ii) We have shown that the remainders are only of N$^4$LO, the very same order at which the diagrams $(c)$ and $(d)$ start to contribute. (iii) At this order, the pion-field dependent contributions of the sum of diagram $(a)$ and $(b)$ indeed cancel against the corresponding contributions of diagram $(c)$ and $(d)$. (iv) We have argued that further subleading orders of the nucleon propagator and vertices will lead to pion-field dependent terms that are at least of N$^5$LO.
These results can be generalized in the following way: as long as the order in the expansion of the nucleon propagator in diagram $(a)$ matches those of diagrams $(b)$, $(c)$ and $(d)$ and as long as the order in the expansion of the $NN\pi$ vertex matches those of the $NN3\pi$ vertex, the following cancellations are bound to happen: first, the cancellation between the $\alpha$-dependent contribution of diagram $(b)$ and the one of diagram $(a)$ that results from the insertion of the last term of Eq. (\[v4pi\]); at this stage the remainder of the $\alpha$-dependent contribution of diagram $(a)$ has now the same order in the two-scale expansion as the $\alpha$-dependent contributions of diagram $(c)$ and $(d)$ calculated with the same input; secondly, since the cancellation is based on covariant input from the (inverse) pion-propagators and since the rest of the input is the same, the sum of these remaining $\alpha$-dependent contributions has to vanish. Of course, at the same order in the chiral expansion, say at N$^n$LO with a fixed $n>4$, the diagrams of Fig. \[loops\] might generate additional $\alpha$-dependent terms resulting from further subleading orders in the expansion of the nucleon propagators and vertices, as it was [e.g.]{} the case at the leading and subleading order in the expansion of the nucleon propagator. Nevertheless, for the same reasons as above, also these additional contributions have to sum to zero. Eventually at an even higher order in the expansion of the nucleon propagator and vertices no more $\alpha$-dependent terms of N$^n$LO can appear in the summation; instead contributions of the next order $n+1$ will arise which again sum to zero and so on. As indicated, our proof linked to the diagrams of Fig.\[loops\] can easily be generalized to other classes of potentially pion-field dependent diagrams as [e.g.]{} given in Fig.\[loops2\]. In summary, the two-scale expansion scheme of Refs.[@pwaves; @mitnorbert] is consistent with pion-field independence to all orders in the expansion.
As by-products of the investigation we could show that the [*parameterizations*]{} of the pion field indeed correspond to [*gauge choices*]{}, and could clarify the structure of the heavy-baryon propagator connected to an on-shell nucleon leg. Contrary to a naive interpretation of the heavy-baryon rules the on-shell residual energy of the external nucleon is of the same order as the kinetic recoil term of the nucleon – in fact, to the very same order, they cancel each other.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Fred Myhrer for fruitful discussions at an early stage of this research.
Reparameterizations of the chiral matrix $U$ {#app:chiral_matrix}
============================================
The theorem that [*on-shell*]{} matrix elements do not dependent on the specific parameterization of the local interpolating field(s) has a long history reaching back to the LSZ reduction formula [@LSZ55] and the work of Haag [@haag58], see also Refs. [@borchers60; @chisholm61; @kamefuchi61; @ruelle62] etc. In the context of non-linear realizations of chiral Lagrangians this general theorem of axiomatic field theory was confirmed in Ref.[@CWZ69]. The more restricted question of the general reparameterizations of the chiral matrix $U$ for the chiral group $SU(2)\times SU(2)$, more specifically, the general reparameterization of the pion field under nonlinear transformations induced by chiral $SU(2)\times SU(2)$ was first studied by Weinberg [@wein68]. From the parity of the pion and the transformation properties of the pion field under vector and axial–vector transformations combined with Jacobi-identity constraints Weinberg could show that the most general redefinition of the nonlinearly realized pion field is of the form $${\pi'}^a = \pi^a g(\pi^2),
\quad
a\in\{1,2,3\}$$ where the $\pi^a$’s are the usual isospin components of the pion field $\pi\equiv\vec\tau\cdot\vec\pi= \sum_{a=1}^3\tau^a \pi^a$ and where $g(\pi^2)$ is regular in $\pi^2={\vec \pi}^2=\sum_{a=1}^3
(\pi^a)^2$. In terms of the chiral matrix and the dimensionful version of the pion field this corresponds to $$U' \equiv \exp\left (\frac{i}{f_\pi} \vec\tau \cdot {\vec{\pi'}}
\right)=
\exp\left (\frac{i}{f_\pi} \vec\tau \cdot \vec\pi\,
g(\pi^2/f_\pi^2)\right) \ .
\label{Umat1}$$ This is the result of Eq.(\[Umat\]) under the additional condition that $g(0)=1$, which follows from fixing the wave function normalization of the pion at tree-level or, in other words, the free-particle part of the Lagrangian (\[LGSS\]), see Refs. [@chisholm61; @kamefuchi61].
The various known parameterizations, the exponential one, the so-called $\sigma$-gauge, the Weinberg one [@wein68], etc. follow from (\[Umat1\]) with the help of the following choices of $g(\pi^2/f_\pi^2)$ functions: $$\begin{aligned}
g(\pi^2/f_\pi^2)&=& 1 \qquad\qquad\qquad\qquad\quad
(\mbox{exponential parameterization}),
\label{exppara}\\
g(\pi^2/f_\pi^2) &=&
\frac{1}{\sqrt{\pi^2/f_\pi^2}}
\arcsin \left(\sqrt{\pi^2/f_\pi^2}\,\right)
= 1 +\frac{\pi^2}{6f_\pi^2}
+ \cdots\ (\sigma\mbox{-gauge}),\label{G-sigma}
\\
g(\pi^2/f_\pi^2) &=& \frac{1}{\sqrt{\pi^2/f_\pi^2}}
\arcsin\left(\frac{\sqrt{\pi^2/f_\pi^2}}
{1+\pi^2/(4 f_\pi^2)}\right)
= 1 -\frac{\pi^2}{12 f_\pi^2}
+ \cdots\nonumber\\
&&
\qquad\qquad\qquad\qquad\qquad\ \ (\mbox{Weinberg parameterization}).
\label{weinpara}\end{aligned}$$ In fact, the transformation (\[Umat1\]) can be simplified by the following rescaling $$U' = \exp\left (\frac{i}{f_\pi} \vec\tau \cdot \vec\pi\,
g(\pi^2/f_\pi^2)\right)
=\exp\biggl( {i} \vec\tau \cdot \hat{\vec\pi}\,
F\Bigl(\sqrt{\pi^2/f_\pi^2}\,\Bigr)
\biggr)
\label{Umat2}$$ where $\hat{\vec \pi}={\vec \pi}/\sqrt{\pi^2}$ is the pion unit vector in isospin-space and $F(x)=x g(x^2)$ is an odd analytic function of the variable $x$ with a normalized first coefficient in the Taylor expansion, $F(x) = x + \sum_{n=2}^\infty c_{2n-1}x^{2n-1}$, see [e.g.]{} [@delorme96; @chanfray96]. In terms of this function the various parameterizations become especially simple [@delorme96; @chanfray96]:[^5] $$\begin{aligned}
F(x) &=& x \qquad\qquad\qquad\qquad\qquad\quad
(\mbox{exponential parameterization}),\\
F(x) &=& \arcsin(x)\qquad\qquad\qquad\quad\;\
(\sigma\mbox{-gauge parameterization}),
\label{F-sigma}\\
F(x) &=& \arcsin\left(x/(1+x^2/4)\right)\qquad
(\mbox{Weinberg parameterization}).\end{aligned}$$ In fact, with the help of the machinery of Ref.[@CWZ69] the various parameterizations can be transformed into each other by the following axial gauge transformations $$U \to U' = U_A(\pi) \,U \,U_A(\pi) \label{Ugauge}$$ in terms of the local $SU(2)$ matrix $
U_A(\pi) =\exp\left((i/{2f_\pi}) \vec\tau \cdot \vec\pi\, \left(
g(\pi^2/f_\pi^2)-1\right) \right)\ .
$ The backtransformation follows then from the inverse gauge transformation $U' \to U = U_A(\pi)^\dagger \,U' \,U_A(\pi)^\dagger$. Transitions between other representations or [*gauges*]{} can be found as compositions of gauge transformations from and to the exponential gauge, say. The $\sigma$-[*gauge*]{} indeed results from a gauge transformation (\[Ugauge\]) of the exponential “gauge” $U=\exp\left(i \vec\tau\cdot\vec \pi/f_\pi\right)$ when the [*gauge choice*]{} (\[G-sigma\]) is inserted into $U_A(\pi)$. In addition, the various parameterizations of the matrix $u=\sqrt{U}$ transform into each other as $$u \to u' = U_A(\pi) \,u \,h(U_A,\pi)^{-1}
= h(U_A,\pi) \,u \, U_A(\pi), \label{ugauge}$$ where $h(U_A,\pi)\in SU(2)_{V}$ is the so-called “compensator” or “hidden” matrix [@CWZ69] which cancels in $U'$ = $u'u'$ = $U_A(\pi) u u U_A(\pi)$ and in the Lagrangian (\[LGSS\]).
Whereas the transformations of the type (\[Umat1\]) or (\[Umat2\]) are $SU(2)$ specific, the gauge transformation as such – whether in the form (\[Ugauge\]) or (\[ugauge\]) – can be generalized to $SU(3)$ with a suitably selected $SU(3)$ gauge matrix $U_A(\pi)\sim 1 + i \frac{\alpha}{2}\pi^3/f_\pi^3 +\cdots$ that does not spoil the wave function normalization at tree level, where $\pi = \sum_{a=1}^8\lambda^a\pi^a$ in terms of the Gell-Mann matrices $\lambda^a$.
On the $1/M$ expansion of the nucleon propagator {#app:propagator}
================================================
In the main section the rules of HBChPT were applied directly. For illustration, we show in this appendix that the same expressions can be recovered by a straight forward expansion of the nucleon propagator. We start from the covariant expression for the nucleon propagator $$iS_N^{\rm cov}(P-q)=i\frac{P\!\!\!/-q\!\!\!/+M}{(P-q)^2-M^2 +i\epsilon} \ ,
\label{covprop}$$ where the momenta are defined in Fig. \[momdef\]. We now want to expand this propagator in powers of $1/M$.
The easiest way to proceed is via the decomposition $$iS_N^{\rm cov}(K)=i\frac{M}{E_K}\sum_s \left\{
\frac{u(\vec K,s)\bar u(\vec K,s)}{K_0-
E_K+i\epsilon}
+\frac{v(-\vec K,s)\bar
v(-\vec K,s)}{K_0+E_K-i\epsilon}\right\} \ ,$$ where $E_K=\sqrt{M^2+{\vec K}^2}$. First observe that the second term, corresponding to the contribution of anti–nucleons, does not propagate in HBChPT. It therefore gets absorbed into local counter terms at the Lagrangian level. The spinors get part of the vertex functions and we may therefore focus on the denominator of the first term. In the kinematics chosen we have $
K = P-q
$. To make contact to the expressions of HBChPT, we write, in accordance with the notation of the main text, $ P^\mu = Mv^\mu + p^\mu$ with $v^\mu =(1,0,0,0)$. Thus $$\begin{aligned}
P_0-q_0-E_{( P-q)} &=& M+p_0-q_0-\sqrt{M^2+(\vec p-\vec q)^2}
\nonumber
\\
&=&-q_0+\frac{2\vec p\cdot \vec q-{\vec q}^2}{2M}
+{\mathcal O}\left(\frac{(\vec p-\vec q)^4}{M^3}, \frac{{\vec p}^4}{M^3}
\right) \ ,
\label{P0q0E}\end{aligned}$$ where the on-shell condition $p_0={\vec p}^2/2M+{\mathcal O}({\vec p}^4/M^3)$ was used in the last step. Observe that $p_0$ has disappeared from the expression (\[P0q0E\]). However, this happens only, if $p_0$ is put into the same order as ${\vec p}^2/2M$, as advocated in the main section. In the power counting relevant for pion production, the pion energies in loops are to be counted as order $p_{\rm thr}$ as in case (a) and (b) of Fig. \[loops\]. Therefore, in line with Eqs. (\[nprop1\]) and (\[subleadingprop\]), the expression for the propagator in HBChPT is simply $$\begin{aligned}
\nonumber iS_N(p-q)&=&\frac{i}{-q_0+i\epsilon}
\left(1-\frac{2\vec p\cdot \vec q-{\vec q}^2}{2M(-q_0+i\epsilon)}
+{\mathcal O}\left(\frac{p^2,p\cdot q,q^2}{M^2}\right)\right)
\\
&=&
\frac{i}{-v\cdot q \!+\!i\epsilon}\left(1+\frac{2 p\cdot q_\perp-q_\perp^2}
{2M(-v\cdot q\!+\!i\epsilon)}
+{\mathcal O}\left(\frac{p^2,p\cdot q,q^2}{M^2}\right)\right)
\ .
\label{sexpand}\end{aligned}$$ The last relation refers to the general “velocity” case ($v^2=1$ and $v_0\geq 1$) with the definition $q_\perp\equiv q-v (v\cdot q)$ and the on-shell condition $v\cdot p =-p^2/2M$ [@scherer1].
Note that the above equations hold even for more general kinematics. In all cases of relevance here, the loop momenta are at least of the order of the pion mass. Thus the components of $q^\mu$ either scale as $p_{\rm thr}$, as used in the previous paragraph, or as $m_\pi$ — as in the integral of Eqs. (\[intermediate\]) and (\[nloadep\]) or in Fig. \[loops\] (c) and (d). Let us stress that also in the latter case the expansion of Eq. (\[sexpand\]) holds, since the other terms of order $m_\pi$, namely $p_0$ and $\vec p^2/2M$, canceled and the remaining recoil terms are suppressed by at least one power of $\chi_{\rm prod}$.
It is also instructive to derive the $1/M$ expansion of the propagator directly from the covariant expression of Eq. (\[covprop\]). Using again the on-shell condition for the incoming nucleon, $P^2=M^2$, we may write $P_0=M\left(1+{\mathcal O}({\vec p}^2/M^2)\right)$ – note that $\vec P\equiv \vec p$. We are interested in the case $|\vec p|\sim p_{\rm thr}$, where $p_{\rm thr}$ was defined below Eq. (\[chiprod\]). Therefore ${\mathcal O}({\vec p}^2/M^2)$ corresponds to ${\mathcal O}(\chi_{\rm prod}^2)$. We thus identify $-i/q_0$ as the leading term for the propagator in accordance with Eq. (\[nprop1\]). All other terms that still appear in the denominator are corrections. After a Taylor expansion to next–to–leading order we get $$\begin{aligned}
&&iS_N^{\rm cov}(P-q)=\frac{i}{-q_0+i\epsilon}\biggl\{ {\textstyle\frac12}
\left(\mathbf 1+\gamma_0\right)
\Bigl(1-
{\frac{2\vec p\cdot\vec q-{\vec q}^2}
{2M(-q_0+i\epsilon)}}
\Bigr)\\
& &\quad -\underbrace{\frac1{2M}\vec \gamma\cdot \left(\vec p-\vec
q\right)}_{\mbox{into vertices}}
\ \ +\underbrace{\frac{q_0}{2M}\,
{\textstyle\frac12}
\left(\mathbf 1-\gamma_0\right)}_{\mbox{effect of anti--nucleon}}
\biggr\}
\left(1
+{\mathcal O}\left(\frac{p^2,p\cdot q,q^2}{M^2}\right)
\right)
\ .\end{aligned}$$ As indicated, this expression contains the leading and next–to–leading piece of the propagator and, in addition, a piece that can give momentum dependence to the vertices (this contribution can be mapped onto the effect of the spinors in the previous derivation), and, finally, a contact term that is the leading term for the effects of the anti–nucleon in the intermediate state.
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[^1]: In this case the large momentum $p_{\rm thr}$ can be removed from the integral by, e.g., a proper choice of the integration variable in full analogy to the discussion given below Eq. (\[firststep\]).
[^2]: In the sigma gauge all vertices can be found in appendix A of Ref. [@ulfbible].
[^3]: The power counting can only account for a parametric suppression of diagrams relative to each other. Obviously the expression of Eq. (\[nloadep\]) can be enhanced artificially by choosing a gauge that corresponds to a very large value $\alpha$. Note that for all standard choices $|\alpha|\le 1/4$ (see Eqs. (\[exppara\])–(\[weinpara\])). For practical purposes, the $\sigma$-gauge is of course the most efficient one, since each $\alpha$-dependent term trivially vanishes individually.
[^4]: Of course the subleading contribution to the propagator can be interpreted as an ${\mathcal O}(q^2/M)$ insertion between two leading-order propagators $(-q_0+i\epsilon)^{-1}$.
[^5]: In $SU(2)$: $\exp\Bigl(i \vec\tau\cdot \hat{\vec\pi}
F\bigl(\!\sqrt{\pi^2/f_\pi^2}\,\bigr)\Bigr)\! =\!
\cos\Bigl(\!F\bigl(\!\sqrt{\pi^2/f_\pi^2}\,\bigr)\Bigr)
+ i \vec\tau \cdot\hat{\vec\pi}
\sin\Bigl(\!F\bigl(\!\sqrt{\pi^2/f_\pi^2}\,\bigr)\Bigr)
$. \[footsu2\]
|
---
abstract: 'We study the restricted families of projections in vector spaces over finite fields. We show that there are families of random subspaces which admit a Marstrand-Mattila type projection theorem.'
address: 'School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia '
author:
- Changhao Chen
title: Restricted families of projections in vector space over finite fields
---
Introduction
============
A fundamental problem in fractal geometry is to determine how the projections affect dimension. Recall the classical Marstrand-Mattila projection theorem: Let $E\subset {\mathbb{R}}^{n}, n\geq2,$ be a Borel set with Hausdorff dimension $s$.
- (dimension part) If $s\leq m$, then the orthogonal projection of $E$ onto almost all $m$-dimensional subspaces has Hausdorff dimension $s$.
- (measure part) If $s>m$, then the orthogonal of $E$ onto almost all $m$-dimensional subspaces has positive $m$-dimensional Lebesgue measure.
In 1954 J. Marstand [@Marstrand] proved this projection theorem in the plane. In 1975 P. Mattila [@Mattila1975] proved this for general dimension via 1968 R. Kaufman’s [@Kaufman] potential theoretic methods. We refer to the recent survey of K. Falconer, J. Fraser, and X. Jin [@Falconer] for more backgrounds.
Recently there has been a growing interest in studying finite field version of some classical problems arising from Euclidean spaces. For instance, there are finite field Kakeya sets (also called Besicovitch sets), see Z. Dvir [@Dvir]; there are finite field Erdős/ Falconer distance problem, see A. Iosevich, M. Rudnev [@IosevichRudnev], T. Tao [@Tao1]; etc. Motivated by the above works, the author [@ChenP] studied the projections in vector spaces over finite fields, and obtained the Marstrand-Mattila type projection theorem in this setting. In this paper, we turn to the restricted families of projections in the vector spaces over finite fields. For more details on projection in vector space over finite fields see [@ChenP]. For more backgrounds on restricted families of projections in Euclidean spaces, we refer to [@Falconer Section 6], [@FOO], [@KOV] and reference therein.
Let $p$ be a prime number, $\mathbb{F}_{p}$ be the finite field with $p$ elements, and ${\mathbb{F}_{p}^{n}}$ be the $n$-dimensional vector space over this field. We use the same notation as in the Euclidean spaces. Let $G(n,m)$ be the collection of all $m$-dimensional linear subspaces of ${\mathbb{F}_{p}^{n}}$, and ${A(n,m)}$ be the family of all $m$-dimensional planes, i.e., the translation of some $m$-dimensional subspace. In the following we show the definition of projections in ${\mathbb{F}_{p}^{n}}$, see [@ChenP] for more details.
Let $E$ be a subset of ${\mathbb{F}_{p}^{n}}$ and $W$ be a non-trivial subspace of ${\mathbb{F}_{p}^{n}}$. Denoted by $\pi^{W}(E)$ the collection of coset of $W$ which intersects $E$, that is $$\pi^{W}(E)=\{x+W: E\cap (x+W) \neq \emptyset, x\in {\mathbb{F}_{p}^{n}}\}.$$ In this paper we are interested in the cardinality of $\pi^{W}(E)$.
For any set $E \subset {\mathbb{F}_{p}^{n}}$ and $W\in G(n,n-m)$ the Lagrange’s group theorem implies $$|\pi^{W}(E)|\leq \min\{|E|, p^{m}\}.$$ Here and in the following let $|J|$ denote the cardinality of set $J$. The author [@ChenP Corollary 1.3] obtained the following Marstrand-Mattila type projection theorem in ${\mathbb{F}_{p}^{n}}$. In fact the following form is often called the size of the exceptional sets of projections.
\[thm:ChenP\] Let $E \subset {\mathbb{F}_{p}^{n}}$ with $|E|=p^{s}$.
\(a) If $s\leq m$ and $t \in (0, s]$, then $$| \{W \in G(n,n-m) : |\pi^{W} (E)| \leq p^{t}/10 \} \leq \frac{1}{2} p^{m(n - m) -(m - t)}.$$
\(b) If $s> m$, then $$| \{W \in G(n,n-m) : |\pi^{W} (E)| \leq p^{m}/ 10 \}| \leq \frac{1}{2} p^{m(n - m) -(s-m)}.$$
We note that $|G(n,m)|\approx p^{m(n-m)}$, see P. Cameron [@Cameron Theorem 6.3]. We write $f\lesssim g$ if there is a positive constant $C$ such that $f\leq Cg$, $f\gtrsim g$ if $g\lesssim f$, and $f\approx g$ if $f\lesssim g$ and $f\gtrsim g$.
In the following, we formulate finite fields version of restricted families of projections. Let $G$ be a subset of $G(n,k)$, then $(\pi^{W})_{W\in G }$ is called a restricted family of projection. The purpose of this paper is looking for subsets $G\subset G(n,k)$ such that $(\pi^{W})_{W\in G }$ admit a Marstrand-Mattila type projection theorem.
By studying the random subsets of $G(n, n-m)$, we obtain the following result.
\[thm:main\] For any $ \min\{m, n-m\} <\alpha \leq m(n-m)$, there exists a subset $G\subset G(n,n-m)$ with $|G|\approx p^{\alpha}$ such that for any $E\subset {\mathbb{F}_{p}^{n}}$, $$\label{eq:eee}
|\{W\in G: |\pi^{W}(E)|\leq N\}|\lesssim |G|N(|E|^{-1}+p^{-m}).$$
Note that for the case $\alpha=m(n-m)$, Theorem \[thm:main\] follows from Theorem \[thm:ChenP\] by choosing $G=G(n,m)$. Thus we consider the case $\min\{m, n-m\} <\alpha < m(n-m)$ only. We immediately have the following Marstrand-Mattila type projection theorem via the special choice $N$ in Theorem \[thm:main\].
For any $ \min\{m, n-m\} <\alpha \leq m(n-m)$, there exists a subset $G\subset G(n,n-m)$ with $|G|\approx p^{\alpha}$ such the following holds. Let $E\subset {\mathbb{F}_{p}^{n}}$ with $|E|=p^{s}$.
\(1) If $|E|\leq p^{m}$ and $t\in (0,s]$, then $$|\{W\in G: |\pi^{W}(E)|\leq p^{t}\}|\lesssim |G|p^{t-s}.$$
\(2) If $|E|> p^{m}$, then for any small $\varepsilon$ $$|\{W\in G: |\pi^{W}(E)|\leq \varepsilon p^{m}\}|\lesssim |G|\varepsilon.$$
For restricted families of projections in Euclidean spaces, the author [@ChenR] obtained that some random subsets of sphere of ${\mathbb{R}}^{3}$ admit a Marstrand-Mattila type projection theorem. For more details, see [@ChenR].
The structure of the paper is as follows. In Section \[sec:p\], we set up some notation and show some lemmas for later use. We prove Theorem \[thm:main\] in Section \[sec:proof\]. In the last Section we given some examples of restricted families of projections which admit a Marstrand-Mattila type theorem in finite fields setting.
Preliminaries {#sec:p}
=============
In this section we show some lemmas for later use.
Outline of the methods
----------------------
In short words, we take a random subset $G\subset G(n,n-m)$, see the random model in Subsection \[sub:rrr\]. Then we estimate the cardinality of “the exceptional sets”, $$\{W\in G: |\pi^{W} (E)|\leq N\},$$ and show that it satisfies our need. To estimate the “exceptional sets”, we adapt the arguments in [@ChenP] to our setting which is a variant of Orponen’s pairs argument [@OrponenA Estimate (2.1)].
Let $W\in G(n,n-m)$ then Lagrange’s group theorem implies that there are $p^{m}$ cosets of $W$. Let $x_{W, j}+W, 1\leq j\leq p^{m}$ be the different cosets of $W$. Let $E\subset {\mathbb{F}}_{p}^{n}$, then $$|E| =\sum_{j=1}^{p^{m}} |E\cap (x_{W, j}+W)|,$$ and the Cauchy-Schwarz inequality implies $$\label{eq:pairss}
|E|^{2}\leq |\pi^{W}(E)|\sum_{j=1}^{p^{m}} |E\cap (x_{W, j}+W)|^{2}.$$ Note that $|E\cap (x_{W, j}+W)|^{2}$ is the amount of pairs of $E$ inside the coset $x_{W, j}+W$. Let $N\leq p^{m}$, define $$\Theta=\{W\in G: |\pi^{W} (E)|\leq N\}.$$ Summing two sides over $W\in \Theta$ in estimate , we obtain $$\label{eq:argument}
|\Theta| |E|^{2} \leq {\mathcal{E}}(E,\Theta')N$$ where ${\mathcal{E}}(E,\Theta')=\sum_{W\in \Theta}\sum_{j=1}^{p^{m}}|E\cap (x_{W, j}+W)|^{2}.$ Thus the left problem is to estimate ${\mathcal{E}}(E, \Theta')$, and we use the doubling counting argument of Murphy and Petridis [@MurphyPetridis Lemma 1] and the discrete Plancherel identity. The above discusses motivated the following definition.
Let $E\subset {\mathbb{F}_{p}^{n}}$ and $ \mathcal{A} \subset A(n,m)$. Define the energy of $E$ on $\mathcal{A}$ as $${\mathcal{E}}(E, \mathcal{A}) =\sum_{W\in \mathcal{A}} |E\cap W|^{2}.$$
We note that ${\mathcal{E}}(E, \mathcal{A})$ is closely related to the incidence identity of Murphy and Petridis [@MurphyPetridis Lemma 1], and the additive energy in additive combinatorics [@TaoVu Chapter 2].
Discrete Fourier transform
--------------------------
In the following we collect some basic facts about Fourier transformation which related to our setting. For more details on discrete Fourier analysis, see Green [@Green], Stein and Shakarchi [@Stein]. Let $f : {\mathbb{F}_{p}^{n}}\longrightarrow \mathbb{C}$ be a complex value function. Then for $\xi \in {\mathbb{F}_{p}^{n}}$ we define the Fourier transform $$\label{eq:dede}
\widehat{f}(\xi)=\sum_{x\in {\mathbb{F}_{p}^{n}}} f(x)e(-x\cdot \xi),$$ where the dot product $ x\cdot\xi $ is defined as $ x_1\xi_1+\cdots +x_n\xi_n$ and $e(-x \cdot \xi)={e^{-\frac{2\pi i x\cdot\xi}{p}}}$. Recall the following Plancherel identity, $$\sum_{\xi \in {\mathbb{F}_{p}^{n}}}|\widehat{f}(\xi)|^{2}=p^{n}\sum_{x\in {\mathbb{F}_{p}^{n}}} |f(x)|^{2}.$$ Specially for the subset $E\subset {\mathbb{F}_{p}^{n}}$, we have $$\sum_{\xi \in {\mathbb{F}_{p}^{n}}} |\widehat{E}(\xi)|^{2}=p^{n}| E|.$$ Here and in the following we use $E$ as characteristic function of the set $E$.
For $W\in G(n,n-m)$, define $$Per(W):=\{x\in {\mathbb{F}_{p}^{n}}: x\cdot w=0, w\in W\}.$$ Note that if $W$ is some subspace in Euclidean space then $Per(W)$ is the orthogonal complement of $W$. Furthermore, unlike in the Euclidean spaces, here $W\cap Per(W)$ can be some non-trivial subspace. However the rank-nullity theorem of linear algebra implies that for any subspace $W \subset{\mathbb{F}_{p}^{n}}$, $$\label{eq:rank}
\dim W+\dim Per(W)=n.$$
The following Lemma \[lem:fff\] of [@ChenP Lemma 2.3] plays an important role in the proof of Lemma \[lem:abstract\] (2). For more details see [@ChenP Lemma 2.3].
\[lem:fff\] Use the above notation. We have $$\label{eq:kk}
\sum_{j=1}^{p^{m}} | E \cap (x_{j}+W)|^{2}=p^{-m}\sum_{\xi\in Per(W)} |\widehat{E}(\xi)|^{2}.$$
We note that the Lemma \[lem:fff\] is the only place in this paper where the prime field $\mathbb{F}_{p}$ is needed. We do not know if the Lemma \[lem:fff\] also holds for vector spaces over general finite fields.
In the following we extend a result of [@ChenP Lemma 3.1] to general subset of ${G(n,n-m)}$. Let $G\subset G(n,n-m)$, define $$\label{eq:define}
G'=\bigcup_{W\in G}\bigcup_{j=1}^{p^{m}}(x_{j,W}+W)$$ where $x_{W, j}+W, 1\leq j\leq p^{m}$ are the cosets of $W$. For each $W$ we simply use $x_{W, j}+W, 1\leq j\leq p^{m}$ to represent the cosets of $W$.
\[lem:abstract\] Let $G$ be a subset of $G(n, n-m)$ with $|G|\gtrsim p^{\beta}$.
\(1) If for any $\xi\neq 0$, $$\label{eq:l11}
|\{W\in G: \xi \in V\}|\lesssim |G| p^{-\beta},$$ then $${\mathcal{E}}(E,G')\lesssim |E||G|+|E|^{2}|G|p^{-\beta}.$$
\(2) If for any $\xi\neq 0$, $$\label{eq:l22}
|\{W\in G: \xi \in Per(W)\}|\lesssim |G| p^{-\beta},$$ then $${\mathcal{E}}(E,G')\lesssim p^{-m}|G|(|E|^{2}+|E|p^{n-\beta}).$$
The claim $(1)$ follows by doubling counting. Recall that we denote by $F(x)$ the characteristic function of the subset $F\subset {\mathbb{F}_{p}^{n}}$. Then $$\begin{aligned}
{\mathcal{E}}(E, G')&= \sum_{V\in G' }|E \cap V|^{2}\\
&=\sum_{V \in G'} \left(\sum_{x\in E}V(x) \right)^{2}\\
&=\sum_{V \in G'} \left(\sum_{x\in E}V(x)+\sum_{x\neq y \in E} V(x)V(y) \right)\\
&\lesssim |E||G|+|E|(|E|-1)|G|p^{-\beta}.
\end{aligned}$$
To establish $(2)$, the Lemma \[lem:fff\] implies $$\begin{aligned}
{\mathcal{E}}(E, G')
&=\sum_{W\in G} \sum_{j=1}^{p^{m}}|E \cap (x_{W, j}+W)|^{2}\\
& =p^{-m}\sum_{W\in G} \sum_{\xi\in Per(W)}|\widehat{E}(\xi)|^{2}\\
& =p^{-m}(|G||E|^{2}+\sum_{W\in G} \sum_{\xi\in Per(W)\backslash \{0\}}|\widehat{E}(\xi)|^{2})\\
&\lesssim p^{-m}(|G||E|^{2}+p^{n}|E||G|p^{-\beta}).
\end{aligned}$$ Thus we finish the proof.
Proof of Theorem \[thm:main\] {#sec:proof}
=============================
Random subsets of ${G(n,n-m)}$ {#sub:rrr}
------------------------------
We start by a description of these random subsets in ${G(n,n-m)}$. Let $0<\delta<1$. We choose each element of ${G(n,n-m)}$ with probability $\delta$ and remove it with probability $1-\delta$, all choices being independent of each other. Let $G=G^{\omega}$ be the collection of these chosen subspaces. Let $\Omega ({G(n,n-m)}, \delta)$ be our probability space which consists of all the possible outcomes of random subspaces.
For the convenience to our use, we formulate the following large deviations estimate. For more background and details on large deviations estimate, see Alon and Spencer [@Alon Appendix A].
\[lem:law of large numbers\] Let $\{X_j\}_{j=1}^N$ be a sequence independent Bernoulli random variables which takes value $1$ with probability $\delta$ and value $0$ with probability $1-\delta$. Then $${\mathbb{P}}( \sum^N_{j=1} X_j \geq 3N\delta )\leq e^{-N\delta}.$$
We also need the following Lemma of [@ChenP Lemma 2.7].
\[lem:c\] Let $\xi$ be a non-zero vector of ${\mathbb{F}_{p}^{n}}$.
\(1) $|\{W\in G(n, k): \xi\in W\}|=|G(n-1, k-1)|$.
\(2) $|\{W\in G(n, k): \xi\in Per(W)\}|=|G(n-1, k)|$.
\[co:cc\] For any $m<\alpha<m(n-m)$, there exists a subset $G\subset {G(n,n-m)}$ such that $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in W\}|\lesssim |G|p^{-m}.$$
We consider the random model $\Omega({G(n,n-m)}, \delta)$ where $\delta=|G(n,m)|^{-1}p^{\alpha}$. First observe that $p^{\alpha}/2 \leq |G| \leq 2 p^{\alpha}$ with high probability ($>1/2$) provided large $p$. This follows by applying Chebyshev’s inequality, which says that $$\label{eq:che}
\begin{aligned}
{\mathbb{P}}(||G| - p^{\alpha}|&> \frac{1}{2}p^{\alpha})\leq \frac{4p^{\alpha}(1-\delta)}{p^{2\alpha}}\\
&\leq \frac{4}{p^{\alpha}}\rightarrow 0 \text{ as } p \rightarrow \infty.
\end{aligned}$$
Let $\xi\neq 0$ and $G_{\xi}:=\{W\in {G(n,n-m)}: \xi \in W\}$. Lemma \[lem:c\] (1) implies that $$|G_{\xi}|=|G(n-1,n-m-1)|\approx p^{m(n-m)-m}.$$ Observe that for $G\in \Omega({G(n,n-m)},\delta)$, $$|\{W\in G: \xi \in W\}|=\sum_{W\in G_{\xi}}{\bf 1}_{G}(W).$$ Thus by Lemma \[lem:law of large numbers\], $${\mathbb{P}}(\sum_{W\in G_{\xi}}{\bf 1}_{G}(W)\geq 3|G(n-1,m)|\delta)\leq e^{-Cp^{\alpha-m}}$$ where $C$ is a positive constant. It follows that $$\begin{aligned}
{\mathbb{P}}(\exists \xi\neq 0, s.t. \sum_{W\in G_{\xi}}&{\bf 1}_{G}(W) \geq 3|G(n-1,m)|\delta)\\
&\leq p^{n}e^{-Cp^{\alpha-m}}\rightarrow 0 \text{ as } p \rightarrow \infty.
\end{aligned}$$ Together with the estimate , we conclude that $G\in \Omega({G(n,n-m)}, \delta)$ satisfies our need with high probability (at least one) provided $p$ is large enough.
\[co:ccc\] For any $n-m<\alpha<m(n-m)$, there exists a subset $G\subset {G(n,n-m)}$ such that $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in Per(W)\}|\lesssim |G|p^{-(n-m)}.$$
We consider the random model $\Omega({G(n,n-m)}, \delta)$ where $\delta=|G(n,m)|^{-1}p^{\alpha}$. For any $\xi\neq 0$, Lemma \[lem:c\] (2) implies that $$|\{W\in G(n, n-m): \xi \in Per(W)\}|=|G(n-1, n-m)|\approx p^{m(n-m)-(n-m)}.$$ Then applying the similar argument to the proof of Corollary \[co:cc\], we obtain that $G\in \Omega({G(n,n-m)}, \delta)$ satisfies our need with high probability provided $p$ is large enough.
Now we intend to apply Lemma \[lem:abstract\] and the above two Corollaries to prove Theorem \[thm:main\].
Suppose $\alpha>m$. By Corollary \[co:cc\] there exists a subset $G\subset {G(n,n-m)}$ such that $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in W\}|\lesssim |G|p^{-m}.$$ Applying Lemma \[lem:abstract\] (1), we obtain that for any $E\subset {\mathbb{F}_{p}^{n}}$, $${\mathcal{E}}(E,G')\lesssim |G|(|E|+|E|^{2}p^{-m}).$$ By estimate we obtain $$|\{W\in G: |\pi^{W}(E)|\leq N\}|\lesssim |G|N(|E|^{-1}+p^{-m}).$$
For the case $\alpha>n-m$. By Corollary \[co:ccc\] there exists a subset $G\subset G(n,n-m)$ with $|G|\approx p^{\alpha}$ and for any $\xi\neq 0$, $$|\{W\in G: \xi \in Per(W)\}|\lesssim |G|p^{n-m}.$$ Applying Lemma \[lem:abstract\] (2), we obtain that for any $E\subset {\mathbb{F}_{p}^{n}}$, $${\mathcal{E}}(E,G')\lesssim |G|(|E|+|E|^{2}p^{-m}).$$ Again by estimate , we obtain $$|\{W\in G: |\pi^{W}(E)|\leq N\}|\lesssim |G|N(|E|^{-1}+p^{-m}).$$ Thus we complete the proof.
Examples
========
We show two examples in the following. For $D\subset {\mathbb{F}_{p}^{n}}$ let $G_{D}$ be the collection of one dimensional subspaces which intersects $D$, i.e., $$G_{D}=\{kx: x\in D, k\in \mathbb{F}_{p}\}.$$
\[exa:1\] Let $
S_{1}=\{(x_{1}, x_{2}, 1)\in {\mathbb{F}}_{p}^{3}: x_{1}^{2}+x_{2}^{2}=1\}$. Then for any $E\subset \mathbb{F}_{p}^{3}$, $$|\{L\in G_{S_{1}}: |\pi^{L}(E)|\leq N\}|\lesssim |S_{1}|N(p^{-2}+|E|^{-1}).$$
A. Iosevich and M. Rudnev [@IosevichRudnev Lemma 2.2] proved that $|S_{1}|\approx p$, and hence $|G_{S_{1}}|\approx p$. Observe that $|W\cap S_{1}|\lesssim 1$ for any $W\in G(3,2)$.
For $\xi\neq 0$ let $Span(\xi)=\{k\xi: k\in \mathbb{F}_{p}\}$. Then $$\{L\in G_{S_{1}}: \xi \in Per(L)\}=G_{S_{1}}\cap Per(Span(\xi)).$$ The rank-nullity theorem implies that $\dim Per(Span(\xi))=2$. Thus $Per(Span(\xi))\in G(3,2)$, and hence we obtain $$|\{L\in G_{S_{1}}: \xi \in Per(L)\}|\lesssim 1.$$ Applying estimate and Lemma \[lem:abstract\] (2) with $\beta=1, m =2$, we finish the proof.
Note that the above example $S_{1}$ can be considered as a finite fields version of curve $$\Gamma=\{\frac{1}{\sqrt{2}}(\cos t, \sin t, 1): t\in [0, 2\pi])\} \subset {\mathbb{R}}^{3}.$$ For more details on restricted families of projections with respect to $\Gamma$ we refer to [@KOV], [@OV]. In the following, we show a finite fields version of curve $$\{(t,t^{2},\cdots, t^{n}): t\in [0,1]\}\subset {\mathbb{R}}^{n}.$$
\[ex:ex\] Let $S=\{( a, a^{2} \cdots, a^{n}): a \in {\mathbb{F}}_{p}\backslash \{0\}\}$. Then $|G_{S}|=p-1$ and for any subset $E\subset {\mathbb{F}_{p}^{n}}$, $$|\{L\in G_{S}: |\pi^{L}(E)|\leq N\}|\lesssim |G_{S}|N(|E|^{-1}+p^{-(n-1)}).$$
For $n=2$ we have $|G_{S}|\approx |G(2,1)|\approx p$, and the claim follows by applying Theorem \[thm:ChenP\]. In the following we fix $n\geq 3$ and let $p$ be a large prime number.
For any $\xi\neq 0$, $$\{L\in G_{S}:\xi \in Per(L) \}=G_{S}\cap Per(Span(\xi)).$$ The rank-nullity theorem implies that $\dim Per(Span(\xi))=n-1$. Observe that any $n$ elements of $S$ form a nonsingular Vandermonde matrix, and hence these $n$ vectors are linear independent. It follows that for any hyperplane $W\in G(n,n-1)$, $$|W\cap S|\leq n-1\lesssim_{n} 1.$$ Therefore we obtain $$|\{L\in G_{S}: \xi \in Per(L) \}|\lesssim_{n}1.$$ Applying estimate and Lemma \[lem:abstract\] (2) with $\beta=1, m=n-1$, we finish the proof.
By the special choices of $N$ in the above two examples, we conclude that Marstrand-Mattila type projection theorem hold for the restricted families $(\pi^{L})_{L\in G_{S_{1}}}$ and $(\pi^{L})_{L\in G_{S}}$.
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---
abstract: 'Nuclear dynamics of giant resonances are investigated with the real-time Skyrme TDHF method. The TDHF equation is explicitly linearized with respect to variation of single-particle wave functions. The time evolution of transition densities are calculated for giant dipole resonances. The time-dependent densities of protons and neutrons suggest that the dynamics of giant dipole resonance in neutron-rich nuclei are significantly different from that in stable nuclei with $N\approx Z$.'
address: 'Institute of Physics and Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan'
author:
- Takashi Nakatsukasa and Kazuhiro Yabana
title: 'Real-time Skyrme TDHF dynamics of giant resonances'
---
Introduction
============
Atomic nuclei exhibit a variety of collective modes of excitation. In particular, the giant resonances always have been of central interest in nuclear structure and reaction studies. The giant resonances correspond to the most fundamental oscillations in nuclei, in the sense that they exhaust major portions of the energy-weighted sum-rule values and that the nucleus strongly absorbs energy from an external field, acting as a whole. Indeed, the giant resonances can be qualitatively described in terms of semiclassical hydrodynamical models [@RS80].
Among many kinds of giant resonances, the isovector giant dipole resonance (GDR) is the most famous and exhausts almost 100 % of the sum-rule value. It is well explained by ordinary hydrodynamical models. This is rather exceptional because quantitative description of other modes requires a treatment as a Fermi liquid [@RS80]. There are two famous hydrodynamical models for GDR: the Goldhaber-Teller (GT) model [@GT48] and the Steinwedel-Jensen (SJ) model [@SJ50]. The GT model predicts $A^{-1/6}$ dependence of the GDR frequencies which arises from the concept that the restoring force is proportional to the nuclear surface area. In contrast, the SJ model relaxes the assumption of incompressibility, which leads to $A^{-1/3}$ dependence of the GDR frequencies. Experimental data are best fitted by a combination of these two [@BF75]: In light nuclei, the data seem to indicate the $A^{-1/6}$ law, while the $A^{-1/3}$ dependence becomes increasingly dominant for increasing values of $A$.
The hydrodynamical models have a close connection to the time-dependent Hartree-Fock (TDHF) theory. In the limit of $\hbar\rightarrow 0$, the TDHF equation goes over into the Vlasov equation. Therefore, without the collision term, the TDHF should provide a microscopic description of an appropriate hydrodynamical model. Recently, we have proposed the real-time TDHF method combined with the absorbing-boundary condition (TDHF+ABC method) for a linear response function [@NY01; @NY05]. This may be regarded as an extension of the continuum random-phase approximation (RPA), made applicable to a deformed system. In this paper, we show real-time dynamics of the TDHF for GDR and discuss how their properties change from stable ($N\approx Z$) to unstable nuclei ($N \gg Z$). As is discussed in the following sections, we take a small-amplitude limit of the TDHF. Although this is equivalent to the RPA, the time-dependent snap shots of the TDHF wave packet may provide an intuitive dynamical picture of the GDR. It should be also noted that the TDHF provides a proper description for low-lying modes as well, for which quantum effects are so important that the semiclassical hydrodynamical models are not applicable.
Linearized TDHF in real time
============================
The HF ground state is assumed to be a Slater determinant which consists of $A$ single-particle orbitals, $\Phi_0(x_1,\cdots,x_A)=\det\{\phi_i(x_j)\}_{i,j=1,\cdots,A}$ with $x=(\vec{r},\sigma,\tau)$. Each single-particle orbital is determined by $$\label{HF-eq}
h[\phi,\phi^*] \phi_i(x) =
\epsilon_i \phi_i(x)
\quad\quad\mbox{ for } i=1,\cdots,A ,$$ where $h[\phi,\phi^*]$ is the single-particle Hamiltonian which depends on $\phi_i(x)$ ($i=1,\cdots,A$) self-consistently. The TDHF equation is obtained by replacing, in Eq. (\[HF-eq\]), $\epsilon_i$ by the time derivative $i\hbar\partial/\partial t$, and $\phi_i(x)$ by the time-dependent wave function $\psi_i(x,t)$. The TDHF equation is now linearized with respect to variation of each single-particle wave function and a time-dependent external field $v(x,t)$. Substituting $\psi_i(x,t)=(\phi_i(x)+\delta\psi_i(x,t))e^{-i\epsilon_i t/\hbar}$ into the TDHF equation, we have $$\label{LTDHF-eq}
i\hbar\frac{\partial}{\partial t} \delta\psi_i(x,t)
= \left( h[\phi,\phi^*] -\epsilon_i\right) \delta\psi_i(x,t)
+ \delta h(t) \phi_i(x)
+ v(x,t) \phi_i(x),$$ where $\delta h(t)\equiv h[\psi,\psi^*] -h[\phi,\phi^*]$ in the first order of $\delta\psi_i(x,t)$. If we put $\delta h(t)=0$, Eq. (\[LTDHF-eq\]) gives unperturbed particle-hole excitations with a fixed single-particle potential in $h[\phi,\phi^*]$. $\delta h(t)$ is nothing but the residual interaction in the language of the energy representation. The second term in the r.h.s. of Eq. (\[LTDHF-eq\]) contains a dynamical effect which comes from variations of the self-consistent one-body potential.
Equation (\[LTDHF-eq\]) is equivalent to the well-known RPA equation in the energy representation. In practice, however, there are some differences, advantages and disadvantages in each method. For instance, the uncertainty in energy, $\Delta E$, is inversely proportional to the period of the time propagation $T$; $\Delta E\sim \hbar/T$. Therefore, when we want to distinguish states nearly degenerate, we need to propagate the wave functions for a long period of time. In this case, the energy representation may be a better choice. On the other hand, when we are interested in a bulk structure of excited states in a wide range of energy, calculations using the time representation becomes more efficient than those with the energy representation. The time-dependent calculation should be suitable for giant resonances, since their energies are rather high and spread over a wide range of energy.
The transition density in the time representation is defined by the density variation from its ground-state value, $$\begin{aligned}
\label{delta-rho}
\delta\rho(x,x';t) &=& \rho(x,x';t)-\rho_0(x,x') \nonumber\\
&=& \sum_{i=1}^A \left\{ \phi_i^*(x) \delta\psi_i(x',t)
+\delta\psi_i^*(x,t)\phi_i(x') \right\} .\end{aligned}$$ In this paper, we are mainly interested in the spin-independent diagonal part of Eq. (\[delta-rho\]); $$\label{delta-rho-diag}
\delta\rho_\tau(\vec{r};t)
= \sum_{i=1}^A \sum_{\sigma}\left\{ \phi_i^*(x) \delta\psi_i(x,t)
+\delta\psi_i^*(x,t)\phi_i(x) \right\} .$$ The expectation value of an operator $\hat{F}(\vec{r},\tau)$ can be expressed as $$\label{Ft}
F(t) = F_0 + \delta F(t)
= F_0 +\sum_\tau\int d^3r \hat{F}(\vec{r},\tau) \delta \rho_\tau(\vec{r};t) ,$$ where $F_0$ is the ground-state expectation value.
The external field $v(x,t)$ in Eq. (\[LTDHF-eq\]) can be chosen according to the purpose of the calculation. In order to calculate the strength distribution in a wide range of energy, an instantaneous external field, $v(x,t)=v(x)\delta(t)$, is suitable, because this excites the system to states in all energies. In contrast, if we adopt an oscillating field with a fixed frequency $\omega$, $v(x,t)=v(x) \cos(\omega t)$, the system is excited to a specific state with $E_{\rm x}=\hbar\omega$. In this way, we can investigate dynamical properties of the specific state in the time-dependent manner.
To calculate the strength function of the operator $\hat{F}(\vec{r},\tau)$, $$S(\hat{F};E) = \sum_{E'} \delta(E-E')
\left|{\langle \Psi_{E'} | } \hat{F} {| \Phi_0 \rangle }\right|^2,$$ we adopt the external field $v(x)$ proportional to $\hat{F}(\vec{r},\tau)$. Then, $S(\hat{F};E)$ can be obtained as the Fourier transform of the expectation value of Eq. (\[Ft\]).
Before showing results, let us discuss a numerical difficulty related to presence of zero modes in nuclei. The “zero mode” means zero-energy modes of excitation associated with the spontaneous symmetry breaking in the HF states, such as translation and rotation. These modes should correspond exactly to the zero energy if the numerical calculation is perfect. However, a small numerical error and approximation may give imaginary energies to these modes. Since the time evolution of wave functions carries all the information of the excited states, the presence of these imaginary-energy modes leads to a kind of numerical instability to prevent performing a long period of the time propagation. Thus, we need to remove components of the zero modes from the time-dependent single-particle wave functions $\delta\psi_i(x,t)$. The zero modes can be constructed by operating the symmetry operator $\hat{P}$ and its conjugate one $\hat{Q}$ to the ground state. For the translational case, $\hat{P}$ is the total momentum operator and $\hat{Q}$ is the center-of-mass coordinate.
Giant dipole resonances in stable and neutron-rich nuclei
=========================================================
We now apply the method to GDR in even-even Be isotopes. We calculated $B(E1)$ distribution for Be isotopes in Ref. [@NY05] using the SIII parameter set. We found that the large deformation splitting in $^8$Be and $^{14}$Be, because of the large quadrupole (prolate) deformation in the ground state ($\beta_2\approx 0.8$). However, the width in $^{14}$Be is much larger than that in $^8$Be. These results are robust and do not depend on the choice of the Skyrme parameter set. We also found that there is a significant low-energy $E1$ strength around $E_{\rm x}=5$ MeV for $^{14}$Be [@NY05]. However, the $E1$ strength and its peak position are rather sensitive to the choice of the parameters. Thus, in this section, we show time-dependent transition densities for the main peak of GDR.
The calculation is performed on the three-dimensional Cartesian coordinate grid space, using the Skyrme energy functional of Ref. [@BFH87]. The Galilean symmetry is respected in this functional including the spin-orbit, Coulomb, and time-odd densities. We adopt the SGII parameter set in the calculation. In order to take account of the single-particle continuum, we use the absorbing potential outside of the interacting region [@NY01; @NY05]. In Fig. \[fig:Be8\_14\], we show the $E1$ oscillator strength distribution for $^{8,14}$Be. Two-peak structure due to the deformation splitting is prominent for both $^8$Be and $^{14}$Be. Hereafter, let us focus our discussion on the peak around $E_{\rm x}=15$ MeV with $K=0$.
![Calculated $E1$ oscillator strength distribution in $^{8,14}$Be. Thin solid and dashed lines indicate the response to dipole fields parallel and perpendicular to the symmetry axis, respectively. Thick line shows the total strength. []{data-label="fig:Be8_14"}](Be8-14.eps){height="0.4\textheight"}
We use a Gaussian-pulse external field, $v(x,t)=M(E1)_{K=0} \cos(\omega t) e^{-\gamma (t-t_0)^2}$, to selectively excite the GDR around $E_{\rm x}=\hbar\omega=15$ MeV, with $\gamma=3$ MeV$/\hbar$ and $t_0=2\ \hbar/$MeV. Then, the spin-independent transition density of Eq. (\[delta-rho-diag\]) is calculated in the 3D coordinate space. It turns out that one of the Steinwedel-Jensen’s assumptions, $\delta\rho_n(\vec{r};t)=-\delta\rho_p(\vec{r};t)$, is approximately satisfied for $^8$Be. In contrast, in $^{14}$Be, we see a large deviation from this property. Figure \[fig:Be14\] shows how $\delta\rho_\tau(\vec{r},t)$ ($\tau=p,n$) evolve in time in the $x$-$z$ plane. The time difference from one panel to the next is $\Delta t = 0.2\ \hbar/$MeV which roughly corresponds to the half period $\pi/\omega$. We see that significant portions of neutrons actually move together with protons. The neutron transition density $\delta\rho_n$ shows a peculiar node structure. In Fig. \[fig:Be14\], the regions of $\delta\rho_p>0$ ($\delta\rho_p<0$) have a large overlap with the those of $\delta\rho_n>0$ ($\delta\rho_n<0$). This means a violation of the property of the SJ model, $\delta\rho_p+\delta\rho_n=0$. Weakly bound neutrons in neutron-rich nuclei seem to be significantly affected by strong attraction between protons and neutrons, and to oscillate in phase with protons’ movement. This is a consequence of the dynamical effect of the time-dependent self-consistent potential, $\delta h(t)$ in Eq. (\[LTDHF-eq\]).
(p1)(p2)(p3)(p4)\
![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](210_p.eps "fig:"){height="24.00000%"} ![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](230_p.eps "fig:"){height="24.00000%"} ![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](250_p.eps "fig:"){height="24.00000%"} ![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](270_p.eps "fig:"){height="24.00000%"}
(n1)(n2)(n3)(n4)\
![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](210_n.eps "fig:"){height="24.00000%"} ![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](230_n.eps "fig:"){height="24.00000%"} ![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](250_n.eps "fig:"){height="24.00000%"} ![Snap shots of calculated $\delta\rho_\tau(\vec{r},t)$ in the $x$-$z$ plane for the $K=0$ peak at $E_{\rm x}=15$ MeV in $^{14}$Be. The upper panels (p1-4) indicate $\delta\rho_p(\vec{r};t)$, while the lower (n1-4) for $\delta\rho_n(\vec{r};t)$. White (black) regions indicate those of $\delta\rho_\tau > 0$ ($\delta\rho_\tau < 0$). The time difference between two neighboring panels is $\Delta t=0.2\ \hbar/$MeV. The two panels at the same column corresponds to the same time $t$. []{data-label="fig:Be14"}](270_n.eps "fig:"){height="24.00000%"}
Summary
=======
We present calculations of the linearized TDHF method in real time for giant dipole resonances in $^{8,14}$Be. These nuclei are calculated to be largely deformed in the ground state. The main GDR peak is split into two peaks with $K=0$ and $K=1$. We show the time-dependent transition densities for $K=0$ peaks. The total density, $\rho_p+\rho_n$, is approximately conserved for $^8$Be, while its conservation is significantly violated in $^{14}$Be. The time evolution of the transition density, $\delta\rho_\tau(t)$, suggests a strong dynamical effect for neutron-rich nuclei, and seems to indicate the mixture of the isoscalar volume-type and the isovector surface-type components.
This work has been supported by the Grant-in-Aid for Scientific Research in Japan (Nos. 17540231 and 17740160). The numerical calculations have been performed at SIPC, University of Tsukuba, at RCNP, Osaka University, and at YITP, Kyoto University.
[9]{} P. Ring and P. Schuck, [*The Nuclear Many-Body Problem*]{} (Springer-Verlag, New York, 1980). M. Goldhaber and E. Teller, Phys. Rev. [**74**]{} (1948) 1046. H. Steinwedel and J. H. D. Jensen, Z. Naturforschung [**5A**]{} (1950) 413. B. L. Berman and S. C. Fultz, Rev. Mod. Phys. [**47**]{} (1975) 713. T. Nakatsukasa and K. Yabana, J. Chem. Phys. [**114**]{} (2001) 2550. T. Nakatsukasa and K. Yabana, Phys. Rev. C [**71**]{} (2005) 024301. P. Bonche and H. Flocard and P. H. Heenen, Nucl. Phys. A [**467**]{} (1987) 115.
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---
address: 'S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy'
author:
- LUCIA SCARDIA
title: 'The nonlinear bending-torsion theory for curved rods as $\Gamma$-limit of three-dimensional elasticity '
---
Introduction
============
This paper is part of a series of recent works concerning the rigorous derivation of lower dimensional models for thin domains from nonlinear three-dimensional elasticity, by means of $\Gamma-$ convergence.
The first result in this direction is due to E. Acerbi, G. Buttazzo and D. Percivale (see [@ABP91]), who deduced a nonlinear model for elastic strings by means of a 3D-1D dimension reduction. The two-dimensional analogue was studied by H. Le Dret and A. Raoult in [@LDR95], where they derived a nonlinear model for elastic membranes. The more delicate case of plates was justified more recently by G. Friesecke, R.D. James and S. Müller in [@FJM02] (see also [@FJMP] for a complete survey on plate theories). The case of shells was considered in [@LDR00] and [@FJMM03].
As for one-dimensional models, nonlinear theories for elastic rods have been deduced by M.G. Mora, S. Müller (see [@MM03], [@MGMM04]) and, independently, by O. Pantz (see [@P02]). In all these results, as in [@ABP91], the beam is assumed to be straight in the unstressed configuration.
In this paper we study the case of a heterogeneous curved beam made of a hyperelastic material. Let $\Omega:=(0,L)\times D$, where $L>0$ and $D$ is a bounded domain in $\mathbb{R}^2$, and let $h>0$. We consider a beam, whose reference configuration is given by $$\widetilde{\Omega}_{h}:= \{\gamma(s) + h\,\xi\,\nu_{2}(s) + h\,\zeta\, \nu_{3}(s) : (s,\xi,\zeta)\in \Omega\},$$ where $\gamma:(0,L)\to \mathbb{R}^{3}$ is a smooth simple curve describing the mid-fiber of the beam, and $\nu_2, \nu_3:(0,L)\to \mathbb{R}^{3}$ are two smooth vectors such that $(\gamma',\nu_2, \nu_3)$ provide an orthonormal frame along the curve. In particular, the cross section of the beam is constant along $\gamma$ and is given by the set $hD$. It is natural to parametrize $\widetilde{\Omega}_{h}$ through the map $$\Psi^{(h)} : \Omega \rightarrow \widetilde{\Omega}_{h}, \quad (s,\xi,\zeta)\mapsto \gamma(s) + h\,\xi\,\nu_{2}(s) + h\,\zeta\,\nu_{3}(s),$$ which is one-to-one for $h$ small enough.
The starting point of our approach is the elastic energy per unit volume $$\label{puv}
\tilde{I}^{(h)}(\tilde{v}):= \frac{1}{h^2}\int_{\widetilde{\Omega}_{h}}
W\big(\big(\Psi^{(h)}\big)^{-1} (x),\nabla \tilde{v}(x)\big) dx$$ of a deformation $\tilde{v} \in W^{1,2}(\widetilde{\Omega}_{h};\mathbb{R}^{3})$. The stored energy density $W:\Omega\times {\mathbb M}^{3\times 3}\to [0,+\infty]$ has to satisfy some natural conditions; i.e.,
- $W$ is frame indifferent: $W(z,RF) = W(z,F)$ for a.e. $z\,\in \Omega$, every $F\in \mathbb{M}^{3\times 3}$, and every $R\in SO(3)$;
- $W(z,F)\geq C\,\mbox{dist}^{2}(F,SO(3))$ for a.e. $z\in \Omega$ and every $F\in\mathbb{M}^{3\times 3}$;
- $W(z,R)=0$ for a.e. $z\in\Omega$ and every $R\in SO(3)$.
For the complete list of assumptions on $W$ we refer to Section 2.
The aim of this work is to study the asymptotic behaviour of different scalings of the energy $\tilde{I}^{(h)}$, as $h\rightarrow 0$, by means of $\Gamma$-convergence (see [@DM93] for a comprehensive introduction to $\Gamma$-convergence). Heuristic arguments suggest that, as in the case of straight beams, energies of order $1$ correspond to stretching and shearing deformations, leading to a *string theory*, while energies of order $h^2$ correspond to bending flexures and torsions keeping the mid-fiber unextended, leading to a *rod theory*.
The main results of the paper are contained in Section $3$, where we identify the $\Gamma$-limit of the sequence of functionals $\big(\tilde{I}^{(h)}/h^2\big)$. We first show a compactness result for sequences of deformations having equibounded energies (Theorem \[compactness\]). More precisely, given a sequence $\big(\tilde{v}^{(h)}\big)\subset W^{1,2}(\widetilde{\Omega}_{h};\mathbb{R}^{3})$ with $\tilde{I}^{(h)}(\tilde{v}^{(h)})/h^2\leq C$, we prove that there exist a subsequence (not relabelled) and some constants $c^{(h)}\in\mathbb{R}^3$ such that $$\begin{aligned}
\tilde{v}^{(h)}\circ \Psi^{(h)} -
c^{(h)} &\rightarrow v \quad \mbox{strongly in}\,\,
W^{1,2}(\Omega;\mathbb{R}^{3}),
\\
\frac{1}{h}\,\partial_\xi\big(\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)&\,\rightarrow d_2 \quad \mbox{strongly in}\,
L^{2}(\Omega;\mathbb{R}^{3}),
\\
\frac{1}{h}\,\partial_\zeta\big(\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)\,&\rightarrow d_3 \quad \mbox{strongly in}\,
L^{2}(\Omega;\mathbb{R}^{3}),\end{aligned}$$ where $(v,d_{2},d_{3})$ belongs to the class $$\begin{aligned}
\mathcal{A}:= \{(v,d_{2},d_{3}) \in W^{2,2}((0, L);\mathbb{R}^{3})\times W^{1,2}((0, L);\mathbb{R}^{3})\times
W^{1,2}((0, L);\mathbb{R}^{3}):\nonumber\\
(v'(s)\,|\,d_{2}(s)\,|\,d_{3}(s))\in SO(3)\,\,\mbox{for a.e.\ } s \mbox{ in}\,\, (0, L)\}.\end{aligned}$$ The key ingredient in the proof is a geometric rigidity theorem proved by G. Friesecke, R.D. James and S. Müller in [@FJM02]. In Theorems \[scithm\] and \[bfa\] we show that the $\Gamma$-limit of the sequence $\big(\tilde{I}^{(h)}/h^2\big)$ is given by $$\label{introd}
I(v,d_2,d_3):=
\left\{
\vspace{.7cm}
\begin{array}{ll}
\displaystyle\frac{1}{2}\int_{0}^{L} Q_{2} \big(s,\big(R^{T}(s)R'(s) -
R_{0}^{T}(s)R_{0}'(s)\big)\big)ds & \mbox{if } \, (v,d_2,d_3)\in \mathcal{A},\\
\displaystyle + \infty & \mbox{otherwise},
\end{array}
\right.$$ where $R := (v'\,|\,d_2\,|\,d_3)$, $R_{0}: = (\gamma'\,|\,\nu_2\,|\,\nu_3)$, and $Q_{2}$ is a quadratic form arising from a minimization procedure involving the quadratic form of linearized elasticity (see (\[Q2\])). We point out that in Theorems \[scithm\] and \[bfa\] we do not require any growth condition from above on the energy density $W$.
We notice that in the limit problem the behaviour of the rod is described by a triple $(v,d_2,d_3)$. The function $v$ represents the deformation of the mid-fiber, which satisfies $|v'|=1$ a.e., because of the constraint $(v'\,|\,d_{2}\,|\,d_{3})\in SO(3)$ a.e.. Therefore, the admissible deformations are only those leaving the mid-fiber unextended. Moreover, the triple $(v,d_2,d_3)$ provides an orthonormal frame along the deformed curve; in particular, $d_2$ and $d_3$ belong to the normal plane to the deformed curve and describe the rotation undergone by the cross section.
Since $R = (v'\,|\,d_2\,|\,d_3)$ is a rotation a.e., the matrix $R^TR'$ is skew-symmetric a.e. and its entries are given by $$(R^TR')_{1k}=-(R^TR')_{k1}=v'\cdot d_k' \quad \hbox{for }k=2,3,\quad
(R^TR')_{23}=-(R^TR')_{32}=d_2\cdot d_{3}'.$$ It is easy to see that the scalar products $v'\cdot d_k'$ are related to curvature and therefore, to bending effects, while $d_2\cdot d_{3}'$ is related to torsion and twist. We remark also that the energy depends explicitly on the reference state of the beam through the quantity $R_0^{T}R'_0$, which encodes informations about the bending and torsion of the beam in the initial configuration.
We notice that, specifying $R_{0} = Id$ in (\[introd\]), we recover the result for straight rods obtained in [@MM03] and [@P02].
The last section of the paper is devoted to the study of lower scalings of the energy. Assuming that the energy density $W$ satisfies a growth condition from above, we prove the $\Gamma$-convergence of the sequence $\big(\tilde{I}^{(h)}\big)$ to a functional corresponding to a string model. Finally we show that intermediate scalings of the energy between $1$ and $h^2$ lead to a trivial $\Gamma$-limit.
Notations and formulation of the problem
========================================
In this section we describe the geometry of the unstressed curved beam. Let $\gamma : [0, L] \rightarrow \mathbb{R}^{3}$ be a simple regular curve of class $C^{2}$ parametrized by the arc-length and let $\tau = \dot{\gamma}$ be its unitary tangent vector. We assume that there exists an orthonormal frame of class $C^{1}$ along the curve. More precisely, we assume that there exists $R_{0}\in C^{1}([0, L]; \mathbb{M}^{3\times 3})$ such that $R_{0}(s)\in SO(3)$ for every $s \in [0, L]$ and $R_{0}(s)\,e_{1} = \tau(s)$ for every $s \in [0, L]$, where $e_i$, for $i=1,2,3$, denotes the i-th vector of the canonical basis of $\mathbb{R}^{3}$ and . We set $\nu_k (s) := R_{0}(s)\,e_{k}$ for $k = 2,3$.
Let $D\subset \mathbb{R}^{2}$ be a bounded open connected set with Lipschitz boundary such that $$\label{dom1}
\int_{D}\xi\,\zeta\, d\xi\, d\zeta = 0$$ and $$\label{dom2}
\int_{D}\xi\,d\xi\,d\zeta = \int_{D}\zeta\,d\xi\,d\zeta = 0,$$ where $(\xi,\zeta)$ stands for the coordinates of a generic point of $D$. Without loss of generality, we can assume $\mathcal{L}^2(D) = 1$. We set $\Omega:= (0, L)\times D$. The reference configuration of the thin beam is given by $$\widetilde{\Omega}_{h}:= \{\gamma(s) + h\,\xi\,\nu_{2}(s) + h\,\zeta\, \nu_{3}(s) : (s,\xi,\zeta)\in \,\Omega\},$$ where $h$ is a small positive parameter. Clearly the curve $\gamma$ and the set $D$ represent the middle fiber and the cross section of the beam, respectively. The set $\widetilde{\Omega}_{h}$ is parametrized by the map $$\Psi^{(h)} : \Omega \rightarrow \widetilde{\Omega}_{h}\,: \quad (s,\xi,\zeta)\mapsto \gamma(s) + h\,\xi\,\nu_{2}(s) + h\,\zeta\,\nu_{3}(s),$$ which is one-to-one for $h$ small enough and of class $C^1$.
We assume that the thin beam is made of a hyperelastic material whose stored energy density $W : \Omega\times\mathbb{M}^{3\times 3} \rightarrow [0, + \infty]$ is a Carathéodory function satisfying the following hypotheses:
- there exists $\delta>0$ such that the function $F\mapsto W(z,F)$ is of class $C^{2}$ on the set\
$\big\{F\in\mathbb{M}^{3\times 3}: \mbox{dist}(F,SO(3)) < \delta\big\}$ for a.e. $z\,\in \Omega$;
- the second derivative $\partial^{2}W/\partial F^{2}$ is a Carathéodory function on the set $$\label{set}
\Omega\times\{F\in \mathbb{M}^{3\times 3}:\,\mbox{dist}(F,SO(3)) < \delta \}$$ and there exists a constant $C_{1} > 0$ such that $$\begin{aligned}
&\bigg|\frac{\partial^{2}W}{\partial F^{2}}(z,F)[G,G]\bigg| \leq C_{1} |\,G\,|^{2}\quad \mbox{for a.e. } z\in \Omega,\, \mbox{every} F \hbox{with} \hbox{dist}(F,SO(3))<\delta\\
& \mbox{and every } G\in \mathbb{M}^{3\times 3}_{sym};\end{aligned}$$
- $W$ is frame indifferent, i.e., $W(z,RF) = W(z,F)$ for a.e. $z\,\in \Omega$, every $F\in \mathbb{M}^{3\times 3}$ and every $R\in SO(3)$;
- $W(z,R)=0$ for every $R\in SO(3)$;
- $\exists$ $C_{2} >\,0$ independent of $z$ such that $W(z,F)\geq C_{2}\,
\mbox{dist}^{2}(F,SO(3))$ for a.e. $z\in \Omega$ and every $F\in\mathbb{M}^{3\times 3}$.
Notice that, since we do not require any growth condition from above, $W$ is allowed to assume the value $+ \infty$ outside the set (\[set\]). Therefore our treatment covers the physically relevant case in which $W = + \infty$ for $\det F < 0$, $W\rightarrow + \infty$ as $\det F \rightarrow 0^+$.
Let $\tilde{v} \in W^{1,2}(\widetilde{\Omega}_{h};\mathbb{R}^{3})$ be a deformation of $\widetilde{\Omega}_{h}$. The elastic energy per unit volume associated to $\tilde{v}$ is defined by $$\label{energypuv}
\tilde{I}^{(h)}(\tilde{v}):= \frac{1}{h^{2}}\int_{\widetilde{\Omega}_{h}}
W\big(\big(\Psi^{(h)}\big)^{-1} (x),\nabla
\tilde{v}(x)\big) dx.$$ The main part of this work is devoted to the study of the asymptotic behaviour as $h\rightarrow 0$ of the sequence of functionals $\tilde{I}^{(h)}/h^{2}$. In the final part we will also discuss the scaling $\tilde{I}^{(h)}/h^{\alpha}$ for $0 \leq \alpha < 2$.
We conclude this section by analysing some properties of the map $\Psi^{(h)}$, which will be useful in the sequel. We will use the following notation: for any function $z\in W^{1,2}(\Omega;\mathbb{R}^3)$ we set $$\nabla_{h}z := \left(\partial_s z\,\Big|\,\frac{1}{h}\,\partial_\xi z\,\Big|\, \frac{1}{h}\,\partial_\zeta z\right).$$ We observe that $\nabla_{h}\Psi^{(h)}$ can be written as the sum of the rotation $R_{0}$ and a perturbation of order $h$, that is, $$\nabla_{h}\Psi^{(h)}(s,\xi,\zeta) = R_{0}(s) + h\,\left(\xi\,\nu'_{2}(s) + \zeta\,\nu'_{3}(s)\right)\otimes e_1.$$ From this fact it follows that, as $h\rightarrow 0$, $$\label{convdet}
\nabla_{h}\Psi^{(h)}(s,\xi,\zeta)\rightarrow R_{0}(s)\quad \mbox{and}\quad\det \big(\nabla_{h}\Psi^{(h)}\big) \rightarrow \det R_{0} = 1 \,\,\,\mbox{uniformly}.$$ This implies that for $h$ small enough $\nabla_{h}\Psi^{(h)}$ is invertible at each point of $\Omega$. Since the inverse of $\nabla_{h}\Psi^{(h)}$ can be written as $$\label{invA}
\big(\nabla_{h}\Psi^{(h)}\big)^{-1}(s,\xi,\zeta) =
R_{0}^{T}(s) - h\,R_{0}^{T}(s)\,\big[(\xi\,\nu'_{2}(s) +
\zeta\,\nu'_{3}(s))\otimes e_1\big] R_{0}^{T}(s) + O(h^{2})$$ with $O(h^{2})/h^{2}$ uniformly bounded, we have also that $\big(\nabla_{h}\Psi^{(h)}\big)^{-1}$ converges to $R_{0}^{T}$ uniformly.
Derivation of the bending-torsion theory for curved rods
========================================================
The aim of this section is the study of the asymptotic behaviour of the sequence of functionals $$\frac{1}{h^{2}}\,\tilde{I}^{(h)}(\tilde{v})= \frac{1}{h^{4}} \int_{\widetilde{\Omega}_{h}} W\big(\big(\Psi^{(h)}\big)^{-1} (x),\nabla \tilde{v}(x)\big) dx$$ under the assumptions (i)-(v) of Section 2.
Compactness
-----------
We will show a compactness result for sequences of deformations having equibounded energy $\tilde{I}^{(h)}/h^2$. A key ingredient in the proof is the following rigidity result, proved by G. Friesecke, R.D. James and S. Müller in [@FJM02].
\[Teorigid\] Let $U$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$, $n\geq
2$. Then there exists a constant $C(U)$ with the following property: for every $u\in W^{1,2}(U;\mathbb{R}^{n})$ there is an associated rotation $R\in SO(n)$ such that $${\left\Vert\nabla u - R\right\Vert}_{L^{2}(U)} \leq C(U){\left\Vert\textnormal{dist}(\nabla
u, SO(n))\right\Vert}_{L^{2}(U)}.$$
The constant $C(U)$ can be chosen independent of $U$ for a family of sets that are Bilipschitz images of a cube (with uniform Lipschitz constants), as remarked in [@FJMM03].
We introduce the class of limiting admissible deformations $$\begin{aligned}
\label{defA}
\mathcal{A}:= \{(v,d_{2},d_{3}) \in W^{2,2}((0, L);\mathbb{R}^{3})\times W^{1,2}((0, L);\mathbb{R}^{3})\times
W^{1,2}((0, L);\mathbb{R}^{3}):\nonumber\\
(v'(s)\,|\,d_{2}(s)\,|\,d_{3}(s))\in SO(3)\,\,\mbox{for a.e. s in}\,\, (0, L)\}.\end{aligned}$$ Now we are ready to state and prove the main result of this subsection.
\[compactness\] Let $\big(\tilde{v}^{(h)}\big)$ be a sequence in $W^{1,2}\big(\widetilde{\Omega}_{h};\mathbb{R}^{3}\big)$ such that $$\label{finite}
\frac{1}{h^2}\,\tilde{I}^{(h)}(\tilde{v}^{(h)})
\leq c < +\infty.$$ Then there exist a triple $(v,d_{2},d_{3})\in \mathcal{A}$, a map $\overline{R}\in W^{1,2}((0, L);\mathbb{M}^{3\times 3})$ with $\overline{R}(s)\in SO(3)$\
for a.e. $s\in [0, L]$, and some constants $c^{(h)}\in\mathbb{R}^3$ such that, up to subsequences, $$\begin{aligned}
\tilde{v}^{(h)}\circ \Psi^{(h)} -
c^{(h)} &\rightarrow v \quad \mbox{strongly in}\,\,
W^{1,2}(\Omega;\mathbb{R}^{3}), \label{teo1}\\
\frac{1}{h}\,\partial_\xi\big(\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)&\,\rightarrow d_2 \quad \mbox{strongly in}\,
L^{2}(\Omega;\mathbb{R}^{3}),\label{teo2}\\
\frac{1}{h}\,\partial_\zeta\big(\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)\,&\rightarrow d_3 \quad \mbox{strongly in}\,
L^{2}(\Omega;\mathbb{R}^{3}),\label{teo25}\\
\nabla\tilde{v}^{(h)}\circ
\Psi^{(h)}&\rightarrow \overline{R}\quad \mbox{strongly
in}\, L^{2}(\Omega;\mathbb{M}^{3\times 3}).\label{teo3}\end{aligned}$$ Moreover, for a.e. $s\in [0, L]$, we have $(v'(s)\,|\,d_{2}(s)\,|\,d_{3}(s)) = \overline{R}(s)\,R_{0}(s)$, where $R_{0} = (\tau\,|\,\nu_{2}\,|\,\nu_{3})$.
Let $\big(\tilde{v}^{(h)}\big)$ be a sequence in $W^{1,2}(\widetilde{\Omega}_{h};\mathbb{R}^{3})$ satisfying (\[finite\]). The assumption (v) on $W$ implies that $$\int_{\widetilde{\Omega}_{h}} \textrm{dist}^{2}\big(\nabla
\tilde{v}^{(h)}(x),SO(3)\big) dx < C\,h^4$$ for a suitable constant $C$. Using the change of variables $\Psi^{(h)}$, we have $$\label{bound2}
\int_{\Omega} \mbox{dist}^{2}\big(\nabla
\tilde{v}^{(h)}\circ\Psi^{(h)},SO(3)\big) \det
\big(\nabla_{h}\Psi^{(h)}\big)
ds\,d\xi\,d\zeta \leq c\,h^{2}.$$ From (\[convdet\]) and the estimate $$\mbox{dist}^{2}(F,SO(3)) \geq \frac{1}{2}\,|\,F\,|^{2} - 3,$$ we get the bound $$\label{uff}
\int_{\Omega}\big|\,\nabla\tilde{v}^{(h)}\circ
\Psi^{(h)}\big|^{2} ds\,d\xi\,d\zeta \leq c.$$ Define the sequence $F^{(h)} :=
\nabla\tilde{v}^{(h)}\circ\Psi^{(h)}$; from (\[uff\]) it follows that there exists a function $F \in
L^{2}(\Omega;\mathbb{M}^{3\times 3})$ such that, up to subsequences, $$\label{Fconv}
F^{(h)} \rightharpoonup F \quad \textrm{weakly in} \,
L^{2}(\Omega;\mathbb{M}^{3\times 3}).$$ Using Theorem \[Teorigid\], we will show that this convergence is in fact strong in $L^{2}$ and that the limit function $F$ is a rotation a.e. depending only on the variable along the mid-fiber and belonging to $W^{1,2}((0, L);\mathbb{M}^{3\times 3})$. The idea is to divide the domain $\widetilde{\Omega}_{h}$ in small curved cylinders, which are images of homotetic straight cylinders through the same Bilipschitz function. Then, we can apply the rigidity theorem to each small curved cylinder with the same constant. In this way we construct a piecewise constant rotation, which is close to the deformation gradient $\nabla\tilde{v}^{(h)}$ in the $L^{2}$ norm. For every small enough $h>0$, let $K_{h}\in
\mathbb{N}$ satisfy $$h \leq \frac{L}{K_{h}} < 2\,h.$$ For every $a\in[0, L)\cap
\dfrac{L}{K_{h}}\,\mathbb{N}$, define the segments $$S_{a,K_{h}}:=
\left\{
\begin{array}{ll}
\vspace{.2cm}
(a, a + 2\,h) & \mbox{if } \,\, a< L-\dfrac{L}{K_{h}},\\
(L - 2\,h,L ) & \mbox{otherwise}.
\end{array}
\right.$$ Now consider the cylinders $C_{a,h}:= S_{a,K_{h}}\times D$ and the subsets of $\widetilde{\Omega}_{h}$ defined by $\widetilde{C}_{a,h}:=
\Psi^{(h)}(C_{a,h})$. Remark that $\widetilde{C}_{a,h}$ is a Bilipschitz image of a cube of size $h$, that is $(a,0,0) + h\,\big((0, 2)\times D \big)$, through the map $\Psi$ defined as $$\Psi : [0, L]\times\mathbb{R}^{2} \rightarrow \mathbb{R}^{3}, \quad (s,y_{2},y_{3})\mapsto \gamma(s) + y_{2}\,\nu_{2}(s) + y_{3}\,\nu_{3}(s).$$ By Theorem \[Teorigid\] we obtain that there exists a constant rotation $\widetilde{R}_{a}^{(h)}$ such that $$\label{rigid}
\int_{\widetilde{C}_{a,h}}\big|\,\nabla\tilde{v}^{(h)}
- \widetilde{R}_{a}^{(h)}\big|^{2} dx \leq c
\int_{\widetilde{C}_{a,h}}\mbox{dist}^{2}(\nabla\tilde{v}^{(h)},SO(3))
dx.$$ The subscript $a$ in $\widetilde{R}_{a}^{(h)}$ is used to remember that the rotation depends on the cylinder $\widetilde{C}_{a,h}$. In particular, since $\Psi^{(h)}\big(\big(a, a +
\frac{L}{K_{h}}\big)\times D\big)\subset \widetilde{C}_{a,h}$, we get $$\label{rigidity}
\int_{\Psi^{(h)}\big(\big(a, a +
\frac{L}{K_{h}}\big)\times
D\big)}\big|\,\nabla\tilde{v}^{(h)} -
\widetilde{R}_{a}^{(h)}\big|^{2} dx \leq c
\int_{\widetilde{C}_{a,h}}\mbox{dist}^{2}(\nabla\tilde{v}^{(h)},SO(3))
dx.$$ Changing variables in the integral on the left-hand side, inequality (\[rigidity\]) becomes $$\begin{aligned}
\int_{\big(a, a + \frac{L}{K_{h}}\big)\times
D}&\big|\,\nabla\tilde{v}^{(h)}\circ\Psi^{(h)} -
\widetilde{R}_{a}^{(h)}\big|^{2}\det
\big(\nabla\Psi^{(h)}\big) ds\,d\xi\,d\zeta \\
&\leq
c\,\int_{\widetilde{C}_{a,h}}\mbox{dist}^{2}\big(\nabla\tilde{v}^{(h)},SO(3)\big)
dx\\
&\leq
c\,\int_{\widetilde{C}_{a,h}}W\big(\big(\Psi^{(h)}\big)^{-1}
(x),\nabla\tilde{v}^{(h)}(x)\big) dx.\end{aligned}$$ Notice that $\det \big(\nabla\Psi^{(h)}\big) =
h^2 \det
\big(\nabla_{h}\Psi^{(h)}\big)$ and, since $\det
\big(\nabla_{h}\Psi^{(h)}\big)\rightarrow 1$ uniformly, $$\label{rigidity2}
\int_{\big(a, a + \frac{L}{K_{h}}\big)\times
D}\big|\,\nabla\tilde{v}^{(h)}\circ\Psi^{(h)} -
\widetilde{R}_{a}^{(h)}\big|^{2} ds\,d\xi\,d\zeta \leq
\frac{c}{h^2}\,\int_{\widetilde{C}_{a,h}}W\big(\big(\Psi^{(h)}\big)^{-1}
(x),\nabla\tilde{v}^{(h)}(x)\big) dx.$$ Now define the map $R^{(h)}: [0, L)\rightarrow SO(3)$ given by $$R^{(h)}(s):= \widetilde{R}_{a}^{(h)} \quad
\mbox{for}\, s\in \Big[a, a + \frac{L}{K_{h}}\Big),\,
a\in [0, L)\cap \frac{L}{K_{h}}\,\mathbb{N}.$$ Summing (\[rigidity2\]) over $a\in [0, L)\cap
\frac{L}{K_{h}}\,\mathbb{N}$ leads to $$\int_{\Omega}\big|\,\nabla\tilde{
v}^{(h)}\circ\Psi^{(h)} -
R^{(h)}\big|^{2}ds\,d\xi\,d\zeta \leq
\frac{c}{h^2}\,\int_{\widetilde{\Omega}_{h}}W\big(\big(\Psi^{(h)}\big)^{-1}
(x),\nabla\tilde{v}^{(h)}(x)\big) dx$$ for a suitable constant independent of $h$. By (\[finite\]) we obtain $$\label{rigid2}
\int_{\Omega}\big|\,\nabla
\tilde{v}^{(h)}\circ\Psi^{(h)} -
R^{(h)}\big|^{2}ds\,d\xi\,d\zeta \leq c\,h^2.$$ Now, applying iteratively estimate (\[rigidity2\]) in neighbouring cubes, one can prove the following difference quotient estimate for $R^{(h)}$: for every $I'\subset\subset [0, L]$ and every $\delta \in \mathbb{R}$ with ${\left|\,\delta\right|} \leq \mbox{dist}(I',\{0, L\})$ $$\label{increm}
\int_{I'}\big|\,R^{(h)}(s + \delta) - R^{(h)}(s)\,\big|^{2} ds
\leq c\,(|\,\delta\,| + h)^{2},$$ with $c$ independent of $I'$ and $\delta$ (see [@MM03], proof of Theorem 2.1). Using the Fréchet-Kolmogorov criterion, we deduce that, for every sequence $(h_{j})\rightarrow 0$, there exists a subsequence of $R^{(h_{j})}$ which converges strongly in $L^{2}(I';\mathbb{M}^{3\times 3})$ to some $\overline{R}\in L^{2}(I';\mathbb{M}^{3\times 3})$, with $\overline{R}(s)\in SO(3)$ for a.e. $s\in I'$. From (\[Fconv\]) and (\[rigid2\]) it follows that $F = \overline{R}$ a.e.. Moreover (\[convdet\]) and (\[bound2\]) imply the convergence of the $L^{2}$ norm of $F^{(h)}$ to the $L^{2}$ norm of $\overline{R}$, hence $$F^{(h)} \rightarrow \overline{R} \quad \mbox{strongly
in} \,L^{2}(\Omega;\mathbb{M}^{3\times 3}).$$ This proves (\[teo3\]), once the regularity of the function $\overline{R}$ is shown. To this aim, divide both sides of the inequality (\[increm\]) by $({\left|\delta\right|} + h)^{2}$ and let $h\rightarrow 0$; then $$\label{regu}
\int_{I'}\frac{{\left|\overline{R}(s + \delta) - \overline{R}(s)\right|}^{2}}{{\left|\delta\right|}^{2}}\,ds \leq c$$ and so $\overline{R} \in W^{1,2}(I';\mathbb{M}^{3\times 3})$. But this holds for every $I'\subset\subset [0, L]$ with a constant independent of the subset $I'$, hence $\overline{R} \in
W^{1,2}((0, L);\mathbb{M}^{3\times 3})$. Now notice that $$\label{successione}
\nabla_{h}\big(\tilde{v}^{(h)}\circ\Psi^{(h)}\big)
= \big(\nabla\tilde{v}^{(h)}\circ\Psi^{(h)}\big)
\nabla_{h}\Psi^{(h)}=
F^{(h)}\nabla_{h}\Psi^{(h)};$$ by (\[convdet\]) and (\[teo3\]) we deduce that $$\label{3.16bis}
\nabla_{h}\big(\tilde{v}^{(h)}\circ\Psi^{(h)}\big)\longrightarrow \overline{R}\,R_{0} \quad
\mbox{strongly in}\,\, L^{2}(\Omega;\mathbb{M}^{3\times 3}).$$ In particular, we have $$\label{3.16ter}
\nabla\big(\tilde{v}^{(h)}\circ\Psi^{(h)}\big)\longrightarrow
\big(\overline{R}\,R_{0}e_1\big)\otimes e_1 \quad
\mbox{strongly in}\,\, L^{2}(\Omega;\mathbb{M}^{3\times 3}).$$ By Poincaré inequality there exist some constants $c^{(h)}\in \mathbb{R}^3$ and a function $v$ in $W^{1,2}(\Omega;\mathbb{R}^{3})$ such that (\[teo1\]) is satisfied. Moreover (\[3.16ter\]) entails that the function $v$ depends only on the variable $s$ in $[0, L]$ and satisfies $v' = \overline{R}\,R_{0}e_1$. Setting $d_{k}:= \overline{R}\,R_{0}e_k$ for $k = 2,3$, we have that $(v, d_2, d_3)\in \mathcal{A}$ and (\[teo2\]), (\[teo25\]) are satisfied by (\[3.16bis\]).
Bound from below
----------------
Let $Q_{3}: \Omega\times\mathbb{M}^{3\times 3}\longrightarrow [0, +\infty)$ be twice the quadratic form of linearized elasticity; i.e., $$Q_{3}(z,G) := \frac{\partial^2 W}{\partial F^2}(z,Id)[G,G]$$ for a.e. $z\in \Omega$ and every $G\in \mathbb{M}^{3\times 3}$. We introduce the quadratic form $Q_{2}: [0, L]\times\mathbb{M}^{3\times 3}_{\textrm{skew}}\rightarrow [0, +\infty)$ defined by $$\label{Q2}
Q_{2}(s,P):= \hspace{-0.4cm}\inf_{\begin{array}{c}
\vspace{-.15cm}
\scriptstyle\hat{\alpha}\in W^{1,2}(D;\mathbb{R}^{3})\\
\scriptstyle\hat{g}\in \mathbb{R}^{3}
\end{array}}\Bigg\{\int_{D}Q_{3}\bigg(s,\xi,\zeta,R_{0}(s)\Bigg(P\,
\Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + \hat{g}\,\bigg|\,\partial_{\xi}\hat{\alpha}\,\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)
R_{0}^{T}(s)\bigg)d\xi\,d\zeta\Bigg\}.$$
\[remark\] It is easy to check that the minimum in (\[Q2\]) is attained; moreover the minimizers depend linearly on $P$, hence $Q_2$ is a quadratic form of $P$. Notice also that if $P\in L^2 ((0,L);\mathbb{M}^{3\times 3})$, then $\hat{\alpha}\in L^2 (\Omega;\mathbb{R}^{3})$ with $\partial_{\xi}\hat{\alpha}, \partial_{\zeta}\hat{\alpha} \in L^2 (\Omega;\mathbb{R}^{3})$, and $\hat{g} \in L^2 ((0,L);\mathbb{R}^{3})$ (see [@MGMM04 Remarks 4.1 - 4.3]).
In the following theorem we prove a lower bound for the energies $\tilde{I}^{(h)}/h^{2}$ in terms of the functional $$\label{funI}
I(v,d_2,d_3):=
\left\{
\vspace{.7cm}
\begin{array}{ll}
\displaystyle\frac{1}{2}\int_{0}^{L} Q_{2} \big(s,\big(R^{T}(s)R'(s) -
R_{0}^{T}(s)R_{0}'(s)\big)\big)ds & \mbox{if } \, (v,d_2,d_3)\in \mathcal{A},\\
\displaystyle + \infty & \mbox{otherwise },
\end{array}
\right.$$ where $R\in W^{1,2}((0, L);\mathbb{M}^{3\times 3})$ denotes the matrix $R:= (v'\,|\,d_2\,|\,d_3)$ and $\mathcal{A}$ is the class defined in (\[defA\]).
\[scithm\] Let $v\in W^{1,2}(\Omega;\mathbb{R}^{3})$ and let $d_2, d_3 \in L^{2}(\Omega;\mathbb{R}^{3})$. Then, for every positive sequence $(h_j)$ converging to zero and every sequence $\big(\tilde{v}^{(h_j)}\big)\subset
W^{1,2}(\widetilde{\Omega}_{h_j};\mathbb{R}^{3})$ such that $$\label{teo5}
\tilde{v}^{(h_j)}\circ \Psi^{(h_j)} \rightarrow v \quad \mbox{strongly in}\,\,
W^{1,2}(\Omega;\mathbb{R}^{3}),$$ $$\label{teo65}
\frac{1}{h_j}\,\partial_\xi\big(\tilde{v}^{(h_j)}\circ
\Psi^{(h_j)}\big)\,\rightarrow d_2\quad
\mbox{strongly in}\,\, L^{2}(\Omega;\mathbb{R}^{3}),$$ $$\label{teo6}
\frac{1}{h_j}\,\partial_\zeta\big(\tilde{v}^{(h_j)}\circ
\Psi^{(h_j)}\big)\,\rightarrow d_3\quad
\mbox{strongly in}\,\, L^{2}(\Omega;\mathbb{R}^{3}),$$ it turns out that $$\label{cl}
I(v,d_2,d_3)\leq\liminf_{j \rightarrow \infty}
\frac{1}{h_{j}^{4}} \int_{\widetilde{\Omega}_{h_j}}
W\big(\big(\Psi^{(h_j)}\big)^{-1} (x), \nabla
\tilde{v}^{(h_j)}(x)\big) dx.$$
In the following, for notational brevity, we will write simply $h$ instead of $h_j$. Let $\big(\tilde{v}^{(h)}\big)$ be a sequence satisfying (\[teo5\]), (\[teo65\]) and (\[teo6\]). We can assume that $$\liminf_{j \rightarrow \infty}
\frac{1}{h_{j}^{4}} \int_{\widetilde{\Omega}_{h_j}}
W\big(\big(\Psi^{(h_j)}\big)^{-1} (x), \nabla
\tilde{v}^{(h_j)}(x)\big) dx \leq C < + \infty,$$ otherwise (\[cl\]) is trivial. Therefore, up to subsequences, (\[finite\]) is satisfied. By Theorem \[compactness\] we deduce that $(v, d_2, d_3)\in \mathcal{A}$, $$\label{rel1}
F^{(h)}:= \nabla \tilde{v}^{(h)}\circ \Psi^{(h)} \longrightarrow \overline{R} \quad \mbox{strongly in } \,L^{2}(\Omega;\mathbb{M}^{3\times 3})$$ with $\overline{R} \in W^{1,2}((0, L);\mathbb{M}^{3\times 3})$, $\overline{R}\in SO(3)$ a.e., and $$\label{rel2}
R:= (v'\,|\,d_2\,|\,d_3) = \overline{R}\,R_{0}.$$ Moreover, as in the proof of Theorem \[compactness\], we can construct a piecewise constant approximation $R^{(h)}: [0, L]\rightarrow SO(3)$ such that $$\label{rot}
\int_{\Omega} \big|F^{(h)} - R^{(h)}\big|^{2} ds\,d\xi\,d\zeta \leq c\,h^{2}$$ and $R^{(h)} \rightarrow \overline{R}$ strongly in $L^{2}(I';\mathbb{M}^{3})$ for every $I'\subset\subset [0, L]$. Define the functions $G^{(h)}: \Omega\rightarrow \mathbb{M}^{3\times 3}$ as $$G^{(h)}:= \frac{1}{h}\Big((R^{(h)})^{T} F^{(h)} - Id\Big) =
\frac{1}{h}\Big((R^{(h)})^{T} \nabla_{h}v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1} - Id\Big).$$ By (\[rot\]) they are bounded in $L^{2}(\Omega;\mathbb{M}^{3\times 3})$, so there exists $G\in L^{2}(\Omega;\mathbb{M}^{3\times 3})$ such that $G^{(h)}\rightharpoonup G$ weakly in $L^{2}(\Omega;\mathbb{M}^{3\times 3})$. We claim that $$\label{liminf}
\liminf_{h\rightarrow 0}\frac{1}{h^4}\int_{\widetilde{\Omega}_{h}} W\Big(\big(\Psi^{(h)}\big)^{-1} (x),\nabla \tilde{v}^{(h)}(x)\Big) dx \geq \frac{1}{2} \int_{\Omega} Q_{3}(s,\xi,\zeta,G) ds\,d\xi\,d\zeta.$$ Performing the change of variables $\Psi^{(h)}$, we have $$\begin{aligned}
\label{newvar}
\frac{1}{h^4}\int_{\widetilde{\Omega}_{h}} W\Big(\big(\Psi^{(h)}\big)^{-1} (x),\nabla \tilde{v}^{(h)}(x)\Big) dx & = \frac{1}{h^2}\int_{\Omega} W\big(s,\xi,\zeta, F^{(h)}\big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \nonumber\\
& = \frac{1}{h^2}\int_{\Omega} W\big(s,\xi,\zeta, \big(R^{(h)}\big)^T F^{(h)}\big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta\end{aligned}$$ where the last equality follows from the frame indifference of $W$. Define the family of functions $$\chi^{(h)}(s,\xi,\zeta):=
\left\{
\begin{array}{ll}
\vspace{.1cm}
\displaystyle 1 & \mbox{in } \,\, \Omega\cap\big\{(s,\xi,\zeta): {\left|G^{(h)}(s,\xi,\zeta)\right|} \leq h^{-\frac{1}{2}}\big\},\\
\displaystyle 0 & \mbox{otherwise}.
\end{array}
\right.$$ From the boundedness of $G^{(h)}$ in $L^2(\Omega;\mathbb{M}^{3\times 3})$ we get that $\chi^{(h)}\rightarrow 1$ boundedly in measure, so that $$\label{convchi}
\chi^{(h)}G^{(h)} \rightharpoonup G \quad \mbox{weakly in}\, L^{2}(\Omega;\mathbb{M}^{3\times 3}).$$ By expanding $W$ around the identity, we obtain that for every $(s,\xi,\zeta) \in \Omega$ and $A\in \mathbb{M}^{3\times 3}$ $$W\big(s,\xi,\zeta, Id + A) = \frac{1}{2}\,\frac{\partial^{2}W}{\partial F^{2}}\,(s,\xi,\zeta, Id + t\,A)[A,A]$$ where $0<t<1$ depends on the point $(s,\xi,\zeta)$ and on $A$. By (\[newvar\]) and by the definition of $G^{(h)}$ $$\begin{aligned}
\frac{1}{h^2}\,\tilde{I}^{(h)}\big(\tilde{v}^{(h)}\big) &=
\frac{1}{h^2}\int_{\Omega} W\big(s,\xi,\zeta, Id + h\,G^{(h)} \big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \nonumber\\
&\geq \frac{1}{h^2}\int_{\Omega} \chi^{(h)} W\big(s,\xi,\zeta,Id + h\,G^{(h)}\big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \nonumber\\
&= \frac{1}{2}\int_{\Omega} \chi^{(h)}\left(\frac{\partial^{2}W}{\partial F^{2}}\,\big(s,\xi,\zeta, Id + h\,t(h)\,G^{(h)}\big)\big[G^{(h)},G^{(h)}\big]\right)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta\label{3.22ter}\end{aligned}$$ where $0 < t(h) < 1$ depends on $(s,\xi,\zeta)$ and on $G^{(h)}$. The last integral in the previous formula can be written as $$\begin{aligned}
&\frac{1}{2}\int_{\Omega} \chi^{(h)}\left(\frac{\partial^{2}W}{\partial F^{2}}\,\big(s,\xi,\zeta, Id + h\,t(h)\,G^{(h)}\big)\big[G^{(h)},G^{(h)}\big]\right)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \nonumber\\
& = \frac{1}{2}\int_{\Omega}\Big( \chi^{(h)}\bigg(\frac{\partial^{2}W}{\partial F^{2}}\,\big(s,\xi,\zeta, Id + h\,t(h)\,G^{(h)}\big)\big[G^{(h)},G^{(h)}\big] - Q_{3}\big(s,\xi,\zeta, G^{(h)}\big)\Big)\Big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \nonumber\\
&\hspace{1cm}+ \frac{1}{2}\int_{\Omega}Q_{3}\big(s,\xi,\zeta,\chi^{(h)}\,G^{(h)}\big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta. \label{3.22quater}\end{aligned}$$ By Scorza-Dragoni theorem there exists a compact set $K\subset \Omega$ such that the function $\partial^{2}W/\partial F^{2}$ restricted to $K\times \overline{B_{\delta}(Id)}$ is continuous, hence uniformly continuous. Since $h\,t(h)\,\chi^{(h)}\,G^{(h)}$ is uniformly small for $h$ small enough, for every $\varepsilon > 0$ we have $$\begin{aligned}
\frac{1}{2}\int_{\Omega}&\chi^{(h)}\Bigg(\frac{\partial^{2}W}{\partial F^{2}}\,\big(s,\xi,\zeta, Id + h\,t(h)\,G^{(h)}\big)\big[G^{(h)},G^{(h)}\big] - Q_{3}\big(s,\xi,\zeta,G^{(h)}\big)\Bigg)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \\
&\geq - \frac{\varepsilon}{2}\int_{K}\chi^{(h)}\big|\,G^{(h)}\big|^{2}\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta
\geq -\,C\,\varepsilon\end{aligned}$$ for $h$ small enough. As for the second integral in (\[3.22quater\]), by (\[convdet\]) and (\[newvar\]) we get $$\label{3.22quinq}
\liminf_{h\rightarrow 0}\frac{1}{2}\int_{\Omega}Q_{3}\big(s,\xi,\zeta,\chi^{(h)}\,G^{(h)}\big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta \geq \frac{1}{2}\int_{\Omega}Q_{3}\big(s,\xi,\zeta,G\big)ds\,d\xi\,d\zeta$$ since $Q_{3}$ is a nonnegative quadratic form. Combining (\[3.22ter\]), (\[3.22quater\]) and (\[3.22quinq\]) we have $$\liminf_{h\rightarrow 0}\frac{1}{h^2}\,\tilde{I}^{(h)}(\tilde{v}^{(h)})
\geq \frac{1}{2}\int_{\Omega}Q_{3}(s,\xi,\zeta,G) ds\,d\xi\,d\zeta \,-\,C\,\varepsilon$$ and, since $\varepsilon$ is arbitrary, (\[liminf\]) is proved. It remains to identify $G$.
Fix $(\xi_0,\zeta_0)\in D$; let $\delta_0 = \delta_{0}(\xi_0,\zeta_0) > 0$ be such that $B_{2\,\delta_0}(\xi_0,\zeta_0)\subset D$ and let $U_0:= (0, L)\times B_{\delta_0}(\xi_0,\zeta_0)$. Fix $t\in \mathbb{R}-\{0\}$, $|\,t\,|\,<\,\delta_0$. For every $(s,\xi,\zeta)\in U_0$ we can define the difference quotients of the functions $G^{(h)}$ with respect to the variables $\xi$ and $\zeta$ along the direction $\tau$, given by $$\left\{
\begin{array}{ll}
H^{(h)}_{t}(s,\xi,\zeta):=& \dfrac{1}{t}\,\Big(G^{(h)}(s,\xi + t,\zeta) - G^{(h)}(s,\xi,\zeta)\Big)\,\tau(s),\\
\mbox{ }\\
K^{(h)}_{t}(s,\xi,\zeta):=& \dfrac{1}{t}\,\Big(G^{(h)}(s,\xi,\zeta + t) - G^{(h)}(s,\xi,\zeta)\Big)\,\tau(s),
\end{array}
\right.$$ and the corresponding difference quotients of the limit function $G$ $$\left\{
\begin{array}{ll}
H_{t}(s,\xi,\zeta):=& \dfrac{1}{t}\,\Big(G(s,\xi + t,\zeta) - G(s,\xi,\zeta)\Big)\,\tau(s),\\
\mbox{ }\\
K_{t}(s,\xi,\zeta):=& \dfrac{1}{t}\,\Big(G(s,\xi,\zeta + t) - G(s,\xi,\zeta)\Big)\,\tau(s).
\end{array}
\right.$$ Since $G^{(h)}\rightharpoonup G$ in $L^{2}(\Omega;\mathbb{M}^{3\times 3})$ and $R^{(h)} \longrightarrow \overline{R}$ boundedly in measure, we have $$\begin{aligned}
\label{convH}
H^{(h)}_{t}&\rightharpoonup H_{t}
\quad \mbox{weakly in} \, L^{2}(U_{0};\mathbb{R}^{3}) \,\, \mbox{and} \nonumber\\
R^{(h)}\,H^{(h)}_{t} &\rightharpoonup \overline{R}\,H_{t}
\quad \mbox{weakly in} \, L^{2}(U_{0};\mathbb{R}^{3}). \end{aligned}$$ In terms of $F^{(h)}$ the left-hand side of (\[convH\]) reads as $$\label{riscritta}
R^{(h)}(s) H^{(h)}_{t}(s,\xi,\zeta) =
\frac{1}{h\,t}\,\Big(F^{(h)}(s,\xi + t,\zeta) - F^{(h)}(s,\xi,\zeta)\Big)\,\tau(s).$$ Now recall that, if we set $ v^{(h)}:= \tilde{v}^{(h)}\circ \Psi^{(h)}$, we have $$\label{VvsF}
\nabla v^{(h)} = F^{(h)} \, \nabla \Psi^{(h)};$$ in particular, taking the first column of the two matrices, we obtain $$F^{(h)}(s,\xi,\zeta)\,\tau(s) = \partial_{s}v^{(h)}(s,\xi,\zeta) - h\,F^{(h)}(s,\xi,\zeta)\,(\xi\,\nu'_{2}(s) + \zeta\,\nu'_{3}(s)).$$ By the last equality and (\[riscritta\]) we get $$\begin{aligned}
\label{lunga}
R^{(h)}(s) H^{(h)}_{t}(s,\xi,\zeta)
=& \,\frac{1}{h\,t}\,\Big(\partial_{s} v^{(h)}(s,\xi + t,\zeta) - \partial_{s} v^{(h)}(s,\xi,\zeta)\Big)\nonumber\\
-&\,\frac{1}{t}\Big((\xi + t)\,F^{(h)}(s,\xi + t,\zeta) - \xi\,F^{(h)}(s,\xi,\zeta)\Big)\,\nu_{2}'(s)\nonumber\\
-&\,\frac{1}{t}\Big(\zeta\,F^{(h)}(s,\xi + t,\zeta) - \zeta\,F^{(h)}(s,\xi,\zeta)\Big)\,\nu_{3}'(s).\end{aligned}$$ For the first term we have $$\begin{aligned}
\frac{1}{h\,t}\,\partial_{s}\Big(v^{(h)}(s,\xi + t,\zeta) - v^{(h)}(s,\xi,\zeta)\Big)
& = \frac{1}{h\,t}\,\partial_{s}\bigg(\int_{\xi}^{\xi + t}\partial_{\xi}v^{(h)}(s,\vartheta,\zeta)\, d\vartheta\bigg)\\
& = \partial_{s}\bigg(\frac{1}{t}\int_{0}^{t}\frac{1}{h}\,\partial_{\xi}v^{(h)}(s,\xi +\vartheta,\zeta)\,d\vartheta\bigg),\end{aligned}$$ so by (\[teo6\]) and (\[rel2\]) $$\label{RHS1}
\frac{1}{h\,t}\,\partial_{s}\Big(v^{(h)}(s,\xi + t,\zeta) - v^{(h)}(s,\xi,\zeta)\Big) \rightharpoonup d'_2 (s) = \partial_{s} (\overline{R}(s)\,\nu_2(s)) \quad\mbox{weakly in}\, W^{-1,2}(U_{0};\mathbb{R}^{3}).$$ By (\[rel1\]) the second term in (\[lunga\]) converges to $$\label{RHS2}
\frac{1}{t}\,\Big((\xi + t)\,\overline{R}(s) - \xi\,\overline{R}(s)\Big)\,\nu_{2}'(s) = \overline{R}(s)\,\nu_2'(s) \quad\mbox{strongly in}\, L^{2}(U_0;\mathbb{R}^{3})$$ and the last term to $$\label{RHS3}
\frac{1}{t}\,\Big(\zeta\,\overline{R}(s) - \zeta\,\overline{R}(s)\Big)\,\nu_{3}'(s) = 0 \quad\mbox{strongly in}\, L^{2}(U_0;\mathbb{R}^{3}).$$ Putting together (\[RHS1\]), (\[RHS2\]), (\[RHS3\]) and (\[convH\]) $$\overline{R}(s)\,H_{t}(s,\xi,\zeta) = \partial_{s} (\overline{R}(s)\,\nu_2(s)) - \overline{R}(s)\,\nu_2'(s) \, \,\mbox{a.e. in } \, U_0$$ and so $$\label{acca}
H_{t}(s,\xi,\zeta) = (\overline{R}(s))^{T}\,\overline{R}'(s)\,\nu_2(s) \,\,\mbox{a.e. in } \, U_0.$$ Repeating the same argument for $K^{(h)}_{t}$ we get $$\label{kappa}
K_{t}(s,\xi,\zeta) = (\overline{R}(s))^{T}\,\overline{R}'(s)\,\nu_3(s) \,\,\mbox{a.e. in } \, U_0.$$ From the last two equalities we deduce that the functions $H_{t}$ and $K_{t}$ depend only on the variable $s$. Moreover, letting $t$ go to $0$ both in (\[acca\]) and in (\[kappa\]), we get that the gradient of $G\,\tau$ w.r.to the variables $(\xi,\zeta)$ depends only on $s$, i.e., $$\label{gradgrad}
\nabla_{(\xi,\zeta)}\big( G(s,\xi,\zeta)\,\tau(s)\big) = (\overline{R}(s))^{T}\,\overline{R}'(s)\,(\nu_2(s)\,|\,\nu_3(s)) \,\mbox{a.e. in } \, U_0.$$ Being this equality valid in $U_0 = (0, L)\times B_{\delta_0}(\xi_0,\zeta_0)$, for an arbitrary $(\xi_0,\zeta_0)\in D$, we can conclude that it holds a.e. in the whole $\Omega$. Since $D$ is connected, we obtain that for a.e. $(s,\xi,\zeta) \in \Omega$ $$G(s,\xi,\zeta)\,\tau(s) = (\overline{R}(s))^{T}\,\overline{R}'(s)\,(\xi\,\nu_2(s) + \zeta\,\nu_3(s)) + g(s)$$ with $g: [0, L]\rightarrow \mathbb{R}^{3}$. Remark that from the previous formula $g\in L^{2}((0, L);\mathbb{R}^{3})$. It remains to identify the components $G(s,\xi,\zeta)\,\nu_{2}(s)$ and $G(s,\xi,\zeta)\,\nu_{3}(s)$. By (\[VvsF\]) we have $$\begin{aligned}
G^{(h)}(s,\xi,\zeta)\,\nu_{2}(s) &=& \frac{1}{h}\,\Big((R^{(h)}(s))^{T} F^{(h)}(s,\xi,\zeta)\,\nu_{2}(s) - \nu_{2}(s)\Big)\\
&=& \frac{1}{h}\,\Big(h^{-1}(R^{(h)}(s))^{T}\partial_{\xi}v^{(h)}(s,\xi,\zeta) - \nu_{2}(s)\Big)\end{aligned}$$ and $$\begin{aligned}
G^{(h)}(s,\xi,\zeta)\,\nu_{3}(s) &=& \frac{1}{h}\,\Big((R^{(h)}(s))^{T} F^{(h)}(s,\xi,\zeta)\,\nu_{3}(s) - \nu_{3}(s)\Big)\\
&=& \frac{1}{h}\,\Big(h^{-1}(R^{(h)}(s))^{T}\partial_{\zeta}v^{(h)}(s,\xi,\zeta) - \nu_{3}(s)\Big),\end{aligned}$$ so, if we define $$\alpha^{(h)}(s,\xi,\zeta):= \frac{1}{h}\,\Big(h^{-1}(R^{(h)})^{T} v^{(h)}(s,\xi,\zeta) - \xi\,\nu_{2}(s) - \zeta\,\nu_{3}(s)\Big)$$ it turns out that $$\label{alfa}
\partial_{\xi}\alpha^{(h)}(s,\xi,\zeta) = G^{(h)}(s,\xi,\zeta)\,\nu_{2}(s)\quad\mbox{and}\quad
\partial_{\zeta}\alpha^{(h)}(s,\xi,\zeta) = G^{(h)}(s,\xi,\zeta)\,\nu_{3}(s).$$ Applying the Poincaré inequality to the functions $\alpha^{(h)}$ for fixed $s$ we obtain that for a.e. $s\in [0, L]$ $$\int_{D}\big|\,\alpha^{(h)}(s,\xi,\zeta) - \alpha_{0}^{(h)}(s)\,\big|^{2}\,d\xi\,d\zeta \leq c\int_{D}\left(\big|\,\partial_{\xi}\alpha^{(h)}(s,\xi,\zeta)\big|^{2} + \big|\,\partial_{\zeta}\alpha^{(h)}(s,\xi,\zeta)\big|^{2}\right)\,d\xi\,d\zeta,$$ where $\alpha_{0}^{(h)}(s):= \int_{D}\alpha^{(h)}(s,\xi,\zeta)\,d\xi\,d\zeta$. Integrating over $[0, L]$, we have $$\big|\big|\alpha^{(h)} - \alpha_{0}^{(h)}\big|\big|^{2}_{L^{2}(\Omega)} \leq c\left(\big|\big|\partial_{\xi}\alpha^{(h)}\big|\big|^{2}_{L^{2}(\Omega)} +\big|\big|\partial_{\zeta}\alpha^{(h)}\big|\big|^{2}_{L^{2}(\Omega)}\right).$$ Since the right-hand side is bounded by (\[alfa\]), there exists a function $\alpha\in L^{2}(\Omega; \mathbb{R}^{3})$ such that, up to subsequences, $$\alpha^{(h)} - \alpha_{0}^{(h)}\rightharpoonup \alpha \quad \mbox{weakly in} \,L^{2}(\Omega; \mathbb{R}^{3}).$$ Moreover, from (\[alfa\]) we conclude that $$\label{alfa1}
\partial_{\xi}\alpha(s,\xi,\zeta) = G(s,\xi,\zeta)\,\nu_{2}(s)\quad\mbox{and}\quad
\partial_{\zeta}\alpha(s,\xi,\zeta) = G(s,\xi,\zeta)\,\nu_{3}(s),$$ therefore $\partial_{\xi}\alpha,\partial_{\zeta}\alpha \in L^{2}(\Omega; \mathbb{R}^{3})$. Now, define the functions $\hat{\alpha}(s,\xi,\zeta):= R_{0}^{T}(s)\,\alpha(s,\xi,\zeta)$ and $\hat{g}(s):= R_{0}^{T}(s)\,g(s)$. Thanks to these definitions and to (\[rel2\]), $G$ can be written as $$\begin{aligned}
\label{tildeG}
G =& \Bigg(\Bigl(R\,R_{0}^{T}\Bigr)^{T} \Bigl(R\,R_{0}^{T}\Bigr)' R_{0}\Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + g\,\bigg|\,\partial_{\xi}\alpha\,
\bigg|\,\partial_{\zeta}\alpha\Bigg)\,R^{T}_{0}\nonumber\\
=&\, R_{0}\Bigg(\Bigl( R^{T}R' + (R_{0}^{T})'R_{0}\Bigr)\Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + \hat{g} \,\bigg|\,\partial_{\xi}\hat{\alpha}\,
\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)\,R_{0}^{T} \nonumber\\
=&\, R_{0}\Bigg(\Bigl( R^{T}R' - R_{0}^{T}R_{0}'\Bigr)\Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + \hat{g}\,\bigg|\,\partial_{\xi}\hat{\alpha}\,
\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)R_{0}^{T},\end{aligned}$$ where the last equality follows from the identity $\big(R_{0}^{T}\big)'R_{0} + R_{0}^{T}R_{0}' = 0$. Combining (\[liminf\]) and (\[tildeG\]), we obtain $$\liminf_{h\rightarrow 0}\frac{1}{h^2}\,\tilde{I}^{(h)}(\tilde{v}^{(h)})
\geq\frac{1}{2}\int_{\Omega} Q_{3}\bigg(s,\xi,\zeta,R_{0}(s)\Bigg(P(s)\Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + \hat{g}\,\bigg|\,\partial_{\xi}\hat{\alpha}\,
\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)R_{0}^{T}(s)\bigg)ds\,d\xi\,d\zeta,$$ with $P(s):= R^{T}(s)R'(s) - R_{0}^{T}(s)R_{0}'(s)$. By the definition of the quadratic form $Q_2$ in (\[Q2\]) we clearly have $\int_{D} Q_{3}(s,\xi,\zeta,G)d\xi\,d\zeta \geq Q_{2}(s,P(s))$, and so $$\liminf_{h\rightarrow 0} \frac{1}{h^4}
\int_{\widetilde{\Omega}_{h}} W\big(\big(\Psi^{(h)}\big)^{-1} (x),
\nabla \tilde{v}^{(h)}(x)\big) dx \geq \frac{1}{2}\int_{0}^{L} Q_{2}
\big(s,\big(R^{T}(s)R'(s) - R_{0}^{T}(s)R_{0}'(s)\big)\big)ds.$$
Bound from above
----------------
In this subsection we show that the lower bound proved in Theorem \[scithm\] is optimal.
\[bfa\] For every sequence of positive $(h_j)$ converging to $0$ and for every $(v,d_2,d_3)\in \mathcal{A}$ there exists a sequence $\big(\tilde{v}^{(h_j)}\big) \subset W^{1,2}\big(\widetilde{\Omega}_{h_j}; \mathbb{R}^{3}\big)$ such that $$\begin{aligned}
\tilde{v}^{(h_j)}\circ\Psi^{(h_j)} &\rightarrow v \quad \mbox{strongly in}\,\,
W^{1,2}(\Omega; \mathbb{R}^{3}),\label{star}\\
\frac{1}{h_j}\,\partial_\xi \big(\tilde{v}^{(h_j)}\circ\Psi^{(h_j)} \big)\,&\rightarrow d_2\quad
\mbox{strongly in}\,\, L^{2}(\Omega; \mathbb{R}^{3}), \label{starr1}\\
\frac{1}{h_j}\,\partial_\zeta \big(\tilde{v}^{(h_j)}\circ\Psi^{(h_j)}\big)\,&\rightarrow d_3\quad
\mbox{strongly in}\,\, L^{2}(\Omega; \mathbb{R}^{3}), \label{star2}\end{aligned}$$ and $$\label{starec}
I(v,d_2,d_3) = \lim_{j \rightarrow \infty}
\frac{1}{h_{j}^4}\int_{\widetilde{\Omega}_{h_j}} W\big(\big(\Psi^{(h_j)}\big)^{-1} (x),
\nabla \tilde{v}^{(h_j)}(x)\big) dx,$$ where the class $\mathcal{A}$ and the functional $I$ are defined in (\[defA\]) and (\[funI\]), respectively.
Let $(v,d_{2},d_{3})\in \mathcal{A}$. Assume in addition that $v\in C^{2}([0,L];\mathbb{R}^{3})$and $d_{2},d_{3}\in C^{1}([0,L];\mathbb{R}^{3})$. Consider the functions $v^{(h)}: \Omega\rightarrow \mathbb{R}^{3}$ defined by $$v^{(h)}(s,\xi,\zeta):= v(s) + h\,\xi\,d_{2}(s) + h\,\zeta\,d_{3}(s) + h\,q(s) + h^{2}\,\beta(s,\xi,\zeta),$$ with $q\in C^{1}([0,L];\mathbb{R}^{3})$ and $\beta\in C^{1}(\overline{\Omega};\mathbb{R}^{3})$. We define $\tilde{v}^{(h)} := v^{(h)}\circ\big(\Psi^{(h)}\big)^{-1}$; these functions clearly satisfy (\[star\]). Moreover, since $$\label{starec2}
\nabla_{h} \big(\tilde{v}^{(h)}\circ\Psi^{(h)}\big) =
\nabla_{h} v^{(h)} = (v'\,|\,d_{2}\,|\,d_{3}) + h\,\big(\xi\,d'_2 + \zeta\,d'_3 + q'\,|\,\partial_\xi\beta\,|\,\partial_\zeta\beta\big) + h^2\partial_s \beta\otimes e_1,$$ also (\[starr1\]) and (\[star2\]) follow easily. In order to prove (\[starec\]), we first observe that, performing the change of variables $(s,\xi,\zeta) = \big(\Psi^{(h)}\big)^{-1}(x)$, we obtain $$\begin{aligned}
\frac{1}{h^2}\,\tilde{I}^{(h)}\big(\tilde{v}^{(h)}\big) =&
\frac{1}{h^2}\,\int_{\Omega} W\big(s,\xi,\zeta,\nabla\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)\det
\big(\nabla_{h}\Psi^{(h)}\big)
ds\,d\xi\,d\zeta \nonumber\\
=& \frac{1}{h^2}\,\int_{\Omega} W\big(s,\xi,\zeta,\nabla_h \big(\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)\,\big(\nabla_h\Psi^{(h)}\big)^{-1}\big)\det
\big(\nabla_{h}\Psi^{(h)}\big)
ds\,d\xi\,d\zeta,\label{flower2}\end{aligned}$$ where the last equality is justified observing that $$\nabla_h\big(\tilde{v}^{(h)}\circ
\Psi^{(h)}\big) = \big(\nabla\tilde{v}^{(h)}\circ
\Psi^{(h)}\big)\,\big(\nabla_h\Psi^{(h)}\big).$$ Then, by the definition of $\tilde{v}^{(h)}$, $$\label{flower}
\frac{1}{h^2}\,\tilde{I}^{(h)}\big(\tilde{v}^{(h)}\big) = \frac{1}{h^2}\,\int_{\Omega} W\big(s,\xi,\zeta,\big(\nabla_h v^{(h)}\big)\,\big(\nabla_h\Psi^{(h)}\big)^{-1}\big)\det
\big(\nabla_{h}\Psi^{(h)}\big)
ds\,d\xi\,d\zeta.$$ Using (\[invA\]) and (\[starec2\]) we get $$\begin{aligned}
\nabla_{h} v^{(h)}\,\big(\nabla_{h}\Psi^{(h)}\big)^{-1} =&\,
R\, R_{0}^{T} + h\,(\xi\,d\,'_{2} + \zeta\,d\,'_{3} + q'\,|\,\partial_{\xi}\beta\,|\,\partial_{\zeta}\beta)\,R_{0}^{T} \\
-& \,h\, R\,R_{0}^{T}\big[(\xi\,\nu'_{2} + \zeta\,\nu'_{3})\otimes e_1\big]\,R_{0}^{T} + O(h^{2}),\end{aligned}$$ where $R = (v'|d_2|d_3)$ and $O(h^{2})/h^2$ is uniformly bounded. Now consider the rotation $\overline{R}(s) = R(s)\, R_{0}^{T}(s)$. Then $$\overline{R}^{T}\nabla_{h} v^{(h)}\,\big(\nabla_{h}\Psi^{(h)}\big)^{-1} =\,
Id + h\,\overline{R}^{T}(\xi\,d\,'_{2} +\,\zeta\,d\,'_{3} + q'\,|\,\partial_{\xi}\beta\,|\,\partial_{\zeta}\beta)\,R_{0}^{T} - \,h \,\big[(\xi\,\nu'_{2} + \zeta\,\nu'_{3})\otimes e_1\big]\,R_{0}^{T} + O(h^{2}).$$ If we define the functions $$B^{(h)}(s,\xi,\zeta):= \,
\frac{1}{h}\bigg(\overline{R}^{T}\, \nabla_{h}v^{(h)}\,\big(\nabla_{h}\Psi^{(h)}\big)^{-1} - Id \bigg),$$ it turns out that $$\begin{aligned}
B^{(h)} &=\, (R_{0}\, R^{T})(\xi\,d\,'_{2} +\zeta\,d\,'_{3} + q'\,|\,\partial_{\xi}\beta\,|\,\partial_{\zeta}\beta)R_{0}^{T}
- \big[(\xi\,\nu'_{2} + \zeta\,\nu'_{3})\otimes e_1\big]\,R_{0}^{T} + O(h)\nonumber\\
&=\,
R_{0}\, R^{T}\Bigg(R'\Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + q' \,\bigg|\,
\partial_{\xi}\beta\,\bigg|\,\partial_{\zeta}\beta\Bigg)R_{0}^{T} - \Bigg[
\Bigg(R_{0}'\Bigg(
\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}
\Bigg)\Bigg)\otimes e_1\Bigg]\,R_{0}^{T} + O(h)\nonumber\\
&=\,
R_{0}\Bigg(\Big(R^{T} R' - R_{0}^{T} R_{0}'\Big)\Biggl(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Biggr) + R^{T}q\,\bigg|\,R^{T}
\partial_{\xi}\beta\,\bigg|\,R^{T}\partial_{\zeta}\beta\Bigg)R_{0}^{T} + O(h)\nonumber\\
&=:\, G_{q,\beta} + O(h)\label{rombo}\end{aligned}$$ where $O(h)/h$ is uniformly bounded. By frame indifference and the definition of $B^{(h)}$, we have $$\begin{aligned}
\frac{1}{h^{2}}\,W\big(s,\xi,\zeta,\nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}) &= \, \frac{1}{h^{2}}\,W\big(s,\xi,\zeta,\overline{R}^{T} \nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}) \\
&= \,\frac{1}{h^{2}}\,W\big(s,\xi,\zeta,Id + h\,B^{(h)}\big).\end{aligned}$$ Using (\[rombo\]) and the expansion of $W$ around the identity, we obtain $$\frac{1}{h^{2}}\,W\big(s,\xi,\zeta, \nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1})\rightarrow \frac{1}{2}\,Q_{3}(s,\xi,\zeta,G_{q,\beta}) \quad\mbox{a.e.}.$$ Moreover, the assumption (ii) gives the uniform bound $$\frac{1}{h^{2}}\,W\big(s,\xi,\zeta, \nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}) \leq
\frac{1}{2}\,C_{1}\,|\,G_{q,\beta}\,|^{2} + C \in\, L^{1}(\Omega),$$ so, by the dominated convergence theorem and by (\[flower\]) we conclude that $$\label{gammasup}
\lim_{h\rightarrow 0}\frac{1}{h^4}\int_{\widetilde{\Omega}_{h}} W\Big(\big(\Psi^{(h)}\big)^{-1} (x),\nabla \tilde{v}^{(h)}(x)\Big) dx
= \frac{1}{2}\int_{\Omega} Q_{3}(s,\xi,\zeta,G_{q,\beta})\,ds\,d\xi\,d\zeta .$$ This holds for every $q\in C^{1}([0,L];\mathbb{R}^{3})$ and for every $\beta\in C^{1}(\overline{\Omega};\mathbb{R}^{3})$.
Consider now the general case. Let $(v,d_{2},d_{3})\in\mathcal{A}$, and let $\hat{\alpha}(s,\cdot)\in W^{1,2}(D;\mathbb{R}^{3})$, $\hat{g}(s)$ be a solution to the minimum problem (\[Q2\]) for $P = R^T R' - R_0^T R'_0$. By Remark \[remark\], $\hat{\alpha}\in L^{2}(\Omega;\mathbb{R}^{3})$ with $\partial_{\xi}\hat{\alpha}, \partial_{\zeta}\hat{\alpha}\in L^{2}(\Omega;\mathbb{R}^{3})$ and $\hat{g}\in L^{2}((0,L);\mathbb{R}^{3})$. In order to conclude the proof it is enough to construct a sequence of smooth deformations converging to $(v,d_{2},d_{3})$, on which the energy $\tilde{I}^{(h)}/h^2$ converges to the right-hand side of (\[gammasup\]) with $q$ and $\beta$ replaced by $R^T \hat{g}$ and $R^T \hat{\alpha}$, respectively. This can be done by repeating the same construction as in [@MM03].
If the rod is made of a homogeneous material, i.e., $W(z,F) = W(F)$, for a.e. $z$ in $\Omega$ and every $F\in \mathbb{M}^{3\times 3}$, then the limiting energy density $Q_2$ is given by the simpler formula $$\label{(a)}
Q_{2}(s,P) = \inf_{\hat{\alpha}\in W^{1,2}(D;\mathbb{R}^{3})}\Bigg\{\int_{D} Q_{3}\bigg(R_{0}(s)\Bigg(P\, \Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg)\,\bigg|\,\partial_{\xi}\hat{\alpha}\,\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)
R_{0}^{T}(s)\bigg)d\xi\,d\zeta\Bigg\}.$$ In other words the optimal choice for $\hat{g}$ in (\[Q2\]) is $\hat{g} = 0$. In order to show this, let $\hat{\alpha}\in W^{1,2}(D;\mathbb{R}^{3})$ and let $\hat{g}\in\mathbb{R}^3$. We introduce the function $$\label{alfetta}
\tilde{\alpha}(s,\xi,\zeta):= \hat{\alpha}(s,\xi,\zeta) - \xi \int_{D}\partial_{\xi}\hat{\alpha}\, d\xi\,d\zeta - \zeta \int_{D}\partial_{\zeta}\hat{\alpha}\, d\xi\,d\zeta.$$ Then, $$\begin{aligned}
R_{0}\,\Bigg(P\, \Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg) + \hat{g}\,\bigg|\,\partial_{\xi}\hat{\alpha}\,\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)\,
R_{0}^{T}
=&\, R_{0}\,\Bigg(P\, \Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg)\,\bigg|\,\partial_{\xi}\tilde{\alpha}\,\bigg|\,\partial_{\zeta}\tilde{\alpha}\Bigg)\,
R_{0}^{T}\\ +& \,R_0\,\left(\hat{g}\,\Big|\,\int_{D}\partial_{\xi}\hat{\alpha}\, d\xi\,d\zeta\,\bigg|\,\int_{D}\partial_{\zeta}\hat{\alpha}\, d\xi\,d\zeta\right)\,R_{0}^{T}\\
=:&\,\, \tilde{G} + Z.\end{aligned}$$ By expanding the quadratic form $Q_{3}$, we have $$\label{Q3}
\int_{D} Q_{3}(G)d\xi\,d\zeta = \int_{D} Q_{3}(\tilde{G})d\xi\,d\zeta + \int_{D} Q_{3}(Z)d\xi\,d\zeta
\geq \int_{D} Q_{3}(\tilde{G})d\xi\,d\zeta,$$ where we used (\[dom2\]), the fact that $\partial_{\xi}\tilde{\alpha}$ and $\partial_{\zeta}\tilde{\alpha}$ have zero average on $D$ and the non negativity of $Q_{3}$. From this inequality the thesis follows immediately.\
Notice that, due to the nontrivial geometry of the body, the limit energy depends on the position over the curve $\gamma$ even for a homogeneous material.
Assume the density $W$ is homogeneous and isotropic, that is, $$W(F) = W(FR) \quad\mbox{for every} \,\, R\in SO(3).$$ Then the quadratic form $Q_{3}$ is given by $$Q_{3}(G) = 2\,\mu\,\bigg|\frac{G + G^{T}}{2}\bigg|^{2} + \lambda\,(\mbox{tr}\, G)^{2}$$ for some constants $\lambda,\mu \,\in \mathbb{R}$. It is easy to show that for all $G \in\mathbb{M}^{3\times 3}$ and $R\in SO(3)$ $$Q_{3}(R\,G\,R^{T}) = Q_{3}(G),$$ and so, formula (\[(a)\]) reduces to $$\begin{aligned}
Q_{2}(P) =&\, \inf_{\hat{\alpha}\in W^{1,2}(D;\mathbb{R}^{3})}\Bigg\{\int_{D} Q_{3} \Bigg(P\, \Bigg(\begin{array}{c}
0\\
\xi\\
\zeta
\end{array}\Bigg)\,\bigg|\,\partial_{\xi}\hat{\alpha}\,\bigg|\,\partial_{\zeta}\hat{\alpha}\Bigg)\,d\xi\,d\zeta\Bigg\}\\
=&\, \frac{1}{2\,\pi}\,\frac{\mu(3\,\lambda + 2\,\mu)}{\lambda + \mu}\,(p_{12}^2 + p_{13}^2) + \frac{\mu}{2\,\pi}\,p_{23}^2,\end{aligned}$$ where the last equality follows from [@MM03 Remark 3.5]. This means that in the case of a homogeneous and isotropic material the quadratic form $Q_2$ is exactly the same as in the case of a straight rod treated in [@MM03].
Assume that the cross section $D$ is a circle of radius $\frac{1}{\sqrt{\pi}}$ centred at the origin. In this case, the quadratic form $Q_2$ can be computed by a pointwise minimization. More precisely, for every $s$ and for every $P$, $$Q_{2}(s,P) = \frac{1}{4\pi}\min_{u,v,w}\left\{Q_{3}\Bigg(R_{0}(s)\,\Bigg(\begin{array}{c}
p_{12} \\
0\\
- p_{23}
\end{array}\,\Bigg|\,u\,
\bigg|\,v \Bigg) R_{0}^{T}(s)\Bigg) + Q_{3}\Bigg(R_{0}(s)\,\Bigg(\begin{array}{c}
p_{13}\\
p_{23}\\
0
\end{array}\,\Bigg|\, v\,
\bigg|\, w\Bigg) R_{0}^{T}(s)\Bigg)\right\}.$$ The proof is completely analogous to [@MM03 Remark 3.6].
Lower scalings of the energy
============================
The content of this section is the study of the asymptotic behaviour of the functionals $\tilde{I}^{(h)}/h^\alpha$ for $0\leq\alpha<2$, as $h\rightarrow 0$. In addition to conditions (i)-(v) of Section $2$ we assume also that $W(z,F) = W(z_{1},F)$ for every $z=(z_{1},z_{2},z_{3})\in\mathbb{R}^{3}$ and every $F\in \mathbb{M}^{3\times 3}$, and that $$\begin{aligned}
(\textnormal{vi})& \,\, \exists \, C_3 > 0 \,\, \mbox{ independent of $z_1$ such that }\,
W(z_{1},F) \leq \,C_3\,\mbox{dist}^{2}(F,SO(3)) \,\, \mbox{for a.e. $z_1$ } \\
&\mbox{ and every} \, F\in\mathbb{M}^{3\times 3}.\end{aligned}$$ It is convenient to write the functionals $\tilde{I}^{(h)}$ as integrals over the fixed domain $\Omega = \big(\Psi^{(h)}\big)^{-1}\big(\tilde{\Omega}_{h}\big)$. Changing variables as in (\[flower2\]) and setting $v:= \tilde{v}\circ \Psi^{(h)}$, we have $$\tilde{I}^{(h)}(\tilde{v}) = \int_{\Omega} W\big(s,\big(\nabla_h v\big)\,\big(\nabla_h\Psi^{(h)}\big)^{-1}\big)\det
\big(\nabla_{h}\Psi^{(h)}\big)
ds\,d\xi\,d\zeta =: \tilde{J}^{(h)}(v).$$ We extend the functional to the space $L^{2}(\Omega;\mathbb{R}^{3})$, setting $$J^{(h)}(v) =
\left\{
\begin{array}{ll}
\vspace{.1cm}
\tilde{J}^{(h)}(v) & \mbox{if } v\in W^{1,2}(\Omega;\mathbb{R}^{3}),\\
\displaystyle + \infty & \mbox{otherwise in} \, L^{2}(\Omega;\mathbb{R}^{3}).
\end{array}
\right.$$ The aim of this section is to determine the $\Gamma$-limit of $J^{(h)}/h^\alpha$,for $0\leq\alpha<2$, as $h\rightarrow 0$, with respect to the strong topology of $L^{2}$.
Derivation of the nonlinear theory for curved strings
-----------------------------------------------------
For this first part we specify $\alpha = 0$, so we are interested in the asymptotic behaviour of the functionals representing the energy per unit volume associated to a deformation of the reference configuration.
\[comp1\] For every sequence $\big(v^{(h)}\big)$ in $L^{2}(\Omega;\mathbb{R}^{3})$ such that $$\label{fin}
J^{(h)}\big(v^{(h)}\big) \leq c < +\infty$$ there exist a function $v\in W^{1,2}((0, L);\mathbb{R}^{3})$ and some constants $c^{(h)}\in\mathbb{R}^3$ such that, up to subsequences, $$v^{(h)} - c^{(h)} \rightharpoonup v \quad \mbox{weakly in }\,
W^{1,2}(\Omega;\mathbb{R}^{3}).$$
Let $\big(v^{(h)}\big)$ be a sequence in $L^{2}(\Omega;\mathbb{R}^{3})$ satisfying (\[fin\]). From the definition of the functional we have immediately that $v^{(h)}\in W^{1,2}(\Omega;\mathbb{R}^{3})$. The assumptions on $W$ and the uniform boundedness of $\big(\nabla_{h}\Psi^{(h)}\big)^{-1}$ and of $\det \big(\nabla_{h}\Psi^{(h)}\big)$ give the boundedness in $L^{2}(\Omega;\mathbb{M}^{3\times 3})$ of $\big(\nabla_{h}v^{(h)}\big)$ and hence of $\big(\nabla v^{(h)}\big)$. Therefore, using the Poincaré inequality $$\big|\big|v^{(h)} - c^{(h)}\big|\big|_{L^{2}(\Omega;\mathbb{R}^{3})} \leq \big|\big|\nabla v^{(h)}\big|\big|_{L^{2}(\Omega;\mathbb{M}^{3\times 3})},$$ where $c^{(h)}\in\mathbb{R}^3$ is the mean value of $v^{(h)}$ over $\Omega$, it turns out that the sequence $v^{(h)} - c^{(h)}$ is bounded in $W^{1,2}(\Omega;\mathbb{R}^{3})$; hence there exists a function $v \in W^{1,2}(\Omega;\mathbb{R}^{3})$ such that, up to subsequences, $$v^{(h)} - c^{(h)}\rightharpoonup v \quad\mbox{weakly in}\,W^{1,2}(\Omega;\mathbb{R}^{3}).$$ Moreover since $\big(\nabla_{h} v^{(h)}\big)$ is bounded in $L^{2}(\Omega;\mathbb{M}^{3\times 3})$, we have $$\partial_{\xi} v^{(h)} \rightarrow 0\quad \mbox{and}\quad \partial_{\zeta} v^{(h)} \rightarrow 0 \quad\mbox{strongly in}\,L^{2}(\Omega;\mathbb{R}^{3}).$$ Therefore the limit function $v$ depends only on the first variable.
\[Gamcon\] Let $I$ be the functional defined as $$\label{Glim}
I(v) =
\left\{
\vspace{.5cm}
\begin{array}{ll}
\displaystyle \int_{0}^{L} W_{0}^{**}(s,v'(s))\,ds & \mbox{if } \, v\in W^{1,2}((0,L);\mathbb{R}^3),\\
\displaystyle + \infty & \mbox{otherwise in }\, L^{2}(\Omega;\mathbb{R}^3),
\end{array}
\right.$$ where $W_{0}^{**}$ is given by the convex envelope of the function $W_{0}: [0, L]\times \mathbb{R}^{3}\rightarrow \mathbb{R}$ defined as $$W_0(s,z):= \inf \big\{ W\left(s,(z\,|\,y_{2}\,|\,y_{3}) R_{0}^{T}(s)\right): y_{2},y_{3} \in \mathbb{R}^{3}\big\}.$$ Then $$\Gamma-\lim_{h\rightarrow 0} J^{(h)} = I,$$ i.e., the following conditions are satisfied:\
(i)(liminf inequality) for every $v\in L^{2}(\Omega;\mathbb{R}^{3})$ and every sequence $\big(v^{(h)}\big)\subset L^{2}(\Omega;\mathbb{R}^{3})$ such that $v^{(h)} \rightarrow v$ strongly in $L^{2}(\Omega;\mathbb{R}^{3})$, it turns out that $$\label{linf}
I(v)\leq\liminf_{h \rightarrow 0} J^{(h)}\big(v^{(h)}\big);$$ (ii)(limsup inequality) for every $v\in L^{2}(\Omega;\mathbb{R}^{3})$ there exists a sequence $\big(v^{(h)}\big)\subset L^{2}(\Omega;\mathbb{R}^{3})$ converging strongly to $v$ in $L^{2}(\Omega;\mathbb{R}^{3})$ such that $$\label{55}
\limsup_{h \rightarrow 0}
J^{(h)}\big(v^{(h)}\big)\leq I(v).$$
Notice that, if $A:= (z\,|\,y_2\,|\,y_3)\,R_0^T$, then $A\,\tau = z$ and $A\,\nu_k = y_k$ for $k = 2,3$. In other words, in the definition of $W_0$, the minimization is done with respect to the normal components of the matrix in the argument of $W$, keeping equal to $z$ the tangential component.
\[abp\] Observe that conditions (iv) and (v) imply that for a.e. $s\in [0, L]$, $$\label{ABP}
W_{0}^{**}(s,z) = 0 \quad \mbox{if and only if} \quad {\left|z\right|}\leq 1,$$ (see [@ABP91]).
(of Theorem \[Gamcon\]) (i) Let $v$ and $v^{(h)}$ be as in the statement. We can assume that $$\liminf_{h \rightarrow 0} J^{(h)}\big(v^{(h)}\big) < + \infty,$$ otherwise (\[linf\]) is trivial. Therefore, up to subsequences, (\[fin\]) is satisfied. From Theorem \[comp1\] we deduce that $v \in W^{1,2}((0,L);\mathbb{R}^3)$ and that the convergence is indeed weak in $W^{1,2}(\Omega;\mathbb{R}^{3})$.\
Now define the function $W_{0}: [0, L]\times \mathbb{R}^{3}\rightarrow \mathbb{R}$ as $$W_{0}(s,z):= \inf \big\{W\left(s,(z\,|\,y_{2}\,|\,y_{3}) R_{0}^{T}(s)\right): y_{2},y_{3} \in \mathbb{R}^{3}\big\}.$$ Due to the coercivity assumptions this function is finite.
Notice that, since $R_{0}\,R_{0}^{T} = Id$, we can write $$W\big(s,\nabla_{h}v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}\big) =
W\Big(s,\nabla_{h}v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}R_{0}R_{0}^{T}\Big)$$ and using the explicit expression of $\big(\nabla_{h}\Psi^{(h)}\big)^{-1}$ given in (\[invA\]), i.e., $$\big(\nabla_{h}\Psi^{(h)}\big)^{-1}(s,\xi,\zeta) =
R_{0}^{T}(s) - h\,R_{0}^{T}(s)\,\big[(\xi\,\nu'_{2}(s) +
\zeta\,\nu'_{3}(s))\otimes e_1\big] R_{0}^{T}(s) + O(h^{2}),$$ we have $$\label{convVp}
\nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}R_{0}e_1 \rightharpoonup v' \quad \mbox{weakly in } \, L^{2}(\Omega;\mathbb{R}^3).$$ So, from the definition of $W_{0}$ $$\begin{aligned}
J^{(h)}\big(v^{(h)}\big) &\geq&
\int_{\Omega} W_{0}\big(s,\nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}R_{0}e_1\big)\,\det \big(\nabla_{h}\Psi^{(h)}\big)\,ds\,d\xi\,d\zeta \\
&\geq& \int_{\Omega} W_{0}^{**}\big(s,\nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}R_{0}e_1\big)\,\det \big(\nabla_{h}\Psi^{(h)}\big)\,ds\,d\xi\,d\zeta.\end{aligned}$$ Now we pass to the $\liminf$ in both sides of the previous inequality, using the uniform convergence of the determinant remarked in (\[convdet\]), and we get $$\begin{aligned}
\liminf_{h\rightarrow 0}\,J^{(h)}\big(v^{(h)}\big)&\geq&
\liminf_{h\rightarrow 0}\int_{\Omega} W_{0}^{**}\big(s,\big(\nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}R_{0}\big)e_1\big)\,\det \big(\nabla_{h}\Psi^{(h)}\big)\,ds\,d\xi\,d\zeta \\
&=& \liminf_{h\rightarrow 0}\int_{\Omega}W_{0}^{**}\big(s,\big(\nabla_{h} v^{(h)}\big(\nabla_{h}\Psi^{(h)}\big)^{-1}R_{0}\big)e_1\big)\,ds\,d\xi\,d\zeta.\end{aligned}$$ Since the functional $$G(u):= \int_{\Omega} W_{0}^{**}(s, u)\,ds\,d\xi\,d\zeta$$ is convex, it is sequentially weakly lower semicontinuous in $L^{2}(\Omega;\mathbb{R}^{3})$; so, by (\[convVp\]) we can conclude that $$\label{supe1}
\liminf_{h\rightarrow 0}\,J^{(h)}\big(v^{(h)}\big) \geq \int_{0}^{L} W_{0}^{**}(s,v'(s))\,ds.$$ (ii) Let $v$ be a function in $W^{1,2}((0, L);\mathbb{R}^3)$, otherwise the bound in (\[55\]) is trivial. Let $w_{2}, w_{3} \in W^{1,2}((0, L);\mathbb{R}^{3})$ be arbitrary functions and consider the functions $v^{(h)}: \Omega\rightarrow \mathbb{R}^{3}$ defined by $$v^{(h)}(s,\xi,\zeta) := v(s) + h\,\xi\,w_{2}(s) + h\,\zeta\,w_{3}(s).$$ Clearly, as $\nabla v^{(h)} = v'\otimes e_1 + h\,\big(\xi\,w'_2 + \zeta\,w'_3\,|\,w_2\,|\,w_3\big)$, we have that $$\label{(b)}
v^{(h)}\rightarrow v\quad \mbox{strongly in}\, W^{1,2}(\Omega;\mathbb{R}^{3}).$$ Now we want to study the behaviour of the sequence $$J^{(h)}\big(v^{(h)}\big) = \int_{\Omega}W\big(s,(\nabla_{h} v^{(h)})\big(\nabla_{h}\Psi^{(h)}\big)^{-1}) \det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta$$ when $h\rightarrow 0$. Notice that the scaled gradient of $v^{(h)}$ satisfies $$\label{almever}
\nabla_{h} v^{(h)} = (v'\,|\,w_2\,|\,w_3) +
h\, (\xi\,w'_2 + \zeta\, w'_3)\otimes e_1 \rightarrow (v'\,|\,w_2\,|\,w_3) \,\, \mbox{a.e.}.$$ So, by (\[convdet\]) and (vi), using the dominated convergence theorem we get $$\begin{aligned}
\lim_{h\rightarrow 0}J^{(h)}\big(v^{(h)}\big) &=\, \lim_{h\rightarrow 0}\int_{\Omega} W\big(s,(\partial_{s} v^{(h)}\,|\,w_{2}\,|\,w_{3})\big(\nabla_{h}\Psi^{(h)}\big)^{-1}) \det \big(\nabla_{h}\Psi^{(h)}\big)\, ds\,d\xi\,d\zeta\\
&=\, \int_{0}^{L} W\big(s,(v'\,|\,w_{2}\,|\,w_{3})\,R_{0}^{T})\, ds.\end{aligned}$$ Up to now we have shown that for every choice of $w_{2}, w_{3} \in W^{1,2}((0, L);\mathbb{R}^{3})$, there exists a sequence $\big(v^{(h)}\big)$ such that (\[(b)\]) is satisfied and $$\lim_{h\rightarrow 0}J^{(h)}\big(v^{(h)}\big) = \int_{0}^{L} W\big(s,(v'\,|\,w_{2}\,|\,w_{3})\,R_{0}^{T}) ds.$$ Therefore, $$\begin{aligned}
\label{densi}
\Gamma-\limsup_{h\rightarrow 0} J^{(h)}(v)&:=\,
\inf\left\{\limsup_{h\rightarrow 0}J^{(h)}\big(u^{(h)}\big):
u^{(h)} \rightarrow v \,\, \mbox{strongly in} \,\, L^{2}(\Omega;\mathbb{R}^{3})\right\}\nonumber\\
&\leq \inf\left\{\int_{0}^{L} W\big(s,(v'\,|\,w_{2}\,|\,w_{3})\,R_{0}^{T})\,ds :
w_{2}, w_{3} \in W^{1,2}((0, L);\mathbb{R}^{3})\right\}\nonumber\\
&= \inf\left\{\int_{0}^{L} W\big(s,(v'\,|\,w_{2}\,|\,w_{3})\,R_{0}^{T})\,ds :
w_{2}, w_{3} \in L^{2}((0, L);\mathbb{R}^{3})\right\},\end{aligned}$$ where the last equality is a consequence of the dominated convergence theorem and of the density of $W^{1,2}((0, L);\mathbb{R}^{3})$ in $L^{2}((0, L);\mathbb{R}^{3})$.
By the measurable selection lemma (see for example [@EkTe]) applied to the Carathéodory function $$g:[0, L]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\rightarrow \mathbb{R},\quad (s,y_{2},y_{3})\mapsto g(s,y_{2},y_{3}):= W\big(s,(v'(s)\,|\,y_{2}\,|\,y_{3})R_{0}^{T}(s))$$ we obtain the existence of two measurable functions $w^{0}_{2}, w^{0}_{3}: [0, L]\rightarrow \mathbb{R}^{3}$ satisfying $$W\big(s,(v'(s)\,|\,w^{0}_{2}(s)\,|\,w^{0}_{3}(s))R_{0}^{T}(s)) =\inf_{y_{2},y_{3}\in \mathbb{R}^{3}} W\big(s,(v'(s)\,|\,y_{2}\,|\,y_{3})R_{0}^{T}(s)) = W_{0}(s,v'(s)).$$ Moreover, from the coerciveness of $W$ it follows that $w^{0}_{2}, w^{0}_{3}$ belong indeed to $L^{2}((0, L);\mathbb{R}^{3})$ and so they are in competition for the infimum in (\[densi\]). Hence, for every $v\in W^{1,2}((0, L);\mathbb{R}^{3})$ we have $$\Gamma-\limsup_{h\rightarrow 0} J^{(h)}(v) \leq \int_{0}^{L} W_{0}(s,v'(s))\,ds =: \tilde{J}(v).$$ Now define the functional $$\label{Glim2}
J(v) =
\left\{
\vspace{.5cm}
\begin{array}{ll}
\displaystyle \tilde{J}(v) & \mbox{if } \, v\in W^{1,2}((0,L);\mathbb{R}^3),\\
\displaystyle + \infty & \mbox{otherwise in }\, L^{2}(\Omega;\mathbb{R}^3);
\end{array}
\right.$$ clearly it turns out that $$\label{darelax}
\Gamma-\limsup_{h\rightarrow 0} J^{(h)}(v) \leq J(v) \quad\mbox{for every} \,\, v\in L^{2}(\Omega;\mathbb{R}^{3}).$$ As the lower semicontinuous envelope of $J$ with respect to the strong topology of $L^{2}(\Omega;\mathbb{R}^{3})$ is given by the functional $I$ (see [@DM93] and [@LDR95 Lemma 5]), the thesis follows immediately from (\[darelax\]).
Intermediate scaling
---------------------
In this subsection we show that scalings of the energy of order $h^\alpha$, with $\alpha\in (0,2)$, lead to a trivial $\Gamma$-limit.
Let $\mathcal{W}_1$ be the class of functions defined as $$\label{defW1M}
\mathcal{W}_1:= \{v\in W^{1,2}((0,L);\mathbb{R}^{3}) : |v'(s)| \leq 1 \,\textnormal{a.e.}\}.$$ For every sequence $\big(v^{(h)}\big)$ in $L^{2}(\Omega;\mathbb{R}^{3})$ such that $$\label{fin2}
\frac{1}{h^\alpha}\,J^{(h)}\big(v^{(h)}\big) \leq c < +\infty$$ there exist a function $v\in \mathcal{W}_{1}$ and some constants $c^{(h)}\in\mathbb{R}$ such that, up to subsequences, $$v^{(h)} - c^{(h)} \rightharpoonup v \quad \mbox{weakly in }\,
W^{1,2}(\Omega;\mathbb{R}^{3}).$$ Moreover, $$\label{(c)}
\Gamma-\lim_{h \rightarrow 0}\, \frac{1}{h^\alpha}\,J^{(h)} =
\left\{
\vspace{.3cm}
\begin{array}{ll}
\vspace{.15cm}
\quad 0 & \textnormal{in} \,\, \mathcal{W}_1,\\
\displaystyle + \infty & \textnormal{otherwise in} \,\, L^{2}(\Omega;\mathbb{R}^{3}).
\end{array}
\right.$$
Let $\big(v^{(h)}\big)$ be such that (\[fin2\]) is satisfied. Then $$\label{boundalfa}
J^{(h)}\big(v^{(h)}\big) < c\,h^{\alpha}.$$ By Theorem \[comp1\] this implies that there exist $v\in W^{1,2}((0,L);\mathbb{R}^{3})$ and some constants $c^{(h)}\in\mathbb{R}$ such that the sequence $v^{(h)} - c^{(h)}$ converges to $v$ weakly in $W^{1,2}(\Omega;\mathbb{R}^{3})$. Moreover by Theorem \[Gamcon\] and by (\[boundalfa\]) $$0 = \liminf_{h\rightarrow 0}J^{(h)}\big(v^{(h)}\big) \geq \int_{0}^{L} W_{0}^{**}(s,v'(s))ds,$$ and this gives the additional condition that ${\left|v'(s)\right|}\leq 1$ for almost every $s\in [0,L]$, thanks to Remark \[abp\]. Therefore $v\in \mathcal{W}_{1}$.\
Let us prove (\[(c)\]). The liminf inequality follows directly from the fact that the energy density $W$ is nonnegative and from the compactness. As for the limsup inequality we first notice that we can restrict our analysis to functions $v\in \mathcal{W}_{1}$, being the other case trivial. Since ${\left|v'(s)\right|}\leq 1$ for a.e. $s\in [0, L]$, there exist two measurable functions $d_{2}, d_{3}: [0, L]\rightarrow \mathbb{R}^3$ such that $$(v'(s)\,|\,d_{2}(s)\,|\,d_{3}(s))\in Co(SO(3))\quad \mbox{for a.e.}\,\, s\in [0,L],$$ where $Co(SO(3))$ denotes the convex hull of $SO(3).$ As first step, we assume in addition that $(v'\,|\,d_{2}\,|\,d_{3})$ is a piecewise constant rotation; for simplicity we can limit ourselves to the case $$(v'(s)\,|\,d_{2}(s)\,|\,d_{3}(s)) =
\left\{
\begin{array}{ll}
\vspace{.15cm}
R_{1} & \mbox{if } s\in [0,s_{0}[,\\
R_{2} & \mbox{if } s\in [s_{0},L]
\end{array}
\right.$$ with $R_{1},R_{2} \in SO(3)$. Now, let $\omega(h)$ be a sequence converging to zero, as $h\rightarrow 0$, and let $P$ be a smooth function $P:[0,1] \longrightarrow SO(3)$, such that $P(0) = R_{1}$ and $P(1) = R_{2}$. Now consider a reparametrization of $P$, denoted by $P^{(h)}$ and given by $$P^{(h)}(s) := P\bigg(\frac{s - s_{0}}{\omega(h)}\bigg).$$ Define the sequence $v^{(h)}: \Omega\rightarrow \mathbb{R}^3$ as $$v^{(h)}(s,\xi,\zeta) :=
\left\{
\begin{array}{lll}
R_{1}\Biggl(\begin{array}{c}
s\\
h\,\xi\\
h\,\zeta
\end{array}\Biggr) & \mbox{on } s\in [0,s_{0}[\times D,\\
\displaystyle\int_{s_{0}}^{s} \big(P^{(h)}\big)(\sigma)e_1\,d\sigma + P^{(h)}(s)\Biggl(\begin{array}{c}
0\\
h\,\xi\\
h\,\zeta
\end{array}\Biggr) + b^{(h)} & \mbox{on } \, \big[s_{0}, s_{0} + \omega(h)\big]\times D,\\
R_{2}\Biggl(\begin{array}{c}
s\\
h\,\xi\\
h\,\zeta
\end{array}\Biggr) + d^{(h)} & \mbox{on } \, \big]s_{0} + \omega(h), L\big]\times D,\\
\end{array}
\right.$$ where the constants $b^{(h)}$ and $d^{(h)}$ are chosen in order to make $v^{(h)}$ continuous. It turns out that the scaled gradient has the following expression: $$\label{Gradve}
\nabla_{h}v^{(h)} = \left\{
\begin{array}{lll}
R_{1} & \mbox{on } \, [0,s_{0}[\times D,\\
P^{(h)}(s) + \Bigg(\big(P^{(h)}\big)'(s)\Bigg(\begin{array}{c}
0\\
h\,\xi\\
h\,\zeta
\end{array}\Bigg)\Bigg)\otimes e_1 & \mbox{on } \, \big[s_{0}, s_{0} + \omega(h)\big]\times D,\\
R_{2} & \mbox{on } \, \big]s_{0} + \omega(h), L\big]\times D;\\
\end{array}
\right.$$ moreover $\nabla_{h}v^{(h)}\rightarrow (v'\,|\,d_{2}\,|\,d_{3})$ strongly in $L^{2}(\Omega;\mathbb{R}^{3})$. In order to evaluate the functional on this sequence we use the fact that, by (v) and (\[convdet\]), $$\label{estimdist}
\frac{1}{h^\alpha}\,J^{(h)}\big(v^{(h)}\big) \leq \frac{c}{h^\alpha}\, \int_{\Omega}\mbox{dist}^2\big(\nabla_{h}v^{(h)}\,
\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\,ds\,d\xi\,d\zeta.$$ From (\[Gradve\]) the integral on the right-hand side of the previous expression can be written as $$\begin{aligned}
\label{3int}
&\int_{0}^{s_0}\int_{D}\mbox{dist}^2\big(R_1\,
\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\,ds\,d\xi\,d\zeta +\,\int_{s_0 + \omega(h)}^{L}\int_{D}\mbox{dist}^2\big(R_2\,
\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\,ds\,d\xi\,d\zeta \nonumber\\
&+\int_{s_0}^{s_0 + \omega(h)}\int_{D}\mbox{dist}^2\big(\nabla_{h}v^{(h)}\,
\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\,ds\,d\xi\,d\zeta.\end{aligned}$$ The first two terms in (\[3int\]) give a contribution of order $h^2$ since, by (\[invA\]), for $i= 1,2$, $$\begin{aligned}
\mbox{dist}^2\big(R_i\,\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big) &\leq h^2\, \mbox{dist}^2\big(R_i\,R_{0}^{T}\,\big[(\xi\,\nu'_{2} +
\zeta\,\nu'_{3})\otimes e_1\big] R_{0}^{T}, SO(3)\big)\\
&\leq \,C\,h^2\, \mbox{dist}^2\big(\big[(\xi\,\nu'_{2} +
\zeta\,\nu'_{3})\otimes e_1\big], SO(3)\big),\end{aligned}$$ so they can be neglected in the computation of the limit of (\[estimdist\]). The only term we have to analyse is the last integral in (\[3int\]). Set $$A^{(h)}(s,\xi,\zeta):= \Bigg(\big(P^{(h)}\big)'\Bigg(\begin{array}{c}
0\\
h\,\xi\\
h\,\zeta
\end{array}\Bigg)\Bigg)\otimes e_1.$$ Using again (\[invA\]) we have that $$\mbox{dist}^2\big(\nabla_{h}v^{(h)}\,\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\leq \,\mbox{dist}^2\,\big(A^{(h)}\,\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\leq \,C\,h^2\,\big(\xi^{2} + \zeta^{2}\big)\, \big|\,\big(P^{(h)}\big)'\,\big|^2,$$ so we get the following estimate: $$\begin{aligned}
\int_{s_0}^{s_0 + \omega(h)}\int_{D}\mbox{dist}^2\big(\nabla_{h}v^{(h)}\,
\big(\nabla_{h}\Psi^{(h)}\big)^{-1}, SO(3)\big)\,ds\,d\xi\,d\zeta &\leq
C\,h^2\,\int_{s_0}^{s_0 + \omega(h)}\,\big|\,\big(P^{(h)}\big)'\,\big|^2\,ds\\
&= \,C \,\frac{h^{2}}{\omega(h)}\int_{0}^{1} \big|\,P'\big|^{2} ds.\end{aligned}$$ Notice that, if we choose $\omega(h)\sim h^{\beta}$, with $0<\beta<2 - \alpha$, also this term can be neglected in (\[estimdist\]), hence $$\lim_{h\rightarrow 0}\frac{1}{h^\alpha}\,J^{(h)}\big(v^{(h)}\big) = 0$$ and this concludes the proof in the case $(v'\,|\,d_{2}\,|\,d_{3})$ is a piecewise constant rotation.
Consider now the general case. Since $(v'\,|\,d_{2}\,|\,d_{3}) \in Co(SO(3))$ a.e., there exists a sequence of piecewise constant rotations $R_j: [0, L]\longrightarrow SO(3)$ such that $R_j \rightarrow (v'\,|\,d_{2}\,|\,d_{3})$ strongly in $L^{2}((0, L);\mathbb{M}^{3\times 3})$. For each element $R_j$ of the sequence we can repeat the same construction done in the previous case and find a sequence $v^{(h)}_j$ whose scaled gradients $\nabla_{h}v^{(h)}_j$ converge to $R_j$ as $h\rightarrow 0$ and such that for every $j$ $$\lim_{h\rightarrow 0}\frac{1}{h^{\alpha}}\int_{\Omega}W(s,\nabla_{h} v_{j}^{(h)}\, \big(\nabla_{h}\Psi^{(h)}\big)^{-1}\big)\det \big(\nabla_{h}\Psi^{(h)}\big) ds\,d\xi\,d\zeta = 0.$$ Now we can choose, for every $j$, an element of the sequence $v^{(h)}_j$, say $v^{(h_j)}_j$, in such a way that $${\left\Vert\nabla_{h_j}v^{(h_j)}_j - R_{j}\right\Vert}_{L^2(\Omega;\mathbb{M}^{3\times 3})} < \frac{1}{j}$$ and $$\frac{1}{h_j^{\alpha}}\int_{\Omega}W(s,\nabla_{h_j} v_{j}^{(h_j)}\, \big(\nabla_{h_j}\Psi^{(h_j)}\big)^{-1}\big)\det \big(\nabla_{h_j}\Psi^{(h_j)}\big) ds\,d\xi\,d\zeta < \frac{1}{j}.$$ These estimates show that the sequence $v^{(h_j)}_j$ converges to $(v'\,|\,d_{2}\,|\,d_{3})$ strongly in $L^{2}((0, L);\mathbb{M}^{3\times 3})$ and that $$\lim_{j\rightarrow \infty}\frac{1}{h_j^{\alpha}}\int_{\Omega}W(s,\nabla_{h_j} v^{(h_j)}\, \big(\nabla_{h_j}\Psi^{(h_j)}\big)^{-1}\big)\det \big(\nabla_{h_j}\Psi^{(h_j)}\big) ds\,d\xi\,d\zeta = 0.$$ This concludes the proof.
<span style="font-variant:small-caps;">Acknowledgments</span>
I would like to thank Maria Giovanna Mora for having proposed to me the study of this problem and for many helpful and interesting suggestions. I would like also to thank Gianni Dal Maso for several stimulating discussions on the subject of this paper.
This work is part of the project “Calculus of Variations" 2004, supported by the Italian Ministry of Education, University, and Research.
[9]{}
Acerbi E., Buttazzo G., Percivale D.: A variational definition of the strain energy for an elastic string. J. Elasticity **25**, 137-148 (1991)
Dal Maso G.: An introduction to $\Gamma$-convergence. Birkhäuser, Boston, 1993
Ekeland I., Temam R.: Convex analysis and variational problems. North-Holland, Amsterdam, 1976
Friesecke G., James R.D., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. **55**, 1461-1506 (2002)
Friesecke G., James R.D., Mora M.G., Müller S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by $\Gamma$-convergence. C. R. Math. Acad. Sci. Paris **336**, 697-702 (2003)
Friesecke G., James R.D., Müller S.: A hierarchy of plate models derived from nonlinear elasticity by $\Gamma$-convergence. MPI-MIS Preprint **7** (2005)
Le Dret H., Raoult A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. (9) **74**, 549-578 (1995)
Le Dret H., Raoult A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. In Mechanics: from theory to computation, 59-84 Springer, New York, 2000
Mora M.G., Müller S.: Derivation of the nonlinear bending-torsion theory for inextensible rods by $\Gamma$-convergence. Calc. Var. Partial Differential Equations **18**, 287-305 (2003)
Mora M.G., Müller S.: A nonlinear model for inextensible rods as a low energy $\Gamma$-limit of three-dimensional nonlinear elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire **21**, 271-293 (2004)
Pantz O.: Le modèle de poutre inextentionnelle comme limite de l’ élasticité non-linéaire tridimensionnelle. Preprint (2002)
|
---
abstract: 'The success of the neutrino mechanism of core-collapse supernovae relies on the supporting action of two hydrodynamic instabilities: neutrino-driven convection and the Standing Accretion Shock Instability (SASI). Depending on the structure of the stellar progenitor, each of these instabilities can dominate the evolution of the gain region prior to the onset of explosion, with implications for the ensuing asymmetries. Here we examine the flow dynamics in the neighborhood of explosion by means of parametric two-dimensional, time-dependent hydrodynamic simulations for which the linear stability properties are well understood. We find that systems for which the convection parameter $\chi$ is sub-critical (SASI-dominated) develop explosions once large-scale, high-entropy bubbles are able to survive for several SASI oscillation cycles. These long-lived structures are seeded by the SASI during shock expansions. Finite-amplitude initial perturbations do not alter this outcome qualitatively, though they can lead to significant differences in explosion times. Supercritical systems (convection-dominated) also explode by developing large-scale bubbles, though the formation of these structures is due to buoyant activity. Non-exploding systems achieve a quasi-steady state in which the time-averaged flow adjusts itself to be convectively sub-critical. We characterize the turbulent flow using a spherical Fourier-Bessel decomposition, identifying the relevant scalings and connecting temporal and spatial components. Finally, we verify the applicability of these principles on the general relativistic, radiation-hydrodynamic simulations of Müller, Janka, & Heger (2012), and discuss implications for the three-dimensional case.'
author:
- |
Rodrigo Fernández$^{1,2,3}$, Bernhard Müller$^4$, Thierry Foglizzo$^5$, Hans-Thomas Janka$^4$\
$^1$ Institute for Advanced Study, Princeton, NJ 08540, USA\
$^2$ Department of Physics, University of California, Berkeley, CA 94720, USA\
$^3$ Department of Astronomy & Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA\
$^4$ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany\
$^5$ Laboratoire AIM, CEA/DSM-CNRS-Université Paris Diderot, IRFU/Service d’Astrophysique, CEA-Saclay F-91191, France
bibliography:
- 'ccsne.bib'
- 'apj-jour.bib'
date: Submitted to MNRAS
title: 'Characterizing SASI- and Convection-Dominated Core-Collapse Supernova Explosions in Two Dimensions'
---
\[firstpage\]
hydrodynamics — instabilities – neutrinos – nuclear reactions, nucleosynthesis, abundances — shock waves – supernovae: general
Introduction
============
In the neutrino mechanism of core-collapse supernovae, a small fraction of the energy emitted in neutrinos by the forming neutron star is deposited in a layer behind the stalled accretion shock, powering its final expansion [@bethe85]. Extensive theoretical work over the last two decades has led to a consensus on the failure of this mechanism in spherically symmetric systems, except for the very lightest stellar progenitors (see, e.g., @janka2012a for a recent review).
Successful neutrino-driven explosions require additional assistance by non-spherical hydrodynamic instabilities that increase the efficiency of neutrino energy deposition. This phenomenon has been observed in numerous two-dimensional (e.g., @herant94 [@burrows95; @janka96; @mezzacappa98; @scheck06; @ohnishi06; @buras06b; @burrows07; @murphy08; @ott08; @marek09; @suwa09; @mueller2012; @couch2013]) as well as three-dimensional (e.g., @iwakami08 [@nordhaus10a; @hanke2012; @burrows2012; @emueller2012; @takiwaki2012; @ott2012b; @couch2012; @dolence2013; @hanke2013]) core-collapse simulations of various levels of sophistication. In addition to assisting the onset of explosion, these instabilities can contribute to the generation of pulsar kicks [@scheck06; @nordhaus10b; @wongwathanarat2010], the spin-up of the forming neutron star [@fryer07; @blondin07a; @blondin07b; @F10] and the seeding of late-time asymmetries [@kifonidis06; @hammer2010; @wongwathanarat2012].
The shock-neutrinosphere cavity is unstable to *convection* driven by the energy deposition from neutrinos emitted in deeper layers (e.g., @bethe90). This process generates kinetic energy on spatial scales comparable to or smaller than the size of the neutrino heating region. Work by @BM03 and @BM06 isolated a distinct, global oscillatory instability of the standing accretion shock that operates independent of neutrino heating, the so-called Standing Accretion Shock Instability (SASI). The driving mechanism involves an unstable cycle of advected and acoustic perturbations trapped within the shock-neutrinosphere cavity (@F07 [@foglizzo09; @guilet2012]). The most unstable modes of the SASI reside on the largest spatial scales. Convection and the SASI are easily distinguishable in the linear regime, but their effects become intertwined in the non-linear turbulent flow that follows the stalling of the bounce supernova shock (e.g., @scheck08).
Recent three-dimensional studies of core-collapse supernova hydrodynamics have found that large-scale oscillation modes of the shock attain smaller amplitudes than in two dimensions [@nordhaus10a; @wongwathanarat2010; @hanke2012; @takiwaki2012; @murphy2012]. This has been interpreted as a consequence of the different behavior of turbulence in two- and three-dimensions [@hanke2012], and has led to the suggestion that the SASI may play a secondary role in the explosion mechanism, if it arises at all [@burrows2012; @burrows2013]. These models have largely focused on a small sample of stellar progenitors, however, and in many cases do not include physical effects that are favorable for the growth of the SASI [@janka2012b].
@mueller2012 followed the collapse and bounce of $8.1$ and $27M_\sun$ progenitors using a two-dimensional, general relativistic hydrodynamic code with energy dependent neutrino transport, finding that differences in progenitor structure lead to very different paths to explosion. In particular, the $27M_\sun$ progenitor evolution is such that the SASI dominates the dynamics throughout the pre-explosion phase. Three-dimensional simulations of the same progenitor, with similar neutrino treatment, display episodic SASI activity, though a successful explosion is not yet obtained [@hanke2013]. @ott2012b evolved the same $27M_\odot$ progenitor with a more approximate neutrino prescription and a higher level of numerical perturbations, initially finding smaller SASI amplitudes than @mueller2012 and @hanke2013, though later confirming SASI activity (C. Ott 2013, private communication). The lack of the same level of numerical perturbations in the @mueller2012 models could mean that convection dominance instead of SASI dominance is dependent not only on the progenitor structure, but also on the details of the initial conditions.
It is the purpose of this paper to investigate some of these issues involving the interplay of SASI and convection, and the implications for successful explosions. In particular, we address the following questions: (1) Is there a fundamental difference between the transition to explosion in SASI- and convection-dominated models? (2) Can finite amplitude perturbations, generated in, e.g., multi-dimensional stellar progenitors (e.g., @arnett2011), tilt the balance in favor of convection in situations that would otherwise be SASI-dominated? (3) Does the SASI play any discernible role in convection-dominated systems? (4) Are there systematic trends in models close to an explosion that shed insight into the operation of each instability?
Our approach is experimental, employing hydrodynamic simulations that model neutrino source terms, the equation of state, and gravity in a parametric way (e.g., @FT09b). This setup has the advantage that its linear stability properties are well understood [@FT09a], allowing the development of model sequences that probe different parameter regimes. Our experimental approach to studying SASI and convection follows similar works [@foglizzo06; @ohnishi06; @scheck08; @FT09b; @burrows2012], to which we relate our findings. To connect with more realistic models, we also test the generality of our analysis results on the simulations of @mueller2012.
The structure of the paper is the following. Section \[s:methods\] describes the numerical models employed and introduces the Spherical Fourier-Bessel decomposition. Section \[s:results\] presents results, separated by exploding and quasi-steady state behavior. A summary and discussion follows in Section \[s:summary\]. Appendix \[s:L0\] addresses the stability of the $\ell=0$ mode in the parametric setup, and Appendix \[s:sfb\_appendix\] provides details about the spherical basis functions for the cases of Dirichlet and Neumann boundary conditions.
Methods {#s:methods}
=======
Parametric Hydrodynamic Simulations {#s:parametric}
-----------------------------------
### Numerical Setup {#s:setup}
The parametric, two-dimensional stalled supernova shock simulations employed for the majority of the analysis follow the setup of @FT09b [@FT09a]. These models have been calibrated to the global linear stability analysis of @F07. The linear analysis has been extended to include the effects of parameterized nuclear dissociation and lightbulb neutrino heating [@FT09b].
In our time-dependent models, the equations of mass, momentum, and energy conservation are solved in spherical polar coordinates ($r,\theta$), subject to the gravity from a point mass $M$ at the origin and parameterized neutrino heating and cooling: $$\begin{aligned}
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) & = & 0\\
\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v}\cdot \nabla)\mathbf{v}
& = & -\frac{1}{\rho}\nabla p - \frac{GM}{r^2}\hat r\\
\frac{\totd e_{\rm int}}{\totd t} -\frac{1}{\rho}\frac{\totd p}{\totd t} & = & Q_\nu.\end{aligned}$$ Here $\rho$, $\mathbf{v}$, $p$, and $e_{\rm int}$ are the fluid density, velocity, pressure, and specific internal energy, respectively. The equation of state is that of an ideal gas with adiabatic index $\gamma$, i.e., $p = (\gamma-1)\rho e_{\rm int}$. To connect with previous studies, the net neutrino source term is set to $$\label{eq:neutrino_source_term}
Q_\nu = \left[ \frac{B}{r^2} - A p^{3/2}\right]
\,e^{-(s/s_{\rm min})^2}\,\Theta(\mathcal{M}_0-\mathcal{M}),$$ where $s$ is the fluid entropy, $\mathcal{M}$ the Mach number, and $\Theta$ the step function. This functional form models heating as a lightbulb, with $B$ a normalization constant proportional to the neutrino luminosity. The cooling function, first introduced by @BM06 and subsequently used by @F07, models electron and positron capture in an optically thin environment $(\propto \rho T^6)$ assuming a radiation-dominated gas $(p\propto T^4)$. The exponential suppression at a low entropy $s_{\rm min}$ is introduced to prevent runaway cooling at the base of the flow, and the cutoff at high Mach number $\mathcal{M}_0=2$ is used to suppress heating and cooling in the upstream flow [@FT09b].
The initial condition consists of a steady-state spherical accretion shock at a radius $r_s$, below which the fluid settles subsonically onto a protoneutron star of radius $r_*$. Given a boundary condition at the shock, the normalization of the cooling function $A$ is determined by demanding that the radial velocity vanishes at $r=r_*$. The upstream flow is supersonic and adiabatic, with zero Bernoulli parameter. The Mach number upstream of the shock is set to $\mathcal{M}_1 = 5$ at a radius $r_{\rm s0}$ equal to the shock radius obtained with zero heating ($B=0$). To connect with previous studies (e.g., @FT09b), the adiabatic index is set to $\gamma=4/3$, even though a more realistic flow would have this index varying within the range $1.4-1.6$. A constant specific energy loss by nuclear dissociation $\varepsilon$ is allowed at the shock, increasing the compression ratio [@thompson00]. The solution is uniquely determined by specifying the ratio $r_*/r_{\rm s0}$, the nuclear dissociation parameter $\varepsilon$, the upstream Mach number $\mathcal{M}_1$, and the heating rate $B$ (see @FT09b for a sample of initial density profiles). In all models, we set $r_*/r_{\rm s0} = 0.4$.
Throughout this paper, we adopt the initial shock radius without heating, $r_{\rm s0}$, the free fall speed at this radius, $v_{\rm ff0}^2 = 2GM/r_{\rm s0}$, and the upstream density $\rho_1$ as the basic system of units. Full-scale simulations yield characteristic values $r_{\rm s0}\simeq 150$ km, $M\simeq 1.3M_\odot$, and $\dot{M}\simeq 0.3M_\sun$ s$^{-1}$, with a resulting free-fall speed $v_{\rm ff0} \simeq 5\times 10^9$ cm s$^{-1}$, dynamical time $t_{\rm ff0} = r_{\rm s0}/v_{\rm ff0} \simeq 3$ ms, and upstream density $\rho_1 \simeq 10^8$ g cm$^{-3}$. Setting the heating term in equation (\[eq:neutrino\_source\_term\]) equal to the approximation from @janka01 commonly used in ‘lightbulb’ heating studies (e.g., @murphy08 [@couch2013]), one obtains a relation between $B$ and the electron neutrino luminosity, $$\label{eq:B_dimensional}
B \simeq 0.009\, L_{\nu_e,52}\, T_{\nu,4}^2\,
\left(\frac{r_{\rm s0}}{150\textrm{ km}}\right)^{1/2}\left(\frac{1.3M_\odot}{M} \right)^{3/2},$$ where $L_{\nu_e,52}$ is the electron neutrino luminosity in units of $10^{52}$ erg s$^{-1}$, and $T_{\nu,4}$ is the neutrinospheric temperature in units of $4$ MeV.
The numerical models are evolved in FLASH3.2 [@dubey2009], with the modifications introduced in @F12. The computational domain covers the radial range $r\in [0.4,7]r_{\rm s0}$, and the full range of polar angles. The radial grid spacing is logarithmic, with 408 cells in radius ($\Delta r/r \simeq 0.7\%$). We use 300 angular cells equispaced in $\cos\theta$, yielding constant volume elements at fixed radius ($\Delta \theta \simeq \Delta r/r$ on the equator). The boundary conditions are reflecting at the polar axis and at the surface of the neutron star, and set to the upstream solution at the outer radial boundary. Accreted material accumulates in the innermost $\sim $ two cells next to the inner boundary.
----------- ------------------------- ---------------------------------- -------------- --------------------------------- ----------- ----------- ------------------------------- --------------------- ------------------------- -------------------------------- ------------------------------- -------------------
[Model]{} [$\varepsilon$]{} [$B$]{} [$\chi_0$]{} [$(\Delta\rho/\rho)_{\rm c}$]{} [$r_g$]{} [$r_s$]{} [Pert.]{} [Ampl.]{} [$\bar{t}_{\rm adv}$]{} [$\bar{\chi}$]{} [$E_{\rm kin,g}/M_{\rm g}$]{} [$t_{\rm exp}$]{}
[ ]{} [($v_{\rm ff0}^2/2$)]{} [($r_{\rm s0} v_{\rm ff0}^3$)]{} [($t_{\rm ff0}$)]{} [($10^{-2}\,v_{\rm ff0}^2$)]{} [($t_{\rm ff0}$)]{}
e0B00 0 0 0 ... ... 1 rand. $\delta \mathbf{v}/v_r$ $10^{-3}$ 8.9 0 ... ...
e0B02 0.002 0.06 0.14 0.90 1.04 9.5 0.69 3.7 ...
e0B04 0.004 0.3 0.07 0.78 1.09 10.2 0.64 2.9 ...
e0B06 0.006 0.6 0.05 0.72 1.14 11.3 0.81 2.7 ...
e0B08 0.008 1.0 0.05 0.69 1.20 13.1 1.06 2.7 ...
e0B10 0.010 1.5 0.04 0.67 1.28 ... ... ... 336
p0B08L1 0 0.008 1.0 0.05 0.69 1.20 $\ell=1$ shell 0.1 12.5 1.2 2.3 ...
p0B10L1 0.010 1.5 0.04 0.67 1.28 ... ... ... 218
p0B10L2 $\ell=2$ shell ... ... ... 376
p0B10R1 rand. $\delta\rho/\rho$ 0.1 ... ... ... 127
p0B10R3 0.3 ... ... ... 241
p0B10G4 $\ell=4$ gain 0.5 ... ... ... 246
p0B10G5 $\ell=5$ gain ... ... ... 122
e3B00 0.3 0 0 ... ... 1 rand. $\delta \mathbf{v}/v_r$ $10^{-3}$ 19.8 0 ... ...
e3B02 0.002 1.5 0.021 0.66 1.06 22.5 0.89 0.9 ...
e3B04 0.004 3.9 0.016 0.60 1.13 25.5 1.33 1.9 ...
e3B06 0.006 7.1 0.013 0.58 1.23 34.8 2.11 2.6 ...
e3B08 0.008 8.0 0.010 0.57 1.25 ... ... ... 223
----------- ------------------------- ---------------------------------- -------------- --------------------------------- ----------- ----------- ------------------------------- --------------------- ------------------------- -------------------------------- ------------------------------- -------------------
### Models Evolved and Initial Perturbations {#s:models}
Based on the linear stability analysis of @foglizzo06, the transition from SASI- to convection-dominated behavior occurs when the parameter $$\label{eq:convection_parameter}
\chi = \int_{r_{\rm g}}^{r\rm s}\,\frac{\textrm{Im}(\omega_{\rm BV})}{|v_r|}\,\totd r,$$ exceeds a critical value of the order of 3. Here $r_g$ is the gain radius, $\omega_{\rm BV}$ is the buoyancy frequency, $$\label{eq:brunt}
\omega^2_{\rm BV} = \frac{GM}{r^2}\left[\frac{1}{\gamma}\frac{\partial \ln p}{\partial r}
-\frac{\partial\ln\rho}{\partial r}\right]$$ and $v_r$ is the radial velocity. This critical value of $\chi$ is the number of e-foldings by which an infinitesimal buoyant perturbation needs to grow to counter advection out of the gain region. Larger heating rates and longer advection times are favorable for the growth of convection, as they increase $\chi$.
A finite-amplitude density perturbation can also overcome the stabilizing effect of advection when $\chi < 3$. The minimum amplitude required for a perturbation to rise buoyantly against the accretion flow is [@thompson00; @scheck08; @FT09b; @dolence2013; @couch2012] $$\label{eq:delta_crit}
\left(\frac{\Delta \rho}{\rho}\right)_{\rm c} \simeq \frac{C_D\, v_2^2}{2l_v\, g_s}$$ where $v_2$ and $g_s$ are the postshock velocity and gravitational acceleration at the shock, respectively, $C_D$ is the drag coefficient of the perturbation ($\simeq 0.5$ for a sphere), and $l_v$ is the ratio of the volume to the cross-sectional area of the perturbation in the direction of gravity ($4/3$ times the radius, for a sphere).
[f1a.eps]{} (18,63)[$\ell$]{} (61,78.5)[[$\ell$]{}]{} (76,80)[[$\ell$]{}]{} (82,88.5)[[$\ell$]{}]{}
[f1b.eps]{} (20,64.5)[[$\ell$]{}]{} (40,78.7)[[$\ell$]{}]{} (70,87.5)[[$\ell$]{}]{} (78,63.5)[$\ell$]{}
We evolve two different sequences of models for which the heating rate $B$ is varied from zero to a value that yields an explosion in 2D, and a third set of models that explores the effect of large-amplitude perturbations on a SASI-dominated background state. All models are summarized in Table \[t:models\].
The first sequence (e0) is such that all models are well within the $\chi<3$ regime, corresponding to a SASI-dominated system. This background flow is obtained by setting the dissociation parameter $\varepsilon$ to zero. The flow is initially perturbed everywhere with random cell-to-cell velocity fluctuations, with an amplitude $0.1\%$ of the local radial velocity.
The second sequence (e3) has the dissociation parameter set to $30\%$ of the gravitational energy at the shock position without heating ($r=r_{\rm s0}$). The larger density jump yields smaller postshock velocities [@FT09a], increasing the value of $\chi$. Most of the models in this sequence lie in the $\chi>3$ regime, and are therefore convection-dominated. This combination of parameters is the same as used in one of the sequences of @FT09b. The same set of initial perturbations as in the e0 sequence are used.
A third set of models (p0) has large-amplitude initial perturbations applied mostly to the exploding model of the e0 sequence. We explore the effect of random cell-to-cell density perturbations in the entire computational domain with an amplitude of $10\%$ and $30\%$, overdense shells in the upstream flow that trigger specific SASI modes (see @FT09a for details), and density perturbations in the gain region which are radially-constant from $r_g$ to $r_s$, but with an angular dependence set by a Legendre polynomial. The latter are aimed at exciting convection by large-scale perturbations. The amplitude is chosen to be $50\%$.
Note that we focus on exploding models that are marginally above the heating rate for explosion, where non-radial instabilities are expected to have the maximum effect. Further increase of the heating rate yields explosions that develop earlier, eventually approaching the spherically-symmetric runaway condition (e.g., Appendix \[s:L0\]).
### Linear Stability Properties {#s:linear_stability}
The linear stability properties of the two sets of background flow configurations (e0 and e3) are shown in Figure \[f:growth\_timescales\]. The growth rates of the fundamental $\ell=1$, and $\ell=2$ modes as a function of heating rate are monotonically decreasing as long as $\chi < 3$. Above $\chi > 3$, modes transition into a non-oscillatory (convective) state with two branches, in line with the results of @yamasaki07. The mode $\ell_{\rm crit}$ that bifurcates at the lowest heating rate ($\chi \simeq 3$) is approximately that for which $2\ell_{\rm crit}$ eddies of size $(r_s-r_g)$ fit into a transverse wavelength [@foglizzo06] $$\label{eq:lmax_def}
\lambda_{\perp,\rm crit} \equiv \frac{\pi (r_s+r_g)}{\sqrt{\ell_{\rm crit}(\ell_{\rm crit}+1)}} \sim 2(r_s - r_g).$$ Modes with larger or smaller $\ell$ bifurcate at higher heating rate.
Figure \[f:growth\_timescales\] also shows two important timescale ratios in the stationary solution as a function of heating rates. The first one is the ratio of advection times in the gain and cooling regions, $t_{\rm adv-g}$ and $t_{\rm adv-c}$, respectively. On the basis of numerical simulations with a realistic EOS, @F12 found that equality between these two timescales at $t=0$ corresponds approximately to the onset of oscillatory instability. Figure \[f:growth\_timescales\] shows that this relation is valid for the $\varepsilon=0$ sequence, losing accuracy when nuclear dissociation is included.
The *instantaneous* value of the ratio of advection to heating timescales in the gain region has for long been used to quantify proximity to an explosion in numerical simulations [@janka98; @thompson00; @thompson05]. @F12 found that equality between these two timescales in the *initial condition* – or equivalently, at the time of shock stalling – marks approximately the subsequent onset of non-oscillatory $\ell=0$ instability in numerical simulations. However, Figure \[f:growth\_timescales\] shows that the point where the linear $\ell=0$ growth rate bifurcates to a non-oscillatory mode lies at a much higher heating rate than the point where $t_{\rm adv-g} = t_{\rm heat-g}$ in both sequences. Nevertheless, it is shown in Appendix \[s:L0\] that non-oscillatory instability still sets in at the heating rate for which these timescales are equal in the initial condition, indicating that the expansion is a non-linear effect[^1].
Spherical Fourier-Bessel Spectral Decomposition {#s:sfb_outline}
-----------------------------------------------
To analyze the properties of the flow accounting for its intrinsic spherical geometry, we employ a spherical Fourier-Bessel expansion to perform various spectral decompositions. This set of functions forms an orthogonal basis of two- or three-dimensional space in spherical coordinates, allowing the expansion of an arbitrary scalar function $f(r,\theta,t)$ in a series of the form $$\label{eq:sfb_2D}
f(r,\theta,t) = \sum_{n,\ell} f_{n\ell}(t) g_\ell(k_{n\ell}r) P_\ell(\cos\theta),$$ where $g_\ell(k_{n\ell}r)$ are the radial basis functions, $k_{n\ell}$ is the radial wave number of order $n$, $P_\ell(\cos\theta)$ are the Legendre polynomials of index $\ell$, and $f_{n\ell}(t)$ are (time-dependent) scalar coefficients. Expansions of this form have previously been used in the context of galaxy redshift surveys (e.g., @fisher1995). Appendix \[s:sfb\_appendix\] contains a detailed description of the expansion method, including the straightforward extension to three-dimensional space. In what follows we provide a brief outline, focusing on the quantities needed to analyze the turbulent flow in our 2D models.
The domain considered is the volume enclosed between two concentric spheres of inner and outer radii $r_{\rm in}$ and $r_{\rm out}$, respectively. These spherical surfaces can be any pairwise combination of the neutrinosphere, gain radius, or shock radius, depending on the particular region to be studied. The radial basis functions $g_\ell(k_{n\ell}r)$ are linear combinations of spherical Bessel functions $j_\ell$ and $y_\ell$, with coefficients chosen to satisfy specific boundary conditions at both interfaces (Appendix \[s:sfb\_appendix\]).
Imposing these boundary conditions generates a set of discrete radial wave numbers $k_{n\ell}$, in analogy with the modes of a membrane in cylindrical coordinates. In addition to its quantum numbers $n$ and $\ell$, these wave numbers depend on the chosen ratio of inner and outer radii $r_{\rm in}/r_{\rm out}$. Appendix \[s:sfb\_appendix\] derives the wave numbers, relative coefficients, and normalization of the radial basis functions for the cases of vanishing (Dirichlet) and zero gradient (Neumann) boundary conditions. For low $n$, $\ell$, and $r_{\rm in}/r_{\rm out}\to 1$, these wave numbers approach $$k_{n\ell}\to \frac{\pi}{(r_{\rm out}-r_{\rm in})}(n+1), \quad (n=0,1,2,...)$$ increasing in value for stronger curvature. The normalized basis functions satisfy the orthogonality relation (equations \[eq:normalization\_condition\_dirichlet\] and \[eq:normalization\_condition\_neumann\]) $$\int_{r_{\rm in}}^{r_{\rm out}}r^2\totd r\, g_{\ell}(k_{n\ell}r)g_{\ell}(k_{m\ell}r) = \delta_{nm}.$$ The coefficients for the spherical Fourier-Bessel expansion in equation (\[eq:sfb\_2D\]) are thus $$f_{n\ell}(t) = \frac{2\ell+1}{2}\int f(r,\theta,t) g_\ell(k_{n\ell}r)\,P_\ell(\cos\theta)\,
r^2dr\,\sin\theta\totd\theta.$$
From Parseval’s identity, $$\label{eq:parseval_2d}
\int |f|^2\, \totd^2 x = \sum_{n,\ell} \frac{2}{2\ell+1}\,|f_{n\ell}|^2,$$ one can define a discrete power spectral density in 2D space $$P_{n\ell} = \frac{2}{2\ell+1}\,|f_{n\ell}|^2.$$ The coefficients $f_{n\ell}(t)$ can also be Fourier analyzed in time, yielding an individual power spectrum for each $(n,\ell)$ mode. Using a Discrete Fourier Transform (DFT) in time, the normalization can be taken to be the time-average of the volume integral of the variable in question (e.g., @NR), $$\begin{aligned}
\frac{1}{N_q}\sum_q \int |f|\, \totd^2 x & = & \frac{1}{N_q^2}\sum_{n\ell q} \frac{2}{2\ell+1}\,|\widehat{f}_{n\ell q}|^2\\
\label{eq:3d_spectrum}
&\equiv & \sum_{n\ell q} \mathcal{P}_{n\ell q},\end{aligned}$$ where $N_q$ is the number of time samples, and $\widehat{f}_{n\ell q}$ is the DFT of $f_{n\ell}(t)$ at frequency $q$.
In practical applications, the series in equation (\[eq:parseval\_2d\]) must be truncated at a finite value of the indices. In our analysis we set these maximum indices to be at most half the number of grid points in the corresponding direction, in analogy with the Nyquist limit in cartesian coordinates.
General Relativistic, Radiation-Hydrodynamic Simulations
--------------------------------------------------------
We use the set of two-dimensional, general-relativistic, radiation-hyrodynamic simulations of @mueller2012 to test the validity of the general principles inferred from the parametric models. The @mueller2012 models follow the evolution of a star of mass $8.1M_\odot$ and metallicity $Z=10^{-4}$ (A. Heger 2013, private communication), and a $27M_\odot$ star of solar metallicity [@woosley02]. The code employed is VERTEX-CoCoNuT [@mueller2010], which treats multi-group neutrino transport using the ‘ray-by-ray-plus’ approach [@rampp02; @bruenn2006; @buras06a].
These two successfully exploding models follow very different paths on their way to runaway expansion. The $8.1M_\odot$ progenitor (model u8.1) becomes dominated by convection shortly after the shock stalls, and remains so until runaway sets in. In contrast, the $27M_\odot$ model (s27) develops a strong SASI throughout the evolution.
Results {#s:results}
=======
Transition to Explosion
-----------------------
We first concentrate on the differences in the transition to explosion introduced by the initial dominance of the SASI or convection. To this end, we focus the discussion on models that bracket the critical heating rate for explosion (Table \[t:models\]). We then discuss the effect of different initial perturbations on exploding models.
![Time-series diagnostics for SASI-dominated models below and above the threshold for explosion (p0B08L1 and p0B10L1, respectively). The $\ell=1$ SASI mode is excited with an overdense shell. The top panels shows $\ell=0$ Legendre coefficient (eq. \[\[eq:legendre\_coeff\]\]) of the shock surface (thick line), as well as minimum and maximum shock radii (thin lines). Middle panels show $\ell=1$ shock Legendre coefficient. Bottom panels show the fraction of the post-shock volume with an entropy higher than a given value (eq. \[\[eq:volume\_fraction\]\]). Note that bubble destruction (sudden decreases in $f_V$ for high entropy) precedes large amplitude sloshings of the shock (as indicated by $a_1$ changing sign). The unit of length is the initial shock radius without heating $r_{\rm s0}$ and the unit of time is the free-fall time at this position ($\sim 3$ ms for a central mass of $1.3M_\sun$ and $r_{\rm s0}\sim 150$ km, §\[s:setup\]).[]{data-label="f:shock_entropy_sasi"}](f2.eps){width="\columnwidth"}
![Entropy at selected times in the evolution of model p0B10L1, which explodes SASI-dominated. Panels show the seeding of perturbations in one hemisphere and bubble disruption on the other (a), loss of coherence of SASI (b), development of first large bubble (c), partial disruption and displacement of bubble (d), and final expansion (e). Compare with Figure \[f:shock\_entropy\_sasi\].[]{data-label="f:shock_entropy_snapshots"}](f3.eps){width="\columnwidth"}
### Interplay of SASI and Convection
The characteristic behavior of models with an early dominance of the SASI is illustrated in Figures \[f:shock\_entropy\_sasi\] and \[f:shock\_entropy\_snapshots\]. Initially, the $\ell=1$ shock Legendre coefficient displays sinusoidal oscillations of exponentially growing amplitude. While in models without heating the SASI grows in amplitude until oscillations saturate while keeping its characteristic period (e.g., @FT09a), in models with significant heating this period increases when the amplitude becomes large, and eventually the regularity of the oscillation is lost.
![Same as Figure \[f:shock\_entropy\_sasi\], but for convection-dominated models that bracket the threshold for explosion (e3B06 and e3B08). Even though $\ell=1$ shock oscillations have no clear periodicity, the relation between destruction of high-entropy bubbles and large amplitude shock sloshings is still present. The unit of length is the initial shock radius without heating $r_{\rm s0}$ and the unit of time is the free-fall time at this position ($\sim 3$ ms for a central mass of $1.3M_\sun$ and $r_{\rm s0}\sim 150$ km, §\[s:setup\]).[]{data-label="f:shock_entropy_conv"}](f4.eps){width="\columnwidth"}
This breakdown of the SASI cycle is due to large-scale, long-lived fluid parcels with enhanced entropy emerging in the post-shock region. These structures are seeded during shock expansions (Figure \[f:shock\_entropy\_snapshots\]a; see also @scheck08). For small shock displacements, these elongated bubbles are shredded by lateral flows inherent in the SASI, and are advected out of the gain region, allowing the advective-acoustic cycle to proceed as in the case without heating. Above a certain amplitude, however, bubbles are able to resist shredding, and the SASI cycle is interrupted. Accretion proceeds then along narrow downflows that circumvent the bubbles (Figure \[f:shock\_entropy\_snapshots\]b).
To quantitatively analyze the interplay between sloshing of the post-shock region and large-scale bubbles, we compare in Figure \[f:shock\_entropy\_sasi\] the evolution of the $\ell=1$ shock Legendre coefficient $a_1$, where $$\label{eq:legendre_coeff}
a_\ell(t) = \frac{2\ell+1}{2}\int_0^{\pi} r_s(\theta,t) P_\ell(\cos\theta)\sin\theta\,\totd \theta,$$ with the fraction of the post-shock volume with entropy higher than a fiducial value $s_0$: $$\label{eq:volume_fraction}
f_V(s>s_0) = \frac{1}{V}\int_{s_0}^\infty \frac{\totd V}{\totd s}\totd s,$$ where the entropy $$s = \frac{1}{\gamma-1}\ln\left[\frac{p}{p_s}\left(\frac{\rho_2}{\rho}\right)^\gamma\right]$$ is defined so that it vanishes below the shock in the initial model ($p_2$ and $\rho_2$ are the initial post-shock pressure and density, respectively; e.g. @F07), and the post-shock volume is defined as $$V(t) = 2\pi\int_{0}^\pi\int_{r_*}^{r_s(\theta,t)}r^2\totd r\, \sin\theta\totd\theta.$$ We use volume instead of mass to minimize the influence of low-entropy downflows. The emergence of peaks in $f_V(t)$ for high values of the entropy is related to the loss of periodicity and eventual halting of shock sloshings, while bubble destruction can allow regular periodicity to emerge again (c.f. Figure \[f:shock\_entropy\_sasi\]c,e in the range $t\in [200,250]t_{\rm ff0}$).
Large-scale bubbles that have halted the SASI can nevertheless be broken when low-entropy downflows bend and flow laterally. This process triggers bubble disruption, and results in their shredding or displacement to the opposite hemisphere (Figure \[f:shock\_entropy\_snapshots\]). Accretion is then able to proceed through the whole hemisphere previously occupied by the bubble, and the shock executes a sloshing (c.f. Figure \[f:shock\_entropy\_sasi\]d,f in the range $t\in[150,200]t_{\rm ff0}$, also Figure \[f:shock\_entropy\_snapshots\]d). The shock retractions are related to a decrease in pressure support triggered by an increase in cooling. The buoyancy of high-entropy bubbles blocks the flow of gas to the cooling region, resulting in a lower amount of cooling per SASI cycle and a loss of periodicity in the shock oscillations.
The key difference between exploding and non-exploding models appears to be whether the system can form entropy perturbations of sufficient size and amplitude. Model p0B10L1 displays such an entropy enhancement at time $t\sim 115t_{\rm ff0}$. This enhancement is perturbed and displaced around time $t\simeq 175t_{\rm ff0}$, triggering a large sloshing of the shock that transitions into runaway expansion. In contrast, model p0B08L1 fails to develop a long-lived structure with entropy higher than $s>1.5$. The transition to explosion for a large enough bubble results from the relative importance of buoyancy and drag forces [@thompson00].
The characteristic evolution of convection-dominated models is illustrated by Figure \[f:shock\_entropy\_conv\]. Entropy enhancements are initially generated by convection. Bubbles grow and merge into large-scale structures, which cause non-linear shock displacements. In non-exploding models, bubbles have a short lifetime, and hence the shock undergoes sloshings of moderate amplitude over a range of temporal frequencies. Note that the destruction of large bubbles can also lead to shock sloshings, but the persistent generation of entropy fluctuations of smaller scale and amplitude prevent the emergence of SASI oscillations with a well-defined periodicity.
For high enough heating rate, large bubbles are able to survive for many eddy turnover times, leading to explosion in a manner similar to that of SASI-dominated models. The role of high-entropy bubbles in convection-dominated models has been documented previously [@dolence2013; @couch2012].
![Time-series diagnostics for exploding models with different initial perturbations (Table \[t:models\]). Top panels show average shock radius, middle panels show $\ell=1$ shock Legendre coefficient, and bottom panel shows fraction of the postshock volume with entropy higher than unity. Note the longer time to explosion and late onset of $\ell=1$ oscillations in models with even $\ell$ perturbations (p0B10L2 and p0B10G4). The unit of length is the initial shock radius without heating $r_{\rm s0}$ and the unit of time is the free-fall time at this position ($\sim 3$ ms for a central mass of $1.3M_\sun$ and $r_{\rm s0}\sim 150$ km, §\[s:setup\]).[]{data-label="f:shock_entropy_pert"}](f5.eps){width="\columnwidth"}
### Effect of Initial Perturbations
The effect of different initial perturbations on the exploding model of the e0 sequence is illustrated in Figure \[f:shock\_entropy\_pert\]. Models with large amplitude random cell-to-cell density perturbations (p0B10R1 and pB10R3) follow the same path as the model where $\ell=1$ is directly perturbed (p0B10L1, Fig. \[f:shock\_entropy\_sasi\]). The model with an $\ell=2$ perturbation (p0B10L2) undergoes a weak convective phase over a number of advection times, during which $\ell=0$ grows and $\ell=2$ saturates at a small amplitude. After a delay of $\sim 100t_{\rm ff0}$, however, $\ell=1$ oscillations of the shock emerge, and the model joins the usual SASI-dominated explosion path. The models with large amplitude density perturbations in the gain region with a fixed $\ell=4$ and $5$ dependence (p0B10G4 and p0B10G5) trigger less regular sloshings of the shock, which however still result in the formation of large-scale bubbles. As with the $\ell=2$ perturbation, an even-$\ell$ convective perturbation takes longer to couple to an $\ell=1$ SASI mode.
The time to explosion appears to be a non-trivial function of the perturbation form and amplitude. Model p0B10R3 has larger amplitude perturbations, yet it hits the outer boundary $100$ dynamical times later than model p0B10R1. Despite the very large amplitude perturbation of model p0B10G4, it explodes later than all models with an odd $\ell$ perturbation. This strong sensitivity to initial conditions has been documented previously by @scheck06.
We emphasize however that we are focusing on models that are barely above the threshold for explosion. Recently, @couch2013c have pointed out the importance of pre-collapse perturbations in tilting the balance towards explosion. Such an effect is likewise only going to make a difference if a model is already close to exploding in the absence of perturbations. For instance, models e0B10 and p0B10L1 differ in the type and amplitude of perturbations, leading to explosions that differ by more than $100$ dynamical times in onset. In contrast, neither of models e0B08 or p0B08L1 explode, despite the fact that they mirror the exact perturbations as the previous two exploding models (the latter having a $10\%$ density perturbation in the form of a thin shell).
From our results it is not obvious that a large enough density perturbation suffices to turn a model for which the background state is SASI-dominated into a convectively dominated model. Note however that our models have $\chi \ll 3$. Previous studies have witnessed more sensitivity to the type of initial perturbation when the $\chi$ parameter at shock stalling is close to or even transiently exceeds criticality [@scheck08; @hanke2013].
![Time- and angle-average profiles of selected quantities for non-exploding models. Top, middle, and bottom panels show models with no heating (e0B00), SASI-dominated close to explosion (e0B08), and convection-dominated close to explosion (e3B06), respectively. Curves correspond to r.m.s. density fluctuation normalized to its mean value at each radius (eqns. \[\[eq:time\_average\]\]-\[\[eq:rms\_fluctuation\]\], solid black), r.m.s. Mach number (solid red), r.m.s. radial velocity (dashed blue), r.m.s. meridional velocity (solid green), and average sound speed (dashed orange). The vertical dotted lines in panel (c) bracket the radial range where the post-shock flow is subsonic and free from strong stratification effects, with $r_{\rm in}$ and $r_{\rm out}$ corresponding to the peak of the average sound speed, and the average of the minimum shock radius minus its r.m.s. fluctuation (c.f. Fig. 8 of @FT09a), respectively.[]{data-label="f:profiles_timeave"}](f6.eps){width="\columnwidth"}
The ‘purity’ of an excited $\ell=1$ SASI mode also depends on whether the background flow allows for unstable harmonics. Figure \[f:growth\_timescales\] shows that the first $\ell=1$ overtone is unstable for all the heating rates in the e0 sequence. This may lead to shock oscillations that are not a clean sinusoid, but which should not be mistaken as an imprint of convection.
Properties of the Quasi-Steady State {#s:quasi-steady}
------------------------------------
We now address the properties of the turbulent flow in the gain region in cases where an explosion is not obtained, focusing on the differences between models where either SASI or convection dominate. We first discuss general properties of the time-averaged flow, and then analyze models using a spherical Fourier-Bessel decomposition in space and a discrete Fourier transform in time.
### Time-Averaged Flow and Convective Stability
![*Top:* Squared buoyancy frequency (eq. \[\[eq:brunt\]\]) as a function of radius for a convection-dominated model close to explosion (e3B06). Curves show initial (red) and time-angle-averaged values (black). *Bottom:* angle-averaged convection parameter as a function of time for the same model (e3B06). The time average value $\langle \chi\rangle$ is much larger than what is obtained when computing this parameter with quantities from the time-averaged flow, $\bar\chi$ (eq. \[\[eq:chi\_mean\_flow\]\]), because $\chi$ is a non-linear function.[]{data-label="f:brunt_comparison"}](f7a.eps "fig:"){width="\columnwidth"} ![*Top:* Squared buoyancy frequency (eq. \[\[eq:brunt\]\]) as a function of radius for a convection-dominated model close to explosion (e3B06). Curves show initial (red) and time-angle-averaged values (black). *Bottom:* angle-averaged convection parameter as a function of time for the same model (e3B06). The time average value $\langle \chi\rangle$ is much larger than what is obtained when computing this parameter with quantities from the time-averaged flow, $\bar\chi$ (eq. \[\[eq:chi\_mean\_flow\]\]), because $\chi$ is a non-linear function.[]{data-label="f:brunt_comparison"}](f7b.eps "fig:"){width="\columnwidth"}
The spatial structure of the quasi-steady-state becomes clear when the flow is averaged in angle and time (e.g., @FT09a). Figure \[f:profiles\_timeave\] shows such a representation for a model without heating (e0B00), as well as SASI- and convection-dominated models close to an explosion (e0B08 and e3B06, respectively). The time- and angle-average of a generic scalar quantity is denoted by $$\label{eq:time_average}
\langle A(r) \rangle = \frac{1}{2(t_{\rm f}-t_{\rm i})}\int_{t_{\rm i}}^{t_{\rm f}}\totd t\,
\int_0^\pi A(r,\theta,t)\,\sin\theta\totd\theta,$$ where $[t_i,t_f]$ is the time interval considered for the average, and the corresponding root-mean-square (r.m.s.) fluctuation is defined as $$\label{eq:rms_fluctuation}
\Delta A_{\rm rms} = \left[ \langle A^2\rangle-\langle A\rangle^2\right]^{1/2}.$$
All three models share a basic general structure. From the inside out, this structure is composed of a narrow cooling layer adjacent to $r_*$, a region of sub-sonic turbulence encompassing part of the cooling layer and part of the (time-averaged) gain region, an extended zone of shock oscillation, and the unperturbed upstream flow.
The most notorious difference among these models lies in the properties of the shock oscillation zone and in the flow around the cooling layer. Models where the SASI dominates have a wider shock oscillation zone than the model where convection is dominant. This can be seen by comparing the minimum and maximum shock radii of the non-exploding models in Figures \[f:shock\_entropy\_sasi\] and \[f:shock\_entropy\_conv\]. Also, in models where the SASI is prominent there is a bump in the r.m.s lateral velocity in the cooling layer, indicating strong shear. This bump is absent in the convection-dominated model.
In contrast, the subsonically turbulent region has very similar properties in the three different models shown in Figure \[f:profiles\_timeave\], with only slight changes in the radial slopes. Characteristic values are $\Delta \rho_{\rm rms}/\langle \rho\rangle \sim 0.25$, r.m.s. Mach number $\sim 0.5$, and $\Delta v_{\rm r,rms}\simeq \Delta v_{\theta,{\rm rms}}\sim 0.15v_{\rm ff0}$. This similarity in time-averaged properties suggests that flows are not very different from each other.
By analogy with convective systems in steady-state (e.g., nuclear burning stars), one can investigate whether the time-averaged system adjusts itself to a state of marginal convective stability. In hydrostatic systems, convection acts to erase destabilizing gradients, whereas the presence of advection in core-collapse supernova flows generates a non-zero entropy gradient in steady-state [@murphy11]. One can nevertheless ask whether the relevant critical parameter for convection is restored to stability in the non-linear regime.
[f8.eps]{} (81.5,1)[$\ell$]{} (1.3,48)
[f9.eps]{} (54,1.4)[[$\ell$]{}]{}
Figure \[f:brunt\_comparison\]a shows the initial and time-averaged squared buoyancy frequency (eq. \[\[eq:brunt\]\]) for model e3B06, which is convection dominated. This model has an initial value of $\chi \simeq 7$ (Table 1). The time-averaged flow is such that the degree of convective instability (negative $\omega_{\rm BV}^2$) is significantly weaker than that in the initial state. The implications for convective stability become clear when the $\chi$ parameter (eq. \[\[eq:convection\_parameter\]\]) is computed for the time-average flow. One way of doing this is simply averaging $\chi$ in time and angle, $\langle \chi\rangle$. However, because this is a non-linear function of the flow variables, the resulting value will not only capture the properties of the mean flow, but it will also include the contribution of turbulent correlations in the pressure, density, and velocity. One can nevertheless still define a convection parameter based on the properties of the mean flow $$\label{eq:chi_mean_flow}
\bar{\chi} = \int \frac{{\rm Im}\left(\langle \omega_{\rm BV}^2\rangle^{1/2}\right)}{\langle v_{\rm r}\rangle}\,\totd r,$$ where the integral extends over regions where $\omega_{\rm BV}^2 <0$.
The difference between these two ways of computing $\chi$ is illustrated in Figure \[f:brunt\_comparison\]b. Shown is the instantaneous angle-averaged value of $\chi$, together with its time average $\langle \chi\rangle$ as well as the convection parameter computed using the mean flow, $\bar\chi$ (eq. \[\[eq:chi\_mean\_flow\]\]). The instantaneous angle-averaged value of $\chi$ achieves very large values as soon as the shock displacement becomes non-linear, similar to the results of @burrows2012, with a time-averaged value $\langle \chi \rangle \sim 50$. The convection parameter from the mean flow is much smaller, however, yielding $\bar\chi \simeq 2$. This small number arises from the small magnitude of the time-average of the squared buoyancy frequency shown in Figure \[f:brunt\_comparison\]a.
Values of $\bar\chi$ for all non-exploding models are shown in Table \[t:models\]. All convection-dominated models satisfy $\bar\chi <3$, which indicates that in quasi-steady-state they adjust to a state of convective sub-criticality (the equivalent of ‘flat’ entropy gradients in hydrostatic systems). The SASI-dominated models maintain $\chi_0 \lesssim \bar{\chi} < 3$, where $\chi_0$ is the value of $\chi$ in the initial condition. This *increase* in the time-averaged value of the convection parameter can arise from the increase in the size of the gain region caused by SASI activity, and from the presence of localized entropy gradients induced by the SASI, which trigger secondary convection (e.g., Figure \[f:shock\_entropy\_snapshots\]a).
It is worth emphasizing that the driving agent matters in characterizing convective motions: secondary convection is qualitatively different from neurino-driven convection in that in the former there are both preferred spatial and temporal scales (entropy perturbations induced by the SASI, and advection time, respectively).
Another aspect of the explosion mechanism that can be probed with the time-averaged flow is the dependence of the turbulent kinetic energy in the gain region on neutrino heating. @hanke2012 found that a good indicator of the proximity of an explosion is the growth of the turbulent kinetic energy on the largest spatial scales. Since the mass in the gain region also increases due to the larger average shock radius, it is worth clarifying the origin of the increase in the kinetic energy. Table \[t:models\] shows the ratio of the total time-averaged turbulent kinetic energy in the gain region to the time-averaged mass in the gain region for non-exploding models. SASI-dominated models are such that this ratio is nearly constant, decreasing slightly when an explosion is closer. Thus larger kinetic energy is due solely to the increase in the mass of the gain region. In contrast, convection-dominated models grow both the specific kinetic energy and the mass in the gain region as an explosion is closer.
### Properties of Turbulence in the Subsonic Region
We now use the spherical Fourier-Bessel expansion to analyze the properties of the turbulence in the subsonic region of the time-averaged flow. Operationally, we define the radial limits of this region ($r_{\rm in}$ and $r_{\rm out}$, §\[s:sfb\_outline\]) to be the peak of the time-averaged sound speed, $\langle c_s\rangle$, and the time-average of the minimum shock radius minus its r.m.s. fluctuation, $r_{\rm out} = \langle r_{\rm s,min}\rangle-\Delta r_{\rm s,min,rms}$, respectively. This definition differs from that of @murphy11 in that we restrict ourselves to radii below the minimum shock position to avoid supersonic flow.
To connect with previous studies, we use the meridional velocity $v_\theta$ as a proxy for the turbulent flow. We do not multiply this velocity by $\sqrt{\rho}$, however, because the density stratification over the extended radial range considered would affect the spectral slopes [@endeve2012]. Thus, the sum of the total power (eq. \[\[eq:3d\_spectrum\]\]) does not approach the total kinetic energy in the subsonic region, but instead it is a measure of the kinetic energy per unit mass.
Figure \[f:sfb\_2d\_mosaic\] shows 2D projections of the 3D space-time spectrum $\mathcal{P}_{n\ell q}$ (eq. \[\[eq:3d\_spectrum\]\]), for pure SASI, SASI-dominated, and convection-dominated models (c.f. Figure \[f:profiles\_timeave\]). Power is maximal at low angular and radial scales, as expected from the inverse turbulent cascade in 2D (e.g., @davidson). Models where the SASI is prominent display two characteristic features: (1) an even-odd pattern in the radial spectrum for $\ell = 0-5$, indicating the presence of discrete modes, and (2) enhanced power around the frequency corresponding to the advection time of the mean flow, $\bar{f}_{\rm adv}\sim 0.1t_{\rm ff0}^{-1}$. The dominance of convection manifests as a broadening of the smoother component of the spatial spectrum to larger $n$ and $\ell$, a near disappearance of the even-odd pattern, and the emergence of power at temporal frequencies below and above $\bar{f}_{\rm adv}$. This behavior of SASI- and convection-dominated models in the frequency-domain is consistent with the results of @mueller2012 and @burrows2012.
Figure \[f:spectra1d\_mosaic\] shows the results of contracting the $\mathcal{P}_{n\ell q}$ array along two dimensions, yielding one dimensional spectra, for models that do not explode. In SASI-dominated models, the normalized power as a function of $n$ shows a characteristic sawtooth shape, which is smoothed to clarify the slope (an example of an non-smoothed spectrum is shown by the gray curve in Figure \[f:spectra1d\_mosaic\]a).
Increasing the heating rate leads to minor changes in the (normalized) radial spectrum in SASI-dominated models. The onset of convection, on the other hand, leads to a shift of power from $n \leq 2$ to $n\geq 3$. The spectral slope at large $n$ is approximately $n^{-2}$. This slope could be attributed to Rayleigh-Taylor turbulence, for which the velocity fluctuations satisfy $\delta v \propto \lambda^{1/2}$, with $\lambda$ the wavelength of the perturbation (e.g., @niemeyer1997 [@ciaraldi2009]). Note however that the wave numbers of the radial basis functions of different $\ell$ are not harmonic with each other (Fig. \[f:zeroes\_eigenfunctions\_dirichlet\]), hence one cannot straightforwardly map radial wavelength into index $n$. Nevertheless, the spacing between wave numbers becomes nearly constant at large $n$, with only a linear shift with $\ell$, motivating the use of $n$ as a differential measure of the turbulent cascade.
The angular spectrum in SASI-dominated models shows a peak at $\ell=2$, and a slope at large $\ell$ indicative of a direct vorticity cascade [@kraichnan1967]. Similar to the radial spectrum, the onset of convection results in the shift of power from $\ell \leq 2$ towards $\ell = 5-10$. The resulting spectral shape has a form similar to that found by @hanke2012, @couch2012, and @dolence2013, who radially averaged the kinetic energy over a thin slice. This shape consists of a shallow curved shape at low $\ell$, transitioning to $\sim\ell^{-3}$ slope at large $\ell$.
The temporal spectrum of the sequence of convection-dominated models is consistent with the results of @burrows2012. At very low heating rates, a prominent peak exists at the advection frequency $\bar{f}_{\rm adv}$, indicating the presence of the SASI. As the heating rate is increased, power increases at frequencies below and above the advection peak. At heating rates close to an explosion, this low-frequency power is comparable or higher than that at $\bar{f}_{\rm adv}$.
In contrast, the SASI-dominated sequence has a dominant peak at the advection frequency for all models. This peak moves to lower frequencies as heating is increased, because the advection time increases given the larger average shock radius (Table \[t:models\]). Also, the peak becomes broader as a likely result of secondary convection being triggered by the SASI. Power at the lowest frequencies still increases with heating rate, but it remains below that in the advection peak by at least a factor of two (in contrast, neutrino-driven convection yields a nearly flat spectrum). Note also that the power at frequencies higher than the advection peak in model e0B08 (SASI-dominated model with the highest heating) is within a factor of two of the convection-dominated model with the highest heating (e3B06).
From Figure \[f:profiles\_timeave\] one can infer the turnover time of large eddies to be $t_{\rm eddy} \sim 2\pi r/\Delta v_{\theta,{\rm rms}}\sim 30t_{\rm ff0}$, yielding a frequency $f_{\rm eddy}\sim 0.03t_{\rm ff0}^{-1}$. Thus, the increase in power at frequencies below the advection time appears to be associated with the evolution of large bubbles in the gain region.
Application to Full-Scale Core-Collapse Models
----------------------------------------------
![Same as Figure \[f:shock\_entropy\_sasi\], but for models u8.1 and s27 of @mueller2012. The fiducial entropies $s_0$ are in units of $k_{\rm B}$ per baryon.[]{data-label="f:shock_entropy_mpa"}](f10.eps){width="\columnwidth"}
![Time- and angle-averaged profiles of selected quantities for models u8.1 and s27 of @mueller2012 (compare with Fig. \[f:profiles\_timeave\]). The free-fall velocity normalization is computed in Newtonian gravity, for gravitational masses $\{1.2,1.35\}M_\sun$ and initial shock radii $\{150,130\}$ km for models {u8.1,s27}, respectively. The vertical dashed lines correspond to the time- and angle-averaged radius for which $\rho = 10^{11}$ g cm$^{-3}$, which we associate with $r_*$. The vertical dotted lines bracket the radial extent of the region used for spectral analysis (see text for details).[]{data-label="f:profiles_timeave_mpa"}](f11.eps){width="\columnwidth"}
Here we analyze the models of @mueller2012 with the same methods used in our parametric models, identifying similarities and differences.
Figure \[f:shock\_entropy\_mpa\] shows the evolution of the $\ell=0$ and $\ell=1$ coefficients for models u8.1 and s27, together with the fraction of the volume with entropy higher than fiducial values $s_0 = \{10,15,18,21,25\}~k_{\rm B}$ per baryon. The $f_V$ diagnostic behaves similarly to exploding parametric models p0B10L1 and e3B08 (Figs. \[f:shock\_entropy\_sasi\] and \[f:shock\_entropy\_conv\]). After a large enough fraction of the postshock volume is occupied by high entropy material, the regular periodicity of shock oscillations in model s27 is modified ($t\simeq 150$ ms). Shock sloshings in this late stage are preceded by partial disruption of bubbles. One notable difference with model p0B10L1 is the emergence of secondary shocks in model s27, which prevent complete disruption of high-entropy bubbles. Runaway expansion in model s27 is preceded by accretion of the Si/O composition interface. Another important difference between both @mueller2012 models and the exploding parametric models is the level of $\ell=0$ oscillations, which is much larger in the exploding gamma-law simulations[^2].
Models u8.1 and s27 both undergo a quasi-stationary phase that precedes runaway expansion. We have analyzed the properties of the time-averaged flow over the interval $[80,130]$ ms and $[70,120]$ ms in models u8.1 and s27, respectively. During these intervals, both the average shock radius and the average neutrinospheric radius $r_*$ (defined as the isodensity surface $\rho=10^{11}$ g cm$^{-3}$) change by less than $20\%$. Figure \[f:profiles\_timeave\_mpa\] shows the resulting profiles of time-averaged quantities, in analogy with Figure \[f:profiles\_timeave\]. Above the neutrinosphere, all quantities behave in the same qualitative way as the parametric models. At densities $\rho=10^{11}$ g cm$^{-3}$ and higher, clear differences are introduced by the existence of a protoneutron star, however. In particular, the density and velocity fluctuations decrease significantly inside $r_*$, whereas Figure \[f:profiles\_timeave\] shows a strong increase in the density perturbation near $r_*$ for parametric models due to the accumulation of mass given the reflecting boundary condition, and a bump in the lateral velocity due to shear in SASI-dominated cases. Nonetheless, the very similar behavior of the system outside $r_*$ shows that a reflecting boundary condition is not a bad approximation.
[f12.eps]{} (56.7,0.7)[$\ell$]{}
We have also computed the convection parameter using the time-averaged flow (eq. \[\[eq:chi\_mean\_flow\]\]). The buoyancy frequency is computed following @mueller2013 but ignoring relativistic corrections[^3]. Model u8.1 has $\bar\chi \simeq 0.7$, consistent with the hypothesis that convection-dominated flow adjusts itself to sub-criticality. This parameter is even smaller ($\bar\chi \simeq 0.4$) in model s27.
Figure \[f:spectra1d\_mosaic\_mpa\] shows one-dimensional spectra of the subsonic region in models u8.1 and s27. The limits of the region are defined to be the saddle point in the time- and angle-averaged sound speed on the inside, and the time-average of the minimum shock radius minus its rms fluctuation on the outside (both radii are shown in Figure \[f:profiles\_timeave\_mpa\] for s27). The radial spectrum of the @mueller2012 models has relatively less power at long wavelengths than the parametric models. At short wavelengths, however, the spectral slope is similar, and the sawtooth pattern of SASI-dominated models is also present in model s27 (the spectrum extends to $n=10$ due to the compactness of the subsonic region relative to the grid spacing). The angular spectrum of model s27 shows the same peak at $\ell=2$ as the SASI-dominated parametric models. Model u8.1 shows a similar curved shape as convection-dominated models, though with some excess at $\ell = 3$ and $\ell=5$. Finally, the temporal spectrum shows clearly the distinction between convection-dominance manifesting more power at low frequencies (u8.1), and SASI-dominance generating a clear peak at the advection time (s27). This temporal frequency behavior was already noted by @mueller2012.
In summary, the general results from parametric models regarding SASI- and convection-dominated flow persist in sophisticated, full-scale core-collapse simulations. The use of nuclear dissociation in parametric models as the control parameter for switching between SASI- and convection-dominance does not prevent qualitative agreement with full-scale models, even though the latter always have dissociation present. This is because the behavior of the flow depends chiefly on the *relative timescales* of the system (c.f. §\[s:linear\_stability\]).
Summary and Discussion {#s:summary}
======================
We have analyzed the transition to explosion in SASI- and convection-dominated core-collapse supernova explosions using parametric, two-dimensional, time-dependent hydrodynamic simulations. These models are such that the linear stability properties are well understood, allowing the exploration of well-differentiated regions of parameter space (Figure \[f:growth\_timescales\], Table \[t:models\]). We have also introduced a spherical Fourier-Bessel decomposition to characterize the properties of turbulence in the sub-sonic region of the flow, to extract signatures of the interplay of SASI and convection. Our main findings are as follows:
1\. – The behavior of SASI-dominated models is characterized by the interplay of shock sloshings and the formation of large-scale, high-entropy structures. These bubbles are seeded by the SASI during shock expansions. Regular sloshing of the shock requires that these bubbles have a short lifetime and/or small entropy enhancements. Regular destruction of high-entropy structures by lateral flows is characteristic of non-exploding models (Figure \[f:shock\_entropy\_sasi\]).
2\. – Models that explode with SASI dominance are able to form large-scale, high-entropy bubbles that survive for a time longer than a characteristic shock oscillation cycle (Figure \[f:shock\_entropy\_sasi\]). Neutrino heating and the inverse turbulent cascade in 2D ensure that these bubbles continue to grow if left undisturbed. Explosion results from the buoyancy of the bubble overcoming the drag force of the upstream flow [@thompson00].
3\. – Convection-dominated models generate similar large-scale entropy structures by consolidating smaller-scale bubbles arising from buoyant activity. Sloshing of the shock occurs whenever large bubbles are destroyed or displaced, just as in SASI-dominated models, but without a dominant periodicity. The transition to explosion also involves the formation and growth of a sufficiently large bubble (Figure \[f:shock\_entropy\_conv\]), as has been documented previously [@dolence2013; @couch2012].
4\. – Initial perturbations with a large amplitude do not alter the qualitative way in which SASI-dominated models explode in two dimensions. The difference in explosion time can be significant, however, and the time to runaway is not a monotonic function of the perturbation amplitude (Figure \[f:shock\_entropy\_pert\]).
5\. – The time-averaged flow in convection-dominated, non-exploding models adjusts itself to a state in which the convection parameter computed from the mean flow (eq. \[\[eq:chi\_mean\_flow\]\]) lies below the critical value for convective instability (Table \[t:models\]). This phenomenon is obscured when an average value of $\chi$ is computed from the instantaneous flow (Figure \[f:brunt\_comparison\]).
6\. – The spherical Fourier-Bessel power in the subsonic, weakly-stratified region is dominated by the largest spatial scales (Figure \[f:sfb\_2d\_mosaic\]). The SASI manifests itself as a characteristic even-odd pattern in the radial direction, and enhanced power at temporal frequencies corresponding to the advection time. Convection generates a smoother component, with power concentrated primarily below the advection frequency. This behavior of the frequency domain is consistent with the results of @burrows2012 and @mueller2012.
7\. – The slope of the angular spectrum is consistent with an inverse turbulent cascade at large $\ell$. Convection-dominated models yield angular spectra that resemble those of @hanke2012, @couch2012, and @dolence2013, while SASI-dominated models show a peak at $\ell=2$. The radial spectrum shows a scaling $n^{-2}$ at large $n$, which could be associated with Rayleigh-Taylor turbulence (e.g., @ciaraldi2009).
8\. – The general results obtained with the parametric models persist when the analysis is repeated on the general relativistic, radiation-hydrodynamic simulations of @mueller2012. In particular, the behavior of the entropy when approaching explosion, and the value of the convection parameter and spectral slopes of the time-averaged flow are in good agreement with the corresponding parametric models.
9\. – The equality between advection and heating times in the gain region at $t=0$ is a good indicator of the onset of non-oscillatory instability in one-dimensional numerical simulations of parametric models (Appendix \[s:L0\]), in agreement with the numerical results of @F12. The fact that this equality occurs for heating rates such that the linear eigenmodes are still oscillatory (Figure \[f:growth\_timescales\]), however, means that the onset of purely growing expansion is a non-linear effect (growth time shorter than the oscillation period). Initial equality between the advection time in the gain and cooling layers is a good indicator of $\ell=0$ instability in some regions of the space of parametric models, but not in others, particularly when nuclear dissociation is included (Figure \[f:growth\_timescales\]). When the recombination energy from alpha particles is not accounted for, the onset of $\ell=0$ instability does not necessarily lead to an explosion (Appendix \[s:L0\]), thus the instability thresholds do not equal explosion criteria for the parametric setup.
Our results show that despite the non-linearity of the flow, clear signatures of the operation of the SASI and convection can be obtained. In particular, the parameter $\chi$ (equation \[eq:convection\_parameter\]) – evaluated at the time where the shock stalls and before hydrodynamic instabilities set in – is a good predictor of whether the system will be SASI- or convection-dominated on its way to explosion.
Despite the different explosion paths obtained when SASI or convection dominate the dynamics at early times, it is not clear that the resulting explosion properties are very different once the process has started. Our results indicate that in both cases, the formation of at least one large-scale, high-entropy bubble is a necessary condition to achieve explosion in two-dimensions. It may be that this degeneracy is triggered by the inverse turbulent cascade inherent in axisymmetric models.
The absence of this inverse cascade in 3D causes the flow to develop more small-scale structure than in 2D (e.g., @hanke2012). Nonetheless, the tendency of bubbles to merge into bigger structures will persist, as this is an intrinsic property of the Rayleigh-Taylor instability [@sharp1984]. In fact, several 3D hydrodynamic studies have observed that prior to explosion, a large-scale asymmetry (often $\ell=1$) develops in a non-oscillatory way [@iwakami08; @couch2012; @dolence2013; @hanke2013]. The difference lies in the fact that the SASI provides seeds for large-scale entropy fluctuations *independent of dimension*, so it can speed up the formation of a large- and hot enough bubble to achieve explosion. Verifying whether this picture is robust requires numerical experiments in 3D.
Even though we have found clear evidence for high entropy bubbles playing a key role in the interplay between SASI and convection and in the onset of explosion, there are many questions that remain to be answered. First, the evolution of the $f_V$ diagnostic suggests that transition to explosion in a multidimensional environment involves a fraction of the gain region volume achieving a certain entropy or positive energy. What is that volume or mass fraction, and what are the required entropy or energy values as a function of the dominant system parameters?
Second, our characterization of large bubble dynamics in the gain region is limited. Processes such as seeding of bubbles by shock displacements, survivability of bubbles due to neutrino heating, buoyancy, and the turbulent cascade, disruption by lateral SASI flows in the linear phase or low-entropy plumes in the non-linear phase, and feedback of these bubbles on SASI modes deserve further study. Preliminary steps in this direction have already been taken (e.g., @guilet09a [@couch2012; @dolence2013; @murphy2012]), though much more work remains if a quantitative understanding – in the form of a predictive explosion criterion – is to be attained.
It is interesting to compare the critical heating rates for explosion in our parametric models and those from lightbulb setups with a full EOS and a time-dependent mass accretion rate (e.g., @nordhaus10a [@hanke2012; @couch2012]). In the former, explosion occurs above (but close to) the $\ell=0$ instability threshold in both SASI- and convection-dominated models, with only a $\sim 10\%$ difference between 1D and 2D (Fig. \[f:growth\_timescales\]). In contrast, the latter models are such that non-spherical instabilities make a larger difference ($\sim 20\%$) relative to the 1D case, with explosion occurring for heating rates below the $\ell=0$ oscillatory instability. Note however that both classes of models neglect the (negative) feedback to the heating rate due to the drop in accretion luminosity when the shock expands, thus the numbers obtained from these models should be treated with caution. The search for a robust and predictive explosion criterion valid for both SASI- and convection-dominated models is a worthwhile pursuit, though outside the scope of the present paper.
The modification of our results by the introduction of a third spatial dimension will be addressed in future work.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Jeremiah Murphy, Sean Couch, Christian Ott, Jérôme Guilet, Adam Burrows, Josh Dolence, Yudai Suwa, Kei Kotake, Ernazar Abdikamalov, and Annop Wongwathanarat for stimulating discussions. The authors thank the Institute for Nuclear Theory at the University of Washington for its hospitality, and the US Department of Energy for partial support during the completion of this work. The anonymous referee provided helpful comments that improved the presentation of the paper. RF is supported by NSF grants AST-0807444, AST-1206097, and the University of California Office of the President. BM and HJ are supported by the Deutsche Forschungsgemeinschaft through the Transregional Collaborative Research Center SBF-TR7 “Gravitational Wave Astronomy" and the Cluster of Excellence EXC 153 “Origin and Structure of the Universe". TF is supported by the grant ANR-10-BLAN-0503 funding the SN2NS project. The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. Parametric models were evolved at the IAS *Aurora* cluster, while the Garching models were evolved on the IBM p690 of the Computer Center Garching (RZG), on the Curie supercomputer of the Grand Équipement National de Calcul Intensif (GENCI) under PRACE grant RA0796, on the Cray XE6 and the NEC SX-8 at the HLRS in Stuttgart (within project SuperN), and on the JUROPA systems at the John von Neumann Institute for Computing (NIC) in Jülich.
On the Stability of the $\ell=0$ SASI mode {#s:L0}
==========================================
Here we compare the predictions from timescale ratio diagnostics with the actual eigenfrequencies of the $\ell=0$ mode in the parametric system.
Figure \[f:shock\_L0\_paper\] shows the shock radius as a function of time for one dimensional (1D) versions of the e0 and e3 sequences shown in Table \[t:models\]. By comparing with Figure \[f:growth\_timescales\], one can see that the oscillatory radial stability thresholds are well captured at this resolution. The initial value of the ratio of advection time in the gain region to advection time in the cooling region is a good indicator of oscillatory radial stability for the $\varepsilon=0$ sequence, but not so much when nuclear dissociation is introduced.
![Shock radius as a function of time for 1D models without dissociation (panel a) and with dissociation (panel b). Curves are labeled by the value of the heating parameter $B$; compare with Figure \[f:growth\_timescales\]. Note that non-oscillatory expansion sets in when $B$ is such that $t_{\rm adv-g}>t_{\rm heat-g}$. The shock expansion saturates in all models with dissociation, and in the model with $B=0.01$ and $\varepsilon=0$.[]{data-label="f:shock_L0_paper"}](fA1.eps){width="\columnwidth"}
The initial ratio of advection to heating times in the gain region is a good predictor of non-oscillatory expansion in the 1D models, in agreement with the numerical results of @F12. Note however that for both sequences, this point lies at a lower heating rate than the bifurcation of the perturbative $\ell=0$ growth rate (Fig. \[f:growth\_timescales\]). Therefore, this runaway expansion is a non-linear effect, likely arising from the fact that the growth time is shorter than the oscillation period (by more than a factor of two in the e0 model when the ratio of advection to heating timescales is unity).
Note also that in contrast to the models of @F12, radial instability does not always lead to runaway expansion. This is clear from the model with $\varepsilon=0$ and $B=0.01$, which saturates. Also, all the models with nuclear dissociation saturate. @FT09b showed that this effect is due to the artificial assumption of constant nuclear dissociation at the shock. Including the recombination energy of alpha particles as the shock expands (which decreases the effective dissociation rate), leads to a runaway as soon as instability sets in.
Spherical Fourier-Bessel Decomposition in between Concentric Shells {#s:sfb_appendix}
===================================================================
{width="49.00000%"} {width="49.00000%"}
In spherical polar coordinates, the general solution to the Helmholtz equation[^4] is a superposition of functions of the form (e.g., @jackson) $$\label{eq:helmholtz_solution}
\left[a_{\ell,m}j_\ell(kr) + b_{\ell,m}y_\ell(kr) \right]\,Y_{\ell}^m(\theta,\phi),$$ where $j_\ell$ and $y_\ell$ are the spherical Bessel functions, $Y_\ell^m$ are the Laplace spherical harmonics, and $\{a_{\ell,m},b_{\ell,m}\}$ are constant coefficients. The wavenumber $k$ and the coefficients are determined once boundary conditions for the problem are imposed at the radial domain boundaries $r_{\rm in}$ and $r_{\rm out}$.
Dirichlet Boundary Conditions
-----------------------------
Requiring that the eigenfunctions vanish at the radial boundaries for all $\{\ell,m\}$ yields the system of equations $$\label{eq:Dirichlet_condition}
\left[\begin{array}{cc} j_\ell(k\,r_{\rm in}) & y_\ell(k\,r_{\rm in})\\
j_\ell(k\,r_{\rm out}) & y_\ell(k\,r_{\rm out})\end{array} \right]
\left( \begin{array}{c} a_{\ell,m}\\ b_{\ell,m}\end{array}\right) = 0.$$ Non-trivial solutions are obtained by setting the determinant of the matrix of coefficients to zero. This condition then defines a discrete set of radial wavenumbers: $$\begin{aligned}
\label{eq:Dirichlet_wavenumbers}
j_\ell(k_{n\ell}\,r_{\rm in})\,y_\ell(k_{n\ell}\,r_{\rm out})
- j_\ell(k_{n\ell}\,r_{\rm out})\,y_\ell(k_{n\ell}\,r_{\rm in}) = 0\\
(n=0,1,2,...)\nonumber\end{aligned}$$ where $n$ labels the roots in increasing magnitude.
Figure \[f:zeroes\_eigenfunctions\_dirichlet\]a shows the first five solutions for $\ell = 1$, as a function of the ratio of boundary radii $r_{\rm in}/r_{\rm out}$. We adopt the convention of labeling the smallest wavenumber by $n=0$, since the eigenfunction has no nodes. For low $\ell$, the relation $$\label{eq:wave_number_approx}
k_{n\ell}\simeq \frac{\pi}{(r_{\rm out}-r_{\rm in})}(n+1)\qquad n=0,1,2...$$ holds approximately, becoming better for $r_{\rm in}/r_{\rm out}\to 1$. Increasing the angular degree increases the value of the wave number relative to equation (\[eq:wave\_number\_approx\]), as shown in Figure \[f:zeroes\_eigenfunctions\_dirichlet\]b.
[fB2.eps]{} (41,83)[[ $\ell$]{}]{} (85,83)[[ $\ell$]{}]{}
Equation (\[eq:Dirichlet\_condition\]) also determines the ratio of coefficients $$\label{eq:Dirichlet_coefficients}
\frac{b_{n\ell m}}{a_{n\ell m}} = -\frac{j_\ell(k_{n\ell}\,r_{\rm in})}{y_\ell(k_{n\ell}\,r_{\rm in})}
= -\frac{j_\ell(k_{n\ell}\,r_{\rm out})}{y_\ell(k_{n\ell}\,r_{\rm out})}.$$ Note that $n$ has been added as an index to the coefficients. The radial eigenfunctions $g_\ell$ are then $$\begin{aligned}
\label{eq:first_formulation}
g_{n\ell}(r) & = & N^{-1/2}_{n\ell}\left[y_\ell(k_{n\ell}\,r_{\rm out})j_\ell(k_{n\ell}\,r)\right.\nonumber\\
&&\left. - j_\ell(k_{n\ell}\,r_{\rm out})y_\ell(k_{n\ell}\,r)\right]\\
& = & \tilde{N}^{-1/2}_{n\ell}\left[y_\ell(k_{n\ell}\,r_{\rm in})j_\ell(k_{n\ell}\,r)\right.\nonumber\\
&&\left. - j_\ell(k_{n\ell}\,r_{\rm in})y_\ell(k_{n\ell}\,r)\right],\end{aligned}$$ where the two formulations differ only by a global (real) phase. The normalization constant is found from the orthogonality condition (Lommel integral). Combining two solutions of the spherical Bessel differential equation, integrating over the radial domain, applying the boundary conditions, and using L’Hôpital’s rule yields $$\begin{aligned}
\label{eq:normalization_condition_dirichlet}
\int_{r_{\rm in}}^{r_{\rm out}} g_\ell(k_{n\ell}\,r)g_\ell(k_{m\ell}\,r)\,r^2\totd r & = &
\frac{\delta_{nm}}{2}\left\{r_{\rm out}^3 \left[g^\prime_\ell(k_{n\ell}\,r_{\rm out})\right]^2\right.\nonumber\\
&&\left.- r_{\rm in}^3 \left[g^\prime_\ell(k_{n\ell}\,r_{\rm in})\right]^2\right\},\end{aligned}$$ where $\delta_{nm}$ is the Kronecker symbol and primes denote derivative respect to the argument. For the first formulation (eq. \[eq:first\_formulation\]), choosing $$\begin{aligned}
N_{n\ell} && = \frac{1}{2}\left\{r_{\rm out}^3
\left[y_\ell(k_{n\ell}\,r_{\rm out})j^\prime_\ell(k_{n\ell}\,r_{\rm out})\right.\right.\nonumber\\
&&\left. - j_\ell(k_{n\ell}\,r_{\rm out})y^\prime_\ell(k_{n\ell}\,r_{\rm out})\right]^2\nonumber\\
&& -r_{\rm in}^3 \left[y_\ell(k_{n\ell}\,r_{\rm out})j^\prime_\ell(k_{n\ell}\,r_{\rm in})\right. \nonumber\\
&&\left.\left. - j_\ell(k_{n\ell}\,r_{\rm out})y^\prime_\ell(k_{n\ell}\,r_{\rm in})\right]^2\right\}\end{aligned}$$ makes the eigenfunctions orthonormal. Figure \[f:sample\_eigenfunctions\] shows two examples of the resulting normalized eigenfunctions in a two dimensional, axisymmetric space.
{width="49.00000%"} {width="49.00000%"}
In three dimensions, the expansion of an arbitrary function $f(r,\theta\,\phi)$ with Dirichlet boundary conditions in the radial interval $[r_{\rm in},r_{\rm out}]$ can be written as $$\label{eq:expansion_functional_form}
f(r,\theta,\phi) = \sum_{n,\ell,m} f_{n\ell m} g_\ell(k_{n\ell}\,r) Y_\ell^m (\theta,\phi),$$ with coefficients given by $$%\label{eq:coefficients_expression}
f_{n\ell m} = \int r^2\totd r\,\totd\Omega\, g_\ell(k_{n\ell}\,r) Y_\ell^{m*}\,F(r,\theta,\phi),$$ where the star denotes complex conjugation. The corresponding Parseval identity is $$\int |F|\, \totd^3 x = \sum_{n,\ell,m} |a_{n\ell m}|^2$$ yielding a three-dimensional spatial power spectrum: $$\label{eq:PSD_3D}
P_{n\ell m} = |a_{n\ell m}|^2.$$
Neumann Boundary Conditions
---------------------------
Requiring that the radial derivative of the eigenfunctions vanish at the boundaries yields the equation for the radial wave numbers $$\begin{aligned}
\label{s:Neumann_wavenumbers}
j^\prime_\ell(k_n\,r_{\rm in})\,y^\prime_\ell(k_n\,r_{\rm out})
- j^\prime_\ell(k_n\,r_{\rm out})\,y^\prime_\ell(k_n\,r_{\rm in})
= 0\\
\quad (n=1,2,3 ...),\nonumber\end{aligned}$$ where the primes again denote derivative respect to the argument. The eigenfunctions are now $$\begin{aligned}
\label{eq:eigenfunctions_neumann_form1}
f_\ell(k_n\,r)& = & M^{-1/2} \left[y^\prime_\ell(k_n\,r_{\rm in})j_\ell(k_n\,r)\right.\nonumber\\
&&\left. - j^\prime_\ell(k_n\,r_{\rm in})y_\ell(k_n\,r)\right]\\
\label{eq:eigenfunctions_neumann_form2}
& = & \tilde{M}^{1/2} \left[y^\prime_\ell(k_n\,r_{\rm out})j_\ell(k_n\,r)\right.\nonumber\\
&& \left. - j^\prime_\ell(k_n\,r_{\rm out})y_\ell(k_n\,r)\right],\end{aligned}$$ and the orthogonality condition reads $$\begin{aligned}
\label{eq:normalization_condition_neumann}
\int_{r_{\rm in}}^{r_{\rm out}} f_\ell(k_n\,r)f_\ell(k_m\,r)\,r^2\totd r = &&\nonumber\\
\frac{\delta_{nm}}{2}
\left\{r_{\rm out}^3 \left[1-\frac{\ell(\ell+1)}{(k_{n\ell}r_{\rm out})^2} \right]f^2_\ell(k_{n\ell}\,r_{\rm out})\right.\nonumber\\
\qquad\left.- r_{\rm in}^3 \left[1 - \frac{\ell(\ell+1)}{(k_{n\ell}r_{\rm in})^2} \right]f^2_\ell(k_{n\ell}\,r_{\rm in}) \right\}.\end{aligned}$$ The normalization constant for equation (\[eq:eigenfunctions\_neumann\_form1\]) is $$\begin{aligned}
M_{n\ell} & = & \frac{1}{2}\left\{
r_{\rm out}^3\left[1 - \frac{\ell(\ell+1)}{(k_{n\ell}r_{\rm out})^2}\right]\right.\times\nonumber\\
&& \left[y_\ell^\prime(k_{n\ell}r_{\rm out})\,j_\ell(k_{n\ell}r_{\rm out})
-j^\prime_\ell(k_{n\ell}r_{\rm out})\,y_\ell(k_{n\ell}r_{\rm out})\right]^2\nonumber\\
&& -r_{\rm in}^3\left[1 - \frac{\ell(\ell+1)}{(k_{n\ell}r_{\rm in})^2}\right]
\left[y_\ell^\prime(k_{n\ell}r_{\rm out})\,j_\ell(k_{n\ell}r_{\rm in})\right.\nonumber\\
&&\left.\left.\qquad\qquad\qquad\quad
-j^\prime_\ell(k_{n\ell}r_{\rm out})\,y_\ell(k_{n\ell}r_{\rm in})\right]^2.
\right\} \end{aligned}$$
The radial wave numbers and eigenfunctions for the first few harmonics and $\ell$ values are shown in Figure \[f:zeroes\_eigenfunctions\_neumann\]. The overall structure of the wave numbers is very similar to the Dirichlet case, with slightly higher values for small ratio of radii and large $\ell$. For fixed harmonic, the eigenfunctions change their shape as $\ell$ is increased, in contrast to the Dirichlet case.
\[lastpage\]
[^1]: For the e0 sequence, the e-folding time for the $\ell=0$ mode is approximately one half of the oscillation period at the heating rate for which $t_{\rm adv-g} = t_{\rm heat-g}$.
[^2]: The difference in shock expansion rate once runaway starts is due to the absence of alpha particle recombination in the parametric models [@FT09b]; this is independent of the level of $\ell=0$ oscillations.
[^3]: The leading order corrections to the buoyancy frequency and sound speed scale like $(c_s/c)^2$ [@mueller2013], which is only a few percent in the gain region.
[^4]: Since the Laplacian operator is Hermitian, its eigenfunctions – solutions to the Helmholtz equation – form a complete orthogonal basis in the Hilbert space $L^2$ [@arfken05].
|
---
abstract: 'Humans naturally perceive a 3D scene in front of them through accumulation of information obtained from multiple interconnected projections of the scene and by interpreting their correspondence. This phenomenon has inspired artificial intelligence models to extract the depth and view angle of the observed scene by modeling the correspondence between different views of that scene. Our paper is built upon previous works in the field of unsupervised depth and relative camera pose estimation from temporal consecutive video frames using deep learning (DL) models. Our approach uses a hybrid learning framework introduced in a recent work called GeoNet, which leverages geometric constraints in the 3D scenes to synthesize a novel view from intermediate DL-based predicted depth and relative pose. However, the state-of-the-art unsupervised depth and pose estimation DL models are exclusively trained/tested on a few available outdoor scene datasets and we have shown they are hardly transferable to new scenes, especially from indoor environments, in which estimation requires higher precision and dealing with probable occlusions. This paper introduces “Indoor GeoNet”, a weakly supervised depth and camera pose estimation model targeted for indoor scenes. In Indoor GeoNet, we take advantage of the availability of indoor RGBD datasets collected by human or robot navigators, and added partial (i.e. weak) supervision in depth training into the model. Experimental results showed that our model effectively generalizes to new scenes from different buildings. Indoor GeoNet demonstrated significant depth and pose estimation error reduction when compared to the original GeoNet, while showing 3 times more reconstruction accuracy in synthesizing novel views in indoor environments.'
author:
- |
Amirreza Farnoosh\
Northeastern University\
Boston, MA, USA\
[farnoosh.a@husky.neu.edu]{}
- |
Sarah Ostadabbas\
Northeastern University\
Boston, MA, USA\
[ostadabbas@ece.neu.edu]{}
bibliography:
- 'paper.bib'
title: |
Indoor GeoNet:\
Weakly Supervised Hybrid Learning for Depth and Pose Estimation
---
Introduction
============
In spite of extensive research in the field of indoor navigation, this problem is still unsolved [@huang2009survey]. In order to travel a path in an unknown or even known indoor scene a map along with a positioning system needs to be provided to the navigator (e.g. a person or a robot). The global navigation satellite system (GNSS) data collected over time provides such information in outdoor scenes [@ojeda2007personal]. However, GNSS signals are usually not available or very weak in indoor places. To compensate for the lack of GNSS inside buildings, information from other sensing modalities such as artificially installed beacons or wearable inertial measurement units (IMUs) are often used for odometry [@farnoosh2018first]. However, data from these sources are usually not reliable over an extended period of time due to the extensive drift caused by accumulation error [@woodman2007introduction]. Besides that, these sensors can only provide low-level information about the scene, and are not able to reveal any other information about the overall 3D structure of the indoor places to be used for scene understanding, dynamic interaction, and ultimately a reliable indoor navigation.
In contrast to the sparse distance-based sensing, information such as depth and relative camera pose can be used together to give a very accurate and detailed representation of an indoor scene [@izadi2011kinectfusion]. These information could facilitate both navigation and dynamic interaction and also help to reconstruct a unified 3D model of the scene for the purpose of map generation of unknown places [@maier2017efficient; @dzitsiuk2017noising] or even adding augmented reality features to the scene for a better interaction experience [@izadi2011kinectfusion]. It can also be used for virtual tours of an indoor scene while the observer looks into the rendered scenes in different views [@eslami2018neural].
In computer vision field, there has been extensive research for indoor odometry, scene understanding, and specifically camera pose, depth, and flow extraction from a moving camera (e.g. robot or head mounted), most of which are recently powered with the advances in deep learning (DL) used in visual simultaneous localization and mapping (vSLAM) works [@cadena2016past]. The use of DL in vSLAM can be separated into two categories of supervised and unsupervised learning, while the former is more studied [@cadena2016past]. The availability of the open-source benchmark datasets from outdoor scenes (e.g. KITTI [@menze2015object]) or indoor scenes (e.g. NYU Depth [@silberman2012indoor], RSM Hallway Dataset [^1], and MobileRGBD [^2]) has been very crucial in the success of DL-based supervised vSLAM models. However, similar to the most cases in supervised learning, the need for large and diverse data/label pairs for training such deep networks is still a limiting factor in this domain. Recently, there have been several works proposed for unsupervised learning of depth and camera pose from video frames that use time order of consequent video frames as their hidden supervision signal, including the GeoNet presented in CVPR2018 [@yin2018geonet]. In particular, GeoNet took advantage of a hybrid learning approach by combining an unsupervised deep learning algorithm and a geometry-based reconstruction equation into a same inference framework. This hybrid learning approach allows to integrate the domain knowledge (i.e. 3D scene geometry constraints) into the framework to suppress physically unfeasible solutions. Although, such unsupervised configurations can be trained on any amount of data without labeling cost, they still fall behind supervised methods in terms of the estimation accuracy, and are hardly generalizable to new scenes which are not seen *apriori* by the network. Inspired by the GeoNet framework, in this paper we propose our “Indoor GeoNet” model, which is a *weakly* supervised hybrid learning approach for camera pose, depth and flow estimation targeted to indoor scenes. Capitalized on the availability of the inexpensive depth sensing (e.g. Microsoft Kinect and Intel RealSense), we introduce the weak supervision by providing a set of groundtruth depth data into the model during the training. This type of supervision is weak due to the fact that the model is only partially supervised on depth data and the camera pose needs to be learned in an unsupervised fashion. We also believe that this kind of weak supervision for indoor scene understanding has recently become viable since the release of several RGBD open-source datasets collected by human or robot navigators such as NYU Depth, and MobileRGBD datasets.
Related Work
============
Supervised Approach to Deep Learning of Scene
---------------------------------------------
In the last few years, there has been several studies for supervised learning of the depth and camera pose [@cadena2016past]. In one of the early works [@eigen2014depth], Eigen et al. proposed a two-level network architecture, in which a coarse global prediction from the first stage was refined locally by a fine-scale network thereafter. This network was trained using a scale-invariant error that compares the final lower-resolution output with its corresponding groundtruth map, and could achieve state-of-the-art results on both NYU Depth and KITTI datasets.
In another work by Fischer et.al [@fischer2015flownet], authors proposed an encoder-decoder architecture for flow prediction in an end-to-end training fashion given datasets consisting of image pairs and their corresponding groundtruth flows. In a follow-up study [@ilg2017flownet], the authors realized that the network performance could be improved if the training data with different properties are presented to the network. Additionally, they proposed a stacked architecture that takes warping of the second image with intermediate optical flow as input for further refinement, as well as a sub-network for improving prediction on small motions, which could obtain state-of-the-art results on a few benchmark datasets, including Sintel [^3], Middlebury [^4], and KITTI datasets.
Early 2018, Liu et al. addressed the problem of novel view synthesis from a single image using an architecture with two sub-networks [@liu2018geometry]. One of the sub-networks, adapted from the work of [@eigen2015predicting], is responsible for pixel-wise prediction of depth and normals from a single image, which was pre-trained and fine-tuned in a supervised manner. These predictions together with the extracted region masks and relative poses are then used by the second sub-network to compute multiple homographies to warp input image into a novel view. The entire network is finally trained end-to-end on pairs of images.
In another recent work, Eslami et al. introduced a probabilistic generative network for 3D rendering of a scene given multiple viewpoints [@eslami2018neural]. This network takes as input images of a scene taken from different viewpoints and their corresponding camera poses, constructs an internal representation, and uses this representation to predict novel views from new query viewpoints. It is trained on pairs of images and their corresponding viewpoints from millions of synthetic scenes. Although, they obtained promising results for these synthetic scenes, they faced computational difficulties when implementing their network on real datasets.
Unsupervised Approach to Deep Learning of Scene
-----------------------------------------------
The unsupervised learning of the depth and camera pose could alleviate or remove the need for expensive data acquisition and labeling process. The common methodology behind all of the recent unsupervised methods for vSLAM is warping one image in pairs of related images (either stereo pairs or consecutive frames in a video) to the other view by leveraging the geometry constraints of the problem, in an approach very similar to the idea of autoencoders.
In one of the earlier work in this topic, Garg et al. proposed an unsupervised deep convolutional neural network (CNN) for single view depth prediction [@garg2016unsupervised]. At training time, they considered pairs of stereo images, and trained the network by warping one view to the other one using the intermediate predicted depth and known inter-view displacement through an image similarity loss. They used Taylor expansion of the geometric warping function to make it differentiable for neural network training. Later, Godard et al. showed that this photometric loss combined with a consistency loss between the disparities produced relative to both the left and right images of a stereo pair, would lead to improved performance and robustness in depth prediction [@godard2017unsupervised].
In another concurrent work, Jason et al. proposed an unsupervised approach to train a CNN for predicting optical flow between two images [@jason2016back]. The network was trained using pairs of temporally consecutive images through a photometric loss between the first image and the inverse warping of the second image as well as a flow smoothness loss term. In a similar approach, Vijayanarasimhan et al. proposed a geometry-aware network that predicts depth, camera pose, and a set of motion masks corresponding to rigid object motions and segmentation given a sequence of consecutive frames in the input [@vijayanarasimhan2017sfm]. This is performed by converting the predictions to optical flow and then warping the frames to each other and considering forward-backward consistency constraints.
Following the same approach, Zhou et al. proposed a network for jointly unsupervised training of a depth CNN and a camera pose estimation network from video sequences in mostly rigid scenes again by leveraging a photometric loss from novel-view synthesis [@zhou2017unsupervised]. They additionally trained an explainability prediction network which outputs a per-pixel soft mask, with which the view synthesis objective is weighted, in order to handle visibility, non-rigidity and other non-modeled factors. Meister et al. proposed using a robust census transform for the photometric loss along with an occlusion-aware loss to mask occluded pixels, whenever there is a large mismatch between estimated forward and backward flows [@meister2017unflow].
In October 2018, Ranjan et al. released their work, which used an adversarial collaboration structure for jointly unsupervised learning of depth, camera motion, optical flow, and motion segmentation from video sequences [@ranjan2018adversarial]. They used two adversaries in their framework, one for static scene reconstruction based on estimated depth and camera motion, and one for moving region reconstructor based on estimated flow. This competition is moderated by a pixel-wise probabilistic motion segmentation network that distributes training data to these adversaries. The moderator itself is trained to segment static and moving regions correctly by taking a consensus between flow of the static scene and moving regions from the two adversaries. The authors argue that since these four fundamental vision problems are coupled, learning them together would result in an enhanced performance.
Last but not least, Zhichao et al. proposed GeoNet, a jointly unsupervised deep network for depth, camera pose and flow estimation given a sequence of video frames [@yin2018geonet]. They broke down the problem of flow estimation into two parts: rigid flow which handles static background, and non-rigid flow which handles moving objects, and assigned two cascaded sub-networks to perform full flow estimation accordingly. In addition, they proposed an adaptive geometric consistency loss inspired by [@godard2017unsupervised] to increase robustness towards outliers and non-Lambertian regions.
In this paper, we utilized the hybrid learning framework of GeoNet, which is trained and tested on videos from outdoor scenes. Specifically, these videos are recorded from a fixed camera mounted on top of a car. We argue that this approach is not well transferable to indoor scenes for several reasons: Firstly, in contrast to outdoor scenes, the relative displacement with respect to the depth range is high in the indoor scenes. Besides that, the outdoor scenes are much wider, and therefore are less affected by camera movement. Secondly, the relative range of camera pose angles is much less in outdoor scenes, however for indoor scenes, because of the limited space, the changes in camera view can be much sharper. In addition, the head mounted cameras are much more affected by distorted movement. Thirdly, depth precision needed for indoor scenes is much higher because of shorter depth ranges in a more detailed scene with more edges. Our proposed “Indoor GeoNet” addressed these issues and provides an efficient hybrid learning framework for accurate pose and depth estimation in indoor scene, based on weak supervision to exploit the advantages of supervised and unsupervised learning in a unified framework.
Building the Indoor GeoNet {#sec:IndoorGeoNet}
==========================
Indoor GeoNet shares the same network structure as original GeoNet [@yin2018geonet], in which two sub-networks called DepthNet and PoseNet predict rigid layout of the observed scene including the depth and relative camera pose. The training samples to the network are temporal consecutive frames $I_i(i = 1 \sim N)$ for $N=3$ or $N=5$ with known camera intrinsics. Typically in a sequence of frames, a reference frame $I_r$ is specified as the reference view, and the other frames are target frames $I_t$. During training, the DepthNet takes the entire sequence concatenated along batch dimension as input. This allows for single view depth prediction at the test time. In contrast, the PoseNet is naturally fed with the entire sequence concatenated along channel dimension, and outputs all of the relative camera poses. This allows the network to learn the connections between different views in a sequence. Fused with the deep structures of DepthNet and PoseNet, rigid scene geometry equations then will be used to warp a target view to the reference view. Unlike the fully unsupervised approach of original GeoNet, Indoor GeoNet takes advantage of the depth supervision to enhance the transferability of the pose and depth learning to indoor scenes.
Geometric-Based Hybrid Learning
-------------------------------
Static scene geometry can be well-defined from patterns of motion of objects, surfaces, and edges in a sequence of ordered images collected from a visual scene. This scene level consistent movement perceived in image plane, known as optical flow, is governed by the relative camera motion between an observer and a scene. Therefore, this rigid optical flow can be completely modeled by a collection of depth maps $D_i$ for frame $I_i$, and the relative camera motion $\mathbf{T}_{r\rightarrow t}=[R|T]$ from reference frame $I_r$ to target frame $I_t$, where $R_{3\times3}$ and $T_{3\times1}$ represent the relative rotation and displacement matrices, respectively. Let $p_r=[X, Y, 1]^T$ denote the homogeneous coordinates of a pixel in the reference view, $D_{p_r}$ be its depth value, $[x, y , D_{p_r}] ^T$ be its corresponding 3D coordinates (referenced on camera pinhole), and $K_{3\times3}$ be the camera intrinsic matrix. Then, $p_r$ in the image plane is:
$$\label{eqn:reference}
p_r = \begin{bmatrix}X\\Y\\1\end{bmatrix} = \frac{1}{D_{p_r}} \mathbf{K} \begin{bmatrix}1 & 0 & 0|0\\0 & 1 & 0|0\\0 & 0 & 1|0\end{bmatrix}
\begin{bmatrix}x\\y\\D_{p_r}\\1\end{bmatrix}\\$$
Moreover, we can obtain the projected coordinates of $p_r$ onto the target view $p_t$, as: $$\label{eqn:projection}
p_t \sim \mathbf{K} \mathbf{T}_{r\rightarrow t}
\begin{bmatrix}x\\y\\D_{p_r}\\1\end{bmatrix} = \mathbf{K} [R|T] \begin{bmatrix}x\\y\\D_{p_r}\\1\end{bmatrix}$$
Rewriting the [Eq. (\[eqn:reference\])]{} will result in $\begin{bmatrix}x\\y\\D_{p_r}\end{bmatrix}= D_{p_r} \mathbf{K}^{-1}p_r$ and merging that with the [Eq. (\[eqn:projection\])]{} will give us the corresponding target pixel coordinates $p_t$ in terms of the reference depth map $D_{p_r}$, reference pixel coordinates $p_r$, and the relative camera motion $[R|T]$, as:
$$\label{eqn:geometry}
p_t \sim \mathbf{K} \Big[D_{p_r} R\mathbf{K}^{-1} p_r + T \Big]$$
Using [Eq. (\[eqn:geometry\])]{}, we can synthesize a novel nearby view from a reference frame in non-occluded regions having the depth map of pixels in the reference view as well as the relative camera pose between the views. Therefore, the DepthNet and PoseNet can be trained together through novel view synthesis between any pairs of training samples.
Weakly Supervised Multi-Objective Training
------------------------------------------
Let us denote consecutive frames $\{I_1, \dots, I_r, \dots, I_N\}$ as a training sequence with the middle frame $I_r$ being the reference view and the rest being the target views, $I_t$’s. Then, $\hat{I}_{t\rightarrow r}$ represents the target view $I_t$ warped to the reference coordinate frame by taking the predicted depth $\hat{D}_r$, the predicted camera transformation matrix $\hat{T}_{r\rightarrow t}$, and the target view $I_t$ as input. In order to train the Indoor GeoNet in a weakly supervised manner, we define a total loss function $\mathcal{L}_{T}$ as the weighted summation of multiple losses as: $$\mathcal{L}_{T} = \sum_{(r,t)} \lambda_{P}\mathcal{L}_{P} + \lambda_{D}\mathcal{L}_{D} + \lambda_{C}\mathcal{L}_{C} + \lambda_{W}\mathcal{L}_{W}$$ where $\mathcal{L}$’s are different loss functions explained in the following sections, $\lambda$’s are the corresponding loss weights, and $(r,t)$ iterates over all possible pairs of reference $I_r$ and target $I_t$ frames.
### Photometric Loss: $\mathcal{L}_{P}$
The DepthNet and PoseNet networks can be trained by minimizing the photometric loss between the synthesized view (warped target view) $\hat{I}_{t\rightarrow r}$ and reference frame $I_r$: $$\mathcal{L}_{P} = \sum_{(r,t)}\sum_{p_r} F_{diss}\big(I_r(p_r), \hat{I}_{t\rightarrow r}(p_r)\big)$$ where $\hat{I}_{t\rightarrow r}(p_r)=I_t(p_t)$, with warping between $p_t$ and $p_r$ obtained from [Eq. (\[eqn:geometry\])]{}, and $F_{diss}(.)$ is a dissimilarity measure between reference and synthesized frame. To obtain $I_t(p_t)$ for estimating the value of $\hat{I}_{t\rightarrow r}(p_r)$, we used the differentiable bilinear sampling mechanism proposed in the spatial transformer networks [@jaderberg2015spatial] that linearly interpolates the values of the 4 neighboring pixels of $p_t$ to approximate $\hat{I}_{t\rightarrow r}(p_r)$ such that: $$\begin{aligned}
\hat{I}_{t\rightarrow r}(p_r) = \sum_{\overset{i\in\{t,b\}}{j\in\{l,r\}}} w^{ij} I_t(p^{ij}_t)
\end{aligned}$$ where $w^{ij}$ is linearly proportional to the spatial proximity between $p_t$ and $p^{ij}_t$, and $\sum_{i,j}w^{ij} = 1$. As far as $F_{diss}(.)$, we adopted the differentiable photometric dissimilarity measure proposed in [@godard2017unsupervised], which has proven to be successful in measuring perceptual image similarity, and handling occlusions: $$\begin{aligned}
&F_{diss}\big(I_r, \hat{I}_{t\rightarrow r}\big)=\\&\alpha \frac{1-\text{SSIM}\big(I_r, \hat{I}_{t\rightarrow r}\big)}{2}
+(1-\alpha)\big\|I_r-\hat{I}_{t\rightarrow r}\big\|_1
\end{aligned}$$ where SSIM denotes the structural similarity index [@wang2004image] and $\alpha$ is taken to be $0.85$.
### Depth Smoothness Loss: $\mathcal{L}_{D}$
The $\mathcal{L}_{P}$ loss function defined in the previous section is non-informative in homogeneous (monotone) regions of the scene where multiple depth maps and relative poses can result in the same warping. Therefore, as proposed in [@yin2018geonet], we used a depth map smoothness loss term $\mathcal{L}_{D}$ weighted per-pixel by image gradients in order to obtain coherent depth maps while allowing depth discontinuities on the edges of the image: $$\mathcal{L}_{D}=\sum_{p_r}|\nabla D_{r}(p_r)| . \big(\exp(-|\nabla I_{r}(p_r)|\big)^T$$ where $\nabla$ is the vector differential operator.
### Forward-Backward Consistency Loss: $ \mathcal{L}_{C}$
We applied a forward-backward consistency check as in [@yin2018geonet] to enhance our predictions. Pixels for which the forward and backward flows (obtained from target to reference warping and vice versa) disagree significantly are considered as possible occluded regions. Therefore, such pixels are excluded from both the photometric loss and the forward-backward flow consistency check, and are defined as $\mathbf{p_r}$ (see [Eq. (\[eqn:consistency\])]{}) . Let us denote $f_{r\rightarrow t}(p_r) = p_r-p_t^{\{D_{p_r}, \mathbf{T}\}}$ as forward flow ($p_t$ is computed using [Eq. (\[eqn:geometry\])]{}), and conversely $f_{t\rightarrow r}(p_r) = p_r^{\{D_t,\mathbf{T}^{-1}\}}-p_t$ as backward flow, and $\Delta f_{t,r}(p_r) = f_{r\rightarrow t}(p_r) - f_{t\rightarrow r}(p_r)$. Then, the geometry consistency is imposed by adding the following loss term: $$\label{eqn:consistency}
\begin{aligned}
\mathcal{L}_{C} &= \sum_{p_r\in \mathbf{p_r}} \big\|\Delta f_{t,r}(p_r)\big\|_1\\
\text{such that}&\\
\mathbf{p_r} &= \big\{p_r: \|\Delta f_{t,r}(p_r)\|_2<\max(\alpha, \beta \|f_{r\rightarrow t}(p_r)\|_2)\big\}
\end{aligned}$$ in which $(\alpha,\beta)$ are set to be $(3.0, 0.05)$. Please note that both the photometric loss $\mathcal{L}_{P}$ and geometric consistency loss $\mathcal{L}_{C}$ are enforced on pixel locations in $\mathbf{p_r}$, where there is little contradiction between forward and backward flow.
### Weak Supervision Loss: $\mathcal{L}_{W}$
In order to enhance the overall performance of the network in prediction of depth and camera pose, we enforced the groundtruth depth maps, $D^{gt}$, by introducing another loss term on pixel locations for which we have the groundtruth depth values: $$\mathcal{L}_{W} = \sum_{i\in r,t}\big\|D_i-D_i^{gt}\big\|_2$$ where $D_i$ and $D^{gt}$ are the predicted and groundtruth depth maps of training samples, respectively.
Experimental Results and Evaluation
===================================
Here, we report the Indoor GeoNet performance in depth and pose estimation as well as novel view reconstruction evaluated using publicly-available RGBD indoor and outdoor datasets. We also compared the performance of our proposed weakly supervised model with the unsupervised version of the model trained on indoor datasets as well as the original GeoNet trained solely on an outdoor dataset.
Indoor GeoNet Architecture and Implementation Details
-----------------------------------------------------
The Indoor GeoNet contains two sub-networks, the DepthNet, and the PoseNet, which construct the novel view synthesis of a rigid scene by leveraging geometric constraints, similar to the original GeoNet structure [@yin2018geonet]. The DepthNet consists of an encoder part and a decoder part. The encoder part has the structure of ResNet50 [@he2016deep] and the decoder part uses deconvolution layers to enlarge predicted depth maps to their original resolution (as input) in a multi-scale scheme. Several skip connections are used between encoder and decoder parts in order to reuse high level or detailed information that was captured in the initial layers for reconstruction process in deconvolution layers. The PoseNet has the same architecture as in [@yin2018geonet], which consists of 8 convolutional layers followed by a global average pooling layer that outputs the 6-DoF camera poses including rotation and translation. We used batch normalization [@ioffe2015batch] and ReLU activation functions [@nair2010rectified] for all of the convolutional layers except the prediction layers. We considered the training sequence length to be $N=5$, and resized all the RGB frames to $144\times256$ pixels, and then trained the network with learning rate of $0.0002$, and batch size of $4$ for $20$ epochs in Tensorflow [@abadi2016tensorflow]. We set the loss weights to be $\lambda_P= \lambda_W =1, \lambda_D=0.5, \lambda_C=0.2$, and used Adam optimizer [@kingma2014adam] with its parameters set as $\beta_1 = 0.9, \beta_2 = 0.999$ for network training.
Evaluation Datasets
-------------------
We performed the performance evaluation and comparison of our Indoor GeoNet using four available indoor and outdoor scene datasets: MobileRGBD, RSM Hallway, and KITTI raw and odometry datasets. Some RGB and depth samples from each datasets are shown in the first and second columns of [Fig. \[fig:depth\]]{}.
MobileRGBD is a corpus dedicated to low-level RGBD algorithms benchmarking on mobile platform. This dataset contains RGB and depth videos taken from a multi-section hallway scene with a moving robot equipped with a Microsoft Kinect v2. Each path is taped several times at different robot angles forming a total of 25 videos each with duration less than 1 minute. The robot information including the odometry data from robot (location coordinates internally reported by robot), and control commands to the robot (linear and angular velocity and stop commands) are also included within this dataset. This information makes this dataset suitable for our evaluation since it gives synchronized RGB, depth, pose and location information. We preprocessed MobileRGBD dataset in order to register images from RGB and depth cameras, since they have different dimensions, aspect ratio, fields of view, and cameras are positioned some distance apart from each other. The resolution for RGB image is $1920\times1080$, however, for depth image is $512\times412$. Therefore, we registered depth images to their RGB counterparts, and warped them into the same size. We preprocessed all of the images to sequences of $5$ consecutive frames with $144\times256$ resolution, forming a total of roughly $3200$ training samples. We left two videos for evaluation of network performance.
RSM Hallway dataset includes videos from hallways of RSM building at the Imperial College London, which can be a proper dataset for our indoor training purposes. This dataset contains videos from 6 hallways, each taken 10 times forming a total of 60 videos with $1280\times720$ RGB resolution. Similar to RGBD dataset, we preprocessed all of the videos to sequences of $5$ consecutive frames with $144\times256$ resolution, forming a total of roughly $18000$ training examples. However, we excluded videos from the first hallway for evaluation purposes.
The KITTI raw and odometry datasets are collected with two high-resolution color and gray-scale video cameras mounted on top of a standard station wagon while it is driving around a city, in rural areas and on highways. For the KITTI raw dataset, accurate groundtruth values for depth is provided by a Velodyne laser scanner. The KITTI odometry dataset consists of 22 stereo sequences, with half of the sequences having groundtruth trajectories and camera pose, which makes it suitable for outdoor camera pose estimation approaches.
Estimation Performance Evaluation
---------------------------------
The weakly supervised Indoor GeoNet (referred to as IndoorGeoNet-WSup) performance is evaluated against two other models, one the original GeoNet trained on KITTI raw and odometry datasets (referred to as GeoNet-UnSup) and the other one the GeoNet trained from scratch in an unsupervised manner on the RSM Hallway dataset (referred to as IndoorGeoNet-UnSup). The performance comparison is done in two aspects: (1) the accuracy of depth and camera pose estimation using different approaches, (2) the reconstruction accuracy of the novel RGB scene synthesis using different approaches. The quantified results are calculated and reported for the datasets with available groundtruth depth and pose labels. Please note that we have both groundtruth depth and camera pose for MobileRGBD dataset, groundtruth depth for KITTI Raw dataset, groundtruth camera pose for KITTI Odometry dataset, while no depth or pose groundtruth for RSM Hallway is available. Based on the availability of the groundtruth depth labels, we chose a subset of MobileRGBD datasets (9 out of 21 videos) in the weak supervision of IndoorGeoNet-WSup initialized by the IndoorGeoNet-UnSup network trained on RSM Hallway dataset.
### Depth and Pose Estimation
We computed the depth and relative camera pose root mean squared error (RMSE) on the test set for those datasets/sequences for which we have the groundtruth values, and reported the errors in [Table \[tbl:depth\]]{} for the three models. Although GeoNet-UnSup works pretty well on the KITTI Raw and Odometry datasets, its performance degrades significantly on indoor datasets compared to the IndoorGeoNet-WSup, proving that the model is not generalizable to the indoor scenes. We also depicted some sample figures of depth prediction for the three models side by side in [Fig. \[fig:depth\]]{} along with groundtruth depth maps (if available) for comparison using sample monocular images from MobileRGBD, RSM Hallway, and KITTI datasets. As seen in this figure, although GeoNet-UnSup predicts satisfactory depth maps for the KITTI dataset, its predicted depth maps for MobileRGBD and RSM Hallway samples, hardly give any information about the general geometry of the scene, and the edges are completely lost. On the other hand, IndoorGeoNet-UnSup gives a fair prediction of the global geometry of the scene on sample images of RSM Hallway (on which the model is trained), however, predicted depth maps completely miss the details. The predictions of this model on the MobileRGBD sample images (not seen by the model during training) show that this model also fails to adapt to a new unseen indoor scene.
As seen in [Table \[tbl:depth\]]{}, with IndoorGeoNet-WSup model, depth and pose errors drop significantly for MobileRGBD dataset as compared to other models, since we are adding the depth supervision. Its predicted depth maps on sample images of MobileRGBD dataset clearly demonstrates the effect of supervision (even weak) on the ability of the network to learn detailed maps as shown in [Fig. \[fig:depth\]]{}. IndoorGeoNet-WSup also shows acceptable depth image estimation on other indoor scene (e.g. RSM Hallway) that were not part of weak supervision. This demonstrates the generalization capability of the proposed IndoorGeoNet-WSup.
### Novel View Reconstruction Estimation
Similar to the case of depth and pose estimation evaluation, we first check the adaptability of original GeoNet-UnSup to the indoor scenes of the RSM Hallway and MobileRGBD datasets. The mean image photometric loss $\mathcal{L}_{P}$, plus the structural similarity index measure between the reference image and the inverse warped target image are reported in table [Table \[tbl:reconst\]]{} for all of the dataset on our three models. Evident from this table, for GeoNet-UnSup, the reconstruction loss increases significantly on the MobileRGBD and RSM Hallway datasets that are not seen by the network during the training, which proves that the network fails to adapt to the indoor scenes. IndoorGeoNet-UnSup gives the lowest reconstruction $\mathcal{L}_{P}$ loss on RSM Hallway dataset (on which its network is trained), however, IndoorGeoNet-WSup also gives a comparable SSIM on this dataset, although it has not seen the dataset during the training. As expected, IndoorGeoNet-WSup gives the best reconstruction results on test set of MobileRGBD dataset with which the network is trained in a weak supervision fashion. We also depicted some sample images of novel view reconstruction on MobileRGBD, RSM Hallway, and KITTI datasets using the three models in [Fig. \[fig:const\]]{}. As seen in this figure, IndoorGeoNet-WSup is able to successfully reconstruct novel nearby view from input images on both sample images of MobileRGBD and RSM Hallway where both depth and pose predictions contribute to the loss. Although GeoNet-UnSup works well on KITTI sample images, it fails to correctly reconstruct the novel view of indoor scenes. Using the IndoorGeoNet-UnSup model, the reconstructed views of RSM Hallway sample images are acceptable, because as we discussed in the previous section, its predicted depth maps give a fair estimation of the global geometry of the scene.
Conclusion {#sec:conclusion}
==========
In this work, we presented a weakly supervised hybrid learning approach to estimate depth data and relative camera pose targeted for indoor scenes. Our approach is inspired by the recent works in unsupervised scene understating, which integrate the scene geometry constraints into the deep learning frameworks. However, the state-of-the-work in this domain are mostly concentrated on outdoor scenes found in datasets such as KITTI and CityScape. Here, we argued that these approaches are harldy transferable to indoor scenes, there is much more variations and more precision with detailed information is needed. In contrast, we proposed “Indoor GeoNet” using a weak supervision in terms of depth to improve both depth and pose predictions for indoor datasets. We believe that such supervision is sensible due to the availability of inexpensive indoor RGB and depth sensors and several open-source indoor datasets. We compared the outcomes of our Indoor GeoNet in terms of depth, camera pose and novel view estimation with the original unsupervied GeoNet models trained on different benchmark datasets. The results revealed that Indoor GeoNet is able to detect more detailed depth maps and also the pose estimation is improved when applied on indoor datasets.
[^1]: http://www.bicv.org/datasets/rsm-dataset/
[^2]: http://mobilergbd.inrialpes.fr/
[^3]: http://sintel.is.tue.mpg.de/
[^4]: http://vision.middlebury.edu/stereo/data/
|
---
abstract: 'In this paper we present in a topological way the construction of the orientable surface with only one end and infinite genus, called *The Infinite Loch Ness Monster*. In fact, we introduce a flat and hyperbolic construction of this surface. We discuss how the name of this surface has evolved and how it has been historically understood.'
address:
- 'Fundación Universitaria Konrad Lorenz. CP. 110231, Bogotá, Colombia.'
- 'Fundación Universitaria Konrad Lorenz. CP. 110231, Bogotá, Colombia.'
author:
- 'John A. Arredondo'
- Camilo Ramírez Maluendas
title: On the Infinite Loch Ness monster
---
Introduction {#Introduction}
============
The term Loch Ness Monster is well known around the world, specially in The Great Glen in the Scottish highlands, a rift valley which contains three important lochs for the region, called Lochy, Oich and Ness. The last one, people believe that a monster lives and lurks, baptized with the name of the loch. The existence of the monster is not farfetched, people say, taking into account that the Loch Ness is deeper than the North Sea and is very long, very narrow and has never been known to freeze (see Figure \[real-mons\]).
\[real-mons\]
*Image by xKirinARTZx, taken from devianart.com*
The earliest report of such a monster appeared in the Fifth century, and from that time different versions about the monster passed from generation to generation [@Ste]. A kind of modern interest in the monster was sparked by 1933 when George Spicer and his wife saw the monster crossing the road in front of their car. After that sighting, hundreds of different reports about the monster have been collected, including photos, portrayals and other descriptions. In spite of this evidence, without a body, a fossil or the monster in person, The Loch Ness Monster is only part of the folklore.
In a different context, in mathematics, the term *Loch Ness Monster* is well known, and not in the folklore. In number theory there is a family of functions called exponential sums, which in general take the form $$s(n)=\sum_{n=1}^N e^{2\pi i f(n)},$$ and for the special case in which $$f(n)=(ln (n))^4$$ the graph of the curve associated to that function is called *Loch Ness Monster*, dubbed to the curve by J. H. Loxton [@Lox], [@Lox1].
\[curve-mons\]
From view of the Kerékjártó’s theorem of classification of noncompact surfaces (*e.g.*, [@Ker], [@Ian]), the *Infinite Loch Ness Monster* is the name of the orientable surface which has infinite genus and only one end [@Val]. Simply, É. Ghys (see [@Ghy]) describes it as the orientable surface obtained from the Euclidean plane which is attached to an infinity of handles (see Figure \[Figure3\]). Or alternatively, from a geometric viewpoint one can think that the Infinite Loch Ness monster is the only orientable surface having infinitely many handles and only one way to go to infinity.
![*The Infinite Loch Ness monster.*[]{data-label="Figure3"}](LNM.pdf "fig:")\
In the seventies, the interest by several authors (*e.g.*, [@Sow], [@Ni], [@Can]) on the qualitative study in the noncompact leaves in foliations of closed manifolds had grown. Ongoing in this line of research considering closed 3-manifolds foliated by surfaces, A. Phillips and D. Sullivan proved that the quasi-isometry types of the surfaces well known as the *Jacob’s ladder*[^1], the *Infinite jail cell windows* [@Spiv p.24], and the *Infinite jangle gym* (see Figure \[Figure4\]) cannot occur in foliations of $S^3$, or in fact in orientable foliation of any manifold with second Betti number zero. Nevertheless, all these surfaces are diffeomorphic to the *Infinite Loch Ness monster* (see [@PSul]). Roughly speaking from the historical point of view, this nomenclature to this topological surface appeared published on *Leaves with Isolated ends in Foliated 3-Manifolds* ([@Can2 1977]), however the authors wedge the term *Infinite Loch Ness monster* to preliminary manuscript of [@PSul], which was published the following year. Under these evident, one can consider to A. Phillips and D. Sullivan as the *Infinite Loch Ness monster*’s parents.
-------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------
![*Surfaces having only one end and infinite genus.*[]{data-label="Figure4"}](jacob.pdf "fig:") ![*Surfaces having only one end and infinite genus.*[]{data-label="Figure4"}](jail.pdf "fig:")
![*Surfaces having only one end and infinite genus.*[]{data-label="Figure4"}](jungle.pdf "fig:")
-------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------
Perhaps the reader has found on the literature other names for this surface with infinite genus and only one end, for example, the *infinite-holed torus* (see [@Spiv p.23]). Figure \[Figure5\].
![*The infinite-holed torus.*[]{data-label="Figure5"}](lochness.pdf "fig:")\
The Infinite Loch Ness monster has also appeared in the area of Combinatory. Its arrival was in 1929 when J. P. Petrie told H. S. M. Coxeter that had found two new infinite regular polyhedra. As soon as J. P. Petrie begun to describe them and H. S. M. Coxeter understood this, the second pointed out a third possibility. Later they wrote a paper calling this mathematical objets the *skew polyhedra* [@Cox1], or also known today as the Coxeter-Petrie polyhedra. Indeed, they are topologically equivalent to the Infinite Loch Ness monster as shown in [@ARV]. Given that from a combinatory view one can think that skew polyhedra are multiple covers of the first three Platonic solids, J. H. Conway and *et. al.* [@Con p.333] called them the *multiplied tetrahedron*, the *multiplied cube*, and the *multiplied octahedron*, and denoted them $\mu T$, $\mu C$, and $\mu O$, respectively. See Figure \[Polyhedra\].
---------------------------------------------------------------------------------------- --------------------------------------------------------------------------------- -- -- --
 
---------------------------------------------------------------------------------------- --------------------------------------------------------------------------------- -- -- --
*Images by Tom Ruen, distributed under CC BY-SA 4.0.* \[Polyhedra\]
In billiards, an interesting area of Dynamical Systems, during 1936 the mathematicians *R. H. Fox* and *R. B. Kershner* [@Fox] (later, used it by A. B. Katok and A. N. Zemljakov [@KZ]) associated to each *billiard* $\phi_P$ coming from an Euclidian compact polygon $P\subseteq \mathbb{E}^2$ a surface $S_{P}$ with structure of translation, which they called *Ueberlagerungsfläche* and means *covered surface*, and a projection map $\pi_p: S_p\to \phi_P$ mapping each geodesic of $S_P$ onto a *billiard trajectory* of $\phi_P$ (see Table \[tabla1\] and Figure \[billar\]). Later, F. Valdez published a paper [@Val], which proved that the surface *Ueberlagerungsfläche* $S_P$ associated to the billiard $\phi_P$, being $P\subseteq \mathbb{E}^2$ a polygon with almost an interior vertex of the form $\lambda \pi$ such that $\lambda$ is a irrational number, is the Infinite Loch Ness monster.
$$\xymatrix{
& *+[F]{ P\subseteq \mathbb{E}^2 }\ar@(u,u)[dr]^{\text{ \,\, \emph{Ueberlagerungsfl\"ache}}} \ar@(u,u)[dl]_{Billiard}& \\
\phi_P & & S_P\ar[ll]_{\pi_P}
}$$
![*Billiard associated to a rectangle triangle.*[]{data-label="billar"}](octagono.pdf "fig:")\
Building the Infinite Loch Ness Monster {#Building}
=======================================
A tame Infinite Loch Ness Monster {#tame}
---------------------------------
An easy and simple way to get an Infinite Loch Ness monster from the Euclidean plane is using the operation well-known as the gluing straight segments. Actually, it consists of drawing two disjoint straight segments $l$ and $l^{'}$ of the same lengths on the Euclidean plane $\mathbb{E}^2$, then we cut along to $l$ and $l^{'}$ turns $\mathbb{E}^2$ into a surface with a boundary consisting of four straight segments (see Figure \[gluemarks\]).
![*Two straight segments on $\mathbb{E}^2$.*[]{data-label="gluemarks"}](pegarmarcas.pdf)
Finally, we glue this segments using translations to obtain a new surface $S,$ which is homeomorphic to the torus pictured by only one point (see Figure \[genus\]). The operation described above is called *gluing the straight segments* $l$ and $l^{'}$ [@RaVa].
----------------------------------------------------------------------------------- --------------------------------------------------------------------------------------
![*Gluing straight segments.*[]{data-label="genus"}](plano_con_un_asa.pdf "fig:") ![*Gluing straight segments.*[]{data-label="genus"}](torus_without_point.pdf "fig:")
*Gluing the two straight segments on $\mathbb{E}^2$.* *Torus pictured by only one point.*
----------------------------------------------------------------------------------- --------------------------------------------------------------------------------------
Note that to build a Loch Ness monster from the Euclidian plane using the gluing straight segments is necessary to draw on it a countable family of straight segment and suitable glue them. It means, we consider $\mathbb{E}^2$ a copy of the Euclidean plane equipped with a fixed origin $\overline{0}$ and an orthogonal basis $\beta= \{e_{1},e_{2}\}$. On $\mathbb{E}^2$ we draw[^2] the countable family of straight segments following: $$\mathcal{L}:= \{l_{i} =((4i-1)e_{1}, \, 4ie_{1}) : \forall i \in \mathbb{N}\} \text{ (see Figure \ref{glue})}.$$
![*Countable family of straight segments $\mathcal{L}$.*[]{data-label="glue"}](marka.pdf)
Now, we cut $\mathbb{E}^2$ along $l_{i}$, for each $i\in \mathbb{N}$, which turns $\mathbb{E}^2$ into a surface with boundary consisting of infinite straight segments. Then, we glue the straight segments $l_{2i-1}$ and $l_{2i}$ as above (see Figure \[Figure3\]). Hence, the surface $S$ comes from the Euclidean plane attached to an infinitely many handles, which appear gluing the countable disjoint straight segments belonged to the family $\mathcal{L}$. In other words, the mathematical object $S$ is the Infinite Loch Ness monster.
From view of differential geometric, the surface $S$ is conformed by two kind of points. The set of *flat points* conformed by all points in $S$ except the ends of the straight segments $l_i$, for every $i\in \mathbb{N}$. To each one of this elements there exist an open isometric to some neighborhood of the Euclidean plane. Since the curvature is invariant under isometries then the curvature in the flat points is equal to zero. The other ones, are called *singular points*, in this case they are the end points of the straight segments $l_i$, for each $i\in\mathbb{N}$. Their respective neighborhood is isometric to cyclic branched covering $2:1$ of the disk in the the Euclidean plane, *i.e.*, they are *cone angle singularity of angle* $4\pi$ (see Figure \[cone\]). The surfaces having this kind of structure are known as *tame translation surfaces* (see *e.g.*, [@PSV]).
![*Cone angle singularity of angle $4\pi$.*[]{data-label="cone"}](punto_conico.pdf)
Hyperbolic Infinite Loch Ness Monster {#hyperbolic}
-------------------------------------
An application of the Uniformization Theorem (see *e.g.*, [@Abi], [@Muc]) ensures the existence of a subgroup $\Gamma$ of the isometries group of the hyperbolic plane $Isom(\mathbb{H})$ acting on the hyperbolic plane $\mathbb{H}$ performing the quotient space $\mathbb{H} / \Gamma$ in a hyperbolic surface homeomorphic to the Infinite Loch Ness monster. In other words, there exist a hyperbolic polygon $P\subseteq \mathbb{H}$, which is suitable identifying its sides by hyperbolic isometries to get the Infinite Loch Ness monster. An easy way to define the polygon $P$ is as follows[^3].
First, we consider the countable family conformed by the disjoint half-circles $\mathcal{C}=\{C_{4n}: n\in\mathbb{Z}\}$ having $C_{4n}$ center in $4n$ and radius equal to one, for every $n\in \mathbb{Z}$. See Figure \[circles\]. In other words, $C_{4n}:=\{z\in \mathbb{H}: |z- 4n|=1\}$
![*Family of half-circles $\mathcal{C}$.*[]{data-label="circles"}](circulos.pdf)
Removing the half-circle $C_{4n}$ of the hyperbolic plane $\mathbb{H}$ we get two connected component, which are called the *inside* of $C_{4n}$ and the *outside* of $C_{4n}$, respectively (see Figure\[inside\]). They are denoted as $\check{C}_{4n}$ and $\hat{C}_{4n}$, respectively.
![*Inside and outside.*[]{data-label="inside"}](inner_outner.pdf)
Hence, our connected hyperbolic polygon $P\subseteq\mathbb{H}$ is the closure of the intersection of the outsides following (see Figure \[poligono\])
$$\label{eq:4}
P:=\overline{\bigcap\limits_{n\in\mathbb{Z}} \hat{C}_{4n}}=\bigcap\limits_{n\in\mathbb{Z}} \{z\in\mathbb{H}: |z-4n|\geq 1\}.$$
![*Family of half-circles $\mathcal{C}$ and hyperbolic polygon $P$.*[]{data-label="poligono"}](poligono.pdf)
The boundary of $P$ is conformed by the half-circle belonged to the family $\mathcal{C}$. Then for every $m\in\mathbb{Z}$ the hyperbolic geodesics $C_{4(4m)}$ and $C_{4(4m+2)}$ are identified as it is shown in Figure \[identificacion\] by some of the following Möbius transformations:
$$\label{eq:5}
\begin{array}{rl}
f_{m}(z) & :=\dfrac{(16m+8)z-(1+16m(16m+8))}{z-16m}\\
f_{m}^{-1}(z) & := \dfrac{-16mz+(1+16m(16m+8))}{-z+(16m+8)}.
\end{array}$$
![*Gluing the side of the hyperbolic polygon $P$.*[]{data-label="identificacion"}](Identificacion.pdf)
Analogously, the hyperbolic geodesics $C_{4(4m+1)}$ and $C_{4(4m+3)}$ are identified as it is shown in Figure \[identificacion\] by the Möbius transformations: $$\label{eq:6}
\begin{array}{rl}
g_{m}(z) & :=\dfrac{(16m+8)z-(1+(16m+4)(16m+8))}{z-(16m+4)}, \\
g_{m}^{-1}(z) & := \dfrac{-(16m+4)z+(1+(16m+4)(16m+8))}{-z+(16m+8)}.
\end{array}$$
Through the Möbius transformations above, the inside of the half-circle $C_{4(4m)}$ (the half-circle $C_{4(4m+1)}$, respectively) is send by the map $f_m(z)$ (the map $g_m(z)$, respectively) into the outside of the half-circle $C_{4(4m+2)}$ (the half-circle $C_{4(4m+3)}$, respectively). Furthermore, the outside of the half-circle $C_{4(4m)}$ (the half-circle $C_{4(4m+1)}$, respectively) is send by $f_{m}(z)$ (the map $g_m(z)$, respectively) into the inside of the half-circle $C_{4(4m+2)}$ (the half-circle $C_{4(4m+3)}$, respectively).
Hence, the hyperbolic surface $S$ get glued the side of the polygon $P$ is the Infinite Loch Ness Monster. From the polygon $P$ we deduce that noncompact quotient space $S$ comes whit a hyperbolic structure having infinite area. Fortunately, the identification defined above takes the pairwise disjoint straight segment in the boundary of $P$ performing into the only one end of the surface $S$.
![*Subregion $P_m$.*[]{data-label="subregion"}](subregion.pdf)
Furthermore, for each integer number $n\in\mathbb{Z}$ we consider the subregion $P_m\subseteq P$, which is gotten by the intersection of $P$ and the strip $\{z\in \mathbb{H}:4(4m)-2 <Re(z)< 4(4m+3)+2\}$ (See Figure \[subregion\]), then restricting to $P_m$ the identification defined above it is turned into a torus with one hole $S_m$ (see Figure \[subsurface\]), which is a subsurface of $S$. Given the elements of the countable family $\{S_m:m\in\mathbb{Z}\}$ are pair disjoint subsurfaces of $S$ then it performs infinite genus in the hyperbolic surface $S$. In other words, $S$ is the Infinite Loch Ness monster.
![*Topological subregion $P_m$ and torus with one hole $S_m$.*[]{data-label="subsurface"}](toro.pdf)
From the analytic point of view, we have built a Fuchsian subgroup $\Gamma$ of $PSL(2,\mathbb{Z})$, where $\Gamma$ is infinitely generated by the set of Möbius transformations $\{f_m(z), g_m(z), f^{-1}_m(z), g^{-1}_m(z): \text{ for all } m\in \mathbb{Z}\}$ (see equations \[eq:5\] and \[eq:6\]), having the subset $P\subseteq \mathbb{H}$ as fundamental domain[^4]. Then $\Gamma$ acts on the hyperbolic plane $\mathbb{H}$. Defining the subset $K\subseteq \mathbb{H}$ as follows, $$\label{eq:7}
K:=\{w\in\mathbb{H}: f(w)=w \text{ for any } f\in \Gamma-\{Id\}\}\subseteq\mathbb{H},$$ the Fuchsian group $\Gamma$ acts freely and properly discontinuously on the open subset $\mathbb{H}-K$. Hence, the quotient space $$\label{eq:8}
S:= (\mathbb{H}-K)/\Gamma$$ is a well-defined hyperbolic surface homeomorphic to the Infinite Loch Ness monster. Moreover, it follows from an application of the Uniformization Theorem that the fundamental group $\pi_1(S)$ of the Infinite Loch Ness monster is isomorphic to $\Gamma$.
[00]{}
[^1]: E. Ghys calls Jacob’s ladder to the surface with two ends and each ends having infinite genus (see [@Ghy]). However, M. Spivak calls this surface the doubly infinite-holed torus (see [@Spiv p.24])
[^2]: Straight segments are given by their ends points.
[^3]: The reader can also found in [@AR] a great variety of hyperbolic polygons that perform hyperbolic surfaces having infinite genus.
[^4]: To deepen in these topics we suggest to reader [@MB], [@KS].
|
---
abstract: 'This report will review the Higgs boson properties: the mass, the total width and the couplings to fermions and bosons. The measurements have been performed with the data collected in $2011$ and $2012$ at the LHC accelerator at CERN by the ATLAS and CMS experiments. Theoretical frameworks to search for new physics are also introduced and discussed.'
address:
- 'INFN, Sezione di Torino, Italy'
- |
Dipartimento di Fisica Teorica, Università di Torino, Italy\
INFN, Sezione di Torino, Italy
author:
- 'Chiara Mariotti[^1]'
- 'Giampiero Passarino[^2]'
bibliography:
- 'HC\_review4.bib'
title: 'Higgs boson couplings: measurements and theoretical interpretation'
---
Higgs boson phenomenology: production and decay \[Sect1\]
=========================================================
From discovery to properties \[Sect2\]
======================================
Analysis of the measurements \[Sect3\]
======================================
The final states \[Sect31\]
---------------------------
The original kappa-framework \[Sect4\]
======================================
Results from Run I \[Sect5\]
============================
The measurement of the mass \[Sect51\]
--------------------------------------
On-shell results \[Sect52\]
---------------------------
Off-shell results, experimental constraints on the width \[Sect53\]
-------------------------------------------------------------------
Theoretical developments \[Sect6\]
==================================
Prospects for Run II \[Sect8\]
==============================
Conclusions \[Conc\]
====================
We acknowledge important discussions with Tiziano Camporesi, Andr[è]{} David and Gino Isidori.
[^1]: chiara.mariotti@cern.ch
[^2]: giampiero@to.infn.it
|
---
abstract: 'The $q$-generalizations of the two fundamental statements of matrix algebra – the Cayley-Hamilton theorem and the Newton relations – to the cases of quantum matrix algebras of an “RTT-” and of a “Reflection equation” types have been obtained in [@NT]–[@IOPS]. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT- and Reflection equation algebras, in the sense that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton identities.'
author:
- |
A. Isaev\
[*Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna,*]{}\
[ *Moscow region, Russia*]{}\
O. Ogievetsky[^1] and P. Pyatov[^2]\
[*Center of Theoretical Physics, Luminy, 13288 Marseille, France*]{}
title: 'Generalized Cayley-Hamilton-Newton identities'
---
= 0.5cm -2cm
Introduction
============
Let $V$ be a vector space and $\R\in {\mathop{\mbox{\rm Aut}}\nolimits}(V\otimes V)$ an $\R$-matrix of Hecke type, that is, $\R$ satisfies the Yang-Baxter equation and Hecke condition, respectively, \_1\_2\_1 &=& \_2\_1\_2,\
\^2 &=& + (q-q\^[-1]{}) . \[hecke\] We use here the matrix notations of [@FRT] (e.g., $\R_1 = \R\otimes\id$, $\R_2 = \id\otimes \R$ in (\[YBE\]) etc.), $\id$ is an identity operator and $q\neq 0$ is a numeric parameter.
In this note we deal with quantum matrix algebras of two types: an RTT-algebra and a Reflection equation (RE) algebra. They are associative unital algebras generated, respectively, by elements of “$q$-matrices” $T = || T^i_j||_{i,j=1,\dots,\dim V}$ and $L = || L^i_j||_{i,j=1,\dots,\dim V}$ subject to relations T\_1T\_2 &=& T\_1T\_2 ,\
L\_1 L\_1 &=& L\_1 L\_1 .
For both these algebras, $q$-versions of the Newton identities and the Cayley-Hamilton theorem have been recently established (see [@NT]–[@IOPS]). The proofs of these two statements given for the $q$-matrix $T$ in [@PS2] and [@IOPS] turn out to be very similar ideologically and technically, which indicates that there should exist a more wide set of identities containing the Newton and the characteristic identities as particular cases. The main object of the present note is a construction of such generalized Cayley-Hamilton-Newton (CHN) identities.
We prove a $q$-version of the CHN identities for the RTT-algebra case. The CHN identities for the RE algebra are presented also. In case when both the RTT- and RE algebras originate from a quasitriangular Hopf algebra, the CHN identities for the $q$-matrix $L$ can be derived from those for the $q$-matrix $T$ by a procedure described in [@IOPS]. An independent proof of the CHN identities for the RE algebra will be given elsewhere.
Note that taking $\R =P$, the permutation matrix, one obtains – from any of the $q$-versions of the CHN theorem – a set of identities for usual matrices with commuting entries. It is worth mentioning that the CHN identities appear to be a new result even for the classical matrix algebra.
Notation
========
0
We shall begin with a brief reminder on the $\R$-matrix technique (a more complete treatment can be found, e.g., in [@G; @GPS]).
Assume that $q$ is not a root of unity, that is $k_q \equiv (q^k-q^{-k})/(q-q^{-1}) \neq 0$ for any $k=2,3,\dots$ .
Given a Hecke $\R$-matrix, one can construct two series of projectors, $A^{(k)}$ and $S^{(k)}$, called $q$-antisymmetrizers and $q$-symmetrizers, respectively. They are defined inductively as A\^[(1)]{}:= ,&& A\^[(k)]{}:=[1k\_q]{} A\^[(k[-]{}1)]{}(q\^[k-1]{}-(k[-]{}1)\_q\_[k[-]{}1]{})A\^[(k[-]{}1)]{} ,\
S\^[(1)]{}:= ,&& S\^[(k)]{}:= [1k\_q]{} S\^[(k[-]{}1)]{}(q\^[1-k]{}+(k[-]{}1)\_q\_[k[-]{}1]{})S\^[(k[-]{}1)]{} .
Further, assume that the $q$-antisymmetrizers fulfil the conditions A\^[(n)]{}=1 , A\^[(n[+]{}1)]{}=0 for some $n$. In this case the corresponding $\R$-matrix is called [*even*]{} and the number $n$ is called the [*height*]{} of the $\R$-matrix.
For an $\R$-matrix of finite height $n$ one introduces the following two matrices := [n\_q q\^n]{} \_[(2 …n)]{} A\^[(n)]{} , && := [n\_q q\^n]{} \_[(1 …n-1)]{} A\^[(n)]{} , Here and below we use notation $\tr_{(i_1\dots i_k)}$ to denote the operation of taking traces in the spaces on places $(i_1\dots i_k)$.
Cayley–Hamilton–Newton identities
=================================
0
Let us consider three sequences of elements in the RTT-algebra: s\_k(T) := && \_[(1…k)]{}(\_1\_2…\_[k-1]{}T\_1 T\_2…T\_k) ,\
\_k(T) := q\^k && \_[(1…k)]{}(A\^[(k)]{}T\_1 T\_2…T\_k) ,\
\_k(T) := q\^[-k]{} && \_[(1…k)]{}(S\^[(k)]{}T\_1 T\_2…T\_k) , k=1,2,… . We also put $s_0(T) =\s_0(T) = \t_0(T) = 1$.
To clarify the meaning of these elements, consider the classical limit $\R = P$. Denote $\{x_a\}$ the spectrum of the semisimple part of an operator $X\in {\mathop{\mbox{\rm Aut}}\nolimits}(V)$. Then the elements $s_k(X)$, $\s_k(X)$, $\t_k(X)$ are symmetric polynomials in $x_a$. Namely, $s_k(X)=\tr X^k =\sum_a x_a^k$ are [*power sums*]{}, $\s_k(X)= \sum_{a_1<\dots <a_k}x_{a_1}\dots x_{a_k}$ are [*elementary symmetric functions*]{}, and $\t_k(X)= \sum_{a_1\leq\dots\leq a_k}x_{a_1}\dots x_{a_k}$ are [*complete symmetric functions*]{}. We keep this notation for the elements $s_k(T)$, $\s_k(T)$, $\t_k(T)$ of the RTT-algebra also.
0.2cm The $q$-version of power sums $s_k(T)$ has been introduced by J.M. Maillet, who established their important property — the commutativity [@M]. Just as in the classical case, the elementary and complete symmetric functions admit an expression in terms of the power sums (see Corollary 2 below) and, hence, the commutativity property extends to any pair of elements of the sets $\{s_k(T)\}$, $\{\s_k(T)\}$, $\{\t_k(T)\}$.
.2cm If $\R$ is an even $\R$-matrix of height $n$, then one has $\s_k(T)=0$ for $k>n$ and the last nontrivial element $\s_n(T)$ is proportional to a quantum determinant of $T$, $det_q T$ \_n(T)=q\^n det\_q T .
Finally, we need an appropriate generalization of the matrix multiplication in the RTT-algebra. Inspired by the definition of the quantum power sums (\[s\]), one can introduce two versions of a $k$-th power of the $q$-matrix $T$ [@IOPS]: T\^ &:=& \_[(1 …k-1)]{} (\_1 \_2 …\_[k-1]{} T\_1 T\_2 …T\_k ) ,\
T\^ &:=& \_[(2 …k)]{} ( \_1 \_2 …\_[k-1]{} T\_1 T\_2 …T\_k ) . We use the superscripts $\underline{k}$ and $\overline{k}$ here for denoting different types of the $k$-th power of matrix $T$. This should not make a confusion with the usual matrix power (one has $T^{\underline{k}}=T^{\overline{k}}=T^k$ in the classical limit only).
In the same manner one can introduce a pair of versions of [*$k$-wedge*]{} ([*$k$-symmetric*]{}) powers of the $q$-matrix $T$, $T^{\underline{\wedge k}}$ and $T^{\overline{\wedge k}}$ ($T^{\underline{{\scriptscriptstyle \cal S}k}}$ and $T^{\overline{{\scriptscriptstyle \cal S}k}}$), removing the last or the first trace in the definition of the elementary (complete) symmetric functions, respectively, T\^ := \_[(1 …k-1)]{} (A\^[(k)]{} T\_1 …T\_k ) &,& T\^ := \_[(2 …k)]{} ( A\^[(k)]{} T\_1 …T\_k ) ,\
T\^ := \_[(1 …k-1)]{} (S\^[(k)]{} T\_1 …T\_k ) &,& T\^ := \_[(2 …k)]{} ( S\^[(k)]{} T\_1 …T\_k ) .
.3cm With these definitions we can formulate the main result. .2cm ).\
[*Let $\R$ be Hecke $R$-matrix. For any $j$, the following identities hold*]{} j\_q T\^ &=& \_[k=0]{}\^[j-1]{}(-1)\^[j-k+1]{}\_k(T) T\^ ,\
j\_q T\^ &=& \_[k=0]{}\^[j-1]{}(-1)\^[j-k+1]{} T\^ \_k(T) ,\
j\_q T\^ &=& \_[k=0]{}\^[j-1]{}\_k(T) T\^ ,\
j\_q T\^ &=& \_[k=0]{}\^[j-1]{}T\^ \_k(T) . . We shall give the details of the proof of the eq.(\[le\]). The relations (\[le2\])–(\[le4\]) can be proved analogously.
For $k=1,\dots ,j-1$ we have \_k(T) T\^&=&q\^[k]{}\_[(1…k)]{} (A\^[(k)]{}T\_[1]{}…T\_k)\_[(k+1…j-1)]{} (R\_[k+1]{}…R\_[j-1]{} T\_[k+1]{}…T\_[j]{})\
&=&q\^k\_[(1…j-1)]{} (A\^[(k)]{}T\_1…T\_k\_[k+1]{}…\_[j-1]{}T\_[k+1]{}…T\_j)\
&=&q\^k\_[(1…j-1)]{} (A\^[(k)]{}\_[k+1]{}…\_[j-1]{} T\_[1]{}…T\_j) We use (\[antis\]) in the form $q^kA^{(k)}=(k+1)_qA^{(k+1)}+k_qA^{(k)}\R_k A^{(k)}$ to rewrite (\[vsp\]) as &&(k+1)\_q\_[(1…j-1)]{}(A\^[(k+1)]{}\_[k+1]{}…\_[j-1]{}T\_1…T\_j)\
&&+ k\_q\_[(1…j-1)]{}(A\^[(k)]{}\_kA\^[(k)]{} \_[k+1]{}…\_[j-1]{}T\_1…T\_j) . In the last term, the right antisymmetrizer $A^{(k)}$ commutes with the expression $R_{k+1}\dots R_{j-1}T_1\dots T_j$, so one can move $A^{(k)}$ through this expression to the right. Next, we can move $A^{(k)}$ to the left using the cyclic property of the trace. Finally, $(A^{(k)})^2=A^{(k)}$ and we obtain \_k(T) T\^&=& (k+1)\_q\_[(1…j-1)]{}(A\^[(k+1)]{}\_[k+1]{}…\_[j-1]{}T\_1…T\_j)\
&&+ k\_q\_[(1…j-1)]{}(A\^[(k)]{}\_[k]{}…\_[j-1]{}T\_1…T\_j) . We have also $\s_0(T) T^{\underline{j}}\; =T^{\underline{j}}$. Taking the alternative sum, we obtain the relation (\[le\]).
------------------------------------------------------------------------
.3cm ([**Newton identities for the RTT-algebra [@PS2]**]{}.)\
[*Let $\R$ be Hecke $R$-matrix. The following iterative relations hold for the elements of the sets $\{s_k(T)\}$, $\{\s_k(T)\}$ and $\{\t_k(T)\}$* ]{} \[qNewton\] q\^[-j]{} j\_q\_j(T)&=&\_[k=1]{}\^[j-1]{}(-1)\^[k-1]{} \_[j-k]{}(T) s\_k(T)+(-1)\^[j-1]{}s\_j(T) ,\
\[qNewton2\] q\^j j\_q\_j(T)&=&\_[k=1]{}\^[j-1]{} \_[j-k]{}(T) s\_k(T)+s\_j(T) ,\
\[qNewton3\] 0 &=& \_[k=0]{}\^[j]{}(-1)\^k q\^[2(j-k)]{}\_[j-k]{}(T) \_k(T) , j=1,2,… . To obtain the eqs. (\[qNewton\]) and (\[qNewton2\]) one just takes the last trace (in the space with number $j$) in (\[le\]) and (\[le3\]), correspondingly. The eq. (\[qNewton3\]) then follows from (\[qNewton\]) and (\[qNewton2\]).
------------------------------------------------------------------------
.3cm ([**Cayley-Hamilton theorem for the RTT-algebra [@IOPS]**]{}).\
[*Let $\R$ be even Hecke $\R$-matrix of rank $n$. The $q$-matrix $T$ satisfies identities* ]{} \_[k=1]{}\^n \_[n-k]{}(T)(-T)\^ + \_n(T) &=&0 ,\
\_[k=1]{}\^n (-T)\^ \_[n-k]{}(T) + \_n(T) &=&0 .
[**Proof.**]{} Let $j=n$ in (\[le\]). We also have $A^{(n)}T_1\dots T_n=A^{(n)}det_q T$. Then, the eq. (\[hc1\]) follows by an application of (\[d-a\]) and (\[qdet\]). The eq. (\[hc2\]) is similarly derived from (\[le2\]).
------------------------------------------------------------------------
.3cm ([**Inverse CHN theorem for the RTT-algebra**]{}).\
[*The formulas inverse to the eqs. (\[le\])–(\[le4\]) are T\^ &=& \_[k=1]{}\^[j]{}(-1)\^[k+1]{}q\^[2(j-k)]{} k\_q \_[j-k]{}(T) T\^ ,\
T\^ &=& \_[k=1]{}\^[j]{}(-1)\^[k+1]{}q\^[2(j-k)]{}k\_q T\^ \_[j-k]{}(T) ,\
T\^ &=& \_[k=1]{}\^[j]{} (-1)\^[j-k]{} q\^[-2(j-k)]{} k\_q \_[j-k]{}(T) T\^ ,\
T\^ &=& \_[k=1]{}\^[j]{} (-1)\^[j-k]{} q\^[-2(j-k)]{} k\_q T\^ \_[j-k]{}(T) .* ]{}
Consider two lower triangular matrices: H&:=&{ H\^j\_k = q\^[2(j-k)]{}\_[j-k]{}(T) jk ; H\^j\_k = 0 } ,\
E&:=&{E\^j\_k = (-1)\^[j-k]{}\_[j-k]{}(T) jk ; E\^j\_k = 0 } . By the eq. (\[qNewton3\]) one has $HE = \id$.
With this notation one rewrites (\[le\]) as $
(-1)^{j+1}j_q\, T^{\underline{\wedge j}} =
\sum_{k=1}^j E^j_k T^{\underline{k}}\ .
$ Then $
T^{\underline{j}} = \sum_{k=1}^j
(-)^{k+1} k_q H^j_k T^{\underline{\wedge k}}\ ,
$ which is equivalent to (\[inv1\]).
The relations (\[inv2\])–(\[inv4\]) are proved similarly.
------------------------------------------------------------------------
.5cm
We conclude by formulating the CHN theorem for the RE algebra.
).\
*Let $\R$ be Hecke $R$-matrix and the $q$-matrix $L$ generate the RE algebra (\[rlrl\]). Then the following identities hold j\_q L\^[j]{} = \_[k=0]{}\^[j-1]{}(-1)\^[j-k+1]{}\_k(L) L\^[j-k]{} , j\_q L\^[[S]{} j]{} = \_[k=0]{}\^[j-1]{}\_k(L) L\^[j-k]{} .*
Here the notation is as follows: $$L^{\wedge k} := \trq_{(2\dots k)}
(A^{(k)}L_{\overline{1}}\dots L_{\overline{k}})\ , \qquad
L^{{\scriptscriptstyle\cal S} k} := \trq_{(2\dots k)}
(S^{(k)}L_{\overline{1}}\dots L_{\overline{k}})$$ are the $k$-wedge and the $k$-symmetric powers of the $q$-matrix $L$, respectively; $L^k$ is the usual matrix power; $\s_k(L) := q^k \trq L^{\wedge k}$ and $\t_k(L) := q^{-k} \trq L^{{\scriptscriptstyle\cal S}k}$ are the elementary and complete symmetric functions on the spectrum of $L$, respectively; $\trq X := \tr (\Dr X)$ is a $q$-trace operation, and $L_{\overline{k}}$ is defined inductively by $$L_{\overline{1}} := L_1\ , \qquad
L_{\overline{k}} := \R_{k-1}L_{\overline{k-1}}\R^{-1}_{k-1}\ .$$
.2cm We are grateful to D. Gurevich and P. Saponov for discussions. This work is supported in part by the grants for promotion of french–russian scientific cooperation: the CNRS grant PICS No. 608 and the RFBR grant No. 98-01-2033. The work of P.P. and A.I. is also partly supported by the RFBR grant No. 97-01-01041.
[99]{} Faddeev L.D., Reshetikhin N.Yu., and Takhtajan L.A.: Algebra i Analiz [*1*]{} no.1 (1989) 178. English translation in: Leningrad Math. J. [*1*]{} (1990) 193. Nazarov M. and Tarasov V.: Publications RIMS [*30*]{} (1994) 459. Pyatov P.N. and Saponov P.A.: J. Phys. A: Math. Gen., [*28*]{} (1995) 4415. Gurevich D.I., Pyatov P.N., Saponov P.A.: Lett. in Math. Phys. [*41*]{} (1997) 255. Pyatov P., Saponov P.: “Newton relations for quantum matrix algebras of $RTT$-type”, Preprint IHEP 96-76 (1996). Isaev A., Ogievetsky O., Pyatov P. and Saponov P.: “Characteristic polynomials for Quantum Matrices”, Preprint CPT-97/P3471 (1997). To appear in Proc. of the Intern. Conf. in memory of V.I. Ogievetsky (Dubna, Russia, 1997): Springer-Verlag, 1998. Gurevich D.I.: Algebra i Analiz [*2*]{} (1990). English translation in: Leningrad Math. J. [*2*]{} (1991) 801. Maillet J.M.: Phys. Lett. [*B245*]{} (1990) 480.
[^1]: On leave of absence from P. N. Lebedev Physical Institute, Theoretical Department, Leninsky pr. 53, 117924 Moscow, Russia
[^2]: On leave of absence from Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia
|
---
abstract: 'In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin in [@CR]. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in [@GX4].'
author:
- Xu Xu
title: '**On the global rigidity of sphere packings on 3-dimensional manifolds**'
---
**MSC (2010):** 52C25; 52C26
**Keywords:** Global rigidity; Sphere packing; Combinatorial scalar curvature
Introduction {#section 1}
============
In his investigation of hyperbolic metrics on 3-manifolds, Thurston ([@T1], Chapter 13) introduced the circle packing with prescribed intersection angles and proved the Andreev-Thurston Theorem, which consists of two parts. The first part is on the existence of circle packing for a given triangulation. The second part is on the rigidity of circle packings, which states that a circle packing is uniquely determined by its discrete Gauss curvature (up to scaling for the Euclidean background geometry). For a proof of Andreev-Thurston Theorem, see [@CL1; @DV; @H; @MR; @S; @T1].
To study the $3$-dimensional analogy of the circle packing on surfaces, Cooper and Rivin [@CR] introduced the sphere packing on 3-dimensional manifolds. Suppose $M$ is a 3-dimensional closed manifold with a triangulation $\mathcal{T}=\{V,E,F,T\}$, where the symbols $V,E,F,T$ represent the sets of vertices, edges, faces and tetrahedra, respectively.
\[definition of sphere packing metric\] A Euclidean (hyperbolic respectively) sphere packing metric on $(M, \mathcal{T})$ is a map $r:V\rightarrow (0,+\infty)$ such that (1) the length of an edge $\{ij\}\in E$ with vertices $i, j$ is $l_{ij}=r_{i}+r_{j}$ and (2) for each tetrahedron $\{i,j,k,l\}\in T$, the lengths $l_{ij},l_{ik},l_{il},l_{jk},l_{jl},l_{kl}$ form the edge lengths of a Euclidean (hyperbolic respectively) tetrahedron.
The condition (2) is called the nondegenerate condition, which makes the space of sphere packing metrics to be a proper open subset of $\mathbb{R}^{|V|}_{>0}$. The space of sphere packing metrics will be denoted by $\Omega$ in the paper.
To study sphere packing metrics, Cooper and Rivin [@CR] introduced the combinatorial scalar curvature $K: V\rightarrow \mathbb{R}$, which is defined as angle deficit of solid angles at a vertex $i$ $$\label{CR curvature}
K_{i}= 4\pi-\sum_{\{i,j,k,l\}\in T}\alpha_{ijkl},$$ where $\alpha_{ijkl}$ is the solid angle at the vertex $i$ of the tetrahedron $\{i,j,k,l\}\in T$ and the summation is taken over all tetrahedra with $i$ as a vertex. The sphere packing metrics have the following local rigidity with respect to the combinatorial scalar curvature $K$.
Suppose $(M, \mathcal{T})$ is a closed $3$-dimensional triangulated manifold. Then a Euclidean or hyperbolic sphere packing metric on $(M, \mathcal{T})$ is locally determined by its combinatorial scalar curvature $K$ (up to scaling for the Euclidean background geometry).
The global rigidity of sphere packing metrics on 3-dimensional triangulated manifolds was conjectured by Cooper and Rivin in [@CR]. In this paper, we solve this conjecture and prove the following result.
\[main theorem rigidity\] Suppose $(M, \mathcal{T})$ is a closed triangulated $3$-manifold.
(1)
: A Euclidean sphere packing metric on $(M, \mathcal{T})$ is determined by its combinatorial scalar curvature $K: V\rightarrow \mathbb{R}$ up to scaling.
(2)
: A hyperbolic sphere packing metric on $(M, \mathcal{T})$ is determined by its combinatorial scalar curvature $K: V\rightarrow \mathbb{R}$.
Although the combinatorial curvature $K$ is a good candidate for the 3-dimensional combinatorial scalar curvature, it has two disadvantages comparing to the smooth scalar curvature on Riemannian manifolds. The first is that it is scaling invariant with respect to the Euclidean sphere packing metrics, i.e. $K(\lambda r)=K(r)$ for $\lambda>0$; The second is that $K_i$ tends to zero as the triangulation of the manifold is finer and finer. Motivated by the observations, Ge and the author [@GX4] introduced a new combinatorial scalar curvature defined as $R_i=\frac{K_i}{r_i^2}$ for 3-dimensional manifolds with Euclidean background geometry, which overcomes the two disadvantages if we take $g_i=r_i^2$ as an analogue of the Riemannian metric tensor for the Euclidean background geometry. This definition can be modified to fit the case of hyperbolic background geometry. We further generalized this definition of combinatorial scalar curvature to the following combinatorial $\alpha$-curvature.
\[definition of alpha curvature\] Suppose $(M, \mathcal{T})$ is a closed triangulated $3$-manifold with a sphere packing metric $r: V\rightarrow (0, +\infty)$ and $\alpha\in \mathbb{R}$. Combinatorial $\alpha$-curvature at a vertex $i\in V$ is defined to be $$\label{definition of R-curvature}
R_{\alpha,i}=\frac{K_i}{s_i^{\alpha}},$$ where $s_i=r_i$ for the Euclidean sphere packing metrics and $s_i=\tanh \frac{r_i}{2}$ for the hyperbolic sphere packing metrics.
When $\alpha=0$, the $0$-curvature $R_0$ is the combinatorial scalar curvature $K$. Combinatorial $\alpha$-curvatures on triangulated surfaces were studied in [@GJ3; @GX2; @GX4; @GX5; @GX3; @GX6; @X]. Using the combinatorial $\alpha$-curvature, we prove the following global rigidity on 3-manifolds.
\[main theorem global rigidity for alpha curvature\] Suppose $(M, \mathcal{T})$ is a closed triangulated $3$-manifold and $\overline{R}$ is a given function defined on the vertices of $(M, \mathcal{T})$.
(1)
: In the case of Euclidean background geometry,
(a)
: if $\alpha\overline{R}\equiv0$, there exists at most one Euclidean sphere packing metric in $\Omega$ with combinatorial $\alpha$-curvature equal to $\overline{R}$ up to scaling.
(b)
: if $\alpha\overline{R}\leq0$ and $\alpha\overline{R}\not\equiv0$, there exists at most one Euclidean sphere packing metric in $\Omega$ with combinatorial $\alpha$-curvature equal to $\overline{R}$.
(2)
: In the case of hyperbolic background geometry, if $\alpha\overline{R}\leq 0$, there exists at most one hyperbolic sphere packing metric in $\Omega$ with combinatorial $\alpha$-curvature equal to $\overline{R}$.
When $\alpha=0$, Theorem \[main theorem global rigidity for alpha curvature\] is reduced to Theorem \[main theorem rigidity\]. When $\alpha=2$, the local rigidity of Euclidean sphere packing metrics with nonpositive constant $2$-curvature was proven in [@GX4]. For $\alpha\in\mathbb{R}$, the local rigidity of Euclidean sphere packing metrics with constant combinatorial $\alpha$-curvature on 3-dimensional triangulated manifolds was proven in [@GX5]. Results similar to Theorem \[main theorem global rigidity for alpha curvature\] were proven for Thurston’s circle packing metrics on surfaces in [@GX4; @GX3] and for inversive distance circle packing metrics on surfaces in [@GJ1; @GJ2; @GJ3; @GX6; @X].
Glickenstein [@G1] introduced a combinatorial Yamabe flow to study the constant curvature problem of $K$. He found that the combinatorial scalar curvature $K$ evolves according to a heat type equation along his flow and showed that the solution converges to a constant curvature metric under some nonsingular conditions. He [@G2] further derived a maximal principle for the curvature along the combinatorial Yamabe flow under certain assumptions on the triangulation. Ge and the author [@GX2; @GX4; @GX5] generalized Cooper and Rivin’s definition of combinatorial scalar curvature and introduced a combinatorial Yamabe flow to deform the sphere packing metrics, aiming at finding the corresponding constant curvature sphere packing metrics on 3-dimensional triangulated manifolds. Ge and Ma [@GM] studied the deformation of combinatorial $\alpha$-curvature on $3$-dimensional triangulated manifolds using a modified combinatorial Yamabe flow.
The paper is organized as follows. In Section \[Section 2\], We give a description of the admissible space of sphere packing metrics for a single tetrahedron. In Section \[Section 3\], we recall Cooper and Rivin’s action functional and extend it to be a convex functional. In Section \[Section 4\], We prove Theorem \[main theorem rigidity\] and Theorem \[main theorem global rigidity for alpha curvature\].
Admissible space of sphere packing metrics for a single tetrahedron {#Section 2}
===================================================================
Suppose $M$ is a 3-dimensional connected closed manifold with a triangulation $\mathcal{T}=\{V,E,F,T\}$. We consider sphere packing metrics as points in $\mathbb{R}^N_{>0}$, where $N=|V|$ denotes the number of vertices. And we use $\mathbb{R}^V$ to denote the set of real functions defined on the set of vertices $V$.
Suppose $r$ is a Euclidean sphere packing metric on $(M, \mathcal{T})$. For any edge $\{ij\}\in E$, let $l_{ij}=r_i+r_j$ and for a tetrahedron $\{ijkl\}\in T$, $l_{ij}, l_{ik}, l_{il}, l_{jk}, l_{jl}, l_{kl}$ can be realized as edge lengths of a Euclidean tetrahedron. Gluing all of these Euclidean tetrahedra in $T$ along the faces isometrically produces a piecewise linear metric on the triangulated manifold $(M, \mathcal{T})$. On this manifold, drawing a sphere $S_i$ centered at vertex $i$ of radius $r_i$ for each vertex $i\in V$, we obtain a Euclidean sphere packing. A hyperbolic sphere packing can be constructed similarly.
As $l_{ij}=r_i+r_j$, it is straightforward to show that the triangle inequalities for $l_{ij}, l_{ik}, l_{jk}$ hold on the face $\{ijk\}\in F$. However, triangle inequalities on the faces are not enough for $l_{ij}, l_{ik}, l_{il}, l_{jk}, l_{jl}, l_{kl}$ to determine a Euclidean or hyperbolic tetrahedron. There are nondegenerate conditions. It is found [@CR; @G1] that Descartes circle theorem, also called Soddy-Gossett theorem, can be used to describe the degenerate case. We state a version obtained in [@M].
An oriented circle is a circle together with an assigned direction of unit normal. The interior of an oriented circle is its interior for an inward pointing normal and its exterior for an outward pointing normal.
A Euclidean (hyperbolic respectively) oriented Descartes configuration consists of $4$ mutually tangent oriented circles in the Euclidean (hyperbolic respectively) plane such that all pairs of tangent circles have distinct points of tangency and the interiors of all four oriented circles are disjoint.
Several Euclidean oriented Descartes configurations are shown in Figure \[Descartes configurations\], where the shadow denotes the interior of a circle.
![Descartes configurations[]{data-label="Descartes configurations"}](circle4.pdf){height="43.00000%" width="100.00000%"}
We allow the Euclidean oriented Descartes configuration to include the straight lines and the hyperbolic oriented Descartes configuration to include the horocycles.
\[Descartes circle theorem\]
(1)
: Given a Euclidean Descartes configuration $C_i, i=1,2,3,4$ such that $C_i$ has radius $r_i$. Then $$\left(\sum_{i=1}^4k_i\right)^2-2\sum_{i=1}^4k_i^2=0,$$ where $k_i=\frac{1}{r_i}$, if $C_i$ is assigned an inward pointing normal, otherwise $k_i=-\frac{1}{r_i}$.
(2)
: Given a hyperbolic Descartes configuration $C_i, i=1,2,3,4$ such that $S_i$ has radius $r_i$. Then $$\label{hyperbolic quadratic}
\begin{aligned}
\left(\sum_{i=1}^4k_i\right)^2-2\sum_{i=1}^4k_i^2+4=0,
\end{aligned}$$ where $k_i=\coth r_i$, if $C_i$ is assigned an inward pointing normal, otherwise $k_i=-\coth r_i$.
There is also a version of Soddy-Gossett theorem for the spherical background geometry. See [@LMW; @M] for Descartes circle theorem and Soddy-Gosset theorems with different background geometries. In this paper, we concentrate on the Euclidean and hyperbolic cases.
For the Euclidean background geometry, Glickenstein [@G1] observed the admissible space of Euclidean sphere packing metrics for a tetrahedron $\{ijkl\}\in T$ to be a Euclidean tetrahedron is $$\Omega^{\mathbb{E}}_{ijkl}=\{(r_i, r_j, r_k, r_l)\in \mathbb{R}^4_{>0}|Q^{\mathbb{E}}_{ijkl}>0\},$$ where $$Q^{\mathbb{E}}_{ijkl}=\left(\frac{1}{r_{i}}+\frac{1}{r_{j}}+\frac{1}{r_{k}}+\frac{1}{r_{l}}\right)^2-
2\left(\frac{1}{r_{i}^2}+\frac{1}{r_{j}^2}+\frac{1}{r_{k}^2}+\frac{1}{r_{l}^2}\right).$$ For the hyperbolic background geometry, we need the following result.
\[nondegenerate condition for hyperbolic\] A non-degenerate hyperbolic tetrahedron with edge lengths $l_{ij}, l_{ik}, l_{il}, l_{jk}, l_{jl}, l_{kl}$ exists if and only if all principal minors of $$\begin{aligned}
\left|
\begin{array}{ccccc}
1 & \cosh l_{ij} & \cosh l_{ik} & \cosh l_{il}\\
\cosh l_{ij} & 1 & \cosh l_{jk} & \cosh l_{jl}\\
\cosh l_{ik} & \cosh l_{jk} & 1 & \cosh l_{kl}\\
\cosh l_{il} & \cosh l_{jl} & \cosh l_{kl} & 1 \\
\end{array}
\right|
\end{aligned}$$ are negative.
Applying Proposition \[nondegenerate condition for hyperbolic\] to hyperbolic sphere packing metrics, we have the admissible space of hyperbolic sphere packing metrics for a tetrahedron $\{ijkl\}\in T$ to be a non-degenerate hyperbolic tetrahedron is $$\Omega^{\mathbb{H}}_{ijkl}=\{(r_i, r_j, r_k, r_l)\in \mathbb{R}^4_{>0}|Q^{\mathbb{H}}_{ijkl}>0\},$$ where $$\begin{aligned}
Q^{\mathbb{H}}_{ijkl}=&\left(\coth r_i+\coth r_j+\coth r_k+\coth r_l\right)^2\\
&-2\left(\coth^2 r_i+\coth^2 r_j+\coth^2 r_k+\coth^2 r_l\right)+4.
\end{aligned}$$
Cooper and Rivin [@CR] called the tetrahedra produced by sphere packing conformal and proved that a tetrahedron is a Euclidean conformal tetrahedron if and only if there exists a unique sphere tangent to all of the edges of the tetrahedron. Moreover, the point of tangency with the edge $\{ij\}$ is of distance $r_i$ to $i$-th vertex. They proved the following lemma on the admissible space of sphere packing metrics for a single tetrahedron.
\[simply connectness of ijkl\] For a Euclidean or hyperbolic tetrahedron $\{ijkl\}\in T$, the admissible spaces $\Omega^{\mathbb{E}}_{ijkl}$ and $\Omega^{\mathbb{H}}_{ijkl}$ are simply connected open subsets of $\mathbb{R}^4_{>0}$.
They further pointed out that $\Omega^{\mathbb{E}}_{ijkl}$ is not convex. For a triangulated $3$-manifold $(M, \mathcal{T})$, the admissible spaces $$\Omega^{\mathbb{E}}=\{r\in \mathbb{R}^N_{>0}|Q_{ijkl}^{\mathbb{E}}>0, \forall \{ijkl\}\in T\}$$ and $$\Omega^{\mathbb{H}}=\{r\in \mathbb{R}^N_{>0}|Q^{\mathbb{H}}_{ijkl}>0, \forall \{ijkl\}\in T\}$$ are open subsets of $\mathbb{R}^N_{>0}$.
We need a good description of the admissible spaces $\Omega^{\mathbb{E}}_{ijkl}$ and $\Omega^{\mathbb{H}}_{ijkl}$ for a single tetrahedron $\{ijkl\}\in T$. If the radii $r_j, r_k, r_l$ of the spheres $S_j, S_k, S_l$ are fixed, Cooper and Rivin [@CR] observed that degeneracy occurs when $r_i$ is large enough so that the sphere $S_i$ is large enough to be tangent to the other three spheres, yet small enough that its center $i$ lies in the plane defined by $j, k$ and $l$. This defines a degenerate set $V_i$ of the sphere packing metrics. The degenerate sets $V_j, V_k$ and $V_l$ can be defined similarly.
We have the following result on the structure of the sets $V_i$, $V_j$, $V_k$, $V_l$ and $\Omega_{ijkl}$. Here and in the rest of the paper, $\Omega_{ijkl}$ denotes $\Omega^{\mathbb{E}}_{ijkl}$ or $\Omega^{\mathbb{H}}_{ijkl}$ according to the background geometry and $\overline{\Omega}_{ijkl}$ denotes the closure of $\Omega_{ijkl}$ in $\mathbb{R}^4_{>0}$.
\[structure of admissible r\] Connected components of $\mathbb{R}^4_{>0}-\Omega_{ijkl}$ are $V_i$, $V_j$, $V_k$ and $V_l$. Furthermore, the intersection of $\overline{\Omega}_{ijkl}$ with any of $V_i$, $V_j$, $V_k$, $V_l$ is a connected component of $\overline{\Omega}_{ijkl}-\Omega_{ijkl}$, which is a graph of a continuous and piecewise analytic function defined on $\mathbb{R}^3_{>0}$.
[**Proof.** ]{}We prove the Euclidean case in details. The hyperbolic case is similar and will be omitted. By symmetry, it suffices to consider $V_i$. It is observed [@G2] that $$\begin{aligned}
Q^{\mathbb{E}}_{ijkl}
=&\frac{1}{r_i}(\frac{1}{r_j}+\frac{1}{r_k}+\frac{1}{r_l}-\frac{1}{r_i})+\frac{1}{r_j}(\frac{1}{r_i}+\frac{1}{r_k}+\frac{1}{r_l}-\frac{1}{r_j})\\
&+\frac{1}{r_k}(\frac{1}{r_i}+\frac{1}{r_j}+\frac{1}{r_l}-\frac{1}{r_k})+\frac{1}{r_l}(\frac{1}{r_i}+\frac{1}{r_j}+\frac{1}{r_k}-\frac{1}{r_l}).
\end{aligned}$$ If $r_i=\min\{r_i, r_j, r_k, r_l\}$ and $Q^{\mathbb{E}}_{ijkl}=0$, then $\frac{1}{r_j}+\frac{1}{r_k}+\frac{1}{r_l}-\frac{1}{r_i}<0$ and $$\label{derivative of Q}
\frac{\partial Q^{\mathbb{E}}_{ijkl}}{\partial r_i}=-\frac{2}{r_i^2}(\frac{1}{r_j}+\frac{1}{r_k}+\frac{1}{r_l}-\frac{1}{r_i})>0.$$ This implies that if $Q^{\mathbb{E}}_{ijkl}=0$, we can always increase $r_i$ to make the tetrahedron nondegenerate. So we just need to analyze the critical degenerate case of $V_i$, where $S_i$ is externally tangent to the other three spheres $S_j, S_k, S_l$ and the center $i$ lies in the plane defined by $j, k$ and $l$. Glickenstein further proved the following result.
\[degenerate lemma\] If $Q^{\mathbb{E}}_{ijkl}\rightarrow 0$ such that none of $r_i, r_j, r_k, r_l$ tend to $0$, then one solid angle tends to $2\pi$ and the others tend to 0. Furthermore, if $r_i$ is the minimum of $r_i, r_j, r_k, r_l$, then the dihedral angles $\beta_{ijkl}, \beta_{ikjl}, \beta_{iljk}$ tend to $\pi$, the dihedral angles $\beta_{jkil}, \beta_{jlik}, \beta_{klij}$ tend to $0$ and the solid angle $\alpha_{ijkl}$ tends to $2\pi$, where $\beta_{ijkl}$ is the dihedral angle along the edge $\{ij\}$.
Proposition \[degenerate lemma\] implies that $V_i, V_j, V_k, V_l$ are the only degenerations that can occur. Furthermore, if $Q^{\mathbb{E}}_{ijkl}=0$ and $r_i<\min\{r_j, r_k, r_l\}$, the center $i$ lies in the interior of the Euclidean triangle $\triangle jkl$, which determines an oriented Descartes configuration consisting of four externally tangent circles $C_i, C_j, C_k, C_l$ in the plane defined by $j, k$ and $l$. See Figure \[circle\].
Suppose $C_i, C_j, C_k, C_l$ are four oriented circles with finite radii and inward pointing normal in the Euclidean plane. If $C_i, C_j, C_k, C_l$ form an oriented Descartes configuration and $r_i<\min\{r_j, r_k, r_l\}$, then $$\label{equation of r_i}
\begin{aligned}
r_i=f(r_j,r_k,r_l):=\left\{
\begin{array}{ll}
\frac{-B+\sqrt{B^2-4AC}}{2A}, & \hbox{$(r_j, r_k, r_l)\in \Omega_{jkl}$;} \\
-\frac{C}{B}, & \hbox{$(r_j, r_k, r_l)\in \overline{\Omega}_{jkl}\setminus\Omega_{jkl}$;} \\
\frac{-B+\sqrt{B^2-4AC}}{2A}, & \hbox{$(r_j, r_k, r_l)\in \mathbb{R}^3_{>0}\setminus \overline{\Omega_{jkl}}$;}
\end{array}
\right.
\end{aligned}$$ where $$\begin{aligned}
A=&2r_jr_kr^2_l+2r_jr_lr^2_k+2r_kr_lr^2_j-r_k^2r_l^2-r_j^2r_l^2-r_j^2r_k^2,\\
B=&2 r_jr_kr_l(r_kr_l+r_jr_l+r_jr_k),\\
C=&-r_j^2r_k^2r_l^2,\\
\Omega_{jkl}=&\{(r_j,r_k,r_l)\in \mathbb{R}^3_{>0}|A>0\}.
\end{aligned}$$
[**Proof.** ]{}By Descartes circle theorem \[Descartes circle theorem\], we have $$\begin{aligned}
Q^{\mathbb{E}}_{ijkl}=\left(\frac{1}{r_{i}}+\frac{1}{r_{j}}+\frac{1}{r_{k}}+\frac{1}{r_{l}}\right)^2 -2\left(\frac{1}{r_{i}^2}+\frac{1}{r_{j}^2}+\frac{1}{r_{k}^2}+\frac{1}{r_{l}^2}\right)=0,
\end{aligned}$$ which is equivalent to the quadratic equation in $r_i$ $$\label{quadratic equation in r_i}
\begin{aligned}
Ar_i^2+Br_i+C=0.
\end{aligned}$$ For the quadratic equation (\[quadratic equation in r\_i\]) in $r_i$, the discriminant is $$\begin{aligned}
\Delta=&B^2-4AC=16r_j^3r_k^3r_l^3(r_j+r_k+r_l),
\end{aligned}$$ which is always positive for $(r_j, r_k, r_l)\in \mathbb{R}^3_{>0}$. Note that $$\begin{aligned}
A=&2r_jr_kr^2_l+2r_jr_lr^2_k+2r_kr_lr^2_j-r_k^2r_l^2-r_j^2r_l^2-r_j^2r_k^2\\
=&(\sqrt{r_jr_k}+\sqrt{r_jr_l}+\sqrt{r_kr_l})(\sqrt{r_jr_k}+\sqrt{r_jr_l}-\sqrt{r_kr_l})\\
&(\sqrt{r_jr_k}-\sqrt{r_jr_l}+\sqrt{r_kr_l})(-\sqrt{r_jr_k}+\sqrt{r_jr_l}+\sqrt{r_kr_l}).
\end{aligned}$$
![circle configurations (the circles with dotted line correspond to the roots of (\[quadratic equation in r\_i\]) rejected)[]{data-label="circle"}](circle2.pdf){height="47.00000%" width="100.00000%"}
(1)
: In the case that $A>0$, we have $\sqrt{r_jr_k}, \sqrt{r_jr_l}, \sqrt{r_kr_l}$ satisfy the triangle inequalities. As $B>0$ and $C<0$, the quadratic equation (\[quadratic equation in r\_i\]) has two roots with different signs and negative sum. Note that $r_i>0$ by Theorem \[Descartes circle theorem\], we have $$r_i=\frac{-B+\sqrt{\Delta}}{2A}>0.$$ The negative root corresponds to the case that the mutually externally tangent circles $C_j, C_k, C_l$ are internally tangent to a circle $C'_i$. See (a) in Figure \[circle\].
(2)
: In the case that $A=0$, i.e. $\sqrt{r_jr_k}+\sqrt{r_jr_l}=\sqrt{r_kr_l}$ or $\sqrt{r_jr_k}+\sqrt{r_kr_l}=\sqrt{r_jr_l}$ or $\sqrt{r_kr_l}+\sqrt{r_jr_l}=\sqrt{r_jr_k}$, we have $$r_i=-\frac{C}{B}>0.$$ This corresponds to the case that there is a straight line tangent to the circles $C_j, C_k, C_l$ on the same side. See (b) in Figure \[circle\].
(3)
: In the other cases, we have $A<0$, $B>0$ and $C<0$ with discriminant $\Delta>0$. Then the quadratic equation (\[quadratic equation in r\_i\]) have two different positive roots $\frac{-B+\sqrt{\Delta}}{2A}$ and $\frac{-B-\sqrt{\Delta}}{2A}$. This corresponds to the case that there are two circles with finite radii externally tangent to $C_j, C_k, C_l$ simultaneously. See (c) in Figure \[circle\]. Without loss of generality, we assume $r_j\leq r_k\leq r_l$. We claim that $\frac{-B-\sqrt{\Delta}}{2A}>r_j$. Then $r_i=\frac{-B+\sqrt{\Delta}}{2A}$ by $r_i<\min\{r_j, r_k, r_l\}$.
To prove the claim, note that $$\begin{aligned}
A=&r_j^2r_k^2r_l^2(\frac{1}{\sqrt{r_j}}+\frac{1}{\sqrt{r_k}}+\frac{1}{\sqrt{r_l}})(\frac{1}{\sqrt{r_j}}+\frac{1}{\sqrt{r_k}}-\frac{1}{\sqrt{r_l}})\\
&(\frac{1}{\sqrt{r_j}}+\frac{1}{\sqrt{r_l}}-\frac{1}{\sqrt{r_k}})(\frac{1}{\sqrt{r_k}}+\frac{1}{\sqrt{r_l}}-\frac{1}{\sqrt{r_j}}).
\end{aligned}$$ $A<0$ implies $\frac{1}{\sqrt{r_j}}>\frac{1}{\sqrt{r_k}}+\frac{1}{\sqrt{r_l}}$. To simplify the notations, set $a=\frac{1}{\sqrt{r_j}}$, $b=\frac{1}{\sqrt{r_k}}$, $c=\frac{1}{\sqrt{r_l}}$, then $a>b+c$.
$\frac{-B-\sqrt{\Delta}}{2A}>r_j$ is equivalent to $$\label{inequality}
\begin{aligned}
& 2 r_jr_kr_l(r_kr_l+r_jr_l+r_jr_k)+\sqrt{16r_j^3r_k^3r_l^3(r_j+r_k+r_l)}\\
&>-2r_j(2r_jr_kr^2_l+2r_jr_lr^2_k+2r_kr_lr^2_j-r_k^2r_l^2-r_j^2r_l^2-r_j^2r_k^2)\\
\Leftrightarrow & \frac{1}{r_j}+\frac{1}{r_k}+\frac{1}{r_l}+2\sqrt{\frac{1}{r_jr_k}+\frac{1}{r_jr_l}+\frac{1}{r_kr_l}}
>r_j(\frac{1}{r_j^2}+\frac{1}{r_k^2}+\frac{1}{r_l^2}-\frac{2}{r_jr_k}-\frac{2}{r_jr_l}-\frac{2}{r_kr_l})\\
\Leftrightarrow & a^2+b^2+c^2+2\sqrt{a^2b^2+a^2c^2+b^2c^2}>\frac{1}{a^2}(a^4+b^4+c^4-2a^2b^2-2a^2c^2-2b^2c^2)\\
\Leftrightarrow & 3a^2(b^2+c^2)+2a^2\sqrt{a^2b^2+a^2c^2+b^2c^2}>(b^2-c^2)^2.
\end{aligned}$$ By $a>b+c$, we have $$3a^2(b^2+c^2)>3(b+c)^2(b^2+c^2)>(b+c)^2(b-c)^2=(b^2-c^2)^2,$$ which implies (\[inequality\]). This completes the proof of the claim and the lemma.
------------------------------------------------------------------------
Note that $\partial \Omega_{jkl}=\overline{\Omega}_{jkl}\setminus \Omega_{jkl}=\{(r_j,r_k,r_l)\in \mathbb{R}^3_{>0}|A=0\}$. It is straightforward to show that, as $(r_j,r_k,r_l)$ tends to a point in $\partial \Omega_{jkl}$, $$\frac{-B+\sqrt{B^2-4AC}}{2A}\rightarrow -\frac{C}{B},$$ which implies that $r_i=f(r_j, r_k, r_l)$ in (\[equation of r\_i\]) is a continuous and piecewise analytic function of $(r_j, r_k, r_l)\in \mathbb{R}^3_{>0}$.
By (\[derivative of Q\]), the degenerate set $V_i$ is $$V_i=\{(r_i,r_j,r_k,r_l)\in \mathbb{R}^4_{>0}|0<r_i\leq f(r_j,r_k,r_l)\},$$ which is a simply connected subset of $\mathbb{R}^4_{>0}$. Similarly, we have $$\begin{aligned}
V_j=&\{(r_i,r_j,r_k,r_l)\in \mathbb{R}^4_{>0}|0<r_j\leq f(r_i,r_k,r_l)\},\\
V_k=&\{(r_i,r_j,r_k,r_l)\in \mathbb{R}^4_{>0}|0<r_k\leq f(r_i,r_j,r_l)\},\\
V_l=&\{(r_i,r_j,r_k,r_l)\in \mathbb{R}^4_{>0}|0<r_l\leq f(r_i,r_j,r_k)\}.
\end{aligned}$$ Therefore, we have $$\mathbb{R}^4_{>0}=\Omega_{ijkl}\cup V_i\cup V_j \cup V_k \cup V_l.$$
We claim that $V_i, V_j, V_k, V_l$ are mutually disjoint. Otherwise that $V_i\cap V_j\neq \emptyset$ and $r=(r_i, r_j, r_k, r_l)\in V_i\cap V_j \subset \mathbb{R}^4_{>0}$. By the geometric meaning of the critical degenerate case, we have $$\label{i in jkl}
A_{\triangle ijk}+A_{\triangle ijl}+A_{\triangle ikl}\leq A_{\triangle jkl}$$ and $$\label{j in ikl}
A_{\triangle ijk}+A_{\triangle ijl}+A_{\triangle jkl}\leq A_{\triangle ikl},$$ where $A_{\triangle ijk}$ denotes the area of the triangle $\{ijk\}\in F$ with edge lengths $l_{ij}=r_i+r_j, l_{ik}=r_i+r_k, l_{jk}=r_j+r_k$. Combining (\[i in jkl\]) with (\[j in ikl\]), we have $A_{\triangle ijk}+A_{\triangle ijl}\leq 0$, which is impossible. So we have $V_i\cap V_j=\emptyset$. This completes the proof for the theorem with Euclidean background geometry.
The proof for the case of hyperbolic background geometry is similar. The boundary of $V_i$ in $\mathbb{R}^4_{>0}$ is given by the function $$\begin{aligned}
\tanh r_i=\left\{
\begin{array}{ll}
\frac{-\mathcal{B}+\sqrt{\mathcal{B}^2-4\mathcal{A}\mathcal{C}}}{2\mathcal{A}}, & \hbox{$(r_j, r_k, r_l)\in\mathbb{R}^3_{>0}\setminus\partial \widetilde{\Omega}_{jkl}$,} \\
-\frac{\mathcal{C}}{\mathcal{B}}, & \hbox{$(r_j, r_k, r_l)\in \partial \widetilde{\Omega}_{jkl}$,}
\end{array}
\right.
\end{aligned}$$ where $$\begin{aligned}
\mathcal{A}=&4\tanh^2 r_j \tanh^2 r_k\tanh^2 r_l+\left(\tanh r_j \tanh r_k+\tanh r_j\tanh r_l+\tanh r_k\tanh r_l\right)^2\\
&-2\left(\tanh^2 r_j \tanh^2 r_k+\tanh^2 r_j \tanh^2 r_l+\tanh^2 r_k \tanh^2 r_l\right),\\
\mathcal{B}=&2\tanh r_j \tanh r_k\tanh r_l(\tanh r_j \tanh r_k+\tanh r_j \tanh r_l+\tanh r_k \tanh r_l),\\
\mathcal{C}=&-\tanh^2 r_j \tanh^2 r_k \tanh^2 r_l,\\
\partial \widetilde{\Omega}_{jkl}=&\{(r_j, r_k, r_l)\in \mathbb{R}^3_{>0}|\mathcal{A}=0\}.
\end{aligned}$$ Set $r_i=g(r_j, r_k, r_l)$, then $g$ is continuous and piecewise analytic. The corresponding degenerate set $V_i$ is $$V_i=\{(r_i, r_j, r_k, r_l)\in \mathbb{R}^4_{>0}|0<r_i\leq g(r_j, r_k, r_l)\}.$$ The rest of the proof for the hyperbolic case is similar to that of the Euclidean case, so we omit the details here.
By Theorem \[structure of admissible r\], the admissible space $\Omega_{ijkl}$ of sphere packing metrics for a single tetrahedron is homotopy equivalent to $\mathbb{R}^4_{>0}$. This provides another proof of Cooper-Rivin’s Lemma \[simply connectness of ijkl\] [@CR] that $\Omega_{ijkl}$ is simply connected.
In the following, we take $\partial_i \Omega_{ijkl}=\overline{\Omega}_{ijkl}\cap V_i$.
Cooper-Rivin’s action functional and its extension {#Section 3}
==================================================
Cooper-Rivin’s action functional
--------------------------------
For a triangulated 3-manifold $(M, \mathcal{T})$ with sphere packing metric $r$, Cooper and Rivin [@CR] introduced the definition (\[CR curvature\]) of combinatorial scalar curvature $K_{i}$ at the vertex $i$ $$K_{i}= 4\pi-\sum_{\{i,j,k,l\}\in T}\alpha_{ijkl},$$ where $\alpha_{ijkl}$ is the solid angle at the vertex $i$ of the tetrahedron $\{i,j,k,l\}\in T$ and the sum is taken over all tetrahedra with $i$ as one of its vertices. Given a single tetrahedron $\{ijkl\}\in T$, we usually denote the solid angle $\alpha_{ijkl}$ at the vertex $v_i$ by $\alpha_i$ for simplicity. $K_i$ locally measures the difference between the volume growth rate of a small ball centered at vertex $v_i$ in $M$ and a Euclidean ball of the same radius. Cooper and Rivin’s definition (\[CR curvature\]) of combinatorial scalar curvature is motivated by the fact that, in the smooth case, the scalar curvature at a point $P$ locally measures the difference of the volume growth rate of the geodesic ball with center $P$ to the Euclidean ball with the same radius [@Be; @LP]. In fact, for a geodesic ball $B(P, r)$ in an n-dimensional Riemannian manifold $(M^n, g)$ with center $P$ and radius $r$, we have the following asymptotical expansion for the volume of $B(P, r)$ $$\text{Vol}(B(P, r))=\omega(n)r^n\left(1-\frac{1}{6(n+2)}R(P)r^2+o(r^2)\right),$$ where $\omega(n)$ is the volume of the unit ball in $\mathbb{R}^n$ and $R(P)$ is the scalar curvature of $(M, g)$ at $P$. From this point of view, Cooper and Rivin’s definition of combinatorial scalar curvature is a good candidate for combinatorial scalar curvature with geometric meaning similar to the smooth case.
\[concave for one tetrahedron\] For a tetrahedron $\{ijkl\}\in T$, set $$\begin{aligned}
\mathcal{S}_{ijkl}=\left\{
\begin{array}{ll}
\sum_{\mu\in \{i,j,k,l\}}\alpha_{\mu}r_{\mu}, & \hbox{Euclidean background geometry} \\
\sum_{\mu\in \{i,j,k,l\}}\alpha_{\mu}r_{\mu}+2 \text{Vol}, & \hbox{hyperbolic background geometry}
\end{array}
\right.
\end{aligned},$$ where Vol denotes the volume of the tetrahedron for the hyperbolic background geometry. Then $$\label{schlafli formula}
d\mathcal{S}_{ijkl}=\sum_{\mu\in \{i,j,k,l\}}\alpha_{\mu}dr_{\mu}=\alpha_idr_i+\alpha_jdr_j+\alpha_kdr_k+\alpha_ldr_l.$$ Furthermore, the Hessian of $\mathcal{S}_{ijkl}$ is negative semi-definite with kernel $\{t(r_i, r_j, r_k, r_l)|t\in \mathbb{R}\}$ for the Euclidean background geometry and negative definite for the hyperbolic background geometry.
It is observed [@CR; @G1] that (\[schlafli formula\]) is essentially the Schläfli formula.
Combining Lemma \[simply connectness of ijkl\] with Lemma \[concave for one tetrahedron\], we have
\[F\_ijkl\] Given a Euclidean or hyperbolic tetrahedron $\{ijkl\}\in T$ and $r_0\in \Omega_{ijkl}$, $$\label{definition of F_ijkl}
F_{ijkl}(r)=\int_{r_0}^r\alpha_idr_i+\alpha_jdr_j+\alpha_kdr_k+\alpha_ldr_l$$ is a well-defined locally concave function on $\Omega_{ijkl}$. Furthermore, $F_{ijkl}(r)$ is strictly concave on $\Omega_{ijkl}\cap\{r_i^2+r_j^2+r_k^2+r_l^2=c\}$ for any $c>0$ in the Euclidean background geometry and strictly concave on $\Omega_{ijkl}$ in the hyperbolic background geometry.
Using Lemma \[concave for one tetrahedron\], we have the following property for the combinatorial scalar curvature.
\[property of Lambda\] ([@CR; @G1; @G2; @R]) Suppose $(M, \mathcal{T})$ is a triangulated 3-manifold with sphere packing metric $r$, $\mathcal{S}$ is the total combinatorial scalar curvature defined as $$\begin{aligned}
\mathcal{S}(r)=\left\{
\begin{array}{ll}
\sum K_ir_i, & \hbox{Euclidean background geometry} \\
\sum K_ir_i-2\text{Vol}(M), & \hbox{hyperbolic background geometry}
\end{array}.
\right.
\end{aligned}$$ Then we have $$d\mathcal{S}=\sum_{i=1}^NK_idr_i.$$ Set $$\label{Matrix Lambda}
\begin{aligned}
\Lambda=\operatorname{Hess}_r\mathcal{S}=
\frac{\partial(K_{1},\cdots,K_{N})}{\partial(r_{1},\cdots,r_{N})}=
\left(
\begin{array}{ccccc}
{\frac{\partial K_1}{\partial r_1}}& \cdot & \cdot & \cdot & {\frac{\partial K_1}{\partial r_N}} \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
{\frac{\partial K_N}{\partial r_1}}& \cdot & \cdot & \cdot & {\frac{\partial K_N}{\partial r_N}}
\end{array}
\right).
\end{aligned}$$ In the case of Euclidean background geometry, $\Lambda$ is symmetric and positive semi-definite with rank $N-1$ and kernel $\{tr|t\in\mathbb{R}\}$. In the case of hyperbolic background geometry, $\Lambda$ is symmetric and positive definite.
We refer the readers to [@G1; @G4] for a nice geometrical explanation of $\frac{\partial K_i}{\partial r_j}$. It should be emphasized that, as pointed out by Glickenstein [@G2], the elements $\frac{\partial K_i}{\partial r_j}$ for $i\sim j$ may be negative, which is different from two-dimensional case.
Extension of Cooper-Rivin’s action functional
---------------------------------------------
For a single Euclidean or hyperbolic tetrahedron $\{ijkl\}\in T$, the solid angle function $\alpha(r)=(\alpha_i(r), \alpha_j(r), \alpha_k(r), \alpha_l(r))$ is defined on the admissible space $\Omega_{ijkl}$. We will extend the solid angle function to $\mathbb{R}^4_{>0}$ continuously by making it to be a constant function on each connected components of $\mathbb{R}^4_{>0}\setminus\Omega_{ijkl}$, which is called a continuous extension by constants in [@L3]. We have the following result.
\[extension of solid angle\] The solid angle function $\alpha=(\alpha_i, \alpha_j, \alpha_k, \alpha_l)$ defined on $\Omega_{ijkl}$ can be extended continuously by constants to a function $\widetilde{\alpha}=(\widetilde{\alpha}_i, \widetilde{\alpha}_j, \widetilde{\alpha}_k, \widetilde{\alpha}_l)$ defined on $\mathbb{R}^4_{>0}$.
[**Proof.** ]{}The extension $\widetilde{\alpha}_i$ of $\alpha_i$ is defined to be $\widetilde{\alpha}_i(r)=2\pi$ for $r=(r_i, r_j, r_k, r_l)\in V_i$ and $\widetilde{\alpha}_i(r)=0$ for $r=(r_i, r_j, r_k, r_l)\in V_\alpha$ with $\alpha\in \{j, k, l\}$. The extensions $\widetilde{\alpha}_j$, $\widetilde{\alpha}_k$, $\widetilde{\alpha}_l$ of $\alpha_j$, $\alpha_k$, $\alpha_l$ are defined similarly.
If $r=(r_i, r_j, r_k, r_l)\in \Omega_{ijkl}$ and $r \rightarrow P$ for some point $P\in \partial_i\Omega_{ijkl}$, then geometrically the tetrahedron $\{ijkl\}$ tends to degenerate with the center $v_i$ of the sphere $S_i$ tends to lie in the geodesic plane defined by $v_j, v_k, v_l$ and the corresponding circle $C_i$ is externally tangent to $C_j, C_k, C_l$ with $i$ in the interior of the triangle $\triangle jkl$. Then by Proposition \[degenerate lemma\], we have $\alpha_i\rightarrow 2\pi, \alpha_j\rightarrow 0, \alpha_k\rightarrow 0$ and $\alpha_l\rightarrow 0$, as $r\rightarrow P\in \partial_i\Omega_{ijkl}$. This implies that the extension $\widetilde{\alpha}$ of $\alpha$ is continuous on $\mathbb{R}^4_{>0}$.
Before going on, we recall the following definition and theorem of Luo in [@L3].
A differential 1-form $w=\sum_{i=1}^n a_i(x)dx^i$ in an open set $U\subset \mathbb{R}^n$ is said to be continuous if each $a_i(x)$ is continuous on $U$. A continuous differential 1-form $w$ is called closed if $\int_{\partial \tau}w=0$ for each triangle $\tau\subset U$.
\[Luo’s convex extention\] Suppose $X\subset \mathbb{R}^n$ is an open convex set and $A\subset X$ is an open subset of $X$ bounded by a $C^1$ smooth codimension-1 submanifold in $X$. If $w=\sum_{i=1}^na_i(x)dx_i$ is a continuous closed 1-form on $A$ so that $F(x)=\int_a^x w$ is locally convex on $A$ and each $a_i$ can be extended continuous to $X$ by constant functions to a function $\widetilde{a}_i$ on $X$, then $\widetilde{F}(x)=\int_a^x\sum_{i=1}^n\widetilde{a}_i(x)dx_i$ is a $C^1$-smooth convex function on $X$ extending $F$.
Combining Theorem \[structure of admissible r\], Lemma \[F\_ijkl\], Lemma \[extension of solid angle\] and Theorem \[Luo’s convex extention\], we have
\[extension of F\_ijkl\] For a Euclidean or hyperbolic tetrahedron $\{i,j,k,l\}\in T$, the function $F_{ijkl}(r)$ defined on $\Omega_{ijkl}$ in (\[definition of F\_ijkl\]) can be extended to $$\label{extended Ricci potential for a tetrahedron}
\widetilde{F}_{ijkl}(r)=\int_{r_0}^r\widetilde{\alpha}_idr_i+\widetilde{\alpha}_jdr_j+\widetilde{\alpha}_kdr_k+\widetilde{\alpha}_ldr_l,$$ which is a $C^1$-smooth concave function defined on $\mathbb{R}^4_{>0}$ with $$\nabla_r\widetilde{F}_{ijkl}=\widetilde{\alpha}^T=(\widetilde{\alpha}_i, \widetilde{\alpha}_j, \widetilde{\alpha}_k, \widetilde{\alpha}_l)^T.$$
Proof of the global rigidity {#Section 4}
============================
Proof of Theorem \[main theorem rigidity\]
------------------------------------------
We first introduce the following definition.
Given a function $f\in \mathbb{R}^V$, if there exists a sphere packing metric $r\in \Omega$ such that $K(r)=f$, then $f$ is called an admissible curvature function.
Theorem \[main theorem rigidity\] is equivalent to the following form.
Suppose $\overline{K}$ is an admissible curvature function on a connected closed triangulated 3-manifold $(M, \mathcal{T})$, then there exists only one admissible sphere packing metric in $\Omega$ with combinatorial scalar curvature $\overline{K}$ (up to scaling in the case of Euclidean background geometry).
[**Proof.** ]{}We only prove the Euclidean sphere packing case and the hyperbolic sphere packing case is proved similarly. Suppose $r_0\in \Omega$ is a sphere packing metric. Define a Ricci potential function $$\label{Ricci potential}
\begin{aligned}
\widetilde{F}(r)=-\sum_{\{ijkl\}\in T}\widetilde{F}_{ijkl}(r_i, r_j, r_k, r_l)+\sum_{i=1}^N(4\pi-\overline{K}_i)r_i,
\end{aligned}$$ where the function $\widetilde{F}_{ijkl}$ is defined by (\[extended Ricci potential for a tetrahedron\]). Note that the second term in the right-hand-side of (\[Ricci potential\]) is linear in $r$ and well-defined on $\mathbb{R}^N_{>0}$. Combining with Lemma \[extension of F\_ijkl\], we have $\widetilde{F}(r)$ is a well-defined $C^1$-smooth convex function on $\mathbb{R}^N_{>0}$. Furthermore, $$\label{gradient of Ricci potential}
\nabla_{r_i}\widetilde{F}=-\sum_{\{ijkl\}\in T}\widetilde{\alpha}_{ijkl}+(4\pi-\overline{K}_i)=\widetilde{K}_i-\overline{K}_i,$$ where $\widetilde{K}_i=4\pi-\sum_{\{ijkl\}\in T}\widetilde{\alpha}_{ijkl}$ is an extension of $K_i=4\pi-\sum_{\{ijkl\}\in T}\alpha_{ijkl}$.
If there are two different sphere packing metrics $\overline{r}_A$ and $\overline{r}_B$ in $\Omega$ with the same combinatorial scalar curvature $\overline{K}$, then $$\nabla \widetilde{F}(\overline{r}_A)=\nabla \widetilde{F}(\overline{r}_B)=0$$ by (\[gradient of Ricci potential\]). Set $$\begin{aligned}
f(t)=&\widetilde{F}((1-t)\overline{r}_A+t\overline{r}_B)\\
=&\sum_{\{ijkl\}\in T}f_{ijkl}(t)+\sum_{i=1}^N(4\pi-\overline{K}_i)[(1-t)\overline{r}_{A, i}+t\overline{r}_{B, i}],
\end{aligned}$$ where $$f_{ijkl}(t)=-\widetilde{F}_{ijkl}((1-t)\overline{r}_{A, i}+t\overline{r}_{B, i}, (1-t)\overline{r}_{A, j}+t\overline{r}_{B, j}, (1-t)\overline{r}_{A, k}+t\overline{r}_{B, k}, (1-t)\overline{r}_{A, l}+t\overline{r}_{B, l}).$$ Then $f(t)$ is a $C^1$-smooth convex function on $[0, 1]$ and $f'(0)=f'(1)=0$. Therefore $f'(t)\equiv 0$ on $[0, 1]$.
Note that $\overline{r}_A\in \Omega$ and $\Omega$ is an open subset of $\mathbb{R}^N_{>0}$, there exists $\epsilon>0$ such that $(1-t)\overline{r}_{A}+t\overline{r}_{B}\in \Omega$ for $t\in [0, \epsilon]$. Hence $f(t)$ is $C^2$ (in fact $C^{\infty}$) on $[0, \epsilon]$. $f'(t)\equiv 0$ on $[0, 1]$ implies that $f''(t)\equiv 0$ on $[0, \epsilon]$. But for $t\in [0, \epsilon]$, we have $$f''(t)
=(\overline{r}_A-\overline{r}_B)\Lambda(\overline{r}_A-\overline{r}_B)^T,$$ where $\Lambda$ is the matrix defined in (\[Matrix Lambda\]). By Lemma \[property of Lambda\], we have $\overline{r}_A=c\overline{r}_B$ for some positive constant $c\in \mathbb{R}$. So there exists only one Euclidean sphere packing metric up to scaling with combinatorial scalar curvature equal to $\overline{K}$.
The proof of Theorem \[main theorem rigidity\] is based on a variational principle introduce by Colin de Verdiere [@DV]. Bobenko, Pinkall and Springborn [@BPS] introduced a method to extend a locally convex function on a nonconvex domain to a convex function and solved affirmably a conjecture of Luo [@L1] on the global rigidity of piecewise linear metrics on surfaces. Using the method of extension, Luo [@L3] proved the global rigidity of inversive distance circle packing metrics for nonnegative inversive distance and the author [@X] proved the global rigidity of inversive distance circle packing metrics when the inversive distance is in $(-1, +\infty)$. The method of extension was further used to study the deformation of combinatorial curvatures on surfaces in [@GJ0; @GJ1; @GJ2; @GJ3; @GX6].
Proof of Theorem \[main theorem global rigidity for alpha curvature\]
---------------------------------------------------------------------
The proof is the same as that of Theorem \[main theorem rigidity\] using a similar defined Ricci potential function. To be more precisely, for the Euclidean sphere packing metrics, define $$\begin{aligned}
F(r)=-\sum_{\{ ijkl\}\in T}F_{ijkl}(r_i, r_j, r_k, r_l)+\int_{r_0}^r\sum_{i=1}^N(4\pi-\overline{R}_ir_i^{\alpha})dr_i
\end{aligned}$$ on $\Omega$, where $r_0\in \Omega$ and $F_{ijkl}$ is defined by (\[definition of F\_ijkl\]). $F(r)$ can be extended to a $C^1$-smooth function $$\widetilde{F}(r)=-\sum_{\{ ijkl\}\in T}\widetilde{F}_{ijkl}(r_i, r_j, r_k, r_l)+\int_{r_0}^r\sum_{i=1}^N(4\pi-\overline{R}_ir_i^\alpha)dr_i$$ defined on $\mathbb{R}^N_{>0}$ by Lemma \[extension of F\_ijkl\], where $\widetilde{F}_{ijkl}$ is defined by (\[extended Ricci potential for a tetrahedron\]).
Following the same arguments as that in the proof of Theorem \[main theorem rigidity\], there exists $\epsilon\in (0,1)$ such that $$\label{second derivative of f_alpha}
\begin{aligned}
f''(t)=(\overline{r}_A-\overline{r}_B)\cdot \operatorname{Hess}_r F \cdot (\overline{r}_A-\overline{r}_B)^T\equiv 0
\end{aligned}$$ for $t\in[0, \epsilon]$, where $$\begin{aligned}
\operatorname{Hess}_r F=\Lambda-\alpha\left(
\begin{array}{ccc}
\overline{R}_1r_1^{\alpha-1} & & \\
& \ddots & \\
& & \overline{R}_Nr_N^{\alpha-1}\\
\end{array}
\right).
\end{aligned}$$
In the case that $\alpha\overline{R}\equiv0$, $f''(t)=(\overline{r}_A-\overline{r}_B) \Lambda (\overline{r}_A-\overline{r}_B)^T=0$ for $t\in [0, \epsilon]$. By Lemma \[property of Lambda\], we have $\overline{r}_A=c\overline{r}_B$ for some positive constant $c\in \mathbb{R}$. So there exists at most one Euclidean sphere packing metric with combinatorial $\alpha$-curvature equal to $\overline{R}$ up to scaling.
In the case that $\alpha\overline{R}\leq 0$ and $\alpha\overline{R}\not\equiv0$, $\operatorname{Hess}_r F$ is positive definite on $\Omega$ by Lemma \[property of Lambda\]. Then (\[second derivative of f\_alpha\]) implies $\overline{r}_A=\overline{r}_B$. Therefore there exists at most one Euclidean sphere packing metric with combinatorial $\alpha$-curvature equal to $\overline{R}$.
For the hyperbolic case, $$\begin{aligned}
F(r)=-\sum_{\{ ijkl\}\in T}F_{ijkl}(r_i, r_j, r_k, r_l)+\int_{r_0}^r\sum_{i=1}^N(4\pi-\overline{R}_i\tanh^\alpha\frac{r_i}{2})dr_i.
\end{aligned}$$ The proof is similar to the Euclidean case, so we omit the details here. \
**Acknowledgements**\
The research of the author is supported by National Natural Science Foundation of China under grant no. 11301402 and Hubei Provincial Natural Science Foundation of China under grant no. 2017CFB681. The author would like to thank Professor Feng Luo for his interest in this work and careful reading of the paper and Professor Kehua Su for the help of Figure \[circle\]. The author also would like to thank the referees for their valuable suggestions that greatly improve the paper.
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(Xu Xu)
$^1$ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China
$^2$ Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, P.R.China
E-mail: xuxu2@whu.edu.cn\
|
---
abstract: 'Molecular dynamics computer simulations of a binary Lennard-Jones glass under shear are presented. The mechanical response of glassy states having different thermal histories is investigated by imposing a wide range of external shear rates, at different temperatures. The stress-strain relations exhibit an overshoot at a strain of around 0.1, marking the yielding of the glass sample and the onset of plastic flow. The amplitude of the overshoot shows a logarithmic behavior with respect to a dimensionless variable, given by the age of the sample times the shear rate. Dynamical heterogeneities having finite lifetimes, in the form of shear bands, are observed as the glass deforms under shear. By quantifying the spatial fluctuations of particle mobility, we demonstrate that such shearbanding occurs only under specific combinations of imposed shear-rate, age of glass and ambient temperature.'
author:
- Gaurav Prakash Shrivastav
- Pinaki Chaudhuri
- Jürgen Horbach
title: 'Heterogeneous dynamics during yielding of glasses: effect of aging'
---
Introduction
============
The mechanical properties of amorphous solids have been harnessed extensively in designing materials which are ubiquitous in our everyday life. However, a complete microscopic understanding of the mechanisms leading to the macroscopic response of these materials is still missing. In order to develop materials with specific functions, it is necessary to have an improved knowledge of these underlying processes. This remains a challenging task.
It is known that the material properties of amorphous solids, such as colloidal or metallic glasses, depend on their history of production, e.g. the cooling rate by which they were quenched from a fluid phase [@glassbook]. This dependence on the history, i.e. the age of the amorphous solid, is an important issue in computer simulations of glasses, especially because the accessible cooling rates in simulations are many orders of magnitudes larger than those accessible in experiments of real systems. With respect to the comparison between simulation and experiment, it is therefore crucial to systematically understand the dependence of structural and dynamic properties on the age of the glassy solid.
It is thus expected that the response of a glass to an external mechanical loading is affected by the age of the glass. If one shears an amorphous solid under a constant strain rate, it is in general transformed into a flowing fluid [@rodneyrev2011; @barratlemaitrerev]. While at sufficiently high strains, the flowing fluid reaches a steady state without any memory of the initial unsheared state, the transient response to the shear is affected by the history of the initial glass state. The characteristic stress-strain relation of a glassy system, in response to an externally applied shear rate, exhibits typically a maximum at a strain of the order of 0.1 [@zausch08]. In numerical simulations of sheared thermal glasses, the amplitude of this maximum is observed to depend on the age of the material, typically growing logarithmically with increasing age [@varnik04; @robbinsprl05].
Moreover, the transient response of glasses to an external shear field is often associated with the occurrence of shear bands, i.e. band-like structures with strain or mobility higher than other regions, observed both in experiments [@schuhrev; @mb08; @divouxrev15; @fs14; @vp11; @bp10; @wilde11; @divouxprl10] and numerical simulations [@vb03; @ch13; @ir14; @gps15; @sf06; @bailey06; @chboc12; @ratul-procaccia-12]. Such spatially localised structures are seen to emerge after the occurrence of the stress overshoot, as the stress relaxes to the steady state value. Also, the formation of these transient shearbands have been observed to be influenced by the thermal history of the glassy state, with states which are obtained by faster cooling being less susceptible to shearband formation. Further, it has been noted that such a spatially heterogeneous response is more likely to occur in the transient regime beyond the stress overshoot, at any given temperature.
The focus of our study are thermal glasses which are often characterized as simple yield stress fluids, e.g. colloids, emulsions. For such materials, the steady state flow curve (i.e. stress vs. imposed shear rate) is a monotonic function [@bp10; @nordstrom2010; @BBDM15].. Thus there are no persistent shear-bands, which would be the case for non-monotonic flow curves [@fs14; @coussot2010; @mb12; @ir14]. In the case of simple yield stress fluids, the transient shearband that emerges are seen to broaden with time and eventually the entire material is fluidized, with the timescale of fluidization depending on the imposed shear-rate or stress [@divouxprl10; @divoux12; @ch13]. Such spatio-temporal fluctuations are also visible during steady flow, both in experiments and simulations [@vb03; @tsamados10; @bp10]. However, in this work, our objective is to characterize the transient spatial heterogeneities, prior to onset of steady flow.
The formation of transient shearbands has been addressed within the scope of various theoretical models. Within the framework of spatially-resolved fluidity and soft-glass-rheology (SGR) models [@fs14; @moorcroft-cates-fielding-11; @moorcroft-fielding-13], the age-dependent spatially heterogeneous response has been obtained, with the occurrence of the stress-overshoot under an applied shear-rate being associated with an instability leading to the formation of these transient heterogeneities. The model recovers the observation that more pronounced and long-lived shearbanding occurs for more aged glassy samples. Similarly, Manning et al. [@manningpre2007; @manningpre2009] also observe various transient heterogeneous states, by analysing a shear-transformation-zone (STZ) model of glassy materials, which depend upon the initial state of the system (characterised by an initial effective temperature) and the imposed shear-rate. Further, they were able to map their results to those obtained from numerical simulations [@shiprl2007]. The same phenemenology has also been reproduced by other mesoscopic models [@jaglajstat; @damienroux2011].
In this work, we address the question how the combination of ambient temperature, applied shear-rate and age of the glass affects the transient response, specifically the observation of spatio-temporal heterogeneities. This has not been systematically studied in earlier numerical simulations. Consequently, we also compare our observations with those from the theoretical models.
To this end, we perform molecular dynamics computer simulations of the Kob-Andersen binary Lennard-Jones (KABLJ) model [@ka94], a well-studied glass former. Amorphous states are prepared by quenching a supercooled liquid to different temperatures below the mode coupling temperature, followed by a relaxation of the sample over a waiting time $t_{\rm w}$. Then, the resulting glass samples are sheared with different shear rates $\dot{\gamma}$. The onset of plastic flow occurs around the location of $\sigma^{\rm max}$, i.e. at a strain $\gamma^{\rm max} =
\dot{\gamma} t^\star \approx 0.1$ (with $t^\star$ the time at which the maximum is obtained), with the appearance of a peak in the stress-strain response. We demonstrate that at the different temperatures $T$, the peak height, $\sigma_{\rm max}$, for all ages and shear-rates, obeys the functional behavior $C(\dot{\gamma}, T) + A(T) {\rm ln}(\dot{\gamma}t_{\rm
w})$ (with $C$ a function depending on $\dot{\gamma}$ and $T$ and $A$ a temperature-dependent amplitude). Note that this finding is consistent with earlier studies [@varnik04; @robbinsprl05]. Further, as we have shown recently [@gps15], transient (but long-lived) shear bands are formed for $\gamma > \gamma^{\rm max}$, provided that shear rate is sufficiently low. We quantify the contrast in spatial mobilities and demonstrate that the extent of spatial heterogeneities is not only dependent on the age of the glass, but also on the ambient temperature.
The rest of the paper is organized as follows. In Sec. \[sec2\] we describe the KABLJ model and the details of the simulation. Then, we present the results for the stress-strain relations and the analysis in terms of mobility maps in Sec. \[sec3\]. Finally, in Sec. \[sec4\], we summarize the results and draw some conclusions.
Model and Methods {#sec2}
=================
We consider a binary mixture of Lennard-Jones (LJ) particles (say A and B) with 80:20 ratio. This is a well-studied glass former. Particles interact via LJ potential which is defined as: $$\begin{aligned}
\label{LJ1}
\textrm{U}^{\textrm{LJ}}_{\alpha\beta}(r) &=&
\phi_{\alpha\beta}(r)-\phi_{\alpha\beta}(R_{c})-\left(r-R_{c}\right)\left.
\frac{d\phi_{\alpha\beta}}{dr}\right|_{r=R_{c}},\nonumber\\
\phi_{\alpha\beta}(r) &=&
4\epsilon_{\alpha\beta}\left[\left(\sigma_{\alpha\beta}/r\right)^{12}-
\left(\sigma_{\alpha\beta}/r\right)^{6}\right]\: r<R_{c},\end{aligned}$$ where $\alpha, \beta = \textrm{A, B}$. The interaction parameters are given by $\epsilon_{\textrm{AA}} = 1.0$, $\epsilon_{\textrm{AB}}
= 1.5\epsilon_{\textrm{AA}}$, $\epsilon_{\textrm{BB}} =
0.5\epsilon_{\textrm{AA}}$, $\sigma_{\textrm{AA}} = 1.0$, $\sigma_{\textrm{AB}} = 0.8\sigma_{\textrm{AA}}$, $\sigma_{\textrm{BB}} =
0.88\sigma_{\textrm{AA}}$, and $R_{c} = 2.5\sigma_{\textrm{AA}}$. Masses of both type of particles are equal, i.e., $m_{\textrm{A}} =
m_{\textrm{B}} = m$. All quantities are expressed in LJ units in which the unit of length is $\sigma_{\textrm{AA}}$, energy is expressed in the units of $\epsilon_{\textrm{AA}}$ and the unit of time is $\sqrt{{m\sigma_{\textrm{AA}}^{2}}/\epsilon_{\textrm{AA}}}$. More details about the model and parameters can be found in Ref. [@ka94].
We perform molecular dynamics (MD) simulation in the $NVT$ ensemble using the package LAMMPS (“Large-scale Atomic/Molecular Massively Parallel Simulator”) [@plimpton95]. The simulations are done for the box geometry having the dimension $20\times20\times80$. Temperature is kept constant via a dissipative particle dynamics (DPD) thermostat [@sk03].
Our method for the preparation of the glass samples is as follows: At a density $\rho=1.2$, we first equilibrate the system at the temperature $T=0.45$, which is in the super-cooled regime. Then, we quench it to the target temperatures $T = 0.2, 0.3, 0.4$, below the mode coupling transition temperature [@ka94]. For exploring the effect of aging on the mechanical response, we sample glassy states having different ages, $t_{\rm w} = 10^{2}, 10^{3}, 3\times10^{3}, 10^{4}, 3\times10^{4}$ and $10^{5}$, as the system evolves after the quench to each target temperature. Using these initial states sampled at different $t_{\rm w}$, we apply shear on $x$-$z$ plane in the direction of $x$ with different constant strain rates $\dot{\gamma} = 10^{-2}, 10^{-3}, 3\times10^{-4}, 10^{-4}, 3\times10^{-5}$ and $10^{-5}$. To simulate a bulk glass under shear, we use Lees-Edwards periodic boundary conditions [@le72].
Results {#sec3}
=======
In order to characterize the mechanical response of the aged amorphous solids, we measured different structural and dynamical properties, both at the macroscopic and local scales. We now discuss these observation in detail.
Macroscopic response
--------------------
### Stress vs strain
When the externally applied shear is imposed on the aging quiescent glass, the deformation response of the material is characterized by measuring the stress, $\sigma_{\rm xz}$, generated in the system with increasing strain ($\dot{\gamma}t$). In Fig. \[fig1\](a), we show how the stress evolves when we impose $\dot{\gamma}=10^{-4}$ on initial states sampled from different ages, at $T=0.2$ . We observe the characteristic stress overshoot [@varnik04], with the peak height ($\sigma^{\rm max}$) increasing for larger $t_{\rm w}$. Across temperatures, within the glassy regime, this characteristic response does not change. However, as expected the material becomes less rigid with increasing temperature. For example, for a fixed age, with increasing temperature, the steady state value of $\sigma_{\rm xz}$ decreases, but still one observes the stress-overshoot, albeit with a lesser value of $\sigma^{\rm max}$; see Fig. \[fig1\](b) for the corresponding data at $T=0.4,0.3,0.2$ for an imposed $\dot{\gamma}=10^{-4}$.
In Fig. \[fig1\](c), we illustrate the variation of $\sigma^{\rm max}$ with age, for a wide range of $\dot{\gamma}$ at $T=0.2$. Thereafter, we demonstrate that at each temperature, the data for $\sigma^{\rm max}$ can be collapsed onto a master curve using the relation $$\sigma^{\rm max} = C(\dot{\gamma}, T)
+ A(T) {\rm ln}(\dot{\gamma}t_{\rm w}),$$ with $A(T)$ a temperature-dependent amplitude that is independent of $\dot{\gamma}$ and $t_{\rm w}$ and $C(\dot{\gamma},T)$ a function that solely depends on $\dot{\gamma}$ and $T$.
Such a logarithmic relationship is motivated by Ree-Eyring’s viscosity theory, modified appropriately to take into account the role of the sample’s age, $t_{\rm w}$, as noted by Varnik et al. [@varnik04]. A similar dependence was also proposed by Rottler et al. [@robbinsprl05] for the same binary LJ mixture subjected to uniaxial strain at a different state-point. Thus our scaling results are consistent with earlier observations. We note that such a logarithmic dependence is not observed in experiments involving other yield stress fluids like colloidal pastes [@derec2003] or carbopol [@divouxsoftmatter2011] where a power-law increase is observed, which is also captured in the numerical simulations of model gels [@parksoft13; @whittle97]. Thus, it seems that the response of dense thermal glasses are different from the low density materials with more complex structures.
Having observed the scale of the stress overshoot for various ages, imposed shear-rates and ambient temperatures, we will later explore whether such observations necessarily lead to the occurrence of transient shearbands.
### Potential Energy
To have a measure of the structural changes in the system, both during the aging process and the onset of flow, we monitor the potential energy ($E_{\rm {Pot}}$) of the system. In Fig. \[fig2\](a), we show how the potential energy decreases when the system is quenched from the supercooled state at $T=0.45$ to the lower temperatures $T=0.2, 0.3, 0.4$. After the fast decrease during the initial aging regime, the potential energy decreases slowly, logarithmically depending on the age $t_{\rm w}$ of the system, as is typically observed during aging.
When the external shear is imposed on the system, the potential energy increases, with the steady state value depending upon the magnitude of the imposed shear-rate; see Fig. \[fig2\](b),(c) for $T=0.2,
0.4$, respectively. Further, we check whether the potential energy of the system measured during the transient regime, prior to the onset of steady flow, is dependent on the initial age of the quiescent glass. As can be seen in Fig. \[fig2\](b), (c), for both the temperatures, the dependence on $t_{\rm w}$ becomes more visible with decreasing shear-rate. This thus indicates that the system evolves through different intermediate structures, dependent on the initial state having different $t_{\rm w}$, before steady states structures start getting explored. We now explore how these age-dependent transient structures are dynamically different. We also note that for the applied shear-rates, the approach to steady state seems faster for $T=0.4$ than compared to $T=0.2$. While in the former case, at long strain, the steady state potential energy, under applied shear, matches at large strains, this is not the case for $T=0.2$. This combination of temperature, shear-rate and age of the sample will be further explored in later sections.
Single particle dynamics
------------------------
### Average MSD
In order to characterise the microscopic dynamical response of the glass, when yielding under shear, we monitor the single-particle dynamics. This is quantified by measuring the non-affine mean-squared displacement (MSD), $\Delta r_{z}^{2}$, in the direction transverse to the applied shear; the data is shown for $\dot{\gamma}=10^{-4}$, at $T=0.2$ (Fig. \[fig3\](a)) and $T=0.4$ (in Fig. \[fig3\](c)). At both temperatures, during early times, particles exhibit ballistic motion, which is then followed by a caging regime. Subsequently, the onset of flow is marked by a super-linear regime, and eventually diffusive dynamics is observed. The onset of the super-linear or “super-diffusive” regime occurs around the stress overshoot in the stress-strain curve, when the stress, building up in the system, is released via the breaking of local cages leading to the subsequent diffusive motion of the particles [@zausch08; @koumakisprl12]. We check how the initial age of the glassy state influences the measured $\Delta r_{z}^{2}$; the corresponding data is plotted in Fig. \[fig3\](a), (c) for the two temperatures. At both temperatures, both in the initial response and in the steady state, there is no visible difference in the measured MSD, for the samples having different ages. Only during the transient regime, where the onset of super-diffusive dynamics occurs, we observe variation with age of the quenched glass, with the effect more visible for $T=0.4$; see Fig. \[fig3\](c). We also note that the onset timescale depends upon age, shifting to longer timescales with increasing $t_{\rm w}$, as is clearly visible.
### Spatially resolved MSD
To further probe the spatial features of the local dynamics during yielding, we divide the simulation box into eight layers of thickness $10\sigma_{\rm AA}$ along the $z$-direction. For each of these layers, we compute the averaged MSD for the particles populating it at $t=0$, i.e. before shear is imposed. In Fig. \[fig3\](b), (d), we show the corresponding data, at $T=0.2,
0.4$, respectively, for an imposed $\dot{\gamma}=10^{-4}$ and age of $t_{\rm w}=10^5$. We see that there is variation in the dynamics across the different layers, the variation being greater at $T=0.2$ than at $T=0.4$. While for $T=0.4$, eventually, the long time dynamics seems to converge to the diffusive limit, within the timescale of observation, that is not the case for $T=0.2$. In the latter case, while in one case, we see the onset of diffusive dynamics at long times, in some of the other slices, the dynamics continues to be sub-diffusive. To summarise, this demonstrates that, during yielding of the glass, at any temperature, dynamics is heterogeneous. However, the extent of heterogeneity and persistence increases with decreasing temperature. We will quantify that in more detail in the subsequent sections.
### MSD maps
To visualise the dynamical heterogeneities, we construct three-dimensional maps of local MSD [@ch13]. These maps are constructed in the following manner. We divide the simulation box into small cubic sub-boxes having linear size of $\sigma_{\textrm{AA}}$. At any time $t$, we calculate the average MSD of the particles located in each sub-box at $t=0$ (unsheared glassy state), which provides a cumulative picture of how yielding proceeds locally starting from the quiescent glass. As discussed earlier, we measure the $z$-component of MSD of each particle. The evolution of the three-dimensional maps with increasing strain, is shown in Fig. \[fig4\] and Fig. \[fig4b\], for different ages of the glassy state at $T=0.2$ and $0.4$, respectively, corresponding to an imposed shear rate of $\dot{\gamma}=10^{-4}$.
First, we focus on the situation at the lower temperature ($T=0.2$); see Fig. \[fig4\]. At early strain ($\dot{\gamma}=0.1$), we observe localised “hot spots” of large mobilities [@clement12; @sentja15]. At a strain of $0.5$, more such local regions of faster dynamics have emerged. In the case of well aged initial state (e.g. $t_{\rm w}=10^5$), we see that the hot spots are organised in the form of a shear band. On the other hand, for a younger initial state, i.e. $t_{\rm w}=10^2$, these hot spots are much more dispersed. At a later strain ($\dot{\gamma}=1$), this difference is further amplified. For the younger sample, large regions of the system have been fluidized, whereas for the older system, enhanced mobility is still localised in the form of a shear band. Thus, the age of the initial glassy state, does influence the spatio-temporal organisation of regions of increased mobility. This is consistent with earlier findings in numerical simulations using different model systems [@sf06] or predictions from theoretical models [@moorcroft-cates-fielding-11].
We now contrast this with the situation at the higher temperature ($T=0.4$); see Fig. \[fig4b\]. At early strain ($\dot{\gamma}=0.1$), we observe the hot spots, for all $t_{\rm w}$; however, there are more in number as compared to the case of $T=0.2$, for any $t_{\rm w}$. With increased strain $\dot{\gamma}=0.5$, we see that the hot spots have proliferated and cover almost the entire domain, including for the most aged sample. Thus, access to increased thermal fluctuations at higher temperatures lead to the tendency for a more homogenised and faster fluidisation of the glassy state under applied shear.
We will now carry out a more quantitative characterisation of this spatio-temporal response of the glassy states, sampled from different temperatures, and aged over different timescales.
### Spatial profile of local mobility and fluctuations
So far, by spatially resolving the single particle MSDs, we demonstrated the existence of heterogeneous dynamics during yielding of the glass and how that depends on the temperature of the glassy state.
Using single particle MSDs, we construct the instantaneous spatial profile $\Delta r^{2}_{z}\left(z\right)/\langle{\Delta r^{2}_{z}}\rangle$, which gives us a measure of fluctuations in local dynamics. The evolution of these profiles, with increasing strain, is shown in Fig. \[fig6\], for two different ages of the glass, viz. $t_{\rm w}=10^2, 10^5$, under an imposed shear-rate of $\dot{\gamma}=10^{-4}$, at $T=0.2$. We observe that the spatial fluctuations are much larger and more persistent in the older sample, consistent with the qualitative discussion, above, involving MSD maps.
Using these spatial profiles, we can now quantify the contrast in local mobilities in the deforming glass, under applied shear, and compare the response across temperatures. In order to do that, we compute the fluctuations in local MSD, $\chi=(\Delta r^{2}_{z}\left(z\right)-\langle{\Delta r^{2}_{z}}\rangle)^2/\langle{\Delta r^{2}_{z}}\rangle^2$ with increasing strain $\dot{\gamma}t$. Such a quantity measures the extent of fluctuations in average mobilities, as characterised by the local MSD, across the regions parallel to the direction of flow. Defined in this manner, $\chi$ would therefore easily capture the contrast in the dynamics during the formation of shear-bands. To note, such a quantity is different from the dynamical susceptibility, $\chi_4$, which is a measure of fluctuations in single particle dynamics, measured within the steady-state ensemble. Here, we are concerned with the fluctuations across regions during the onset of flow from a quiescent state.
The corresponding data, for temperatures $T=0.2, 0.3, 0.4$, across ages $t_{\rm w}=10^2, 10^3, 10^4, 10^5$, for a range of imposed shear-rates, viz. $\dot{\gamma}=10^{-2}, 10^{-3}, {10^{-4}}$, are shown in Fig. \[fig7\]. Typically, $\chi$ behaves non-monotonically with increasing strain, with the maximal spatial fluctuations ($\chi^{\rm
max}$) occuring at a strain which corresponds to the super-linear regime in the MSD (see Fig. \[fig3\]); i.e., once yielding of the glass has occurred. Further, for a fixed $\dot{\gamma}$, the magnitude of $\chi^{\rm max}$ increases with age, i.e. the contrast in mobilities across a sample becomes largest for a more aged sample. Importantly, we also note that $\chi^{\rm max}$ also starts increasing when we shear the glasses with smaller and smaller shear-rates; e.g. see Fig. \[fig7\](a)-(c), for the variation of $\chi^{\rm max}$ with age and imposed shear-rate. With decreasing temperature, this spatial fluctuation increases even further.
To get a more complete picture of the variation of these spatial fluctuations, across age and temperature, we plot the corresponding contour maps for $\chi(t_{\rm w},\dot{\gamma})$; these are shown in Fig. \[fig8\]. From the contour maps, one can infer how strong and persistent the spatial fluctuations are with varying temperature, age and imposed shear-rate. It is evident that for large shear-rates ($\dot{\gamma}=10^{-2}$), these fluctuations are negligible at any age of the glassy state sampled at any of the temperatures; see Fig. \[fig8\](a),(d),(h). On the other hand, at smaller shear-rates ($\dot{\gamma}=10^{-4}$), large fluctuations are visible at all temperatures. However, the age of the sample then becomes a factor in determining the degree of fluctuations and persistence with increasing strain. At a lower temperature, $T=0.2$, these are prominent over the range of ages we investigated and also persistent over large strains; see Fig. \[fig8\](i). On the other hand, at higher temperatures, $T=0.4$ (Fig. \[fig8\](c)), that scale of fluctuations is only slightly visible at very large ages over a small strain window. The range of ages and strain-interval over which similar fluctuations are seen broadens out at an intermediate temperature of $T=0.3$ (Fig. \[fig8\](f)) and, even more extensively, at $T=0.2$. This trend is also evident at an intermediate shear-rate of $\dot{\gamma}=10^{-3}$, with the scale of fluctuations being less but visible (Fig. \[fig8\](h)); howevever, it is important to note that the fluctuations are nearly negligible at the higher temperature $T=0.4$ at this shear-rate (Fig. \[fig8\](b)). Thus, we demonstrate, that the scale of transient dynamical heterogeneities depend on a combination of temperature of the glass, its age and the imposed shear-rate. Since the quantity we compute, $\chi(t_{\rm w},\dot{\gamma})$, measures the contrast in mobilities in layers parallel to the flow direction, the conclusions drawn from the contour maps allow us to infer that the propensity to shear-band during yielding is determined on how rigid the glass is and how small/large the applied shear-rate is.
### Time evolution of shear-bands
Finally, we study how a transient shear-band, once formed, spatially evolves with time. Such analysis can only be done for the case where a well-defined shear-band can be identified. Thus, we focus at the case of $T=0.2$ and small shear-rate ($\dot{\gamma}=10^{-4}$), where we observe transient shear-bands for a range of $t_{\rm w}$.
In order to further quantify and characterise the spatio-temporal evolution of the shear-band, we define a region to be mobile or not, by setting a threshold $\mu_{th}=0.02$ on the local $\Delta r^{2}_{z}$. As can be seen in Fig. \[fig3\](a), such a choice of $\mu_{th}$ is larger than the plateau value in the MSD and thus corresponds to motions beyond cage-breaking. We then define the local mobility $\psi$ as $$\begin{aligned}
\label{orderparam}
\psi = \begin{cases} 1 &\mbox{if } \mu \ge \mu_{th}, \\
0 & \mbox{otherwise},\end{cases}\end{aligned}$$ where $\mu$ is the average MSD of particles in a sub-box. Following this convention, we digitize the whole system into mobile and immobile regions. An example of the mobility map, thus constructed, is shown in Fig. \[fig9\](a).
We quantify the extent of fluidisation in the system by measuring the fraction of mobile cells, $p$, at any given instant. The evolution of $p$ as a function of strain, for an imposed shear of $\dot{\gamma}=10^{-4}$, is shown in Fig. \[fig9\](b). We observe that independent of age ($t_{\rm w}$), the sudden initial increase of $p$ occurs nearly at the same strain. However, the subsequent increase in $p$ does depend upon the age and thus, the fraction of system which is mobilised at $\dot{\gamma}t=0.2$ is substantially larger for $t_{\rm w}=10^2$ as compared to $t_{\rm w}=10^5$.
Further, we try to identify the spatial organisation of these mobile cells. By locating contiguous layers of mobile cells, we identify the formation of the shear band and, thereafter, by marking the interfaces of this band, we measure how the band-width, $\xi_{\textrm{b}}$, evolves with time. For different ages of the initial glassy state, this time evolution is shown in Fig. \[fig9\](c), for a fixed $\dot{\gamma}=10^{-4}$. We see that $\xi_{\textrm{b}}$ initially grows quickly and then eventually it reaches a regime where the data can be fitted with $\xi_{\textrm{b}} \sim t^{1/2}$, implying that the propagating interface of the shear band has a diffusive motion. This diffusive regime is not extensively dependent on $t_{\rm w}$, although samples having largest $t_{\rm w}$ do seem to display a slower motion of the spreading interface.
Summary and conclusions {#sec4}
=======================
To summarize, we have used numerical simulations of a model glass former, to probe the mechanical response of amorphous solids, having different aging histories, by imposing a wide range of external shear rates. The response is studied at different temperatures below the mode coupling temperature, in order to ascertain how thermal fluctuations affect the spatio-temporal response. We underline, here, that the numerical simulations are done by integrating the Newton’s equations of motions using Lees-Edwards periodic boundary conditions and the local DPD thermostat. This, thus, allows for unhindered spatio-temporal fluctuations as the glassy state responds to the applied shear, without introducing any biases or suppressing any fluctuations, which would be the case if one were using walls [@vb03] or integrating the SLLOD equations along with some global thermostat (like Nose-Hoover) [@shiprl2007]. Therefore, qualitatively, the nature of dynamical heterogeneities observed could be different from the earlier works.
The macroscopic response of the glass is characterized by the occurrence of a overshoot in the stress-strain curves, observed for the range of shear-rates and ages that we have studied, at the different temperatures. Consistent with earlier works, we find that, at each temperature, the peak height can be scaled on to a master-curve as a function of $\dot{\gamma}t_{\rm w}$. The corresponding potential energies of the system, measured during the deformation process, show a dependence on age with decreasing shear rate. This implies that the structures visited during the transient regime, prior to the onset of steady flow, start becoming different, depending upon the initial history. We also observe that for lower temperatures, it takes longer for potential energies to reach the steady state value at smaller shear-rates, reflecting the role of thermal fluctuations in affecting the duration of the transient regime for these parameters.
This is further revealed, by measuring dynamical quantities at the microscopic scale. Our probe of choice is the time evolution of non-affine mean squared displacement (MSD) of the particles relative to the initial quiescent state. Spatial profiles of the MSD show that for a fixed imposed shear and age, the transient dynamics is more heterogeneous and more long-lived at lower temperatures. This is illustrated by constucting sptatially resolved three-dimensional maps which provide a handle to visualize the extent of dynamical heterogeneity as the glassy states respond to the imposed shear. Such maps demonstrate that the heterogeneities are spatially localised in the form of shearbands, for old enough systems at low temperatures.
We quantify the extent of dynamical heterogeneity organised in the form of shear-bands, by measuring the degree of fluctuations, $\chi$, in local mobilities across regions oriented parallel to the direction of imposed flow. Thus, large values in $\chi$ reflect more propensity of the system to exhibit shear-banding. We do such measurements by scanning across entire range of shear-rates and thermal histories explored by us. We observe that the fluctuations are maximum around the yielding strain and the peak in fluctuations increase with decreasing shear-rate as well as larger ages. Further by constructing contour maps of $\chi (t_{\rm w}, \dot{\gamma}t)$ for varying shear-rates at different temperatures, we demonstrate that the emergence of significant shear-banding depends upon the combination of temperature, age and imposed shear-rate. The contrast in local mobilites is most prominent at low shear-rates and low temperatures, where the age of the system does not seem to matter, over the range explored by us. With increasing temperature, the age of the sample starts to determine whether transient shear-banding will be visible or not. For even higher temperatures, one has to go to lower shear-rates and even older systems in order for such largely localized heteroegeneities to be visible. Also, in the cases where shear-bands can be identified, we observe that the spreading of the shear-band across the system seems to depend on age, albeit weakly.
Thus, our study shows that the transient shearbanding, with sharply contrasting mobilities across regions, only occur for specific combinations of temperature, shear-rate and age, as discussed above. Note that, for most of the glassy samples that we study, the macroscopic response under an imposed strain-rate is characterized by the stress overshoot. Yet, the transient shear-bands are only visible in only a subset of the cases we study. Thus, the occurrence of a stress overshoot does not necessarily lead to the transient shearbands, quantified via the mobilities relative to the initial quiescent state, as predicted by the spatially resolved fluidity and SGR models [@fs14; @moorcroft-cates-fielding-11; @moorcroft-fielding-13]. On the other hand, as predicted by these and other models [@manningpre2007; @manningpre2009; @jaglajstat; @damienroux2011], we do observe that the propensity to shearband is more in the case of increasingly aged samples. However, the choice of disorder distributions in these models (e.g. distribution of yielding thresholds), with changing age, remains arbitrary, and it would be useful to obtain these as inputs from atomistic simulations.
The transient shear-banding observed in our numerical simulations as well as other, bears resemblance to dynamical heterogeneities observed in thermal glasses [@ludormp11], except that in the case of a sheared glass, strong anisotropies come into play leading to the spatial localisation. In the case of supercooled liquids, the scale of dynamical heterogeneities increases as the temperature is decreased towards the putative glass transition temperature. Similarly, in the case of applied shear, the heterogeneous dynamics is enhanced as one approaches vanishingly small shear-rates, with the yield stress being identified as a critical threshold [@bocquet-prl-2009; @divoux12; @chenprl2016]. Also, it is possible that the regime of imposed shear-rates over which the critical threshold influences flow [@ludo-jpcm-03; @chboc12] could change with temperature, with decreasing thermal fluctuations extending this regime over longer scales. This would rationalise why with increasing temperatures, the contrast in heterogeneities progressively decreases, for a fixed shear-rate. Over and above, the aging effects are more prominent at higher temperatures, with the glass remembering the history from where it was quenched over longer aging timescales [@ludo-jpcm-03; @ludormp11]. At low temperature, thus, structural arrest and therefore rigidity emerges at shorter timescale. Hence, the mechanical response of the glass, having the same age and same imposed shear-rate, would be different across temperatures. This interplay of different timescales therefore brings about the complex response as a function of all these control parameters. Eventually, it would be useful to quantify the exact ranges in these control parameters, where we expect transient but prominent shearbanding to be visible.
The transient dynamical heterogeneities that we track and analyse are similar to measurements done via confocal microscopy in colloidal glasses [@vp11] or scattering measurements in granular systems [@clement12]. This is different from using velocity profiles as a diagnostic tool, which capture plasticity over short timescales. Further, in our case, because of periodic boundary conditions, the shear bands can emerge anywhere in the system, unlike many cases where initial mobile regions happen to occur near confining surfaces [@divouxsoftmatter2011; @vp11]. However, similar to experiments, we do observe the propagation of the mobile front to eventually fluidize the system. [**Acknowledgement:**]{}\
We achnowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the priority programme SPP 1594 (Grant No. HO 2231/8-2). PC thanks FOR 1394 for financial support during visit to University of Düsseldorf.
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abstract: |
Let $\Pc \subset \RR^N$ be an integral convex polytope of dimension $d$ and $\delta(\Pc)
= (\delta_0, \delta_1, \ldots, \delta_d)$ its Ehrhart $\delta$-vector. It is known that $
\sum_{j=0}^{i} \delta_{d-j}
\leq
\sum_{j=0}^{i} \delta_{j+1}
$ for each $0 \leq i \leq [(d-1)/2]$. A $\delta$-vector $\delta(\Pc)
= (\delta_0, \delta_1, \ldots, \delta_d)$ is called shifted symmetric if $
\sum_{j=0}^{i} \delta_{d-j}
=
\sum_{j=0}^{i} \delta_{j+1}
$ for each $0 \leq i \leq [(d-1)/2]$, i.e., $\delta_{d-i} = \delta_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$. In this paper, some properties of integral convex polytopes with shifted symmetric $\delta$-vectors will be studied. Moreover, as a natural family of those, $(0,1)$-polytopes will be introduced. In addition, shifted symmetric $\delta$-vectors with (0,1)-vectors are classified when $\sum_{i=0}^d\delta_i \leq 5$.
address: 'Akihiro Higashitani, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Akihiro Higashitani
title: ' Shifted symmetric $\delta$-vectors of convex polytopes '
---
Introduction {#introduction .unnumbered}
============
An *integral* convex polytope is a convex polytope any of whose vertices has integer coordinates. Let $\Pc \subset \RR^N$ be an integral convex polytope of dimension $d$ and $$i(\Pc,n) = |n\Pc \cap \ZZ^N|,
\, \, \, \, \,
n = 1, 2, 3, \ldots.$$ Here $n\Pc = \{ n\alpha : \alpha \in \Pc \}$ and $|X|$ is the cardinality of a finite set $X$. The systematic study of $i(\Pc,n)$ originated in the work of Ehrhart [@Ehrhart], who established the following fundamental properties:
1. $i(\Pc,n)$ is a polynomial in $n$ of degree $d$. (Thus in particular $i(\Pc,n)$ can be defined for [*every*]{} integer $n$.)
2. $i(\Pc,0) = 1$.
3. (loi de réciprocité) $( - 1 )^d i(\Pc, - n)
= |n(\Pc - \partial \Pc) \cap \ZZ^N|$ for every integer $n > 0$.
We say that $i(\Pc,n)$ is the [*Ehrhart polynomial*]{} of $\Pc$. We refer the reader to [@StanleyEC pp. 235–241] and [@HibiRedBook Part II] for the introduction to the theory of Ehrhart polynomials.
We define the sequence $\delta_0, \delta_1, \delta_2, \ldots$ of integers by the formula $$\begin{aligned}
\label{delta}
(1 - \lambda)^{d + 1}
\left[ 1 + \sum_{n=1}^{\infty} i(\Pc,n) \lambda^n \right]
= \sum_{i=0}^{\infty} \delta_i \lambda^i.\end{aligned}$$ It follows from the basic fact (0.1) and (0.2) on $i(\Pc,n)$ together with a fundamental result on generating function ([@StanleyEC Corollary 4.3.1]) that $\delta_i = 0$ for every $i > d$. We say that the sequence $$\delta(\Pc)
= (\delta_0, \delta_1, \ldots, \delta_d)$$ which appears in Eq.(\[delta\]) is the [*$\delta$-vector*]{} of $\Pc$. Alternate names of $\delta$-vector are for example [*Ehrhart $\delta$-vector*]{}, [*Ehrhart $h$-vector*]{} or [*$h^*$-vector*]{}.
Thus $\delta_0 = 1$ and $\delta_1 = |\Pc \cap \ZZ^N| - (d + 1)$. Let $\partial \Pc$ denote the boundary of $\Pc$ and $$i^*(\Pc,n) = |n(\Pc - \partial \Pc) \cap \ZZ^N|,
\, \, \, \, \,
n = 1, 2, 3, \ldots.$$ By using (0.3) one has $$\begin{aligned}
\label{deltadual}
\sum_{n=1}^{\infty} i^*(\Pc,n) \lambda^n
= \frac
{\sum_{i=0}^{d} \delta_{d-i} \lambda^{i+1}}
{(1 - \lambda)^{d + 1}}.\end{aligned}$$ In particular, $$\delta_d = |(\Pc - \partial \Pc) \cap \ZZ^N|.$$ Hence $\delta_1 \geq \delta_d$. Moreover, each $\delta_i$ is nonnegative ([@StanleyDRCP]). In addition, if $(\Pc - \partial \Pc) \cap \ZZ^N$ is nonempty, then one has $\delta_1 \leq \delta_i$ for every $1 \leq i \leq d - 1$ ([@HibiLBT]).
When $d = N$, the leading coefficient $(\sum_{i=0}^{d}\delta_i)/d!$ of $i(\Pc,n)$ is equal to the usual volume of $\Pc$ ([@StanleyEC Proposition 4.6.30]). In general, the positive integer $\vol(\Pc) = \sum_{i=0}^{d}\delta_i$ is said to be the [*normalized volume*]{} of $\Pc$.
It follows from Eq.(\[deltadual\]) that $$\max\{ i : \delta_i \neq 0 \}
= d+1- \min\{ i :
i(\Pc - \partial \Pc) \cap \ZZ^N \neq \emptyset \}.$$
Recently, $\delta$-vectors of integral convex polytopes have been studied intensively. (For example, see [@Payne],[@Staple1] and [@Staple2].)
There are two well-known inequalities of $\delta$-vectors. Let $s = \max\{ i : \delta_i \neq 0 \}$. Stanley [@StanleyJPAA] shows the inequalities $$\begin{aligned}
\label{Stanley}
\delta_0 + \delta_1 + \cdots + \delta_i
\leq \delta_s + \delta_{s-1} + \cdots + \delta_{s-i},
\, \, \, \, \,
0 \leq i \leq [s/2]\end{aligned}$$ by using the theory of Cohen–Macaulay rings. On the other hand, the inequalities $$\begin{aligned}
\label{Hibi}
\delta_{d} + \delta_{d-1} + \cdots + \delta_{d-i}
\leq \delta_1 + \delta_2 + \cdots + \delta_i
+ \delta_{i+1},
\, \, \, \, \,
0 \leq i \leq [(d-1)/2]\end{aligned}$$ appear in [@HibiLBT Remark (1.4)]. The above two inequalities are generalized in [@Staple1].
A $\delta$-vector $\delta(\Pc)
= (\delta_0, \delta_1, \ldots, \delta_d)$ is called [*symmetric*]{} if the equalities hold in Eq. (\[Stanley\]) for each $0 \leq i \leq [s/2]$, i.e., $\delta_i = \delta_{s-i}$ for each $0 \leq i \leq [s/2]$. The $\delta$-vector $\delta(\Pc)$ of $\Pc$ is symmetric if and only if the Ehrhart ring [@HibiRedBook Chapter X] of $\Pc$ is Gorenstein. A combinatorial characterization for the $\delta$-vector to be symmetric is studied in [@HibiCombinatorica] and [@DeNegriHibi].
We say that a $\delta$-vector $\delta(\Pc)
= (\delta_0, \delta_1, \ldots, \delta_d)$ is [*shifted symmetric*]{} if the equalities hold in Eq. (\[Hibi\]) for each $0 \leq i \leq [(d-1)/2]$, i.e., $\delta_{d-i} = \delta_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$. It seems likely that an integral convex polytope with a shifted symmetric $\delta$-vector is quite rare. Thus it is reasonable to sutudy a property of and to find a natural family of integral convex polytopes with shifted symmetric $\delta$-vectors. In section 2, some characterizations of an integral convex polytope with a shifted symmetric $\delta$-vector are given. Concretely, in Theorem \[faces1\], it is shown that integral convex polytopes with shifted symmetric $\delta$-vectors have a special property. Moreover, as a generalization of an integral convex polytope with a shifted symmetric $\delta$-vector, an integral simplicial polytope any of whose facet has the normalized volume 1 is considered in section 2. In section 3, a family of $(0,1)$-polytopes with shifted symmetric $\delta$-vectors is presented. These shifted symmetric $\delta$-vectors are $(0,1)$-vectors. In addition, by using those examples, we classify shifted symmetric $\delta$-vectors with (0,1)-vectors when $\sum_{i=0}^d\delta_i \leq 5$ in section 4.
Review on the computation of the $\delta$-vector of a simplex
=============================================================
We recall from [@HibiRedBook Part II] the well-known combinatorial technique how to compute the $\delta$-vector of a simplex.
- Given an integral $d$-simplex $\Fc \subset \RR ^N$ with the vertices $v_0, v_1, \ldots, v_d$, we set $\widetilde \Fc=\left\{(\a,1)\in \RR^{N+1} \, : \, \a \in \Fc \right\}$. And $
\partial \widetilde \Fc=\left\{(\a,1)\in \RR^{N+1} \, : \, \a \in \partial \Fc \right\}
$ is its boundary.
- Let $\Cc(\widetilde \Fc)=\Cc=
\{r \beta \, : \, \beta \in \widetilde \Fc,0 \leq r \in \QQ \}
\subset \RR^{N+1}.$ Its boundary is $
\partial \Cc = \left\{r\beta \, : \, \beta \in \partial \widetilde \Fc,0 \leq r \in \QQ \right\}.
$
- Let $S$ (resp. $S^*$) be the set of all points $\a \in \Cc \cap \ZZ^{N+1}$ (resp. $\a \in (\Cc- \partial \Cc) \cap \ZZ^{N+1}$) of the form $
\a= \sum_{i=0}^{d}r_i(v_i,1),
$ where each $r_i \in \QQ$ with $0 \leq r_i<1$ (resp. with $0<r_i \leq 1$).
- The degree of an integer point $(\a,n) \in \Cc$ is $
\deg(\a,n):=n.
$
\[computation\] [*(a)*]{} Let $\delta_i$ be the number of integer points $\a \in S$ with $\deg \a=i$. Then $$1+ \sum_{n=1}^{\infty}i(\Fc,n) \lambda^n=\frac{\delta_0+\cdots+\delta_d\lambda^d}{(1-\lambda)^{d+1}}.$$
[*(b)*]{} Let $\delta_i^*$ be the number of integer points $\a \in S^*$ with $\deg\a=i$. Then $$\sum_{n=1}^{\infty}i^*(\Fc,n) \lambda^n=\frac{\delta_1^*\lambda+\cdots+\delta_{d+1}^*\lambda^{d+1}}{(1-\lambda)^{d+1}}.$$
[*(c)*]{} One has $\delta_i^* = \delta_{(d+1)-i}$ for each $1 \leq i \leq d+1$.
We say that a $\delta$-vector $\delta(\Pc)
= (\delta_0, \delta_1, \ldots, \delta_d)$ is [*shifted symmetric*]{} if $\delta_{d-i} = \delta_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$. Since $\delta_d = \delta_1$, an integral convex polytope with a shifted symmetric $\delta$-vector is always a $d$-simplex.
The followings are some examples of a simplex with a shifted symmetric $\delta$-vector.
Let ${\bf e}_i$ denote the $i$th canonical unit coordinate vector of $\RR^d$.
\[example\]
*(a) We define $v_i \in \RR^d,i=0,1,\ldots,d$ by setting $v_i = {\bf e}_i$ for $i=1,\ldots,d$ and $v_0 = (-e,\ldots,-e)$, where $e$ is a nonnegative integer. Let $\Pc=\con\{v_0,v_1,\ldots,v_d\}$. Then one has $\vol(\Pc) = ed + 1$ by using an elementary linear algebra. When $e=0$, it is clear that $\delta(\Pc)=(1,0,0,\ldots,0)$. When $e$ is positive, we know that $$\frac{j}{ed+1}\sum_{i=1}^d(v_i,1)+\frac{(e-j)d+1}{ed+1}(v_0,1) = (j-e,j-e,\ldots,j-e,1)$$ and $0 < \frac{j}{ed+1}, \frac{(e-j)d+1}{ed+1} < 1$ for every $1 \leq j \leq e$. Then Lemma \[computation\] says that $\delta_1,\delta_d \geq e$. Since $\delta_i \geq \delta_1$ for $1 \leq i \leq d-1$ and $\vol(\Pc)=ed+1$, we obtain $\delta(\Pc)=(1,e,e,\ldots,e)$.*
\(b) Let $d \geq 3$. We define $v_i \in \RR^d,i=0,1,\ldots,d$ by setting $v_i = {\bf e}_i$ for $i=1,\ldots,d$ and $v_0 = (e,\ldots,e)$, where $e$ is a positive integer. Let $\Pc=\con\{v_0,v_1,\ldots,v_d\}$. Then one has $\vol(\Pc) = ed - 1$ by using an elementary linear algebra. And we know that $$\frac{j}{ed-1}\sum_{i=1}^d(v_i,1)+\frac{(e-j)d-1}{ed-1}(v_0,1) = (e-j,e-j,\ldots,e-j,1)$$ and $0 < \frac{j}{ed-1}, \frac{(e-j)d-1}{ed-1} < 1$ for every $1 \leq j \leq e-1$. Thus $\delta_1,\delta_d \geq e-1$ by Lemma \[computation\]. In addition we know that $$\frac{ke+j}{ed-1}\sum_{i=1}^d(v_i,1)+\frac{(e-j)d-1-k}{ed-1}(v_0,1)
= (e-j,e-j,\ldots,e-j,k+1)$$ and $0 <\frac{ke+j}{ed-1}, \frac{(e-j)d-1-k}{ed-1}< 1$ for every $0 \leq j \leq e-1$ and $1 \leq k \leq d-2$. Hence $\delta(\Pc)=(1,e-1,e,e,\ldots,e,e-1)$.
Some characterizations of an integral convex polytope with a shifted symmetric $\delta$-vector
==============================================================================================
In the first half of this section, two results of an integral convex polytope with a shifted symmetric $\delta$-vector are given. And in the latter half of this section, we generalize an integral convex polytope with a shifted symmetric $\delta$-vector.
\[faces1\] Let $\Pc$ be a $d$-simplex whose vertices are $v_0,v_1,\ldots,v_d \in \RR^d$ and $S \; (S^*)$ the set which appears in section $1$. Then the following conditions are equivalent:
1. $\delta(\Pc)$ is shifted symmetric;
2. the normalized volume of all facets of $\Pc$ is equal to $1$;
3. each element $(\a,n) \in S \backslash \{(0,\ldots,0,0)\}$ has a unique expression of the form: $$\label{form}
(\a,n)=\sum_{j=0}^{d}r_j(v_j,1) \;\; \text{with} \;\; 0<r_j<1 \;\; \text{for} \;\; j=0,1,\ldots,d,$$ where $\a \in \ZZ^d$ and $n \in \ZZ$.
[**((i) $\Leftrightarrow$ (iii))**]{} If each element $(\a,n) \in S \backslash \{(0,\ldots,0,0)\}$ has the form (\[form\]), each element $(\alpha',n') \in S^* \backslash \{(\sum_{j=0}^dv_j,\ldots,\sum_{j=0}^dv_j,d+1)\}$ also has the same form (\[form\]). This implies that $\d(\Pc)$ is shifted symmetric. On the other hand, suppose that $\delta(\Pc)$ is shifted symmetric. Let $\min\{i:\delta_i \not= 0, i>0 \} = s_1$ and $\delta_{s_1}=m_1(\not=0)$. Then one has $d+1-\max\{i:\delta_i \not= 0 \} = s_1$ and both $S$ and $S^*$ have the $m_1$ elements with degree $s_1$. If an element $(\alpha',s_1) \in S^*$ does not have the form (\[form\]), there is $0 \leq j \leq d$ with $r_j=1$, say, $r_0=r_1=\cdots=r_a=1$ and $0 < r_{a+1},r_{a+2},\ldots,r_d < 1$. Then $S$ has an element $(\a'-v_0-v_1-\cdots-v_a,s_1-a-1)\not=(0,\ldots,0,0)$, a contradiction. Thus each element $(\a',s_1) \in S^*$ has the form (\[form\]). If we set $\min\{i:\delta_i\not=0,i>s_1 \} = s_2$, the same discussions can be done as above. Thus each element $(\b',s_2) \in S^*$ has the form (\[form\]). Hence each element $(\alpha',n') \in S^* \backslash \{(\sum_{j=0}^dv_j,\ldots,\sum_{j=0}^dv_j,d+1)\}$ has the form (\[form\]), that is to say, each element $(\a,n) \in S \backslash \{(0,\ldots,0,0)\}$ has the form (\[form\]).
[**((ii) $\Leftrightarrow$ (iii))**]{} Let $\delta(\Pc)=(\delta_0,\delta_1,\ldots,\delta_d) \in \ZZ^{d+1}$ be the $\delta$-vector of $\Pc$ and $\delta(\Fc)=(\delta_0',\delta_1',\ldots,\delta_{d-1}') \in \ZZ^d$ the $\delta$-vector of a facet $\Fc$ of $\Pc$. Then one has $\delta_i' \leq \delta_i$ for $0 \leq i \leq d-1$. If there is a facet $\Fc$ with $\vol(\Fc)\not=1$, say, its vertices are $v_0,v_1,\ldots,v_{d-1}$, there exists an element $(\a,n) \in S$ with $\a=\sum_{j=0}^{d-1}r_jv_j+0 \cdot v_d$ and $n > 0$. This implies that there exists an element of $S \backslash \{(0,\ldots,0,0)\}$ which does not have the form (\[form\]). On the other hand, suppose that there exists an element $(\a,n) \in S \backslash \{(0,\ldots,0,0)\}$ which does not have the form (\[form\]), i.e., $(\a,n)=\sum_{j=0}^{d}r_j(v_j,1)$ and there is $0 \leq j \leq d$ with $r_j=0$, say, $r_d=0$. Then the normalized volume of the facet whose vertices are $v_0,v_1,\ldots,v_{d-1}$ is not equal to 1.
Let $\Pc$ be a $d$-simplex. If $\vol(\Pc)=p$ with a prime number $p$ and $$\min\{ i:\delta_i \not=0 ,i>0 \} = d+1- \max\{ i:\delta_i \not= 0 \},$$ then $\delta(\Pc)$ is shifted symmetric.
The elements of $S$ form a cyclic group of prime order. Then every non-zero element of $S$ is a generator. Thus it can be considered whether $S \backslash \{(0,\ldots,0,0)\}$ is disjoint from $\partial S$, which case satisfies the condition of Theorem \[faces1\] (iii), or $S$ is contained in a facet of $\partial S$, where $\partial S$ is a cyclic group generated by the vertices of a facet of $\Pc$. In the latter case, let $x \in S$ be an element of the maximal degree deg$(x)$, and let $-x$ denote its inverse. Since $S$ is contained in $\partial S$, deg$(x) + $deg$(-x) < d + 1$, which contradicts the assumption. Therefore, since Theorem \[faces1\], $\d(\Pc)$ is shifted symmetric.
Recall that an integral convex polytope with a shifted symmetric $\delta$-vector is always a simplex. Then we expand the definition of shifted symmetric to an integral simplicial polytope. We study an integral simplicial polytope $\Pc$ any of whose facet has the normalized volume 1. When $\Pc$ is a simplex, its $\delta$-vector is shifted symmetric by Theorem \[faces1\].
Let $h(\Delta(\Pc))=(h_0,h_1,\ldots,h_d)$ denote the $h$-vector of the boundary complex of $\Pc$. (See, [@HibiRedBook Part I].) Then the following is a well-known fact about a lower bound of the $h$-vector for a simplicial $(d-1)$-sphere.
\[lb\][*([@bar1],[@bar2])*]{} The $h$-vector $h(\Delta(\Pc))=(h_0,h_1,\ldots,h_d)$ of a simplicial $(d-1)$-sphere satisfies $h_1 \leq h_i$ for every $1 \leq i \leq d-1$.
Now, all of $h$-vectors of simplicial $(d-1)$-spheres satisfying the lower bound, i.e., $h_1 = h_i$ for every $1 \leq i \leq d-1$, are given by $h$-vectors of the boundary complexes of simplicial polytopes any of whose facet has the normalized volume 1. In fact,
For an arbitrary positive integer $h_1$, there exists a $d$-dimensional simplicial polytope $\Pc$ any of whose facet has the normalized volume 1 and whose $h$-vector of the boundary complex coincides with $(1,h_1,\ldots,h_1,1) \in \ZZ^{d+1}$.
Let $d=2$. A convex polygon is always simplicial. And each facet of an integral polygon has the normalized volume 1 if and only if there is no integer point in its boundary except its vertices. Hence, for an arbitrary positive integer $h_1$, we can say that there exists an integral polygon with $h_1 + 2$ vertices any of whose facet has the normalized volume 1.
We assume when $d \geq 3$. Let $\Pc$ be the $d$-dimensional integral convex polytope whose vertices $v_i \in \RR^d$, $i=0,1,\ldots,d+n$, are of the form: $$\begin{aligned}
v_i=
\begin{cases}
(0,\ldots,0) \;\;\;\; &\text{for} \;\;\;i=0, \\
{\bf e}_i &\text{for} \;\;\;i=1,2,\ldots,d, \\
(c_j,\ldots,c_j,j) &\text{for} \;\;\;i=d+1,d+2,\ldots,d+n,
\end{cases}\end{aligned}$$ where $n$ is a positive even number, $j=i-d$ and $c_j=n+\frac{(n-j)(j-1)}{2}$.
*First step.* We prove that $\Pc$ is a simplicial convex polytope. We define the $\{(n+1)(d-1)+2\}$ convex hulls by setting $$\begin{cases}
\Fc_i:=\con\{v_0,v_1,\ldots,v_{i-1},v_{i+1},\ldots,v_{d}\} \;\;\;\; \text{for} \;\;\;i=1,\ldots,d, \\
\Fc':=\con\{v_1,\ldots,v_{d-1},v_{d+1}\}, \\
\Gc_{i,j}:=\con\{v_1,\ldots,v_{i-1},v_{i+1},\ldots,v_{d-1},v_{d+j},v_{d+j-1}\} \;\;\;\;\;
\text{for} \;\;\; i=1,\ldots,d-1,j=2,\ldots,n, \\
\Gc'_i:=\con\{v_1,\ldots,v_{i-1},v_{i+1},\ldots,v_{d-1},v_d,v_{d+n}\} \;\;\;\;\; \text{for} \;\;\; i=1,\ldots,d-1,
\end{cases}$$ and the followings are the equations of the hyperplanes containing the above convex hulls: $$\begin{cases}
\Hc_i \supset \Fc_i: -x_i=0 \;\;\; \text{for} \;\;\; i=1,\ldots,d, \\
\Hc' \supset \Fc': \sum_{k=1}^{d-1}x_k-(n(d-1)-1)x_d=1, \\
\Ic_{i,j} \supset \Gc_{i,j}: c'_j\sum_{k=1}^{i-1}x_k-(1-(d-2)c'_j)x_i+
c'_j\sum_{k=i+1}^{d-1}x_k+(j-\frac{n+2}{2})x_d =c_j' \\
\text{for} \;\;\; i=1,\ldots,d-1,j=2,\ldots,n, \;\;\; \text{where} \;\;\;
c'_j=((j-1)c_j-jc_{j-1})=\frac{j^2-j+n}{2}, \\
\Ic'_i \supset \Gc'_i: n\sum_{k=1}^{i-1}x_k-(n(d-1)-1)x_i+n\sum_{k=i+1}^dx_k=n
\;\;\; \text{for} \;\;\; i=1,\ldots,d-1.
\end{cases}$$
We prove that these $\{(n+1)(d-1)+2\}$ convex hulls are all facets of $\Pc$. If we write $\Hc \subset \RR^d$ for the hyperplane defined by the equation $a_1x_1+\cdots+a_dx_d = b$, then we write $\Hc^{(+)} \subset \RR^d$ for the closed half-space defined by the inequality $a_1x_1+\cdots+a_dx_d \leq b$.
- Let $\Pc_{n+1}=\con\{ v_0,\ldots,v_d \}$. Then one has $\Pc_{n+1} = (\bigcap_{i=1}^d\Hc_i^{(+)}) \cap (x_1+\cdots+x_d \leq 1)$.
- Let $\Pc_k=\con\{ \Pc_{k+1} \union \{v_{d+k}\} \}$ for $k=n,n-1,\ldots,1$. Then it can be shown easily that $$\Pc_k = (\bigcap_{i=1}^d\Hc_i^{(+)}) \cap (\bigcap_{i=1}^{d-1} \Ic_i^{(+)'})
\cap (\bigcap_{\begin{subarray}{c}1 \leq i \leq d-1 \\ k+1 \leq j \leq n \end{subarray}}
\Ic_{i,j}^{(+)}) \cap
(kx_1+\cdots+kx_{d-1}-(c_k(d-1)-1)x_d \leq k).$$
Then one has $\Pc_1=\con\{ \Pc_2 \union \{v_{d+1}\} \}=\Pc$. Thus we obtain the following equality: $$\Pc=(\bigcap_{i=1}^d\Hc_i^{(+)}) \cap (\bigcap_{i=1}^{d-1}\Ic_i^{(+)'}) \cap
(\bigcap_{\begin{subarray}{c}1 \leq i \leq d-1 \\ 2 \leq j \leq n \end{subarray}}
\Ic_{i,j}^{(+)})\cap \Hc'^{(+)}.$$ Hence we can say that $\Fc_i,\Fc',\Gc_{i,j}$ and $\Gc_i'$ are all facets of $\Pc$ and they are $(d-1)$-simplices.
*Second step.* We prove that the normalized volume of each facet of $\Pc$ is equal to 1. One has $\vol(\Fc_i)=1$ for $i=1,\ldots,d$ since $\vol(\Pc_{n+1})=1$ and one has $\vol(\Gc_i')=1$ for $1 \leq i \leq d-1$ since $\con\{v_1,\ldots,v_d,v_{d+n}\}$ is a simplex with a shifted symmetric $\delta$-vector by Examples \[example\](b). And one has $\vol(\Fc')=1$ since $\vol(\con\{v_0,v_1,\ldots,v_{d-1},v_{d+1}\})=1$. When we consider $\vol(\Gc_{i,j})$, we are enough to prove that $\vol(\Gc_{d-1,j})=1$ by the symmetry.
For a $(d-1)$-simplex $\Gc_{d-1,j}$, we consider the elements of the set $S$ which appears in section $1$: $$(\a_1,\ldots,\a_d,r)=
r_1(v_1,1)+\cdots+r_{d-2}(v_{d-2},1)+r_{d-1}(v_{d+j},1)+r_d(v_{d+j-1},1),$$ where $(\a_1,\ldots,\a_d,r) \in \ZZ^{d+1}$ and $0 \leq r_i < 1$ for $0 \leq i \leq d$. Then one has $$(\a_1,\ldots,\a_d) =
(r_1+r_{d-1}c_j+r_dc_{j-1},\ldots,r_{d-2}+r_{d-1}c_j+r_dc_{j-1},
r_{d-1}c_j+r_dc_{j-1},r_{d-1}j+r_d(j-1)).$$ Since $r_1=\a_1-\a_{d-1} \in \ZZ$, we obtain $r_1=0$. Similary we obtain $r_2=\cdots=r_{d-2}=0$. Hence we can rewrite $(\a_1,\ldots,\a_d,r)=r_{d-1}(v_{d+j},1)+r_d(v_{d+j-1},1)$. It then follows that $$\vol(\Gc_{d-1,j})=\vol(\con\{v_{d+j},v_{d+j-1}\})=
\vol(\con\{(c_j,j),(c_{j-1},j-1)\})$$ Since $r_{d-1}j+r_d(j-1) \in \ZZ$ and $r_{d-1}+r_d \in \ZZ$, one has $r_{d-1}=r_d=0$. This implies that $\vol(\con\{(c_j,j),(c_{j-1},j-1)\})=1$. Thus $\vol(\Gc_{d-1,j})=1$.
*Third step.* By the first step and the second step, $\Pc$ is a $d$-dimensional simplicial polytope with $\{(n+1)(d-1)+2\}$ facets and $(d+n+1)$ vertices any of whose facet has the normalized volume 1. Hence, by Lemma \[lb\], one has $h(\Delta(\Pc))=(1,n+1,\ldots,n+1,1)$ for a positive even number $n$. Thus, when $h_1$ is odd and $h_1 \geq 3$, we know that there exists a simplicial polytope with $h(\Delta(\Pc))=(1,h_1,\ldots,h_1,1)$ any of whose facet has the normalized volume 1. When $h_1=1$, it is clear that $h(\Delta(\Pc_{n+1}))=(1,1,\ldots,1)$. When $h_1$ is even and $h_1 \geq 2$, let $\Pc'=\con\{v_1,\ldots,v_{d+n}\}$. Then we can verify that $\Pc'$ is a simplicial polytope with $h(\Delta(\Pc'))=(1,n,\ldots,n,1)$ any of whose facet has the normalized volume 1.
A family of $(0,1)$-polytopes with shifted symmetric $\delta$-vectors
=====================================================================
In this section, a family of $(0,1)$-polytopes with shifted symmetric $\delta$-vectors is studied. We classify completely the $\delta$-vectors of those polytopes. Moreover, we consider when those $\delta$-vectors are both symmetric and shifted symmetric.
Let $d=m+n$ with positive integers $m$ and $n$. We study the $\delta$-vector of the integral convex polytope $\Pc \subset \RR^d$ whose vertices are of the form: $$\begin{aligned}
\label{vertex}
v_i =
\begin{cases}
{\bf e}_i + {\bf e}_{i+1} + \cdots + {\bf e}_{i+m-1} \;\;\;\; &i=1,\ldots,d, \\
(0,\ldots,0) &i=0,
\end{cases}\end{aligned}$$ where ${\bf e}_{d+i} = {\bf e}_i$.
The normalized volume of $\Pc$ is equal to the absolute value of the determinant of the circulant matrix $$\begin{aligned}
\label{matrix}
\begin{vmatrix}
v_1 \\
\vdots \\
v_{d}
\end{vmatrix}. \end{aligned}$$ This determinant (\[matrix\]) can be calculated easily. In fact,
\[m\] When $(m,n)=1$, the determinant (\[matrix\]) is equal to $\pm m$. And when $(m,n)\not=1$, the determinant (\[matrix\]) is equal to $0$. Here $(m,n)$ is the greatest common divisor of $m$ and $n$.
A proof of this proposition can be given by the formula of the determinant of the circulant matrix. Thus one has $\vol(\Pc)=m$ when $(m,n)=1$.
In this section, we assume only the case of $(m,n)=1$. Hence $(m,d)=1$.
For $j = 1,2,\ldots,d-1,$ let $q_j$ be the quotient of $jm$ divided by $d$ and $r_j$ its remainder i.e., one has the equalities $$jm=q_jd + r_j \quad \text{for} \quad j=1,2,\ldots,d-1.$$ It then follows from $(m,d)=1$ that $$0 \leq q_j \leq m-1, 1 \leq r_j \leq d-1$$ and $$r_j \not= r_{j'} \text{ if } j\not=j'$$ for every $1 \leq j,j' \leq d-1$. In addition, for $k=1,2,\ldots,m-1$, let $j_k \in \{1,2,\ldots.d-1\}$ be the integer with $r_{j_k}=k$, i.e., one has the equalities $$j_km=q_{j_k}d+r_{j_k}=q_{j_k}d+k \quad \text{for} \quad k=1,2,\ldots,m-1.$$ Then $q_{j_k} > 0$. Thus one has $$1 \leq q_{j_k},r_{j_k} \leq m-1$$ for every $1 \leq k \leq m-1$.
For an integer $a$, let $\overline{a}$ denote the residue class in $\ZZ/d\ZZ$.
\[main\] Let $\Pc$ be the integral convex polytope whose vertices are of the form (\[vertex\]) and $\d(\Pc)=(\delta_0,\delta_1,\ldots,\delta_d)$ its $\delta$-vector. For each $1 \leq i \leq d$, one has $\overline{im} \in \{ \overline{1},\overline{2},\ldots,\overline{m-1} \}$ if and only if one has $\delta_i=1$. Moreover, $\d(\Pc)$ is shifted symmetric, i.e., $\delta_{i+1}=\delta_{d-i}$ for each $0 \leq i \leq [(d-1)/2]$.
By using the above notations, we obtain $$\begin{aligned}
\frac{q_{j_k}}{m}\left\{(v_1,1)+(v_2,1)+\cdots+(v_d,1)\right\}+\frac{r_{j_k}}{m}(v_0,1)
&=(q_{j_k},\ldots,q_{j_k},j_k) \in \ZZ^{d+1} \end{aligned}$$ and $0 < \frac{q_{j_k}}{m},\frac{r_{j_k}}{m} < 1$ for every $1 \leq k \leq m-1$. Then Lemma \[computation\] guarantees that one has $\delta_{j_k} \geq 1$ for $k=1,\ldots,m-1$. Considering $\sum_{i=0}^d\delta_i=m$ by Proposition \[m\], it turns out that $\delta(\Pc)$ coincides with $$\begin{aligned}
\delta_i=
\begin{cases}
1 \qquad i=0,j_1,j_2,\ldots,j_{m-1}, \\
0 \qquad otherwise.
\end{cases}\end{aligned}$$ Now $\overline{im} \in \{ \overline{1},\overline{2},\ldots,\overline{m-1} \}$ is equivalent with $i \in \{j_1,\ldots,j_{m-1}\}$. Therefore one has $\delta_i=1$ if and only if $\overline{im} \in \{ \overline{1},\overline{2},\ldots,\overline{m-1} \}$ for each $1 \leq i \leq d$.
In addition, by virtue of Theorem \[faces1\], $\d(\Pc)$ is shifted symmetric, as required.
\[corollary\] Let $\Pc$ be the integral convex polytope whose vertices are of the form (\[vertex\]) and $\d(\Pc)=(\delta_0,\delta_1,\ldots,\delta_d)$ its $\delta$-vector. Then $\d(\Pc)$ is symmetric, i.e., $\delta_i=\delta_{s-i}$ for each $0 \leq i \leq [s/2]$ if and only if one has $d \equiv m-1 \pmod{m}$.
Let $p$ be the quotient of $d$ divided by $m$ and $r$ its remainder, i.e., one has $d=mp+r$. And let $j_t=\min\{j_1,j_2,\ldots,j_{m-1}\}$. On the one hand, one has $j_tm=d+t$. On the other hand, one has $(p+1)m=d+m-r$ and $1 \leq m-r \leq m-1$. It then follows from Theorem \[main\] that $p+1=j_t=\min\{i:\delta_i\not=0,i>0\}$. Hence $d-p = s = \max\{ i : \delta_i \neq 0 \}$ since $\d(\Pc)$ is shifted symmetric.
When $d \equiv m-1 \pmod{m}$, i.e., $r=m-1$, we can obtain the equalities $$\begin{aligned}
d-p=mp+r-p&=&mp+m-1-p=(m-1)(p+1). \end{aligned}$$ In addition, for nonnegative integers $l(p+1)$, $l=1,2,\ldots,m-1$, the following equalities hold: $$\overline{l(p+1)m}=\overline{l(mp+m)}=\overline{l(mp+m-1)+l}=\overline{ld+l}=
\overline{l} \in \{ \overline{1},\overline{2},\ldots,\overline{m-1} \}.$$ Thus it turns out that $\d(\Pc)$ coincides with $$\begin{aligned}
\delta_i=
\begin{cases}
1 \quad i=0,p+1,2(p+1),\ldots,(m-1)(p+1), \\
0 \quad otherwise,
\end{cases}\end{aligned}$$ by Theorem \[main\]. It then follows that $$\delta_{k(p+1)}=\delta_{(m-1-k)(p+1)}=\delta_{s-k(p+1)}=1$$ for every $0 \leq k \leq m-1$ and $$\delta_i=\delta_{s-i}=0$$ for every $0 \leq i \leq s$ with $i \not= k(p+1)$, $k=0,1,\ldots,m-1$. These equalities imply that $\d(\Pc)$ is symmetric.
Suppose that $\d(\Pc)$ is symmetric. Our work is to show that $r=m-1$. Then one has $$\delta_0=\delta_s=\delta_{d-p}=\delta_{(m-1)(p+1)}=1.$$ Since $\d(\Pc)$ is also shifted symmetric, one has $\delta_{(m-1)(p+1)}=\delta_{p+1}$. Hence one has $\delta_{p+1}=\delta_{(m-2)(p+1)}=\delta_{2(p+1)}=\cdots=\delta_{[(m-1)/2](p+1)}$=1 since $\d(\Pc)$ is both symmetric and shifted symmetric. When $m$ is odd, one has $\frac{d-p}{2}=\frac{m-1}{2}(p+1)$ since $\d(\Pc)$ is symmetric. Thus $r=m-1$. When $m$ is even, one has $\frac{d+1}{2}=\frac{m}{2}(p+1)$ since $\d(\Pc)$ is shifted symmetric. Thus $r=m-1$.
Therefore $\d(\Pc)$ is symmetric if and only if $d \equiv m-1 \pmod{m}$.
Classifications of shifted symmetric $\d$-vectors with (0,1)-vectors
====================================================================
In this section, we will classify all the possible shifted symmetric symmetric $\d$-vectors with (0,1)-vectors when $\sum_{i=0}^d\delta_i \leq 5$ by using the examples in the previous section.
In [@smallvolume], the possible $\delta$-vectors of integral convex polytopes are classified completely when $\sum_{i=0}^d\delta_i \leq 3$.
\[small\] Let $d \geq 3$. Given a finite sequence $(\delta_0, \delta_1, \ldots, \delta_d)$ of nonnegative integers, where $\delta_0 = 1$ and $\delta_1 \geq \delta_d$, which satisfies $\sum_{i=0}^{d} \delta_i \leq 3$, there exists an integral convex polytope $\Pc \subset \RR^d$ of dimension $d$ whose $\delta$-vector coincides with $(\delta_0, \delta_1, \ldots, \delta_d)$ if and only if $(\delta_0, \delta_1, \ldots, \delta_d)$ satisfies all inequalities [*(\[Stanley\])*]{} and [*(\[Hibi\])*]{}.
As an analogy of Lemma \[small\], we classify shifted symmetric symmetric $\delta$-vectors with (0,1)-vectos when $\sum_{i=0}^d\delta_i =4$ or 5.
Now, in what follows, a sequence $(\delta_0, \delta_1, \ldots, \delta_d)$ with each $\delta_i \in \{ 0, 1 \}$, where $\delta_0 = 1$, which satisfies all inequalities (\[Stanley\]) and all equalities (\[Hibi\]) together with $\sum_{i=0}^{d} \delta_i = 4$ or 5 will be considered.
At first, we consider the case of $\sum_{i=0}^{d} \delta_i = 4$. Let $\delta_{m_1}=\delta_{m_2}=\delta_{m_3}=1$ with $1 \leq m_1 < m_2 < m_3 \leq d$. Let $p_1=m_1-1$, $p_2=m_2-m_1-1$, $p_3=m_3-m_2-1$ and $p_4=d-m_3$. By $\delta_{i+1}=\delta_{d-i}$ for $0 \leq i \leq [(d-1)/2]$, one has $p_1=p_4$ and $p_2=p_3$. Moreover, by (\[Stanley\]), one has $p_1 \geq p_2.$ Thus, $$\begin{aligned}
\label{eqeq}
p_1 \geq p_2 \geq 0, \;\;\; 2p_1+2p_2=d-3. \end{aligned}$$ Our work is to construct an integral convex polytope $\Pc$ with dimension $d$ whose $\delta$-vector coincides with $$\d(\Pc)=(1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_2},
1,\underbrace{0,\ldots,0}_{p_2},1,\underbrace{0,\ldots,0}_{p_1})$$ for an arbitrary integer $1 \leq m_1 < m_2 < m_3 \leq d$ satisfying the conditions (\[eqeq\]). When $p_1=p_2=0$, it is easy to construct it by Examples \[example\] (a). When $p_1=p_2>0$, if we set $d=4p_1+3$ and $m=4$, then the $\delta$-vector of the integral convex polytope whose vertices are of the form (\[vertex\]) coincides with $\d(\Pc)=(1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_1},
1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_1})$ by virtue of Corollary \[corollary\].
\[first\] Let $d=4k+3$, $l \geq 1$ and $d'=d+2l$. There exists an integral simplex $\Pc \subset \RR^{d'}$ of dimension $d'$ whose $\delta$-vector coincides with $$\begin{aligned}
\label{d1}
(1,\underbrace{0,\ldots,0}_{k+l},1,\underbrace{0,\ldots,0}_{k},
1,\underbrace{0,\ldots,0}_{k},1,\underbrace{0,\ldots,0}_{k+l})
\in \ZZ^{d'+1}.\end{aligned}$$
When $l=1$, if we set $d=4k+5$ and $m=4$, then the $\delta$-vector of the integral convex polytope whose vertices are of the form (\[vertex\]) coincides with (\[d1\]). When $l \geq 2$, let $v_0',v_1',\ldots,v_{4k+2l+3}' \in \RR^{4k+2l+3}$ be the vertices as follows: $$\begin{aligned}
v_i'=
\begin{cases}
(v_1,\underbrace{1,1,\ldots,1,1}_{2l-2}), \quad &i=1, \\
(v_i,\underbrace{0,1,\ldots,0,1}_{2l-2}), &i=2,3, \\
(v_i,\underbrace{0,0,\ldots,0,0}_{2l-2}), &i=4,\ldots,4k+5, \\
{\bf e}_i, &i=4k+6,\ldots,4k+2l+3, \\
(0,\ldots,0), &i=0,
\end{cases}\end{aligned}$$ where $v_1,\ldots,v_{4k+5}$ are of the form (\[vertex\]) with $d=4k+5$ and $m=4$. Then a simple computation enables us to show that $$\begin{aligned}
\begin{vmatrix}
v_1' \\
\vdots \\
v_{4k+2l+3}'
\end{vmatrix}
=
\begin{vmatrix}
& & & & & & \\
& &\text{\Huge{A}} & & &\text{\huge{*}}& \\
& & & & & & \\
& & & &1& & \\
& &\text{\huge{0}} & & &\ddots & \\
& & & & & &1
\end{vmatrix}
=4, \end{aligned}$$ where $A$ is the determinant (\[matrix\]) with $d=4k+5$ and $m=4$. One has $$\begin{aligned}
\frac{3}{4}(v_0',1)+\frac{1}{4}\{(v_1',1)+\cdots+(v_{4k+5}',1)\}
+\frac{3}{4}\{(v_{4k+6}',1)+(v_{4k+8}',1)+\cdots+(v_{4k+2l+2}',1)\} \\
+\frac{1}{4}\{(v_{4k+7}',1)+(v_{4k+9}',1)+\cdots+(v_{4k+2l+3}',1)\}
= (1,1,\ldots,1,k+l+1)\end{aligned}$$ and $$\begin{aligned}
\frac{1}{2}\{(v_0',1)+(v_1',1)+\cdots+(v_{4k+2l+3}',1)\}
= (\underbrace{2,\ldots,2}_{4k+5},\underbrace{1,2,\ldots,1,2}_{2l-2},2k+l+2). \end{aligned}$$ Hence $\delta_{k+l+1}=\delta_{2k+l+2}=\delta_{3k+l+3}=1$, as required.
Next, we consider the case of $\sum_{i=0}^{d} \delta_i = 5$. Let $\delta_{m_1}=\cdots=\delta_{m_4}=1$ with $1 \leq m_1 < \cdots < m_4 \leq d$. Let $p_1=m_1-1$, $p_2=m_2-m_1-1$, $p_3=m_3-m_2-1$,$p_4=m_4-m_3-1$ and $p_5=d-m_4$. By $\delta_{i+1}=\delta_{d-i}$ for $0 \leq i \leq [(d-1)/2]$, one has $p_1=p_5$ and $p_2=p_4$. Moreover, by (\[Stanley\]), one has $p_1 \geq p_2,p_3.$ Thus, $$\begin{aligned}
\label{eqeqeq}
p_1 \geq p_2,p_3 \geq 0, \;\;\; 2p_1+2p_2+p_3=d-4. \end{aligned}$$ Our work is to construct an integral convex polytope $\Pc$ with dimension $d$ whose $\delta$-vector coincides with $$\d(\Pc)=(1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_2},
1,\underbrace{0,\ldots,0}_{p_3},1,\underbrace{0,\ldots,0}_{p_2},1,\underbrace{0,\ldots,0}_{p_1})$$ for an arbitrary integer $1 \leq m_1 < \cdots < m_4 \leq d$ satisfying the conditions (\[eqeqeq\]). When $p_1=p_2=p_3=0$, it is easy to construct it by Examples \[example\] (a). When $p_1=p_2=p_3>0$, if we set $d=5p_1+4$ and $m=5$, then the $\delta$-vector of the integral convex polytope whose vertices are (\[vertex\]) coincides with $\d(\Pc)=(1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_1},
1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_1},1,\underbrace{0,\ldots,0}_{p_1})$ by virtue of Corollary \[corollary\].
\[second\] Let $d=5k+4$ and $l > 0$.\
[*(a)*]{} Let $d'=d+2l$. There exists an integral simplex $\Pc \subset \RR^{d'}$ of dimension $d'$ whose $\delta$-vector coincides with $$\begin{aligned}
\label{d2}
(1,\underbrace{0,\ldots,0}_{k+l},1,\underbrace{0,\ldots,0}_{k},
1,\underbrace{0,\ldots,0}_{k},1,\underbrace{0,\ldots,0}_{k},
1,\underbrace{0,\ldots,0}_{k+l})
\in \ZZ^{d'+1}.\end{aligned}$$ [*(b)*]{} Let $d'=d+3l$. There exists an integral simplex $\Pc \subset \RR^{d'}$ of dimension $d'$ whose $\d$-vector coincides with $$\begin{aligned}
\label{d3}
(1,\underbrace{0,\ldots,0}_{k+l},1,\underbrace{0,\ldots,0}_{k},
1,\underbrace{0,\ldots,0}_{k+l},1,\underbrace{0,\ldots,0}_{k},
1,\underbrace{0,\ldots,0}_{k+l})
\in \ZZ^{d'+1}.\end{aligned}$$ [*(c)*]{} Let $d'=d+4l$. There exists an integral simplex $\Pc \subset \RR^{d'}$ of dimension $d'$ whose $\d$-vector coincides with $$\begin{aligned}
\label{d4}
(1,\underbrace{0,\ldots,0}_{k+l},1,\underbrace{0,\ldots,0}_{k+l},
1,\underbrace{0,\ldots,0}_{k},1,\underbrace{0,\ldots,0}_{k+l},
1,\underbrace{0,\ldots,0}_{k+l})
\in \ZZ^{d'+1}.\end{aligned}$$
A proof can be done as the similar way of Lemma \[first\].
\(a) When $l=1$, if we set $d=5k+6$ and $m=5$, then the $\delta$-vector of the integral convex polytope whose vertices are of the form (\[vertex\]) coincides with (\[d2\]). When $l \geq 2$, let $v_0',v_1',\ldots,v_{5k+2l+4}' \in \RR^{5k+2l+4}$ be the vertices as follows: $$\begin{aligned}
v_i'=
\begin{cases}
(v_1,\underbrace{1,1,\ldots,1,1}_{2l-2}), \quad &i=1, \\
(v_i,\underbrace{0,1,\ldots,0,1}_{2l-2}), &i=2,3,4, \\
(v_i,\underbrace{0,0,\ldots,0,0}_{2l-2}), &i=5,\ldots,5k+6, \\
{\bf e}_i, &i=5k+7,\ldots,5k+2l+4, \\
(0,\ldots,0), &i=0,
\end{cases}\end{aligned}$$ where $v_1,\ldots,v_{5k+6}$ are of the form (\[vertex\]) with $d=5k+6$ and $m=5$. Then a simple computation enables us to show that $\begin{vmatrix}
v_1' \\
\vdots \\
v_{5k+2l+4}'
\end{vmatrix}
=5. $ One has $$\begin{aligned}
\frac{4}{5}(v_0',1)+\frac{1}{5}\{(v_1',1)+\cdots+(v_{5k+6}',1)\}
+\frac{4}{5}\{(v_{5k+7}',1)+(v_{5k+9}',1)+\cdots+(v_{5k+2l+3}',1)\} \\
+\frac{1}{5}\{(v_{5k+8}',1)+(v_{5k+10}',1)+\cdots+(v_{5k+2l+4}',1)\}
= (1,1,\ldots,1,k+l+1)\end{aligned}$$ and $$\begin{aligned}
\frac{3}{5}(v_0',1)+\frac{2}{5}\{(v_1',1)+\cdots+(v_{5k+6}',1)\}
+\frac{3}{5}\{(v_{5k+7}',1)+(v_{5k+9}',1)+\cdots+(v_{5k+2l+3}',1)\} \\
+\frac{2}{5}\{(v_{5k+8}',1)+(v_{5k+10}',1)+\cdots+(v_{5k+2l+4}',1)\}
= (\underbrace{2,\ldots,2}_{5k+6},\underbrace{1,2,\ldots,1,2}_{2l-2},2k+l+2). \end{aligned}$$ Hence $\delta_{k+l+1}=\delta_{2k+l+2}=\delta_{3k+l+3}=\delta_{4k+l+4}=1$, as required.
\(b) When $l=1$, if we set $d=5k+7$ and $m=5$, then the $\delta$-vector of the integral convex polytope whose vertices are of the form (\[vertex\]) coincides with (\[d3\]). When $l \geq 2$, let $v_0',v_1',\ldots,v_{5k+3l+4}' \in \RR^{5k+3l+4}$ be the vertices as follows: $$\begin{aligned}
v_i'=
\begin{cases}
(v_i,\underbrace{1,1,1,\ldots,1,1,1}_{3l-3}), \quad &i=1,2, \\
(v_i,\underbrace{0,1,1,\ldots,0,1,1}_{3l-3}), &i=3,4, \\
(v_i,\underbrace{0,0,0,\ldots,0,0,0}_{3l-3}), &i=5,\ldots,5k+7, \\
{\bf e}_i, &i=5k+8,\ldots,5k+3l+4, \\
(0,\ldots,0), &i=0,
\end{cases}\end{aligned}$$ where $v_1,\ldots,v_{5k+7}$ are of the form (\[vertex\]) with $d=5k+7$ and $m=5$. Then a simple computation enables us to show that $\begin{vmatrix}
v_1' \\
\vdots \\
v_{5k+3l+4}'
\end{vmatrix}
=5. $ One has $$\begin{aligned}
\frac{3}{5}(v_0',1)+\frac{1}{5}\{(v_1',1)+\cdots+(v_{5k+7}',1)\}
+\frac{3}{5}\{(v_{5k+8}',1)+(v_{5k+11}',1)+\cdots+(v_{5k+3l+2}',1)\} \\
+\frac{1}{5}\{(v_{5k+9}',1)+(v_{5k+10}',1)+\cdots+(v_{5k+3l+3}',1)+(v_{5k+3l+4}',1)\}
= (1,1,\ldots,1,k+l+1)\end{aligned}$$ and $$\begin{aligned}
\frac{1}{5}(v_0',1)+\frac{2}{5}\{(v_1',1)+\cdots+(v_{5k+7}',1)\}
+\frac{1}{5}\{(v_{5k+8}',1)+(v_{5k+11}',1)+\cdots+(v_{5k+3l+2}',1)\} \\
+\frac{2}{5}\{(v_{5k+9}',1)+(v_{5k+10}',1)+\cdots+(v_{5k+2l+3}',1)+(v_{5k+2l+4}',1)\} \\
= (\underbrace{2,\ldots,2}_{5k+7},\underbrace{1,2,2,\ldots,1,2,2}_{3l-3},2k+l+2). \end{aligned}$$ Hence $\delta_{k+l+1}=\delta_{2k+l+2}=\delta_{3k+2l+3}=\delta_{4k+2l+4}=1$, as required.
\(c) When $l=1$, if we set $d=5k+8$ and $m=5$, then the $\delta$-vector of the integral convex polytope whose vertices are of the form (\[vertex\]) coincides with (\[d4\]). When $l \geq 2$, let $v_0',v_1',\ldots,v_{5k+4l+4}' \in \RR^{5k+4l+4}$ be the vertices as follows: $$\begin{aligned}
v_i'=
\begin{cases}
(v_i,\underbrace{1,1,1,1,\ldots,1,1,1,1}_{4l-4}), \quad &i=1,2,3, \\
(v_4,\underbrace{0,1,1,1,\ldots,0,1,1,1}_{4l-4}), &i=4, \\
(v_i,\underbrace{0,0,0,0,\ldots,0,0,0,0}_{4l-4}), &i=5,\ldots,5k+8, \\
{\bf e}_i, &i=5k+9,\ldots,5k+4l+4, \\
(0,\ldots,0), &i=0,
\end{cases}\end{aligned}$$ where $v_1,\ldots,v_{5k+8}$ are of the form (\[vertex\]) with $d=5k+8$ and $m=5$. Then a simple computation enables us to show that $\begin{vmatrix}
v_1' \\
\vdots \\
v_{5k+4l+4}'
\end{vmatrix}
=5. $ One has $$\begin{aligned}
\frac{2}{5}(v_0',1)+\frac{1}{5}\{(v_1',1)+\cdots+(v_{5k+8}',1)\}
+\frac{2}{5}\{(v_{5k+9}',1)+(v_{5k+13}',1)+\cdots+(v_{5k+4l+1}',1)\} \\
+\frac{1}{5}\{(v_{5k+10}',1)+(v_{5k+11}',1)+(v_{5k+12}',1)+\cdots
+(v_{5k+4l+2}',1)+(v_{5k+4l+3}',1)+(v_{5k+4l+4}',1)\} \\
= (1,1,\ldots,1,k+l+1)\end{aligned}$$ and $$\begin{aligned}
\frac{4}{5}(v_0',1)+\frac{2}{5}\{(v_1',1)+\cdots+(v_{5k+8}',1)\}
+\frac{4}{5}\{(v_{5k+9}',1)+(v_{5k+13}',1)+\cdots+(v_{5k+4l+1}',1)\} \\
+\frac{2}{5}\{(v_{5k+10}',1)+(v_{5k+11}',1)+(v_{5k+12}',1)+\cdots
+(v_{5k+4l+2}',1)+(v_{5k+4l+3}',1)+(v_{5k+4l+4}',1)\} \\
= (2,2,\ldots,2,2k+2l+2). \end{aligned}$$ Hence $\delta_{k+l+1}=\delta_{2k+2l+2}=\delta_{3k+2l+3}=\delta_{4k+3l+4}=1$, as required.
[**Acknowledgements**]{}
The author would like to thank Prof. T. Hibi for helping me in writing this paper.
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|
---
author:
- '<span style="font-variant:small-caps;">L</span>eonid Polterovich'
title: Symplectic intersections and invariant measures
---
We present a method, based on symplectic topology, which enables us to detect invariant measures with “large" rotation vectors for a class of Hamiltonian flows. The method is robust with respect to $C^0$-perturbations of the Hamiltonian. Our results (Theorem \[thm-sympint\] and Theorem \[thm-sympint-1\]) are applicable even in the absence of homologically non-trivial closed orbits of the flow, see Section \[sec-disc\] for a discussion and an example.
The first three sections of the note deal with the case of autonomous Hamiltonians where the formulations and the approach are more transparent. The story starts (Section \[sec-1\]) with a “rigid" configuration of subsets of a symplectic manifold provided by symplectic intersections theory. In Section \[sec-pb\] the geometric setup is reformulated in the language of function theory on symplectic manifolds by using a version of Poisson bracket invariants introduced in [@BEP]. With this reformulation, the main result is proved by an elementary ergodic argument.
The last section contains a generalization to the case of non-autonomous Hamiltonians.
From symplectic intersections to invariant measures {#sec-1}
===================================================
Let $M$ be a closed manifold, and $v$ be a smooth vector field on $M$ generating a flow $\phi_t$. For an invariant Borel probability measure $\mu$ of $\phi_t$ define its [*rotation vector*]{} $\rho(\mu,v) \in H_1(M,{{\mathbb{R}}})$ by $$\langle a, \rho(\mu,v)\rangle := \int_M \alpha(v) d\mu\;\; \forall a \in H^1(M,{{\mathbb{R}}})\;,$$ where $\alpha$ is (any) closed 1-form representing $a$ (see [@Schw]).
Suppose now that $(M,\omega)$ is a closed symplectic manifold. Denote by ${{\hbox{\it Symp} }}_0(M,\omega)$ the identity component of the symplectomorphism group of $M$. Given a path $\{\phi_t\}$, $t\in [0,1]$, $\phi_0=\id$ of symplectomorphisms, denote by $v_t$ the corresponding vector field. By the Cartan formula, $\lambda_t := i_{v_t}\omega$ is a closed $1$-form on $M$. The cohomology class $\int_0^1 [\lambda_t]$ is called [*the flux*]{} of the path $\{\phi_t\}$ and is denoted by $\operatorname{Flux}(\{\phi_t\})$. The flux does not change under a homotopy of $\{\phi_t\}$ with fixed endpoints [@MS]. A diffeomorphism $\theta \in Symp_0(M,\omega)$ is called Hamiltonian if it can be represented as the time one map of a path with vanishing flux. Hamiltonian diffeomorphisms form a group denoted by ${{\hbox{\it Ham\,}}}(M,\omega)$.
Our main result involves a pair of compact subsets $X,Y \subset M$ with the following properties:
- $Y$ cannot be displaced from $X$ by any Hamiltonian diffeomorphism: $\theta(Y) \cap X \neq \emptyset$ for every $\theta \in {{\hbox{\it Ham\,}}}(M,\omega)$;
- There exists a path $\{\psi_t\}$, $t\in [0,1]$, $\psi_0=\id$ of symplectomorphisms so that $\psi_1$ displaces $Y$ from $X$: $\psi_1 (Y) \cap X =\emptyset$.
Put $X':= \psi_1(Y), a:= \operatorname{Flux}(\{\psi_t\})$.
\[thm-sympint\] For every $F\in C^\infty(M)$ with $$\label{eq-ineq-main}
F|_{X} \leq 0,\;\;F|_{X'} \geq 1$$ the Hamiltonian flow $\{\phi_t\}$ of $F$ possesses an invariant measure $\mu$ with $$\label{eq-mainineq}
|\langle a, \rho(\mu,\operatorname{sgrad}F)\rangle| \geq 1 \;.$$
Theorem \[thm-sympint\] extends with minor modifications to certain non-compact symplectic manifolds, see Remark \[rem-noncomp\] below. An generalization to the case of non-autonomous Hamiltonian flows is given in Section \[sec-non-auton\].
Theorem \[thm-sympint\] is deduced from a more general statement involving so called Poisson bracket invariants in Section \[sec-pb\].
In certain situations Property (P1) can be detected by methods of “hard" symplectic topology.
\[exam-1\][Let $X=Y=L$ be a Lagrangian torus in the standard symplectic torus $(M={{\mathbb{T}}}^{2n}={{\mathbb{R}}}^{2n}/{{\mathbb{Z}}}^{2n}, \omega= dp \wedge dq)$ given by $$\label{eq-lag-1}
L:=\{p_1=...=p_n=0\}\;.$$ The fact that $X$ is non-displaceable from itself is a basic particular case of Arnold’s Lagrangian intersections conjecture proved in the 1980ies by various authors including Chaperon, Hofer, Laudenbach-Sikorav, Floer (see [@F] and references therein). Let $\psi_t$ be the translation by $t/2$ in the direction of $p_1$-axis. We see that $X' = \psi_1(Y)=L'$, where $$\label{eq-lag-2}
L':= \{p_1=1/2, p_2=...=p_n=0\}$$ is disjoint from $X$. The flux $a$ of $\{\psi_t\}$, $t\in [0,1]$ equals $\frac{1}{2} \cdot [dq_1] \in H^1({{\mathbb{T}}}^{2n},{{\mathbb{R}}})$. We conclude that the Hamiltonian flow of any Hamiltonian $F$ on ${{\mathbb{T}}}^{2n}$ with $F|_{X} \leq 0$, $F|_{X'} \geq 1$ possesses an invariant measure $\mu$ with $$|\langle [dq_1], \rho(\mu,\operatorname{sgrad}F)\rangle| \geq 2\;.$$ Furthermore, suppose that $n=1$ and fix $\epsilon >0$. Take a function $F$ of the form $F= u(p_1)$, so that $$|dq_1(\operatorname{sgrad}F)|= |du/dq_1| \leq 2+\epsilon\;.$$ Thus every invariant measure $\mu$ of the corresponding Hamiltonian flow satisfies $$|\langle [dq_1], \rho(\mu,\operatorname{sgrad}F)\rangle| \leq 2+\epsilon\;.$$ This shows that the conclusion of Theorem \[thm-sympint\] is sharp. ]{}
\[exam-2\]
The previous example can be generalized as follows. We keep notations $L$ and $L'$ for the Lagrangian tori in ${{\mathbb{T}}}^{2n}$ given by and . Let $(N,\Omega)$ be a closed symplectic manifold. Let $A,B \subset N$ be a pair of compact subsets such that
[**$(\spadesuit)$**]{} $B \times L$ cannot be displaced from $A \times L$ by a Hamiltonian diffeomorphism of $(M = N \times {{\mathbb{T}}}^{2n}, \Omega \oplus dp \wedge dq)$.
Put $X=A \times L, Y= B \times L$ and observe that the translation by $t/2$ in the direction of $p_1$-axis sends $Y$ to $X'= B \times L'$, so that $X \cap X' = \emptyset$. Thus $X$ and $Y$ satisfy Properties (P1),(P2).
Property $\spadesuit$ holds, for instance, when $A,B$ are Lagrangian submanifolds of $N$ with non-vanishing Floer homology: $HF(A,B) \neq 0$.
Another example of $\spadesuit$ is follows: assume that $(N,\Omega)$ splits as $$N= N_1 \times ... \times N_k,\;\, \Omega=\Omega_1 \oplus ... \oplus \Omega_k$$ and $$A=B= C_1 \times ... \times C_k\;,$$ where $C_j$ is the codimension one skeleton of a sufficiently fine triangulation of $N_j$. Observe that the sets $X$ and $Y$ in this case could be quite singular. More generally, one can take $A=B$ to be a heavy subset of $N$, see [@EP].
Invariant measures vs. periodic orbits {#sec-disc}
======================================
It is instructive to discuss Theorem \[thm-sympint\] in the context of the following informal principle which nowadays is confirmed in various situations by tools of “hard" symplectic topology: certain robust restrictions on the $C^0$-profile of the Hamiltonian function may yield meaningful information about homologically non-trivial closed orbits of the Hamiltonian flow (see e.g. [@LG; @BPS; @Lee; @W; @Ni]). Such a restriction in Theorem \[thm-sympint\] is given by inequalities . Observe that every $T$-periodic orbit representing a non-trivial homology class $b \in H_1(M,{{\mathbb{Z}}})$ determines an invariant measure with the rotation vector $b/T$. Thus it is natural to ask the following question: [*Can one, under assumptions of Theorem \[thm-sympint\], deduce existence of a closed orbit of the Hamiltonian flow so that the corresponding rotation vector satisfies inequality ?* ]{} As the next example shows, the answer is in general negative.
\[exam-noorb\]
Let $M$ be the symplectic torus ${{\mathbb{T}}}^4={{\mathbb{R}}}^4/{{\mathbb{Z}}}^4$ equipped with the symplectic form $$\omega= dp_1\wedge dq_1 +\gamma dp_2 \wedge dq_1 + dp_2 \wedge dq_2\;,$$ where $\gamma$ is an [*irrational*]{} number. As in Example \[exam-1\] above, consider Lagrangian torus $L=\{p=0\}$ and its image $L'$ under the shift by $1/2$ in $p_1$-direction. Take a Hamiltonian $F(p,q) = \sin^2 (\pi p_1)$, so that $F=0$ on $L$ and $F=1$ on $L'$. Exactly as in the case of the standard symplectic form, Floer theory guarantees that Properties (P1) and (P2) hold for $X=Y=L$ and $X'=L'$. Therefore Theorem \[thm-sympint\] detects an invariant measure with non-vanishing rotation number of the Hamiltonian flow of $F$.
Of course, this measure can be seen explicitly. One readily checks that the Hamiltonian vector field of $F$ is parallel to $$\frac{\partial}{\partial q_1}-\gamma \cdot \frac{\partial}{\partial q_2}\;.$$ In particular, the restriction of the Hamiltonian flow to every invariant torus $\{p_1=c_1,p_2=c_2\}$, $c_1\neq 0,1/2$ carries a quasi-periodic motion and possesses unique invariant measure with non-vanishing rotation number. The crux of this example is that the only closed orbits of the flow are fixed points lying on hypersurfaces $\{p_1=0\}$ and $\{p_1=1/2\}$. In particular, the flow does not have homologically non-trivial closed orbits.
The discussion is continued in Remark \[rem-Lalonde\] below.
Poisson bracket invariants {#sec-pb}
==========================
For a closed 1-form $\alpha$ on $M$ define its locally Hamiltonian vector field $\operatorname{sgrad}\alpha$ by $i_{\operatorname{sgrad}\alpha} \omega = \alpha$. With this notation, the Hamiltonian vector field of a function $F$ is $\operatorname{sgrad}F:= \operatorname{sgrad}(-dF)$, mind the minus sign. For a function $F$ and a closed $1$-form $\alpha$ their Poisson bracket is given by $$\{F,\alpha\}=dF(\operatorname{sgrad}\alpha) = \alpha(\operatorname{sgrad}F)\;.$$ The next definition is a variation on the theme of [@BEP]. We write $||F||$ for the uniform norm $\max_M |F|$.
\[def-pb\][Let $X$ and $X'$ be a pair of disjoint closed subsets of $M$. For a cohomology class $a \in H^1(M,{{\mathbb{R}}})$ put $$pb^a(X,X'):= \inf ||\{F,\alpha\}||\;,$$ where the infimum is taken over all $F\in C^\infty(M)$ with $F|_{X} \leq 0$, $F|_{X'} \geq 1$ and all closed 1-forms $\alpha$ representing $a$. ]{}
This invariant is non-trivial, for instance, in the following situation. Let $X,Y \subset M$ be a pair of compact subsets satisfying Properties (P1) and (P2) of Section \[sec-1\]: $Y$ cannot be displaced from $X$ by a Hamiltonian diffeomorphism, but, on the other hand, it can be displaced from $X$ by a symplectomorphism $\psi_1$ which is the end-point of a symplectic path $\{\psi_t\}$ with the flux $a \in H^1(M,{{\mathbb{R}}})$. Put $X'=\psi_1(Y)$.
\[prop-exammain\] Under above assumptions, $pb^a(X,X') \geq 1$.
Take any closed $1$-form $\alpha$ representing $a$, and denote by $\theta_t$, $t \in [0,1]$ the corresponding locally Hamiltonian flow. Note that $\operatorname{Flux}(\{\psi_t^{-1}\theta_t\}) =0$, and thus the diffeomorphism $\psi_1^{-1}\theta_1$ is Hamiltonian. Therefore $\psi_1^{-1}\theta_1(X) \cap Y \neq \emptyset$, and thus $\theta_1(X) \cap X' \neq \emptyset$. The latter yields existence of a point $x \in X$ such that $x':= \theta_1 x \in X'$. Given any function $F\in C^\infty(M)$ with $F|_{X} \leq 0$, $F|_{X'} \geq 1$, we have that $$1 \leq F(x')-F(x) = \int_0^1 \{F,\alpha\}(\theta_tx)\;dt\;.$$ Thus $||\{F,\alpha\}||\geq 1$, which proves the proposition.
The main result of the present section is as follows.
\[thm-main\] Assume that $pb^{a}(X,X')=p >0$. Then
- For every $F\in C^\infty(M)$ with $F|_{X} \leq 0$, $F|_{X'} \geq 1$, the Hamiltonian flow $\{\phi_t\}$ of $F$ possesses an invariant measure $\mu$ with $$\label{eq-mainineq-1}
|\langle a, \rho(\mu,\operatorname{sgrad}F)\rangle| \geq p \;.$$
- For every closed $1$-form $\alpha$ representing $a$ the corresponding locally Hamiltonian flow possesses a chord of time length $\leq 1/p$ joining $X$ and $X'$.
In view of Proposition \[prop-exammain\], Theorem \[thm-sympint\] immediately follows from Theorem \[thm-main\](i).
Part (ii) of Theorem \[thm-main\] is proved exactly as in [@BEP Section 4.1].
[**Proof of Theorem \[thm-main\](i):**]{} Suppose that $pb^a(X,X')=p >0$. Take any function $F\in C^\infty(M)$ with $F|_{X} \leq 0$, $F|_{X'} \geq 1$ and any $1$-form $\alpha$ representing the class $a$. Denote by $\phi_t$ the Hamiltonian flow generated by $F$. For a point $x \in M$ and a number $T >0$ denote by $\mu_{x,T}$ a probability measure on $M$ given by $$\int H d\mu_{x,T}:= \frac{1}{T} \cdot \int_0^T H(\phi_t x)dt\;\; \forall H \in C(M).$$ Observe, following the standard proof of the Bogolyubov-Krylov theorem (see e.g. Chapter 1, Section 8 of [@CFS]), that if a sequence $\mu_{x_i,T_i}$ with $T_i \to +\infty$ as $i \to +\infty$ weakly converges to a measure $\mu$, this measure is invariant under the flow $\phi_t$.
Following a suggestion by Michael Entov, apply now the averaging trick similar to the one used in [@BEP]. For $T>0$ consider the averaged form $$\alpha_T := \frac{1}{T} \cdot \int_0^T \phi_t^* \alpha\;dt\;.$$ Note that $\alpha_T$ is still a closed form in the class $a$. Observe that $$|\{F,\alpha_T\}(x)| = \Big{|}\int \alpha(\operatorname{sgrad}F) d\mu_{x,T}\Big{|}\;.$$ Since $pb^a(X,X') = p$, there exists a point $x_T \in M$ such that $$\label{eq-rotation-vsp}
\Big{|}\int \alpha(\operatorname{sgrad}F) d\mu_{x_T,T}\Big{|} \geq p\;.$$ Compactness yields existence of a sequence $T_i \to +\infty$ so that the sequence of measures $\mu_i := \mu_{x_{T_i},T_i}$ weakly converges to a measure $\mu$ on $M$ which, by the above discussion, is invariant under the flow $\phi_t$. Using weak convergence and we get that $$|\rho(\mu,\operatorname{sgrad}F)| = \lim_{i \to \infty} \Big{|}\int \alpha(\operatorname{sgrad}F) d\mu_i \Big{|} \geq p\;,$$ as required.
We conclude this section with a couple of remarks.
\[rem-Lalonde\]
$\;$
[**\[rem-Lalonde\].1.**]{} It would be interesting to explore further examples of pairs $X,X' \subset M$ with $pb^a(X,X') >0$ for some $a \in H^1(M,{{\mathbb{R}}})$. A promising pool of such examples is given by disjoint Lagrangian submanifolds $X$ and $X'$ which can be joined by pseudo-holomorphic annuli persisting under Lagrangian (not necessarily Hamiltonian!) isotopies of $X$ and $X'$ keeping these submanifolds disjoint. In this case one can deduce positivity of $pb^a(X,X')$ by using a method of [@BEP Section 1.6]. The latter is based on the study of obstructions to deformations of the symplectic form $\omega$ given by $\omega_s = \omega +sdF \wedge \alpha$, where $F$ is a function which vanishes near $X$ and equals $1$ near $X'$, and $\alpha$ is a closed $1$-form representing $a$. The cohomology class $a$ is chosen in such a way that it does not vanish when evaluated on (any) boundary component of the annulus.
[**\[rem-Lalonde\].2.**]{} Let us mention that annuli with boundaries on $X$ and $X'$ satisfying a non-homogeneous Cauchy-Riemann equation play a crucial role in the Gatien-Lalonde approach to homologically non-trivial closed orbits of Hamiltonians satisfying inequalities (see [@LG; @Lee]). This brings us to a new facet of the discussion in Section \[sec-disc\] on invariant measures vs. periodic orbits. Assume that an appropriately defined relative Gromov-Witten invariant responsible for the count of pseudo-holomorphic annuli with boundaries on $X$ and $X'$ does not vanish. It sounds plausible that under this assumption one can establish existence of closed orbits with the rotation vector satisfying inequality directly by the Gatien-Lalonde method.
[**\[rem-Lalonde\].3.**]{} Let us mention finally that count of pseudo-holomorphic annuli appears in a number of recent papers on the Fukaya category, see [@Abouzaid; @Seidel]. It would be interesting to understand its applicability in our context.
\[rem-noncomp\] [Theorem \[thm-sympint\] extends with minor modifications to certain non-compact symplectic manifolds. We start with a compactly supported function $F$ on $M$ with $F|_X \leq 0$, $F|_{X'} \geq 1$, where $X$ and $X'$ are disjoint compact subsets. Fix a closed $1$-form $\alpha$ representing a cohomology class $a \in H^1(M,{{\mathbb{R}}})$. Assume that the locally Hamiltonian flow $\psi_t$ of $\alpha$ is well defined for all times. Suppose now that $\psi_1(X)$ cannot be displaced from $X'$ by any Hamiltonian diffeomorphism. Arguing exactly as in the compact case, we get that the Hamiltonian flow of $F$ possesses an invariant measure $\mu$ with compact support whose rotation vector satisfies $$|\langle a, \rho(\mu,\operatorname{sgrad}F) \rangle| \geq 1\;.$$ A meaningful example is given by $M=T^*{{\mathbb{T}}}^n$ equipped with canonical coordinates $(p,q)$ and the standard symplectic form $dp \wedge dq$, $X$ – the zero section, $X'$–the Lagrangian torus $\{p=v\}$ with $v \neq 0$ and $\alpha = vdq$. Let us mention that for fiber-wise convex Hamiltonians on cotangent bundles invariant measures with non-vanishing rotation vectors have been studied in the framework of Aubry-Mather theory [@Mather]. ]{}
Non-autonomous Hamiltonian flows {#sec-non-auton}
================================
In this section we present a generalization of Theorem \[thm-sympint\] to general, not necessarily autonomous, Hamiltonian diffeomorphisms. Let $M$ be a connected symplectic manifold (either open or closed). Let $F: M \times S^1 \to {{\mathbb{R}}}$ a compactly supported Hamiltonian function which is time-periodic with period $1$. Denote by $\phi_t$ the corresponding Hamiltonian flow. The time periodicity of $F$ yields $\phi_{t+1} = \phi_t \phi$, where $\phi=\phi_1$ is the time one map of the flow. In what follows we write $F_t(x) = F(x,t)$.
For an invariant compactly supported Borel probability measure $\mu$ of $\phi$ the [*rotation vector*]{} $\rho(\mu,\phi)$ is a compactly supported homology class in $H_{1,c}(M,{{\mathbb{R}}})$ defined by $$\langle a, \rho(\mu,\phi)\rangle := \int_0^1 \int_M \langle \alpha,\operatorname{sgrad}F_t \rangle (\phi_tx) d\mu(x)dt$$ $$=\int_M \left( \int_{\gamma_x}\alpha \right) \;d\mu(x) \;\; \forall a \in H^1(M,{{\mathbb{R}}})\;,$$ where $\alpha$ is (any) closed 1-form representing $a$ and $\gamma_x$ stands for the trajectory $\{\phi_t x\}, t \in [0,1]$.
We claim that the rotation vector $\rho(\mu,\phi)$ depends only on the time one map $\phi$ but not on the specific Hamiltonian $F$ generating $\phi$. Indeed, it is a standard fact of Floer theory that if $F'$ is another Hamiltonian such that the corresponding Hamiltonian flow $\phi'_t$ satisfies $\phi'_1 =\phi$, the orbits $\gamma_x= \{\phi_tx\}$ and $\gamma'_x=\{\phi'_tx\}$, $t \in [0,1]$ are homotopic with fixed end points for every $x \in M$ and hence $$\int_{\gamma_x}\alpha = \int_{\gamma'_x} \alpha\;.$$
Let us note that when the Hamiltonian $F$ is autonomous, $\rho(\mu,\phi)$ coincides with $\rho(\mu, \operatorname{sgrad}F)$ as defined in Section \[sec-1\].
Consider the extended phase space $$(N,\Omega):= (M \times T^*S^1, \omega + dr \wedge ds)\;,$$ where $r$ and $s(\text{mod}\; 1)$ are canonical coordinates on $T^*S^1$. For a set $X \subset M$ define its [*stabilization*]{} $$\text{stab}(X):= X \times \{r=0\} \subset N\;.$$
The next result involves a pair of compact subsets $X,Y \subset M$ which satisfy the following properties (cf. properties (P1) and (P2) in Section \[sec-1\]):
- $\text{stab}(Y)$ cannot be displaced from $\text{stab}(X)$ by any Hamiltonian diffeomorphism of $N$.
- There exists a closed $1$-form $\alpha$ on $M$ whose locally Hamiltonian flow $\{\psi_t\}$ is defined for all $t \in {{\mathbb{R}}}$ (this is a non-trivial assumption in the case when $M$ is non-compact) so that $\psi_1$ displaces $Y$ from $X$.
Put $X':= \psi_1(Y)$. Let $a$ be the cohomology class of $\alpha$.
\[thm-sympint-1\] For every $F\in C^\infty(M \times S^1)$ with $$F_t|_{X} \leq 0,\;\;F_t|_{X'} \geq 1 \;\; \forall t \in {{\mathbb{R}}}$$ the time one map $\phi$ of the Hamiltonian flow $\{\phi_t\}$ generated by $F$ possesses an invariant measure $\mu$ with $$|\langle a, \rho(\mu,\phi)\rangle| \geq 1 \;.$$
Let us mention that properties (Q1) and (Q2) hold true in Examples \[exam-1\], \[exam-2\], \[exam-noorb\] and in Remark \[rem-noncomp\].
The proof of Theorem \[thm-sympint-1\] starts with the Hamiltonian suspension construction: We pass to the extended phase space $N$ and look at the [*autonomous*]{} Hamiltonian flow generated by $H(x,r,s)= F(x,s) +r$. This flow encodes the original dynamics of $\phi$ on $M$ (cf. the proof of Theorem 1.12 in [@BEP]). Then we argue along the lines of the autonomous case considered in Theorem \[thm-sympint\]. Let us mention however that since $H$ is not compactly supported, Theorem \[thm-sympint\] is not directly applicable even after the modification described in Remark \[rem-noncomp\], which makes the proof below somewhat more involved.
The proof is divided into several steps. Denote by $\pi: N \to M$ the natural projection and put $\beta:= \pi^*\alpha$.
[Step 1:]{} Consider a Hamiltonian $H(x,r,s) = F(x,s) +r$ on $N$. It generates the flow $$h_t(x(0),r(0),s(0)) = (x(t),r(t),s(t))\;,$$ where $$x(t)= \phi_{s(0)+t}\phi_{s(0)}^{-1},\; r(t) = r(0)-\int_0^t \frac{\partial F}{\partial s}(x(t),s(t)) dt,\; s(t) = s(0) +t\;.$$ Observe that $h_t$ commutes with the symplectic ${{\mathbb{R}}}$-action $$S_c: N \to N, (x,r,s) \to (x,r+c,s)\;.$$ Fix $T >0$ and put $$\alpha_T = \frac{1}{T}\cdot \int_0^T h_t^*\beta\;.$$ Observe that $$S_c^*h_t^*\beta = h_t^*S_c^*\beta = h_t^*\beta$$ for all $c$ and $t$, and hence $$\label{eq-nonaut-complete}
S_c^*\alpha_T = \alpha_T\;.$$
[Step 2:]{} We claim that the flow $\operatorname{sgrad}\alpha_T$ is defined for all times $t \in {{\mathbb{R}}}$. Indeed, by the vector field $\operatorname{sgrad}\alpha_T$ descends to the manifold $N':= N/{{\mathbb{Z}}}$ where the action of ${{\mathbb{Z}}}$ is given by the integer shifts $S_k, k \in {{\mathbb{Z}}}$ along the $r$-direction. Furthermore, for $x$ outside a sufficiently large compact in $M$ we have that $h_t(x,r,s)= (x,r, s+t)$ and hence $\alpha_T = \beta = \pi^*\alpha$. It follows, by our assumption on $\alpha$, that the trajectories of $\operatorname{sgrad}\alpha_T$ cannot escape to infinity in finite time on $N'$. Thus the flow of $\operatorname{sgrad}\alpha_T$ is defined for all times on $N'$, and therefore it lifts to a well defined flow $\theta_t$, $t \in {{\mathbb{R}}}$ on $N$. The claim follows.
[Step 3:]{} Since $[\alpha_T]= [\beta] \in H^1(N,{{\mathbb{R}}})$, the flow $$g_t: = (\psi_t \times \id)^{-1} \theta_t : N \to N$$ is Hamiltonian. By Property (Q1) $g_t (\text{stab}(X)) \cap \text{stab}(Y) \neq \emptyset$ which yields $$\theta_1(\text{stab}(X)) \cap \text{stab}(X') \neq \emptyset\;.$$ Thus there exists a point $z \in \text{stab}(X)$ such that $\theta_1(z) \in \text{stab}(X')$. Observe that $H \leq 0$ on $\text{stab}(X)$ and $H \geq 1$ on $\text{stab}(X')$. Since $$1 \leq {H}(\theta_1z)-{H}(z) = \int_0^1 \{H,\alpha_T\}(\theta_tz)\;dt\;,$$ there exists $t' \in [0,1]$ with $$|\{{H},\alpha_T\}(\theta_{t'} z)| \geq 1\;.$$ Put $y=\theta_{t'}z$.
Note that $S_c^*H=H+c$ and $S_c^*\alpha_T = \alpha_T$ by . It follows that $S_c^*\{H,\alpha_T\} = \{H,\alpha_T\}$, and hence $$|\{{H},\alpha_T\}(S_cy)| \geq 1\;\; \forall c \in {{\mathbb{R}}}\;.$$ Therefore there exists a point $y_T := (x_T,0, s_T)$ such that $$\label{eq-nonauton-brack}
|\{{H},\alpha_T\}(y_T)| \geq 1\;.$$ As we shall see below, it matters that the $r$-coordinate of $y_T$ vanishes.
[Step 4:]{} We claim that all the orbits $\gamma_T := \{h_ty_T\}$, $t \in [0,T]$ lie in some compact $Q \subset N$ for all $T >0$. Denote by $K \subset M \times S^1$ the support of $F(x,s)$. The set $$X:= (M \setminus K) \times {{\mathbb{R}}}\subset N$$ is invariant under $h_t$. Moreover, on $X$ we have that $\{H,\alpha_T\}(z) = \{r,\beta\} = 0$ which violates . It follows that $\gamma_T \subset K \times {{\mathbb{R}}}$. Furthermore, write $h_ty_T = (x^t,r^t,s^t)$. By the energy conservation law, $$F(x^t,s^t) + r^t = F(x_T,s_T) +0\;,$$ which yields an upper bound $$|r_t| \leq C:= \max F - \min F \;\;\forall t\;.$$ The claim follows with $Q= K \times [-C,C]$.
[Step 5:]{} Define a measure $\nu_T$ on $N$ by $$\int G d\nu_T = \frac{1}{T} \int_0^T G(h_t y_T)\; dt\;\; \forall G \in C(N)\;.$$ By Step 4, these measures are supported in the compact subset $Q$. Hence, after passing to a subsequence $T_k \to +\infty$, they weakly converge to a measure, say $\nu$ on $N$. The standard Bogolyubov-Krylov argument shows that $\nu$ is $h_t$-invariant. Furthermore, by $$\Big{|} \int \beta(\operatorname{sgrad}H) d\nu_T \Big{|} = |\{H,\alpha_T\}(y_T)| \geq 1\;,$$ and hence $$\label{eq-nonaut-main}
\Big{|} \int \beta(\operatorname{sgrad}H) d\nu \Big{|} \geq 1\;.$$
[Step 6:]{} Consider the flow $$g_t: M \times S^1 \to M \times S^1,\; (x,s) \to (\phi_{s+t}\phi_s^{-1}x, s+t)\;.$$ Denote by $\tau: N \to M \times S^1$ the natural projection. Since $\tau h_t = g_t$, the push-forward measure $\sigma:= \tau_*\nu$ is invariant under $g_t$. Further, $\beta(\operatorname{sgrad}H) = \alpha (\operatorname{sgrad}F_s)$ is independent on $r$. Therefore inequality yields $$\label{eq-nonaut-main-1}
\Big{|} \int \alpha(\operatorname{sgrad}F_s) d\sigma \Big{|} \geq 1\;.$$
[Step 7:]{} Invariant measures of the flow $g_t$ have quite a simple structure. To see this, introduce diffeomorphisms $A$ and $B$ of $M \times {{\mathbb{R}}}$ given by $$A: (x,s) \mapsto (\phi_sx, s), \;\; B: (x,s) \mapsto (\phi^{-1}x, s+1)\;,$$ and the translation $R_t$ of $M \times {{\mathbb{R}}}$, $$R_t: (x,s) \mapsto (x,s+t)\;.$$ Let $$\widetilde{g}_t (x,s) = (\phi_{s+t}\phi_s^{-1}x, s+t)$$ be the lift of the flow $g_t$ to the cover $M \times {{\mathbb{R}}}$. Borel probability measures $\sigma$ on $M \times S^1$ are in one to one correspondence with $R_1$-invariant Borel measures $\widetilde{\sigma}$ on $M \times {{\mathbb{R}}}$ satisfying $\widetilde{\sigma}(M \times [0,1)) =1$. The measure $\sigma$ is $g_t$-invariant if and only if $\widetilde{\sigma}$ is $\widetilde{g}_t$-invariant. Observe that $\widetilde{g}_t = AR_tA^{-1}$. Thus every invariant measure $\widetilde{\sigma}$ of $\widetilde{g}_t$ has the form $A_*\overline{\mu}$, where $\overline{\mu}$ is an invariant measure of the flow $R_t$. Note that $\overline{\mu}$ is necessarily of the form $\mu \otimes ds$, where $ds$ is the Lebesgue measure on ${{\mathbb{R}}}$ and $\mu$ a measure on $M$.
The measure $A_*\overline{\mu}$ is $R_1$-invariant if and only if $$A^{-1}_*R_{1*}A_*\overline{\mu}= \overline{\mu}\;.$$ Since $A^{-1}R_1A = B$, we have that $B_*(\mu \otimes ds) = (\mu \otimes ds)$, which is equivalent to the fact that $\mu$ is $\phi$-invariant.
Returning back to $M \times S^1$, we conclude that every $g_t$-invariant Borel probability measure $\sigma$ on $M \times S^1$ satisfies $$\int_{M \times S^1} G(x,s)\; d\sigma(x,s) = \int_0^1 \int_M G(\phi_s x,s)\;d\mu(x)ds\;\;\forall G \in C(M \times S^1),$$ where $\mu$ is a $\phi$-invariant Borel probability measure on $M$.
[Step 8:]{} Apply the conclusion of Step 7 to the measure $\sigma$ constructed in Step 6. Inequality reads $$\Big{|} \int_0^1 \int \langle \alpha, \operatorname{sgrad}F_s\rangle (\phi_sx)\; d\mu(x)ds \Big{|} \geq 1\;,$$ where the measure $\mu$ on $M$ is $\phi$-invariant. By definition, this means that $|\langle a, \rho(\mu,\phi)\rangle|>1$. This completes the proof.
[**Acknowledgements.**]{} This note was inspired by Hofer’s 2013 Aisenstadt Lectures in the CRM, Montreal. I am grateful to Michael Entov for comments which led to a drastic simplification of the original proof (based on [@Sul]) of the main theorem, as well as to a generalization to the non-autonomous case. I thank Lev Buhovsky and Helmut Hofer for useful discussions.
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---------------------------------
Leonid Polterovich
School of Mathematical Sciences
Tel Aviv University
Tel Aviv 69978, Israel
polterov@runbox.com
---------------------------------
|
---
abstract: |
In a recent paper we investigated the internal space of Bessel functions associated with their orders.We found a formula (new) unifying Bessel functions of integer and of real orders. In this paper we study the deformed exterior derivative system $H=d_{\lambda }$ on the puctured plane as a tentative to understand the origin of the formula and find that indeed similar formula occurs.This is no coincidence as we will demonstrate that generating functions of integer order reduced Bessel functions and of real orders are respectively eigenstates of the usual exterior derivative and its deformation .As a direct consequence we rediscover the unifying formula and learn that the system linear in $d_{\lambda }$ is related to Bessel theory much as the system quadratic in ($d_{\lambda }+d_{\lambda }^{*}$) is related to Morse theory
[mekhfi@hotmail.com]{}
author:
- |
**MEKHFI.M**\
[Laboratoire de Physique Mathématique,Es-senia 31100 Oran **ALGERIE**]{}\
[and ]{}\
[Int’ Centre for theoretical Physics ,Trieste 34100 **ITALY**]{}
date: 2000 April 18
title: '**WITTEN DEFORMED EXTERIOR DERIVATIVE AND BESSEL FUNCTIONS**'
---
Introduction
============
Early studies \[e.g.,[@kn:Truesdell],[@kn:Infeld] \] proposed a unifying scheme for special functions showing that some of these functions may originate from the same structure .For Bessel functions of concern here , generating functions of integer orders are representation ”states” of derivative and integral operators of arbitrary orders.More precisely we have the ”inner” structure
$$\begin{aligned}
\partial _{|m|} &=&\frac{\partial }{z\partial z}.\,\frac{\partial }{%
z\partial z}..........\frac{\partial }{z\partial z}. \nonumber \\
\partial _{-|m|} &=&\int zdz.\,\int zdz..............\int zdz. \nonumber \\
\partial _{m}\Phi (z,t) &=&(-\ t)^{-m}\,\,\Phi (z,t)\,\,,m\epsilon Z
\label{eq:1} \\
\Phi (z,t) &=&\sum_{n=-\infty }^{n=\infty }\phi _{n}(z)\ t^{n} \nonumber\end{aligned}$$
$\ $
where we extend the index m to negative values by introducing the symbol $\int dz$ to denote a “truncated ” primitive i.e. in defining the integral we omit the constant of integration $\int \frac{df}{dz}dz=f$ and where $\phi _{n}(z)$ stands for the reduced Bessel function $\phi _{n}(z)=%
\frac{J_{n}(z)}{z^{n}}$ of integer order. For the polynomials such as Hermite and Laguerre for instance ,the generating functions only involve the realization of the set N of positive integers,with slight modifications of the derivative operators to account for the conventions used in defining these polynomials.It is important to note that although this common structure only set up the z-dependance of the generating functions ,it is the “ dynamical ” part of the scheme so to say.The t dependence is simply set by imposing some given desired properties.For Bessel functions for example we require a “symmetry” between positive and negative integer indices that is $J_{-n}=(-1)^{n}J_{n}$ ,while for the polynomials it is the natural property of orthonormality that is invoked.
In a recent paper[@kn:Wissale] we intuitively applied a mechanism to generate real numbers out of integers in order to unify ( reduced ) Bessel functions and showed by direct analytic computation that indeed , Bessel functions fit into the scheme and therefore integer orders are mapped to real orders ( $\lambda $ is real ) through the formula $$\frac{J_{n+\lambda }(z)}{z^{n+\lambda }}=exp\left[ -\lambda \sum_{m\epsilon
Z/(0)}\,\frac{\,\partial _{m}}{m}\right] \frac{J_{n}(z)}{z^{n}} \label{eq:2}$$
Let us summarize the mechanism to see how it works to convert an integer into a real.Suppose we are given an abstract state $\mid n\rangle
\,\,n\epsilon Z$ and a set of raising $\Pi _{m},m>0$ and lowering $m<0$ operators .Then it is easy to show , given that data , that the state $\mid
n+\lambda >$ is related to the state $\mid n>$ through the following formula
$$\mid n+\lambda >=exp\left[ -\lambda \sum_{m\epsilon Z/(0)}\frac{(-1)^{m}\Pi
_{m}}{m}\right] \mid n> \label{eq:3}$$
Fourier transforming the $\mid n>$ state as $$\mid n>=\int_{-\pi }^{\pi }\frac{d\theta }{2\pi }e^{in\theta }\mid \theta >
\label{eq:4}$$
where the $\Pi _{m}$ operators act on the $\mid \theta >$ state by simple multiplication by the factor $e^{im\theta }$ we get
$$\begin{aligned}
\mid n+\lambda > &=&\int_{-\pi }^{\pi }exp\left[ -\lambda \sum_{m\epsilon
Z/(0)}\frac{(-1)^{m}\Pi _{m}}{m}\right] \,\,e^{in\theta }\mid \theta >\frac{%
d\theta }{2\pi } \nonumber \\
&=&\int_{-\pi }^{\pi }e^{i(n+\lambda )\theta }\mid \theta >\frac{d\theta }{%
2\pi } \label{eq:5}\end{aligned}$$
Compare states in \[eq:4\] to states in \[eq:5\]. In deriving the last line use have been made of the known formula
$$\sum_{m=1}^{\infty }(-1)^{m}\frac{sinm\theta }{m}=-\frac{\theta }{2}%
,\,\,\,-\pi <\theta <\pi \, \label{eq:6'}$$
Formula \[eq:2\] is a new formula ( to be added to the huge literature on Bessel functions ) which is shown to apply to Neumann and Hankel functions as well [@kn:Mohsine].It is to be noted that although formula \[eq:2\] is shown to be true ,we didn’t know why the above mechanism should apply to Bessel functions.Our guess of the above relation was based on the following correspondence [@kn:Wissale]
$$\begin{aligned}
\phi _{n}(z) &\Leftrightarrow &\mid n> \\
\phi _{n+\lambda \text{ }}(z) &\Leftrightarrow &\mid n+\lambda >\end{aligned}$$
A hint to this correspondence came from the fact that we indeed have a set of raising and lowering operators $\Pi _{m}=(-1)^{m}\partial
_{m}\,\,m\epsilon N$ [@kn:watson] [@kn:smirnov]
$$(-1)^{m}\frac{d^{m}}{(zdz)^{m}}\phi _{n}(z)=\phi _{n+m}(z)\hspace{5mm}%
m\epsilon N,n\epsilon Z \label{eq:6}$$
and for negative $m^{^{\prime }s}$ we just have to replace derivative operators by integral operators defined in \[eq:1\].That this correspondence works is quite intriguing and therefore further investigations of it are needed .In this paper we answer the point. In section 2 we will study the deformed exterior derivative on the punctured plane and will realize that the converting( or deforming) mechanism is closely tied to the deformation of the exterior derivative .In section 3 we will demonstrate directly that real order reduced Bessel functions are the deformed versions of integer order reduced Bessel functions in the same fashion as $d_{\lambda }$ is the deformed version of the exterior derivative $d$ .As a consequence formula \[eq:2\] will show up more elegantly.
Deformed exterior derivative on $R^{2}/(0)$
===========================================
Let $M$ be a Riemannian manifold of dimension n .let $V_{p}$ $p=0,1....n$ be the space of $p$-forms .Let $d$ and $d^{*\text{ }}$ be the usual exterior derivative which define the De Rham cohomology of $M$ and its adjoint.Let $V
$ be a smooth function $V:M\rightarrow R$ $or\ C$ ( called prepotential in the language of topological quantum field theories ) and $\lambda $ a real number .Define
$$d_{\lambda }[V]=e^{-\lambda V}d\,\,e^{\lambda V} \label{eq:7}$$
Evidently we have $d_{\lambda }^{2}=d_{\lambda }^{*2}=0$. E.Witten[@kn:witten] had shown that $V$ plays the role of a Morse function and his consideration of the system
$$H_{\lambda }=d_{\lambda }d_{\lambda }^{*}+d_{\lambda }^{*}d_{\lambda }
\label{eq:7'}$$
had led to a new proof of Morse inequalities.Let us note at this point that there exists another version of $d$-deformation which is related to the fixed point theorems for Killing vector fields
$$d_{s}=d+s\ iK$$ where s is an arbitrary number and where $K$ is a killing vector field-the infinitesimal generator of an isometry of $M$.In this context $K$ is regarded as an operator $iK$ on differential forms acting by interior multiplication and hence maps a $p$-form into a $(p-1)$-form .Since we are interested in functions ($0$-form) such a deformation is not relevant as $%
d_{s}$ coincide with $d$ on the space of functions.In this section and the subsequent section we investigate the simpler system
$$H=d_{\lambda }$$ and show that it gives informations on the index structure of Bessel functions.Let us note at once that the above system is topological in the sense that $d_{\lambda }$ ,like $d$ or $d_{s}$, can be defined purely in terms of differential topology without choosing a metric in $M$ .Now to proceed we need to know the appropriate form of $V.$The system in \[eq:7’\] has also been investigated , in another context , by Baulieu et all [@kn:baulieu] to get informations on topological invariants.Their analysis of topological quantum mechanics on the punctured plane $R^{2}/(0)$ had selected the prepotential $V=k\theta $ which we later generalized .We have shown [@kn:Nadia]that the most general prepotential compatible with the topology of the punctured plane ( first homotopy group $\sim $ $Z$ ) has necessary the form.
$$V(\theta )=k\theta +\phi (\theta ) \label{eq:8}$$
where $\theta $ is the polar angle on the plane , k a constant and $\phi (\theta )$ any function but periodic , (recall that the polar angle $%
\theta $ is not a periodic function ).It is that form \[eq:8\] that we plug into $d_{\lambda \text{ }}$.On the restricted space of functions which depend only on the angle ,the exterior derivative simplifies to $d=d\theta
\partial _{\theta }\ $( there is no r dependance on which d acts ) .Inserting the specific form of the prepotential $V$ into \[eq:7\] and rewriting the twisted operator as $d_{\lambda }=d\theta \partial _{\theta
}^{\lambda }$ we find
$$\partial _{\theta }^{\lambda }=e^{-\lambda \phi }\partial _{\theta
}\,\,e^{\lambda \phi }+\lambda k=\partial _{\theta }+\lambda k+\lambda
\partial _{\theta }\phi$$ Fourier transforming the periodic function $\phi (\theta )$ $$\partial _{\theta }\phi (\theta )=i\sum_{m\epsilon Z/0}\rho _{m}\
e^{-im\theta }$$ we get $$\partial _{\theta }^{\lambda }=\partial _{\theta }+i\lambda \sum_{m\epsilon
Z}\rho _{m}e^{-im\theta }\noindent \label{eq:9}$$ where we inserted the constant $k=i\rho _{0}$ into the sum .In the punctured plane the operator $\partial _{\theta }$ and $\partial _{\theta }^{\lambda }$ have the natural interpretation respectively of the winding number operator and the effective or perturbed winding number operator .We thuswrite them as $W=-i\partial _{\theta }$ and $\,W_{\lambda }=-i\partial _{\theta
}^{\lambda }$.$\,$We also introduce the operator $\Pi _{m}=e^{im\theta }$ with evident action on the basis $\mid n>$ defined in \[eq:4\] . For the operator $W$ and $\Pi _{m}$ we have $W\mid n>=n\mid n>$ and $\Pi _{m}\mid
n>=\mid n+m>$.In this new basis \[eq:9\] takes the form
$$W_{\lambda }=W+\lambda \sum_{m\epsilon Z}\rho _{m}\Pi _{m}$$
This is an example of a very simple topological quantum mechanical system where $W_{\lambda }$ is the perturbed hamiltonian ,$\Pi _{m}$ a set of operators responsible for the interactions, $\lambda \rho _{m}$ a set of coupling constants and $W$ is the unperturbed hamiltonian .The eigenstates of the effective winding number which we denote $\mid n,\lambda $ $,\rho >$ are shown to be related to the unperturbed one ,through the formula [@kn:Najoua] $$\mid n,\lambda \text{ },\rho >=exp\left[ -\lambda \sum_{m\epsilon Z/(0)}%
\frac{\rho _{m}\Pi _{m}}{m}\right] \mid n> \label{eq:10}$$ Hermiticity of $W_{\lambda }$ restricts the real “ spectral ” function $\rho $ to be symmetric $\rho _{m}=\rho _{-m}$ .Comparing with the previous result \[eq:3\] we see that our application of the formalism to Bessel functions requires the simple choice of the function $\rho
_{m}=(-1)^{m}$ .
Relation of $\phi _{n+\lambda }\ $to $d_{\lambda }$
===================================================
To show the relation , first write the generating functions of integer and of real orders
$$\begin{aligned}
\Phi (z,t) &=&\sum_{n=-\infty }^{n=\infty }\phi _{n}(z)\ t^{n} \nonumber \\
t^{-\lambda }\ \Phi (z,t) &=&\sum_{n=-\infty }^{n=\infty }\phi _{n+\lambda
}(z)\ t^{n} \label{eq:11}\end{aligned}$$
We have learned in the particular case of section 2 that eigenstates of $%
d_{\lambda }$ ( $W_{\lambda }\ $) are deformed versions of the eigenstates of $d$ ( $W$ )and that the deformation consists in converting the index $n$ into $n+\lambda $ .We therefore have to look for eigenstates of $d$ and of $%
d_{\lambda }$ .The generating function $\Phi (z,t)$ $\ $is an eigenstate of the exterior derivative by use of the recursion formula ( fixed t ) $$d\ \Phi (z,t)=(-\frac{t}{z})^{-1}\ \Phi (z,t)\ dz$$ For the eigenstate of $d_{\lambda }$ , the function of interest to look at is $e^{-\lambda V}\Phi (z,t)$ .We will show that ,with an appropriate $V$ ,it is indeed an eigenstate of the deformed exterior derivative and in the same time generating function of real order Bessel functions.The judicious choice of the operator $V$ so as to identify this function with the generating function for real orders (remember that the same crucial point of which $V$ to choose arose in the last section) can be shown to be [^1]
$$V=\sum_{m\epsilon Z/0}\frac{\partial _{m}}{m}$$
In fact we have
$$\begin{aligned}
e^{-\lambda V}\Phi &=&\exp \left[ -\lambda \sum_{m\epsilon Z/0}\frac{%
\partial _{m}}{m}\ \right] \Phi \nonumber \\
&=&\exp \left[ -\lambda \sum_{m\epsilon Z/0}\frac{(-1)^{m}(t)^{-m}}{m}%
\right] \Phi \label{eq:12} \\
&=&t^{-\lambda \ }\Phi (z,t) \nonumber\end{aligned}$$
This is the generating function of real order Bessel functions.To come to the second line we applied the recursion formula \[eq:1\] and from the second line to the last we put $t=e^{i\theta \text{ }}$and made use of \[eq:6’\].Then acting by $d_{\lambda }$ on $e^{-\lambda V}\Phi (z,t)$ we find
$$d_{\lambda }(e^{-\lambda V}\Phi )=e^{-\lambda V}d\Phi =(-t)^{-1}dz\
e^{-\lambda V}(z\ \Phi (z,t))=\digamma (z,t,\lambda )\ dz\ e^{-\lambda V}\Phi
\label{eq:13}$$
The function $\digamma (z,t,\lambda )$ is a more involved expression which we will not work out as this is not necessary .Inspection of the third term in \[eq:13\] shows that $e^{-\lambda V}(z\ \Phi )\backsim \ \Phi \backsim
e^{-\lambda V}\Phi \ $where the last proportionality comes from the result in \[eq:12\] .
Thus using the fact that the generating function of integer orders $\Phi
(z,t)$ is an eigenstate of $d$ ,we show that the function $e^{-\lambda
V}\Phi $ is indeed an eigenstate of $d_{\lambda }$ and generating function of real orders .
Expanding both sides of \[eq:12\] as in \[eq:11\] the above relationship extends to Bessel functions themselves as the operator $V$ acts only on the $%
z$ variable.Hence we recover the unifying formula
$$\phi _{n+\lambda }(z)=\exp \left[ -\lambda \sum_{m\epsilon Z/0}\frac{%
\partial _{m}}{m}\right] \phi _{n}(z)$$
The method outlined in this section has the advantage of giving a new check to the unifying formula ,in addition it shades light on the inner structure of Bessel functions showing that the modern concept of deformed or twisted exterior derivative ,first introduced by Witten ( which has been at the origin of the launching of topological field theories ) is already encoded in the index structure of Bessel functions .
> **Acknowledgment**
I would like to personally thank the head of the high energy section at the **Abdus Salam** international centre for theoretical physics ICTP Dr S.Randjbar -Daemi for inviting me to the centre scientific activities at various occasions.
[99]{} **C.TRUESDELL** An essay Toward a Unified Theory of Special Functions ,Ann.of Math Studies **N 18** ,Princeton University Press ,Princeton N.J.,(1948)
**L.INFELD** and** T.HULL** The Factorization method ,Rev.Mod.Phys **23** (1951) pp 21-68
**M.MEKHFI** Unification of Bessel functions of different orders. Int ’ Journal of Theoretical Physics **Vol.39**,No.4,(2000) hep-th/9512159 V2
**M.MEKHFI** Mapping integer order Neumann’s functions to real orders .Submitted to Journal of physics A : Mathematical and general
**G.N.WATSON** A Treatrise On Bessel Functions Cambrige University Press Second Edition
See for instance **V.I.SMIRNOV** A course on Higher Mathematics ,Vol III ,Part 2, Pergamon Press (1964).
**E.WITTEN** Supersymmetry and Morse theory J.Diff.Ge **17** , 661 (1982)
**L.BAULIEU and E.RABINIVICI** Physics Letters **B,316**,93,(1993)
**M.MEKHFI** Cohomological Quantum Mechanics And Calculability Of Observables Mod.Phys.Lett.**A11** :2065-2082 ,(1996)
**M.MEKHFI** Invariants Of Topological Quantum Mechanics. Int ’ Journal of Theoretical Physics **Vol.3**,No.8,(1996)
[^1]: In choosing a ( differential ) operator for $V$ instead of a simple function as in \[eq:7\] we have implicitly generalized the deformed exterior derivative on flat space.This is enough for our purpose .We do not however, know , the expression of the generalized $d_{\lambda }$ in the case of a general manifod $M$.Such expression should be defined so as to be covariant and not to depend on the metric on $M$ like $d=dzD_{z}=dz\partial _{z}.$
|
---
abstract: 'Primordial fluctuations in the relative number densities of particles, or isocurvature perturbations, are generally well constrained by cosmic microwave background (CMB) data. A less probed mode is the compensated isocurvature perturbation (CIP), a fluctuation in the relative number densities of cold dark matter and baryons. In the curvaton model, a subdominant field during inflation later sets the primordial curvature fluctuation $\zeta$. In some curvaton-decay scenarios, the baryon and cold dark matter isocurvature fluctuations nearly cancel, leaving a large CIP correlated with $\zeta$. This correlation can be used to probe these CIPs more sensitively than the uncorrelated CIPs considered in past work, essentially by measuring the squeezed bispectrum of the CMB for triangles whose shortest side is limited by the sound horizon. Here, the sensitivity of existing and future CMB experiments to correlated CIPs is assessed, with an eye towards testing specific curvaton-decay scenarios. The planned CMB Stage 4 experiment could detect the largest CIPs attainable in curvaton scenarios with more than 3$\sigma$ significance. The significance could improve if small-scale CMB polarization foregrounds can be effectively subtracted. As a result, future CMB observations could discriminate between some curvaton-decay scenarios in which baryon number and dark matter are produced during different epochs relative to curvaton decay. Independent of the specific motivation for the origin of a correlated CIP perturbation, cross-correlation of CIP reconstructions with the primary CMB can improve the signal-to-noise ratio of a CIP detection. For fully correlated CIPs the improvement is a factor of $\sim$$2-$3.'
author:
- Chen He
- Daniel Grin
- Wayne Hu
bibliography:
- 'chen\_spires.bib'
title: Compensated isocurvature perturbations in the curvaton model
---
Introduction
============
The measured cosmic microwave background (CMB) anisotropy power spectra [@Hinshaw:2012aka; @Ade:2015lrj] are consistent with adiabatic primordial fluctuations, initial conditions for which the relative [particle number densities]{} are spatially constant. Adiabatic perturbations arise in the simplest inflationary models, where a single field drives inflation and sets the amplitude of perturbations in all species. If fluctuations in particle densities or quantum numbers observed today are actually set by fluctuations in more than one field, some fraction of the primordial fluctuations may be isocurvature (also known as entropy) perturbations, for which there are initial fluctuations in the relative [particle number densities]{}.
The second field may be an axion [@Linde:1984ti] (and thus a dark-matter candidate), a curvaton [@Linde:1984ti; @Linde:1996gt; @Langlois:2000ar; @Lyth:2002my] (a field that is energetically subdominant during inflation but later sets the density fluctuations in standard-model species), or alternatively, inflation itself may be driven by multiple fields with different couplings to standard-model particles [@Linde:1984ti]. Fluctuations very similar to isocurvature fluctuations may also arise in topological-defect models [@Axenides:1983hj; @Brandenberger:1993by; @Langlois:2000ar; @Lalak:2007vi].
Isocurvature fluctuations are defined by the entropy fluctuation $$S_{i\gamma}=\frac{\delta n_{i}}{n_{i}}-\frac{\delta n_{\gamma}}{n_{\gamma}}$$ between a species $i$ and the photons ($\gamma$); where $n_{i}$ denotes the background number density of the species; $\delta n_{i}$ its spatial fluctuation; and $i\in \left\{b,c,\nu,\gamma\right\}$, where $b$ denotes baryons, $c$ denotes cold dark matter (CDM), and $\nu$ denotes neutrinos. These isocurvature modes individually leave an imprint on the temperature and polarization power spectra of the CMB [@Langlois:2000ar] and are highly constrained by current data [@Ade:2015lrj].
There is one joint combination of isocurvature fluctuations that largely escapes constraints. If $$S_{c\gamma} = -\frac{\rho_b}{\rho_c} S_{b\gamma}, \quad S_{\nu\gamma}=0,$$ then the density perturbations carried by the two isocurvature modes cancel in this combination when both the baryons and CDM are nonrelativistic. This is called a compensated isocurvature perturbation (CIP).
At linear order, CIPs only affect observables through the difference in the baryon and CDM pressure and hence on scales comparable to the baryonic Jeans length [@Gordon:2002gv; @Barkana:2005xu; @2007PhRvD..76h3005L; @Gordon:2009wx; @2011JCAP...10..028K]. For the CMB, these scales are deep into the damping tail and the regime of secondary anisotropy dominance, as well as beyond the beam scale of any foreseeable CMB experiment. CIPs thus do not induce an observable effect on the CMB at linear order [@2007PhRvD..76h3005L; @Lewis:2002nc; @Gordon:2002gv].
There are potentially observable signatures on the $21$ cm signature of neutral hydrogen at very high redshifts (in absorption). Sufficiently sensitive measurements for a CIP detection, however, will require a futuristic space-based $21$ cm experiment with a baseline that dwarfs that of ongoing/upcoming 21 cm efforts like MWA/LOFAR/PAPER/SKA by an order of magnitude [@Gordon:2009wx].
On the other hand, since CIPs modulate the photon-baryon and baryon-CDM ratios, they do impact the CMB at higher order. By modulating these quantities in space, CIPs change the two-point correlations between CMB multipole moments in a way that allows their reconstruction [@Grin:2011tf]. This fact was applied to the WMAP 9-year data in Ref. [@Grin:2013uya] to set upper limits on the CIP amplitude independently of their origin. Similar limits follow from measurements of the gas fraction in massive galaxy clusters [@Holder:2009gd]. In these prior works, the CIP was not assumed to be correlated with the dominant adiabatic fluctuation. Here we consider an early-universe mechanism that generates CIPs correlated with adiabatic fluctuations, yielding another detectable signature.
Correlated isocurvature fluctuations arise naturally in the curvaton model, in which the curvaton, a subdominant field during inflation later seeds the observed primordial curvature fluctuations [@Lyth:2002my; @Gupta:2003jc; @Gordon:2002gv; @2009JCAP...11..003E]. As different species and quantum numbers may be generated by, before, or after curvaton decay, there are mismatches in their number densities which lead to isocurvature fluctuations including correlated CIPs.
We assess the sensitivity of CMB anisotropy measurements to correlated CIPs generated in various curvaton-decay scenarios and find that a CMB Stage 4 [@Abazajian:2013oma] experiment could yield a detection of the largest such CIPs with more than 3$\sigma$ significance. The significance could improve to $11\sigma$ if polarized foregrounds and systematics can be modeled sufficiently to make a cosmic-variance limited measurement out to multipoles of $l=4000$. More generally, we find that cross-correlation of CIP reconstructions with the primary CMB can improve the signal-to-noise ratio for detection of fully correlated CIPs by a factor of $\sim$2$-$3 depending on the specific experiment.
We establish that our reconstruction method [@Grin:2011tf] relies on a separate-universe (SU) approximation, limiting its use to angular scales $L\lsim100$. This has little impact for the signal-to-noise ratio of CIP searches using completed CMB experiments, but ultimately limits CIP reconstruction from nearly cosmic-variance limited future CMB polarization experiments. We correct numerical errors in the reconstruction noise curves of Ref. [@Grin:2011tf]; these errors are ultimately negligible on the scales where the SU approximation is valid. We also update CIP estimators to include sample variance from CMB $B$-mode polarization, as well as covariance between CIP estimators based on off-diagonal correlations between different pairs of observables (e.g. temperature, $E$-mode polarization, and $B$-mode polarization).
We begin in Sec. \[sec:curvaton\] by reviewing the predictions for the amplitude of isocurvature perturbations and their correlations with the adiabatic mode in nine curvaton-decay scenarios. In Sec. \[sec:reconstruction\], we examine the tools for CIP reconstruction and compute updated reconstruction noise spectra based on the methods from Ref. [@Grin:2011tf]. In Sec. \[sec:sigma\] we determine the sensitivity of future CMB experiments to curvaton-inspired correlated CIPs. We assess the improvements in signal-to-noise ratio made possible by cross-correlating the CIP reconstruction with CMB temperature and polarization maps. We conclude in Sec. \[sec:conclude\]. In Appendix \[sec:wprop\], we show that our reconstruction methods are limited to CIP modes that are larger than the sound horizon at recombination. In Appendix \[sec:correct\_noise\] we discuss differences with the reconstruction results of Ref. [@Grin:2011tf].
Correlated CIPs in Curvaton Models {#sec:curvaton}
==================================
General considerations
----------------------
The curvaton $\sigma$ is a light spectator scalar field during inflation and starts to oscillate when the Hubble scale $H$ approaches the curvaton mass $m_{\sigma}$ shortly before or after the inflaton $\phi$ decays into radiation $R$. Once the curvaton starts to oscillate, it redshifts like matter and comes to contribute a larger and larger fraction of the energy density, thus generating curvature fluctuations [@Mollerach:1989hu; @Mukhanov:1990me; @Linde:1996gt; @Moroi:2001ct; @Lyth:2001nq; @Moroi:2002rd; @Lyth:2002my].
In general both the curvaton $\sigma$ and inflaton $\phi$ contribute to the curvature fluctuations on constant total density slicing $\zeta$. Depending on how dominant the curvaton is when it decays, as quantified by [r\_D]{}= |\_D, the relative contribution of inflaton and curvaton contributions to the total curvature varies, and is given by =\_= (1-[r\_D]{})\_+ [r\_D]{}\_, where $\zeta_{i}$ is the curvature perturbation on constant density $\rho_i$ slicing or equivalently the energy density perturbation $\delta \rho_i/3(\rho_i+p_i)$ on spatially flat slicing. When applied to particle components, $\zeta_i$ is also the particle number density perturbation on spatially flat slicing. Thus, $i \in \{ \sigma,\phi, b,c,\nu,\gamma \}$.
The curvaton can also generate isocurvature fluctuations [@Mollerach:1989hu; @Linde:1996gt; @Lyth:2001nq; @Lyth:2003ip; @Lemoine:2006sc] $$S_{ij} = 3 (\zeta_i-\zeta_j),$$ depending on how various particle numbers were generated. If they were created before curvaton decay, then they inherit the inflaton’s fluctuations $\zeta_\phi$. If they were generated by curvaton decay, they inherit the curvaton’s fluctuations $\zeta_{\sigma}$. If they were created from the thermal plasma after the curvaton decay, they inherit the total curvature perturbation $\zeta$. In summary [@Lyth:2001nq; @Lyth:2003ip], $$\zeta_{i} =
\left\{
\begin{array}{ll}
\zeta_\phi, & \text{before decay},\\
\zeta_\sigma , & \text{by decay},\\
\zeta, & \text{after decay}.
\end{array}
\right.$$ Once generated, these curvature fluctuations remain constant outside the horizon [@Bardeen:1980kt; @Mukhanov:1990me; @Malik:2004tf].
We are interested, in particular, in the baryon ($b$) and cold dark matter ($c$) isocurvature fluctuations around the time of recombination. We assume that lepton number is not related to curvaton physics, allowing us to neglect neutrino isocurvature perturbations [@Lyth:2001nq; @Lyth:2003ip]. We thus do not distinguish between photons $\gamma$ and the total radiation.
The remaining two isocurvature modes $S_{b\gamma}$ and $S_{c\gamma}$ can be reorganized into a CIP mode and a CDM isocurvature mode, called the effective mode since it now carries all of the nonrelativistic matter isocurvature fluctuations, none of the baryon isocurvature fluctuations and only part of the CDM isocurvature fluctuations.[^1] Specifically, we split each curvature fluctuation as $$\zeta_i = \zeta_i^{{{\rm CIP}}}+ \zeta_i^{{{\rm eff}}} ,$$ where by definition the CIP mode satisfies the compensation conditions $\delta \rho_b^{{\rm CIP}}= - \delta \rho_c^{{\rm CIP}}$ and $\delta\rho_\gamma^{{\rm CIP}}=0$ or equivalently $$\begin{aligned}
\zeta_c^{{{\rm CIP}}} & {\equiv}& - \frac{f_b}{1-f_b} \zeta_b^{{{\rm CIP}}} , \nonumber\\
\zeta_\gamma^{{{\rm CIP}}} & {\equiv}& 0,\end{aligned}$$ and the effective mode carries the adiabatic fluctuations and CDM isocurvature fluctuations but no baryon isocurvature fluctuations $$\zeta_b^{{\rm eff}}\equiv \zeta_\gamma^{{{\rm eff}}} = \zeta.\label{eq:eff_def_impl}$$
Here the baryon fraction is $$\begin{aligned}
f_b = \frac{\rho_b}{\rho_b+\rho_c},
\end{aligned}$$ and we have assumed that the CIP mode is defined by compensation after both the baryons and CDM become nonrelativistic.
We define the entropy perturbation carried by the two modes as $$S_{ij}^{X}=3(\zeta_{i}^{X}-\zeta_{j}^{X}),$$ where $X\in \left\{{{\rm eff}},{\rm CIP}\right\}$. Equation (\[eq:eff\_def\_impl\]) then implies that the effective mode carries only CDM isocurvature fluctuations, as $$\begin{aligned}
S_{b\gamma}^{{\rm eff}}&=& 0.\end{aligned}$$ From these relations, we have $$\begin{aligned}
S_{b\gamma}^{{\rm CIP}}&=& S_{b\gamma} ,\nonumber\\
S_{c\gamma}^{{\rm eff}}&=& S_{c\gamma} + \frac{f_b}{1-f_b} S_{b\gamma}.\end{aligned}$$ Together $S_{c\gamma}^{{\rm eff}}$ and $S_{b\gamma}^{{\rm CIP}}$ give an alternate representation of the isocurvature modes $S_{c\gamma}$ and $S_{b\gamma}$. The benefit of this representation is that because of the compensating baryon and CDM entropy fluctuations, the CIP mode corresponds to zero total isocurvature in nonrelativistic species \[$S_{m\gamma}^{\rm CIP}\equiv f_{b}S_{b\gamma}^{{{\rm CIP}}}+(1-f_{b})S_{c\gamma}^{{{\rm CIP}}}=0$\], and is unmeasurable in linear theory, while carrying all of the modulation of the baryon-photon ratio (since $S_{b\gamma}^{\rm eff}=0$), thus inducing potentially observable changes to CMB anisotropy properties at second order [@Holder:2009gd; @Gordon:2009wx].
Now let us consider the values of $S_{c\gamma}^{{\rm eff}}$ and $S_{b\gamma}^{{\rm CIP}}$ for the nine baryon, CDM isocurvature scenarios, obtained by specifying whether or not the baryon number and CDM are set before, by, or after curvaton decay. We use the notation $(b_{x},c_{y})$, where $x,y\in\left\{{\rm before,~by, ~after}\right\}$, $b$ denotes baryon number, and $c$ denotes CDM.
Two curvaton-decay scenarios are of particular interest to CIPs. For the case when the baryon number is created by curvaton decay, $\zeta_{b} = \zeta_{\sigma}$ and CDM is created before, $\zeta_{c} = \zeta_{\phi}$, $$\begin{aligned}
\frac{S_{c\gamma}^{{\rm eff}}}{\zeta_\sigma-\zeta_\phi}&= 3 \frac{ f_b - {r_D}}{1- f_b} ,
\nonumber\\
\frac{S_{b\gamma}^{{\rm CIP}}}{\zeta_\sigma-\zeta_\phi} &= 3(1-{r_D}), \qquad (b_{{\rm by}}, c_{{\rm before}}),\end{aligned}$$ and for baryon number created before and CDM by the decay, $$\begin{aligned}
\frac{S_{c\gamma}^{{\rm eff}}}{\zeta_\sigma-\zeta_\phi} &= 3\frac{1-f_b-{r_D}}{1-f_b},
\nonumber\\
\frac{S_{b\gamma}^{{\rm CIP}}}{\zeta_\sigma-\zeta_\phi} & = - 3 {r_D},
\qquad (b_{{\rm before}}, c_{{\rm by}}) .\end{aligned}$$ In these two cases, $S_{c\gamma}^{{\rm eff}}$ can be made small by canceling the ${r_D}$ and $f_b$ terms while leaving $S_{b\gamma}^{{\rm CIP}}$ relatively large. The other cases are given in Table \[tab:curvatoncases\].
In all cases, the isocurvature modes are proportional to $\zeta_\sigma-\zeta_\phi$. This implies that the cross-correlation between the curvature and isocurvature modes share a universal correlation amplitude regardless of the curvaton-decay scenario. For example, for the CIP mode $$\begin{aligned}
R &\equiv
\frac{ P_{S_{b\gamma}^{{\rm CIP}}\zeta}}{\sqrt{ P_{\zeta \zeta} P_{S_{b\gamma}^{{\rm CIP}}S_{b\gamma}^{{\rm CIP}}}}} \\
&= \pm
\frac{ (1-{r_D}) P_{\zeta_\phi \zeta_\phi}-{r_D}P_{\zeta_\sigma\zeta_\sigma} }
{\sqrt{
(P_{\zeta_\phi\zeta_\phi}+ P_{\zeta_\sigma \zeta_\sigma}) [ (1-{r_D})^2 P_{\zeta_\phi \zeta_\phi} + {r_D}^2 P_{\zeta_\sigma \zeta_\sigma }]}} ,\nonumber\end{aligned}$$ where $$\begin{aligned}
\langle \zeta_\sigma ^*({{\bm k}}) \zeta_\sigma({{\bm k}}')\rangle =& (2\pi)^3 \delta({{\bm k}}-{{\bm k}}') P_{\zeta_\sigma \zeta_\sigma}(k), \nonumber\\
\langle \zeta_\phi^*({{\bm k}}) \zeta_\phi({{\bm k}}')\rangle = &(2\pi)^3 \delta({{\bm k}}-{{\bm k}}') P_{\zeta_\phi \zeta_\phi}(k), \nonumber\\
\langle {S_{b\gamma}^{{\rm CIP}}}^*({{\bm k}}) \zeta({{\bm k}}')\rangle =& (2\pi)^3 \delta({{\bm k}}-{{\bm k}}') P_{S_{b\gamma}^{{\rm CIP}}\zeta}(k), \end{aligned}$$ and we have used the fact that the curvaton and inflaton fluctuations are uncorrelated, $$P_{\zeta\zeta} = (1-{r_D})^2 P_{\zeta_\phi \zeta_\phi} + {r_D}^2 P_{\zeta_\sigma \zeta_\sigma}.
\label{eqn:Pzetatotal}$$ If either the curvaton or the inflaton dominates the total curvature, the CIP mode is fully correlated ($R=\pm 1$), as is the CDM-isocurvature mode $S_{c\gamma}^{{\rm eff}}$. The sign of the correlation depends on the decay scenario.
In fact, independently of curvaton domination or decay scenario, $S_{b\gamma}^{{\rm CIP}}$ and $S_{c\gamma}^{{\rm eff}}$ are always fully correlated and cannot be considered independently. Whereas individually the CIP mode implies both a photon-baryon fluctuation and a CDM-baryon fluctuation, it can no longer be considered in isolation from the effective CDM isocurvature mode. This can lead to counterintuitive results when considering other representations of the isocurvature modes and determining their joint observational effects.
For example, in the $(b_{{\rm before}},c_{{\rm before}})$ case, both modes are present and in fact set the total $S_{bc}=0$. Obviously, this scenario cannot be tested through a spatial modulation of the baryon-CDM ratio. Nonetheless, the joint set of modes can be described by a CIP mode which carries $S_{bc}^{{{\rm CIP}}}$ and a fully correlated CDM isocurvature mode where $S_{bc}^{{\rm eff}}=-S_{bc}^{{\rm CIP}}$. For the purposes of the tests in this paper, that is the more useful description, since CMB observables depend mainly on the photon-baryon modulation of quantities like the sound speed and damping scale of the plasma. In this representation, both the adiabatic and effective modes propagate in the presence of a CIP-modulated baryon-photon ratio. Similarly, there are cases \[($b_{\rm after},c_{\rm by})$ and ($b_{\rm after},c_{\rm before})$\] where $S_{bc}\neq 0$, but the effective and adiabatic modes do not see a CIP-induced modulation of the baryon-photon ratio. Ultimately (as we see below), the most interesting cases are those where the CIP mode is much larger than the effective mode, due to observational bounds on the latter, making these subtleties largely irrelevant.
[cc|cc|cc]{}
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Baryons & CDM & $\dfrac{S_{c\gamma}^{{\rm eff}}}{\zeta_\sigma-\zeta_\phi}$ & $\dfrac{S_{b\gamma}^{{\rm CIP}}}{\zeta_\sigma-\zeta_\phi}$ & $\dfrac{S_{c\gamma}^{{\rm eff}}}{\zeta}$ & $A=\dfrac{S_{b\gamma}^{{\rm CIP}}}{\zeta}$\
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[**by**]{} & [**before**]{} & $- 3 \dfrac{ {r_D}-f_b}{1- f_b}$ & $3(1-{r_D})$ & $-\dfrac{3}{{r_D}} \dfrac{{r_D}-f_b}{1-f_b}$ & $\dfrac{1-f_b}{f_b} \bigg( 3 + \dfrac{S_{c\gamma}^{{\rm eff}}}{\zeta} \bigg)$\
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[**before**]{} & [**by**]{} & $3\dfrac{1-f_b-{r_D}}{1-f_b}$ & $- 3 {r_D}$ & $ \dfrac{3}{{r_D}} \dfrac{ 1 - f_b - {r_D}}{1-f_b}$ & $ -3$\
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by & after & $3 f_b \dfrac{1-{r_D}}{1-f_b}$ & $3(1-{r_D}) $ & $3\dfrac{ f_b }{{r_D}} \dfrac{1-{r_D}}{1-f_b} $ & $ \left( \dfrac{1}{f_b} -1 \right)\dfrac{S_{c\gamma}^{{\rm eff}}}{\zeta}$\
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after & by & $3 (1-{r_D})$ & $0$ & $3 \left( \dfrac{1}{{r_D}} - 1\right)$ & $0$\
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[*before*]{} & [*after*]{} & $ -3 \dfrac{f_b }{1-f_b}{r_D}$ & $- 3 {r_D}$ & $ -3 \dfrac{f_b }{1-f_b}$ & $-3$\
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[*after*]{} & [*before*]{} & $ -3{r_D}$ & 0 & $-3$ & $0$\
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[*before*]{} & [*before*]{} & $-3 \dfrac{{r_D}}{1-f_b}$ & $- 3 {r_D}$ & $ -3 \dfrac{1}{1-f_b}$ & $-3$\
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by & by &$ 3 \dfrac{1-{r_D}}{1-f_b} $ & $3(1- {r_D})$ & $ \dfrac{3}{r_D} \dfrac{1-r_D}{1-f_b}$ & $(1-f_b) \dfrac{S_{c\gamma}^{{\rm eff}}}{\zeta}$
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\
after & after & $0$ & $0$ & $0$ & $0$
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\
Observational considerations for curvaton domination {#sec:cdom}
----------------------------------------------------
In forthcoming sections, we will consider the limit of fully correlated CIP modes ($R\approx\pm 1$), a case that results if the curvaton completely dominates the total curvature fluctuation $\zeta \approx
{r_D}\zeta_\sigma$. The inflaton fluctuations obey the usual relationship to tensor modes = ,\[eq:inf\_const\] where $r$ is the tensor-to-scalar ratio and $\epsilon$ is the slow-roll parameter from inflation. By comparing with Eq. (\[eqn:Pzetatotal\]), we see that the curvaton contribution to the total curvature $\zeta$ is dominant over the inflaton contribution as long as $ r \ll 16 \epsilon/ (1-{r_D})^2$. Even if gravitational waves are detected near the current upper limit of $r \sim 0.1$, curvaton curvature domination can still be a good approximation for sufficiently large $\epsilon/(1- {r_D})^2$. The remaining inflaton contribution would then cause a small decorrelation of CIP modes which we ignore in the following sections.
Under the assumption that the inflaton fluctuations are negligible, there are tight constraints on the CDM-isocurvature fraction $S_{c\gamma}^{{\rm eff}}/\zeta$ that then limit the CIP amplitude $$A \equiv \frac{S_{b\gamma}^{{\rm CIP}}}{\zeta}$$ in each of the nine scenarios. The two-sided 95% CL constraints from the Planck 2015 temperature and low-$l$ polarization analysis of totally anticorrelated and correlated isocurvature modes with no tensors combine to imply [@Ade:2015lrj] $$-0.080 < \frac{S_{c\gamma}^{{\rm eff}}}{\zeta} < 0.042 \quad (\text{TT+lowP}).$$ The asymmetric errors reflect the fact that there is a mild preference for anticorrelated CDM isocurvature modes in the Planck TT data [@Ade:2013uln]. The standard adiabatic $\Lambda$CDM model predicts power in excess of the observations at low multipole moment which can be canceled by such a mode. This preference would strengthen if existing upper limits to the amplitude of a primordial gravitational wave background are saturated in the future by a primordial $B$-mode detection [@Kawasaki:2014fwa]. The preliminary Planck 2015 high-$l$ polarization, however, disfavors the anticorrelated scenario and leads to the bounds [@Ade:2015lrj] $$-0.028 < \frac{S_{c\gamma}^{{\rm eff}}}{\zeta} < 0.036 \quad (\text{TT,TE,EE+lowP})$$ without tensors.
Predictions for the various scenarios simplify in this curvaton-dominated limit. The largest CIP amplitude is obtained if baryon number is created by the decay and CDM before $$\begin{aligned}
\frac{S_{c\gamma}^{{\rm eff}}}{\zeta} & = -\frac{3}{{r_D}} \frac{{r_D}-f_b}{1-f_b}, \nonumber\\
A & =\frac{1-f_b}{f_b} \left( 3 + \frac{S_{c\gamma}^{{\rm eff}}}{\zeta} \right), \quad (b_{{\rm by}}, c_{{\rm before}}).\end{aligned}$$ Note that the CDM mode can be anticorrelated in this case but only satisfies observational bounds if the decay fraction is tuned to near the baryon fraction ${r_D}\approx f_b$. The observational bound on $S_{c\gamma}^{{\rm eff}}/\zeta$ implies $A$ $\approx$ $3(1$ $-f_b)/f_b$ $\approx 16.5$. The converse case gives $$\begin{aligned}
\frac{S_{c\gamma}^{{\rm eff}}}{\zeta} &= \frac{3}{{r_D}} \frac{ 1 - f_b - {r_D}}{1-f_b} ,
\
\\
A & = -3, \quad (b_{{\rm before}}, c_{{\rm by}})\nonumber\end{aligned}$$ and again allows anticorrelation and can satisfy observational bounds if ${r_D}$ is tuned to $1-f_b$.
Table \[tab:curvatoncases\] lists the other cases. For the $(b_{\rm by},c_{\rm after})$ and $(b_{\rm after},c_{\rm by})$ scenarios, $S_{c\gamma}^{{\rm eff}}/\zeta>0$, and to satisfy observational bounds, the CIP amplitude is either proportionately small [($|A|\sim 10^{-2}$)]{} or vanishing, [respectively]{}. [The $(b_{\rm before},c_{\rm after})$ and $(b_{\rm after},c_{\rm before})$ scenarios cannot satisfy observational bounds on $S_{c\gamma}^{{\rm eff}}/\zeta$ and are hence ruled out. For the simultaneous scenarios, $(b_{\rm before},c_{\rm before})$ cannot satisfy observational bounds, $(b_{\rm by}, c_{\rm by})$ predicts small CIP modes, and $(b_{\rm after},c_{\rm after})$ predicts no isocurvature modes.]{}
In summary, the two cases that produce substantial CIP modes are the [$(b_{\rm by},c_{\rm before})$ and $(b_{\rm before},c_{\rm by})$ scenarios]{}, which predict $A \approx 3 (1-f_b)/f_b \approx 16.5$ and $A= -3$, respectively. Interestingly, these are also the only two scenarios where the CDM isocurvature mode can cancel the excess low multipole power in the Planck TT data.
CIP Reconstruction {#sec:reconstruction}
==================
CIPs leave observable imprints on the CMB. In this section, we review the method for CIP reconstruction introduced in Ref. [@Grin:2011tf] and point out the limitations imposed by its use of a separate-universe approximation, as further illustrated in Appendix \[sec:wprop\]. Reconstruction methods and results are general, and do not depend on whether or not CIPs are generated in a curvaton scenario. Cross-correlations between the CIPs and the adiabatic mode do depend on the model. Though the techniques again do not depend on the level of correlation, for results we assume fully correlated CIPs as appropriate for a curvaton-dominated scenario.
In Sec. \[sec:SU\], we discuss the separate-universe response of off-diagonal short-wavelength CMB two-point correlations to the presence of a long-wavelength CIP mode. This response is calculated by varying background cosmological parameters, as shown in Sec. \[sec:calcresponse\]. Each two-point correlation function represents a noisy measurement of the CIP mode which we combine to form the minimum variance estimator in Sec. \[ssec:mve\]. We discuss its noise properties in Sec. \[sec:recnoise\]. For correlated CIP modes motivated by the curvaton scenarios of Sec. \[sec:curvaton\], the reconstruction can be correlated with the CMB fields themselves to enhance detection, as discussed further in Sec. \[sec:sigma\].
Separate-universe approximation {#sec:SU}
-------------------------------
The CIP mode $S_{b\gamma}^{{\rm CIP}}$ represents a modulation of the baryon-photon ratio that is compensated by CDM so as to cancel its purely gravitational effects. Consequently, it leaves no imprint on the CMB to linear order. At second order, other modes, including the dominant adiabatic mode, propagate on a perturbed background where quantities such as the photon-baryon sound speed and damping scale are spatially modulated. A fixed CIP mode breaks statistical homogeneity and hence statistical isotropy in the CMB, and so the CIP can be reconstructed from the correlations between different CMB temperature and polarization multipoles that it induces.
In Ref. [@Grin:2011tf], an approximation for characterizing these off-diagonal correlations was applied, based on what amounts to a separate-universe approximation [@Wands:2000dp]. Since the CIP mode does not evolve, its impact can be characterized by a change in cosmological parameters, so long as its wavelength is sufficiently large compared with the scale over which the modes propagate. In Appendix \[sec:wprop\], we show that for CMB anisotropy shortly after recombination, this requires the CIP mode to be larger than the sound horizon at that time. This amounts to the limit $L\lsim 100$, where $L$ is the multipole index of the CIP projected onto the surface of last scattering during recombination. Use of the expressions derived here beyond this domain of validity will bias the associated CIP estimators, an issue we discuss further in Sec. \[ssec:mve\] and Appendix \[sec:wprop\].
For modes that satisfy this approximation, we can treat the CIP mode shortly after recombination as a shift in the background $$\begin{aligned}
\Omega_b &\rightarrow \Omega_b ( 1 + \Delta) ,\nonumber\\
\Omega_c & \rightarrow \Omega_c - \Omega_b \Delta,
\label{eqn:separate}\end{aligned}$$ where $$\Delta({\hat{\bm n}}) = S_{b\gamma}^{{\rm CIP}}({{\bm x}}=D_*{\hat{\bm n}}),
\label{eqn:angularCIP}$$ ${\hat{\bm n}}$ is the direction on the sky, and $D_*$ is the distance to the CMB last-scattering surface during recombination. This angular field can be decomposed into multipole moments () = \_[LM]{} \_[LM]{} Y\_[LM]{}(),\[eq:proj\] so that the restriction on the wavelength of the CIP may be considered as a low-pass filter where the effects are characterized out to $L \lesssim 100$. Note that this restriction justifies the use of a single distance in Eq. (\[eqn:angularCIP\]) rather than an average over the finite width of the recombination era.
In linear theory, the impact of background parameters on CMB power spectra $$\langle X^{*}_{l'm'}Z_{lm} \rangle =
\delta_{l l'}\delta_{mm'}C_{l}^{XZ},
\label{eqn:twoptdiag}$$ are characterized by transfer functions $$\begin{aligned}
C_l^{XZ} = \frac{2}{\pi}\int k^{2}dk T_{l}^X(k) T_l^{Z}(k)P_{\zeta\zeta}(k)
\label{eqn:Cl}\end{aligned}$$ that are given by integral solutions to the Einstein-Boltzmann equations of radiative transfer. Here $X$ and $Z$ are any of the CMB temperature and polarization fields $T,E,B$. Given observational bounds on the CDM isocurvature mode, to a good approximation we can set $S_{c\gamma}^{{\rm eff}}=0$ when evaluating the transfer functions in Eq. (\[eqn:Cl\]). In the curvaton model with $S_{b\gamma}^{{\rm CIP}}= A \zeta$, the $\Delta$ field is correlated with the CMB fields through their joint dependence on $\zeta$. For the case where the curvaton dominates $\zeta$ and in a flat cosmology, the cross power-spectrum of $\Delta$ and the CMB fields $C_l^{X\Delta}$ as well as the auto power-spectrum $C_l^{\Delta\Delta}$ are described by Eq. (\[eqn:Cl\]) with $$T_l^{\Delta}(k) =A j_l(k D_*)$$ and can be numerically evaluated in $\textsc{camb}$ [@Lewis:2002nc] using Eq. (\[eqn:Cl\]).
Note that even in the presence of CIP modes, which are themselves statistically isotropic, two-point correlations are characterized by the diagonal form of Eq. (\[eqn:twoptdiag\]) as long as $\langle \ldots \rangle$ is understood to be the ensemble average over realizations of all modes. In fact, in the curvaton model, the ensemble average over realizations of $\zeta$ automatically includes the CIP and adiabatic modes.
Nonetheless, it is useful to artificially separate the two and consider the response of CMB fields to a fixed realization of the CIP mode. In the SU approximation, this fixed CIP mode is treated as simply a change in cosmological parameters from Eq. (\[eqn:separate\]) that varies across the sky [@Grin:2013uya]. This variation modulates the statistical properties of the CMB modes according to the transfer functions. The utility of this split is that it exposes the fact that there are many pairs of CMB multipoles where $l,l' \gg L$ that can be used to estimate the realization of $\Delta_{LM}$ on our sky. We denote an average over CMB modes with the CIP mode fixed as $\langle \ldots \rangle_{{\rm CMB}}$. This average can be thought of as an average over the subset of $\zeta$ modes that are smaller in wavelength than the sound horizon in the presence of fixed longer-wavelength $\zeta$ modes.
\[tab:response\]
------ ---------------------------------------------------------- --------------
$XZ$ $S^{L, XZ}_{l l'}$ $l + l' + L$
$TT$ $(C^{T,dT}_{l'} + C^{T,dT}_{l}) K^{L}_{ll'}$ even
$EE$ $(C^{E,dE}_{l'} + C^{E,dE}_{l}) H^{L}_{ll'}$ even
$EB$ $-i (C^{E,dE}_{l'} + C^{B,dB}_{l})H^{L}_{ll'}$ odd
$TB$ $-i C^{T,dE}_{l'} H^{L}_{ll'}$ odd
$TE$ $(C^{T,dE}_{l'} H^{L}_{ll'} + C^{E,dT}_{l} K^{L}_{ll'}$) even
$BB$ $(C^{B,dB}_{l'} + C^{B,dB}_{l}) H^{L}_{ll'}$ even
------ ---------------------------------------------------------- --------------
: The response function $S^{L, XZ}_{l l'}$ of the various two-point observables in Eq. (\[eq:response\]).
The product of the source and modulation fields in real space leads to a convolution in harmonic space. Hence, it connects CMB multipole moments of different $l,m$ in the same manner as a three-point function, yielding $$\begin{aligned}
\label{eq:response}
\langle X^{*}_{l'm'}Z_{lm} \rangle_{{\rm CMB}}= & \, \delta_{l l'} \delta_{m m'} C_{l}^{XZ}
\notag\\
& + \sum_{LM} \Delta_{LM} \xi^{LM}_{lm, l' m'} S^{L, XZ}_{l l'},
\end{aligned}$$ where $$\begin{aligned}
\xi^{LM}_{l m l' m'}
= & \, (-1)^{m} \sqrt{\frac{(2L+1)(2l+1)(2l'+1)}{4\pi}}
\notag \\
& \times \wigner{l}{-m}{L}{M}{l'}{m'}, \end{aligned}$$ and the response functions $S^{L,XZ}_{ll'}$ are given in Table \[tab:response\] with $$\begin{aligned}
C_l^{X,dZ} = \frac{2}{\pi}\int k^{2}dk T_{l}^X(k) \frac{d T_l^{Z}}{d\Delta}(k)P_{\zeta\zeta}(k)
\label{eqn:Clderiv}\end{aligned}$$ and $$\begin{aligned}
K^{L}_{ll'} \equiv & \wigner{l}{0}{L}{0}{l'}{0}, \nonumber\\
H^{L}_{ll'} \equiv & \wigner{l}{2}{L}{0}{l'}{-2}, \nonumber\end{aligned}$$ which are Wigner 3$j$ coefficients. The response for intrinsic $B$ modes is new to this work and may provide extra information on CIP modes should they be detected in the future.
In the presence of a fixed long-wavelength CIP mode, statistical isotropy of short-wavelength CMB fields is therefore broken. Statistical isotropy is of course restored once the full ensemble average over the random realizations of the CIP mode is taken. Given the correlation of $\Delta$, $\zeta$ and the $T,E$ CMB fields, a full ensemble average induces a three-point correlation in the CMB which correlates long-wavelength modes to short-wavelength power, i.e. a squeezed bispectrum.
This correlation provides a way of detecting the CIP mode from a noisy two-point reconstruction of $\Delta$ as long as the correlation coefficient R\_L\^[X]{} = \[eq:rxdef\], shown in Fig. \[fig:R\], remains large. The sign of the correlation oscillates due to acoustic oscillations in temperature and polarization, whereas the level of correlation depends on the difference in projection effects between the fields.
![\[fig:R\] Correlation coefficients $R_L^{T\Delta}$ and $R_L^{E\Delta}$ between the CIP and the CMB temperature and polarization fields, respectively, as a function of multipole $L$ for $A>0$. The sign of the correlation oscillates due to acoustic oscillations in temperature and polarization, whereas the level of correlation depends on the difference in projection effects between the fields. ](corr_coeff.pdf)
Response of CMB anisotropies to CIP modes {#sec:calcresponse}
-----------------------------------------
Calculating the response functions in Eq. (\[eqn:Clderiv\]) requires varying cosmological parameters to mimic the effect of the CIP within the separate- universe approximation following Ref. [@Grin:2011tf]. While Eq. (\[eqn:separate\]) provides a prescription for the main effect of compensating variations of the background baryon and cold dark matter densities, there are a number of subtleties that arise from the treatment of the CIP mode as an angular field $\Delta({\hat{\bm n}})$ rather than a three-dimensional field that varies along the line of sight. CMB temperature and polarization anisotropies that are generated at reionization (and thus after recombination) break this approximation. For these sources of CMB anisotropies, the use of the SU approximation is limited by the horizon scale at the given time rather than the sound horizon shortly after recombination.
By varying $\Omega_b$ and $\Omega_c$ in the transfer functions, we implicitly include a reionization response to the CIP that depends on what other parameters are held fixed. Even if we assume that the reionization optical depth $\tau$ is held fixed when varying parameters in the transfer function, there is still an effect on the shape of the polarization spectra due to the implied modulation of the redshift of reionization $z_r$. To assess the impact of reionization, we try two other prescriptions that attempt to remove this sensitivity. The first case is to simply adopt a model with no reionization, and the second is to neglect reionization in evaluating Eq. (\[eqn:Clderiv\]) and then restore it using the analytic damping envelope of Ref. [@Hu:1996mn]. All three prescriptions yield similar (at the $10\%$$-$$20\%$ level) results for the sensitivity of the CMB to correlated CIPs in Sec. \[sec:sigma\]. We conclude that that reionization only causes a small ambiguity for the detectability of CIPs. If they are in the future detected and measured precisely, then a more detailed prescription will be required. For simplicity, we adopt here the constant $\tau$ prescription.
Similarly, the CMB fields from shortly after recombination are gravitationally lensed by large-scale structure in the foreground. Gravitational lensing of the CMB also produces off-diagonal two-point correlations in the presence of a fixed large-scale lensing potential. Given differences in the response function, it is in principle possible to disentangle lensing from CIP effects internally to the CMB. Likewise, external delensing of the CMB can help remove the contamination. These topics will need to be addressed in the future but are beyond the scope of the present work. They will degrade somewhat the forecasts for CIP detectability due to loss of degenerate modes. In the following sections, we treat gravitational lensing effects as a source of additional Gaussian noise only and continue to use $C_l$ to denote the unlensed CMB power spectrum.
Throughout this work, we use a flat $\Lambda$CDM cosmology consistent with the 2013 Planck results [@Ade:2013zuv].[^2] Here $\Omega_b = 0.049$ and $\Omega_c = 0.268$, around which we calculate the CIP response, and the adiabatic scalar power spectrum with amplitude $A_s = 2.215\times10^{-9}$, spectral index $n_s = 0.9624$, the reionization optical depth $\tau = 0.0925$, neutrino mass of a single species $m_{\nu}= 0.06$ $\mathrm{eV}$, and Hubble constant $h$ = 0.6711, which we hold fixed. We assume that the tensor modes (parameterized by the tensor-to-scalar ratio $r$) are negligible, and thus that there are no intrinsic $B$ modes.
Minimum-variance CIP estimator {#ssec:mve}
------------------------------
Each pair of CMB fields $ X^{*}_{l'm'}Z_{lm}$ provides an estimate of a CIP mode $\Delta_{LM}$ that satisfies the triangle inequality, $ | l-l'| \le L \le l +l '$ and $M = m-m'$. Any single pair, however, is highly noisy due to the sample variance of the Gaussian random CMB fields and instrument noise. Here, we also include the change in the power spectra due to lensing as an additional noise source. Including all noise sources means that we replace Eq. (\[eqn:twoptdiag\]) with $$\langle X^{*}_{l'm'}Z_{lm} \rangle =
\delta_{l l'}\delta_{mm'}\tilde C_{l}^{XZ},$$ where \[eq:Cl\_def\] \_l\^[XZ]{} &=& C\_l\^[XZ]{} + C\_l\^[XZ,[lens]{}]{} + N\_l\^[XZ]{}, with $\tilde N_l^{XZ}$ as the measurement noise power in the sky maps and $\delta C_l^{XZ,{\rm lens}}$ as the change in the power spectrum due to lensing, which we treat as noise. Furthermore, given a $XZ$ field pairing, the $X^{*}_{l'm'}Z_{lm}$ and $Z^{*}_{l'm'}X_{lm}$ estimators have correlated noise. Likewise, the different field pairings $XZ$ and $X'Z'$ are also correlated. Following the mathematically identical lensing treatment in Ref. [@Okamoto:2003zw] (see also Ref. [@Namikawa:2011cs]), we optimize the weighting of the estimators to minimize the CMB reconstruction noise. Optimal weighting generalizes the familiar inverse-variance weighting to inverse-covariance weighting for covarying estimators. Differences with the results of Ref. [@Grin:2011tf] are discussed in Appendix \[sec:correct\_noise\].
It is convenient to break the inverse-covariance weighting into two steps. Since we want to examine each $XZ$ pair individually, we first consider the $2\times 2$ covariance of its multipole pairing. We can write a general estimator as $$\hat{\Delta}_{LM}^{XZ}=\sum_{lm l'm'} X^{*}_{l'm'}Z_{lm}\xi^{LM}_{lm l'm'}W_{L l l'}^{XZ},\label{eq:est_form}$$ where $\xi^{LM}_{lm l'm'}$ enforces the triangle inequality and $W_{L l l'}^{XZ}$ are the weights to be determined. The variance of an unbiased estimator due to Gaussian CMB noise becomes
$$\begin{aligned}
\left \langle |\hat{\Delta}_{LM}^{XZ}-\Delta_{LM}|^{2}\right \rangle_{\rm CMB}=\sum_{l l'}&G_{l l'}W_{L l l'}^{XZ}\left\{\tilde{C}_{l'}^{XX}\tilde{C}_{l}^{ZZ}W_{L l l'}^{XZ*} +(-1)^{l+l'+L}\tilde{C}_{l'}^{XZ}\tilde{C}_{l}^{XZ}W_{Ll' l}^{XZ*}\right\},\label{eq:variance_general}\end{aligned}$$
where we have used Wick contractions and identities for the Wigner coefficients, and $$G_{l l'}\equiv \frac{(2l+1)(2l'+1)}{4\pi}.$$ Note that the covariance of the multipole permutation gives rise to the second term. We minimize the variance by taking a derivative of Eq. (\[eq:variance\_general\]) with respect to $W_{L l l'}^{XZ}$, imposing the constraint of an unbiased estimator with a Lagrange multiplier $\lambda$, obtaining $$\begin{aligned}
\tilde{C}_{l'}^{XX}\tilde{C}_{l}^{ZZ}W_{Ll l'}^{XZ*}+\left(-1\right)^{l+l'+L}\tilde{C}_{l'}^{XZ}\tilde{C}_{l}^{XZ}W_{Ll' l}^{XZ*}+\lambda S_{l l'}^{L,XZ}=0,\label{eq:con_a} \end{aligned}$$ This provides the relative weights $W_{Ll l'}^{XZ*} \propto g_{Ll l'}^{XZ*}$, where $$\begin{aligned}
g_{L l l'}^{XZ}=\frac{S_{l l'}^{L,XZ*}\tilde{C}_{l}^{XX}\tilde{C}_{l'}^{ZZ}-\left(-1\right)^{l+l'+L}S_{l' l}^{L,XZ*}\tilde{C}_{l}^{XZ}\tilde{C}_{l'}^{XZ}}{\tilde{C}_{l'}^{XX}\tilde{C}_{l}^{ZZ}\tilde{C}_{l}^{XX}\tilde{C}_{l'}^{ZZ}-\left(\tilde{C}_{l}^{XZ}\tilde{C}_{l'}^{XZ}\right)^{2}}.
\label{eq:single_noise}\end{aligned}$$ Note that this takes the form of the inverse $2\times 2$ covariance weight as expected.
The normalization comes from the requirement that the estimator be unbiased so that $$\begin{aligned}
\hat{\Delta}_{LM}^{XZ}&= N_{L}^{XZ} \sum_{lm l'm'}X_{l'm'}^{*}Z_{lm}g_{L l l'}^{XZ}\xi^{LM}_{lm l'm'},\label{eq:single_estimator}\\
\left[N_{L}^{XZ}\right]^{-1}&=\sum_{l l'}G_{l l'}S_{l l'}^{L,XZ}g_{L l l'}^{ XZ}.\label{eq:single_norm}\end{aligned}$$ The normalization factor is also the variance of the estimator itself, $$\left \langle |\hat{\Delta}_{LM}^{XZ}-\Delta_{LM}|^{2} \right \rangle _{\mathrm{CMB}}=N_{L}^{XZ}.\label{eq:var}$$
Next, we can combine the various $\alpha =XZ$ field pairs to find the total minimum-variance estimator by again inverse-covariance weighting the individual estimators \_[LM]{}&=&\_ w\_L\^\_[LM]{}\^,w\^\_L=N\_[L]{}\^\_( \_[L]{}\^[-1]{})\^[,]{}, \^[-1]{} . \[eq:full\_estimator\] The estimator covariance-matrix $\mathcal{M}_{L}$ is at every $L$ a rank-$2$ tensor over observable pairs. The indices $\alpha$ and $\beta$ take values over labels for pairs of observables, that is, $\alpha,\beta\in \left\{TT,EE,TE,BT,BE\right\}$. Using Eq. (\[eq:single\_estimator\]), and identities of Wigner coefficients, we obtain an expression for the matrix elements $\mathcal{M}_{L}^{\alpha,\beta}$:$$\begin{aligned}
\mathcal{M}_{L}^{XZ, X'Z'}=N_{L}^{XZ}N_{L}^{X'Z'}\sum_{l l'}G_{l l'}g_{L l l'}^{XZ}\left[ \tilde{C}_{l'}^{XX'}\tilde{C}_{l}^{ZZ'}g_{L l l'}^{X'Z'*}+\left(-1\right)^{l+l'+L}\tilde{C}_{l'}^{XZ'}\tilde{C}_{l}^{X'Z}g_{L l' l}^{X'Z'*} \right]. \label{eq:covmat_total}\end{aligned}$$ The total-estimator variance is again the normalization factor $N_{L}^{\Delta \Delta}$. It is straightforward to check that Eqs. (\[eq:single\_norm\]) and (\[eq:var\]) can be recovered from Eq. (\[eq:covmat\_total\]) by restriction to a single pair ($X=X'$, $Z=Z'$).
Data FWHM Noise $f_{\rm sky}$
-------------------- ------ --------- ---------------
WMAP V band 21 434 0.65
WMAP W band 13 409 0.65
Planck 143 [GHz]{} 7.1 37, 78 0.65
Planck 217 [GHz]{} 5.0 54, 119 0.65
ACTPol 1.4 8.9 0.097
SPT-3G 1.1 2.5 0.06
CMB-S4 3.0 1.0 0.50
CVL 0.0 0.0 1
: Instrument noise parameters for illustrative experiments [@wmapnoise; @Planck:2013cta; @2010SPIE.7741E..1SN; @sievers; @2013ApJ...765L..32K; @crawford; @Abazajian:2013oma]: full-width half-max (FWHM) of the beam (in arcmin), noise for temperature measurements $\delta_{TT}$ (in $\mu$K$~{\rm arcmin}$) and, sky fraction $f_{\rm sky}$ used for cosmological analysis. For Planck, we indicate temperature and polarization noise separately as described in the text. For reconstruction, we minimum variance weight the V and W bands for WMAP and the 143 and 217 GHz channels for Planck. The CVL case is full sky and has no instrument noise by definition.
\[tab:instrument\_noise\]
![Reconstruction noise curves $N_{L}^{XZ}$ for CMB-S4 obtained from the CIP estimator of Eq. (\[eq:single\_estimator\]) with the indicated pair of observables $XZ$. The shaded region represents $L(L+1)C_{L}^{\Delta \Delta}/(2\pi)$ signals that are excluded by WMAP 9-year data (Ref. [@Grin:2013uya], see Sec. \[sec:sigma\]). This bound comes from limits to the auto-correlation power spectrum $C_{L}^{\Delta \Delta}$ of CIPs and is thus conservative for models with correlated CIPs. Where the curves intersect this bound, the estimator noise and CIP sample variance of the $|A|=808$ model that saturates it are equal. The approximate domain of validity of the SU approximation $L \lesssim 100$ is indicated by the arrow. Unless otherwise specified we assume, the estimators employ CMB multipoles up to $l_{\rm CMB}=2500$ throughout. []{data-label="fig:recon_noise"}](nc2.pdf){width="3.5"}
\[fig:nc\]
Reconstruction noise {#sec:recnoise}
--------------------
For reconstruction noise forecasts $N_L^{XZ}$ from the CMB fields $XZ$, we can use the $\tilde N_l^{XZ}$ CMB noise power specifications (real or projected) of various experiments. We parameterize it as N\_[l]{}\^[XZ]{} &=& \_[XZ]{}\^2 e\^[l(l+1)\_[FWHM]{}\^[2]{}/82]{},\[eq:instrument\_noise\] with $\delta_{XZ}^2$ as the detector noise covariance assumed to be zero if $X \neq Z$, and $\theta_{\rm FWHM}$ as the full-width half-max of an approximately Gaussian beam. Table \[tab:instrument\_noise\] gives the specifications for WMAP [@wmapnoise], [Planck]{} [@Planck:2013cta], ACTPol [@2010SPIE.7741E..1SN; @sievers], SPT-3G [@2013ApJ...765L..32K; @crawford], and CMB Stage 4 (CMB-S4 henceforth) [@Abazajian:2013oma] experiments. The WMAP and [Planck]{} missions have concluded, but because WMAP data have not yet been used to search for correlated CIPs, and because no CIP reconstruction yet exists from [Planck]{} data, we “predict” in those cases as well. For all but the Planck case, we take $\delta_{EE}^2=\delta_{BB}^2 = 2 \delta_{TT}^2$. For Planck, not all HFI bolometers have polarization sensitivity. To forecast temperature noise, we use all $3S + 4P$ and $4S + 4P$ bolometers from the 143 and 217 GHz channels respectively, where $S$ denotes an unpolarized spider-web bolometer and $P$ a polarized bolometer. We also calculate the ideal reconstruction noise for the (zero instrument-noise) hypothetical cosmic-variance-limited (CVL) case as the ultimate limit.
The results for CMB-S4 are shown in Fig. \[fig:recon\_noise\] (see Appendix \[sec:correct\_noise\] for other experiments). For all cases, we generate reconstruction noise curves using a maximum observed CMB multipole index $l_{{\rm CMB}}=2500$. Beyond this point, foregrounds dominate the TT spectrum. To assess the possible effect of CMB foreground subtraction and lower contamination in the polarization spectra, we explore increasing the limit in Sec. \[sec:sigma\]. Also shown in Fig. \[fig:recon\_noise\] is the bound of $|A|<808$ on the CIP signal power converted from Ref. [@Grin:2013uya] for the fiducial cosmology. This bound comes from limits to the auto-correlation power spectrum $C_{L}^{\Delta \Delta}$ of CIPs. It is thus valid but could be further improved if $\Delta$ and the primordial curvature $\zeta$ are correlated, as we discuss further in Sec. \[sec:sigma\].
The CIP estimator used in all these forecasts, Eq. (\[eq:single\_estimator\]), is derived under the SU approximation. Naive application of Eq. (\[eq:response\]) outside its regime of validity can thus significantly bias estimates of the CIP amplitude $\Delta_{LM}$. In the context of the simple toy model in Appendix \[sec:wprop\], we can compute the response exactly and estimate the reduced sensitivity and bias beyond the SU approximation. Using those results, we estimate that the reconstruction noise curves are accurate for $L\lsim 100$. Moreover, the sharply reduced response beyond this point in the toy model implies that we lose little information by simply restricting ourselves to these $L$ values.
The CIP constraints of Ref. [@Grin:2013uya] are unaffected by the limitations of the SU approximation, as the WMAP signal-to-noise ratio (S/N) for CIPs is dominated by scales $L\lsim 10$. Planck will also not run up against the limitations of the SU estimator. On the other hand, future experiments will not be as sensitive to CIPs as naively estimated in the SU approach, as additional information from polarization-based estimators is severely reduced when the $L\lsim 100$ limit is imposed. We discuss further the limitations to the detection of curvaton-generated CIPs in Sec. \[sec:sigma\].
It is instructive to develop some intuition for the shapes of the reconstruction-noise curves in Fig. \[fig:recon\_noise\]. From the geometry of converting $E$ into $B$ alone, we can understand the relative slopes of $N_{L}$ between $B$-based and non-$B$-based estimators of $\Delta_{LM}$. The difference is easiest to see in the flat-sky approximation. In order to generate a $B$ mode from the CIP modulation of an $E$ mode, the modulation must change the direction of the mode relative to the polarization direction. In the squeezed limit where $l,l' \gg L$, the flat-sky correspondence gives Fourier modes where ${{\bm l}}\parallel
{{\bm l}}'$ and so does not generate a $B$ mode. In the full-sky formulas this comes about because to good approximation [@Hu:2000ee] $$\begin{aligned}
H_{l l'}^{L}&\approx & F(l,L,l')
\begin{cases}
\sin (2\phi), & l+l'+L~{\rm odd}, \\
\cos (2\phi), & l+l'+L~{\rm even} ,
\end{cases}\end{aligned}$$ where $\phi$ is the angle between the $l$ and $l'$ sides of a triangle with side lengths $\{ l,L,l' \}$ upon which $F$ also depends. In the squeezed limit, $\sin(2\phi) \propto L$ and $\cos(2\phi) \approx 1$, which explains why $N^{XB}_{L}/N_{L}^{XE}\propto L^{-2}$, as we can see is the case in Fig. \[fig:recon\_noise\]. We also see in Fig. \[fig:recon\_noise\] that for $L \lesssim 100$, the noise spectra of estimators that do not involve $B$ are white. This behavior is to be expected, given that the prominent features in the CMB carry the scale of the acoustic peaks so that the noise for a modulation by a smaller $L$ does not depend on $L$. Finally, $N_L^{TE}$ changes slope at roughly the sound horizon scale $L \approx 100$. This is because of the acoustic phase difference between $T$ and $E$. The two terms in the response $C_l^{T,dE}$ and $C_l^{E,dT}$ are opposite in sign and roughly out of phase by half a period $\Delta l \approx 100$. So for $l$, $l'$ that differ by less than half a period, the response remains small, while for $l$, $l'$ separated by more than a period, the response grows. Note, however, that the $L \gtrsim 100$ required for the latter is the region for which the SU approximation fails, so this enhancement of the response is not of practical value.
Correlated CIP Forecasts {#sec:sigma}
========================
We now forecast constraints on the amplitude $A$ of totally correlated (or anticorrelated) CIP modes for the various experiments in Table \[tab:instrument\_noise\]. We thus restrict ourselves to the curvaton-dominated limit described in Sec. \[sec:cdom\]. We use Fisher information matrix techniques in Sec. \[sec:fisher\], and then discuss the dependence of these results on various aspects of the data and assumptions in Sec. \[sec:dependencies\].
Fisher errors {#sec:fisher}
-------------
The Fisher information matrix $F_{ij}$ forecasts the inverse-covariance matrix of a set of parameters $p_i$, including $A$, on which the auto- and cross-spectra pairs $\alpha=XZ$ of the observed CMB and reconstructed CIP fields, $\{ X,Z \} \in \{T,E,B,\Delta\}$ depend. Under the assumption of Gaussian statistics for these underlying fields, the Fisher information matrix can be approximated as F\_[ij]{} = \_[L\_]{}\^[L\_]{} (2L+1) [f\_[sky]{}]{}\_[,]{} ([C]{}\^[-1]{}\_[L]{} )\^ , \[eq:fisher\] where ${\cal C}_L$ is the covariance matrix for an individual $LM$ mode in the power spectrum estimator \^[XX’,ZZ’]{}\_[L]{} =[ \_L\^[XZ]{} \_[L]{}\^[X’Z’]{} + \_L\^[XZ’]{} \_[L]{}\^[X’Z]{}]{}, and the ${f_{\rm sky}}$ factor roughly accounts for the reduction in the $2L+1$ independent $M$ modes due to the sky cut. Unless otherwise specified, we always employ the reconstruction-noise power spectrum $N{_L^{\Delta\Delta}}$ for a $\Delta$ reconstructed from CMB multipoles up to $l_{\rm{CMB}} = 2500$.
Data $2\sigma_A$
-------- -------------
WMAP 152
Planck 43.3
ACTPol 40.2
SPT-3G 38.2
CMB-S4 10.3
CVL 6.5
: $2\sigma_A$ detection threshold for $A$ given all auto- and cross-spectra (see Sec. \[sec:fisher\] for assumptions). Cross-spectra allow considerable improvement from the current bounds, and CMB-S4 is able to probe the largest prediction for the curvaton model ($A\approx16.5$) at more than $3\sigma$ significance. []{data-label="tab:2sigma"}
In our case, we are interested in the parameter $p_i= A$. Information on $A$ is contained in the auto-spectrum of the reconstruction $C_L^{\Delta\Delta}\propto A^2$, the cross-spectra with the CMB $T$ and $E$ fields $C_L^{T\Delta}$, $C_L^{E\Delta} \propto A$, and in principle $C_L^{BB}$. We choose to neglect the information coming from $BB$, because CIP $B$-mode power will be swamped by the lensing spectrum by a factor of $\sim 10^{2}$ for the WMAP9 allowed model shown in Fig. \[fig:recon\_noise\] [@Grin:2011tf].
Since other cosmological parameters that define the curvature power spectrum $\Delta_\zeta^2$ and matter content are well determined, we quote $$\sigma^2_A \equiv \frac{1}{F_{AA}},$$ which is the forecasted error with all other parameters fixed. We evaluate the derivatives in Eq. (\[eq:fisher\]) at the fiducial $\Lambda$CDM model defined in Sec. \[sec:calcresponse\]. For $A$ we choose the value for which a $2\sigma_A$ detection is possible which avoids problems with defining the Fisher matrix at $A=0$ discussed below. For the multipole ranges we choose $L_{\rm min} ={\rm max}({f_{\rm sky}}^{-1/2},2)$, due to the sky cut, and $L_{\rm max}=100$, due to the breakdown of the SU approximation, unless otherwise specified.
Results for these fiducial choices are given in Table \[tab:2sigma\]. Note that for the [CMB]{}-S4 experiment we find that the curvaton-motivated value of $A\approx16.5$ \[realized in the scenario ($b_{\rm by}, c_{\rm before}$)\] can in principle be detected at more than $3\sigma$ statistical significance. Even for currently available data from WMAP and Planck, the expected limits are substantially stronger than those determined in Ref. [@Grin:2013uya] for WMAP. We shall see that in large part these improvements are due to the addition of cross-spectra for correlated CIP modes.
Forecast dependencies {#sec:dependencies}
---------------------
Now let us examine the dependence of these results for the CIP detection threshold on various aspects of the data and Fisher matrix assumptions: the maximum and minimum CIP multipole $L_{\rm max}$ and $L_{\rm min}$, the maximum CMB multipole used in the CIP reconstruction $l_{\rm CMB}$, the impact of including CIP and CMB cross-correlations, the fiducial CIP amplitude $A$, and the impact of CMB polarization measurements on reconstruction and cross-correlation. In each case, we vary the assumptions one at a time from the fiducial choices in Sec. \[sec:fisher\].
![\[fig:lmax\] 2$\sigma$ detection threshold as a function of the maximum CIP multipole $L_{\rm{max}}$ from all spectra for Planck, ACTPol, SPT-3G, and CMB-S4 as a function of $L_{\rm{max}}$. The shaded band for $L_{\rm max}>100$ represents the limit of the SU approximation which we take in all other results. Other parameters are set to the fiducial choices of Sec. \[sec:fisher\]. ](2sigma_vs_Lmax2.pdf)
We begin with the multipole ranges. Recall that the fiducial CIP maximum multipole $L_{\rm max}=100$ is chosen to correspond to the angular scale across which sound waves have traveled by the end of the recombination era. As discussed in Appendix \[sec:wprop\], longer-wavelength CIP modes or smaller multipoles can be considered as SU variations in cosmological parameters. Since the breakdown of this approximation occurs within a factor of a few of this scale (see Fig. \[fig:bias\]), we show in Fig. \[fig:lmax\] the dependence of the $2\sigma$ detection threshold on $L_{\rm{max}}$.
For the Planck experiment, whose CIP reconstruction is dominated by $TT$ estimators, the dependence on $L_{\rm max}$ is very mild, consistent with the nearly white-noise spectrum of estimators that do not involve $B$ modes shown in Fig. \[fig:recon\_noise\]. For future experiments that have good polarization sensitivity, the noise curves of $BT$ and $BE$ can cross the others near $L=100$. Were it not for the breakdown of the SU approximation, the implied limits on $A$ would thereafter improve substantially, yielding far better constraints with polarization reconstruction and with cross-correlation than without. As discussed below, improvements from cross-correlation depend strongly on the multipole at which the signal is extracted. With the fiducial $L_{\rm max}=100$, polarization reconstruction yields improvements in $N_{L}^{\Delta \Delta}$ of order unity rather than orders of magnitude.
![\[fig:lmin\] 2$\sigma$ detection threshold as a function of the minimum CIP multipole $L_{\rm{min}}$ from all auto- and cross-correlations for Planck, ACTPol, SPT-3G, and CMB-S4. Other parameters are set to the fiducial choices of Sec. \[sec:fisher\]. ](2sigma_vs_Lmin.pdf)
Given that non-$B$-mode based CIP reconstruction has the highest signal-to-noise ratio at the lowest multipoles, it is also interesting to examine the dependence of $2\sigma_A$ on $L_{\rm min}$ with $L_{\rm max}=100$ (see Fig. \[fig:lmin\]). In addition to variations due to the details of the survey geometry, real experiments are limited by systematics, $1/f$ noise, and foreground subtraction that can compromise their ability to probe small multipoles. As expected, the experiments that are the least dependent on $B$ modes are the most affected by $L_{\rm min}$. For Planck, setting $L_{\rm min}=10$ degrades limits by a factor of 1.6, while for CMB-S4, this choice degrades limits by a factor of $\sim1.3$.
![\[fig:lcmbmax\] 2$\sigma$ detection threshold for the CVL case as a function of the maximum CMB multipole $l_{\rm{CMB}}$ used in reconstruction. Other parameters are set to the fiducial choices of Sec. \[sec:fisher\]. ](2sigma_vs_lcmbmax_cvl_linear.pdf)
We have also assumed that CIP reconstruction will be limited to CMB multipoles smaller than $l_{\mathrm{CMB}}=2500$ beyond which the primary anisotropy is severely Silk damped. For $TT$-based reconstruction, foreground contamination will make information in higher multipoles difficult to extract regardless of instrument sensitivity. This is not necessarily the case for polarization [@Crites:2014nma], and so in Fig. \[fig:lcmbmax\], we plot the dependence on $l_{\mathrm{CMB}}$ for an ideal CVL experiment. By $l_{\rm CMB}=4000$, the detection threshold improves to $2\sigma_A\approx 3$. This level of sensitivity would begin testing the second largest predicted amplitude of the curvaton dominated scenarios $A=-3$ \[realized in the scenario $(b_{\rm before}, c_{\rm by})$\] and models with admixtures of inflaton fluctuations. Recall that these two largest cases are also the only ones where the effective CMB isocurvature mode can cancel large-angle $TT$ power.
Next we examine the impact of CIP cross-correlation with the CMB and the choice of the fiducial value of $A$. In the reconstruction noise-dominated regime, cross-correlation helps extract the signal from the noise. In principle, we can evaluate this improvement by comparing Fisher matrix errors utilizing only the auto-spectrum ($\alpha = \Delta\Delta$) in Eq. (\[eq:fisher\]) with the full result. There is one important subtlety of Fisher errors that we must address first. Given that $C_L^{\Delta\Delta} \propto A^2$, it is clear from Eq. (\[eq:fisher\]) that $F_{AA}\propto A^2$ or $\sigma_A \propto A^{-1}$, which diverges as $A\rightarrow 0$. On the other hand, the corresponding limit on $A^2$, $\sigma_{A^2}$ remains finite: $$\sigma_{A^2} = 2 | A | \sigma_A .$$ Of course a finite upper limit on $A^2$ implies a finite limit on $A$ as well, in spite of the Fisher estimate. The Cramer-Rao bound only guarantees that the Fisher estimate gives the best possible errors of an [*unbiased*]{} estimator. When the data only provide an upper limit, such an estimator can be substantially suboptimal. For this reason, we choose for our fiducial signal the point where $A=2\sigma_A$, which represents a low significance detection rather than an upper limit.
To ensure that our Fisher estimates of $\sigma_A$ from auto-correlation are not misleading, consider an alternate definition of the error on $A$ that corresponds to a mapping of the upper limit on $A^2$: $$\begin{aligned}
\label{eq:tilde_sigma}
\tilde\sigma_A &\equiv & { \sqrt{A^2+\sigma_{A^2}}-|A|} \nonumber\\
&=& { \sqrt{A^2+2 |A| \sigma_A}-|A|}.\end{aligned}$$ This quantity remains finite at $A\rightarrow 0$ as expected. In Fig. \[fig:sigma\_vs\_A\], we compare $\sigma_A$ and $\tilde{\sigma}_A$ of the $\Delta\Delta$ auto-spectrum for the [CMB-S4]{} experiment. The two estimates agree well as long as $|A| \gtrsim 2\sigma_A(A)$ (unshaded region). This is the reason we quote our primary results as the value of $A$ at the detection threshold $A=2\sigma_A$.
We also show here the results obtained if *only* cross-spectra ($T\Delta$ and $E\Delta$) are used, for comparison with the total. In the $A\rightarrow 0$ limit, they dominate the Fisher information leading $\sigma_{A}$ to be finite and independent of the fiducial $A$ value in this limit.
Also interesting is the signal-dominated large-$A$ regime where $C_L^{\Delta\Delta} \gg N_L^{\Delta\Delta}$. Although the cross-spectra alone provide worse limits than the auto-spectrum alone, the total is better than what one would expect by summing their independent information content. This is because having the auto- and cross- correlation helps eliminate the sample variance of the Gaussian random curvature fluctuations $\zeta$. Indeed, if the CIP and CMB modes were perfectly correlated, sample variance could be eliminated entirely.
For the S4 experiment, Fig. \[fig:sigma\_vs\_A\] shows that the cross-spectra improve the detection threshold for $A$ by a factor of 2.3. It is interesting to trace this improvement back to the level of correlation between the CIP and CMB modes. By assuming the noise-dominated regime $C_L^{\Delta\Delta} \ll N_L^{\Delta\Delta}$, as appropriate for a first detection, we can approximate the auto-spectra errors as \[eq:sigma\_DD\] \^[-2]{}\_A |\_ \_L f\_[sky]{} ( )\^2 and compare this to the cross-spectrum where $X \in T,E$: \[eq:sigma\_XD\] \^[-2]{}\_A |\_[X]{} \_L ( [2L+1]{} ) f\_[sky]{} (R\_L\^[X]{})\^2, where we recall that $R_L^{X\Delta}$ is the cross-correlation coefficient shown in Fig. \[fig:R\]. As expected, cross-correlation is more important when the reconstruction signal-to-noise ratio $C_L^{\Delta\Delta}/N_L^{\Delta\Delta}$ is smaller.
For a detection threshold $A = 2\sigma_A$, we can estimate this ratio and hence how the improvement scales with experimental assumptions. For the auto-spectrum this [threshold occurs]{} when \~, where $L_\Delta$ is a representative multipole, [roughly the $L$ value by which the Fisher sum accumulates half its total value]{}. For this level of signal, the cross-spectra would give better constraints by a factor of \[eq:improvement\_cross\_auto\] \~L\_ f\_[sky]{}\^[1/2]{} (R\_L\^[X]{})\^2. Given that the correlation coefficient averaged over a sufficiently large range in $L$ is always of order unity, we can now see that the improvement due to adding the cross-correlation depends on very few aspects of the experiment. For an experiment whose CIP reconstruction is dominated by CVL $TT$ measurements like Planck, the typical CIP multipole in the signal is $L \lesssim 10$ and the improvement is limited (1.9 for Planck). For an experiment with good polarization sensitivity and sky coverage, the improvement can be larger due to both the higher $L$ out to which the signal can be detected and the addition of the $E\Delta$ cross-spectrum. In fact, improvements are ultimately limited by the SU approximation $L \lesssim 100$. For the S4 experiment, the improvement of a factor of 2.3 from the cross-spectra comes partially from $E\Delta$. Without $E\Delta$, the 2$\sigma_A$ threshold goes from 10 to 15, and hence polarization plays a significant role in making a $3\sigma$ detection of $A \approx 16.5$ possible.
![\[fig:sigma\_vs\_A\] Fisher error $\sigma_{A}$ vs $A$ for CMB-S4 from combinations of auto- ($\Delta\Delta$) and cross-spectra ($T\Delta$, $E\Delta$). Also shown is $\tilde\sigma_A$, the Fisher error implied from $\sigma_{A^2}$ for auto-spectra from Eq. (\[eq:tilde\_sigma\]). The two auto-spectra analyses agree well for $A> 2\sigma_A$, which defines the regime where we can meaningfully compare various Fisher results. Cross-spectra increasingly dominate the total at low signal-to-noise ratio $A/\sigma_A$ but also improve the total result at high signal-to-noise ratio by reducing sample variance. All other parameters are set according to Sec. \[sec:fisher\].](sigma_vs_A_cmbs42.pdf)
Finally, it is interesting to compare these forecasted constraints with existing CMB constraints to CIPs, which come from $C_{L}^{\Delta\Delta}$ alone and do not apply information from cross-correlations of $T/E$ with $\Delta.$ The latest WMAP 95% C.L. upper limit on a scale-invariant spectrum of CIPs is $L(L+1)C{_L^{\Delta\Delta}}\leq 0.011$ [@Grin:2013uya]. For correlated CIP modes, this corresponds to an upper limit of $A^2 \leq (808)^2$. The Fisher forecast from the auto-spectrum predicts that the $2\sigma$ detection threshold in $A^2$ is $A^2 = 2\sigma_{A^2}
= (469)^2$. Note that in the Fisher approximation a 2$\sigma$ detection in $\sigma_{A^2}=2 A^2$ implies $\sigma_A = 4 A$ (for example, the 2$\sigma$ threshold in $A$ for WMAP is $A = 264$). The Fisher value in this case underestimates the actual errors by a factor of $3.0$ in $A^2$ or $1.7$ in $A$, which should be borne in mind when considering forecasted errors. We have isolated the root of this discrepancy to the difference between the forecast instrument noise obtained from Eq. (\[eq:instrument\_noise\]) and the true WMAP instrument noise at map level.
The Fisher auto-result can also be compared with the signal-to-noise forecast in Ref. [@Grin:2011tf] for a scale-invariant CIP in the WMAP 7-year data release. There, to be maximally conservative, instrument noise was computed assuming a single differencing assembly. The resulting forecast, $S/N = 300 \Delta_{\rm cl}^2$, corresponds to $\sigma_{A^2} = 1.8\times10^5$ for the correlated CIP in the $N{_L^{\Delta\Delta}}\gg C{_L^{\Delta\Delta}}$ regime. Our Fisher forecast $\sigma_{A^2}=~4.2\times10^4$ is a factor of 4.3 lower, mostly due to lower values of $N{_L^{\Delta\Delta}}$ calculated for the full multiple-differencing-assembly, 9-year experiment.
Conclusions {#sec:conclude}
===========
In the curvaton model, quantum fluctuations of a spectator field during inflation seed the primordial curvature perturbation $\zeta$ after inflation and in the process can produce correlated isocurvature fluctuations from its decay. In some curvaton-decay scenarios, the usual adiabatic and total matter isocurvature perturbations are accompanied by relatively unconstrained compensated isocurvature perturbations (CIPs) between baryons and dark matter. In the curvaton model, CIPs are correlated with the adiabatic fluctuations with amplitude given by $S_{b\gamma}^{\rm CIP}=A\zeta$.
The most interesting (and observationally allowed) scenarios are those where baryon number is generated by curvaton decay, while cold dark matter is generated before; or where baryon number is generated before curvaton decay, and cold dark matter is generated directly by curvaton decay. These cases yield CIP amplitudes of $A\approx 16.5$ and $A = -3$, respectively. By modulating the propagation of acoustic waves during the tightly coupled epoch, CIPs induce detectable off-diagonal two-point correlations in the CMB [@Grin:2011tf]. The correlation with the dominant adiabatic mode means that the cross-power spectrum between the CIP estimators and the CMB fields themselves can in principle be used to probe very small values of $A$ where the auto-correlation is too noisy for detection.
In this work, we obtain the expected amplitude of fully correlated CIPs in the different curvaton-decay scenarios relevant to dark matter and baryon number production. The sensitivity to $A$ of seven different CMB experiments and the ideal cosmic-variance-limited (CVL) case was computed using the Fisher information and applying a more refined calculation of CIP reconstruction noise than past work [@Grin:2011tf]. We find that the validity of the implicit separate-universe approximation made in previous work [@Grin:2011tf] requires a cut $L\lesssim100$ on the multipole index of the reconstructed CIP multipole moments $\hat{\Delta}_{LM}$. While this cut does not affect existing limits to CIPs like Ref. [@Grin:2013uya], it is important for predictions of future sensitivity, particularly for precise future CMB polarization experiments.
Large-scale CMB temperature anisotropies are correlated with the large-scale primordial curvature perturbation $\zeta$, and so cross-correlating the reconstructed CIP with CMB temperature maps can improve the detection threshold for $A$ by a factor of $1.7$-$2.7$ depending on the experiment. The smallest values in this range apply for $TT$-dominated experiments such as WMAP or Planck, and we expect that the upper limits to $A$ from Ref. [@Grin:2013uya] would improve by a factor of $\simeq 1.7$ if those CIP maps were cross-correlated with large-scale temperature maps. For a CVL experiment out to multipoles $l<2500$, the improvement by a factor $\sim 2.7$ is largely independent of the instrument details and most sensitive to $L_{\Delta}$, the multipole below which the majority of the S/N comes from. Since the $TT$ estimator noise has a steeper slope in $L$ than the $BT$ estimator, polarization-dominated experiments will naturally have S/N up to a larger $L_{\Delta}$.
The planned CMB-S4 experiment will approach the cosmic-variance limit for polarization. As a result, it could detect the $A\approx16.5$ scenario (the largest value attainable in curvaton CIP scenarios) with more than $3\sigma$ significance. If polarized foregrounds are negligible or can be removed so that CIP reconstruction can be performed with $l_{\rm CMB}\sim4000$ [@Crites:2014nma], the sensitivity to $A$ of a CVL experiment will dramatically improve. In the cosmic-variance limit, this would allow the $A\approx16.5$ scenario to be detected with $\sim 11\sigma$ significance, and possibly test the second largest CIP scenario $|A| = 3$, as well as models with admixtures of inflaton fluctuations.
A detection of fully correlated CIPs could discriminate between the different curvaton-decay scenarios. The largest correlation $A \approx16.5$ arises in the $(b_{\rm by}, c_{\rm before})$ scenario, where the baryon number is created *by* the curvaton decay and the CDM number *before* the decay. This is the last observationally permitted scenario in which dark matter is produced before curvaton decay, as the other cases are already ruled out by the matter isocurvature constraints. A detection of $A\approx16.5$ would provide strong support for the $(b_{\rm by},c_{\rm before})$ curvaton scenario, in which the dark matter must be produced before curvaton decay, pointing us towards novel dark-matter production mechanisms prior to curvaton decay. This would also hint that baryon number generation is connected to the physics of a spectator field during inflation. This case would also predict a level of local non-Gaussianity of $f_{\rm nl}\approx 6$ [@Sasaki:2006kq] that might be used to confirm a measurement from CIPs. Indeed, the Planck temperature ($f_{\rm nl} = 2.5\pm5.7$, 68% C.L.) and preliminary polarization $(f_{\rm nl} = 0.8\pm5.0$, 68% C.L.) constraints are already close to this predicted amplitude [@Ade:2015ava]. If $A\approx 16.5$ is ruled out by either means, we would know that in the curvaton model, dark matter is either directly produced by curvaton decay or (thermally, from the relativistic plasma) after curvaton decay.
Challenges remain for future work, in particular, a precise evaluation of biases in correlated CIP measurements from off-diagonal correlations induced by weak gravitational lensing, and the generalization of the expressions here to models where inflaton and curvaton contributions to $\zeta$ are more comparable. Our work may pave the way for future CMB measurements to uncover the physics of curvaton decay.
C.H. and W.H. were supported by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli, by U.S. Dept. of Energy contract DE-FG02-13ER41958 and NASA ATP NNX15AK22G. DG is funded at the University of Chicago by a National Science Foundation Astronomy and Astrophysics Postdoctoral Fellowship under Award NO. AST-1302856. We thank B. Benson, T. Crawford, A. L. Erickcek, C. Gordon, M. Kamionkowski, M. LoVerde, A. Manzotti, S. Meyer, J. Sievers, K. Sigurdson, and T. L. Smith for useful discussions. We are especially grateful to A. L. Erickcek and T. L. Smith for a thorough reading of the manuscript.
Wave Propagation in an Inhomogeneous Medium {#sec:wprop}
============================================
Adiabatic acoustic waves in the CMB propagate on a background that is spatially modulated by the presence of the CIP mode $S_{b\gamma}^{\mathrm{CIP}}({{\bm x}})$. Here we present a simplified model of this system to test the domain of validity of the separate-universe (SU) approximation introduced in Ref. [@Grin:2011tf].
In this simple model, we consider an acoustic wave in fractional temperature fluctuations $T$ propagating in a medium with temporally constant, but spatially modulated sound speed $$\ddot T - c_s^2 \left[1 + \frac{d \ln c_s^2}{d \Delta}\Delta({{{\bm x}}}) \right]\nabla^2 T = 0.
\label{eqn:wave}$$ Note that the qualitative difference between this model and the SU model used in Eq. (\[eqn:Clderiv\]) is that the Taylor approximation [for $c_{s}^{2}(\mathbf{x})$ is employed for the medium itself, rather than for the observables after the acoustic mode has propagated through the medium.]{} Since the medium is only weakly inhomogeneous, we can solve the wave equation by iteration. By expanding $$T = T_0 + T_1 + \cdots,$$ Eq. (\[eqn:wave\]) becomes $$\begin{aligned}
\ddot T_0 - c_s^2 \nabla^2 T_0 &= 0, \nonumber\\
\ddot T_1 - c_s^2 \nabla^2T_1 &= c_s^2 \frac{d \ln c_s^2}{d \Delta} \Delta({{\bm x}}) \nabla^2 T_0,
\end{aligned}$$ where overdots are derivatives with respect to the time variable $\eta$. For the zeroth-order solution, we take $T_0 = -\zeta/5$, $\dot T_0=0$ as the initial condition and solve for the Fourier modes $$T_0({{\bm k}},s) =- \frac{1}{5} \zeta({{\bm k}}) \cos(k s),$$ where the sound horizon is $$s(\eta) = c_s \eta .$$
Given the solutions to the homogeneous equation, the solution for $T_1$ is $$\begin{aligned}
T_1({{\bm k}}, s) = & \frac{1}{5} \frac{d \ln c_s^2}{d \Delta} \int \frac{d^3 k'}{(2\pi)^3}
\Delta({{\bm k}}-{{\bm k}}') \zeta({{\bm k}}')
\\ & \times \frac{k'^2}{k^2 - k'^2} \left[
\cos(k' s) - \cos(k s)\right]. \notag
\end{aligned}$$ Note that $T_1$ responds as an oscillator subject to an external force given by the modulation and the unmodulated solution. In particular when $k'=k$ the oscillator is driven at its natural frequency leading to an enhanced response.
The two-point correlations with fixed modulating mode can likewise be expanded to first order in the modulation $$\begin{aligned}
\langle T({{\bm k}})T( {{\bm k}}') \rangle_T \approx & \langle T_0({{\bm k}})T_0( {{\bm k}}') \rangle_T +
R(k,k') \Delta({{{\bm K}}}),
\label{eqn:toyoffdiag}\end{aligned}$$ where ${{\bm K}}= {{\bm k}}+ {{\bm k}}'$. The unmodulated piece is given by $$\langle T_0({{\bm k}})T_0( {{\bm k}}') \rangle_T = (2\pi)^3 \delta({{\bm k}}+ {{\bm k}}')\cos^2(k s) \frac{P_{\zeta\zeta}(k)}{25},$$ and the modulation response function is given by $$\begin{aligned}
\label{eq:R}
R(k,k') =& - \frac{d \ln c_s^2}{d \Delta}\frac{ P_{\zeta\zeta}(k')}{25} \frac{k'^2}{k^2 - k'^2} \cos(k's)
\\ & \times \left [\cos(k' s) - \cos(k s)\right] + {\rm perm.},\notag
\end{aligned}$$ where the permutation refers to $k \leftrightarrow k'$. This response should be compared with the SU approximation, where the Taylor expansion is performed on the solution rather than the medium $$T^{\rm SU}({{\bm x}},s) =T_0({{\bm x}},s) + \frac{d T_0}{d \Delta}({{\bm x}},s) \Delta({{\bm x}}),$$ and gives off-diagonal correlations of the form of Eq. (\[eqn:toyoffdiag\]) but with the SU response function $$\begin{aligned}
R^{\rm SU}(k,k') &= &\frac{P_{\zeta\zeta}(k)}{25} \cos(k s)\frac{d\cos (ks) }{d \Delta}
+ {\rm perm.}\\
\label{eqn:Rsu}
&=&- \frac{d \ln c_s^2}{d \Delta} \frac{P_{\zeta\zeta}(k)}{25} \frac{ks}{4} \sin(2ks) + {\rm perm.}\nonumber\end{aligned}$$ Note that $-\cos(k s)/5$ plays the role of the transfer function.
A comparison shows that the two are only equal in the limit $k' \rightarrow k$. To keep track of the differences, let us define (k’-k)s and $x= k s$. We can rewrite the response \[Eq. (\[eq:R\])\] as $$\begin{aligned}
\label{eq:Reps}
R(k,k') =& \frac{d \ln c_s^2}{d \Delta}\frac{ P_{\zeta\zeta}(k')}{25} \frac{(x+\epsilon)^2}{x^2 - (x+\epsilon)^2} \sin(\epsilon/2)
\\ & \times[\sin(2x +3\epsilon/2) - \sin(\epsilon/2)]+ {\rm perm}.\notag
\end{aligned}$$ It is clear that the validity of the SU approximation requires $|\epsilon|\ll 1$. It is not sufficient for [the]{} wavelength of the acoustic mode to be much smaller than the modulating mode ($k/K \gg 1$). The criterion for a coherent driving of the oscillator is that the phase error introduced by $\epsilon$ itself be small.
Now, let us consider the relevant case for reconstruction where $k \gg K$ or $x \gg \epsilon$ and there are many pairs of acoustic modes (satisfying $|\mathbf{k}|\simeq |\mathbf{k}'|\simeq k$) that can be used to measure the modulation. If $P_{\zeta\zeta}$ is a featureless power law, the response can be simplified considerably, yielding $$\label{eq:Rlim}
R(k,k') \approx - \frac{d \ln c_s^2}{d \Delta} \frac{P_{\zeta\zeta}(k)}{25} \frac{ks}{2} \frac{\sin\epsilon}{\epsilon} {\sin(2ks+\epsilon) } ,$$ which leads to both a damping of the response and a decoherence in the phase for $|\epsilon| > 1$.
Next, consider the impact of the reduced response on the estimator of the modulation mode. In the SU approximation, we can use Eq. (\[eqn:toyoffdiag\]) to obtain the minimum-variance CIP estimator:$$\begin{aligned}
\hat \Delta({{{\bm K}}})&=&N_{K}^{\rm SU}\int \frac{ d^{3} k}{(2\pi)^3} \frac{T({{{\bm k}}})T({{{\bm k}}'})R^{\rm SU}(k,k')}{\tilde P_{TT}(k)\tilde P_{TT}(k')},\nonumber\\
\left(N_{K}^{\rm SU}\right)^{-1}&=& \int \frac{ d^{3} k}{(2\pi)^3} \frac{\left[R^{\rm SU}(k,k')\right]^{2}}{\tilde P_{TT}(k)\tilde P_{TT}(k')}, \label{eq:toy_estimator}\end{aligned}$$ where ${{\bm k}}+ {{\bm k}}' = {{\bm K}}$ and $$\tilde P_{TT}(k) =\cos^2(ks) \frac{P_{\zeta\zeta}(k)}{25} + N_{TT}(k),$$ with $N_{TT}$ as the noise power spectrum from measurement errors. Beyond the SU approximation, this estimator is biased: $$\begin{aligned}
b_{K}(Kr_{s})&\equiv &\frac{ \langle \hat{\Delta}{({{\bm K}})} \rangle_T}{\Delta({{\bm K}})}\nonumber\\&=&N_{K}^{\rm SU}\int \frac{ d^{3} k}{(2\pi)^3} \frac{R(k,k')R^{\rm SU}(k,k')}{\tilde P_{TT}(k)\tilde P_{TT}(k')}.\label{eq:bexact}\end{aligned}$$
To estimate this bias, we can first determine $\epsilon$ for each pair of modes that satisfies ${{\bm k}}+ {{\bm k}}' = {{\bm K}}$. Defining ${{\bm k}}\cdot {{{\bm K}}} = \mu k K$ and assuming $k \gg K$, we have $$\epsilon \approx - K s \mu.$$ Note that $\epsilon$ does not depend on $k$, but only on the angle of ${{\bm k}}$ with ${{\bm K}}$.
We can approximate the minimum variance estimator in Eq. (\[eq:toy\_estimator\]) by ignoring variations in the weights due to the $k$ dependence of $\tilde P_{TT}(k)$, in particular due to the unphysical zeros in power which would be filled in by the Doppler effect and projections in a real observable. Given the difference of the true- and separate-universe responses, this estimator would be biased as $$\begin{aligned}
b_K &\approx& \frac{ \langle \int_{-1}^{1} d\mu
\frac{\sin \epsilon}{\epsilon} {\sin (2x + \epsilon)}[\sin(2 x) + \sin(2 x + 2\epsilon)]
\rangle_x}
{ \langle \frac{1}{2} \int_{-1}^{1} d\mu [\sin(2 x) + \sin(2 x + 2\epsilon)]^2
\rangle_x }
\nonumber\\
&=& \frac{2 {\rm Si}( 2 K s) }{2 Ks + \sin(2 K s)},\label{eq:bk}\end{aligned}$$ where $\langle ...\rangle_x$ denotes an average of a cycle of the oscillation in $x$.
Near $K s \approx 1$, the estimator becomes slightly positively biased, $b_K>1$, but quickly falls to $b_K \ll 1$ for larger values. Thus, the estimator essentially low-pass filters the modulation field, allowing through modes that are larger than the sound horizon. We show the behavior of $b_K$ in Fig. \[fig:bias\] (left), calculated using both Eq. (\[eq:bexact\]) (shown as a range for different noise models, and variety of $k_{\rm max}/K \gg 1$) and the analytic approximation Eq. (\[eq:bk\]). The former is nearly independent of assumptions and agrees very well with the latter except for a small range around $Ks \sim 3$. Also shown is the result of evaluating Eq. (\[eq:bexact\]), with $N_{TT}=0$ for definiteness, but with the same cycle-averaged assumption $\tilde P_{TT}(k)\to \langle \tilde P_{TT}(k) \rangle_{x}$ that was made in the analytic approximation. The agreement with the analytic approximation shows that the overshoot around $Ks \sim 3$ is due to this average. With this average, the unphysical zero crossings of the sound wave are eliminated in both Eqs. (\[eq:bexact\]) and (\[eq:bk\]), and in that sense is closer to a physical model than the full calculation of Eq. (\[eq:bexact\]). In any case, the analytic model captures the main feature of the bias which is a sharp cutoff in sensitivity for $Ks \gg 1$.
![Reconstruction bias in $K$ (left) and in $L$ (right) space. Left: Analytic expression of Eq. (\[eq:bk\]) is in excellent agreement with the evaluation of Eq. (\[eq:bexact\]) using the same cycle-averaged replacement assumption $\tilde P_{TT}(k)\to \langle \tilde P_{TT}(k) \rangle_{x}$. The gray range shows the full calculation for a wide family of Poisson noise power spectra and $k_{\rm max}$. Right: Projected bias using the analytic expression from Eq. (\[eq:toy2dbias\]) with $\theta_*=0.01$, showing that $L<100$ is nearly unbiased.[]{data-label="fig:bias"}](bias.pdf){width="3.4"}
For angular projections of the acoustic waves and the modulation mode, this reduced response and the corresponding bias is somewhat larger. In the flat-sky approximation, the angular modulation mode is related to the spatial one at an epoch $s$ as $$\Delta({{\bm L}}) = \frac{1}{D^2} \int \frac{ d K_\parallel}{2\pi}\Delta({{\bm K}}_\perp= {{\bm L}}/D, K_\parallel)
e^{i k_\parallel D},$$ where $D$ is the distance to the observed surface, and $\parallel$ and $\perp$ denote directions with respect to the line of sight. Defining $T({{\bm l}})$ analogously, we again have off-diagonal correlations of angular moments $$\begin{aligned}
\langle T({{\bm l}}) T({{\bm l}}') \rangle &=& \langle T_0({{\bm l}}) T_0({{\bm l}}') \rangle_T
+\frac{1}{D^4} \int \frac{ d k_\parallel}{2\pi} \int \frac{ d k_\parallel'}{2\pi} e^{i K_\parallel D}
\nonumber\\
&&\times R(k,k') \Delta({{\bm L}}/D,K_\parallel),
\label{eqn:angular}\end{aligned}$$ where ${{\bm L}}={{\bm l}}+ {{\bm l}}'$ and $$\begin{aligned}
\langle T_0({{\bm l}}) T_0( {{\bm l}}') \rangle_T &= &(2\pi)^2 \delta({{\bm l}}+ {{\bm l}}')
\int \dfrac{ dk_\parallel}{2\pi} \cos^2 (k s) \nonumber\\
&&\times \dfrac{ P_{\zeta\zeta}(l/D,k_\parallel)}{25 D^2}.
\label{eqn:T0angular}\end{aligned}$$ Note that here we take the angular observable as the value of the local temperature field on the surface. For the real case of CMB temperature anisotropy, the Doppler effect also contributes and suffers even greater projection effects.
In the separate-universe approximation for $R$ in Eq. (\[eqn:Rsu\]), Eq. (\[eqn:angular\]) reduces to $$\langle T({{\bm l}}) T({{\bm l}}') \rangle_T = \langle T_0({{\bm l}}) T_0({{\bm l}}') \rangle_T + R^{\rm SU}(l,l') \Delta({{\bm L}}).
\label{eqn:angularsu}$$ The angular response function $$\begin{aligned}
R^{\rm SU} (l,l') &=& - \frac{d \ln c_s^2}{d \Delta} \int \frac{d k_\parallel}{2\pi}\frac{ks}{4} \sin(2ks) \frac{P_{\zeta\zeta}(l/D,k_\parallel)}{25 D^2}
\nonumber\\
&&+ {\rm perm.}\
\end{aligned}$$ is again the derivative of the transfer function implied by Eq. (\[eqn:T0angular\]). Likewise, the SU approximation holds for triangles in the integrals where $|\epsilon | \ll 1$. Due to the projection, however, even if the projected 2D triangles in the transverse plane satisfy the analogous criteria, the 3D triangles that compose the estimators may not, since $K \ge K_\perp$.
In the exact expression of Eq. (\[eqn:angular\]), the off-diagonal angular multipole pairs are no longer simply proportional to the projected modulation mode $\Delta({{\bm L}})$. Thus, it is not possible to evaluate the bias in the reconstruction itself. Instead, the bias appears in the auto- and cross-correlations of the reconstructed mode. Since the expressions are cumbersome, we instead estimate projection effects by calculating the bias that would result if the auto-correlation were constructed by the projection of biased estimators of $\Delta({{\bm K}})$.
For a nearly scale-invariant spectrum [($K^3 P_{\Delta\Delta} \approx $ const.)]{}, $$\begin{aligned}
C_{L}^{\Delta\Delta} &=& \frac{1}{D^2}\int \frac{d K_\parallel}{2\pi} P^{\Delta\Delta}(
L/D,K_\parallel) \nonumber\\
&\approx& \frac{2\pi}{D^2} \frac{ K^3 P^{\Delta\Delta}
}{2\pi^2} \int_{K_{\perp}}^{\infty} \frac {d K}{K^2 \sqrt{K^2 - K_\perp^2}}
\nonumber\\
&=& \frac{2\pi}{L^2 } \frac{ K^3 P^{\Delta\Delta}
}{2\pi^2} .\end{aligned}$$ Thus, the angular power spectrum gets contributions from modes with $y=K/K_\perp>1$ as $$\frac{d \ln C_L^{\Delta\Delta}}{d y} = p(y) \approx \frac{1}{y^2 \sqrt{y^2-1}}.$$
We therefore estimate projection effects by weighting $b_K^2$ accordingly: $$b_L^{2}(L\theta_*)= \int_1^\infty dy\, p(y) b_K^2( Ks= y L \theta_*) .\label{eq:toy2dbias}$$ Here $\theta_* = s/D$ is the projected acoustic scale. In the $\Lambda$CDM cosmology, $\theta_* \approx 0.01$, and so $L\theta_*=1$ for $L=100$. The resulting projected bias is shown in Fig. \[fig:bias\] (right). Clearly the bump near $L\theta_{*}=1$ arises from the bump in the three-dimensional bias plot, Fig. \[fig:bias\] (left). For this toy model of CMB acoustic waves, we see that $L=100$ is a scale at which to safely truncate all estimators of $\Delta$. Since this is only a toy model, we expect that this estimate is only accurate to order unity corrections and explore sensitivity to variations in $L_{\rm max}$ in the main text.
Improved reconstruction noise curves {#sec:correct_noise}
====================================
The CIP reconstruction methods introduced in Ref. [@Grin:2011tf] are valid as long as $L\lesssim 100$, as discussed in Appendix \[sec:wprop\]. Aside from changes to precise instrumental noise properties and best-fit cosmological parameters, the curves shown for $N_{L}^{TT}$ and $N_{L}^{EE}$ in Ref. [@Grin:2011tf] are thus valid for all $L\lesssim 100$. We find, however, that numerical errors were made in Ref. [@Grin:2011tf], [affecting the shape of $N{{_L^{\Delta\Delta}}}$ at scales $L\gsim 100$; the limit to CIP reconstruction ($L\lesssim 100$) imposed by the SU approximation, however, means that these errors are of no practical significance, and have no bearing on the validity of existing CMB limits to CIPs [@Grin:2013uya].]{}
Due to an erroneous index in the code employed there, however, a swap took place between the indices $l$ and $l'$ when evaluating Eqs. (\[eq:single\_noise\]) and (\[eq:single\_norm\]) for $N_{L}^{TB}$ and $N_{L}^{EB}$. For the last two experimental cases considered there \[the nearly cosmic-variance-limited (CVL) EPIC mission concept and the actual cosmic-variance limit\], the analytic damping envelope reionization prescription of Ref. [@Hu:1996mn], was employed, rather than the constant-$\tau$ prescription employed elsewhere in Ref. [@Grin:2011tf]. For $L\lesssim 100$, we find that together these two errors lead to errors of $\Delta N_{L}^{XB}/N_{L}^{XB}\lesssim 10^{-2}$ when $X\in\left\{T,E\right\}$ for *all* experiments considered in Ref. [@Grin:2011tf], and so these errors are negligible on scales where the SU approximation is valid.
The curves shown there for $N_{L}^{XY}$ were actually numerically obtained using inverse-variance weighting of different multipole pairs, instead of the correct inverse-covariance weighting. Inverse-variance weights are correct (and agree with inverse-covariance weights) for reconstruction noise from the pairs $TT$, $EE$, $TB$, and $EB$, for which the observable members of a pair are either totally correlated or uncorrelated in the absence of an isotropy-breaking realization of the CIP field. For $TE$, however, the neglect of covariance between $T$ and $E$ leads to incorrect behavior. Additionally, denominators in expressions for $\left(N_{L}^{TE}\right)^{-1}$ were evaluated as if $l=l'$ even when this was not the case. For $L\lesssim 100$, we find that this leads to errors of $\Delta N_{L}^{TE}/N_{L}^{TE}\lesssim 10^{-1}$ for *all* experiments considered in Ref. [@Grin:2011tf], and so these errors are negligible on scales where the SU approximation is valid.
Calculations in Ref. [@Grin:2011tf] were sped up by using permutation symmetries inside sums to simplify evaluations of $N_{L}^{XY}$. For $N_{L}^{TB}$, $N_{L}^{TE}$, and $N_{L}^{EB}$, summands were erroneously multiplied by a factor of $1/2$ when $l\geq |L-l'|$, but we find this leads to errors of $\Delta N_{L}^{TE}/N_{L}^{TE}\lesssim 2 \times 10^{-1}$ when $L<10$ and $\Delta N_{L}^{TE}/N_{L}^{TE}\lesssim 5 \times 10^{-2}$ when $10<L<1000$, and so these errors are small on scales where the SU approximation is valid.
For clarity and future reference, correct reconstruction noise curves $N_{L}^{XY}$ are shown in [Fig. \[fig:nc3\] for the Planck, ACTPol, and SPT-3G experiments, as well as for the CVL case. Reconstruction noise curves for CMB-S4 are shown earlier in this paper, in Fig. \[fig:nc\].]{} Total reconstruction noise curves are obtained using the full inverse-covariance weighting of different estimators \[See Eqs. (\[eq:full\_estimator\]) and (\[eq:covmat\_total\])\] rather than the inverse-variance weighted sum used in Ref. [@Grin:2011tf].[^3]
Using these reconstruction noise curves, we evaluate the signal-to-noise ratio for a detection of a scale-invariant spectrum of CIPs (with damping from projection as in Ref. [@Grin:2011tf] for scales below the thickness of the recombination era) using only $\Delta \Delta$ auto-correlations, analogously to Ref. [@Grin:2011tf], but using Planck 2013 parameter values [@Ade:2013uln]. Results are shown in Tables \[tab:autotable\_noproject\] and \[tab:autotable\_project\] for a variety of experiments and for the CVL case, with and without the SU domain of validity imposed, and with/without damping from projection. We now discuss the results qualitatively.
[ccccc]{} Data & $\mathcal{C}$ (SU limit, improved $N_{L}^{\Delta \Delta}$)& $\mathcal{C}$ (SU limit, old $N_{L}^{\Delta \Delta}$)& $\mathcal{C}$ (no SU limit, improved $N_{L}^{\Delta \Delta}$)& $\mathcal{C}$ (no SU limit, old $N_{L}^{\Delta \Delta}$)\
Planck & 8.73 & 8.58 & 8.73 & 8.58\
ACTPol & 13.8 & 13.6 & 13.8&13.9\
SPT-3G & 19.2 & 19.1 &19.6&23.6\
CMB-S4 & 115 & 104 &117&128\
CVL & 171 & 155&193&226
When erroneous reconstruction noise curves are used (with regard to the errors enumerated above), the spurious improvement obtained by neglecting the SU approximation can be as high as $\sim 50\%$ (in the CVL case). If, on the other hand, the improved reconstruction noise curves are used, the spurious improvement falls to $5\%$ for the CVL case. Put another way, if signal-to-noise ratio is evaluated including only modes for which the SU approximation is valid, the difference between the old and new noise curve codes is negligible.
[ccccc]{} Data & $\mathcal{C}$ (SU limit, improved $N_{L}^{\Delta \Delta}$)& $\mathcal{C}$ (SU limit, old $N_{L}^{\Delta \Delta}$)& $\mathcal{C}$ (no SU limit, improved $N_{L}^{\Delta \Delta}$)& $\mathcal{C}$ (no SU limit, old $N_{L}^{\Delta \Delta}$)\
Planck & 9.17 & 9.00 & 9.17 & 9.01\
ACTPol & 14.1 & 13.9 & 14.1&14.8\
SPT-3G & 19.7 & 19.6 &20.1&32.7\
CMB-S4 & 120 & 110 &123&174\
CVL & 179 & 162&213&321
If CIP projection damping is neglected (that is, a scale-invariant power spectrum of CIPs is assumed on all scales), the spurious improvement obtained by neglecting the SU approximation can be as high as $100\%$ (in the CVL) if the erroneous reconstruction noise curves are used. If, on the other hand, the improved reconstruction noise curves are used, the spurious improvement falls to $\sim 20\%$ for the CVL case. In all cases, the spurious improvement in signal-to-noise (for high $L_{\rm max}$) is caused by the polarization-driven flattening in the reconstruction noise curves at high $L$.
We recommend using the reconstruction noise curves and tables here for future CIP forecasts in the auto-only case. The differences between these different scenarios are much more dramatic if information from cross-correlations between $\Delta$ and $T$ or $E$ is included, as is the case in the rest of the paper.
[^1]: This split between CIPs and an effective mode was introduced in Ref. [@Gordon:2002gv] with the opposite convention of an effective baryon isocurvature, rather than the CDM isocurvature mode. The latter has since become standard (e.g. [@Ade:2015lrj]).
[^2]: Specifically, we use baseline model 2.13 from [Grid\_limit68](http://wiki.cosmos.esa.int/planckpla/images/9/9b/Grid_limit68.pdf).
[^3]: In practice, this distinction only matters for $L$ ranges where the identity of the lowest-noise individual CIP estimator (e.g. $TT$, $EE$, $TE$, $TB$, or $EB$) is transitioning from one observable pair to another.
|
---
abstract: 'The quality factors of modes in nearly identical GaAs and Al$_{0.18}$Ga$_{0.82}$As microdisks are tracked over three wavelength ranges centered at 980nm, 1460nm, and 1600nm, with quality factors measured as high as 6.62$\times$10$^5$ in the 1600-nm band. After accounting for surface scattering, the remaining loss is due to sub-bandgap absorption in the bulk and on the surfaces. We observe the absorption is, on average, 80percent greater in AlGaAs than in GaAs and in both materials is 540percent higher at 980nm than at 1600nm.'
author:
- 'C. P. Michael'
- 'K. Srinivasan'
- 'T. J. Johnson'
- 'O. Painter'
- 'K. H. Lee'
- 'K. Hennessy'
- 'H. Kim'
- 'E. Hu'
title: 'Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities'
---
In recent semiconductor cavity QED experiments involving self-assembled III-V quantum dots (QDs), Rabi splitting of the spontaneous emission line from individual QD excitonic states has been measured for the first time [@nat432-197; @nat432-200]. Potential application of these devices to quantum networks [@prl78-3221] and cryptography [@nat420-762] over long-haul silica fibers has sparked interest in developing QD-cavity systems with efficient light extraction operating in the telecommunication bands at 1300nm and 1550nm [@jjap44-L620]. The demonstration of vacuum-Rabi splitting in this system, a result of coupling a QD to localized optical modes of a surrounding microresonator, has been greatly aided by prior improvements to the design and fabrication of semiconductor microcavities [@nat425-944; @apl86-111105; @oe14-3472]. At the shorter wavelengths involved in these Rabi splitting experiments (740–1200nm), the optical quality factors ($Q$) of the host AlGaAs microcavities were limited to $Q \approx $ 2$\times$10$^4$—corresponding to a loss rate comparable with the coherent QD-cavity coupling rate. Further reduction of optical loss would increase the relative coherence of the QD-cavity system and would allow greater coupling efficiency to the cavity mode.
In previous measurements of wavelength-scale AlGaAs microdisk resonators, we have demonstrated $Q$-factors up to 3.6$\times$10$^5$ in the $1400$-nm band [@apl86-151106] and attributed the improved performance to an optimized resist-reflow and dry-etching technique, which produces very smooth sidewalls [@oe13-1515]. Subsequently, we have also measured Al$_{0.3}$Ga$_{0.7}$As microdisks with similar quality factors between 1200nm and 1500nm; however, these disks exhibit a significant unexpected decrease in $Q$ at shorter wavelengths ($\lambda{_{\mathrm{o}}} \approx 852$nm) [@qels2005-Kartik]. In related work on silicon microdisks, methods have been developed to specifically measure and characterize losses due to material absorption and surface scattering [@apl85-3693; @oe13-1515; @apl88-131114]. In this Letter we study the properties of GaAs and Al$_{0.18}$Ga$_{0.82}$As microdisks across three wavelength bands centered at 980nm, 1460nm, and 1600nm. After estimating and removing the surface-scattering contribution to the cavities losses, we find the remaining absorption, composed of losses in the bulk and on the surfaces, depends significantly on both wavelength and material composition.
Within the microdisk resonators studied here, optical loss can be separated into three main components: intrinsic radiation of the whispering-gallery modes (WGMs), scattering from roughness at the air-dielectric interface due to fabrication imperfections, and absorption at the surface or in the bulk of the semiconductor material. The measured total intrinsic $Q{_{\mathrm{i}}}$ is given by $$1/Q{_{\mathrm{i}}} = 1/Q{_{\mathrm{rad}}} + 1/Q{_{\mathrm{ss}}} + 1/Q{_{\mathrm{a}}},
\label{totalQ}$$ where $Q{_{\mathrm{rad}}}$, $Q{_{\mathrm{ss}}}$, and $Q{_{\mathrm{a}}}$ describe cavity losses to radiation, surface scattering, and absorption, respectively. As in the devices studied here for disk diameters (thicknesses) $\gg$$\lambda{_{\mathrm{o}}}/n{_{\mathrm{d}}}$ ($>$$\lambda{_{\mathrm{o}}}/2n{_{\mathrm{d}}}$) where $\lambda{_{\mathrm{o}}}$ is the free-space wavelength and $n{_{\mathrm{d}}}$ is the refractive index of the disk material, microdisk cavities support a large number of modes with very low radiation loss. For all the microdisk modes studied here, the calculated $Q{_{\mathrm{rad}}}$ is $\gtrsim$10$^6$ and typically is $>$10$^8$.
The samples were fabricated from high-quality heterostructures grown by molecular beam epitaxy (MBE) on a GaAs substrate. Two different samples were grown: a “GaAs” sample containing a 247-nm GaAs disk layer, and an “AlGaAs” sample with a 237-nm Al$_{0.18}$Ga$_{0.82}$As disk layer. In both samples the disk layer was grown nominally undoped (background doping levels $n_p \lesssim 10^{15}$cm$^{-3}$) and was deposited on a 1.6-$\mu$m Al$_{0.7}$Ga$_{0.3}$As sacrificial layer. Microdisks with a radius of $\sim$3.4$\mu$m were defined by electron-beam lithography and etched in a 55 percent (by volume) HBr solution containing 3.6g of K$_2$Cr$_2$O$_7$ per litre [@apl75-1908]. The disks were partially undercut by etching the sacrificial layer in 8percent HF acid for 45s, prior to e-beam resist removal.
Passive measurements of the cavity $Q{_{\mathrm{i}}}$ were performed using an evanescent coupling technique employing a dimpled fiber-taper waveguide [@apl85-3693; @prb70-081306R; @CM_dimple_taper]. The dimpled taper was mounted to a three-axis 50-nm-encoded stage and positioned in the near field of the resonator. Using the taper position to vary the coupling, the WGMs of the microdisks were excited using three swept tunable laser sources (linewidth $< 5$MHz, covering 963–993nm, 1423–1496nm, and 1565–1625nm). By weakly loading the cavity, the resonance linewidth $\delta\lambda$ \[see Fig. \[SEM\_samples\](b)\] is a good measure for the intrinsic quality factor ($Q{_{\mathrm{i}}} = \lambda{_{\mathrm{o}}}/\delta\lambda$).
![Measured (a) $Q{_{\mathrm{i}}}$ and (b) $\Delta\lambda$ for the (,) GaAs TE- and TM-polarized microdisk modes and (,) Al$_{0.18}$Ga$_{0.82}$As TE and TM modes, respectively. In (c) and (d), connected points represent calculated bounds on (c) $Q{_{\mathrm{ss}}}$ and (d) $Q{_{\mathrm{a}}}$ for each family of modes. The data were compiled from two disks of each material.[]{data-label="Qs_graph"}](Qi_split_Qss_Qa.eps)
High-resolution linewidth scans were calibrated to $\pm0.01$pm accuracy using a fiber-based Mach-Zehnder interferometer. A set of polarization-controlling paddle wheels was used to selectively couple to TE-like or TM-like microdisk modes. Each family of modes (same radial order $p$) was identified by comparing the coupling behavior and free spectral range (FSR) to finite-element method (FEM) models [@apl88-131114].
The measured $Q{_{\mathrm{i}}}$s for all observed modes are summarized in Fig. \[Qs\_graph\](a). Modes with $Q{_{\mathrm{i}}}$s dominated by radiation loss, [i.e.]{} the measured $Q{_{\mathrm{i}}}$ is near $Q{_{\mathrm{rad}}}$ calculated using FEM simulations, are omitted. In the 1600-nm band, the high-$Q$ TE modes are $p =$ 1–4 in GaAs and $p=$ 1–3 in AlGaAs; all TM modes are radiation limited in this band. Near 1460nm, the TE$_{p=1-4}$ and TM$_{p=1}$ modes in both materials are detectable and not radiation limited. In the 980-nm range, identifying modes becomes more difficult: at this wavelength families through TE$_{p=8}$ and TM$_{p=7}$ are not radiation limited, and significant spectral overlap between the modes causes Fano-like resonance features [@pr124-1866]. In addition, we are unable to couple to the lowest order modes of both polarizations ($p\approx$ 1–3) because they are poorly phase matched to the fiber taper.
Despite efforts to produce perfectly smooth side walls, Figs. 1(c,d) indicate that significant surface roughness is still present. Surface roughness on microdisk resonators backscatters light between the degenerate WGMs, which breaks their degeneracy and results in the measured mode-splitting ($\Delta\lambda$) shown in Fig. \[SEM\_samples\](b). Following the theory developed in Refs. [@oe13-1515] and [@ol21-1390], $\Delta\lambda$ and $Q{_{\mathrm{ss}}}$ are both dependent on the characteristic volume of the scatterer ($V{_{\mathrm{s}}}$): $$\begin{aligned}
\Delta\lambda & = & \frac{\pi^{3/4}}{\sqrt{2}}\lambda{_{\mathrm{o}}}V{_{\mathrm{s}}}(n{_{\mathrm{d}}}^2-n{_{\mathrm{o}}}^2)\sum_{\hat{\eta}}\overline{u}{_{\mathrm{s}}}(\hat{\eta}) \\
Q{_{\mathrm{ss}}} & = & \frac{\lambda{_{\mathrm{o}}}^3}{\pi^{7/2}n{_{\mathrm{o}}}(n{_{\mathrm{d}}}^2-n{_{\mathrm{o}}}^2)^2V{_{\mathrm{s}}}^2\sum_{\hat{\eta}}\overline{u}{_{\mathrm{s}}}(\hat{\eta})G(\hat{\eta})}\end{aligned}$$ where $n{_{\mathrm{d}}}$ and $n{_{\mathrm{o}}}$ are the indices of refraction of the disk and surrounding medium, respectively, $\overline{u}{_{\mathrm{s}}}(\hat{\eta})$ is the spatially-averaged $\hat{\eta}$-polarized normalized electric field energy at the disk edge, and $G(\hat{\eta}) = \{2/3,2,4/3\}$ is a geometrical factor weighting the radiation contribution from the $\hat{\eta} = \{\hat{r},\hat{\phi},\hat{z}\}$ polarizations. The mode field profiles are calculated by the FEM. For FEM models in the 980-nm span, we treat all measured TE (TM) modes as TE$_{p=7}$ (TM$_{p=6}$) because the appropriate field parameters do not vary significantly between radial orders.
We employ two separate measurements to find rough bounds on $Q{_{\mathrm{ss}}}$. First, we use the average doublet splitting for each family to find the average ${\langle}V{_{\mathrm{s}}}{\rangle}_{p}$ sampled by each mode and then calculate the $Q{_{\mathrm{ss}}}$ associated with each family. $$\textrm{Splitting Method:\ \ } \Delta\lambda \Rightarrow {\langle}V{_{\mathrm{s}}}{\rangle}_{p} \Rightarrow Q{_{\mathrm{ss}}}$$ Second, we statistically analyze the roughness of the disk edges in high resolution SEM images [@apl85-3693]. Fitting the autocorrelation of the roughness to a Gaussian, the roughness amplitude ($\sigma{_{\mathrm{r}}}$) and correlation length ($L{_{\mathrm{c}}}$) give the “statistical” scatterer volume ($\overline{V}{_{\mathrm{s}}} = \sigma{_{\mathrm{r}}}t\sqrt{RL{_{\mathrm{c}}}}$ where $t$ and $R$ are the disk’s thickness and radius) for each disk, which is used to estimate $Q{_{\mathrm{ss}}}$. $$\textrm{Statistical Method:\ \ } \sigma{_{\mathrm{r}}},L{_{\mathrm{r}}} \Rightarrow \overline{V}{_{\mathrm{s}}} \Rightarrow Q{_{\mathrm{ss}}}$$ The average $\{\sigma{_{\mathrm{r}}},L{_{\mathrm{c}}}\}$ for the GaAs and AlGaAs disks are $\{0.6,38.7\}$nm and $\{1.8,29.4\}$nm, respectively. Because each mode will not sample all of the disk’s physical irregularities, the roughness estimated by the statistical analysis is slightly greater than the roughness calculated from the doublet splittings. Hence, the doublet splitting places an upper bound and more accurate value for $Q{_{\mathrm{ss}}}$ \[Fig. \[Qs\_graph\](c)\]. The statistical analysis gives a lower bound, although neither bound is strict in the theoretical sense.
Through Eq. (\[totalQ\]), $Q{_{\mathrm{ss}}}$ and $Q{_{\mathrm{rad}}}$ are removed from the measured $Q{_{\mathrm{i}}}$ to obtain limits on $Q{_{\mathrm{a}}}$ \[Fig. \[Qs\_graph\](d)\]. To relate cavity losses to material properties, the material absorption rate ($\gamma_{\mathrm{a},p}$) for the $p{^{\mathrm{th}}}$ mode is given by $\gamma_{\mathrm{a},p} = 2\pi{}c/\lambda{_{\mathrm{o}}}Q{_{\mathrm{a}}}$ [^1]. We weight each measured doublet equally and average over all families in a band to determine an average $\overline\gamma{_{\mathrm{a}}}$. Table \[gamma\_table\] compiles the average absorption rates for both GaAs and Al$_{0.18}$Ga$_{0.82}$As across the three wavelength ranges.
Method: Splitting Statistical
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- ----------------------------------------------------------------------------------
Sample
$\begin{array}{l} \mbox{GaAs} \\ \quad @\ 980\,\mbox{nm}\,\ \end{array} $ $\Bigg\{ \begin{array}{c} \mbox{all modes} \\ \mbox{TE} \\ \mbox{TM} \end{array} $ $ \begin{array}{c} 3.47\pm0.59 \\ 2.94\pm1.24 \\ 4.08\pm0.91 \end{array} $ $ \begin{array}{c} 3.29\pm0.75 \\ 2.61\pm1.28 \\ 4.06\pm0.89 \end{array} $
$\begin{array}{l} \mbox{AlGaAs} \\ \quad @\ 980\,\mbox{nm}\,\ \end{array} $ $\Bigg\{ \begin{array}{c} \mbox{all modes} \\ \mbox{TE} \\ \mbox{TM} \end{array} $ $ \begin{array}{c} 5.84\pm0.13 \\ 5.77\pm1.61 \\ 6.03\pm1.56 \end{array} $ $ \begin{array}{c} 4.00\pm0.91 \\ 3.44\pm2.31 \\ 5.39\pm1.73 \end{array} $
$\begin{array}{l} \mbox{GaAs} \\ \quad @\ 1460\,\mbox{nm} \end{array} $ $\Bigg\{ \begin{array}{c} \mbox{all modes}\\ \mbox{TE}_{p=1} \\ \mbox{TM}_{p=1} \end{array} $ $\begin{array}{c} 0.942\pm0.696 \\ 0.514\pm0.085 \\ 0.495\pm0.089 \end{array} $ $\begin{array}{c} 0.888\pm0.692 \\ 0.444\pm0.108 \\ 0.467\pm0.095 \end{array} $
$\begin{array}{l} \mbox{AlGaAs} \\ \quad @\ 1460\,\mbox{nm} \end{array} $ $\Bigg\{ \begin{array}{c} \mbox{all modes} \\ \mbox{TE}_{p=1} \\ \mbox{TM}_{p=1} \end{array} $ $\begin{array}{c} 1.73\pm0.50 \\ 1.32\pm0.22 \\ 2.30\pm0.49 \end{array} $ $\begin{array}{c} 1.43\pm0.76 \\ 0.882\pm0.380 \\ 2.39\pm0.40 \end{array} $
GaAs @ 1600nm – TE $0.507\pm0.186$ $0.460\pm0.185$
AlGaAs@1600nm– TE $0.968\pm0.179$ $0.629\pm0.173$
: Summary of material absorption rates.[]{data-label="gamma_table"}
The average rates are 540percent larger at 980nm than at 1600nm and 80percent greater in AlGaAs than in GaAs.
The measured absorption may be due to a number of sources. Although nonlinear-absorption-induced optical bistability was measured for internal cavity energies as low as 106aJ, the losses reported in Table \[gamma\_table\] were all taken at input powers well below the nonlinear absorption threshold. Free carrier absorption can also be neglected given the nominally undoped material and relatively short wavelengths studied here [@pr114-59]. The Urbach tail makes a small contribution in the 980-nm band ($\leq$15percent) and is negligible otherwise [@pr114-59]. This leaves deep electron (hole) traps as the major source contributing to bulk material absorption in the measured microdisks. Similar wavelength dependent absorption has been observed in photocurrent measurements of MBE-grown AlGaAs waveguides [@ieee-jqe33-933] and attributed to sub-bandgap trap levels associated with vacancy complexes and oxygen incorporation during growth [@jap61-5062]. Given the high surface-volume ratio of the microdisks, another possible source of loss is surface-state absorption. The sensitivity to absorption from surface-states can be quantified by the $p{^{\mathrm{th}}}$ mode’s energy overlap with the disk’s surface, $\Gamma'{_{\mathrm{p}}}$; TM modes are more surface-sensitive than TE modes [@apl88-131114] whereas both polarizations are almost equally sensitive to the bulk. The calculated surface overlap ratio is $\Gamma'{_{\mathrm{TM}}}/\Gamma'{_{\mathrm{TE}}} \approx 2.65$ for $p=1$ modes in the 1460-nm band, where all surfaces of the disk (top, bottom, and etched edge) are treated equally. For these modes the measured absorption ratio is $\gamma{_{\mathrm{a,TM}}}/\gamma{_{\mathrm{a,TE}}} = 1.74\pm0.47$ ($0.96\pm0.23$) in the AlGaAs (GaAs) microdisks, which indicates the presence of significant surface-state absorption in the AlGaAs resonators and dominant bulk absorption in the GaAs disks. In the 980-nm band, the data are consistent with bulk absorption \[$\gamma{_{\mathrm{a,TM}}}/\gamma{_{\mathrm{a,TE}}}=1.05\pm0.40$ ($1.39\pm0.66$) for the AlGaAs (GaAs) devices\] although the results are less conclusive due to the larger scatter in the data.
In summary, after accounting for radiation and surface scattering losses, we measure greater sub-bandgap absorption in Al${_{\mathrm{0.18}}}$Ga${_{\mathrm{0.82}}}$As microdisks than in similar GaAs resonators, and the absorption in both materials decreases towards longer wavelengths. From the polarization dependence of the measured optical loss, we infer that both surface states and bulk states contribute to the residual absorption in these structures. Our results imply that reductions in the optical loss of AlGaAs-based microphotonics, especially at the shorter wavelengths $< 1$$\mu$m and in high Al content alloys, will require further study and reduction of deep level traps, and that surface passivation techniques [@apl64-1911] will also likely be important.
Two authors (C.P.M. and K.S., respectively) would like to thank the Moore Foundation and the Hertz Foundation for their graduate fellowship support. K. H. L. thanks the Wingate Foundation. H. K. has been supported by NSF grant No. 0304678.
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[^1]: The commonly used absorption coefficient ($\alpha$) depends on both the material absorption rate and the group velocity of the cavity mode ($\alpha = \gamma{_{\mathrm{a}}}/v{_{\mathrm{g}}}$).
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abstract: 'Interference alignment(IA) is mostly achieved by coding interference over multiple dimensions. Intuitively, the more interfering signals that need to be aligned, the larger the number of dimensions needed to align them. This dimensionality requirement poses a major challenge for IA in practical systems. This work evaluates the necessary and sufficient conditions on channel structure of a 3 user interference channel(IC) to make perfect IA feasible within limited number of channel extensions. It is shown that if only one of interfering channel coefficients can be designed to a specific value, interference would be aligned perfectly at all receivers.'
author:
- |
Zainalabedin Samadi, Vahid Tabatabavakili and Farzan Haddadi\
Dept. of Elec. Eng., Iran University of Sceince and Technology Tehran, Iran\
{z.samadi}@elec.iust.ac.ir\
{vakily, haddadi}@iust.ac.ir
title: '[On feasibility of perfect interference alignment in interference networks]{}'
---
Interference Channels, Interference Alignment, Degrees of Freedom, Generic Channel Coefficients, Vector Space.
Introduction
============
One of the recent strategies to deal with interference is interference alignment. In a multiuser channel, the interference alignment method puts aside a fraction of the available dimension at each receiver for the interference and then adjusts the signaling scheme such that all interfering signals are squeezed in the interference subspace. The remaining dimensions are dedicated for communicating the desired signal, keeping it free from interference.
Cadambe and Jafar [@Cadam08], proposed the linear interference alignment (LIA) scheme for IC and proved that this method is capable of reaching optimal degrees of freedom of this network. The optimal degrees of freedom for a $K$ user IC is obtained in the same paper to be $K/2$. The proposed scheme in [@Cadam08] is applied over many parallel channels and achieves the optimal degrees of freedom as the signal-to-noise ratio (SNR) goes to infinity. Nazer et. al. [@Nazer12], proposed the so called ergodic interference alignment scheme to achieve 1/2 interference-free ergodic capacity of IC at any signal-to-noise ratio. This scheme is based on a particular pairing of the channel matrices. The scheme needs roughly the same order of channel extension compared with [@Cadam08], to achieve optimum performance. The similar idea of opportunistically pairing two channel instances to cancel interference has been proposed independently by [@Sang13] as well. However, ergodic interference alignment scheme considers only an special pairing of the channel matrices and does not discuss the general structure of the paired channels suitable for interference cancelation.
This paper addresses the general relationship between the paired channel matrices suitable for canceling interference, assuming linear combining of paired channel output signals. Using this general pairing scheme, to align interference at receiver, proposed scheme significantly lowers the required delay for interference to be canceled.
From a different standpoint, this paper obtains the necessary and sufficient feasibility conditions on channel structure to achieve total DoF of the IC using limited number of channel extension. So far, Interference alignment feasibility literature have mainly focused on network configuration, see [@Ruan] and references therein. To ease some of interference alignment criteria by using channel structure, [@Leejan09] investigates degrees of freedom for the partially connected ICs where some arbitrary interfering links are assumed disconnected. In this channel model, [@Leejan09] examines how these disconnected links are considered on designing the beamforming vectors for interference alignment and closed-form solutions are obtained for some specific configurations. In contrast, our work evaluates the necessary and sufficient conditions on channel structure of an IC to make perfect interference alignment possible with limited number of channel extensions.
System Model {#sysmod}
============
![K user IC Model.[]{data-label="figure:KUser"}](KUserIC.jpg)
Consider the $K$ user IC consisting of $K$ transmitters and $K$ receivers each equipped with a single antenna, as shown in Fig. \[figure:KUser\]. Each transmitter wishes to communicate with its corresponding receiver. All transmitters share a common bandwidth and want to achieve the maximum possible sum rate along with a reliable communication. Channel output at the $k^{\textrm{th}}$ receiver and over the $t^{\textrm{th}}$ time slot is characterized by the following input-output relationship : $$\begin{aligned}
{\bf Y}^{[k]}(t)={\bf H}^{[k1]}(t){\bf X}_p^{[1]}(t)+{\bf H}^{[k2]}(t){\bf X}_p^{[2]}(t) \cdots \nonumber \\+{\bf H}^{[kK]}(t){\bf X}_p^{[K]}(t)+{\bf Z}^{[k]}(t), \end{aligned}$$ where $k\in\{1, \ldots, K\}$ is the user index, $t\in \mathbb{N}$ is the time slot index, ${\bf Y}^{[k]}(t)$ is the output signal vector of the $k^{\textrm{th}}$ receiver, ${\bf X}_p^{[k]} (t)$ is the transmitted precoded signal vector of the $k^{\textrm{th}}$ transmitter which will shortly be defined, ${\bf H}^{[kj]} (t)$, $j\in\{1, \ldots, K\}$ is the fading factor of the channel from the $ j^{\textrm{th}}$ transmitter to the $k^{\textrm{th}}$ receiver over $t^{\textrm{th}}$ time slot, and ${\bf Z}^{[k]}(t)$ is the additive white Gaussian noise at the $k^{\textrm{th}}$ receiver. The noise terms are all assumed to be drawn from a Gaussian independent and identically distributed random process with zero mean and unit variance. It is assumed that all transmitters are subject to a power constraint $P$.
The channel gains are bounded between a positive minimum value and a finite maximum value to avoid degenerate channel conditions (e.g. the case of all channel coefficients being equal or a channel coefficient being zero or infinite). Assume that the channel knowledge is causal and is available globally, i.e. over the time slot $t$, every node knows all channel coefficients ${\bf H}^{[kj]} (\tau), \forall j, k \in \{1, \ldots, K\}, \tau \in \{1, \ldots, t\}$. Hereafter, time index is omitted for the sake of simplicity.
Linear Interference Alignment Limitation
========================================
Degrees-of-freedom region for a $K$ user IC, with the system model discussed in section \[sysmod\], has been derived in [@Cadam08] as follows, $$\begin{aligned}
\mathcal{D}= \left \{ {\bf d} \in \mathbb{R}_+^K: d_i+ d_j \leq 1, \; 1 \leq i, j\leq K \right \},
\label{dofreg}\end{aligned}$$ and the number of DoF achieved by $K$ user IC is obtained to be $K/2$. Following corollay describes the only DoF vector, ${\bf d}$, that achieves total number of DoF.
\[cor1\] The only DoF vector that achieves total number of DoF of an IC is $$\begin{aligned}
d_i=\frac{1}{2}, \forall 1\leq i \leq K.
\label{optdof}\end{aligned}$$
DoF point mentioned in (\[optdof\]) is on the vertex of the DOF region. Since other vertices ${\bf e}_i$ has DOF less than $K/2$. Because DOF region is convex, it is straightforward to see that $d_i=\frac{1}{2}, \forall 1\leq i \leq K$ is the only point that produces the largest total DOF.
Consider a three user IC. Assuming channel coefficients to be generic, i.e. the channel coefficients are time varying and are drawn from continuous independent distributions, [@Cadam08] has shown that optimal degrees of freedom for a three user IC cannot be achieved over limited number of channel usage. To maintain continuity of presentation, a short review is presented here.
Consider using $2n$ time slots of the channel, according to \[cor1\], achieving optimal degrees of freedom implies that $n$ degrees of freedom should be achieved for each of the transmitters. The signal vector at the $k$’th receiver can be stated as $$\begin{aligned}
{\bf Y}^{[k]}&{}={}&{\bf H}^{[k1]}{\bf X}_p^{[1]}+{\bf H}^{[k2]}{\bf X}_p^{[2]}\nonumber \\ &&{+}\: {\bf H}^{[k3]}{\bf X}_p^{[3]}+{\bf Z}^{[k]},\end{aligned}$$ where ${\bf X}_p^{[k]}$ is a $2n\times1$ column vector which is obtained by coding the transmitted symbols over $2n$ time slots of the channel, as will be explained below. ${\bf Y}^{[k]}$ and ${\bf Z}^{[k]}$ represent the $2n$ symbol extension of $y^{[k]}$ and $z^{[k]}$, respectively. ${\bf H}^{[kj]}$ is a diagonal $2n\times 2n$ matrix which represents the $2n$ symbol extension of the channel as shown in (\[topeq\]) at the top of the next page.
$${\bf H}^{[kj]}\equiv \left [\begin{array}{c c c c} h^{[kj]}(2n(t-1)+1) & 0 & \cdots & 0\\ 0 & h^{[kj]}(2n(t-1)+2) & \cdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & h^{[kj]}(2nt) \end {array}\right ]
\label{topeq}$$
In the extended channel, message $ W_1$ at transmitter $1$ is encoded to $n$ independent streams $x_m^{[1]}, m=1, \ldots, n $ and sent along the vector ${\bf v}_m^{[1]} $ so that ${\bf X}_p^{[1]}$ can be written as $$\begin{aligned}
{\bf X}_p^{[1]} ={\bf V}^{[1]} {\bf X}^{[1] },\end{aligned}$$ where ${\bf X}^{[1]}$ is a $n\times 1$ column vector comprised of transmitted symbols $x_m^{[1]}, m=1, \ldots, n $, ${\bf V}^{[1]} $ is a $2n \times n$ dimensional precoding matrix comprised of the vectors ${\bf v}_m^{[1]}, m=1, \ldots, n $ as its columns. In a similar way, $W_2$ and $W_3$ are encoded by transmitters $2$ and $3$, respectively and sent to the channel as: $$\begin{aligned}
{\bf X}_p^{[2]} ={\bf V}^{[2]} {\bf X}^{[2] },\end{aligned}$$ $$\begin{aligned}
{\bf X}_p^{[3]} ={\bf V}^{[3]} {\bf X}^{[3] }.\end{aligned}$$ The received signal at the $k'$th receiver can be evaluated to be $$\begin{aligned}
{\bf Y}^{[k]}&{}={}&{\bf H}^{[k1]}{\bf V}^{[1]} {\bf X}^{[1] }+{\bf H}^{[k2]}{\bf V}^{[2]} {\bf X}^{[2] } \nonumber \\ && {+}\:{\bf H}^{[k3]}{\bf V}^{[3]} {\bf X}^{[3] }+{\bf Z}^{[k]}.\end{aligned}$$
Receiver $i$ cancels the interference by zero forcing all ${\bf V}^{[j]}, j\not = i$ to decode $W_i$. At receiver $1$, $n$ desired streams are decoded after zero forcing the interference from transmitters $ 2$ and $3$. To achieve $ n$ dimensions free of interference from the $2n$ dimensional received signal vector ${\bf Y}^{[1]} $, the dimension of the interference signal should not be more than $n$. This can be realized by perfectly aligning the received interference from transmitters $2$ and $3$ at the receiver $1$, i.e. $$\begin{aligned}
\textrm{span} \left ( {\bf H}^{[12]} {\bf V}^{[2]} \right )=\textrm{span} \left ( {\bf H}^{[13]} {\bf V}^{[3]} \right ),
\label{SE1} \end{aligned}$$ where $\textrm{span}({\bf A})$ denotes the column space of matrix ${\bf A}$. At the same time, receiver $2$ zero forces the interference from ${\bf X}^{[1]}$ and ${\bf X}^{[3]}$. To achieve $n$ dimensions free of interference, we will have: $$\begin{aligned}
\textrm{span} \left ( {\bf H}^{[21]} {\bf V}^{[1]} \right )=\textrm{span} \left ( {\bf H}^{[23]} {\bf V}^{[3]} \right ).
\label{SE2} \end{aligned}$$ In a similar way, ${\bf V}^{[1]}$ and ${\bf V}^{[2]}$ should be designed in a way to satisfy the following condition: $$\begin{aligned}
\textrm{span} \left ( {\bf H}^{[31]} {\bf V}^{[1]} \right )=\textrm{span} \left ( {\bf H}^{[32]} {\bf V}^{[2]} \right ).
\label{SE3} \end{aligned}$$ Hence, ${\bf V}^{[1]}$, ${\bf V}^{[2]}$ and ${\bf V}^{[3]}$ should be chosen to satisfy (\[SE1\]), (\[SE2\]) and (\[SE3\]). Note that the channel matrices $ {\bf H}^{[ji]}$ are full rank almost surely. Using this fact, (\[SE1\]) and (\[SE2\]) imply that $$\begin{aligned}
\textrm{span} \left ( {\bf V}^{[1]} \right )=\textrm{span} \left ( {\bf T} {\bf V}^{[1]} \right ),
\label{CAE1} \end{aligned}$$ where $$\begin{aligned}
{\bf T}= ( {\bf H}^{[13]} ) ^{-1} {\bf H}^{[23]} ( {\bf H}^{[21]})^{-1} {\bf H}^{[12]} ( {\bf H}^{[32]} ) ^{-1} {\bf H}^{[31]}.
\label{TM}\end{aligned}$$
If ${\bf V}^{[1]}$ could be designed to satisfy this criteria, according to (\[SE2\]) and (\[SE3\]), we can obtain ${\bf V}^{[2]}$ and ${\bf V}^{[3]}$ using $$\begin{aligned}
{\bf V}^{[2]} =\left ( {\bf H}^{[32]} \right )^{-1} {\bf H}^{[31]} {\bf V}^{[1]},
\label{CAE2} \end{aligned}$$ $$\begin{aligned}
{\bf V}^{[3]} =\left ( {\bf H}^{[23]} \right )^{-1} {\bf H}^{[21]} {\bf V}^{[1]}.
\label{CAE3} \end{aligned}$$
(\[CAE1\]) implies that there is at least one eigenvector of ${\bf T}$ in $\textrm{span} \left ( {\bf V}^{[1]} \right )$. Since all channel matrices are diagonal, the set of eigenvectors for all channel matrices, their inverse and product are all identical to the set of column vectors of the identity matrix, i.e. the vectors of the from ${\bf e}_k=[0 \; 0 \; \cdots \; 1 \; \cdots \; 0]^T$. Since ${\bf e}_k$ exists in $\textrm{span} \left ( {\bf V}^{[1]} \right )$, (\[SE1\])-(\[SE3\]) imply that $$\begin{aligned}
&{}& {\bf e}_k \in \textrm{span} \left ( {\bf H}^{[ij]} {\bf V}^{[j]} \right ), \quad \forall i, j \in \{1, 2, 3\}. \end{aligned}$$ Thus, at receiver $1$, the desired signal $ {\bf H}^{[11]} {\bf V}^{[1]} $ is not linearly independent of the interference signal, ${\bf H}^{[12]} {\bf V}^{[2]}$, and hence, receiver $1$ cannot fully decode $W_1$ solely by zero forcing the interference signal. Therefore, if the channel coefficients are completely random and generic, we cannot obtain $3/2$ degrees of freedom for the three user single antenna IC through LIA schemes.
Perfect Interference Alignment Feasibility Conditions
=====================================================
In previous section, If the objective was to align interference at two of the receivers, receivers $1$ and $2$ for instance, it could be easily attained using (\[SE1\]) and (\[SE2\]). Though, as discussed above, (\[SE3\]) which refers to interference alignment criteria at receiver $3$, cannot be satisfied simultaneously with (\[SE1\]) and (\[SE2\]). Instead, assume channel matrices which contribute to interference at receiver $3$ would be of a form that already satisfies (\[SE3\]), interference alignment would then be accomplished.
We can wait for the specific form of the channel to happen. The question we intend to answer in the following is that what is the necessary and sufficient condition on channel structure to make perfect interference alignment feasible in finite channel extension.
The following theorem summarizes the main result of this paper.
\[3usertheo\] In a three user IC, the necessary and sufficient condition for the perfect interference alignment to be feasible in finite channel extension is to have the following structure on the channel matrices: $$\begin{aligned}
&{}&{\bf T}= ( {\bf H}^{[13]} ) ^{-1} {\bf H}^{[23]} ( {\bf H}^{[21]})^{-1} {\bf H}^{[12]} ( {\bf H}^{[32]} ) ^{-1} {\bf H}^{[31]}\nonumber\\&&{=}\:{\bf P} \left [ \begin{array}{c c c} \tilde{{\bf T}} & 0 & 0 \\ 0 & \tilde{{\bf T}}& 0 \\ 0 & 0 & f(\tilde{{\bf T}}) \end{array} \right ] {\bf P}^T,
\label{CAEm}\end{aligned}$$ where ${\bf P}$ is a $2n \times 2n$ permutation matrix, $\tilde{{\bf T}}$ is an arbitrary $n_1 \times n_1$ diagonal matrix with nonzero diagonal elements, and with $n_1$ in the range $1 \leq n_1 \leq n$, and $f({\bf X})$ is a mapping whose domain is an arbirary $n_1 \times n_1$ diagonal matrix and range is a $2(n-n_1) \times 2(n-n_1)$ diagonal matrix ${\bf Y}=f({\bf X})$ whose set of diagonal elements is a subset of diagonal elements of ${\bf X}$.
Theorem \[3usertheo\] simply states that matrix ${\bf T}$ has no unique diagonal element.
\[lemma1\] Assuming that $ {\bf V}^{[1]}$ is of rank $n$, (\[CAE1\]) implies that $n$ eigenvectors of ${\bf T}$ lie in $\textrm{span} \left ( {\bf V}^{[1]} \right )$.
From (\[CAE1\]) we conclude that there exists a $n \times n$ dimensional matrix ${\bf Z}$ such that $$\begin{aligned}
{\bf T} {\bf V}^{[1]}={\bf V}^{[1]}{\bf Z}.\end{aligned}$$ Assume that ${\bf u}$ is an eigenvector of ${\bf Z} $ i.e., ${\bf Z}{\bf u}=\gamma{\bf u}$ where $\gamma$ is its corresponding eigenvalue, then ${\bf V}^{[1]} {\bf u} \not = 0$ and we can write: $$\begin{aligned}
{\bf T} {\bf V}^{[1]}{\bf u}={\bf V}^{[1]}{\bf Z}{\bf u}=\gamma {\bf V}^{[1]}{\bf u}.\end{aligned}$$ Then ${\bf V}^{[1]}{\bf u}$, which is in $\textrm{span} \left ( {\bf V}^{[1]} \right )$, is an eigenvector of ${\bf T}$. Since ${\bf Z}$ has $n$ orthogonal eigenvectors, then $n$ orthogonal eigenvectors of ${\bf T}$ lie within $\textrm{span} \left ( {\bf V}^{[1]} \right )$.
$\textrm{span} \left ( {\bf V}^{[1]} \right )$ should not contain any vector of the form ${\bf e}_i$, and since $\textrm{span} \left ( {\bf V}^{[1]} \right )$ has dimension $n$, it should have $n$ basis vectors of the form $ {\bf v}\tilde{{\bf T}}=\sum_{i=1}^{2n}\alpha_i {\bf e}_i, \quad j=1, \ldots, n$, where at least $2$ of $\alpha_i$’s are nonzero. Let’s call vectors with this form as non ${\bf e}_i$ vectors. Since $n$ of ${\bf T}$’s eigenvectors lie in $\textrm{span} \left ( {\bf V}^{[1]} \right )$, the matrix ${\bf T}$ should have at least $n$ non ${\bf e}_i $ eigenvectors. Note that this requirement is necessary not sufficient. Assuming that ${\bf S}=[{\bf s}]$ is a matrix consisted of non ${\bf e}_i $ eigenvectors of ${\bf T}$ as its columns, it is concluded that $\textrm{span} \left ( {\bf V}^{[1]} \right ) \in \textrm{span} \left ( {\bf S} \right )$.
\[lemma2\] ${\bf T}$ has no unique diagonal element, i.e., if $t_l$ is the $l$’th diagonal element of ${\bf T}$, there is at least one $t_k, k=1, \ldots, 2n, k \neq l$ for which $t_l=t_k$.
It is easy to see that if ${\bf s}_1= {\bf e}_i + {\bf e}_j , \quad i,j=1, \ldots, n, i \neq j$ is an eigenvector of ${\bf T}$, then $t_i = t_j$. If $t_l$ is unique, this implies that non ${\bf e}_i $ eigenvectors of ${\bf T}$ do not contain ${\bf e}_l$, and hence, ${\bf e}_l \in \textrm{kernel} \left ( {\bf S} \right )$, where $ \textrm{kernel} \left ( {\bf S} \right )$ denotes the null space of columns of matrix ${\bf S}$. Thus, $ {\bf e}_l \in \textrm{kernel} \left ({\bf V}^{[1]} \right )$ because $\textrm{span} \left ( {\bf V}^{[1]} \right ) \in \textrm{span} \left ( {\bf S} \right )$. Since all channel matrices are diagonal, using (\[CAE1\])-(\[CAE3\]), ${\bf e}_j \in \textrm{kernel}({\bf V}^{[1]})$ implies that $$\begin{aligned}
&{}& {\bf e}_j \in \textrm{kernel} \left ( {\bf H}^{[ij]} {\bf V}^{[j]} \right ), \quad \forall i, j \in \{1, 2, 3\}.\end{aligned}$$
Thus, at receiver $1$, the total dimension of the desired signal $ {\bf H}^{[11]} {\bf V}^{[1]}$ plus interference from undesired transmitters, ${\bf H}^{[1j]} {\bf V}^{[j]}, j \neq 1$, is less than $2n$, and desired signals are not linearly independent from the interference signals, and hence, receiver $1$ can not fully decode $W_1$ solely by zeroforcing the interference signal.
Lemma \[lemma2\] conlcludes the proof of necessary part of theorem \[3usertheo\]. The sufficient part is easily proved by noting the fact that the matrix ${\bf T} $ with the form given in (\[CAEm\]) has $L \geq n$ non ${\bf e}_i $ eigenvectors ${\bf r}_i, i=1, \ldots, L$ with the property that $$\begin{aligned}
{\bf e}_k \not \in \textrm{span}({\bf R}), \quad k=1, \ldots, 2n,
\label{spnprp}\end{aligned}$$ and $$\begin{aligned}
{\bf e}_k \not \in \textrm{kernell}({\bf R}), \quad k=1, \ldots, 2n,
\label{krnlprp}
\end{aligned}$$ where ${\bf R} $ is defined as a $2n \times L$ matrix consisted of ${\bf r}_i$’s as its columns. Every $n$ subset of these eigenvectors can be considered as the columns of user $1$ transmit precoding matrix ${\bf V}^{[1]}$. ${\bf V}^{[2]}$ and ${\bf V}^{[3]}$ can be designed using (\[CAE2\]) and (\[CAE3\]).
As an example, assume that, using $3$ extension of the channel, $6 \times 6$ diagonal matrix ${\bf T}$ has the following form, $$\begin{aligned}
{\bf T}=\textrm{Diag}(1, 2, 1, 2, 1, 2)
\label{Texpl}\end{aligned}$$ which has the form given in (\[CAEm\]) with $n=3, n_1=1$, ${\bf P}={\bf I}_6$, where ${\bf I}_6$ is $6 \times 6$ identity matrix, $\tilde{{\bf T}}=\textrm{Diag}(1, 2)$, and $f(\tilde{{\bf T}})=\tilde{{\bf T}}$. The matrix ${\bf R}$ for this example case can be obtained as $$\begin{aligned}
{\bf R}=\left [ \begin{array}{c c c c} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{array} \right ].\end{aligned}$$ Note that this choice for the set of non ${\bf e}_i$ eigenvectors of ${\bf T}$ defined in (\[Texpl\]), which satisfies (\[spnprp\]) and (\[krnlprp\]), is not unique. Every $6 \times 3$ matrix ${\bf V}^{[1]} \in \textrm{span}({\bf R})$ can be considered as the user $1$ transmit precoding matrix. ${\bf V}^{[2]}$ and ${\bf V}^{[3]}$ can be obtained using (\[CAE2\]) and (\[CAE3\]).
For the rest of the paper, every matrix ${\bf U}$ which can be written in the form of (\[CAEm\]), with the same permutation matrix ${\bf P}$ and mapping function $f({\bf X})$, would be stated as ${\bf U}={\bf T}_P$. It can easily be seen that if ${\bf U}={\bf T}_P$ and ${\bf V}={\bf T}_P$, so is ${\bf U}^{-1}={\bf T}_P$ and ${\bf U} {\bf V}={\bf T}_P$.
If the condition (\[CAEm\]) is true with the following form $$\begin{aligned}
&{}&{\bf T}={\bf P} \left [ \begin{array}{c c} \tilde{{\bf T}} & 0 \\ 0 & \tilde{{\bf T}} \end{array} \right ] {\bf P}^T,
\label{CAEms}\end{aligned}$$ where $\tilde{{\bf T}}$ is an an arbitrary $n \times n$ diagonal matrix, ${\bf V}^{[1]}$ can be designed as $$\begin{aligned}
{\bf V}^{[1]}={\bf P}^T \left[ \begin{array}{c} {\bf I}_n \\ {\bf I}_n \end{array} \right ],
\label{vdesig}\end{aligned}$$ where ${\bf P}$ is the same permutation matrix used in (\[CAEms\]) and ${\bf I}_n$ is the $n \times n$ identity matrix. ${\bf V}^{[1]}$ can also be designed as any other $2n \times n$ matrix having the same column vector subspace with (\[vdesig\]). ${\bf V}^{[2]}$ and ${\bf V}^{[3]}$ are determined accordingly using (\[CAE2\]) and (\[CAE3\]), respectively.
Assuming channel aiding condition with the form given in (\[CAEms\]), consider the special case of ${\bf H}^{[ij]} ={\bf T}_P, \quad \forall i, j, \quad i \not = j$, then ${\bf T}={\bf T}_P$ and the channel aiding condition is already satisfied. The case of ${\bf H}^{[ij]} ={\bf T}_P, i \not = j$ is the condition to satisfy the requirement of ergodic interference alignment in [@Nazer12], therefore, ergodic interference alignment is the special case of the scheme presented in this paper.
Conclusion
==========
Channel aiding conditions obtained in this paper can be considered as the perfect Interference alignment feasibility conditions on channel structure. Stated conditions on channel structure are not exactly feasible, assuming generic channel coefficients. Approximation can be used and its effect on residual interference can be analyzed. Overall, this paper aims at reducing the required dimensionality and signal to noise ratio for exploiting degrees of freedom benefits of interference alignment schemes.
[10]{} \[1\][\#1]{} url@rmstyle \[2\][\#2]{}
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abstract: 'is a principled framework for making efficient use of limited experimental resources. Unfortunately, its applicability is hampered by the difficulty of obtaining accurate estimates of the of an experiment. To address this, we introduce several classes of fast estimators by building on ideas from amortized variational inference. We show theoretically and empirically that these estimators can provide significant gains in speed and accuracy over previous approaches. We further demonstrate the practicality of our approach on a number of end-to-end experiments.'
author:
- |
Adam Foster^^[^1] Martin Jankowiak^^ Eli Bingham^^ Paul Horsfall^^\
**Yee Whye Teh**^^ **Tom Rainforth**^^ **Noah Goodman**^^\
^^Department of Statistics, University of Oxford, Oxford, UK\
^^Uber AI Labs, Uber Technologies Inc., San Francisco, CA, USA\
^^Stanford University, Stanford, CA, USA\
`adam.foster@stats.ox.ac.uk`
bibliography:
- 'references.bib'
title: Variational Bayesian Optimal Experimental Design
---
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge research funding from Uber AI Labs. MJ would like to thank Paul Szerlip for help generating the sprites used in the Mechanical Turk experiment. AF would like to thank Patrick Rebeschini, Dominic Richards and Emile Mathieu for their help and support. AF gratefully acknowledges funding from EPSRC grant no. EP/N509711/1. YWT’s and TR’s research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 617071.
[^1]: Part of this work was completed by AF during an internship with Uber AI Labs.
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abstract: 'Making use of a droplet-epitaxial technique, we realize nanometer-sized quantum ring complexes, consisting of a well-defined inner ring and an outer ring. Electronic structure inherent in the unique quantum system is analyzed using a micro-photoluminescence technique. One advantage of our growth method is that it presents the possibility of varying the ring geometry. Two samples are prepared and studied: a single-wall ring and a concentric double-ring. For both samples, highly efficient photoluminescence emitted from a single quantum structure is detected. The spectra show discrete resonance lines, which reflect the quantized nature of the ring-type electronic states. In the concentric double–ring, the carrier confinement in the inner ring and that in the outer ring are identified distinctly as split lines. The observed spectra are interpreted on the basis of single electron effective mass calculations.'
author:
- 'T. Kuroda'
- 'T. Mano'
- 'T. Ochiai'
- 'S. Sanguinetti'
- 'K. Sakoda'
- 'G. Kido'
- 'N. Koguchi'
title: 'Optical transitions in quantum-ring complexes'
---
Introduction
============
Recent progress in nanofabrication technology allows the simulation of novel atomic physical phenomena on an artificial platform, such as presence of $\delta$-function-like density of states on quantum dots (QD) [@GCL95; @BGL99], realization of molecular-orbital state on spatially coupled QDs [@VGN95; @BHH01], and formation of nanometer-sized quantum rings [@MCB93; @GRS97; @MN05; @MKS05], which are nanoscopic analogues of benzene. Among them, fabrication of semiconductor quantum rings has triggered strong interest in realization of quantum topological phenomena, which are expected in a small systems with simply-connected geometry [@AB59; @B84]. The Aharonov-Bohm (AB) effect, which engenders so-called *persistent current* [@BIL83], has been explored for various types of mesoscopic rings, based on metals [@metal_rings] and semiconductors [@MCB93; @semicon_rings], using magnetic and transport experiments. As an optical, *i.e.* *non-contact*, approach, Lorke *et al.* first observed far-infrared optical response in self-assembled quantum rings, revealing a magneto-induced change in the ground state from angular momentum $l = 0$ to $l = -1$, with a flux quantum piercing the interior [@LLG00]. Later, Bayer *et al.* reported pronounced AB-type oscillation appearing in the resonance energy of a charged exciton confined in a single lithographic quantum ring [@BKH03]; furthermore, Ribeiro *et al.* observed the AB signature in type-II quantum dots, in which a heavy hole travels around a dot [@RGC04].
Although magneto-conductance characteristics have garnered considerable attention, optical manifestation of the AB effect has remained a controversial subject [@RR00]. In this regime, both an electron and a hole are excited simultaneously; the net charge inside a ring decreases to zero. Because of the charge neutrality, the loop current associated with a magnetic flux must vanish, engendering the absence of the AB effect for a tightly bound electron-hole pair (*exciton*). Several theories have been proposed that a composite nature of excitons allows for a non-vanishing AB effect in a small sufficient ring [@RR00], and in excited state emissions [@HZL01]. On the other hand, a negative prediction has also been reported in which the AB effect can hardly take place in more-realistic rings with finite width [@SU01]. A large difference in trajectories for an electron and a hole is necessary to exhibit the AB effect on neutral excitation [@GUK02].
 Atomic force microscope images in 250 $\times$ 250 nm$^2$ area for (a) a GaAs quantum ring (QR), and (b) a concentric double ring (DQR), grown by droplet epitaxy. After ring formation, they are covered by an Al$_{0.3}$Ga$_{0.7}$As layer for optical experimentation.](figure1.eps){width="7.5cm"}
The present study examines the optical transition in *strain-free* self-assembled GaAs quantum rings in a zero magnetic field. We have recently reported self-production of nanometer-sized GaAs rings on (Al,Ga)As by means of a droplet epitaxial technique [@MN05; @MKS05]. Because of their good cyclic symmetry, together with high tunability of the ring size and shape, the present system is expected to open a new route to implement the AB effect within the optical regime. Both the ground-state transition and the excited-state transition are identified by single quantum ring photoluminescence (PL). The spectra are found to be in good agreement with results of single-carrier calculation.
The paper is organized as follows. In Sec. II, we briefly explain the sample preparation and the procedure of the optical experiment. In Sec. III, we present low temperature PL spectroscopy of single quantum rings. Section IV presents the theoretical results; we discuss the experimental data in terms of this model in Sec. V. Our conclusions are summarized briefly in Sec. VI.
Experimental Procedure
======================
sample preparation
------------------
GaAs quantum rings were grown on Al$_{0.3}$Ga$_{0.7}$As using modified droplet epitaxy [@KTC91]. In this growth, cation (Ga) atoms are supplied solely in the initial stage of growth, producing nanometer-sized droplets of Ga clusters. After formation of the Ga droplets, anion atoms (As) are supplied, leading to crystallization of the droplets into GaAs nanocrystals. In contrast to the other methods to fabricate QDs, such as Stranski-Krastanow growth, this technique can produce *strain-free* quantum dots based on lattice-matched heterosystems. In addition to these characteristics, we recently found that it has a high controllability of the crystalline shape: When we irradiate the Ga droplets with an As beam of sufficiently high intensity, typically $2 \times 10^{-4}$ Torr beam equivalent pressure (BEP) at 200 $^{\circ}$C, the crystalline shape becomes cone-like, following the shape of the original droplet [@WKG00; @WTGK01]. When we reduce the As intensity to $1 \times 10^{-5}$ Torr BEP, the QD becomes ring-like, with a well-defined center hole [@MN05]. Further reduction of the As flux, down to $2 \times 10^{-6}$ Torr BEP, produces the striking formation of unique concentric double-rings: an inner ring and an outer one [@MKS05]. The rings show a good circular symmetry, whereas small elongation is found along the \[0-11\] direction (5% for the inner ring and 8% for the outer ring).
Two samples are used in the experiment: a GaAs quantum ring of 40 nm diameter with 15 nm height, (abbreviated to QR, hereafter) and concentric double-rings consisting of an inner ring of 40 nm diameter and 6 nm height, and an outer ring of 80 nm diameter with 5 nm height (abbreviated to DQR). In the growth of these two rings, the same conditions were applied to the initial deposition of Ga droplets (1.75 monolayer (ML) of Ga at 0.05 ML/s to the surface of a Al$_{0.3}$Ga$_{0.7}$As substrate at 300 $^{\circ}$C). Thus, the mean volume for each structure is expected to be equivalent for QR and DQR, whereas its crystalline shape differs drastically. After ring formation, they were capped by an Al$_{0.3}$Ga$_{0.7}$As barrier of 100 nm thickness, following rapid thermal annealing (RTA; 750 $^{\circ}$C for 4 min). The ring shape before capping is characterized by atomic force microscopy (AFM), as shown in Fig. \[fig1\] (see also the averaged cross-section of DQR presented in Fig. \[eh\_level\](c)). The ring density is $1.3 \times 10^{8}$ cm$^{-2}$ for both samples, allowing the capture of the emission from a single structure using a micro-objective setup.
We would like to stress the difference in growth process between these quantum rings and In(Ga)As rings, whose growth was previously reported [@GRS97; @GG03]. The ring formation of the latter case is associated with partial capping of a thin GaAs layer on InAs QDs, which were originally made by the Stranski-Krastanow method. Subsequent annealing results in the morphological change from island-like QDs to ring-like nanocrystals. In contrast, the present rings are formed at the crystalline stage of GaAs. The ring shape in this case is determined by the flux intensity of As beams. After the formation of rings, they are capped by a thick (Al,Ga)As layer. Later we apply RTA to improve their optical characteristics. Note that the final RTA processing does not modify the nanocrystalline ring shape, according to the negligible interdiffusion of Ga and As at a GaAs/(Al,Ga)As heterointerface at the relevant temperature. [@SK86]
![\[macroPL\]Far-field emission spectra of the sample with (a) QRs and (b) DQRs at 5 K plotted on a logarithmic scale. The excitation density is 50 mW/cm$^2$.](figure2.eps){width="6.5cm"}
optical arrangement
-------------------
In the PL experiment, we used a continuous wave He-Ne laser as an excitation source. The laser emitted 544 nm wavelength light, corresponding to 2.28 eV in energy. The excitation beam from the laser was obliquely incident to the sample. It was loosely focused by a conventional lens (30-cm focal length) into an elliptical spot of $0.5 \times 1.4$ mm$^{2}$. Emission from the sample was collected by an aberration-corrected objective lens of N.A. (numerical aperture) = 0.55. Combination of the objective lens and a pinhole (50-$\mu$m diameter) at the focal plane allowed the capture of emissions inside a small spot of 1.2-$\mu$m diameter. For this spot size, $1.3$ rings were expected to lie in the focus on average. The position of detection was translated laterally so that single quantum structures were captured individually. For this purpose, we moved the objective lens with sub-micrometer precision using piezo transducers. During translation of the spot, the condition of excitation was kept unchanged because the excitation area was sufficiently larger than the area that covered the entire translation. The emission passing through the pinhole was introduced into an entrance slit of a polychromator of 32-cm focal length. After being spectrally dispersed, it was recorded by a charged-couple device detector with 0.8-meV resolution. In advance of the micro-objective measurement, we used a conventional PL setup to observe the spectra of the ring ensemble. All experiments were performed at 3.8 K.
Before describing experimental results, we mention the carrier dynamics associated with photoexcitation of the present condition. For our laser beam, photocarriers are produced initially in the barrier, whose band gap is 1.95 eV. After diffusion, the carriers are captured by the quantum rings. The captured carriers then relax into the lower lying quantum ring levels where they radiatively recombine. Our previous study showed the recombination lifetime in GaAs/(Al,Ga)As QDs as $\sim$ 400 ps, whereas the characteristic time of intra-dot relaxation was much shorter – less than 30 ps – depending on excitation density [@SWT02; @KSG02]. Because of the rapid intra-dot relaxation, we can expect that an electron and a hole recombine after they are in quasi-equilibrium. The quantized levels are occupied by carriers according to the Fermi distribution. Similar dynamics are expected in the ring system.
Experimental Results
====================
PL from the ensemble of quantum rings
-------------------------------------
We present the far-field PL spectrum of the sample with QR in Fig. \[macroPL\](a). It comprises several spectral components. The sharp line at 1.49 eV is assigned by impurity-related emissions from the GaAs substrate. Because of a thin deposited layer of the sample (500 nm thickness), the excitation beam is expected to reach the substrate, thereby producing strong emissions. The broad emission band at 1.544 eV in the center energy is attributed to recombination of an electron and a hole, which are confined in GaAs QRs. The spectrum is broadened by 28 meV in full-width-at-half-maximum (FWHM). The line broadening is caused by the size distribution of the rings. Several emission components ranging from 1.7 eV to 1.95 eV are assigned by recombinations in the (Al,Ga)As barrier. The spectral tail to lower energy suggests the presence of impurities and imperfections in the barrier layer. Note that the excitation density at this measurement is quite low (50 mW/cm$^2$). For that reason, the impurity-related signal should be relatively emphasized.
The PL spectrum of the sample with DQR is presented in Fig. \[macroPL\](b). The signals that are related to the GaAs substrate and to the barrier are identical to those of QR. The emission band at 1.628 eV in center energy originates from recombination of the ensemble of DQRs. It is broadened by 49 meV in FWHM. We find that the PL energy of DQR is higher than that of QR. The energy shift reflects the small height of DQR. In our rings, stronger confinement is induced along the growth direction than in the lateral in-plane direction. Thus, the reduction in height, associated with formation of DQR, enhances their confinement energy, causing a blue shift in the PL spectrum.
Spectroscopy of a single quantum ring (QR)
------------------------------------------
![\[fig\_QR\]Emission spectra for a single GaAs QR. Three examples – a, b, and c – are presented. Their respective excitation densities were, from bottom to top, 1, 10, and 30 W/cm$^2$. Spectra are normalized to their maxima and offset for clarity.](figure3.eps){width="5.5cm"}
In Fig. \[fig\_QR\] we show the PL spectra of three different quantum rings, QR a, -b, and -c, and their dependence on excitation intensity. In QR a at low excitation, we find a single emission line appearing at 1.569 eV, which results from recombination of an electron and a hole, both occupying the ground state of the ring. With increasing excitation intensity, a new emission line, indicated by an arrow, emerges at 1.582 eV. Further increase in excitation density causes saturation in the intensity of the original line along with a nonlinear increase in the new line. Superlinear dependence of the emission intensity suggests that the satellite line comes from the electron–hole recombination from an excited level of the ring. Thus, the energy difference between the the ground and the excited state in the QR is 13 meV.
In addition to the state-filling feature associated with photoinjection, we find the ground-state emission being shifted to low energy. It is a signature of multi-carrier effects. In the presence of multiple carriers inside a ring, their energy levels are modified by the Coulomb interaction among carriers. Because the optical transition energy is mainly renormalized according to the exchange correction, the many-carrier effects results in spectral red shift of the emission, depending on the number of carriers. Similar features have been observed in numerous quantum dot systems including GaAs dots [@KSG02] and In(Ga)As rings [@WSH00]. Note that the relevant multiplet is not spectrally resolved in our rings, but it leads to the red shift of the broad emission spectra. Moreover, the biexcitonic emission is expected to contribute to the low energy tail of the spectra because the biexciton binding energy was found to be $\sim$0.8 meV in our droplet-epitaxial GaAs dots [@KSG02; @KKS05]. At high excitation, we also find spectral broadening, which is attributed to an incoherent collision process that occurs among carriers.
Identical spectral features are found in QR b. The ground-state emission is observed at 1.537 eV, whereas the excited-state emission appears at 1.546 eV. The energy spacing between these two lines is 9 meV. Although dependence on excitation density is quite similar for QR a and QR b, the energy levels are slightly different because of a small dispersion of the ring shape and size. We note that the linewidth of the ground state emission is also different between QR a (2.2 meV FWHM) and QR b (3.4 meV). Moreover, these are considerably large compared with that known for self-assembled QDs. We ascribe the line broadening to spectral diffusion, *i.e.*, an effect of the local environment that surrounds each QR: Our samples are expected to contain a relatively large density of imperfections and excess dopants, associated with the low-temperature growth of the sample. It causes low-frequency fluctuation in the local field surrounding QRs, which is due to the carrier hopping inside the barrier. This leads to efficient broadening of the PL spectra, which depend on the local environment of each ring. Detailed examination of the origin of line broadening is studied in droplet-epitaxial GaAs quantum dots [@KKS05].
In QR c, we find a shoulder structure on the ground-state emission, which suggests the split in the relevant level caused by lateral elongation, and/or structural imperfection inherent in this ring. Broken symmetry in the ring shape causes degeneracy lift in the energy level, leading to observation of the doublet spectra. [@footnote1] Apart from this split structure, the same spectral characteristics are found in QR c. The energy difference from the original line to the satellite is observed to 11 meV, which is between the value of QR a and that of QR b.
![\[fig\_spatial\]Position dependence of micro PL spectra in GaAs QR at (a) 1.2 W/cm$^{2}$ and (b) 36 W/cm$^{2}$. From top to bottom the position of detection is moved from 0 to 3.8 $\mu$m in steps of 0.32 $\mu$m. Spectra are vertically offset for clarity.](figure4.eps){width="8cm"}
Spatial dependence of the PL spectrum is shown in Fig. \[fig\_spatial\], where the position of detection is laterally translated on the sample in steps of 0.32 $\mu$m. Figure \[fig\_spatial\](a) presents the results obtained at low excitation. They exhibit a single line associated with the ground-state emission from a single QR, depending on the position of detection. Lateral broadening of the emission is estimated as $\sim$1.2 $\mu$m FWHM, which is consistent with the spatial resolution of our micro-objective setup. At high excitation, the spectra change into multiplets, as shown in Fig. \[fig\_spatial\](b). They show the same lateral profile with those obtained at low excitation, confirming the multiplet being emitted from a single QR, and not from multiple QRs with different energies. The energy split from the ground-state emission to the first-excited-state emission is observed to be 9 meV for this QR.
spectroscopy on a single concentric double-rings (DQR)
------------------------------------------------------
![\[fig\_DQR\]Emission spectra for concentric quantum double-rings, DQR a and -b. Their respective excitation densities were, from bottom to top, 1, 10, and 30 W/cm$^2$. Spectra are normalized to their maxima and offset for clarity. ](figure5.eps){width="5.5cm"}
We present the PL spectra of two concentric quantum double-rings – DQR a and DQR b – in Fig. \[fig\_DQR\]. Similarly to the case of QR, the spectra consist of discrete lines, *i.e.*, a main peak following a satellite one, which is at the high energy side of the main peak. The former is associated with recombination of carriers in the ground state, whereas the latter is from the excited states. The energy difference between the ground-state line and the excited-state one is 7.2 meV in DQR a and 8.5 meV in DQR b. We point out that, in contrast to the QR case, we observe the satellite peak even at the lowest excitation. For the lowest excitation intensity, we can estimate the carrier population inside a ring to be less than 0.1, according to the carrier capturing cross-section determined for GaAs QD, which was prepared with the same epitaxial technique [@KSG02]. Observation of the excited state emission suggests reduction of carrier relaxation from the excited level to the ground level. That feature will be discussed later.
At high excitation, we find that several additional lines are superimposed on the spectra, as shown by the broken arrows. Presence of these contributions implies the presence of fine energy structures in DQR.
Calculation for single-carrier levels
=====================================
{width="15cm"}
Quantity Units GaAs Al$_{0.3}$Ga$_{0.7}$As
-------------------------------- --------- ------- ------------------------
CB effective mass *m*$_0$ 0.067 0.093
VB effective mass (heavy hole) *m*$_0$ 0.51 0.57
CB band offset meV
VB band offset meV
: \[table1\]Material parameters used in the effective mass calculation for the conduction band (CB) and the valence band (VB)
We evaluate the energy levels of the ring in the framework of a single-band effective-mass envelope model [@MB94; @CH00]. In calculation, the actual shape measured by AFM is adopted as the potential of quantum confinement; for simplicity, the ring is assumed to hold a cylindrical symmetry. The technique employed in this section follows Ref. describing the exact diagonalization of the effective-mass Hamiltonian. Note that, in our *lattice-matched* GaAs/(Al,Ga)As rings, strain effects are negligible. For that reason, the simple effective-mass approach is expected to provide accurate energy levels. This presents a contrast to the case of Stranski-Krastanow grown dots, where the electronic structure is modified strongly by complex strain effects [@SGB99]. The versatility of the present method is seen in Refs. and , showing good agreement between the asymmetric PL lineshape in a GaAs/(Al,Ga)As QD ensemble and the calculation, taking into account the morphologic distribution of dots.
We also notice that the present calculation neglects Coulomb interaction between an electron and a hole. Because our rings are sufficiently small that confinement effects are dominant, the Coulomb interaction can be treated as a constant shift in the transition energies, independent of the choice of an electron state and the hole state. In following discussion, we are interested in relative energy shifts from the ground-state transition to the excited-state one, and dependence of the exciton binding energy on relevant (single-carrier) transition should be sufficiently below the experimental accuracy. Thus, we restrict ourselves to calculate the single carrier energy levels, discarding Coulomb correlation effects.
Our approach to the problem is to enclose the nanocrystal inside a large cylinder of radius $R_{c}$ and height $Z_{c}$, on the surface of which the wavefunction vanishes. Care should be taken to set $R_c$ and $Z_c$ away from the ring, so that the eigen values are almost independent of their choice. Taking into account the rotational symmetry of the Hamiltonian, and for this boundary condition, the wave function, $\Phi_L$, where $L(=0, \pm1, \cdots)$ is the azimuthal quantum number, is expanded in terms of a complete set of the base functions, $\xi ^{L}_{i,j}$, formed by products of Bessel functions of integer order $L$ and sine functions of $z$, $$\begin{aligned}
\Phi _L (z,r,\theta ) = \sum_{i,j>0} A^{L}_{i,j} \xi ^{L} _{i,j}(z,r,\theta ),\\
\xi ^{L} _{i,j}(z,r,\theta) = \beta ^{L}_{i} J_{L}(k^L_{i}r) e^{iL\theta} \sin (K_{j}z),\end{aligned}$$ where $k^L_i R_c$ is the $i$ th zero of the Bessel function of integer order $J_{L}(x)$, $K_j = \pi j / Z_c$, and $\beta ^L _i$ is appropriate normalization factors, *i.e.*, $$\beta ^L_i = \sqrt{ \frac {2}{\pi Z_{c} R_{c}^{2}}} \frac{2}{|J_{L-1}(k^{L}_{i}R_c)-J_{L+1}(k^{L}_{i}R_c)|}.$$ In advance of calculation, we have prepared a Hamiltonian that includes a potential term in cylindrical coordinates. Then, we calculated its matrix elements through numerical integration with $\xi ^{L} _{i,j}$ over $r$ and $z$. Finally, the eigen states were obtained with diagonalizing matrix. For the present calculation, we have taken into account 35 Bessels and 35 sine functions as the base functions for each value of $L$, and $R_{c} (Z_{c}) = 120 (20)$ nm. Material parameters used in calculation are summarized in Table \[table1\].
A series of single carrier levels of QR is shown in Fig. \[eh\_level\](a). Because the system has cylindrical symmetry, each carrier level is specified by the principal (*radial*) quantum number [@footnote2] $N (=1, 2, \cdots)$, and an azimuthal quantum number $L$, corresponding to the angular momentum. Two levels with $\pm L$ are degenerated at zero magnetic field. In Fig. \[eh\_level\](a), the carrier levels belonging to each radial quantum number are aligned vertically with those of a different angular momentum. We find that a vertical ($L$-dependent) sequence of quantized levels shows a typical signature of ring-type confinement. For an ideal ring with infinitesimal width, being treated as a one-dimensional system with translational periodicity, the level series is expressed as $$E_{L}=\frac{\hbar ^2}{2m^{*}R^2}L^2,$$ where $R$ and $m^{*}$ respectively represent the radius of the ring and the carrier mass. We find that the bilinear dependence of the level series, shown in Eq. (4), is reflected clearly in the line sequence in Fig. \[eh\_level\](a). The energy levels in DQR are shown in Fig. \[eh\_level\](b). As a result of the smaller height, the quantization energies are larger than those of QR. The level sequence of $N =1$ is more densely populated than that of $N = 2$, suggesting a large difference in carrier trajectories between the two levels. According to Eq. (4), the situation corresponds to the large effective value of $R$ for $N =1$. The fact is confirmed by the wavefunctions shown in Fig. \[eh\_level\](c). This figure illustrates the envelope wavefunction of an electron with various values of $N$. They are of zero angular momentum. We find that the electron of $N = 1$ is confined mainly in the outer ring. That of $N = 2$ is in the inner ring, and that of $N = 3$ is situated in both rings. That differential confinement engenders remarkable changes in their trajectory. The amount of penetration for the electron of $N=1$ to the inner ring is found to be $\sim$0.1, whereas that of $N=2$ to the outer ring is $\sim$0.05.
Discussion
==========
We have evaluated the oscillator strength for transitions between each electron-hole (*e-h*) level to determine a consistency between the emission spectra and the theoretical levels. The magnitude is proportional to the overlap of corresponding (envelope) wavefunctions. Note that, because of cylindrical symmetry, optical transition is not allowed for an electron and a hole with different angular momentum. Moreover, we have determined the transition strengths for the *e-h* pair with different $N$s as less than 1/10 smaller than that with the same $N$s. We can therefore infer that the electron, specified by a pair of $N$ and $L$, recombines only with the hole of the same $N$ and $L$. This selection rule allows us to describe each optical transition simply by $(N, L)$.
![\[pl\_vs\_cal\] (a) A series of optical transition energies in QR, obtained by the calculation. The PL spectrum of QR a at 15 W/cm$^2$ is shown in the inset. (b) The energies of optical transitions in DQR, together with the PL spectrum of DQR a at 10 W/cm$^2$ for comparison.](figure7.eps){width="5.5cm"}
A series of transition energies for QR, obtained by the procedure described above, are shown in Fig. \[pl\_vs\_cal\](a). For comparison, the emission spectra of QR a are plotted as an example of experimental data. The main peak and the high-energy satellite in the observed spectra are assigned respectively by the recombination of the *e-h* pair in the lowest state, $(N, L)=(1, 0)$, and that of the first excited *radial* state, $(2, 0)$. The split between the two transitions is deduced to be 13.1 meV, which agrees with the energy shift obtained by experiments. Note that the emissions associated with high angular momenta are not present in the data, which suggests rapid relaxation of angular momentum, whose process is quite faster than transition between radial quantization levels or recombination between an electron and a hole. A possible origin for fast angular momentum relaxation is structural asymmetry of the ring, which results from elongation, impurity, and surface roughness. In this case, angular momentum does not represent a good quantum number, and scattering between different $L$ levels efficiently occurs.
We show the comparison between the experimental spectra of DQR and the results of calculation in Fig. \[pl\_vs\_cal\](b). As in the case of QR, the main PL peak and the satellite one are explained respectively in terms of the transition of $(N,L)=(1,0)$ and that of $(2,0)$. The energy split deduced from calculation is 8.8 meV, which agrees with the experimental value. It is noteworthy that, in DQR, the wavefunction of $N=1$ is localized mainly in the outer ring, whereas that of $N=2$ is localized in the inner ring. Thus, the two peaks in the observed spectra come from the two rings, which consist of a DQR. In this connection, it is noteworthy that the excited-state emission in our experiment appears even when the carrier population is less than one. This presence of the excited state emission constitutes direct evidence for the carrier confinement into the two rings. Tunneling probability between the inner ring and the outer one is not very large, engendering the observation of the excited-state emission. Note also that the emission of the outer ring is more intense than that of the inner ring. The effect is attributable to their different surface areas, which affects the efficiency of the carrier injection from a barrier.
Finally, we would like to discuss the validity of the theoretical treatment which neglects the Coulomb correlation between an electron and a hole. A limitation of the validity takes place when an exciton binding energy is fairly large compared to a single-carrier split energy, i.e., 13.1 meV for QR, and 8.8 meV for DQR. In this case the Coulomb correlation admixes various single-carrier levels. We can roughly evaluate the exciton binding energy of the rings in comparison with that of GaAs/(Al,Ga)As quantum wells (QWs). This is because the carrier quantization of our rings is mainly associated with vertical confinement, and the lateral dimension is larger than the exciton Bohr diameter. According to numerous attempts on the study of GaAs/(Al,Ga)As QWs [@MK85], the QW confinement enhances the exciton binding energy from a bulk value of 3.7 meV to $\lesssim$10 meV at $\sim$6-nm thick QWs. A smaller thickness results in weaker exciton binding due to the carrier penetration to a barrier. These results support the exciton binding energy being smaller than or at most comparable to the single-carrier split energy of the rings. The observed spectral doublet, therefore, directly reflects the single-carrier levels. The situation presents a clear contrast to that of Stranski-Krastanow grown QDs, in which the dot dimension is quite smaller than our droplet-epitaxial nanostructures, so that the exciton binding energy reaches $\sim$ 30 meV.
Conclusions
===========
We have used a remarkable change in quantum dot shape through droplet epitaxial growth to fabricate unique semiconductor quantum ring complexes. Electronic structures of the quantum rings are identified using an optical, non-contact approach. In the small ring-like system, carriers are quantized along two orthogonal degrees of freedom – radial motion and rotational motion. The latter corresponds to angular momentum. The optical transition takes place on recombination of an electron and a heavy hole, which are in the grand state of the ring, and in the excited radial state. In concentric double quantum rings, emission originating from the outer ring and that from the inner ring are observed distinctly. Results of effective-mass calculations well reproduce the emission spectra applied to a single quantum ring.
We believe that the present ring system will contribute to a deeper understanding of quantum interference effects in a non-simply connected geometry. In this connection, we would like to point out that our concentric double-rings are a good candidate to realize in-plane polarization for carriers, producing a robust Aharonov-Bohm feature in neutral excitonic transition [@GUK02]. Magnet-optical experiments using these quantum rings are now in progress.
We are grateful to Drs. J. S. Kim, T. Noda, K. Kuroda, and Prof. M. Kawabe for their fruitful discussions. We would like to thank K. Kurakami for his experimental assistance. T. K. acknowledges a support of a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (15710076).
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---
abstract: |
In this paper, we obtain the analytical solutions of two kinds of transcendental equations with numerous applications in college physics by means of Lagrange inversion theorem, and rewrite them in the form of ratio of rational polynomials by second order Padé approximant afterwards from a practical and instructional perspective. Our method is illustrated in a pedagogical manner for the purpose that students at the undergraduate level will be beneficial. The approximate formulas introduced in the paper can be applied to abundant examples in physics textbooks, such as Fraunhofer single slit diffraction, Wien’s displacement law and Schrödinger equation with single or double $\delta$ potential. These formulas, consequently, can reach considerable accuracies according to the numerical results, therefore they promise to act as valuable ingredients in the standard teaching curriculum.
**Keywords**: Padé approximant, Lambert $W$ function, Schrödinger equation, Fraunhofer single slit diffraction, Wien’s displacement law
address: |
$^1$ Department of Physics, Renmin University of China, Beijing, 100872, China\
$^2$ School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu, 225002, China\
$^3$ School of Physical Science and Technology, Yangzhou University, Yangzhou, Jiangsu, 225002, China
author:
- 'Qiang Luo$^{1}$, Zhidan Wang$^{2}$ and Jiurong Han$^{3}$'
title: A Padé approximant approach to two kinds of transcendental equations with applications in physics
---
Introduction
============
Transcendental equations of certain types frequently emerge in seemingly unrelated branches of physics, and the roots of these equations appear in a great deal of applications. Furthermore, the ascertainment of zeros of equations is a problem commonly encountered in a broad spectrum of scientific applications. A wide variety of root finding algorithms[@url-rootfinding] are available to approximate the solutions to any desired degree of accuracy though, exact analytical solutions to physical problems, which provide better insights into the physical significances of associated parameters than purely numerical solutions, are always desirable and preferable.
The approximation formula, however, plays an unique role in teaching physics since it fills the gap between analytical approach and numerical solution. Frustratedly, many useful approximation techniques, aiming to solve special physical problems, are not really familiar to college students. Quite recently, Kevin Rapedius presented complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle by means of Siegert approximation method[@approSiegert]. Augusto Beléndez *et al* obtained a simple but highly accurate approximate expression for the period of a simple pendulum at the aid of Kidd-Fogg approximate formula[@approTaylor]. In this paper, we intend to draw attention to the Lagrange inversion theorem[@Lagrangeinv; @url-Lagrange] and Padé approximant[@padeLagevin] by two kinds of transcendental equations that have made their appearances in a variety of applications. The first kind of equation is \[tanxcotx\] $$\begin{aligned}
\tan x= \kappa x\label{tanx},\\
\cot x= \kappa x\label{cotx},\end{aligned}$$ which, splendidly, can be used to find the correlation to the spring oscillations due to the non-ignorable mass of spring[@springAJP], to determine the positions of maxima of Fraunhofer single slit diffraction[@diffraction] and the eigenvalues of infinite square potential well with a residual $\delta$-function interaction[@doubledelta; @chinese]. The second one is $$\label{LambertW}
W(x)e^{W(x)}=x$$ where $W(x)$ is the celebrated Lambert $W$ function[@Winitzki], which has undergone a renewed interest during the past three decades and owns a diversity of multidisciplinary applications. Numerous quantities, such as Wien’s displacement constant[@wienpeak], can be expressed in closed form in terms of the Lambert $W$ function. Here we refer the motivated readers to [@lambertCS; @lambertQS] and references therein for more information.
The main purpose of this paper is to provide a pedagogical derivation of the analytical solutions of transcendental equations and with the help of Lagrange inversion theorem, and bother Padé approximant subsequently to express them in element forms because of the cumbersome terms of Taylor series expansions. In comparison to other approaches, our formulas are rather transparent and forward, without losing the precision apparently. The mathematical tools are explained briefly in section \[sec2\], while the formulas are given afterwards. In section \[sec3\] and \[sec4\] we intend to inspire undergraduate students by some typical examples mentioned above after the derivation of the approximate formulas. Section \[sec5\] is devoted to our conclusions.
Mathematical preliminary: methods and formulas {#sec2}
==============================================
The Lagrange inversion theorem[@Lagrangeinv] is a remarkable tool famous for its ability to give explicit formulas where other approaches run into stone walls. Lagrange-Bürmann formula[@url-Lagrange], a special case of Lagrange inversion theorem, has found many applications, such as evaluating roots of certain transcendental equations and obtaining expansions of a function in powers of a related but different function.
Let the function $f(z)$ be analytic in some neighborhood of the original point $z=0$ of the complex plane with $f(0)\neq0$ and satisfy the equation[@Lagrangeinv] $$\label{funcw}
w=\frac{z}{f(z)}.$$ There exists two positive numbers $a$ and $b$ such that for $\vert w\vert<a$ the equation has just one solution in the domain $\vert z\vert<b$. Lagrange-Bürmann formula tells us that the unique solution is an analytical function of $w$ with[@url-Lagrange] $$\label{BurmannFormula}
z=\sum_{n=1}^{\infty}\frac{w^n}{n!}\Big[\frac{d^{n-1}}{dz^{n-1}}\Big(f(z)\Big)^n\Big]_{z=0}.$$
In contract to Taylor expansion, Padé approximat[@padeLagevin] has abundant applications in physics because of its fast convergence speed and elegant form. Padé approximant is a type of rational fraction approximation to the value of a function. This structure of approximant enables an effective reconstruction of function’s singularities over the whole range using its series expansion obtained for small values of its variable. The $[p,q]$ Padé approximant denotes a fraction of polynomial $P_p(x)=\sum_np_nx^n$ of degree at most $p$, and polynomial $Q_q(x)=\sum_nq_nx^n$ of degree at most $q$ $$\label{padepq}
[p,q]\equiv \frac{P_p(x)}{Q_q(x)}.$$ The fraction consisting of polynomials $P_p(x)$ and $Q_q(x)$ which approximates function $f(x)=\sum_nf_nx^n$ is determined by the equation[@padeLagevin] $$\label{padepq}
f(x) - \frac{P_p(x)}{Q_q(x)}=\mathcal{O}(x^{p+q+1})$$ where the symbol $\mathcal{O}(x^{n})$ stands for the value of the order $x^{n}$. $q_0=1$ is always assumed for convenience. It is known that in many cases a higher accuracy of approximation is achieved for fraction of polynomials of identical degree, thus the degrees of both numerator and denominator are set to be 2 hereafter.
Selected applications of transcendental equation {#sec3}
=================================================
Padé approximant to
--------------------
We plan to hunt for approximate formulas relating to by the Lagrange inversion theorem and Padé approximant in the current section, while the approximate formula of will be provided in the next section.
The analytical solutions to are absent until now. From the perspective of graphical method, the solution to (or ) is equivalent to the solution of the pair of equations $y=\kappa x$ and tangent function $y=\tan x$(or cotangent function $y=\cot x$). The discussion will be limited to non-negative solutions because physical quantities involved in current paper only relate to positive roots. Suppose that the zero in the domain $\big[0,\frac{1}{2}\pi\big)$ is denoted by $x_0^{\pm}$ while the zero in the domain $\big[(n-\frac{1}{2})\pi,(n+\frac{1}{2})\pi\big)$ is denoted by $x_n^{\pm}$ with integer $n\geq1$ for and respectively.
Let us concentrate on the first zero of firstly. The first zero of is trivial(i.e. $x_0^{+}=0$) for any $\kappa$, while situation is far more complicated for . The $[2,2]$ Padé approximant yields[@arxiv-2] $$\label{cotpadeapproxbig}
x_0^{-} \approx \frac{1}{\sqrt{\kappa}}\frac{1+\frac{1291}{4044}\kappa^{-1}+\frac{103}{5593}\kappa^{-2}}{1+\frac{655}{1348}\kappa^{-1}+\frac{255}{3704}\kappa^{-2}}
\approx \frac{1}{\sqrt{\kappa}}\frac{\kappa^{2}+0.3192\kappa+0.0184}{\kappa^{2}+0.4859\kappa+0.0688}
$$ for big $\kappa$, while for small $\vert\kappa\vert$ it is $$\label{cotpadeapproxsmall}
x_0^{-} \approx \frac{\pi}{2}\frac{1+2\kappa+\frac{\pi^2}{12}\kappa^2}{1+3\kappa+(2+\frac{\pi^2}{12})\kappa^2}.$$
We pass now to consider the other roots. By means of Lagrange-Bürmann formula , we arrive at(see appendix A for detail) \[ctanseries\] $$\begin{aligned}
x_n^{+}=\alpha_n\phi_{-}\big(1/\alpha_n\big)\label{tanseries},\\
x_n^{-}=\beta_n\phi_{+}\big(1/\beta_n\big)\label{cotseries},\end{aligned}$$ where $\alpha_n=(n+1/2)\pi$, $\beta_n=n\pi$, and $\phi_{\pm}(x)$ are defined as $$\label{fztaylar}
\phi_{\pm}(x)=1\pm\frac{1}{\kappa}x^2-\frac{3\kappa\pm1}{3\kappa^3}x^4+\mathcal{O}(x^5).$$ Since $\phi_{\pm}(x)$ are even functions, the $[2,2]$ Padé approximant to turns out to be $$\label{fzpade}
\phi_{\pm}(x)\approx\frac{(6\kappa\pm1)x^2\pm3\kappa^2}{(3\kappa\pm1)x^2\pm3\kappa^2}.$$ Specially, if $\kappa=1$, the solutions to are \[ctanpadeapprox\] $$\begin{aligned}
x_n^{+} \approx \alpha_n\frac{3\alpha_n^2-5}{3\alpha_n^2-2}\label{tanpadeapprox},\\
x_n^{-} \approx \beta_n\frac{3\beta_n^2+7}{3\beta_n^2+4}\label{cotpadeapprox}.\end{aligned}$$
We mention here that Frankel[@Frankel] once obtained a fairly accurate approximation solution to in terms of the Taylor series expansion of $\arctan x$. He reached at[@Frankel] $$\label{tanFrankel}
x_n^{+} \approx \alpha_n-\big(1+\frac{1}{\alpha_n^2}\big)\mbox{arccot}\alpha_n.$$ We note that is sightly different from the original version derived by Frankel himself because of the fact that $\arctan x+\mbox{arccot} x={\pi}/{2}$ if $x>0$. Motivated by his inspiring thoughts, we manage to find the counterpart formula for , namely $$\label{cotFrankel}
x_n^{-} \approx \beta_n+\frac{1+\beta_n^2}{2+\beta_n^2}\mbox{arccot}\beta_n.$$ Detailed comparisons among our formulas and Frankel’s formula will be made to end this subsection. and show the accurate values $x_n^{\pm}$ and the absolute errors calculated by (9), (12), and with $\kappa=1$.
[|\*[10]{}[c|]{}]{} \*[$n$]{} & \*[Exact values]{} & \*[$\frac{x_n^{+}}{\pi}$]{} &\
& & & Padé & Frankel & Taylor\
1 &4.49340946 &1.43029665 &0.20508427 &0.45855420 &0.40225822\
2 &7.72525184 &2.45902403 &0.01474265 &0.03420796 &0.02977709\
3 &10.90412166 &3.47088972 &0.00268424 &0.00629142 &0.00546478\
4 &14.06619391 &4.47740858 &0.00075749 &0.00178279 &0.00154718\
5 &17.22075527 &5.48153665 &0.00027654 &0.00065221 &0.00056576\
6 &20.37130296 &6.48438713 &0.00011965 &0.00028254 &0.00024503\
7 &23.51945250 &7.48647425 &0.00005841 &0.00013804 &0.00011969\
8 &26.66605426 &8.48806870 &0.00003121 &0.00007378 &0.00006397\
9 &29.81159879 &9.48932662 &0.00001788 &0.00004229 &0.00003667\
10 &32.95638904 &10.49034444 &0.00001084 &0.00002564 &0.00002222\
[|\*[10]{}[c|]{}]{} \*[$n$]{} & \*[Exact values]{} & \*[$\frac{x_n^{-}}{n\pi}$]{} &\
& & & Padé & Frankel & Taylor\
1 &3.42561846 &1.09040822 &-0.36000169 &-0.18196111 &-0.87179656\
2 &6.43729818 &1.02452783 &-0.01575732 &-0.00868217 &-0.03331845\
3 &9.52933441 &1.01109378 &-0.00222643 &-0.00124921 &-0.00458176\
4 &12.64528722 &1.00627998 &-0.00054192 &-0.00030606 &-0.00110449\
5 &15.77128487 &1.00403118 &-0.00017970 &-0.00010180 &-0.00036460\
6 &18.90240996 &1.00280399 &-0.00007269 &-0.00004125 &-0.00014712\
7 &22.03649673 &1.00206211 &-0.00003376 &-0.00001918 &-0.00006823\
8 &25.17244633 &1.00157982 &-0.00001736 &-0.00000987 &-0.00003505\
9 &28.30964285 &1.00124880 &-0.00000965 &-0.00000549 &-0.00001947\
10 &31.44771464 &1.00101185 &-0.00000571 &-0.00000325 &-0.00001151\
It can be distinguished that the formulas based on Padé approximant have an advantage over Frankel’s formula sightly, but the advantage will be extended with the increase of the order of Padé approximant. Besides, both tables tell us that Padé approximant is around twice as precise as Taylor series expansion, which, of course, suggests the superiority of Padé approxiant.
Selected applications of transcendental equation {#selected-applications-of-transcendental-equation}
-------------------------------------------------
### Effect of spring mass on the frequency of oscillator {#effect-of-spring-mass-on-the-frequency-of-oscillator .unnumbered}
The oscillation of a spring-mass system, where a spring is suspended vertically and a mass $m$ is hung from the bottom end of the spring, is a rather classical topic commonly studied theoretically and experimentally in introductory physics courses. The consideration of the correlation to the spring oscillation due to the non-ignorable mass of spring has led to many papers, and the empirical law that 1/3 the mass of the spring should be added to the mass of the hanging object is frequently-quoted(see reference [@springAJP] and references therein). The motion of this celebrated system is governed by[@url-springmass] $$\label{definiteproblem}
\left\{
\begin{array}{lcr}
u_{tt}-a^2u_{xx}=0, \\
u\vert_{x=0}=0, \big(u_x+\frac{m}{Ys}u_{tt}\big)\vert_{x=l}=0, \\
u\vert_{t=0}=\frac{A_0}{l}x,u_t\vert_{t=0}=0,
\end{array}
\right.$$ where $Y,s$ and $\rho$ are the Young’s modulus, cross-sectional area and mass density of the spring, and $a=\sqrt{Y/\rho}$ is the wave speed in the spring. The solution to is $$\label{definitesolution}
u(x,t)=\sum_{n=1}^{\infty}\frac{\frac{2\rho s}{\sqrt{\rho^2s^2+\lambda_n m^2}}}{l\lambda_n\big(l+\frac{m\rho s}{\rho^2s^2+\lambda_n m^2}\big)}A_0\cos(\sqrt{\lambda_n}at)\sin(\sqrt{\lambda_n}x)$$ where $\lambda_n$ satisfies the transcendental equation $$\label{lambdan}
\cot(\sqrt{\lambda}l)=\frac{m}{m_0}\sqrt{\lambda}l.$$ Taking into account the Hook’s law in the language of Young’s modulus, which is defined as the ratio of the stress(force per unit area) along an axis to the strain(ratio of deformation over initial length) along that axis in the range of stress, we obtain the relation $Y=kl/s$ where $k$ is the stiffness of ideal spring, and therewith becomes $$\label{omegan}
\cot\Big(\omega\sqrt{\frac{m_0}{k}}\Big)=\frac{m}{m_0}\Big(\omega\sqrt{\frac{m_0}{k}}\Big)$$ where the angular frequency $\omega=a\sqrt{\lambda}$. After a trivial trick, the frequency of the spring turns out to be $$\label{oemgam}
\omega=\sqrt{\frac{k}{m+\xi m_0}}$$ where $$\label{equationseries}
\left\{
\begin{array}{lcr}
\xi=\frac{1}{\eta^2}-r \\
\cot(\eta)=r\eta
\end{array}
\right.$$ with $r=m/m_0$. It can be concluded from that the motion of the spring is the superposition of infinite numbers of simple harmonic vibrations. However, the amplitudes of all the other vibrations are much smaller than the principal vibration’s. Many authors, such as Christensen[@springAJP], showed explicitly that the small $r$ case necessarily transitions to the large $r$ case so that the lowest normal mode is always the dominant one. In this occasion, the fundamental frequency of the spring can be calculated by or subsequently.
Now let us consider two extreme cases. If $r$ is large enough, then $\eta\approx\frac{1}{\sqrt{r}}(1-\frac{1}{6r})$, therefore $\xi_{\min}=\lim\limits_{r\rightarrow\infty}r[(1-\frac{1}{6r})^{-2}-1]=\frac{1}{3}$. On the contrary, $\xi_{\max}=\lim\limits_{r\rightarrow0}[\frac{4}{\pi^2}(1-r)^{-2}-r]=\frac{4}{\pi^2}$ since $\eta\approx\frac{\pi}{2}(1-r)$ when $r$ is a small number. Specially, if the mass of the spring and hung object are equal, namely $r=1$, one obtain $\eta\approx0.86$ from and therefore find the effective mass coefficient is $\xi\approx0.35$, which is in accordance with the fact that $\frac{1}{3}\leq\xi\leq\frac{4}{\pi^2}$ and satisfies the empirical law mentioned above.
### Fraunhofer single slit diffraction {#fraunhofer-single-slit-diffraction .unnumbered}
One of the most long-standing problems of classical optics is the various types of diffraction. The theory of Fraunhofer single slit diffraction predicts that the spatial pattern of light intensity on the viewing screen by a light wave passing through a single rectangular-shaped slit is given by[@diffraction] $$\label{GuangQiang}
I=I_0\frac{\sin^2u}{u^2}$$ where $u=\pi b\sin\theta/\lambda$ and $I_0$ is the light intensity at $\theta=0^{\circ}$. The first and second derivative of $I$ with respect to $u$(or $\theta$ more precisely) are badly needed in order to find the maxima and minima of the diffraction pattern. The minima occur when $u=n\pi$, $n=\pm1,\pm2,\cdots$, while the maxima exist on condition that $$\label{lighttan}
\tan u=u.$$ The trivial zero of , i.e. $u=0$ corresponds to the primary maximum of the diffraction pattern, while the non-trivial zeros indicate the secondary maxima. We see that the secondary maxima are not exactly half way between any two adjacent minima, they occur slightly earlier and move toward the center with increasing shift $u$. Figure below shows the variation of the intensity distribution with the distance.
{height="6.0cm"}
Furthermore, intensities of these secondary maxima are much less than primary maximum and fall off rapidly as moving outwards. The relative intensity of the $n$-th secondary maximum to the primary maximum can be determined from as $$\label{XDGQ}
\frac{I_n}{I_0}=\frac{1}{2}\big(\frac{9}{3\alpha_n^2-2}-\frac{1}{\alpha_n^2}\big)$$ where $\alpha_n=(n+1/2)\pi$. For example, the relative intensity of the first three secondary maxima are only $4.7\%$, $1.7\%$ and $0.8\%$ of the principal maximum, respectively.
### Schrödinger equation with single $\delta$ potential {#schrödinger-equation-with-single-delta-potential .unnumbered}
We shall discuss a problem encountered in many introduction course of quantum mechanics in detail. Now let’s consider a microscopic particle confined in an 1-dimensional infinite square well in $0<x<a$, and assume that a $\delta$-function interaction is added in the middle of the interval with constant strength $\gamma$, whose sign indicates the interaction is attractive or repulsive. Therefore, the potential of the particle is[@chinese; @arxiv-2] $$\label{potential}
V(x)=
\cases
{\gamma\delta(x-a/2), &$0<x<a$,\\
+\infty, &\mbox{otherwise}}.$$
The wavefunctions can be classified according to their symmetries under the transformation $\psi_n(x)=\pm\psi_n(a-x)$ since the Hamiltonian is symmetric with respect to the transformation $x\rightarrow a-x$. Thereby only the even parity eigen-states $\psi_n(x)$, which satisfy the Schrödinger equation $$\label{schrodinger}
-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x)+\gamma\delta\big(x-\frac{a}{2}\big)\psi(x)=E\psi(x),$$ are influenced by the additional interaction. At point $x=a/2$ the wavefunction should be continuous but its derivative makes a jump proportional to the strength of the $\delta$-function interaction, i.e. $$\label{schrodingerjump}
\psi'\big(\frac{a}{2}+\epsilon\big)-\psi'\big(\frac{a}{2}-\epsilon\big)=\frac{2m\gamma}{\hbar^2}\psi\big(\frac{a}{2}\big)$$ with $\epsilon\rightarrow0$. In general, the energy $E_n$ relates to the quasi wave-vector $k_n$, that satisfies the energy eigenvalues condition: $$\label{schrodingereig}
\tan\big(\frac{ka}{2}\big)=-\frac{b}{a}\big(\frac{ka}{2}\big)$$ where $b\equiv2\hbar^2/m\gamma$ is the characteristic length of $\delta$-potential. The energy are positive if the interaction is repulsive, situation will be more complicated if the interaction is attractive. In the case of $\gamma<0$, ground state energy for instance, will go through a process from positive to negative with increasing intensity of the interaction. However, unlike the other eigenstates, the $E=0$ energy eigenstate of the Schrödinger equation is not a sinusoidal function. Instead, the time-independent Schrödinger equation simplifies to ${d^2\psi}/{dx^2}=0$ inside the well, and yields $\psi=Ax+B$, where $A$ and $B$ are constants[@arxiv-2]. The energy eigenstate has the proper discontinuity in its slope at the middle point of the well such that[@chinese] $$\label{psi}
\psi(x)=
\cases{
2\sqrt{\frac{3}{a^3}}x, &$0\leq x<a/2$,\\
2\sqrt{\frac{3}{a^3}}\big(a-x\big), &$a/2<x\leq a$.
}$$ The combination of and yields the critical intensity $\gamma_0=-\frac{2\hbar^2}{ma}$, and therefore is reduced to $$\label{schrodingereig0}
\tan\big(\frac{ka}{2}\big)=\frac{ka}{2}.$$ , readily, tells us that the solution of satisies the identity $\big(k_na/2\big)^2-\big(n\pi/2\big)^2\approx-2\big(1+{2}/{3(n\pi)^2}\big)$ with $n=3,5,7,\cdots$ approximately, then the even-parity energy levels of the system considered are $$\label{energydelta}
E_n^{\bf{e}}=\frac{\hbar^2k_n^2}{2m}
\cases{
=E_1^{(0)}+\frac{\pi^2}{4}\frac{\gamma_{0}}{a}, &$n=1$,\\
\approx E_n^{(0)}+2\Big(1+\frac{2}{3(n\pi)^2}\Big)\frac{\gamma_{0}}{a}, &$n=3,5,7,\cdots$
}$$ where $E_n^{(0)}=\frac{n^2\pi^2\hbar^2}{2ma^2}$ is the energy of one-dimensional infinity square well without $\delta$-potential. The errors of the energy obtained from , which can be inferred from , are rather small. Furthermore, suggests us that the $\delta$-potential has a more significant influence on ground state energy than the excited state energy since the prefactor $\pi^2/4(\approx2.5)$ for $n=1$ is larger than $2\big(1+{2}/{3(n\pi)^2}\big)\approx2$ that for $n=3,5,7,\cdots$.
Selected applications of transcendental equation {#sec4}
=================================================
Padé approximant to
--------------------
The Lambert $W$ function is a multivalued function with complex variable though, of special relevance to scientific applications are the solutions when the argument is purely real number. The two real solutions of are the branches $W_0$ and $W_{-1}$ where $W_0$ is the principal branch of the $W$ function.
![Two real branches of the Lambert $W$ function. Red line: $W_0(x)$ called the principal branch is defined for $-1/e < x < +\infty$. Blue line: $W_{-1}(x)$ is defined for $-1/e < x< 0$. The two branches meet at the green point $(-1/e,-1)$[@arxiv-1].](figLambert.eps){width="6.5cm"}
The principal branch $W_0$ is analytical at $x=0$ and its Taylor series expansion[@lambertCS; @lambertQS] $$\label{powLambert}
W_0(x)=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}x^n=x\Big(1-x+\frac{3}{2}x^2-\frac{8}{3}x^3+\frac{125}{24}x^4+\mathcal{O}(x^5)\Big)$$ can be obtained in light of Lagrange inversion theorem. The $[2,2]$ Padé approximant reads $$\label{pade1a}
W_0^{(\bf{I})}(x)=x\frac{1+\frac{19}{10}x+\frac{17}{60}x^2}{1+\frac{29}{10}x+\frac{101}{60}x^2}+\mathcal{O}(x^5).$$ After rounding the fractional coefficients by the special method introduced in [@padeLagevin], is reduced to $$\label{pade1b}
\overline{W}_0^{(\bf{I})}(x)=x\frac{3+6x+x^2}{3+9x+5x^2}+\mathcal{O}(x^5).$$ which, of course, is not only more elegant but isn’t accomplished at the sacrifice of accuracy. In fact, the substitution $x\rightarrow\ln(1+x)$ of the pre-factor is a good succedaneum since the slope of $W(x)$ around zero is no more than 1. We therefore arrive at $$\label{pade2a}
W_0^{(\bf{II})}(x)=\ln(1+x)\frac{1+\frac{123}{40}x+\frac{21}{10}x^2}{1+\frac{143}{40}x+\frac{713}{240}x^2}+\mathcal{O}(x^5)$$ if we utilize the fact that $\ln(1+x)=\sum_n(-1)^{n+1}x^n/n$. Readers can refer to the appendix below for more details. Similar method impels result such that $$\label{pade2b}
\overline{W}_0^{(\bf{II})}(x)=\ln(1+x)\frac{2+6x+4x^2}{2+7x+6x^2}+\mathcal{O}(x^5).$$
shows the precision of the two formulas, i.e. and introduced above. It can be concluded that Padé formula of type is superior to type overwhelmingly, especially for positive argument. Therefore, we recommend that is the perfect approximate formula.
![The curves of the logarithm of the absolute errors($\Delta=\lg\big(\vert\frac{W_0^{A}(x)-W_0^{E}(x)}{W_0^{E}(x)}\vert\big)$) regarding to the two different formulas in the interval $-1/e<x<1$ are plotted where $W_0^{A}(x)$ and $W_0^{E}(x)$ are the approximate and exact values of the $W$ function. The dashed line and the solid line, from the top down, represent Padé formula of type and respectively.[]{data-label="figError"}](lambert2way.eps){width="7.0cm"}
Last but not the least, let us focus on an equation relating to Lambert $W$ function. The equation is $$\label{ExponLinear}
e^{-cx}=a(x-b)$$ where $a,b,c$ are constants. A great deal of quantities, such as the decay constant of an exponentially decaying process[@wienLamEJP] and the time constant of a process subject to a linear resistive force[@Winitzki], satisfy the same equation as shown above. The solution to is $$\label{ExponLinearSolution}
x=b+\frac{1}{c}W\big(\frac{c}{a}e^{-cb}\big).$$
Selected applications of transcendental equation {#selected-applications-of-transcendental-equation-1}
-------------------------------------------------
### Schrödinger equation with double $\delta$ potential {#schrödinger-equation-with-double-delta-potential .unnumbered}
Let’s extend to the case of a particle in a double delta function well which can be used to describe the behaviour of electronic terms of $\mbox{H}_2^{+}$. Mathematically, the potential is described by[@doubledelta] $$\label{potentialdouble}
V(x)=-\gamma\Big(\delta(x+a)+\delta(x-a)\Big)$$ where $\gamma,a>0$. We wouldn’t repeat the tedious derivation again, instead, we recommend the readers to some professional literatures, such as [@doubledelta] for details. Here we just sketch the general conclusions. The potential is an even function, so all the solutions can be expressed as a linear combination of even and odd solutions, namely the so-called even parity and odd parity. The bound energy of the particle, in either situation, can be expressed as $$\label{deltaenergydouble}
E^{\pm}=-\frac{\hbar^2(k^{\pm})^2}{2m}$$ where the wavevector $k^{\pm}$ is the solution to equation $$\label{deltapm}
k=\frac{1}{2b}(1\pm e^{-2ka})$$ with $b\equiv\frac{\hbar^2}{2m\gamma}$, and the plus and minus sign correspond to the even parity and odd parity solution, respectively. Thanks to we manage to obtain the bound state energy $$\label{deltaenergydoubleLambert}
E^{\pm}=-\frac{\hbar^2}{8ma^2}\Big[\frac{a}{b}+W\big(\pm\frac{a}{b}e^{-\frac{a}{b}}\big)\Big]^2.$$ Therefore, it can be inferred that the bound state energy always exists for even parity, while for odd parity it only exists on condition that $a>b$. In other words, there’re two bound states if the intensity of the interaction $\gamma>\frac{\hbar^2}{2ma}$, otherwise there exists only one bound state. Our approximation formula is sufficient to determine the energy since the argument of the Lambert $W$ function in is bounded from $-1/e$ to $1/e$ for both cases. For example, if $a=2b$, the bound state energy are $-0.6148\frac{\hbar^2}{ma^2}$ and $-0.3176\frac{\hbar^2}{ma^2}$ respectively, while the accuracy values presented in [@doubledelta] are $-0.614782\frac{\hbar^2}{ma^2}$ and $-0.317454\frac{\hbar^2}{ma^2}$ respectively. The fact that the relative errors are $0.003\%$ and $0.05\%$ or so indicates that our approximate results are in glorious agreement with the exact ones. The exchange energy $\Delta E$ defined as the difference between $E^{+}$ and $E^{-}$ can be determined subsequently.
### Wien’s displacement law {#wiens-displacement-law .unnumbered}
Planck’s seminal work for blackbody radiation, whose result conflicted dramatically with classical mechanics, opened the era of quantum theory. The concept of blackbody radiation, along with the associated Stefan-Boltzmann law and Wien’s displacement law, is a crucial pillar of modern physics. The Planck spectral distribution is given by[@wienLamEJP] $$\label{Planck}
\rho(\lambda,T)=\frac{8\pi hc}{\lambda^5}\cdot\frac{1}{e^{hc/\lambda kT}-1}$$ where $\lambda$ is the wavelength, $T$ is the temperature, $c$ is the speed of light, and $h$ and $k$ are the Planck and Boltzmann constants respectively. Wien’s displacement law gives the wavelength at which has the maximal intensity. To find the extremum of the specific intensity, it’s necessary for us to take the derivative of with respect to wavelength $\lambda$ and utilize the often-used unitless variable $x=hc/\lambda kT$. Apparently, we have the fancy equation $$\label{peakequation}
(5-x)e^{x}=5$$ whose non-trivial zero is of great interest. In order to solve analytically, several attempts have been made to express the root in integral representation and series representation over the past one century. Siewert and Burniston found that the analytical solution of relates to the canonical solution of the Riemann problem[@LambertSiewert], and Siewert applied it to Wien’s displacement law to observe that[@wienNoteint] $$\label{peakint1}
x_0=4\exp\Big(-\frac{1}{\pi}\int_0^{\infty}\Big[\arctan\Big(\frac{\pi}{\ln 5-5-t-\ln t}\Big)-\pi\Big]\frac{dt}{t+5}\Big).$$ Luck and Stevens, at the meantime, presented another integral representation by virtual of Cauchy’s integral theorem and some basic concepts of complex integration[@wienSIAM], i.e. $$\label{peakint2}
x_0=5\frac{\int_0^{2\pi}w(\theta)e^{i3\theta}d\theta}{\int_0^{2\pi}w(\theta)e^{i2\theta}d\theta}$$ where $w(\theta)=\frac{1}{5}\frac{1}{(1-e^{i\theta})e^{5e^{i\theta}}-1}$. On the other hand, Andersen[@wienNoteseries] and Vial[@wienLamEJP] suggested that only the first three terms of the Taylor series expansion in can reach a satisfactory precision.
As a matter of fact, is nothing but with $a=1/5$, $b=5$ and $c=1$. Therefore Wien’s displacement law can be obtained with an elegant expression for displacement constant $b={hc}/{kx_0}$ where $$\label{x0Lambert}
x_0=5+W_0(-5e^{-5}).$$ It can be distinguished from , and that the nontrivial zero expressed by Lambert $W$ function pushes the mathematical structure of the law into the most comfortable territory, and the calculation based on it is especially accuracy. Furthermore, our approximation formula gives $x_0=4.965114231797$, which owns ten decimal places precision compared to the accuracy value $x_0=4.965114231744$. The relative error, without doubt, is rather small, and it is of the order $10^{-11}$. What’s more, even the simplified version can uncover four decimal places precision.
Conclusion {#sec5}
==========
In spite of the fact that the solutions to many transcendental equations, such as and , can be expressed analytically in power series or named special functions, but it’s not really convenience for instructional purpose in classroom. This paper has provided a detailed illustration of how Lagrange inversion theorem and Padé approximant can be applied to solve transcendental equations analytically, and express the root(s) of those equations in the form of ratio of rational polynomials. Furthermore, our approximation method should be quite powerful in handling with the non-transparent aspect of some rigourous results, which could eventually be expressed in more brief forms, without losing the physical interpretations. Traditional graphical method, however, is not only troublesome but too oversimplified to reach desired accuracy. While all of the drawbacks can be overcome by the built-in algorithms in the softwares such as Matlab and Mathematica, the mathematical softwares tend to give numerical results, or just make the so-called closed-form solutions be lengthy, which are useless in practice. Our method, therefore, is superior in teaching because it is easy to handle and owns satisfactory accuracy. An excellent agreements were achieved between our approximate results and analytical formulas, which prove that those formulas are rather precise in practice, and therefore can act as valuable ingredients in the standard teaching curriculum.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank two anonymous referees for helpful suggestions on improving the manuscript. Q. Luo would especially express his appreciation to Professor Q.H. Liu, working at Hunan University, for his selfless help during the manuscript preparation.
Appendix A {#appA .unnumbered}
==========
we begin the process of putting the equation into the form required by the Lagrange inversion formula firstly. The asymptotic expression $x_n^{+}\sim(n+\frac{1}{2})\pi$ holds as $n\rightarrow\infty$. Let us set $z=w^{-1}-x$ where $w=\alpha_n^{-1}$, then is reduced to $$\label{derive}
\tan(x)=\tan\Big(\big(n+\frac{1}{2}\big)\pi-z\Big)=\cot(z)=\kappa(w^{-1}-z).$$ Therefore, can be arranged to with $$\label{fztanx}
f_{+}(z)=\frac{z(\cos z+ \kappa z\sin z)}{\kappa\sin z}.$$ Similarly, we can get the expression for $f_{-}$. We thus can obtain the approximate formulas to according to since $f_{\pm}(z)$ is analytical around $z=0$ and $f(0)=1/\kappa$.
Appendix B {#appB .unnumbered}
==========
The calculation of $W_0^{(\bf{II})}(x)$, where the pre-factor is $\ln(1+x)$ rather than $x$, is not really straightforward. To begin with, let us define the auxiliary function $$\label{Mfun}
M(x)=\frac{W(x)}{\ln(1+x)}=\sum_{n=0}^{\infty}a_nx^n$$ where $a_n$ are the Taylor expansion coefficients. if we utilize the fact that $\ln(1+x)=\sum_n(-1)^{n+1}x^n/n$ and the equation , we find that $$\label{MfunTaylor}
M(x)=1-\frac{1}{2}x+\frac{11}{12}x^2-\frac{43}{24}x^3+\frac{2651}{720}x^4+\mathcal{O}(x^5)$$ after rearranging the coefficients of the same power. We therefore can get the second order Padé approximant of the Lambert $W$ function around zero as is shown in by virtue of the Padé approximant of $$\label{MfunPade}
M(x)=\frac{1+\frac{123}{40}x+\frac{21}{10}x^2}{1+\frac{143}{40}x+\frac{713}{240}x^2}+\mathcal{O}(x^5).$$
References {#references .unnumbered}
==========
[99]{} <http://www.karenkopecky.net/Teaching/eco613614/Notes_RootFindingMethods.pdf> Rapedius K 2011 Calculating resonance positions and widths using the Siegert approximation method *Eur. J. Phys.* 32 1199-1211 Beléndez A Arribas E Márquez A Ortuno M and Gallego S 2011 Approximate expressions for the period of a simple pendulum using a Taylor series expansion *Eur. J. Phys.* 32 1303-1310
Abramowitz M and Stegun I A 1972 *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables* (New York: Dover) <http://en.wikipedia.org/wiki/Lagrange_inversion_theorem> Cohen A 1991 A Padé approximation to the inverse Langevin function *Rheologica Acta* 30 270-273
Christensen J 2004 An improved calculation of the mass for the resonant spring pendulum *Am. J. Phys.* 72 818-828го Gan K K and Law A T 2009 Measuring slit width and separation in a diffraction experiment *Eur. J. Phys.* 30 1271-1276 Scott T C Babb J F Dalgarno A Morgan J D 1993 The calculation of exchange forces: General results and specific models *J. Chem. Phys.* 99 2841-2855 Wang X and Liu Q H 2014 Spectrum of infinitely deep well decorated with a Dirac delat function at the center *Coll. Phys.* 33 13-15 Houari A 2013 Additional applications of the Lambert $W$ function in physics *Eur. J. Phys.* 34 695-702 Stewart S M 2012 Spectral Peaks and Wien’s Displacement Law *J. Thermophys. Heat Tran.* 26 689-692 Caillol J M 2000 Some applications of the Lambert $W$ function to classical statistical mechanics *J. Phys. A: Math. Gen.* 36 10431 Valluri S R Gil M Jeffrey D J Basu S 2009 The Lambert $W$ function and quantum statistics *J. Math. Phys.* 50 102103
Markushin V E Rosenfelder R Schreiber A W The $W_t$ transcendental function and quantum mechanical applications arXiv:0104019v2 Frankel S 1937 Complete approximate solutions of the equation $x=\tan x$ *National mathematics magazine* 11 177-182 <http://physics.citadel.edu/syost/spring.pdf> Veberic D Having function with Lambert $W(x)$ function arXiv:1003.1628v1 Vial A 2012 Fall with linear drag and Wien’s displacement law: approximate solution and Lambert function *Eur. J. Phys.* 33 751-755 Siewert C E and Burniston E E 1973 Exact analytical solutions of $ze^{z}=a$ *J. Math. Anal. Appl.* 43 626-632го Siewert C E 1981 An exact expression for the wien displeacement constant *J. Quant. Spectrosc. Radiat. Transfer* 26 467-467 Luck R and Stevens J W 2002 Explicit solutions for transcendental equations *SIAM review* 44 227-233 Andersen T B 1982 An exact expression for the wien displeacement constant *J. Quant. Spectrosc. Radiat. Transfer* 27 663-664
|
[**[<span style="font-variant:small-caps;">Bruhat-Tits buildings and analytic geometry</span>]{}**]{}
<span style="font-variant:small-caps;">Bertrand Rémy, Amaury Thuillier and Annette Werner</span>
------------------------------------------------------------------------
[**Abstract:**]{} This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.
[**Keywords:**]{} algebraic group, valued field, Berkovich analytic geometry, Bruhat-Tits building, compactification.
------------------------------------------------------------------------
[**Résumé :**]{} Ce texte introduit les immeubles de Bruhat-Tits associés aux groupes réductifs sur les corps valués et explique comment les réaliser et les compactifier au moyen de la géomérie analytique de Berkovich. Il contient une discussion détaillée du cas du groupe spécial linéaire. En outre, nous donnons une description intrinsèque des immeubles de Bruhat-Tits en géométrie analytique non archimédienne.
[**Mots-clés :**]{} groupe algébrique, corps valué, géométrie analytique au sens de Berkovich, immeuble de Bruhat-Tits, compactification.
------------------------------------------------------------------------
[**AMS classification (2000):**]{} 20E42, 51E24, 14L15, 14G22.
------------------------------------------------------------------------
Introduction {#s - intro .unnumbered}
============
This paper is mainly meant to be a survey on two papers written by the same authors, namely [@RTW1] and [@RTW2]; it also contains some further developments which we found useful to mention here. The general theme is to explain what the theory of analytic spaces in the sense of Berkovich brings to the problem of compactifying Bruhat-Tits buildings.
[**1.**]{} [*Bruhat-Tits buildings*]{}.— The general notion of a building was introduced by J. Tits in the 60ies [@TitsICM], [@Lie456 Exercises for IV.2]. These spaces are cell complexes, required to have some nice symmetry properties so that important classes of groups may act on them. More precisely, it turned out in practice that for various classes of algebraic groups and generalizations, a class of buildings is adapted in the sense that any group from such a class admits a very transitive action on a suitable building. The algebraic counterpart to the transitivity properties of the action is the possibility to derive some important structure properties for the group.
This approach is particularly fruitful when the class of groups is that of simple Lie groups over non-Archimedean fields, or more generally reductive groups over non-Archimedean valued fields – see Sect. \[s - Bruhat-Tits general\]. In this case the relevant class of buildings is that of Euclidean buildings (\[ss - Euclidean buildings\]). [*This is essentially the only situation in building theory we consider in this paper*]{}. Its particularly nice features are, among others, the facts that in this case the buildings are (contractible, hence simply connected) gluings of Euclidean tilings and that deep (non-positive curvature) metric arguments are therefore available; moreover, on the group side, structures are shown to be even richer than expected. For instance, topologically the action on the buildings enables one to classify and understand maximal compact subgroups (which is useful to representation theory and harmonic analysis) and, algebraically, it enables one to define important integral models for the group (which is again useful to representation theory, and which is also a crucial step towards analytic geometry).
One delicate point in this theory is merely to prove that for a suitable non-Archimedean reductive group, there does exist a nice action on a suitable Euclidean building: this is the main achievement of the work by F. Bruhat and J. Tits in the 70ies [@BT1a], [@BT1b]. Eventually, Bruhat-Tits theory suggests to see the Euclidean buildings attached to reductive groups over valued fields (henceforth called [*Bruhat-Tits buildings*]{}) as non-Archimedean analogues of the symmetric spaces arising from real reductive Lie groups, from many viewpoints at least.
[**2.**]{} [*Some compactification procedures*]{}.— Compactifications of symmetric spaces were defined and used in the 60ies; they are related to the more difficult problem of compactifying locally symmetric spaces [@Satake2], to probability theory [@Furst], to harmonic analysis... One group-theoretic outcome is the geometric parametrization of classes of remarkable closed subgroups [@Moore]. For all the above reasons and according to the analogy between Bruhat-Tits buildings and symmetric spaces, it makes therefore sense to try to construct compactifications of Euclidean buildings.
When the building is a tree, its compactification is quite easy to describe [@SerreArbres]. In general, and for the kind of compactifications we consider here, the first construction is due to E. Landvogt [@La]: he uses there the fact that the construction of the Bruhat-Tits buildings themselves, at least at the beginning of Bruhat-Tits theory for the simplest cases, consists in defining a suitable gluing equivalence relation for infinitely many copies of a well-chosen Euclidean tiling. In Landvogt’s approach, the equivalence relation is extended so that it glues together infinitely many compactified copies of the Euclidean tiling used to construct the building. Another approach is more group-theoretic and relies on the analogy with symmetric spaces: since the symmetric space of a simple real Lie group can be seen as the space of maximal compact subgroups of the group, one can compatify this space by taking its closure in the (compact) Chabauty space of all closed subgroups. This approach is carried out by Y. Guivarc’h and the first author [@GuiRem]; it leads to statements in group theory which are analogues of [@Moore] (e.g., the virtual geometric classification of maximal amenable subgroups) but the method contains an intrinsic limitation due to which one cannot compactify more than the set of vertices of the Bruhat-Tits buildings.
The last author of the present paper also constructed compactifications of Bruhat-Tits buildings, in at least two different ways. The first way is specific to the case of the general linear group: going back to Bruhat-Tits’ interpretation of Goldman-Iwahori’s work [@GoldmanIwahori], it starts by seeing the Bruhat-Tits building of ${\rm GL}({\rm V})$ – where ${\rm V}$ is a vector space over a discretely valued non-Archimedean field – as the space of (homothety classes of) non-Archimedean norms on ${\rm V}$. The compactification consists then in adding at infinity the (homothety classes of) non-zero non-Archimedean seminorms on ${\rm V}$. Note that the symmetric space of ${\rm SL}_n({\bf R})$ is the set of normalized scalar products on ${\bf R}^n$ and a natural compactification consists in projectivizing the cone of positive nonzero semidefinite bilinear forms: what is done in [@Wer04] is the non-Archimedean analogue of this; it has some connection with Drinfeld spaces and is useful to our subsequent compactification in the vein of Satake’s work for symmetric spaces. The second way is related to representation theory [@Wer07]: it provides, for a given group, a finite family of compactifications of the Bruhat-Tits building. The compactifications, as in E. Landvogt’s monograph, are defined by gluing compactified Euclidean tilings but the variety of possibilities comes from exploiting various possibilities of compactifying equivariantly these tilings in connection with highest weight theory.
[**3.**]{} [*Use of Berkovich analytic geometry*]{}.— The compactifications we would like to introduce here make a crucial use of Berkovich analytic geometry. There are actually two different ways to use the latter theory for compactifications.
The first way is already investigated by V. Berkovich himself when the algebraic group under consideration is split [@Ber1 Chap. 5]. One intermediate step for it consists in defining a map from the building to the analytic space attached to the algebraic group: this map attaches to each point $x$ of the building an affinoid subgroup ${\rm G}_x$, which is characterized by a unique maximal point $\vartheta(x)$ in the ambient analytic space of the group. The map $\vartheta$ is a closed embedding when the ground field is local; a compactification is obtained when $\vartheta$ is composed with the (analytic map) associated to a fibration from the group to one of its flag varieties. One obtains in this way the finite family of compactifications described in [@Wer07]. One nice feature is the possibility to obtain easily maps between compactifications of a given group but attached to distinct flag varieties. This enables one to understand in combinatorial Lie-theoretic terms which boundary components are shrunk when going from a “big” compactification to a smaller one.
The second way mimics I. Satake’s work in the real case. More precisely, it uses a highest weight representation of the group in order to obtain a map from the building of the group to the building of the general linear group of the representation space which, as we said before, is nothing else than a space of non-Archimedean norms. Then it remains to use the seminorm compactification mentioned above by taking the closure of the image of the composed map from the building to the compact space of (homothety classes of) seminorms on the non-Archimedean representation space.
For a given group, these two methods lead to the same family of compactifications, indexed by the conjugacy classes of parabolic subgroups. One interesting point in these two approaches is the fact that the compactifications are obtained by taking the closure of images of equivariant maps. The construction of the latter maps is also one of the main difficulties; it is overcome thanks to the fact that Berkovich geometry has a rich formalism which combines techniques from algebraic and analytic geometry (the possibility to use field extensions, or the concept of Shilov boundary, are for instance crucial to define the desired equivariant maps).
[**Structure of the paper.**]{} In Sect. 1, we define (simplicial and non-simplicial) Euclidean buildings and illustrate the notions in the case of the groups ${\rm SL}_n$; we also show in these cases how the natural group actions on the building encode information on the group structure of rational points. In Sect. 2, we illustrate general notions thanks to the examples of spaces naturally associated to special linear groups (such as projective spaces); this time the notions are relevant to Berkovich analytic geometry and to Drinfeld upper half-spaces. We also provide specific examples of compactifications which we generalize later. In Sect. 3, we sum up quickly what we need from Bruhat-Tits theory, including the existence of integral models for suitable bounded open subgroups; following the classical strategy, we first show how to construct a Euclidean building in the split case by gluing together Euclidean tilings, and then how to rely on Galois descent arguments for non-necessarily split groups. In Sect. 4, we finally introduce the maps that enable us to obtain compactifications of Bruhat-Tits buildings (these maps from buildings to analytifications of flag varieties have been previously defined by V. Berkovich in the split case); a variant of this embedding approach, close to Satake’s ideas using representation theory to compactify symmetric spaces, is also quickly presented. At last, Sect. 5 contains a new result, namely an intrinsic characterization of the image of the embedding we use, from Bruhat-Tits building to the analytification of the group; this gives a new description of the building in terms of multiplicative norms on the coordinate rings of the group.
[**Acknowledgements.**]{} We warmly thank the organizers of the summer school “Berkovich spaces” held in Paris in July 2010. We are grateful to the referee for many comments, corrections and some relevant questions, one of which led to Proposition 5.11. Finally, we thank Christophe Cornut for an interesting (electronic) discussion which prompted us to formulate and prove Theorem 5.7.
[**Conventions.**]{} In this paper, as in [@Ber1], valued fields are assumed to be non-Archimedean and complete, the valuation ring of such a field $k$ is denoted by $k^\circ$, its maximal ideal is by $k^{\circ\circ}$ and its residue field by $\widetilde{k} =
k^\circ/k^{\circ \circ}$. Moreover a *local field* is a non-trivially valued non-Archimedean field which is locally compact for the topology given by the valuation (i.e., it is complete, the valuation is discrete and the residue field is finite).
Buildings and special linear groups {#s - SL(n) Bruhat-Tits}
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We first provide a (very quick) general treatment of Euclidean buildings; general references for this notion are [@RousseauGrenoble] and [@WeissAffine]. It is important for us to deal with the simplicial as well as the non-simplicial version of the notion of a Euclidean building because compactifying Bruhat-Tits buildings via Berkovich techniques uses huge valued fields. The second part illustrates these definitions for special linear groups; in particular, we show how to interpret suitable spaces of norms to obtain concrete examples of buildings in the case when the algebraic group under consideration is the special linear group of a vector space. These spaces of norms will naturally be extended to spaces of (homothety classes of) seminorms when buildings are considered in the context of analytic projective spaces.
Euclidean buildings {#ss - Euclidean buildings}
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Euclidean buildings are non-Archimedean analogues of Riemannian symmetric spaces of the non-compact type, at least in the following sense: if ${\rm G}$ is a simple algebraic group over a valued field $k$, Bruhat-Tits theory (often) associates to ${\rm G}$ and $k$ a metric space, called a Euclidean building, on which ${\rm G}(k)$ acts by isometries in a “very transitive” way. This is a situation which is very close to the one where a (non-compact) simple real Lie group acts on its associated (non-positively curved) Riemannian symmetric space. In this more classical case, the transitivity of the action, the explicit description of fundamental domains for specific (e.g., maximal compact) subgroups and some non-positive curvature arguments lead to deep conjugacy and structure results – see [@MaubonGrenoble] and [@ParadanGrenoble] for a modern account. Euclidean buildings are singular spaces but, by and large, play a similar role for non-Archimedean Lie groups ${\rm G}(k)$ as above.
### Simplicial definition {#sss - simplicial}
The general reference for building theory from the various “discrete” viewpoints is [@AbramenkoBrown]. Let us start with an affine reflection group, more precisely a [*Coxeter group of affine type*]{} [@Lie456]. The starting point to introduce this notion is a locally finite family of hyperplanes – called [*walls*]{} – in a Euclidean space \[[**loc. cit.**]{}, V §1 introduction\]. An affine Coxeter group can be seen as a group generated by the reflections in the walls, acting properly on the space and stabilizing the collection of walls \[[**loc. cit.**]{}, V §3 introduction\]; it is further required that the action on each irreducible factor of the ambient space be via an infinite [*essential*]{} group (no non-zero vector is fixed by the group).
\[ex - simplicial apartments\]
- The simplest (one-dimensional) example of a Euclidean tiling is provided by the real line tesselated by the integers. The corresponding affine Coxeter group, generated by the reflections in two consecutive vertices (i.e., integers), is the infinite dihedral group ${\rm D}_\infty$.
- The next simplest (irreducible) example is provided by the tesselation of the Euclidean plane by regular triangles. The corresponding tiling group is the Coxeter group of affine type $\widetilde{{\rm A}_2}$; it is generated by the reflections in the three lines supporting the edges of any fundamental triangle.
Note that Poincaré’s theorem is a concrete source of Euclidean tilings: start with a Euclidean polyhedron in which each dihedral angle between codimension 1 faces is of the form ${\pi \over m}$ for some integer $m \geqslant 1$ (depending on the pair of faces), then the group generated by the reflections in these faces is an affine Coxeter group [@Maskit IV.H.11].
In what follows, $\Sigma$ is a Euclidean tiling giving rise to a Euclidean reflection group by Poincaré’s theorem (in Bourbaki’s terminology, it can also be seen as the natural geometric realization of the Coxeter complex of an affine Coxeter group, that is the affinization of the Tits’ cone of the latter group [@Lie456]).
\[defi - simplicial building\] Let $(\Sigma, W)$ be a Euclidean tiling and its associated Euclidean reflection group. A [(discrete) Euclidean builiding]{} of type $(\Sigma, W)$ is a polysimplicial complex, say $\mathcal{B}$, which is covered by subcomplexes all isomorphic to $\Sigma$ – called the [apartments]{} – such that the following incidence properties hold.
1. Any two cells of $\mathcal{B}$ lie in some apartment.
2. Given any two apartments, there is an isomorphism between them fixing their intersection in $\mathcal{B}$.
The cells in this context are called [facets]{} and the group $W$ is called the [Weyl group]{} of the building $\mathcal{B}$. The facets of maximal dimension are called [alcoves]{}.
The axioms of a Euclidean building can be motivated by metric reasons. Indeed, once the choice of a $W$-invariant Euclidean metric on $\Sigma$ has been made, there is a natural way the define a distance on the whole building: given any two points $x$ and $x'$ in $\mathcal{B}$, by [(SEB 1)]{} pick an apartment $\mathbb{A}$ containing them and consider the distance between $x$ and $x'$ taken in $\mathbb{A}$; then [(SEB 2)]{} implies that the so–obtained non-negative number doesn’t depend on the choice of $\mathbb{A}$. It requires further work to check that one defines in this way a distance on the building (i.e., to check that the triangle inequality holds [@Parreau Prop. II.1.3]).
\[rk - metric motivation\] The terminology “polysimplicial” refers to the fact that a building can be a direct product of simplicial complexes rather than merely a simplicial complex; this is why we provisionally used the terminology “cells” instead of “polysimplices” to state the axioms (as already mentioned, cells will henceforth be called facets – alcoves when they are top-dimensional).
Let us provide now some examples of discrete buildings corresponding to the already mentioned examples of Euclidean tilings.
\[ex - simplicial buildings\]
- The class of buildings of type $({\bf R}, {\rm D}_\infty)$ coincides with the class of trees without terminal vertex (recall that a tree is a $1$-dimensional simplicial complex – i.e., the geometric realization of a graph – without non-trivial loop [@SerreArbres]).
- A $2$-dimensional $\widetilde{{\rm A}_2}$-building is already impossible to draw, but roughly speaking it can be constructed by gluing half-tilings to an initial one along [walls]{} (i.e., fixed point sets of reflections) and by iterating these gluings infinitely many times provided a prescribed “shape” of neighborhoods of vertices is respected – see Example \[ex - SL3bis\] for further details on the local description of a building in this case.
It is important to note that axiom (ii) does [*not*]{} require that the isomorphism between apartments extends to a global automorphism of the ambient building. In fact, it may very well happen that for a given Euclidean building $\mathcal{B}$ we have ${\rm Aut}(\mathcal{B}) = \{1 \}$ (take for example a tree in which any two distinct vertices have distinct valencies). However, J. Tits’ classification of Euclidean buildings [@TitsCome] implies that in dimension $\geqslant 3$ any irreducible building comes – via Bruhat-Tits theory, see next remark – from a simple algebraic group over a local field, and therefore admits a large automorphism group. At last, note that there do exist 2-dimensional exotic Euclidean buildings, with interesting but unexpectedly small automorphism groups [@Barre].
\[rk - BrT\] In Sect. \[s - Bruhat-Tits general\], we will briefly introduce Bruhat-Tits theory. The main outcome of this important part of algebraic group theory is that, given a semisimple algebraic group $ {\rm G}$ over a local field $k$, there exists a discrete Euclidean building $\mathcal{B} = \mathcal{B}({\rm G},k)$ on which the group of rational points ${\rm G}(k)$ acts by isometries and [strongly transitively]{} (i.e., transitively on the inclusions of an alcove in an apartment).
Let ${\rm G}$ as above be the group ${\rm SL}_3$. Then the Euclidean building associated to ${\rm SL}_3$ is a Euclidean building in which every apartment is a Coxeter complex of type $\widetilde{{\rm A}_2}$, that is the previously described $2$-dimensional tiling of the Euclidean space ${\bf R}^2$ by regular triangles. Strong transitivity of the ${\rm SL}_3(k)$-action means here that given any alcoves (triangles) $c, c'$ and any apartments $\mathbb{A}, \mathbb{A}'$ such that $c \subset \mathbb{A}$ and $c' \subset \mathbb{A}'$ there exists $g \in {\rm SL}_3(k)$ such that $c'=g.c$ and $\mathbb{A}'=g.\mathbb{A}$.
The description of the apartments doesn’t depend on the local field $k$ (only on the Dynkin diagram of the semisimple group in general), but the field $k$ plays a role when one describes the combinatorial neighborhoods of facets, or small metric balls around vertices. Such subsets, which intersect finitely many facets when $k$ is a local field, are known to be realizations of some (spherical) buildings: these buildings are naturally associated to semisimple groups (characterized by some subdiagram of the Dynkin diagram of ${\rm G}$) over the residue field $\widetilde{k}$ of $k$.
\[ex - SL3bis\] For ${\rm G}={\rm SL}_3$ and $k={\bf Q}_p$, each sufficiently small ball around a vertex is the flag complex of a $2$-dimensional vector space over ${\bf Z}/p{\bf Z}$ and any edge in the associated Bruhat-Tits building is contained in the closure of exactly $p+1$ triangles. A suitably small metric ball around any point in the relative interior of an edge can be seen as a projective line over ${\bf Z}/p{\bf Z}$, that is the flag variety of ${\rm SL}_2$ over ${\bf Z}/p{\bf Z}$.
### Non-simplicial generalization {#sss - non-simplicial}
We will see, e.g. in \[ss - closed embedding\], that it is often necessary to understand and use reductive algebraic groups over valued fields for [*non-discrete*]{} valuations even if in the initial situation the ground field is discretely valued. The geometric counterpart to this is the necessary use of non-discrete Euclidean buildings. The investigation of such a situation is already covered by the fundamental work by F. Bruhat and J. Tits as written in [@BT1a] and [@BT1b], but the intrinsic definition of a non-discrete Euclidean building is not given there – see [@TitsCome] though, for a reference roughly appearing at the same time as Bruhat-Tits’ latest papers.
The definition of a building in this generalized context is quite similar to the discrete one (\[sss - simplicial\]) in the sense that it replaces an atlas by a collection of “slices” which are still called [*apartments*]{} and turn out to be maximal flat (i.e., Euclidean) subspaces once the building is endowed with a natural distance. What follows can be found for instance in A. Parreau’s thesis [@Parreau].
Let us go back to the initial question.
\[q - non-simplicial\] Which geometry can be associated to a group ${\rm G}(k)$ when ${\rm G}$ is a reductive group over $k$, a (not necessarily discretely) valued field?
The answer to this question is a long definition to swallow, so we will provide some explanations immediately after stating it.
The starting point is again a $d$-dimensional Euclidean space, say $\Sigma_{\rm vect}$, together with a finite group $\overline W$ in the group of isometries ${\rm Isom}(\Sigma_{\rm vect}) \simeq {\rm O}_d({\bf R})$. By definition, a [*vectorial wall*]{} in $\Sigma_{\rm vect}$ is the fixed-point set in $\Sigma_{\rm vect}$ of a reflection in $\overline{W}$ and a [*vectorial Weyl chamber*]{} is a connected component of the complement of the union of the walls in $\Sigma_{\rm vect}$, so that Weyl chambers are simplicial cones.
Now assume that we are given an affine Euclidean space $\Sigma$ with underlying Euclidean vector space $\Sigma_{\rm vect}$. We have thus ${\rm Isom}(\Sigma) \simeq {\rm Isom}(\Sigma_{\rm vect}) \ltimes \Sigma_{\rm vect}\simeq {\rm O}_d({\bf R}) \ltimes {\bf R}^d$. We also assume that we are given a group $W$ of (affine) isometries in $\Sigma$ such that the vectorial part of $W$ is $\overline W$ and such that there exists a point $x \in \Sigma$ and a subgroup ${\rm T} \subset {\rm Isom}(\Sigma)$ of translations satisfying $W = W_x \cdot {\rm T}$; we use here the notation $W_x = {\rm Stab}_W(x)$. A point $x$ satisfying this condition is called *special*.
\[defi - non-simplicial building\] Let $\mathcal{B}$ be a set and let $\mathcal{A} = \{f : \Sigma \to \mathcal{B} \}$ be a collection of injective maps, whose images are called [apartments]{}. We say that $\mathcal{B}$ is a [Euclidean building]{} of type $(\Sigma,W)$ if the apartments satisfy the following axioms.
- The family $\mathcal{A}$ is stable by precomposition with any element of $W$ (i.e., for any $f \in \mathcal{A}$ and any $w \in W$, we have $f \circ w \in \mathcal{A}$).
- For any $f,f' \in \mathcal{A}$ the subset $\mathcal{C}_{f,f'}= f'^{-1} \bigl( f(\Sigma) \bigr)$ is convex in $\Sigma$ and there exists $w \in W$ such that we have the equality of restrictions $(f^{-1} \circ f') \mid_{\mathcal{C}_{f,f'}} = w \mid_{\mathcal{C}_{f,f'}}$.
- Any two points of $\mathcal{B}$ are contained in a suitable apartment.
At this stage, there is a well-defined map $d : \mathcal{B}\times \mathcal{B} \to {\bf R}_{\geqslant 0}$ and we further require:
- Given any (images of) Weyl chambers, there is an apartment of $X$ containing sub-Weyl chambers of each.
- Given any apartment $\mathbb{A}$ and any point $x \in \mathbb{A}$, there is a $1$-lipschitz retraction map $r = r_{x,\mathbb{A}} : \mathcal{B} \to \mathbb{A}$ such that $r \mid_\mathbb{A} = {\rm id}_\mathbb{A}$ and $r^{-1}(x) = \{x \}$.
The above definition is taken from [@Parreau II.1.2]; in these axioms a [*Weyl chamber*]{} is the affine counterpart to the previously defined notion of a [*Weyl chamber*]{} and a “sub-Weyl chamber” is a translate of the initial Weyl chamber which is completely contained in the latter.
A different set of axioms is given in G. Rousseau’s paper [@RousseauGrenoble §6]. It is interesting because it provides a unified approach to simplicial and non-simplicial buildings via incidence requirements on apartments. The possibility to obtain a non-discrete building with Rousseau’s axioms is contained in the model for an apartment and the definition of a facet as a filter. The latter axioms are adapted to some algebraic situations which cover the case of Bruhat-Tits theory over non-complete valued fields – see [@RousseauGrenoble Remark 9.4] for more details and comparisons.
In this paper we do not use the plain word “chamber” though it is standard terminology in abstract building theory. This choice is made to avoid confusion: alcoves here are chambers (in the abstract sense) in Euclidean buildings and parallelism classes of Weyl chambers here are chambers (in the abstract sense) in spherical buildings at infinity of Euclidean buildings [@WeissAffine Chap. 8], [@AbramenkoBrown 11.8].
It is easy to see that, in order to prove that the map $d$ defined thanks to axioms [(EB 1)-(EB 3)]{} is a distance, it remains to check that the triangle inequality holds; this is mainly done by using the retraction given by axiom [(EB 5)]{}. The previously quoted metric motivation (Remark \[rk - metric motivation\]) so to speak became a definition. Note that the existence of suitable retractions is useful to other purposes.
The following examples of possibly non-simplicial Euclidean buildings correspond to the examples of simplicial ones given in Example \[ex - simplicial buildings\].
\[ex - non-simplicial buildings\]
- Consider the real line $\Sigma = {\bf R}$ and its isometry group ${\bf Z}/2{\bf Z} \ltimes {\bf R}$. Then a Euclidean building of type $({\bf R}, {\bf Z}/2{\bf Z} \ltimes {\bf R})$ is a real tree – see below.
- For a $2$-dimensional case extending simplicial $\widetilde{{\rm A}_2}$-buildings, a model for an apartment can be taken to be a maximal flat in the symmetric space of ${\rm SL}_3({\bf R})/{\rm SO}(3)$ acted upon by its stabilizer in ${\rm SL}_3({\bf R})$ (using the notion of singular geodesics to distinguish the walls). There is a geometric way to define the Weyl group and Weyl chambers (six directions of simplicial cones) in this differential geometric context – see [@MaubonGrenoble] for the general case of arbitrary symmetric spaces.
Here is a (purely metric) definition of real trees. It is a metric space $({\rm X},d)$ with the following two properties:
- it is [*geodesic*]{}: given any two points $x, x' \in {\rm X}$ there is a (continuous) map $\gamma : [0;d] \to {\rm X}$, where $d = d(x,x')$, such that $\gamma(0)=x$, $\gamma(d)=x'$ and $d\bigl(\gamma(s),\gamma(t) \bigr) = \, \mid\! s-t \!\mid$ for any $s,t \in [0;d]$;
- any geodesic triangle is a tripod (i.e., the union of three geodesic segments with a common end-point).
\[rk - asymptotic cones\] Non-simplicial Euclidean buildings became more popular since recent work of geometric (rather than algebraic) nature, where non-discrete buildings appear as asymptotic cones of symmetric spaces and Bruhat-Tits buildings [@KleLee].
The remark implies in particular that there exist non-discrete Euclidean buildings in any dimension, which will also be seen more concretely by studying spaces of non-Archimedean norms on a given vector space – see \[ss - SL(n) Bruhat-Tits\].
\[rk - BrT bis\] Note that given a reductive group ${\rm G}$ over a valued field $k$, Bruhat-Tits theory “often” provides a Euclidean building on which the group ${\rm G}(k)$ acts strongly transitively in a suitable sense (see Sect. \[s - Bruhat-Tits general\] for an introduction to this subject).
### More geometric properties {#sss - geometry of buildings}
We motivated the definitions of buildings by metric considerations, therefore we must mention the metric features of Euclidean buildings once these spaces have been defined. First, a Euclidean building always admits a metric whose restriction to any apartment is a (suitably normalized) Euclidean distance [@RousseauGrenoble Prop. 6.2]. Endowed with such a distance, a Euclidean building is always a geodesic metric space as introduced in the above metric definition of real trees (\[sss - non-simplicial\]).
[*Recall that we use the axioms [(EB)]{} from Def. \[defi - non-simplicial building\] to define a building; moreover we assume that the above metric is complete.*]{} This is sufficient for our purposes since we will eventually deal with Bruhat-Tits buildings associated to algebraic groups over complete non-Archimedean fields.
Let $(\mathcal{B}, d)$ be a Euclidean building endowed with such a metric. Then $(\mathcal{B}, d)$ satisfies moreover a remarkable non-positive curvature property, called the [*${\rm CAT}(0)$-property*]{} (where “CAT” seems to stand for Cartan-Alexandrov-Toponogov). Roughly speaking, this property says that geodesic triangles are at least as thin as in Euclidean planes. More precisely, the point is to compare a geodesic triangle drawn in $\mathcal{B}$ with “the” Euclidean triangle having the same edge lengths. A geodesic space is said to have the [*${\rm CAT}(0)$-property*]{}, or to [*be*]{} CAT(0), if a median segment in each geodesic triangle is at most as long as the corresponding median segment in the comparison triangle drawn in the Euclidean plane ${\bf R}^2$ (this inequality has to be satisfied for all geodesic triangles). Though this property is stated in elementary terms, it has very deep consequences [@RousseauGrenoble §7].
One first consequence is the uniqueness of a geodesic segment between any two points [@BriHae Chap. II.1, Prop. 1.4].
The main consequence is a famous and very useful fixed-point property. The latter statement is itself the consequence of a purely geometric one: any bounded subset in a complete, CAT(0)-space has a unique, metrically characterized, circumcenter [@AbramenkoBrown 11.3]. This implies that if a group acting by isometries on such a space (e.g., a Euclidean building) has a bounded orbit, then it has a fixed point. This is the [*Bruhat-Tits fixed point lemma*]{}; it applies for instance to any compact group of isometries.
Let us simply mention two very important applications of the Bruhat-Tits fixed point lemma (for simplicity, we assume that the building under consideration is discrete and locally finite – which covers the case of Bruhat-Tits buildings for reductive groups over local fields).
- The Bruhat-Tits fixed point lemma is used to classify maximal bounded subgroups in the isometry group of a building. Indeed, it follows from the definition of the compact open topology on the isometry group ${\rm Aut}(\mathcal{B})$ of a building $\mathcal{B}$, that a facet stabilizer is a compact subgroup in ${\rm Aut}(\mathcal{B})$. Conversely, a compact subgroup has to fix a point and this point can be sent to a point in a given fundamental domain for the action of ${\rm Aut}(\mathcal{B})$ on $\mathcal{B}$ (the isometry used for this conjugates the initial compact subgroup into the stabilizer of a point in the fundamental domain).
- Another consequence is that any Galois action on a Bruhat-Tits building has “sufficiently many” fixed points, since a Galois group is profinite hence compact. These Galois actions are of fundamental use in Bruhat-Tits theory, following the general idea – widely used in algebraic group theory – that an algebraic group ${\rm G}$ over $k$ is nothing else than a split algebraic group over the separable closure $k^s$, namely ${\rm G} \otimes_k k^s$, together with a semilinear action of ${\rm Gal}(k^s/k)$ on ${\rm G} \otimes_k k^s$ [@Borel AG §§11-14].
Arguments similar to the ones mentioned in 1. imply that, when $k$ is a local field, there are exactly $d+1$ conjugacy classes of maximal compact subgroups in ${\rm SL}_{d+1}(k)$. They are parametrized by the vertices contained in the closure of a given alcove (in fact, they are all isomorphic to ${\rm SL}_{d+1}(k^\circ)$ and are all conjugate under the action of ${\rm GL}_{d+1}(k)$ by conjugation).
One can make 2. a bit more precise. The starting point of Bruhat-Tits theory is indeed that a reductive group ${\rm G}$ over any field, say $k$, splits – hence in particular is very well understood – after extension to the separable closure $k^s$ of the ground field. Then, in principle, one can go down to the group ${\rm G}$ over $k$ by means of suitable Galois action – this is one leitmotiv in [@BoTi]. In particular, Borel-Tits theory provides a lot of information about the group ${\rm G}(k)$ by seeing it as the fixed-point set ${\rm G}(k^s)^{{\rm Gal}(k^s/k)}$. When the ground field $k$ is a valued field, then one can associate a Bruhat-Tits building $\mathcal{B} = \mathcal{B}({\rm G},k^s)$ to ${\rm
G} \otimes_k k^s$ together with an action by isometries of ${\rm
Gal}(k^s/k)$. The Bruhat-Tits building of ${\rm G}$ over $k$ is contained in the Galois fixed-point set $\mathcal{B}^{{\rm
Gal}(k^s/k)}$, but this is inclusion is strict in general: the Galois fixed-point set is bigger than the desired building [@RousseauOrsay III]; this point is detailed in \[ss-Galois\]. Still, this may be a good first approximation of Bruhat-Tits theory to have in mind. We refer to \[sss - descent and functoriality\] for further details.
The ${\rm SL}_n$ case {#ss - SL(n) Bruhat-Tits}
---------------------
We now illustrate many of the previous notions in a very explicit situation, of arbitrary dimension. Our examples are spaces of norms on a non-Archimedean vector space. They provide the easiest examples of Bruhat-Tits buildings, and are also very close to spaces occurring in Berkovich analytic geometry. In this section, we denote by ${\rm V}$ a $k$-vector space and by $d+1$ its (finite) dimension over $k$.
*Note that until Remark \[rk - extended building\] we assume that $k$ is a local field.*
### Goldman-Iwahori spaces {#sss - GI spaces}
The materiel of this subsection is classical and could be find, for example, in [@Weil2].
We are interested in the following space.
\[defi - GI\] The [Goldman-Iwahori]{} space of the $k$-vector space ${\rm V}$ is the space of non-Archimedean norms on ${\rm V}$; we denote it by $\mathcal{N}({\rm V},k)$. We denote by $\mathcal{X}({\rm V},k)$ the quotient space $\displaystyle {\mathcal{N}({\rm V},k) \big/ \sim}$, where $\sim$ is the equivalence relation which identifies two homothetic norms.
To be more precise, let $\parallel \cdot \parallel$ and $\parallel \cdot \parallel'$ be norms in $\mathcal{N}({\rm V},k)$. We have $\parallel \cdot \parallel \sim \parallel \cdot \parallel'$ if and only if there exists $c > 0$ such that $\parallel\! x \!\parallel \, = \, c \parallel\! x \!\parallel'$ for all $x \in {\rm V}$. In the sequel, we use the notation $[\cdot]_\sim$ to denote the class with respect to the homothety equivalence relation.
\[ex - norm\] Here is a simple way to construct non-Archimedean norms on ${\rm V}$. Pick a basis $\mathbf{e} = (e_0, e_1, \dots, e_d)$ in ${\rm V}$. Then for each choice of parameters $\underline c = (c_0, c_1, \dots, c_d) \in {\bf R}^{d+1}$, we can define the non-Archimedean norm which sends each vector $x = \sum_i \lambda_i e_i$ to $\max_i \{\exp(c_i) \mid\! \lambda_i \!\mid \}$, where $\mid \cdot \mid$ denotes the absolute value of $k$. We denote this norm by $\parallel \cdot \parallel_{\mathbf{e},{\underline c}}$.
We also introduce the following notation and terminology.
\[defi - adapted\]
- Let $\parallel \cdot \parallel$ be a norm and let $\mathbf{e}$ be a basis in ${\rm V}$. We say that $\parallel \cdot \parallel$ is [diagonalized]{} by $\mathbf{e}$ if there exists $\underline c \in {\bf R}^{d+1}$ such that $\parallel \cdot \parallel = \parallel \cdot \parallel_{\mathbf{e},{\underline c}}$; in this case, we also say that the basis $\mathbf{e}$ is [adapted]{} to the norm $\parallel \cdot \parallel$.
- Given a basis $\mathbf{e}$, we denote by $\widetilde{\mathbb{A}_{\mathbf{e}}}$ the set of norms diagonalized by $\mathbf{e}$:
$\widetilde{\mathbb{A}_{\mathbf{e}}} = \{ \parallel \cdot \parallel_{\mathbf{e},{\underline c}} \, : \, \underline c \in {\bf R}^{d+1} \}$.
- We denote by ${\mathbb{A}_{\mathbf{e}}}$ the quotient of $\widetilde{\mathbb{A}_{\mathbf{e}}}$ by the homothety equivalence relation: $\displaystyle {\mathbb{A}_{\mathbf{e}}} = {\widetilde{\mathbb{A}_{\mathbf{e}}} / \sim}$.
Note that the space $\widetilde{\mathbb{A}_{\mathbf{e}}}$ is naturally an affine space with underlying vector space ${\bf R}^{d+1}$: the free transitive ${\bf R}^{d+1}$-action is by shifting the coefficients $c_i$ which are the logarithms of the “weights” $\exp(c_i)$ for the norms $\parallel \cdot \parallel_{\mathbf{e},{\underline c}} : \sum_i \lambda_i e_i \mapsto \max_{0 \leqslant i \leqslant d}\{\exp(c_i) \mid\! \lambda_i \!\mid \}$. Under this identification of affine spaces, we have: $\displaystyle {\mathbb{A}_{\mathbf{e}}} \simeq {{\bf R}^{d+1} / {\bf R}(1,1,\dots, 1)} \simeq {\bf R}^d$.
\[rk - GI apartment\] The space $\mathcal{X}({\rm V},k)$ will be endowed with a Euclidean building structure (Th.\[th - GI building general\]) in which the spaces ${\mathbb{A}_{\mathbf{e}}}$ – with $\mathbf{e}$ varying over the bases of ${\rm V}$ – will be the apartments.
The following fact can be generalized to more general valued fields than local fields but is [*not*]{} true in general (Remark \[rk - Berko and split norms\]).
\[prop - adapted basis\] Every norm of $\mathcal{N}({\rm V},k)$ admits an adapted basis in ${\rm V}$.
[*Proof*]{}.— Let $\parallel \cdot \parallel$ be a norm of $\mathcal{N}({\rm V},k)$. We prove the result by induction on the dimension of the ambient $k$-vector space. Let $\mu$ be any non-zero linear form on ${\rm V}$. The map ${\rm V}-\{0 \} \to {\bf R}_+$ sending $y$ to $\displaystyle {\mid\! \mu(y) \!\mid \over \parallel\! y \!\parallel}$ naturally provides, by homogeneity, a continuous map $\phi : \mathbf{P}({\rm V})(k) \to {\bf R}_+$. Since $k$ is locally compact, the projective space $\mathbf{P}({\rm V})(k)$ is compact, therefore there exists an element $x \in {\rm V}-\{ 0 \}$ at which $\phi$ achieves its supremum, so that
$(*)$ $\displaystyle {\mid\! \mu(z) \!\mid \over \mid\! \mu(x) \!\mid}\parallel\! x \!\parallel \leqslant \parallel\! z \!\parallel$ for any $z \in {\rm V}$.
Let $z$ be an arbitrary vector of ${\rm V}$. We write $\displaystyle z = y + {\mu(z) \over \mu(x)} x$ according to the direct sum decomposition ${\rm V} = {\rm Ker}(\mu) \oplus k x$. By the ultrametric inequality satisfied by $\parallel \cdot \parallel$, we have
$(**)$ $\displaystyle \parallel\! z \!\parallel \leqslant \max \{\parallel\! y \!\parallel; {\mid\! \mu(z) \!\mid \over \mid\! \mu(x) \!\mid}\parallel\! x \!\parallel \}$
and
$(***)$ $\displaystyle \parallel\! y \!\parallel \leqslant \max \{\parallel\! z \!\parallel; {\mid\! \mu(z) \!\mid \over \mid\! \mu(x) \!\mid}\parallel\! x \!\parallel \}$.
Inequality $(*)$ says that $\displaystyle \max \{\parallel\! z \!\parallel; {\mid\! \mu(z) \!\mid \over \mid\! \mu(x) \!\mid}\parallel\! x \!\parallel \} = \parallel\! z \!\parallel$, so $(***)$ implies $\parallel\! z \!\parallel \geqslant \parallel\! y \!\parallel$. The latter inequality together with $(*)$ implies that $\displaystyle \parallel\! z \!\parallel \geqslant \max \{\parallel\! y \!\parallel; {\mid\! \mu(z) \!\mid \over \mid\! \mu(x) \!\mid}\parallel\! x \!\parallel \}$. Combining this with $(**)$ we obtain the equality $\displaystyle \parallel\! z \!\parallel = \max \{\parallel\! y \!\parallel; {\mid\! \mu(z) \!\mid \over \mid\! \mu(x) \!\mid}\parallel\! x \!\parallel \}$. Applying the induction hypothesis to ${\rm Ker}(\mu)$, we obtain a basis adapted to the restriction of $\parallel \cdot \parallel$ to ${\rm Ker}(\mu)$. Adding $x$ we obtain a basis adapted to $\parallel \cdot \parallel$, as required (note that $ {\mu(z) \over \mu(x)}$ is the coordinate corresponding to the vector $x$ in any such basis). $\square$
Actually, we can push a bit further this existence result about adapted norms.
\[prop - adapted basis - 2\] For any two norms of $\mathcal{N}({\rm V},k)$ there is a basis of ${\rm V}$ simultaneously adapted to them.
[*Proof*]{}.— We are now given two norms, say $\parallel \cdot \parallel$ and $\parallel \cdot \parallel'$, in $\mathcal{N}({\rm V},k)$. In the proof of Prop. \[prop - adapted basis\], the choice of a non-zero linear form $\mu$ had no importance. In the present situation, we will take advantage of this freedom of choice. We again argue by induction on the dimension of the ambient $k$-vector space.
By homogeneity, the map ${\rm V}-\{0 \} \to {\bf R}_+$ sending $y$ to $\displaystyle {\parallel\! y \!\parallel \over \parallel\! y \!\parallel'}$ naturally provides a continuous map $\psi : \mathbf{P}({\rm V})(k) \to {\bf R}_+$. Again because the projective space $\mathbf{P}({\rm V})(k)$ is compact, there exists $x \in {\rm V}-\{ 0 \}$ at which $\psi$ achieves its supremum, so that
$\displaystyle {\parallel\! y \!\parallel \over\parallel\! x \!\parallel} \leqslant {\parallel\! y \!\parallel' \over\parallel\! x \!\parallel'}$ for any $y \in {\rm V}$.
Now we endow the dual space ${\rm V}^*$ with the operator norm $\parallel \cdot \parallel^*$ associated to $\parallel \cdot \parallel$ on ${\rm V}$. Since ${\rm V}$ is finite-dimensional, by biduality (i.e. the normed vector space version of ${\rm V}^{**} \simeq {\rm V}$), we have the equality $\displaystyle \parallel\! x \!\parallel = \sup_{\mu \in {\rm V}^*-\{ 0\}} {\mid\! \mu(x) \!\mid \over \parallel\! \mu \!\parallel^*}$. By homogeneity and compactness, there exists $\lambda \in {\rm V}^*-\{0 \}$ such that $\displaystyle \parallel\! x \!\parallel = {\mid\! \lambda(x) \!\mid \over \parallel\! \lambda \!\parallel^*}$. For arbitrary $y \in {\rm V}$ we have $\mid\! \lambda(y) \!\mid\, \leqslant \, \parallel\! y \!\parallel \cdot \parallel\! \lambda \!\parallel^*$, so the definition of $x$ implies that
$\displaystyle {\mid\! \lambda(y) \!\mid \over \mid\! \lambda(x) \!\mid} \leqslant {\parallel\! y \!\parallel \over\parallel\! x \!\parallel}$ for any $y \in {\rm V}$.
In other words, we have found $x \in {\rm V}$ and $\lambda \in {\rm V}^*$ such that
$\displaystyle {\mid\! \lambda(y) \!\mid \over \mid\! \lambda(x) \!\mid} \leqslant {\parallel\! y \!\parallel \over\parallel\! x \!\parallel}\leqslant {\parallel\! y \!\parallel' \over\parallel\! x \!\parallel'}$ for any $y \in {\rm V}$.
Now we are in position to apply the arguments of the proof of Prop. \[prop - adapted basis\] to both $\parallel \cdot \parallel$ and $\parallel \cdot \parallel'$ to obtain that $\displaystyle \parallel\! z \!\parallel = \max \{\parallel\! y \!\parallel; {\mid\! \lambda(z) \!\mid \over \mid\! \lambda(x) \!\mid}\parallel\! x \!\parallel \}$ and $\displaystyle \parallel\! z \!\parallel' = \max \{\parallel\! y \!\parallel'; {\mid\! \lambda(z) \!\mid \over \mid\! \lambda(x) \!\mid}\parallel\! x \!\parallel' \}$ for any $z \in {\rm V}$ decomposed as $z = x + y$ with $y \in {\rm Ker}(\lambda)$. It remains then to apply the induction hypothesis (i.e., that the desired statement holds in the ambient dimension minus 1). $\square$
### Connection with building theory {#sss - GI building}
It is now time to describe the connection between Goldman-Iwahori spaces and Euclidean buildings. As already mentioned, the subspaces ${\mathbb{A}_{\mathbf{e}}}$ will be the apartments in $\mathcal{X}({\rm V},k)$ (Remark \[rk - GI apartment\]).
Let us fix a basis $\mathbf{e}$ in ${\rm V}$ and consider first the bigger affine space $\widetilde{\mathbf{A}_{\mathbf{e}}} = \{ \parallel \cdot \parallel_{\mathbf{e},{\underline c}} \, : \, \underline c \in {\bf R}^{d+1} \} \simeq {\bf R}^{d+1}$. The symmetric group $\mathcal{S}_{d+1}$ acts on this affine space by permuting the coefficients $c_i$. This is obviously a faithful action and we have another one given by the affine structure. We obtain in this way an action of the group $\mathcal{S}_{d+1} \ltimes {\bf R}^{d+1}$ on $\widetilde{\mathbf{A}_{\mathbf{e}}}$ and, after passing to the quotient space, we can see ${\mathbb{A}_{\mathbf{e}}}$ as the ambient space of the Euclidean tiling attached to the affine Coxeter group of type $\widetilde{{\rm A}_d}$ (the latter group is isomorphic to $\mathcal{S}_{d+1} \ltimes {\bf Z}^d$). The following result is due to Bruhat-Tits, elaborating on Goldman-Iwahori’s investigation of the space of norms $\mathcal{N}({\rm V},k)$ [@GoldmanIwahori].
\[th - GI building simplicial\] The space $\displaystyle \mathcal{X}({\rm V},k) = \displaystyle {\mathcal{N}({\rm V},k) / \sim}$ is a simplicial Euclidean building of type $\widetilde{{\rm A}_d}$, where $d+1 = {\rm dim}({\rm V})$; in particular, the apartments are isometric to ${\bf R}^d$ and the Weyl group is isomorphic to $\mathcal{S}_{d+1} \ltimes {\bf Z}^d$.
[*Reference*]{}.— In [@BT1a 10.2] this is stated in group-theoretic terms, so one has to combine the quoted statement with \[[**loc. cit.**]{}, 7.4\] in order to obtain the above theorem. This will be explained in Sect. \[s - Bruhat-Tits general\]. $\square$
The 0-skeleton (i.e., the vertices) for the simplicial structure corresponds to the [*$k^\circ$-lattices*]{} in the $k$-vector space ${\rm V}$, that is the free $k^\circ$-submodules in ${\rm V}$ of rank $d+1$. To a lattice $\mathcal{L}$ is attached a norm $\parallel \cdot \parallel_\mathcal{L}$ by setting $\parallel\! x \!\parallel_\mathcal{L} = \inf \{\mid\! \lambda \!\mid \, :\, \lambda \in k^\times$ and $\lambda^{-1} x \in \mathcal{L} \}$. One recovers the $k^\circ$-lattice $\mathcal{L}$ as the unit ball of the norm $\parallel \cdot \parallel_\mathcal{L}$.
\[rk - extended building\] Note that the space $\mathcal{N}({\rm V},k)$ is an extended building in the sense of [@TitsCorvallis]; this is, roughly speaking, a building to which is added a Euclidean factor in order to account geometrically for the presence of a center of positive dimension.
Instead of trying to prove this result, let us mention that Prop. \[prop - adapted basis - 2\] says, in our building-theoretic context, that any two points are contained in an apartment. In other words, this proposition implies axiom (SEB 1) of Def. \[defi - simplicial building\]: it is the non-Archimedean analogue of the fact that any two real scalar products are diagonalized in a suitable common basis (Gram-Schmidt).
Now let us skip the hypothesis that $k$ is a local field. If $k$ is a not discretely valued, then it is not true in general that every norm in $\mathcal{N}({\rm V},k)$ can be diagonalized in some suitable basis. Therefore we introduce the following subspace:
$\mathcal{N}({\rm V},k)^{\rm diag} = \{$norms in $\mathcal{N}({\rm V},k)$ admitting an adapted basis$\}$.
\[rk - Berko and split norms\] We will see (Remark \[rk-approximation\]) that the connection between Berkovich projective spaces and Bruhat-Tits buildings helps to understand why $\mathcal{N}({\rm V},k) \,\mathbf{-}\, \mathcal{N}({\rm V},k)^{\rm diag} \neq \varnothing$ if and only if the valued field $k$ is [not]{} maximally complete (one also says spherically complete).
Thanks to the subspace $\mathcal{N}({\rm V},k)^{\rm diag}$, we can state the result in full generality.
\[th - GI building general\] The space $\displaystyle \mathcal{X}({\rm V},k) = \displaystyle {\mathcal{N}({\rm V},k)^{\rm diag} / \sim}$ is a Euclidean building of type $\widetilde{{\rm A}_d}$ in which the apartments are isometric to ${\bf R}^d$ and the Weyl group is isomorphic to $\mathcal{S}_{d+1} \ltimes \Lambda$ where $\Lambda$ is a translation group, which is discrete if and only if so is the valuation of $k$.
[*Reference*]{}.— This is proved for instance in [@Parreau III.1.2]; see also [@BT1b] for a very general treatment. $\square$
\[ex - GI real tree\] For $d=1$, i.e. when ${\rm V} \simeq k^2$, the Bruhat-Tits building $\mathcal{X}({\rm V},k) = \displaystyle {\mathcal{N}({\rm V},k)^{\rm diag} / \sim}$ given by Theorem \[th - GI building general\] is a tree, which is a (non-simplicial) real tree whenever $k$ is not discretely valued.
### Group actions {#sss - actions on GI}
After illustrating the notion of a building thanks to Goldman-Iwahori spaces, we now describe the natural action of a general linear group over the valued field $k$ on its Bruhat-Tits building. We said that buildings are usually used to better understand groups which act sufficiently transitively on them. We therefore have to describe the ${\rm GL}({\rm V},k)$-action on $\mathcal{X}({\rm V},k)$ given by precomposition on norms (that is, $g.\parallel \cdot \parallel \, = \, \parallel \cdot \parallel \circ \, g^{-1}$ for any $g \in {\rm GL}({\rm V},k)$ and any $\parallel \cdot \parallel \in \mathcal{N}({\rm V},k)$). Note that we have the formula
$g.\parallel \cdot \parallel_{\mathbf{e},{\underline c}} = \parallel \cdot \parallel_{g.\mathbf{e},{\underline c}}$.
We will also explain how this action can be used to find interesting decompositions of ${\rm GL}({\rm V},k)$. Note that the ${\rm GL}({\mathrm{V}},k)$-action on $\mathcal{X}({\mathrm{V}},k)$ factors through an action by the group ${\rm PGL}({\mathrm{V}},k)$.
[*For the sake of simplicity, we assume that $k$ is discretely valued until the rest of this section*]{}.
We describe successively: the action of monomial matrices on the corresponding apartment, stabilizers, fundamental domains and the action of elementary unipotent matrices on the buildings (which can be thought of as “foldings” of half-apartments fixing complementary apartments).
First, it is very useful to restrict our attention to apartments. Pick a basis $\mathbf{e}$ of ${\rm V}$ and consider the associated apartment $\mathbb{A}_\mathbf{e}$. The stabilizer of $\mathbb{A}_\mathbf{e}$ in ${\rm GL}({\rm V},k)$ consists of the subgroup of linear automorphisms $g$ which are [*monomial*]{} with respect to $\mathbf{e}$, that is whose matrix expression with respect to $\mathbf{e}$ has only one non-zero entry in each row and in each column; we denote ${\rm N}_\mathbf{e} = {\rm Stab}_{{\rm GL}({\rm V},k)}(\mathbb{A}_\mathbf{e})$. Any automorphism in ${\rm N}_\mathbf{e}$ lifts a permutation of the indices of the vectors $e_i$ ($0 \leqslant i \leqslant d)$ in $\mathbf{e}$. This defines a surjective homomorphism ${\rm N}_\mathbf{e}\twoheadrightarrow \mathcal{S}_{d+1}$ whose kernel is the group, say ${\rm D}_\mathbf{e}$, of the linear automorphisms diagonalized by $\mathbf{e}$. The group ${\rm D}_\mathbf{e} \cap {\rm SL}({\rm V},k)$ lifts the translation subgroup of the (affine) Weyl group $\mathcal{S}_{d+1} \ltimes {\bf Z}^d$ of $\mathcal{X}({\rm V},k)$. Note that the latter translation group consists of the translations contained in the group generated by the reflections in the codimension 1 faces of a given alcove, therefore this group is (of finite index but) smaller than the “obvious” group given by translations with integral coefficients with respect to the basis $\mathbf{e}$. For any $\underline{\lambda} \in (k^{\times})^n$, we have the following “translation formula”:
$\underline{\lambda}.\parallel \cdot \parallel_{\mathbf{e},{\underline c}} = \parallel \cdot \parallel_{\mathbf{e},(c_i - \log \mid\! \lambda_i \!\mid)_i}$,
\[ex - respect type tree\] When $d=1$ and when $k$ is local, the translations of smallest displacement length in the (affine) Weyl group of the corresponding tree are translations whose displacement length along their axis is equal to twice the length of an edge.
The fact stated in the example corresponds to the general fact that the ${\rm SL}({\rm V},k)$-action on $\mathcal{X}({\rm V},k)$ is [*type*]{} (or [*color*]{})[*-preserving*]{}: choosing $d+1$ colors, one can attach a color to each [*panel*]{} (= codimension 1 facet) so that each color appears exactly once in the closure of any alcove; a panel of a given color is sent by any element of ${\rm SL}({\rm V},k)$ to a panel of the same color. Note that the action of ${\rm GL}({\mathrm{V}},k)$, hence also of ${\rm PGL}({\mathrm{V}},k)$, on $\mathcal{X}({\rm V},k)$ is not type-preserving since ${\rm PGL}({\mathrm{V}},k)$ acts transitively on the set of vertices.
It is natural to first describe the isotropy groups for the action we are interested in.
\[prop - stabilizers\] We have the following description of stabilizers:
${\rm Stab}_{{\rm GL}({\rm V},k)}(\parallel \cdot \parallel_{\mathbf{e},{\underline c}})
= \{g \in {\rm GL}({\rm V},k) : {\rm det}(g) = 1$ and $\log(\mid\! g_{ij} \!\mid) \leqslant c_j - c_i \}$,
where $[ g_{ij} ]$ is the matrix expression of ${\rm GL}({\rm V},k)$ with respect to the basis $\mathbf{e}$.
[*Reference*]{}.— This is for instance [@Parreau Cor. III.1.4]. $\square$
There is also a description of the stabilizer group in ${\rm SL}({\mathrm{V}},k)$ as the set of matrices stabilizing a point with respect to a tropical matrix operation [@Wer11 Prop. 2.4].
We now turn our attention to fundamental domains. Let $x$ be a vertex in $\mathcal{X}({\rm V},k)$. Fix a basis $\mathbf{e}$ such that $x = [\parallel \cdot \parallel_{\mathbf{e},{\underline 0}}]_\sim$. Then we have an apartment $\mathbb{A}_\mathbf{e}$ containing $x$ and the inequations
$c_0 \leqslant c_1 \leqslant \dots \leqslant c_d$
define a Weyl chamber with tip $x$ (after passing to the homothety classes). The other Weyl chambers with tip $x$ contained in $\mathbb{A}_\mathbf{e}$ are obtained by using the action of the spherical Weyl group $\mathcal{S}_{d+1}$, which amounts to permuting the indices of the $c_i$’s (this action is lifted by the action of monomial matrices with coefficients $\pm 1$ and determinant 1).
Accordingly, if we denote by $\varpi$ a uniformizer of $k$, then the inequations
$c_0 \leqslant c_1 \leqslant \dots \leqslant c_d$ and $c_d - c_0 \leqslant -\log \mid\! \varpi \!\mid$
define an alcove (whose boundary contains $x$) and any other alcove in $\mathbb{A}_\mathbf{e}$ is obtained by using the action of the affine Weyl group $\mathcal{S}_{d+1} \ltimes {\bf Z}^d$.
\[prop - fundamental domains\] Assume $k$ is local. We have the following description of fundamental domains.
- Given a vertex $x$, any Weyl chamber with tip $x$ is a fundamental domain for the action of the maximal compact subgroup ${\rm Stab}_{{\rm SL}({\rm V},k)}(x)$ on $\mathcal{X}({\rm V},k)$.
- Any alcove is a fundamental domain for the natural action of ${\rm SL}({\rm V},k)$ on the building $\mathcal{X}({\rm V},k)$.
If we abandon the hypothesis that $k$ is a local field and assume the absolute value of $k$ is surjective (onto ${\bf R}_{\geqslant 0}$), then the ${\rm SL}({\rm V},k)$-action on $\mathcal{X}({\rm V},k)$ is transitive.
[*Sketch of proof*]{}.— (ii) follows from (i) and from the previous description of the action of the monomial matrices of ${\rm N}_\mathbf{e}$ on $\mathbb{A}_\mathbf{e}$ (note that ${\rm SL}({\rm V},k)$ is type-preserving, so a fundamental domain cannot be strictly smaller than an alcove).
(i). A fundamental domain for the action of the symmetric group $\mathcal{S}_{d+1}$ as above on the apartment $\mathbb{A}_\mathbf{e}$ is given by a Weyl chamber with tip $x$, and the latter symmetric group is lifted by elements in ${\rm Stab}_{{\rm SL}({\rm V},k)}(x)$. Therefore it is enough to show that any point of the building can be mapped into $\mathbb{A}_\mathbf{e}$ by an element of ${\rm Stab}_{{\rm SL}({\rm V},k)}(x)$. Pick a point $z$ in the building and consider a basis $\mathbf{e'}$ such that $\mathbb{A}_\mathbf{e'}$ contains both $x$ and $z$ (Prop. \[prop - adapted basis - 2\]). We can write $x = \, \parallel \cdot \parallel_{\mathbf{e},0} \, = \, \parallel \cdot \parallel_{\mathbf{e'},{\underline c}}$, with weights $\underline c$ in $\log \mid\! k^\times \!\mid$ since $x$ is a vertex. After dilation, if necessary, of each vector of the basis $\mathbf{e'}$, we may – and shall – assume that $\underline c = 0$. Pick $g \in {\rm SL}({\rm V},k)$ such that $g.\mathbf{e}=\mathbf{e'}$. Since $\mathbf{e}$ and $\mathbf{e'}$ span the same lattice $L$ over $k^\circ$, which is the unit ball for $x$ (see comment after Th. \[th - GI building simplicial\]), we have $g.L=L$ and therefore $g$ stabilizes $x$. We have therefore found $g \in {\rm Stab}_{{\rm SL}({\rm V},k)}(x)$ with $g.\mathbb{A}_\mathbf{e} = \mathbb{A}_\mathbf{e'}$, in particular $g^{-1}.z$ belongs to $\mathbb{A}_\mathbf{e}$. $\square$
\[rk - geometric Cartan\] Point (i) above is the geometric way to state the so-called Cartan decomposition: ${\rm SL}({\rm V},k) = {\rm Stab}_{{\rm SL}({\rm V},k)}(x) \cdot \overline{{\rm T}^+} \cdot {\rm Stab}_{{\rm SL}({\rm V},k)}(x)$, where $\overline{{\rm T}^+}$ is the semigroup of linear automorphisms $t$ diagonalized by $\mathbf{e}$ and such that $t.x$ belongs to a fixed Weyl chamber in $\mathbb{A}_\mathbf{e}$ with tip $x$. The Weyl chamber can be chosen so that $\overline{{\rm T}^+}$ consists of the diagonal matrices whose diagonal coefficients are powers of some given uniformizer with the exponents increasing along the diagonal. Let us recall how to prove this by means of elementary arguments [@PlatonovRapinchuk §3.4 p. 152]. Let $g \in {\rm SL}({\rm V},k)$; we pick $\lambda \in k^\circ$ so that $\lambda g$ is a matrix of ${\rm GL}({\rm V},k)$ with coefficients in $k^\circ$. By interpreting left and right multiplication by elementary unipotent matrices as matrix operations on rows and columns, and since $k^\circ$ is a principal ideal domain, we can find $p,p' \in {\rm SL}_{d+1}(k^\circ)$ such that $p^{-1}\lambda g p'^{-1}$ is a diagonal matrix (still with coefficients in $k^\circ$), which we denote by $d$. Therefore, we can write $g = p \lambda^{-1}d p'$; and since $g$, $p$ and $p'$ have determinant 1, so does $t=\lambda^{-1}d$. It remains to conjugate $\lambda^{-1}d$ by a suitable monomial matrix with coefficients $\pm 1$ and determinant 1 in order to obtain the desired decomposition.
At the beginning of this subsection, we described the action of linear automorphisms on an apartment when the automorphisms are diagonalized by a basis defining the apartment. One last interesting point is the description of the action of elementary unipotent matrices (for a given basis). The action looks like a “folding” in the building, fixing a suitable closed half-apartment.
More precisely, let us introduce the elementary unipotent matrices $u_{ij}(\nu) = {\rm id}+ \nu {\rm E}_{ij}$ where $\nu \in k$ and ${\rm E}_{ij}$ is the matrix whose only non-zero entry is the $(i,j)$-th one, equal to 1.
\[prop - folding\] The intersection $\widetilde{\mathbb{A}_{\mathbf{e}}} \cap u_{ij}(\lambda).\widetilde{\mathbb{A}_{\mathbf{e}}}$ is the half-space of $\widetilde{\mathbb{A}_{\mathbf{e}}}$ consisting of the norms $\parallel \cdot \parallel_{\mathbf{e},{\underline c}}$ satisfying $c_j - c_i \geqslant \log \mid\! \lambda \!\mid$. The isometry given by the matrix $u_{ij}(\lambda)$ fixes pointwise this intersection and the image of the open half-apartment $\widetilde{\mathbb{A}_{\mathbf{e}}} \mathbf{-} \{\parallel\! \cdot \!\parallel_{\mathbf{e},{\underline c}} : c_j - c_i \geqslant \log \mid\! \lambda \!\mid \}$ is (another half-apartment) disjoint from $\widetilde{\mathbb{A}_{\mathbf{e}}}$.
[*Proof*]{}.— In the above notation, we have $u_{ij}(\nu)(\sum_i \lambda_i e_i) = \sum_{k \neq i} \lambda_k e_k + (\lambda_i + \nu\lambda_j) e_i$ for any $\nu \in k$.
First, we assume that we have $u_{ij}(\lambda).\parallel \cdot \parallel_{\mathbf{e},{\underline c}} = \parallel \cdot \parallel_{\mathbf{e},{\underline c}}$. Then, applying this equality of norms to the vector $e_j$ provides $e^{c_j} = \max \{ e^{c_j} ; e^{c_i} \mid\! \lambda \!\mid \}$, hence the inequality $c_j - c_i \geqslant \log \mid\! \lambda \!\mid$.
Conversely, pick a norm $\parallel\! \cdot \!\parallel_{\mathbf{e},{\underline c}}$ such that $c_j - c_i \geqslant \log \mid\! \lambda \!\mid$ and let $x = \sum_i \lambda_i e_i$. By the ultrametric inequality, we have $e^{c_i}\mid\! \lambda_i-\lambda\lambda_j\!\mid \leqslant \max \{ e^{c_i}\mid\! \lambda_i \!\mid ; e^{c_i} \mid\! \lambda \!\mid \mid\! \lambda_j \!\mid \}$, and the assumption $c_j - c_i \geqslant \log \mid\! \lambda \!\mid$ implies that $e^{c_i}\mid\! \lambda_i-\lambda\lambda_j\!\mid \leqslant \max \{ e^{c_i}\mid\! \lambda_i \!\mid ; e^{c_j}\mid\! \lambda_j \!\mid\}$, so that $e^{c_i}\mid\! \lambda_i-\lambda\lambda_j\!\mid \leqslant \max_{1 \leqslant \ell \leqslant d} e^{c_\ell} \mid\! \lambda_\ell \!\mid$. Therefore we obtain that $u_{ij}(\lambda).\parallel x \parallel_{\mathbf{e},{\underline c}} \leqslant \parallel x \parallel_{\mathbf{e},{\underline c}}$ for any vector $x$. Replacing $\lambda$ by $-\lambda$ and $x$ by $u_{ij}(-\lambda).x$, we finally see that the norms $u_{ij}(\lambda).\parallel \cdot \parallel_{\mathbf{e},{\underline c}}$ and $\parallel \cdot \parallel_{\mathbf{e},{\underline c}}$ are the same when $c_j - c_i \geqslant \log \mid\! \lambda \!\mid$. We have thus proved that the fixed-point set of $u_{ij}(\lambda)$ in $\widetilde{\mathbb{A}_{\mathbf{e}}}$ is the closed half-space ${\rm D}_\lambda = \{\parallel\! \cdot \!\parallel_{\mathbf{e},{\underline c}} : c_j - c_i \geqslant \log \mid\! \lambda \!\mid \}$.
It follows from this that $\widetilde{\mathbb{A}_{\mathbf{e}}} \cap u_{ij}(\lambda).\widetilde{\mathbb{A}_{\mathbf{e}}}$ contains ${\rm D}_\lambda$. Assume that $\widetilde{\mathbb{A}_{\mathbf{e}}} \cap u_{ij}(\lambda).\widetilde{\mathbb{A}_{\mathbf{e}}} \supsetneq {\rm D}_\lambda$ in order to obtain a contradiction. This would provide norms $\parallel \cdot \parallel$ and $\parallel \cdot \parallel'$ in $\widetilde{\mathbb{A}_{\mathbf{e}}} - {\rm D}_\lambda$ with the property that $u_{ij}(\lambda).\parallel \cdot \parallel=\parallel \cdot \parallel'$. But we note that a norm in $\widetilde{\mathbb{A}_{\mathbf{e}}}-{\rm D}_\lambda$ is characterized by its orthogonal projection onto the boundary hyperplane $\partial {\rm D}_\lambda$ and by its distance to $\partial {\rm D}_\lambda$. Since $u_{ij}(\lambda)$ is an isometry which fixes ${\rm D}_\lambda$ we conclude that $\parallel \cdot \parallel = \parallel \cdot \parallel'$, which is in contradiction with the fact that the fixed-point set of $u_{ij}(\lambda)$ in $\widetilde{\mathbb{A}_{\mathbf{e}}}$ is exactly ${\rm D}_\lambda$. $\square$
Special linear groups, Berkovich and Drinfeld spaces {#s - SL(n) Berkovich-Drinfeld}
====================================================
We ended the previous section by an elementary construction of the building of special linear groups over discretely valued non-Archimedean field. The generalization to an arbitrary reductive group over such a field is significantly harder and requires the full development of Bruhat-Tits, which will be the topic of Section \[s - Bruhat-Tits general\]. Before diving into the subtelties of buildings construction, we keep for a moment the particular case of special linear groups and describe a realization of their buildings in the framework of Berkovich’s analytic geometry, which leads very naturally to a compactification of those buildings. The general picture, namely Berkovich realizations and compactifications of general Bruhat-Tits buildings will be dealt with in Sect. \[s - general compactifications\]).
Roughly speaking understanding the realization (resp. compactification) described below of the building of a special linear group amounts to understanding (homothety classes of) norms on a non-Archimedean vector space (resp. their degenerations), using the viewpoint of multiplicative seminorms on the corresponding symmetric algebra.
A useful reference for Berkovich theory is [@Temkin]. [*Unless otherwise indicated, we assume in this section that $k$ is a local field*]{}.
Drinfeld upper half spaces and Berkovich affine and projective spaces {#ss - Drinfeld}
---------------------------------------------------------------------
Let ${\rm V}$ be a finite-dimensional vector space over $k$, and let ${\mathrm{S}}^{\bullet}{\mathrm{V}}$ be the symmetric algebra of ${\mathrm{V}}$. It is a graded $k$-algebra of finite type. Every choice of a basis $v_0, \ldots, v_d$ of ${\rm V}$ induces an isomorphism of ${\mathrm{S}}^{\bullet}{\mathrm{V}}$ with the polynomial ring over $k$ in $d+1$ indeterminates. The affine space $\mathbf{A}({\mathrm{V}})$ is defined as the spectrum $\mathrm{Spec}({\mathrm{S}}^{\bullet}{\mathrm{V}})$, and the projective space $\mathbf{P}({\mathrm{V}})$ is defined as the projective spectrum $\mathrm{Proj}({\mathrm{S}}^{\bullet}{\mathrm{V}})$. These algebraic varieties give rise to analytic spaces in the sense of Berkovich, which we briefly describe below.
### Drinfeld upper half-spaces in analytic projective spaces {#sss - Drinfeld}
As a topological space, the Berkovich affine space $\mathbf{A}({\mathrm{V}})^{{\mathrm{an}}}$ is the set of all multiplicative seminorms on ${\mathrm{S}}^{\bullet}{\mathrm{V}}$ extending the absolute value on $k$ together with the topology of pointwise convergence. The Berkovich projective space $\mathbf{P}({\mathrm{V}})^{\rm an}$ is the quotient of $\mathbf{A}({\mathrm{V}})^{{\mathrm{an}}} - \{0\}$ modulo the equivalence relation $\sim$ defined as follows: $\alpha \sim \beta$, if and only if there exists a constant $c>0$ such that for all $f$ in ${\mathrm{S}}^n {\mathrm{V}}$ we have $\alpha(f) = c^n \beta(f)$. There is a natural ${\rm PGL}({\rm V})$-action on $\mathbf{P}({\mathrm{V}})^{\rm an}$ given by $g \alpha = \alpha \circ g^{-1}$. From the viewpoint of Berkovich geometry, Drinfeld upper half-spaces can be introduced as follows [@Ber3].
We denote by $\Omega$ the complement of the union of all $k$-rational hyperplanes in $\mathbf{P}({\mathrm{V}})^{\rm an}$. The analytic space $\Omega$ is called Drinfeld upper half space.
Our next goal is now to mention some connections between the above analytic spaces and the Euclidean buildings defined in the previous section.
### Retraction onto the Bruhat-Tits building {#sss - retraction}
Let $\alpha$ be a point in $\mathbf{A}({\mathrm{V}})^{\rm an}$, i.e. $\alpha$ is a multiplicative seminorm on ${\mathrm{S}}^{\bullet}{\mathrm{V}}$. If $\alpha$ is not contained in any $k$-rational hyperplane of $\mathbf{A}({\mathrm{V}})$, then by definition $\alpha$ does not vanish on any element of ${\mathrm{S}}^1 {\mathrm{V}}= {\mathrm{V}}$. Hence the restriction of the seminorm $\alpha$ to the degree one part ${\mathrm{S}}^1 {\mathrm{V}}= {\mathrm{V}}$ is a norm. Recall that the Goldman-Iwahori space $\mathcal{N}({\rm V},k)$ is defined as the set of all non-Archimedean norms on ${\rm V}$, and that $\mathcal{X}({\mathrm{V}},k)$ denotes the quotient space after the homothety relation (\[sss - GI spaces\]). Passing to the quotients we see that restriction of seminorms induces a map $$\tau: \Omega \longrightarrow \mathcal{X}({\rm V},k).$$ If we endow the Goldman-Iwahori space $\mathcal{N}({\rm V},k)$ with the coarsest topology, so that all evaluation maps on a fixed $v \in {\mathrm{V}}$ are continuous, and $\mathcal{X}({\mathrm{V}},k)$ with the quotient topology, then $\tau$ is continuous. Besides, it is equivariant with respect to the action of ${\rm PGL}({\rm V},k)$. We refer to [@RTW2 §3] for further details.
### Embedding of the building (case of the special linear group) {#sss - embedding PGL}
Let now $\gamma$ be a non-trivial norm on ${\rm V}$. By Proposition \[prop - adapted basis\], there exists a basis $e_0, \ldots, e_{d}$ of ${\mathrm{V}}$ which is adapted to $\gamma$, i.e. we have
$\gamma\bigl(\sum_{i} \lambda_i e_i\bigr) = \max_i \{\exp(c_i) | \lambda_i | \}$
for some real numbers $c_0, \ldots, c_d$. We can associate to $\gamma$ a multiplicative seminorm $j(\gamma) $ on ${\mathrm{S}}^{\bullet}{\mathrm{V}}$ by mapping the polynomial $\sum_{I= (i_0, \ldots, i_d)} a_I e_0^{i_0} \ldots e_d^{i_d}$ to $\max_I \{|a_I| \exp(i_0 c_0 + \ldots + i_d c_d)\}$. Passing to the quotients, we get a continuous map $$j: \mathcal{X}({\mathrm{V}},k) \longrightarrow \Omega$$ satisfying $\tau\bigl(j(\alpha)\bigr) = \alpha$.
Hence $j$ is injective and is a homeomorphism onto its image. Therefore the map $j$ can be used to realize the Euclidean building $\mathcal{X}({\rm V},k)$ as a subset of a Berkovich analytic space. This observation is due to Berkovich, who used it to determine the automorphism group of $\Omega$ [@Ber3].
\[rk-approximation\] In this remark, we remove the assumption that $k$ is local and we recall that the building $\mathcal{X}({\rm V},k)$ consists of homothety classes of *diagonalizable* norms on ${\rm V}$ (Theorem \[th - GI building general\]). Assuming ${\rm dim}({\rm V})=2$ for simplicity, we want to rely on analytic geometry to prove the existence of non-diagonalizable norms on ${\rm V}$ for some $k$.
The map $j : \mathcal{X}({\rm V},k) \rightarrow \mathbf{P}^1({\rm V})^{\rm an}$ can be defined without any assumption on $k$. Given any point $x \in \mathcal{X}({\rm V},k)$, we pick a basis $\mathbf{e} = (e_0,e_1)$ diagonalizing $x$ and define $j(x)$ to be the multiplicative norm on ${\rm S}^\bullet({\rm V})$ mapping an homogeneous polynomial $f = \sum_\nu a_\nu e_0^{\nu_0} e_1^{\nu_1}$ to $\max_{\nu} \{|a_\nu| \cdot |e_0|(x)^{\nu_0} \cdot |e_1|(x)^{\nu_1}\}$. We do not distinguish between $\mathcal{X}({\rm V},k)$ and its image by $j$ in $\mathbf{P}({\rm V})^{\rm an}$, which consists only of points of types 2 and 3 (this follows from [@Temkin 3.2.11]).
Let us now consider the subset $\Omega'$ of $\Omega = \mathbf{P}({\rm V})^{\rm an} - \mathbf{P}({\rm V})(k)$ consisting of multiplicative norms on ${\rm S}^\bullet({\rm V})$ whose restriction to ${\rm V}$ is diagonalizable. The map $\tau$ introduced above is well-defined on $\Omega'$ by $\tau(z) = z_{|{\rm V}}$. This gives a continuous retraction of $\Omega'$ onto $\mathcal{X}({\rm V},k)$. The inclusion $\Omega' \subset \Omega$ is strict in general, i.e. if $k$ is not local. For example, assume that $k = \mathbf{C}_p$ is the completion of an algebraic closure of $\mathbf{Q}_p$; this non-Archimedean field is algebraically closed but not spherically complete. In this situation, $\Omega$ contains a point $z$ of type 4 [@Temkin 2.3.13], which we can approximate by a sequence $(x_n)$ of points in $\mathcal{X}({\rm V},k)$ (this is the translation of the fact that $z$ corresponds to a decreasing sequence of closed balls in $k$ with empty intersection [@Temkin 2.3.11.(iii)]). Now, if $z \in \Omega'$, then $\tau(z)=\tau~(\lim x_n) = \lim~\tau(x_n) = \lim~x_n$ and therefore $z$ belongs to $\mathcal{X}({\rm V},k)$. Since the latter set contains only points of type 2 or 3, this cannot happen and $z \notin \Omega'$; in particular, the restriction of $z$ to ${\rm V}$ produces a norm which is not diagonalizable.
A first compactification {#ss - seminorm compactification}
------------------------
Let us now turn to compactification of the building $\mathcal{X}({\rm
V},k)$. We give an outline of the construction and refer to [@RTW2 § 3] for additional details. The generalization to arbitrary reductive groups is the subject of \[ss-maps\_to\_flags\]. Recall that we assume that $k$ is a local field.
### The space of seminorms {#sss - embedding seminorms}
Let us consider the set $\mathcal{S}({\mathrm{V}},k)$ of non-Archimedean seminorms on ${\mathrm{V}}$. Every non-Archimedean seminorm $\gamma$ on ${\rm V}$ induces a norm on the quotient space ${\mathrm{V}}/ \mathrm{ker}(\gamma)$. Hence using Proposition \[prop - adapted basis\], we find that there exists a basis $e_0, \ldots, e_d$ of ${\mathrm{V}}$ such that $\alpha\bigl(\sum_{i}
\lambda_i e_i\bigr) = \max_i \{r_i \mid\! \lambda_i \!\mid \}$ for some non-negative real numbers $r_0, \ldots, r_d$. In this case we say that $\alpha$ is diagonalized by ${{\bf e}}$. Note that in contrast to Definition \[defi - adapted\] we do no longer assume that the $r_i$ are non-zero and hence exponentials.
We can extend $\gamma$ to a seminorm $j(\gamma)$ on the symmetric algebra ${\mathrm{S}}^\bullet {\mathrm{V}}\simeq k[e_0,\ldots, e_d]$ as follows:
$j(\gamma)\Bigl(\sum_{I= (i_0,\ldots, i_d)} a_I e_0^{i_0} \ldots e_d^{i_d}\Bigr) = \max \{|a_I| r_0^{i_0} \ldots r_d^{i_d}\}$.
We denote by $\overline{\mathcal{X}}({\mathrm{V}},k)$ the quotient of $\mathcal{S}({\mathrm{V}},k) - \{0\}$ after the equivalence relation $\sim$ defined as follows: $\alpha \sim \beta$ if and only if there exists a real constant $c$ with $\alpha = c \beta$. We equip $\mathcal{S}({\mathrm{V}},k)$ with the topology of pointwise convergence and $\overline{\mathcal{X}}(V,k)$ with the quotient topology. Then the association $\gamma \mapsto j(\gamma)$ induces a continuous and ${\rm PGL}({\mathrm{V}},k)$-equivariant map $$j: \overline{\mathcal{X}}({\mathrm{V}},k) \rightarrow \mathbf{P}({\mathrm{V}})^{\rm an}$$ which extends the map $j: \mathcal{X}({\mathrm{V}},k) \rightarrow \Omega$ defined in the previous section.
### Extension of the retraction onto the building {#sss - extension of retraction}
Moreover, by restriction to the degree one part ${\mathrm{S}}^1 {\mathrm{V}}= {\mathrm{V}}$, a non-zero multiplicative seminorm on ${\mathrm{S}}^\bullet {\mathrm{V}}$ yields an element in $\mathcal{S}({\mathrm{V}},k) - \{Ê0 \}$. Passing to the quotients, this induces a map $$\tau: \mathbf{P}({\mathrm{V}})^{\rm an} \longrightarrow \overline{\mathcal{X}}({\mathrm{V}},k)$$ extending the map $\tau: \Omega \rightarrow \mathcal{X}({\mathrm{V}},k) $ defined in section \[ss - Drinfeld\].
As in section \[ss - Drinfeld\], we see that $\tau \circ j$ is the identity on $\overline{\mathcal{X}}({\mathrm{V}},k)$, which implies that $j$ is injective: it is a homeomorphism onto its (closed) image in $\mathbf{P}({\mathrm{V}})^{\rm an}$. Since $\mathbf{P}({\mathrm{V}})^{\rm an}$ is compact, we deduce that the image of $j$, and hence $\overline{\mathcal{X}}({\mathrm{V}},k)$, is compact. As $\mathcal{X}({\mathrm{V}},k)$ is an open subset of $\overline{\mathcal{X}}({\mathrm{V}},k)$, the latter space is a compactification of the Euclidean building $\mathcal{X}({\mathrm{V}},k)$; it was studied in [@Wer04].
### The strata of the compactification {#sss - strata}
For every proper subspace ${\rm W}$ of ${\rm V}$ we can extend norms on ${\rm V}/{\rm W}$ to non-trivial seminorms on ${\rm V}$ by composing the norm with the quotient map ${\mathrm{V}}\rightarrow {\rm V}/{\rm W}$. This defines a continuous embedding $$\mathcal{X}({\mathrm{V}}/{\mathrm{W}},k) \rightarrow \mathcal{\overline{X}}({\mathrm{V}},k).$$ Since every seminorm on ${\rm V}$ is induced in this way from a norm on the quotient space after its kernel, we find that $\overline{\mathcal{X}}({\rm V},k)$ is the disjoint union of all Euclidean buildings $\mathcal{X}({\rm V}/{\rm W},k)$, where ${\rm W}$ runs over all proper subspaces of ${\rm V}$. Hence our compactification of the Euclidean building $\mathcal{X}({\mathrm{V}},k)$ is a union of Euclidean buildings of smaller rank.
Topology and group action {#ss - topology}
--------------------------
We will now investigate the convergence of sequences in $\overline{\mathcal{X}}({\rm V},k)$ and deduce that it is compact. We also analyze the action of the group ${\rm SL}({\rm V},k)$ on this space.
### Degeneracy of norms to seminorms and compactness
Let us first investigate convergence to the boundary of $\mathcal{X}({\mathrm{V}}, k)$ in $\overline{\mathcal{X}} ({\mathrm{V}},k) = (\mathcal{S}({\mathrm{V}},k) \backslash \{0\})/ \sim$. We fix a basis ${{\bf e}}= ( e_0, \ldots, e_d)$ of ${\mathrm{V}}$ and denote by $\mathbb{A}_{{{\bf e}}} $ the corresponding apartment associated to the norms diagonalized by ${{\bf e}}$ as in Definition \[defi - adapted\]. We denote by ${\overline{\mathbb{A}}_{{\bf e}}}\subset \overline{\mathcal{X}}(V,k)$ all classes of [*seminorms*]{} which are diagonalized by ${{\bf e}}$.
We say that a sequence $(z_n)_n$ of points in ${\overline{\mathbb{A}}_{{\bf e}}}$ is distinguished, if there exists a non-empty subset ${\mathrm{I}}$ of $\{0,\ldots, d\}$ such that:
- For all $i \in I$ and all $n$ we have $z_n(e_i) \neq 0$.
- for any $i,j \in {\mathrm{I}}$, the sequence $\left(\frac{z_n (e_j)}{z_n(e_i)}\right)_n$ converges to a positive real number;
- for any $i \in {\mathrm{I}}$ and $j \in \{0,\ldots, d\} - {\mathrm{I}}$, the sequence $\left(\frac{z_n(e_j)}{z_n(e_i)}\right)_n$ converges to $0$.
Here we define $\left(\frac{z_n(e_i)}{z_n(e_j)}\right)_n$ as $\left(\frac{x_n(e_i)}{x_n(e_j)}\right)_n$ for an arbitrary representative $x_n \in \mathcal{S}({\mathrm{V}},k)$ of the class $z_n$. Note that this expression does not depend on the choice of the representative $x_n$.
\[distinguished\] Let $(z_n)_n$ be a distinguished sequence of points in ${\overline{\mathbb{A}}_{{\bf e}}}$. Choose some element $i \in I$. We define a point $z_\infty$ in $\mathcal{S}({\mathrm{V}},k)$ as the homothety class of the seminorm $x_\infty$ defined as follows: $$x_\infty(e_j) = \left\{ \begin{array}{ll}
\lim_n \left(\frac{z_n(e_j)}{z_n(e_i)}\right) & \mbox{ if } j \in {\mathrm{I}}\\
0 & \textrm{ if } j \notin {\mathrm{I}}\end{array} \right.$$ and $x_\infty(\sum_j a_j e_j) = \mbox{max}|a_j| x_\infty (e_j)$. Then $z_\infty$ does not depend on the choice of $i$, and the sequence $(z_n)_n$ converges to $z_\infty$ in $\overline{\mathcal{X}}({\mathrm{V}},k)$.
***Proof***. Let $x_n$ be a representative of $z_n$ in $\mathcal{S}({\mathrm{V}},k)$. For $i,j$ and $\ell$ contained in $I$ we have
$$\lim_n \left(\frac{x_n(e_j)}{x_n(e_\ell)}\right) = \lim_n \left(\frac{x_n(e_j)}{x_n(e_i)}\right) \lim_n \left(\frac{x_n(e_i)}{x_n(e_\ell)}\right),$$ which implies that the definition of the seminorm class $z_\infty$ does not depend on the choice of $i \in I$.
The convergence statement is obvious, since the seminorm $x_n$ is equivalent to $ (x_n(e_i))^{-1} x_n$. $\Box$
Hence the distinguished sequence of norm classes $(z_n)_n$ considered in the Lemma converges to a seminorm class whose kernel $W_I$ is spanned by all $e_j$ with $j \notin I$. Therefore the limit point $z_\infty$ lies in the Euclidean building $\mathcal{X}({\mathrm{V}}/W_I)$ at the boundary.
Note that the preceding Lemma implies that ${\overline{\mathbb{A}}_{{\bf e}}}$ is the closure of ${\mathbb{A}_{{{\bf e}}}}$ in $\overline{\mathcal{X}}({\mathrm{V}},k)$. Namely, consider $z \in {\overline{\mathbb{A}}_{{\bf e}}}$, i.e. $z$ is the class of a seminorm $x$ on ${\mathrm{V}}$ which is diagonalizable by ${{\bf e}}$. For every $n$ we define a norm $x_n$ on ${\mathrm{V}}$ by $$x_n(e_i) = \left\{ \begin{array}{ll} x(e_i), & \mbox{ if } x(e_i) \neq 0\\
\frac{1}{n}, & \mbox{ if } x(e_i) = 0 \end{array} \right.$$ and $$x_n(\sum_i a_i e_i) = \max_i |a_i| x_n(e_i).$$
Then the sequence of norm classes $x_n= [z_n]_\sim$ in ${\mathbb{A}_{{{\bf e}}}}$ is distinguished with respect to the set $I = \{i: x(e_i) \neq 0\}$ and it converges towards $z$.
We will now deduce from these convergence results that the space of seminorms is compact. We begin by showing that ${\overline{\mathbb{A}}_{{\bf e}}}$ is compact.
\[subsequences\]Let $(z_n)_n$ be a sequence of points in ${\overline{\mathbb{A}}_{{\bf e}}}$. Then $(z_n)_n$ has a converging subsequence.
***Proof***. Let $x_n$ be seminorms representing the points $z_n$. By the box principle, there exists an index $i \in \{0, \ldots, d\}$ such that after passing to a subsequence we have $$x_n(e_i) \geqslant x_n(e_j) \mbox{ for all } j = 0, \ldots, d,
n\geqslant 0.$$ In particular we have $x_n(e_i) > 0$. For each $j = 0, \ldots, d$ we look at the sequence $$\beta(j)_n = \frac{x_n(e_j)}{x_n(e_i)}$$ which lies between zero and one. In particular, $\beta(i)_n = 1$ is constant.
After passing to a subsequence of $(z_n)_n$ we may – and shall – assume that all sequences $\beta(j)_n$ converge to some $\beta(j)$ between zero and one. Let $I$ be the set of all $j= 0, \ldots, n$ such that $\beta(j) > 0$. Then a subsequence of $(z_n)_n$ is distinguished with respect to $I$, hence it converges by Lemma \[distinguished\].$\Box$
Since ${\overline{\mathbb{A}}_{{\bf e}}}$ is metrizable, the preceding proposition shows that ${\overline{\mathbb{A}}_{{\bf e}}}$ is compact.
We can now describe the ${\rm SL}({\rm V},k)$-action on the seminorm compactification of the Goldman-Iwahori space of ${\rm V}$. As before, we fix a basis $\mathbf{e} = (e_0, \ldots, e_n)$.
Let $o$ be the homothety class of the norm on ${\mathrm{V}}$ defined by $$\left|\sum_{i=0}^{d} a_i e_i \right|(o) = \max_{0\leqslant i
\leqslant d} |a_i|$$ and let $${\rm P}_o = \{g \in {\rm SL}({\mathrm{V}},k) \ ; \ g \cdot o \sim o\}$$ be the stabilizer of $o$. It follows from Proposition \[prop - stabilizers\] that ${\rm P}_o= {\rm SL}_{d+1}(k^0)$ with respect to the basis ${{\bf e}}$.
\[closedchamber\] The map ${\rm P}_o \times {\overline{\mathbb{A}}_{{\bf e}}}\rightarrow \overline{\mathcal{X}}({\mathrm{V}},k)$ given by the ${\rm SL}({\mathrm{V}},k)$-action is surjective.
***Proof***. Let $[x]_\sim$ be an arbitrary point in $\overline{\mathcal{X}}({\mathrm{V}},k)$. The seminorm $x$ is diagonalizable with respect to some basis ${{\bf e}}'$ of ${\mathrm{V}}$. A similar argument as in the proof of Proposition \[prop - fundamental domains\] shows that there exists an element $h \in {\rm P}_o$ such that $hx$ lies in ${\overline{\mathbb{A}}_{{\bf e}}}$ (actually $hx$ lies in the closure, taken in the seminorm compactification, of a Weyl chamber with tip $o$). $\Box$
The group ${\rm P}_o$ is closed and bounded in ${\rm SL}({\rm V},k)$, hence compact. Since $\overline{\mathbb{A}}_{\mathbf{e}}$ is compact by Proposition \[subsequences\], the previous Lemma proves that $\overline{\mathcal{X}}({\rm V},k)$ is compact.
### Isotropy groups {#ss - stabilizers}
Let $z$ be a point in $\overline{\mathcal{X}}({\mathrm{V}},k)$ represented by a seminorm $x$ with kernel ${\rm W} \subset {\mathrm{V}}$. By $\overline{x}$ we denote the norm induced by $x$ on the quotient space ${\mathrm{V}}/{\rm W}$. By definition, an element $g \in {\rm PGL}({\mathrm{V}},k)$ stabilizes $z$ if and only if one (and hence any) representative $h$ of $g$ in ${\rm GL}({\mathrm{V}},k)$ satisfies $h x \sim x$, i.e. if and only if there exists some $\gamma > 0$ such that $$(\ast) \quad \quad x (h^{-1} (v)) = \gamma x(v) \mbox{ for all } v \in {\mathrm{V}}.$$ This is equivalent to saying that $h$ preserves the subspace ${\rm W}$ and that the induced element $\overline{h}$ in ${\rm GL}({\mathrm{V}}/{\rm W},k)$ stabilizes the equivalence class of the norm $\overline{x}$ on ${\mathrm{V}}/{\rm W}$. Hence we find $$\mbox{Stab}_{{\rm PGL}({\mathrm{V}},k)}(z) = \{ h \in {\rm GL}({\mathrm{V}},k): h \mbox{ fixes the subspace }{\rm W} \mbox{ and }
\overline{h}\overline{x} \sim \overline{x}\} / k^\times.$$
Let us now assume that $z$ is contained in the compactified apartment ${\overline{\mathbb{A}}_{{\bf e}}}$ given by the basis ${{\bf e}}$ of ${\mathrm{V}}$. Then there are non-negative real numbers $r_0, r_1, \ldots, r_d $ such that $$x ( \sum_i a_i e_i) = \max_i \{r_i |a_i|\}.$$ The space ${\rm W}$ is generated by all vectors $e_i$ such that $r_i = 0$. We assume that if $r_i$ and $r_j$ are both non-zero, the element $r_j/r_i$ is contained in the value group $|k^\ast|$ of $k$. In this case, if $h$ stabilizes $z$, we find that $ \gamma = x (h^{-1} e_i) / r_i $ is contained in the value group $|k^\ast|$ of $k$, i.e. we have $\gamma = |\lambda|$ for some $\lambda \in k^\ast$. Hence $(\lambda h) x = x$. Therefore in this case the stabilizer of $z$ in ${\rm PGL}({\mathrm{V}},k)$ is equal to the image of $$\{ h \in {\rm GL}({\mathrm{V}},k): h \mbox{ fixes the subspace } {\rm W} \mbox{ and }
\overline{h}\overline{x} = \overline{x}\}$$ under the natural map from ${\rm GL}({\mathrm{V}},k)$ to ${\rm PGL}({\mathrm{V}},k)$.
Assume that $z$ is contained in the closed Weyl chamber $\overline{\mathcal{C}} = \{[x]_\sim \in {\overline{\mathbb{A}}_{{\bf e}}}: x(e_0) \leqslant x(e_1) \leqslant \ldots \leqslant x(e_d) \}$, i.e. using the previous notation we have $ r_0 \leqslant r_1 \leqslant \ldots \leqslant r_d $. Let $d-\mu$ be the index such that $r_{d-\mu} = 0$ and $r_{d-\mu+1} > 0$. (If $z$ is contained in ${\mathbb{A}_{{{\bf e}}}}$, then we put $\mu = d+1$. ) Then the space $W$ is generated by the vectors $e_i$ with $i \leqslant d-\mu$. We assume as above that $r_j / r_i$ is contained in $|k^\ast|$ if $i>
d-\mu$ and $j > d-\mu$. Writing elements in ${\rm GL}({\mathrm{V}})$ as matrices with respect to the basis ${{\bf e}}$, we find that ${\rm Stab}_{{\rm PGL}({\mathrm{V}},k)}(z)$ is the image of $$\left\{ \left( \begin{array}{ll}
{\rm A} & {\rm B} \\ 0 & {\rm D} \end{array} \right) \in {\rm GL}_{d+1}(k) : {\rm D} = (\delta_{ij}) \in {\rm SL}_{\mu}(k) \mbox{ with } |\delta_{ij}| \leqslant r_j/r_i \mbox{ for all } i,j \leqslant \mu. \right\}$$ in ${\rm PGL}({\mathrm{V}},k)$.
***Proof***. This follows directly from the previous considerations combined with Proposition \[prop - stabilizers\] which describes the stabilizer groups of norms.$\Box$
The isotropy groups of the boundary points can also be described in terms of tropical linear algebra, see [@Wer11 Prop. 3.8].
Bruhat-Tits theory {#s - Bruhat-Tits general}
==================
We provide now a very short survey of Bruhat-Tits theory. The main achievement of the latter theory is the existence, for many reductive groups over valued fields, of a combinatorial structure on the rational points; the geometric viewpoint on this is the existence of a strongly transitive action of the group of rational points on a Euclidean building. Roughly speaking, one half of this theory (the one written in [@BT1a]) is of geometric and combinatorial nature and involves group actions on Euclidean buildings: the existence of a strongly transitive action on such a building is abstractly shown to come from the fact that the involved group can be endowed with the structure of a valued root group datum. The other half of the theory (the one written in [@BT1b]) shows that in many situations, in particular when the valued ground field is local, the group of rational points can be endowed with the structure of a valued root group datum. This is proved by subtle arguments of descent of the ground field and the main tool for this is provided by group schemes over the ring of integers of the valued ground field. Though it concentrates on the case when the ground field is local, the survey article [@TitsCorvallis] written some decades ago by J. Tits himself is still very useful. For a very readable introduction covering also the case of a non-discrete valuation, we recommend the recent text of Rousseau [[@RousseauGrenoble]]{}.
Reductive groups {#ss - Reductive groups}
----------------
We introduce a well-known family of algebraic groups which contains most classical groups (i.e., groups which are automorphism groups of suitable bilinear or sesquilinear forms, possibly taking into account an involution, see [@Weil] and [@KMRT]). The ground field here is not assumed to be endowed with any absolute value. The structure theory for rational points is basically due to C. Chevalley over algebraically closed fields [@Bible], and to A. Borel and J. Tits over arbitrary fields [@BoTi] (assuming a natural isotropy hypothesis).
### Basic structure results {#sss - structure reductive}
We first need to recall some facts about general linear algebraic groups, up to quoting classical conjugacy theorems and showing how to exhibit a root system in a reductive group. Useful references are A. Borel’s [@Borel], Demazure-Gabriel’s [@DemazureGabriel] and W.C. Waterhouse’s [@Waterhouse] books.
[*Linear algebraic groups*]{}.— By convention, unless otherwise stated, an “algebraic group” in what follows means a “linear algebraic group over some ground field”; being a linear algebraic group amounts to being a smooth affine algebraic group scheme (over a field). Any algebraic group can be embedded as a closed subgroup of some group ${\rm GL}({\rm V})$ for a suitable vector space over the same ground field (see [@Waterhouse 3.4] for a scheme-theoretic statement and [@Borel Prop. 1.12 and Th. 5.1] for stronger statements but in a more classical context).
Let ${\rm G}$ be such a group over a field $k$; we will often consider the group ${\rm G}_{k^a}= {\rm G}\otimes_k k^a$ obtained by extension of scalars from $k$ to an algebraic closure $k^a$.
[*Unipotent and diagonalizable groups*]{}.— We say that $g \in {\rm G}(k^a)$ is [*unipotent*]{} if it is sent to a unipotent matrix in some ([*a posteriori*]{} any) linear embedding $\varphi : {\rm G}\hookrightarrow {\rm GL}({\rm V})$: this means that $\varphi(g) - {\rm id}_{\rm V}$ is nilpotent. The group ${\rm G}_{k^a}$ is called [*unipotent*]{} if so are all its elements; this is equivalent to requiring that the group fixes a vector in any finite-dimensional linear representation as above [@Waterhouse 8.3].
The group ${\rm G}$ is said to be a [*torus*]{} if it is connected and if ${\rm G}_{k^a}$ is *diagonalizable*, which is to say that the algebra of regular functions $\mathcal{O}({\rm G}_{k^a})$ is generated by the characters of ${\rm G}_{k^a}$, i.e., $\mathcal{O}({\rm G}_{k^a}) \simeq k^a[{\rm X}({\rm G}_{k^a})]$ [@Borel §8]. Here, ${\rm X}({\rm G}_{k^a})$ denotes the finitely generated abelian group of characters ${\rm G}_{k^a}\to {\bf G}_{{\bf m}, k^a}$ and $k^a[{\rm X}({\rm G}_{k^a})]$ is the corresponding group algebra over $k^a$. A torus ${\rm G}$ defined over $k$ (also called a $k$-[*torus*]{}) is said to be [*split over $k$*]{} if the above condition holds over $k$, i.e., if its coordinate ring $\mathcal{O}({\rm G})$ is the group algebra of the abelian group ${\rm X}^*({\rm G}) = {\rm Hom}_{k,{\bf Gr}}({\rm G}, \mathbf{G}_{m,k})$. In other words, a torus is a connected group of simultaneously diagonalizable matrices in any linear embedding over $k^a$ as above, and it is $k$-split if it is diagonalized in any linear embedding defined over $k$ [@Waterhouse §7].
[*Lie algebra and adjoint representation*]{}.— One basic tool in studying connected real Lie groups is the Lie algebra of such a group, that is its tangent space at the identity element [@Borel 3.5]. In the context of algebraic groups, the definition is the same but it is conveniently introduced in a functorial way [@Waterhouse §12].
\[def - Lie algebra\] Let ${\rm G}$ be a linear algebraic group over a field $k$. The [Lie algebra]{} of ${\rm G}$, denoted by $\mathcal{L}({\rm G})$, is the kernel of the natural map ${\rm G}(k[\varepsilon]) \to {\rm G}(k)$, where $k[\varepsilon]$ is the $k$-algebra $k[X]/(X)$ and $\varepsilon$ is the class of $X$; in particular, we have $\varepsilon^2=0$.
We have $k[\varepsilon] = k \oplus k\varepsilon$ and the natural map above is obtained by applying the functor of points ${\rm G}$ to the map $k[\varepsilon]\to k$ sending $\varepsilon$ to $0$. The bracket for $\mathcal{L}({\rm G})$ is given by the commutator (group-theoretic) operation [@Waterhouse 12.2-12.3].
For ${\rm G}={\rm GL}({\rm V})$, we have $\mathcal{L}({\rm G}) \simeq {\rm End}({\rm V})$ where ${\rm End}({\rm V})$ denotes the $k$-vector space of all linear endomorphisms of ${\rm V}$. More precisely, any element of $\mathcal{L}\bigl({\rm GL}({\rm V})\bigr)$ is of the form ${\rm id}_{\rm V} + u \varepsilon$ where $u \in {\rm End}({\rm V})$ is arbitrary. The previous isomorphism is simply given by $u \mapsto {\rm id}_{\rm V} + u \varepsilon$ and the usual Lie bracket for ${\rm End}({\rm V})$ is recovered thanks to the following computation in ${\rm GL}({\rm V},k[\varepsilon])$: $[{\rm id}_{\rm V} + u \varepsilon, {\rm id}_{\rm V} + u' \varepsilon]= {\rm id}_{\rm V} + (uu'-u'u) \varepsilon$ – note that the symbol $[.,.]$ on the left hand-side stands for a commutator and that $({\rm id}_{\rm V} + u \varepsilon)^{-1} = {\rm id}_{\rm V} - u \varepsilon$ for any $u \in {\rm End}({\rm V})$.
An important tool to classify algebraic groups is the adjoint representation [@Borel 3.13].
\[def - adjoint representation\] Let ${\rm G}$ be a linear algebraic group over a field $k$. The [adjoint representation]{} of ${\rm G}$ is the linear representation ${\rm Ad} : {\rm G} \to {\rm GL}\bigl(\mathcal{L}({\rm G})\bigr)$ defined by ${\rm Ad}(g) = {\rm int}(g) \!\mid_{\mathcal{L}({\rm G})}$ for any $g \in {\rm G}$, where ${\rm int}(g)$ denotes the conjugacy $h \mapsto ghg^{-1}$ – the restriction makes sense since, for any $k$-algebra ${\rm R}$, both ${\rm G}({\rm R})$ and $\mathcal{L}({\rm G}) \otimes_k {\rm R}$ can be seen as subgroups of ${\rm G}({\rm R}[\varepsilon])$ and the latter one is normal.
In other words, the adjoint representation is the linear representation provided by differentiating conjugacies at the identity element.
For ${\rm G}={\rm SL}({\rm V})$, we have $\mathcal{L}({\rm G}) \simeq \{u \in {\rm End}({\rm V}) : {\rm tr}(u) = 0\}$ and ${\rm Ad}(g).u = gug^{-1}$ for any $g \in {\rm SL}({\rm V})$ and any $u \in \mathcal{L}({\rm G})$. In this case, we write sometimes $\mathcal{L}({\rm G}) = \mathfrak{sl}({\rm V})$.
[*Reductive and semisimple groups*]{}.— The starting point for the definition of reductive and semisimple groups consists of the following existence statement [@Borel 11.21].
\[propdef - radicals\] Let ${\rm G}$ be a linear algebraic group over a field $k$.
- There is a unique connected, unipotent, normal subgroup in ${\rm G}_{k^a}$, which is maximal for these properties. It is called the [unipotent radical]{} of ${\rm G}$ and is denoted by $\mathcal{R}_u({\rm G})$.
- There is a unique connected, solvable, normal subgroup in ${\rm G}_{k^a}$, which is maximal for these properties. It is called the [radical]{} of ${\rm G}$ and is denoted by $\mathcal{R}({\rm G})$.
The statement for the radical is implied by a finite dimension argument and the fact that the Zariski closure of the product of two connected, normal, solvable subgroups is again connected, normal and solvable. The unipotent radical is also the unipotent part of the radical: indeed, in a connected solvable group (such as $\mathcal{R}({\rm G})$), the unipotent elements form a closed, connected, normal subgroup [@Waterhouse 10.3]. Note that by their very definitions, the radical and the unipotent radical depend only on the $k^a$-group ${\rm G}_{k^a}$ and not on the $k$-group ${\rm G}$.
\[def - reductive and ss groups\] Let ${\rm G}$ be a linear algebraic group over a field $k$.
- We say that ${\rm G}$ is [reductive]{} if we have $\mathcal{R}_u({\rm G}) = \{1 \}$.
- We say that ${\rm G}$ is [semisimple]{} if we have $\mathcal{R}({\rm G}) = \{1 \}$.
\[ex - reductive groups\] For any finite-dimensional $k$-vector space ${\rm V}$, the group ${\rm GL}({\rm V})$ is reductive and ${\rm SL}({\rm V})$ is semisimple. The groups ${\rm Sp}_{2n}$ and ${\rm SO}(q)$ (for most quadratic forms $q$) are semisimple.
If, taking into account the ground field $k$, we had used a rational version of the unipotent radical, then we would have obtained a weaker notion of reductivity. More precisely, it makes sense to introduce the [*rational unipotent radical*]{}, denoted by $\mathcal{R}_{u,k}({\rm G})$ and contained in $\mathcal{R}_u({\rm G})$, defined to be the unique maximal connected, unipotent subgroup in ${\rm G}$ [*defined over $k$*]{}. Then ${\rm G}$ is called [*$k$-pseudo-reductive*]{} if we have $\mathcal{R}_{u,k}({\rm G}) = \{1 \}$. This class of groups is considered in the note [@BoTiCRAS], it is first investigated in some of J. Tits’ lectures ([@Tits9192] and [@Tits9293]). A thorough study of pseudo-reductive groups and their classification are written in B. Conrad, O. Gabber and G. Prasad’s book [@CGP] (an available survey is for instance [@RemyBBK]).
[*In the present paper, we are henceforth interested in reductive groups*]{}.
[*Parabolic subgroups*]{}.— The notion of a parabolic subgroup can be defined for any algebraic group [@Borel 11.2] but it is mostly useful to understand the structure of rational points of reductive groups.
\[def - parabolic subgroup\] Let ${\rm G}$ be a linear algebraic group over a field $k$ and let ${\rm H}$ be a Zariski closed subgroup of $G$. The subgroup ${\rm H}$ is called [parabolic]{} if the quotient space ${\rm G}/{\rm H}$ is a complete variety.
It turns out [*a posteriori*]{} that for a parabolic subgroup ${\rm H}$, the variety ${\rm G}/{\rm H}$ is actually a projective one; in fact, it can be shown that ${\rm H}$ is a parabolic subgroup if and only if it contains a [*Borel subgroup*]{}, that is a maximal connected solvable subgroup [@Borel 11.2].
\[ex - parabolics for GL\] For ${\rm G}={\rm GL}({\rm V})$, the parabolic subgroups are, up to conjugacy, the various groups of upper triangular block matrices (there is one conjugacy class for each “shape” of such matrices, and these conjugacy classes exhaust all possibilities).
The completeness of the quotient space ${\rm G}/{\rm H}$ is used to have fixed-points for some subgroup action, which eventually provides conjugacy results as stated below [@DemazureGabriel IV, §4, Th. 3.2].
[*Conjugacy theorems*]{}.— We finally mention a few results which, among other things, allow one to formulate classification results independent from the choices made to construct the classification data (e.g., the root system – see \[sss - RS and RD\] below) [@Borel Th. 20.9].
\[th - conjugacy\] Let ${\rm G}$ be a linear algebraic group over a field $k$. We assume that ${\rm G}$ is reductive.
- Minimal parabolic $k$-subgroups are conjugate over $k$, that is any two minimal parabolic $k$-subgroups are conjugate by an element of ${\rm G}(k)$.
- Accordingly, maximal $k$-split tori are conjugate over $k$.
For the rational conjugacy of tori, the reductivity assumption can be dropped and simply replaced by a connectedness assumption; this more general result is stated in [@CGP C.2]. In the general context of connected groups (instead of reductive ones), one has to replace parabolic subgroups by [*pseudo-parabolic*]{} ones in order to obtain similar conjugacy results [@CGP Th. C.2.5].
### Root system, root datum and root group datum {#sss - RS and RD}
The notion of a root system is studied in detail in [@Lie456 IV]. It is a combinatorial notion which encodes part of the structure of rational points of semisimple groups. It also provides a nice uniform way to classify semisimple groups over algebraically closed fields up to isogeny, a striking fact being that the outcome does not depend on the characteristic of the field [@Bible].
In order to state the more precise classification of reductive groups up to isomorphism (over algebraically closed fields, or more generally of split reductive groups), it is necessary to introduce a more subtle notion, namely that of a [*root datum*]{}:
\[def - root datum\] Let ${\rm X}$ be a finitely generated free abelian group; we denote by ${\rm X}\,\,\check{}$ its ${\bf Z}$-dual and by $\langle \cdot, \cdot \rangle$ the duality bracket. Let $R$ and $R\,\,\check{}$ be two finite subsets in ${\rm X}$ and ${\rm X}\,\,\check{}$, respectively. We assume we are given a bijection $\,\,\check{} : \alpha \mapsto \alpha\,\,\check{}$ from $R$ onto $R\,\,\check{}$. We have thus, for each $\alpha \in R$, endomorphisms
$s_\alpha : x \mapsto x - \langle x, \alpha\,\,\check{} \,\, \rangle \alpha$ and $s_\alpha\check{} : x\,\check{}\, \mapsto x\,\,\check{}\, - \langle \alpha, x\,\check{}\,\, \rangle \alpha\,\,\check{}$
of the groups ${\rm X}$ and ${\rm X}\,\,\check{}$, respectively. The quadruple $\Psi = ({\rm X}, R, {\rm X}\,\,\check{}, R\,\,\check{}\,)$ is said to be a [root datum]{} if it satisfies the following axioms:
- For each $\alpha \in R$, we have $\langle \alpha, \alpha\,\,\check{} \,\, \rangle=2$.
- For each $\alpha \in R$, we have $s_\alpha(R)=R$ and $s_\alpha\check{}\,(R\,\,\check{}\,) = R\,\,\check{}$.
This formulation is taken from [@Springer]. The elements of $R$ are called [roots]{} and the reflections $s_\alpha$ generate a finite group ${\rm W}$ of automorphisms of ${\rm X}$, called the *Weyl group* of $\Psi$.
Let ${\rm Q}$ denote the subgroup of ${\rm X}$ generated by $R$. Up to introducing ${\rm V} = {\rm Q} \otimes_{\bf Z} {\bf R}$ and choosing a suitable ${\rm W}$-invariant scalar product on ${\rm V}$, we can see that $R$ is a root system in the following classical sense:
\[def - root system\] Let ${\rm V}$ be a finite-dimensional real vector space endowed with a scalar product which we denote by $\langle \cdot, \cdot \rangle$. We say that a finite subset $R$ of ${\rm V} - \{0\}$ is a *root system* if it spans ${\rm V}$ and if it satisfies the following two conditions.
- To each $\alpha \in R$ is associated a reflection $s_\alpha$ which stabilizes $R$ and switches $\alpha$ and $-\alpha$.
- For any $\alpha, \beta \in R$, we have $s_\alpha(\beta) - \beta \in \mathbf{Z}\alpha$.
The Weyl group of $\Psi$ is identified with the group of automorphisms of ${\rm V}$ generated by the euclidean reflections $s_\alpha$.
Let $R$ be a root system. For any subset $\Delta$ in $R$, we denote by $R^+(\Delta)$ the set of roots which can be written as a linear combination of elements of $\Delta$ with non-negative integral coefficients. We say that $\Delta$ is a [*basis*]{} for the root system $R$ if it is a basis of ${\rm V}$ and if we have $R = R^+(\Delta) \sqcup R^-(\Delta)$, where $R^-(\Delta) = - R^+(\Delta)$. Any root system admits a basis and any two bases of a given root system are conjugate under the Weyl group action [@Lie456 VI.1.5, Th. 2]. When $\Delta$ is a basis of the root system $R$, we say that $R^+(\Delta)$ is a [*system of positive roots*]{} in $R$; the elements in $\Delta$ are then called [*simple roots*]{} (with respect to the choice of $\Delta$). The [*coroot*]{} associated to $\alpha$ is the linear form $\alpha^\vee$ on ${\rm V}$ defined by $\beta - s_\alpha(\beta) = \alpha^\vee(\beta) \alpha$; in particular, we have $\alpha^\vee(\alpha)=2$.
\[ex - RS of type A\] Here is a well-known concrete construction of the root system of type ${\rm A}_n$. Let ${\bf R}^{n+1} = \bigoplus_{i=0}^n {\bf R}\varepsilon_i$ be equipped with the standard scalar product, making the basis $(\varepsilon_i)$ orthonormal. Let us introduce the hyperplane ${\rm V} = \{ \sum_i \lambda_i \varepsilon_i : \sum_i \lambda_i = 0 \}$; we also set $\alpha_{i,j}= \varepsilon_i - \varepsilon_j$ for $i \neq j$. Then $R = \{\alpha_{i,j} : i \neq j\}$ is a root system in ${\rm V}$ and $\Delta = \{\alpha_{i,i+1} : 0 \leqslant i \leqslant n-1\}$ is a basis of it for which $R^+(\Delta) = \{\alpha_{i,j} : i < j\}$. The Weyl group is isomorphic to the symmetric group $\mathcal{S}_{n+1}$; canonical generators leading to a Coxeter presentation are for instance given by transpositions $i \leftrightarrow i+1$.
Root systems in reductive groups appear as follows. The restriction of the adjoint representation (Def. \[def - adjoint representation\]) to a maximal $k$-split torus ${\rm T}$ is simultaneously diagonalizable over $k$, so that we can write:
$\mathcal{L}({\rm G}) = \bigoplus_{\varphi \in {\rm X}^*({\rm T})} \mathcal{L}({\rm G})_\varphi$ where $\mathcal{L}({\rm G})_\varphi = \{v \in \mathcal{L}({\rm G}) : {\rm Ad}(t).v = \varphi(t)v$ for all $t \in {\rm T}(k)\}$.
The normalizer ${\rm N}={\rm N}_{\rm G}({\rm T})$ acts on ${\rm X}^*({\rm T})$ via its action by (algebraic) conjugation on ${\rm T}$, hence it permutes algebraic characters. The action of the centralizer ${\rm Z}={\rm Z}_{\rm G}({\rm T})$ is trivial, so the group actually acting is the finite quotient ${\rm N}(k)/{\rm Z}(k)$ (finiteness follows from rigidity of tori [@Waterhouse 7.7], which implies that the identity component ${\rm N}^\circ$ centralizes ${\rm T}$; in fact, we have ${\rm N}^\circ = {\rm Z}$ since centralizers of tori in connected groups are connected).
$R = R({\rm T},{\rm G}) = \{\varphi \in {\rm X}^*({\rm T}) : \mathcal{L}({\rm G})_\varphi \neq \{0 \}\}$.
It turns out that [@Borel Th. 21.6]:
1. the ${\bf R}$-linear span of $R$ is ${\rm V} = {\rm Q} \otimes_{\bf Z} {\bf R}$, where ${\rm Q} \subset {\rm X}^\ast({ \rm T})$ is generated by ${R}$;
2. there exists an ${\rm N}(k)/{\rm Z}(k)$-invariant scalar product ${\rm V}$;
3. the set $R$ is a root system in ${\rm V}$ for this scalar product;
4. the Weyl group ${\rm W}$ of this root system is isomorphic to ${\rm N}(k)/{\rm Z}(k)$.
Moreover one can go further and introduce a root datum by setting ${\rm X} = {\rm X}^*({\rm T})$ and by taking ${\rm X}\,\,\check{}$ to be the group of all 1-parameter multiplicative subgroups of ${\rm T}$. The roots $\alpha$ have just been introduced before, but distinguishing the coroots among the cocharacters in ${\rm X}\,\,\check{}$ is less immediate (over algebraically closed fields or more generally in the split case, they can be defined by means of computation in copies of ${\rm SL}_2$ attached to roots as in Example \[ex - root group datum GL(n)\] below). We won’t need this but, as already mentioned, in the split case the resulting quadruple $\Psi = ({\rm X}, R, {\rm X}\,\,\check{}, R\,\,\check{}\,)$ characterizes, up to isomorphism, the reductive group we started with (see [@SGA3] or [@Springer Chap. 9 and 10]).
One of the main results of Borel-Tits theory [@BoTi] about reductive groups over arbitrary fields is the existence of a very precise combinatorics on the groups of rational points. The definition of this combinatorial structure – called a [*root group datum*]{} – is given in a purely group-theoretic context. It is so to speak a collection of subgroups and classes modulo an abstract subgroup ${\rm T}$, all indexed by an abstract root system and subject to relations which generalize and formalize the presentation of ${\rm SL}_n$ (or of any split simply connected simple group) over a field by means of elementary unipotent matrices [@Steinberg]. This combinatorics for the rational points ${\rm G}(k)$ of an isotropic reductive group ${\rm G}$ is indexed by the root system $R({\rm T}, {\rm G})$ with respect to a maximal split torus which we have just introduced; in that case, the abstract group ${\rm T}$ of the root group datum can be chosen to be the group of rational points of the maximal split torus (previously denoted by the same letter!). More precisely, the axioms of a root group datum are given in the following definition, taken from [@BT1a 6.1][^1].
\[def - RD\] Let $R$ be a root system and let ${\rm G}$ be a group. Assume that we are given a system $\bigl( {\rm T}, ({\rm U}_\alpha, {\rm M}_\alpha)_{\alpha \in R}\bigr)$ where ${\rm T}$ and each ${\rm U}_\alpha$ is a subgroup in ${\rm G}$, and each ${\rm M}_\alpha$ is a right congruence class modulo ${\rm T}$. We say that this system is a [root group datum]{} of type $R$ for ${\rm G}$ if it satisfies the following axioms:
- For each $\alpha \in R$, we have ${\rm U}_\alpha \neq \{1 \}$.
- For any $\alpha,\beta \in R$, the commutator group $[{\rm U}_\alpha, {\rm U}_\beta]$ is contained in the group generated by the groups ${\rm U}_\gamma$ indexed by roots $\gamma$ in $R \cap ({\bf Z}_{>0}\alpha + {\bf Z}_{>0}\beta)$.
- If both $\alpha$ and $2 \alpha$ belong to $R$, we have ${\rm U}_{2\alpha}\subsetneq {\rm U}_\alpha$.
- For each $\alpha \in R$, the class ${\rm M}_\alpha$ satisfies ${\rm U}_{-\alpha} {\bf -} \{Ê1 \}Ê\subset {\rm U}_\alpha {\rm M}_\alpha {\rm U}_\alpha$.
- For any $\alpha,\beta \in R$ and each $n \in {\rm M}_\alpha$, we have $n{\rm U}_\beta n^{-1} = {\rm U}_{s_\alpha(\beta)}$.
- We have ${\rm T}{\rm U}^+ \cap {\rm U}^- = \{1 \}$, where ${\rm U}^\pm$ is the subgroup generated by the groups ${\rm U}_\alpha$ indexed by the roots $\alpha$ of sign $\pm$.
The groups ${\rm U}_\alpha$ are called the [root groups]{} of the root group datum.
This list of axioms is probably a bit hard to swallow in one stroke, but the example of ${\rm GL}_n$ can help a lot to have clearer ideas. We use the notation of Example \[ex - RS of type A\] (root system of type ${\rm A}_n$).
\[ex - root group datum GL(n)\] Let ${\rm G}= {\rm GL}_{n+1}$ and let ${\rm T}$ be the group of invertible diagonal matrices. To each root $\alpha_{i,j}$ of the root system $R$ of type ${\rm A}_n$, we attach the subgroup of elementary unipotent matrices ${\rm U}_{i,j} = {\rm U}_{\alpha_{i,j}}= \{ {\rm I}_n + \lambda {\rm E}_{i,j} : \lambda \in k\}$. We can see easily that ${\rm N}_{\rm G}({\rm T}) = \{$monomial matrices$\}$, that ${\rm Z}_{\rm G}({\rm T}) = {\rm T}$ and finally that ${\rm N}_{\rm G}({\rm T})/{\rm Z}_{\rm G}({\rm T}) \simeq \mathcal{S}_{n+1}$. Acting by conjugation, the group ${\rm N}_{\rm G}({\rm T})$ permutes the subgroups ${\rm U}_{\alpha_{i,j}}$ and the corresponding action on the indexing roots is nothing else than the action of the Weyl group $\mathcal{S}_{n+1}$ on $R$. The axioms of a root group datum follow from matrix computation, in particular checking axiom [(RGD4)]{} can be reduced to the following equality in ${\rm SL}_2$:
$
\begin{pmatrix}1&0\\1&1\end{pmatrix} = \begin{pmatrix}1&1\\0&1\end{pmatrix} \left(\begin{array}{lr}0&-1\\1&0\end{array}\right)\begin{pmatrix}1&1\\0&1\end{pmatrix}
$.
We can now conclude this subsection by quoting a general result due to A. Borel and J. Tits (see [@BT1a 6.1.3 c)] and [@BoTi]).
\[th - RD in isotropic red gps\] Let ${\rm G}$ be a connected reductive group over a field $k$, which we assume to be $k$-isotropic. Let ${\rm T}$ be a maximal $k$-split torus in ${\rm G}$, which provides a root system $R = R({\rm T},{\rm G})$.
- For every root $\alpha \in R$ the connected subgroup ${\rm U}_\alpha$ with Lie algebra $\mathcal{L}({\rm G})_\alpha$ is unipotent; moreover it is abelian or two-step nilpotent.
- The subgroups ${\rm T}(k)$ and ${\rm U}_\alpha(k)$, for $\alpha \in R$, are part of a root group datum of type $R$ in the group of rational points ${\rm G}(k)$.
Recall that we say that a reductive group is [*isotropic over $k$*]{} if it contains a non-central $k$-split torus of positive dimension (the terminology is inspired by the case of orthogonal groups and is compatible with the notion of isotropy for quadratic forms [@Borel 23.4]). Note finally that the structure of a root group datum implies that (coarser) of a Tits system (also called BN-pair) [@Lie456 IV.2], which was used by J. Tits to prove, in a uniform way, the simplicity (modulo center) of the groups of rational points of isotropic simple groups (over sufficiently large fields) [@TitsSimple].
### Valuations on root group data {#sss - DRV}
Bruhat-Tits theory deals with isotropic reductive groups over valued fields. As for Borel-Tits theory (arbitrary ground field), a substantial part of this theory can also be summed up in combinatorial terms. This can be done by using the notion of a [*valuation*]{} of a root group datum, which formalizes among other things the fact that the valuation of the ground field induces a filtration on each root group. The definition is taken from [@BT1a 6.2].
\[def - V\] Let ${\rm G}$ be an abstract group and let $\bigl( {\rm T}, ({\rm U}_\alpha, {\rm M}_\alpha)_{\alpha \in R}\bigr)$ be a root group datum of type $R$ for it. A [valuation]{} of this root group datum is a collection $\varphi = (\varphi_\alpha)_{\alpha \in R}$ of maps $\varphi_\alpha : {\rm U}_\alpha \to {\bf R}\cup \{\infty \}$ satisfying the following axioms.
- For each $\alpha \in R$, the image of $\varphi_\alpha$ contains at least three elements.
- For each $\alpha \in R$ and each $\ell \in {\bf R}\cup \{\infty \}$, the preimage $\varphi_\alpha^{-1}([\ell;\infty])$ is a subgroup of ${\rm U}_\alpha$, which we denote by ${\rm U}_{\alpha, \ell}$; moreover we require ${\rm U}_{\alpha,\infty} = \{1 \}$.
- For each $\alpha \in R$ and each $n \in {\rm M}_\alpha$, the map $u \mapsto \varphi_{-\alpha}(u) - \varphi_\alpha(nun^{-1})$ is constant on the set ${\rm U}_{-\alpha}^* = {\rm U}_{-\alpha}-\{1\}$.
- For any $\alpha,\beta \in R$ and $\ell,\ell' \in {\bf R}$ such that $\beta \not\in -{\bf R}_+\alpha$, the commutator group $[{\rm U}_{\alpha,\ell}, {\rm U}_{\beta,\ell'}]$ lies in the group generated by the groups ${\rm U}_{p\alpha + q\beta, p\ell+q\ell'}$ where $p,q \in {\bf Z}_{>0}$ and $p\alpha + q\beta \in R$.
- If both $\alpha$ and $2 \alpha$ belong to $R$, the restriction of $2\varphi_\alpha$ to ${\rm U}_{2\alpha}$ is equal to $\varphi_{2\alpha}$.
- For $\alpha \in R$, $u \in {\rm U}_\alpha$ and $u', u'' \in {\rm U}_{-\alpha}$ such that $u'uu'' \in {\rm M}_\alpha$, we have $\varphi_{-\alpha}(u') = -\varphi_\alpha(u)$.
The geometric counterpart to this list of technical axioms is the existence, for a group endowed with a valued root group datum, of a Euclidean building (called the [*Bruhat-Tits building*]{} of the group) on which it acts by isometries with remarkable transitivity properties [@BT1a §7]. For instance, if the ground field is discretely valued, the corresponding building is simplicial and a fundamental domain for the group action is given by a maximal (poly)simplex, also called an [*alcove*]{} (in fact, if the ground field is discretely valued, the existence of a valuation on a root group datum can be conveniently replaced by the existence of an affine Tits system [@BT1a §2]). As already mentioned, the action turns out to be strongly transitive, meaning that the group acts transitively on the inclusions of an alcove in an apartment (Remark \[rk - BrT\] in \[sss - simplicial\]).
Bruhat-Tits buildings {#ss - Bruhat-Tits buildings}
---------------------
The purpose of this subsection is to roughly explain how Bruhat-Tits theory attaches a Euclidean building to a suitable reductive group defined over a valued field. This Bruhat-Tits building comes equipped with a strongly transitive action by the group of rational points, which in turn implies many interesting decompositions of the group. The latter decompositions are useful for instance to doing harmonic analysis or studying various classes of linear representations of the group. We roughly explain the descent method used to perform the construction of the Euclidean buildings, and finally mention how some integral models are both one of the main tools and an important outcome of the theory.
### Foldings and gluing {#sss - gluing}
We keep the (connected) semisimple group ${\rm G}$, defined over the (now, complete valued non-Archimedean) field $k$ but from now on, [*we assume for simplicity that $k$ is a local field (i.e., is locally compact) and we denote by $\omega$ its discrete valuation*]{}, normalized so that $\omega(k^\times)=\mathbf{Z}$. Hence $\omega(\cdot) = - {\rm log}_q |\cdot|$, where $q >1$ is a generator of the discrete group $|k^\times|$.
We also assume that ${\rm G}$ contains a $k$-split torus of positive dimension: this is an isotropy assumption over $k$ already introduced at the end of \[sss - RS and RD\] (in this situation, this algebraic condition is equivalent to the topological condition that the group of rational points ${\rm G}(k)$ is non-compact [@PrasadSMF]). In order to associate to ${\rm G}$ a Euclidean building on which ${\rm G}(k)$ acts strongly transitively, according to [@TitsCorvallis] we need two things:
1. a model, say $\Sigma$, for the apartments;
2. a way to glue many copies of $\Sigma$ altogether in such a way that they will satisfy the incidence axioms of a building (\[sss - simplicial\]).
[*Model for the apartment*]{}.— References for what follows are [@TitsCorvallis §1] or [@La Chapter I]. Let ${\rm T}$ be a maximal $k$-split torus in ${\rm G}$ and let ${\rm X}_*({\rm T})$ denote its group of 1-parameter subgroups (or [*cocharacters*]{}). As a first step, we set $\Sigma_{\rm vect}= {\rm X}_{*}({\rm T}) \otimes_{\bf Z} {\bf R}$.
\[prop-affine\_apartment\] There exists an affine space $\Sigma$ with underlying vector space $\Sigma_{\rm vect}$, equipped with an action by affine transformations $\nu : {\rm N}(k) = {\rm N}_{\rm G}({\rm T})(k) \to {\rm Aff}(\Sigma)$ and having the following properties.
- There is a scalar product on $\Sigma$ such that $\nu\bigl( {\rm N}(k) \bigr)$ is an affine reflection group.
- The vectorial part of this group is the Weyl group of the root system $R = R({\rm T}, {\rm G})$.
- The translation (normal) subgroup acts cocompactly on $\Sigma$, it is equal to $\nu\bigl({\rm Z}(k)\bigr)$ and the vector $\nu(z)$ attached to an element $z \in {\rm Z}(k)$ is defined by $\chi\bigl(\nu(z)\bigr) = -\omega \bigl( \chi(z) \bigr)$ for any $\chi \in {\rm X}^*({\rm T})$.
If we go back to the example of ${\rm GL}({\rm V})$ acting by precomposition on the space of classes of norms $ \mathcal{X}({\rm V},k)$ as described in \[ss - SL(n) Bruhat-Tits\], we can see the previous statement as a generalization of the fact, mentioned in \[sss - actions on GI\], that for any basis $\mathbf{e}$ of ${\rm V}$, the group ${\rm N}_{\rm e}$ of monomial matrices with respect to ${\bf e}$ acts on the apartment $\mathbb{A}_{\bf e}$ as $\mathcal{S}_d \ltimes {\bf Z}^d$ where $d = {\rm dim}({\rm V})$.
[*Filtrations and gluing*]{}.— Still for this special case, we saw (Prop. \[prop - folding\]) that any elementary unipotent matrix $u_{ij}(\lambda) = {\rm I}_d + \lambda {\rm E}_{ij}$ fixes pointwise a closed half-apartment in $\mathbb{A}_{\bf e}$ bounded by a hyperplane of the form $\{c_i - c_j =$ constant$\}$ (the constant depends on the valuation $\omega(\lambda)$ of the additive parameter $\lambda$), the rest of the apartment $\mathbb{A}_{\bf e}$ associated to ${\bf e}$ being “folded” away from $\mathbb{A}_{\bf e}$.
In order to construct the Bruhat-Tits building in the general case, the gluing equivalence will impose this folding action for unipotent elements in root groups; this will be done by taking into account the “valuation” of the unipotent element under consideration. What formalizes this is the previous notion of a valuation for a root group datum (Definition \[def - V\]), which provides a filtration on each root group. For further details, we refer to the motivations given in [@TitsCorvallis 1.1-1.4]. It is not straightforward to perform this in general, but it can be done quite explicitly when the group ${\rm G}$ is [*split*]{} over $k$ (i.e., when it contains a maximal torus which is $k$-split). For the general case, one has to go to a (finite, separable) extension of the ground field splitting ${\rm G}$ and then to use subtle descent arguments. The main difficulty for the descent step is to handle at the same time Galois actions on the split group and on its “split” building in order to descend the ground field both for the valuation of the root group datum and at the geometric level (see \[sss - descent and functoriality\] for slightly more details).
Let us provisionally assume that ${\rm G}$ is split over $k$. Then each root group ${\rm U}_\alpha (k)$ is isomorphic to the additive group of $k$ and for any such group ${\rm U}_\alpha(k)$ we can use the valuation of $k$ to define a decreasing filtration $\{{\rm U}_{\alpha}(k)_\ell \}_{\ell \in {\bf Z}}$ satisfying:
$\bigcup_{\ell \in {\bf Z}} {\rm U}_{\alpha}(k)_\ell = {\rm
U}_\alpha(k) \hskip3mm {\rm and} \hskip3mm \bigcap_{\ell \in {\bf Z}} {\rm U}_{\alpha}(k)_\ell = \{1 \},$
and further compatibilities, namely the axioms of a valuation (Def. \[def - V\]) for the root group datum structure on ${\rm G}(k)$ given by Borel-Tits theory (Th. \[th - RD in isotropic red gps\]) – the latter root group datum structure in the split case is easy to obtain by means of Chevalley bases [@Steinberg] (see remark below). For instance, in the case of the general linear group, this can be merely done by using the parameterizations
$(k,+) \simeq {\rm U}_{\alpha_{i,j}}(k) = \{{\rm id}+ \lambda {\rm E}_{i,j} : \lambda \in k\}$.
\[rk-filtrations\_split\] Let us be slightly more precise here. For a split group ${\rm G}$, each root group ${\rm U}_\alpha$ is $k$-isomorphic to the additive group $\mathbf{G}_a$, and the choice of a Chevalley basis of ${\rm Lie}({\rm G})$ determines a set of isomorphisms $\{p_\alpha : {\rm U}_\alpha \rightarrow \mathbf{G}_a\}_{\alpha \in R}$. It is easily checked that the collection of maps $$\xymatrix{\varphi_\alpha : {\rm U}_\alpha(k) \ar@{->}[r]^{p_\alpha} & \mathbf{G}_a(k) \ar@{->}[r]^\omega & \mathbf{R}}$$ defines a valuation on the root group datum $({\rm T}(k),({\rm U}_\alpha(k),{\rm M}_\alpha))$.
For each $\ell \in \mathbf{R}$, the condition $|p_{\alpha}| \leqslant q^{-s}$ defines an *affinoid* subgroup ${\rm U}_{\alpha,s}$ in ${\rm U}_{\alpha}^{\rm an}$ such that ${\rm U}_{\alpha}(k)_\ell = {\rm U}_{\alpha,s}(k)$ for any $s \in (\ell-1, \ell]$. The latter identity holds after replacement of $k$ by any finite extension $k'$, *as long as we normalize the valuation of $k'$ in such a way that is extends the valuation on $k$*. This shows that Bruhat-Tits filtrations on root groups, in the split case at this stage, comes from a decreasing, exhaustive and separated filtration of ${\rm U}_{\alpha}^{\rm an}$ by affinoid subgroups $\{{\rm U}_{\alpha,s}\}_{s \in \mathbf{R}}$.
Let us consider again the apartment $\Sigma$ with underlying vector space $\Sigma_{\rm vect}= {\rm X}_*({\rm T}) \otimes_{\bf Z} {\bf R}$. We are interested in the affine linear forms $\alpha + \ell$ ($\alpha \in R$, $\ell \in {\bf Z}$). We fix an origin, say $o$, such that $(\alpha + 0)(o) = 0$ for any root $\alpha \in R$. We have “level sets” $\{\alpha + \ell = 0 \}$ and “positive half-spaces” $\{\alpha + \ell \geqslant 0 \}$ bounded by them.
For each $x \in \Sigma$, we set ${\rm N}_x = {\rm Stab}_{{\rm G}(k)}(x)$ (using the action $\nu$ of Prop. \[prop-affine\_apartment\]) and for each root $\alpha$ we denote by ${\rm U}_{\alpha}(k)_x$ the biggest subgroup ${\rm U}_{\alpha}(k)_\ell$ such that $x \in \{\alpha + \ell \geqslant 0\}$ (i.e. $\ell$ is minimal for the latter property). At last, we define ${\rm P}_x$ to be the subgroup of ${\rm G}(k)$ generated by ${\rm N}_x$ and by $\{{\rm U}_{\alpha}(k)_x \}_{\alpha \in R}$. We are now in position to define a binary relation, say $\sim$, on ${\rm G}(k) \times \Sigma$ by:
$(g,x) \sim (h,y)$ $\Longleftrightarrow$ there exists $n \in {\rm N}_{\rm G}(T)(k)$ such that $y=\nu(n).x$ and $g^{-1}hn \in {\rm P}_x$.
[*Construction of the Bruhat-Tits buildings*]{}.— This relation is exactly what is needed in order to glue together copies of $\Sigma$ and to finally obtain the desired Euclidean building.
\[th - gluing\] The relation $\sim$ is an equivalence relation on the set ${\rm G}(k) \times \Sigma$ and the quotient space $\displaystyle {\mathcal{B}} = {\mathcal{B}}({\rm G},k) = \left({\rm G}(k) \times \Sigma\right) / \sim$ is a Euclidean building whose apartments are isomorphic to $\Sigma$ and whose Weyl group is the affine reflection group $W = \nu\bigl( {\rm N}(k) \bigr)$. Moreover the ${\rm G}(k)$-action by left multiplication on the first factor of ${\rm G}(k) \times \Sigma$ induces an action of ${\rm G}(k)$ by isometries on ${\mathcal{B}}({\rm G},k)$.
[*Notation*]{}.— According to Definition \[defi - non-simplicial building\], copies of $\Sigma$ in $\mathcal{B}({\rm G},k)$ are called *apartments*; they are the maximal flat (i.e., euclidean) subspaces. Thanks to ${\rm G}(k)$-conjugacy of maximal split tori \[th - conjugacy\], apartments of $\mathcal{B}({\rm G},k)$ are in bijection with maximal split tori of ${\rm G}$. Therefore, we will speak of the *apartment of a maximal split torus* ${\rm S}$ of ${\rm G}$ and write ${\rm A}({\rm S},k)$. By construction, this is an affine space under the $\mathbf{R}$-vector space ${\rm Hom}_{\bf Ab}({\rm X}^*({\rm S}),\mathbf{R})$.
[*Reference*]{}.— As already explained, the difficulty is to check the axioms of a valuation (Def \[def - V\]) for a suitable choice of filtrations on the root groups of a Borel-Tits root group datum (Th. \[th - RD in isotropic red gps\]). Indeed, the definition of the equivalence relation $\sim$, hence the construction of a suitable Euclidean building, for a valued root group datum can be done in this purely abstract context [@BT1a §7]. The existence of a valued root group datum for reductive groups over suitable valued (not necessarily complete) fields was announced in [@BT1a 6.2.3 c)] and was finally settled in the second IHÉS paper (1984) by F. Bruhat and J. Tits [@BT1b Introduction and Th. 5.1.20]. $\square$
One way to understand the gluing equivalence relation $\sim$ is to see that it prescribes stabilizers. Actually, it can eventually be proved that [*a posteriori*]{} we have:
$\Sigma^{{\rm U}_{\alpha,\ell}(k)} = \{\alpha + \ell \geqslant 0 \}$ and ${\rm Stab}_{{\rm G}(k)}(x) = {\rm P}_x$ for any $x \in {\mathcal{B}}$.
A more formal way to state the result is to say that to each valued root group datum on a group is associated a Euclidean building, which can be obtained by a gluing equivalence relation defined as above [@BT1a §7].
\[SL-building\] In the case when ${\rm G}={\rm SL}({\rm V})$, it can be checked that the building obtained by the above method is equivariantly isomorphic to the Goldman-Iwahori space $\mathcal{X}({\mathrm{V}},k)$ [@BT1a 10.2].
### Descent and functoriality {#sss - descent and functoriality}
Suitable filtrations on root groups so that an equivalence relation $\sim$ as above can be defined do not always exist. Moreover, even when things go well, the way to construct the Bruhat-Tits building is not by first exhibiting a valuation on the root group datum given by Borel-Tits theory and then by using the gluing relation $\sim$. As usual in algebraic group theory, one has first to deal with the split case, and then to apply various and difficult arguments of descent of the ground field. Bruhat and Tits used a two-step descent, allowing a fine description of smooth integral models of the group associated with facets. A one-step descent was introduced by Rousseau in his thesis [[@RousseauOrsay]]{}, whose validity in full generality now follows from recent work connected to Tits’ Center Conjecture ([[@Struyve]]{}).
[*Galois actions*]{}.— More precisely, one has to find a suitable (finite) Galois extension $k'/k$ such that ${\rm G}$ splits over $k'$ (or, at least, [*quasi-splits*]{} over $k'$, i.e. admits a Borel subgroup defined over $k'$) and, which is much more delicate, which enables one:
1. to define a ${\rm Gal}(k'/k)$-action by isometries on the “(quasi)-split” building ${\mathcal{B}}({\rm G},k')$;
2. to check that a building for ${\rm G}(k)$ lies in the Galois fixed point set ${\mathcal{B}}({\rm G},k')^{{\rm Gal}(k'/k)}$.
Similarly, the group ${\rm G}(k')$ of course admits a ${\rm Gal}(k'/k)$-action.
\[rk - enough fixed points\] Recall that, by completeness and non-positive curvature, once step 1 is settled we know that we have sufficiently many Galois-fixed points in ${\mathcal{B}}({\rm G},k')$ (see the discussion of the Bruhat-Tits fixed point theorem in \[sss - geometry of buildings\]).
F. Bruhat and J. Tits found a uniform procedure to deal with various situations of that kind. The procedure described in [@BT1a 9.2] formalizes, in abstract terms of buildings and group combinatorics, how to exhibit a valued root group datum structure (resp. a Euclidean building structure) on a subgroup of a bigger group with a valued root group datum (resp. on a subspace of the associated Bruhat-Tits building). The main result [@BT1a Th. 9.2.10] says that under some sufficient conditions, the restriction of the valuation to a given sub-root group datum “descends” to a valuation and its associated Bruhat-Tits building is the given subspace. These sufficient conditions are designed to apply to subgroups and convex subspaces obtained as fixed-points of “twists” by Galois actions (and they can also be applied to non-Galois twists “à la Ree-Suzuki”).
[*Two descent steps*]{}.— As already mentioned, this needn’t work over an arbitrary valued field $k$ (even when $k$ is complete). Moreover F. Bruhat and J. Tits do not perform the descent in one stroke, they have to argue by a two step descent.
The first step is the so-called [*quasi-split*]{} descent [@BT1b §4]. It consists in dealing with field extensions splitting an initially quasi-split reductive group. The Galois twists here (of the ambient group and building) are shown, by quite concrete arguments, to fit in the context of [@BT1a 9.2] mentioned above. This is possible thanks to a deep understanding of quasi-split groups: they can even be handled via a presentation (see [@Steinberg] and [@BT1b Appendice]). In fact, the largest part of the chapter about the quasi-split descent [@BT1b §4] is dedicated to another topic which will be presented below (\[sss - models\]), namely the construction of suitable integral models (i.e. group schemes over $k^\circ$ with generic fiber ${\rm G}$) defined by geometric conditions involving bounded subsets in the building. The method chosen by F. Bruhat and J. Tits to obtain these integral models is by using a linear representation of ${\rm G}$ whose underlying vector space contains a suitable $k^\circ$-lattice, but they mention themselves that this could be done by Weil’s techniques of group chunks. Since then, thanks to the developments of Néron model techniques [@BLR], this alternative method has been written down [@La].
The second step is the so-called [*étale*]{} descent [@BT1b §5]. By definition, an étale extension, in the discretely valued case (to which we stick here), is unramified with separable residual extension; let us denote by $k^{\rm sh}$ the maximal étale extension of $k$. This descent step consists in considering situations where the semisimple $k$-group ${\rm G}$ is such that ${\rm G} \otimes_k k^{\rm sh}$ is quasi-split (so that, by the first step, we already have a valued root group datum and a Bruhat-Tits building for ${\rm G}(k^{\rm sh})$, together with integral structures). Checking that this fits in the geometric and combinatorial formalism of [@BT1a 9.2] is more difficult in that case. In fact, this is the place where the integral models over the valuation ring $k^\circ$ are used, in order to find a suitable torus in ${\rm G}$ which become maximal split in ${\rm G} \otimes_k k'$ for some étale extension $k'$ of $k$ [@BT1b Cor. 5.1.12].
\[rk-filtrations\] In the split case, we have noticed that the Bruhat-Tits filtrations on rational points of root groups come from filtrations by affinoid subgroups (\[rk-filtrations\_split\]). This fact holds in general and can be checked as follows: let $k'/k$ be a finite Galois extension splitting ${\rm G}$ and consider a maximal torus ${\rm T}$ of ${\rm G}$ which splits over $k'$ and contains a maximal split torus ${\rm S}$. The canonical projection ${\rm X}^*({\rm T} \otimes_k k')
\rightarrow {\rm X}^*({\rm S} \otimes_k k') \tilde{=} {\rm X}^*({\rm S})$ induces a surjective map $$p: R({\rm T}
\otimes_k k',{\rm G} \otimes_k k') \longrightarrow R({\rm S},{\rm G}) \cup \{0\}$$ and there is a natural $k'$-isomorphism $$\prod_{\beta \in p^{-1}(\alpha)} {\rm U}_\beta \times \prod_{\beta \in p^{-1}(2\alpha)} {\rm U}_\beta \simeq {\rm U}_\alpha \otimes_k k'$$ for any ordering of the factor.
*A posteriori*, Bruhat-Tits two-step descent proves that any maximal split torus ${\rm S}$ of ${\rm G}$ is contained in a maximal torus ${\rm T}$ which splits over a finite Galois extension $k'/k$ such that ${\rm Gal}(k'/k)$ fixes a point in the apartment of ${\rm T} \otimes_k k'$ in $\mathcal{B}({\rm G},k')$. *If the valuation on $k'$ is normalized in such a way that it extends the valuation on $k$*, then, for any $\ell \in \mathbf{R}$, the affinoid subgroup $$\prod_{\beta \in p^{-1}(\alpha)} {\rm U}_{\beta,\ell}
\times \prod_{\beta \in p^{-1}(2\alpha)} {\rm U}_{\beta,2\ell}$$ of the left hand side corresponds to an affinoid subgroup of the right hand side which does not depend on the ordering of the factors and is preserved by the natural action of ${\rm Gal}(k'|k)$; this can be checked by using calculations in [@BT1a 6.1] at the level of $k''$ points, for any finite extension $k''/k'$. By Galois descent, we obtain an affinoid subgroup ${\rm U}_{\alpha,\ell}$ of ${\rm U}_{\alpha}^{\rm an}$ such that $${\rm U}_{\alpha,\ell}(k) = {\rm
U}_\alpha(k) \cap \left(\prod_{\beta \in p^{-1}(\alpha)} {\rm U}_{\beta,\ell}(k')
\times \prod_{\beta \in p^{-1}(2\alpha)}{\rm U}_{\beta,2\ell}(k') \right).$$ By [@BT1b 5.1.16 and 5.1.20], the filtrations $\{{\rm U}_{\alpha,\ell}(k)\}_{\ell
\in \mathbf{R}}$ are induced by a valuation on the root group datum $\left({\rm
S}(k), \{{\rm U}_\alpha (k)\} \right)$.
Let us finish by mentioning why this two-step strategy is well-adapted to the case we are interested in, namely that of a semisimple group ${\rm G}$ defined over a complete, discretely valued field $k$ with perfect residue field $\widetilde{k}$: thanks to a result of R. Steinberg’s [@SerreGalois III, 2.3], such a group is known to quasi-split over $k^{\rm sh}$. Compactifications of Bruhat-Tits buildings fit in this more specific context for ${\rm G}$ and $k$. Indeed, the Bruhat-Tits building ${\mathcal{B}}({\rm G},k)$ is locally compact if and only if so is $k$, see the discussion of the local structure of buildings below (\[sss - models\]). Note finally that the terminology “henselian” used in [@BT1b] is a well-known algebraic generalization of “complete” (the latter “analytic” condition is the only one we consider seriously here, since we will use Berkovich geometry).
[*Existence of Bruhat-Tits buildings*]{}.— Here is at last a general statement on existence of Bruhat-Tits buildings which will be enough for our purposes; this result was announced in [@BT1a 6.2.3 c)] and is implied by [@BT1b Th. 5.1.20].
\[th - existence of BT buildings\] Assume that $k$ is complete, discretely valued, with perfect residue field. The root group datum of ${\rm G}(k)$ associated with a split maximal torus admits a valuation satisfying the conditions of Definition \[def - V\].
Let us also give now an example illustrating both the statement of the theorem and the general geometric approach characterizing Bruhat-Tits theory.
\[ex - hermitian trees\] Let $h$ be a Hermitian form of index $1$ in three variables, say on the vector space ${\rm V} \simeq k^3$. We assume that $h$ splits over a quadratic extension, say $E/k$, so that ${\rm SU}({\rm V},h)$ is isomorphic to ${\rm SL}_3$ over $E$, and we denote ${\rm Gal}(E/k) = \{1;\sigma \}$. Then the building of ${\rm SU}({\rm V},h)$ can be seen as the set of fixed points for a suitable action of the Galois involution $\sigma$ on the $2$-dimensional Bruhat-Tits building of type $\widetilde {\rm A}_2$ associated to ${\rm V} \otimes_k E$ as in \[ss - SL(n) Bruhat-Tits\]. If $k$ is local and if $q$ denotes the cardinality of the residue field, then the Euclidean building ${\mathcal{B}}({\rm SU}({\rm V},h),k)$ is a locally finite tree: indeed, it is a Euclidean building of dimension $1$ because the $k$-rank of ${\rm SU}({\rm V},h)$, i.e. the dimension of maximal $k$-split tori, is $1$. The tree is homogeneous of valency $1+q$ when $E/k$ is ramified, in which case the type of the group is [C]{}-[BC]{}${}_1$ in Tits’ classification [@TitsCorvallis p. 60, last line]. The tree is semi-homogeneous of valencies $1+q$ and $1+q^3$ when $E/k$ is unramified, and then the type is ${}^2\! {\rm A}_2'$ [@TitsCorvallis p. 63, line 2]. For the computation of the valencies, we refer to \[sss - models\] below.
[*Functoriality*]{}.— For our purpose (i.e. embedding of Bruhat-Tits buildings in analytic spaces and subsequent compactifications), the existence statement is not sufficient. We need a stronger result than the mere existence; in fact, we also need a good behavior of the building with respect to field extensions.
\[th - functoriality of BT buildings\] Whenever $k$ is complete, discretely valued, with perfect residue field, the Bruhat-Tits building ${\mathcal{B}}({\rm G},K)$ depends functorially on the non-Archimedean extension $K$ of $k$.
More precisely, let us denote by ${\rm G}-\mathbf{Sets}$ the category whose objets are pairs $(K/k,{\rm X})$, where $K/k$ is a non-Archimedean extension and ${\rm X}$ is a topological space endowed with a continuous action of ${\rm G}(K)$, and arrows $(K/k,{\rm X}) \rightarrow (K'/k,{\rm X}')$ are pairs $(\iota, f)$, where $\iota$ is an isometric embedding of $K$ into $K'$ and $f$ is a ${\rm G}(K)$-equivariant and continous map from ${\rm X}$ to ${\rm X}'$. We see the building of ${\rm G}$ as a *section* $\mathcal{B}({\rm G}, -)$ of the forgetful functor $${\rm G}-\mathbf{Sets} \longrightarrow \left( \begin{array}{c} {\rm non-Archimedean} \\ {\rm extensions }\ K/k \end{array} \right).$$
.— It is explained in [@RTW1 1.3.4] how to deduce this from the general theory.
One word of caution is in order here. If $k'/k$ is a Galois extension, then there is a natural action of ${\rm Gal}(k'/k)$ on $\mathcal{B}({\rm G},k')$ by functoriality and the smaller building $\mathcal{B}({\rm G},k)$ is contained in the Galois-fixed point set in ${\mathcal{B}}({\rm G},k')$. In general, this inclusion is strict, even when the group is split [@RousseauOrsay III] (see also [**5.2**]{}). However, one can show that there is equality if the extension $k'/k$ is *tamely ramified* \[[**loc. cit.**]{}\] and [@Prasad].
We will need to have more precise information about the behavior of apartments. As above, we assume that $k$ is complete, discretely valued and with perfect residue field.
Let ${\rm T}$ be a maximal torus of ${\rm G}$ and let $k_{\rm T}$ be the minimal Galois extension of $k$ (in some fixed algebraic closure) which splits ${\rm T}$. We denote by $k_{\rm T}^{\rm ur}$ the maximal unramified extension of $k$ in $k_{\rm T}$.
The torus ${\rm T}$ is *well-adjusted* if the maximal split subtori of ${\rm T}$ and ${\rm T} \otimes_k k_{\rm T}^{\rm ur}$ are maximal split tori of ${\rm G}$ and ${\rm G} \otimes_k k_{\rm T}^{\rm ur}$.
\[functor-apartment\] 1. Every maximal split torus ${\rm S}$ of ${\rm G}$ is contained in a well-adjusted maximal torus ${\rm T}$.\
2. Assume that ${\rm S}$ and ${\rm T}$ are as above, and let $K/k$ be any non-Archimedean field extension which splits ${\rm T}$. The embedding $\mathcal{B}({\rm G},k) \hookrightarrow \mathcal{B}({\rm G},{\rm K})$ maps ${\rm A}({\rm S},k)$ into ${\rm A}({\rm T},{\rm K})$.
*Proof*. 1. For each unramified finite Galois extension $k'/k$, we can find a torus ${\rm S}' \subset {\rm G}$ which contains ${\rm S}$ and such that ${\rm S}' \otimes_k k'$ is a maximal split torus of ${\rm G} \otimes_k k'$ [@BT1b Corollaire 5.1.12]. We choose a pair $(k',{\rm S}')$ such that the rank of ${\rm S}'$ is maximal, equal to the relative rank of ${\rm G} \otimes_k k^{\rm ur}$; this means that ${\rm S}' \otimes_k k''$ is a maximal split torus of ${\rm G} \otimes_k k''$ for any unramified extension $k''/k$ containing $k'$.
The centralizer of ${\rm S}' \otimes_k k'$ in ${\rm G} \otimes_k k'$ is a maximal torus of ${\rm G} \otimes_k k'$, hence ${\rm T} = {\rm Z}({\rm S}')$ is a maximal torus of ${\rm G}$. By construction, ${\rm S}'$ splits over $k_{\rm T}^{\rm ur}$ and ${\rm S'} \otimes_k k_{\rm T}^{\rm ur}$ is a maximal split torus of ${\rm G} \otimes_k k_{\rm T}^{\rm ur}$. Since ${\rm S} \subset {\rm S}'$, this proves that ${\rm T}$ is well-adjusted.
2\. We keep the same notation as above. The extension $K/k$ contains $k_{\rm T}$, hence it is enough by functoriality to check that the embedding $\mathcal{B}({\rm G},k) \hookrightarrow \mathcal{B}({\rm G},k_{\rm T})$ maps ${\rm A}({\rm S},k)$ into ${\rm A}({\rm T},k_{\rm T})$.
Let us consider the embeddings $$\mathcal{B}({\rm G},k) \hookrightarrow \mathcal{B}({\rm G},k_{\rm T}^{\rm ur}) \hookrightarrow \mathcal{B}({\rm G},k_{\rm T}).$$ The first one maps ${\rm A}({\rm S},k)$ into ${\rm A}({\rm S}',k_{\rm T}^{\rm ur})$ by By [@BT1b Proposition 5.1.14] and the second one maps ${\rm A}({\rm S}',k_{\rm T}^{\rm ur})$ into ${\rm A}({\rm T},k_{\rm T})$ by [@RousseauOrsay Théorème 2.5.6], hence their composite has the required property. $\Box$
### Compact open subgroups and integral structures {#sss - models}
In what follows, we maintain the previous assumptions, in particular the group ${\rm G}$ is semisimple and $k$-isotropic. The building ${\mathcal{B}}({\rm G}, k)$ admits a strongly transitive ${\rm G}(k)$-action by isometries. Moreover it is a [*labelled*]{} simplicial complex in the sense that, if $d$ denotes the number of codimension 1 facets (called [*panels*]{}) in the closure of a given alcove, we can choose $d$ colors and assign one of them to each panel in ${\mathcal{B}}({\rm G}, k)$ so that each color appears exactly once in the closure of each alcove. For some questions, it is convenient to restrict oneself to the finite index subgroup ${\rm G}(k)^\bullet$ consisting of the color-preserving (or [*type-preserving*]{}) isometries in ${\rm G}(k)$.
[*Compact open subgroups*]{}.— For any facet $F \subset {\mathcal{B}}({\rm G},k)$ we denote by ${\rm P}_F$ the stabilizer ${\rm Stab}_{{\rm G}(k)}(F)$: it is a bounded subgroup of ${\rm G}(k)$ and when $k$ is local, it is a compact, open subgroup. It follows from the Bruhat-Tits fixed point theorem (\[sss - geometry of buildings\]) that the conjugacy classes of maximal compact subgroups in ${\rm G}(k)^\bullet$ are in one-to-one correspondence with the vertices in the closure of a given alcove. The fact that there are usually several conjugacy classes of maximal compact subgroups in ${\rm G}(k)$ makes harmonic analysis more delicate than in the classical case of real Lie groups. Still, for instance thanks to the notion of a special vertex, many achievements can also be obtained in the non-Archimedean case [@Macdonald]. Recall that a point $x \in {\mathcal{B}}({\rm G},k)$ is called [*special*]{} if for any apartment $\mathbb{A}$ containing $x$, the stabilizer of $x$ in the affine Weyl group is the full vectorial part of this affine reflection group, i.e. is isomorphic to the (spherical) Weyl group of the root system $R$ of ${\rm G}$ over $k$.
[*Integral models for some stabilizers*]{}.— In what follows, we are more interested in algebraic properties of compact open subgroups obtained as facet stabilizers. The following statement is explained in [@BT1b 5.1.9].
\[th - integral structures\] For any facet $F \subset {\mathcal{B}}({\rm G},k)$ there exists a smooth $k^\circ$-group scheme $\mathcal{G}_F$ with generic fiber ${\rm G}$ such that $\mathcal{G}_F(k^\circ) = {\rm P}_F$.
As already mentioned, the point of view of group schemes over $k^\circ$ in Bruhat-Tits theory is not only an important tool to perform the descent, but it is also an important outcome of the theory. Here is an example. The “best” structure [*a priori*]{} available for a facet stabilizer is only of topological nature (and even for this, we have to assume that $k$ is locally compact). The above models over $k^\circ$ provide an algebraic point of view on these groups, which allows one to define a filtration on them leading to the computation of some cohomology groups of great interest for the congruence subgroup problem, see for instance [@PraRag1] and [@PraRag2]. Filtrations are also of great importance in the representation theory of non-Archimedean Lie groups, see for instance [@MoyPrasad1] and [@MoyPrasad2].
[*Closed fibres and local combinatorial description of the building*]{}.— We finish this brief summary of Bruhat-Tits theory by mentioning quickly two further applications of integral models for facet stabilizers.
First let us pick a facet $F \subset {\mathcal{B}}({\rm G}, k)$ as above and consider the associated $k^\circ$-group scheme $\mathcal{G}_F$. As a scheme over $k^\circ$, it has a closed fibre (so to speak obtained by reduction modulo $k^{\circ\circ}$) which we denote by $\overline{\mathcal{G}_F}$. This is a group scheme over the residue field $\widetilde{k}$. It turns out that the rational points $\overline{\mathcal{G}_F}(\widetilde{k})$ have a nice combinatorial structure (even though the $\widetilde{k}$-group $\overline{\mathcal{G}_F}$ needn’t be reductive in general); more precisely, $\overline{\mathcal{G}_F}(\widetilde{k})$ has a Tits system structure (see the end of \[sss - RS and RD\]) with finite Weyl group. One consequence of this is that $\overline{\mathcal{G}_F}(\widetilde{k})$ admits an action on a spherical building (a [*spherical building*]{} is merely a simplicial complex satisfying the axioms of Def. \[defi - simplicial building\] with the Euclidean tiling $\Sigma$ replaced by a spherical one). The nice point is that this spherical building naturally appears in the (Euclidean) Bruhat-Tits building ${\mathcal{B}}({\rm G}, k)$. Namely, the set of closed facets containing $F$ is a geometric realization of the spherical building of $\overline{\mathcal{G}_F}(\widetilde{k})$ [@BT1b Prop. 5.1.32]. In particular, for a complete valued field $k$, the building ${\mathcal{B}}({\rm G}, k)$ is locally finite if and only if the spherical building of $\overline{\mathcal{G}_F}(\widetilde{k})$ is actually finite for each facet $F$, which amounts to requiring that the residue field $\widetilde{k}$ be finite. Note that a metric space admits a compactification if, and only if, it is locally compact. Therefore from this combinatorial description of neighborhoods of facets, we see that [*the Bruhat-Tits building ${\mathcal{B}}({\rm G},k)$ admits a compactification if and only if $k$ is a local field*]{}.
\[rk - parahorics\] Let us assume here that $k$ is discretely valued. This is the context where the more classical combinatorial structure of an (affine) Tits system is relevant [@Lie456 IV.2]. Let us exhibit such a structure. First, a parahoric subgroup in ${\rm G}(k)$ can be defined to be the image of $(\mathcal{G}_F)^\circ (k^\circ)$ for some facet $F$ in ${\mathcal{B}}({\rm G},k)$, where $(\mathcal{G}_F)^\circ$ denotes the identity component of $\mathcal{G}_F$ [@BT1b 5.2.8]. We also say for short that a parahoric subgroup is the connected stabilizer of a facet in the Bruhat-Tits building ${\mathcal{B}}({\rm G}, k)$. If ${\rm G}$ is simply connected (in the sense of algebraic groups), then the family of parahoric subgroups is the family of abstract parabolic subgroups of a Tits system with affine Weyl group [@BT1b Prop. 5.2.10]. An Iwahori subgroup corresponds to the case when $F$ is a maximal facet. At last, if moreover $k$ is local with residual characteristic $p$, then an Iwahori subgroup can be characterized as the normalizer of a maximal pro-$p$ subgroup and an arbitrary parahoric subgroup as a subgroup containing an Iwahori subgroup.
Finally, the above integral models provide an important tool in the realization of Bruhat-Tits buildings in analytic spaces (and subsequent compactifications). Indeed, the fundamental step (see Th. \[thm - subgroup\]) for the whole thing consists in attaching injectively to [*any*]{} point $x \in {\mathcal{B}}({\rm G},K)$ an affinoid subgroup ${\rm G}_x$ of the analytic space ${\rm G}^{\rm an}$ attached to ${\rm G}$, and the definition of ${\rm G}_x$ makes use of the integral models attached to vertices. But one word of caution is in order here since the connexion with integral models avoids all their subtleties! For our construction, only smooth $k^\circ$-group schemes $\mathcal{G}_F$ which are *reductive* are of interest; this is not the case in general, but one can easily prove the following statement: [*given a vertex $x \in \mathcal{B}({\rm G},k)$, there exists a finite extension $k'/k$ such that the ${k'}^{\circ}$-group scheme $\mathcal{G}'_x$, attached to the point $x$ seen as a vertex of $\mathcal{B}({\rm G},k')$, is a Chevalley-Demazure group scheme over ${k'}^\circ$*]{}. In this situation, one can define $({\rm G} \otimes_k k')_x$ as the *generic fibre* of the formal completion of $\mathcal{G}'_x$ along its special fibre; this is a $k'$-affinoid subgroup of $({\rm G} \otimes_k k')^{\rm an}$ and one invokes descent theory to produce a $k$-affinoid subgroup of ${\rm G}^{\rm an}$.
### A characterization of apartments {#sss - characterization_apartments}
For later use, we end this section on Bruhat-Tits theory by a useful characterization of apartments inside buildings.
Given a torus ${\rm S}$ over $k$, we denote by ${\rm S}^1(k)$ the maximal bounded subgroup of ${\rm S}(k)$. It is the subgroup of ${\rm S}(k)$ defined by the equations $|\chi|=1$, where $\chi$ runs over the character group of ${\rm S}$.
\[characterization\_apartments\] Let ${\rm S}$ be a maximal split torus and let $x$ be a point of $\mathcal{B}({\rm G},k)$. If the residue field of $k$ contains at least four elements, then the following conditions are equivalent:
- $x$ belongs to the apartment ${\rm A}({\rm S},k)$;
- $x$ is fixed under the action of ${\rm S}^1(k)$.
*Proof*. Condition (i) always implies condition (ii). With our hypothesis on the cardinality of the residue field, the converse implication holds by [@BT1b Proposition 5.1.37]. $\Box$
Buildings and Berkovich spaces {#s - general compactifications}
==============================
As above, we consider a semisimple group ${\rm G}$ over some non-Archimedean field $k$. In this section, we explain how to realize the Bruhat-Tits building $\mathcal{B}({\rm G},k)$ of ${\rm G}(k)$ in non-Archimedean analytic spaces deduced from ${\rm G}$, and we present two procedures that can be used to compactify Bruhat-Tits buildings in full generality; as we pointed out before, the term “compactification” is abusive if $k$ is not a local field (see the discussion before Remark \[rk - parahorics\]).
Assuming that $k$ is locally compact, let us describe very briefly those two ways of compactifying a building. The first is due to V. Berkovich when ${\rm G}$ is split [@Ber1 Chap. V] and it consists in two steps:
1\. to define a closed embedding of the building into the analytification of the group (\[ss - closed embedding\]);
2\. to compose this closed embedding with an analytic map from the group to a (compact) flag variety (\[ss-maps\_to\_flags\]).
By taking the closure of the image of the composed map, we obtain an equivariant compactification which admits a Lie-theoretic description (as expected). For instance, there is a convenient description of this ${\rm G}(k)$-topological space (convergence of sequences, boundary strata etc.) by means of invariant fans in $\left({\rm X}_*({\rm S}) \otimes_{\mathbf{Z}} \mathbf{R}, {\rm W} \right)$, where ${\rm X}_*({\rm S})$ denotes the cocharacter group of a maximal split torus ${\rm S}$ endowed with the natural action of the Weyl group ${\rm W}$ (\[ss - fans\]). The finite family of compactifications obtained in this way is indexed by ${\rm G}(k)$-conjugacy classes of parabolic subgroups.
These spaces can be recovered from a different point of view, using representation theory and the concrete compactification $\overline{\mathcal{X}}({\rm V},k)$ of the building $\mathcal{X}({\rm V},k)$ of ${\rm SL}({\rm V},k)$ which was described in Section 2. It mimics the original strategy of I. Satake in the case of symmetric spaces [@Satake1]: we pick a faithful linear representation of ${\rm G}$ and, relying on analytic geometry, we embed $\mathcal{B}({\rm G},k)$ in $\mathcal{X}({\rm V},k)$; by taking the closure in $\overline{\mathcal{X}}({\rm V},k)$, we obtain our compactification.
**Caution** — 1. We need some functoriality assumption on the building with respect to the field: in a sense which was made precise after the statement of Theorem \[th - functoriality of BT buildings\], this means that $\mathcal{B}({\rm G}, -)$ is functor on the category of non-Archimedean extensions of $k$.
As explained in [@RTW1 1.3.4], these assumptions are fulfilled if $k$ quasi-splits over a tamely ramified extension of $k$. This is in particular the case is $k$ is discretely valued with perfect residue field, or if ${\rm G}$ is split.
2\. There is no other restriction on the non-Archimedean field $k$ considered in 4.1. From 4.2 on, we assume that $k$ is *local*. In any case, the reader should keep in mind that non-local extensions of $k$ do always appear in the study of Bruhat-Tits buildings from Berkovich’s point of view (see Proposition 4.2).
The references for the results quoted in this section are [@RTW1] and [@RTW2].
Realizing buildings inside Berkovich spaces {#ss - closed embedding}
-------------------------------------------
Let $k$ be a field which is complete with respect to a non-trivial non-Archimedean absolute value. We fix a semisimple group ${\mathrm{G}}$ over $k$. Our first goal is to define a continuous injection of the Bruhat-Tits building $\mathcal{B}({\mathrm{G}},k)$ in the Berkovich space ${\mathrm{G}}^{\rm an}$ associated to the algebraic group ${\mathrm{G}}$. Since ${\mathrm{G}}$ is affine with affine coordinate ring $\mathcal{O}({\mathrm{G}})$, its analytification consists of all multiplicative seminorms on $\mathcal{O}({\mathrm{G}})$ extending the absolute value on $k$ [@Temkin].
### Non-Archimedean extensions and universal points {#sss - n-A extensions}
We will have to consider infinite non-Archimedean extensions of $k$ as in the following example.
\[ex - field\] Let $\mathbf{r} = (r_1, \ldots, r_n)$ be a tuple of positive real numbers such that $r_1^{i_1} \ldots r_n^{i_n} \notin |k^\times|$ for all choices of $(i_1, \ldots, i_n) \in \mathbf{Z}^n - \{0\}$. Then the $k$-algebra $$k_{\mathbf{r}} = \left\{\sum_{I = (i_1 \ldots, i_n)} a_I x_1^{i_1}\ldots x_n^{i_n} \in k[[x_1^{\pm 1},\ldots, x_n^{\pm 1}]] \ ; \ |a_I| r_1^{i_1} \ldots r_n^{i_n} \rightarrow 0 \mbox{ when }|i_1| + \ldots + |i_n| \rightarrow \infty \right\}$$ is a non-Archimedean field extension of $k$ with absolute value $|f| = \max_I \{|a_I| r_1^{i_1} \ldots r_n^{i_n} \}$.
We also need to recall the notion of a *universal* point [^2]. Let $z$ be a point in ${\rm G}^{\rm an}$, seen as a multiplicative $k$-seminorm on $\mathcal{O}({\rm G})$. For a given non-Archimedean field extension $K/k$, there is a natural $K$-seminorm $||.|| = z \otimes 1$ on $\mathcal{O}({\rm G}) \otimes_k K$, defined by $$||a|| = \inf \max_i |a_i(z)|\cdot |\lambda_i|$$ where the infimum is taken over the set of all expressions $\sum_i a_i \otimes \lambda_i$ representing $a$, with $a_i \in \mathcal{O}({\rm G})$ and $\lambda_i \in K$. The point $z$ is said to be *universal* if, for any non-Archimedean field extension $K/k$, the above $K$-seminorm on $\mathcal{O}({\rm G}) \otimes_k K$ is multiplicative. One writes $z_K$ for the corresponding point in ${\rm G}^{\rm an} \widehat{\otimes}_k K$. We observe that this condition depends only on the completed residue field $\mathcal{H}(z)$ of ${\rm G}^{\rm an}$ at $z$.
\[Rk-universal\] 1. Obviously, points of ${\rm G}^{\rm an}$ coming from $k$-rational points of ${\rm G}$ are universal. 2. Let $x \in {\rm G}^{\rm an}$ be universal. For any finite Galois extension $k'/k$, the canonical extension $x_{k'}$ of $x$ to ${\rm G}^{\rm an} \otimes_k k'$ is invariant under the action of ${\rm Gal}(k'/k)$: indeed, the $k'$-norm $x \otimes 1$ on $\mathcal{O}({\rm G}) \otimes_k k'$ is Galois invariant. 3. If $k$ is algebraically closed, Poineau proved that every point of ${\rm G}^{\rm an}$ is universal [@Poineau Corollaire 4.10].
### Improving transitivity {#sss-transitivity}
Now let ${\mathrm{G}}^{\rm an}$ be the Berkovich analytic space associated to the algebraic group ${\mathrm{G}}$. Our goal is the first step mentioned in the introduction, namely the definition of a continuous injection $$\vartheta: \mathcal{B}({\mathrm{G}},k) \longrightarrow {\mathrm{G}}^{\rm an}.$$ We proceed as follows. For every point $x$ in the building $\mathcal{B}({\mathrm{G}},k)$ we construct an affinoid subgroup ${\mathrm{G}}_x$ of ${\mathrm{G}}^{\rm an}$ such that, for any non-Archimedean extension $K/k$, the subgroup ${\rm G}_x(K)$ of ${\rm G}(K)$ is precisely the stabilizer of $x$ in the building over $K$. Then we define $\vartheta(x)$ as the (multiplicative) seminorm on $\mathcal{O}({\mathrm{G}})$ defined by taking the maximum over the compact subset ${\rm G}_x$ of ${\rm G}^{\rm an}$.
If the Bruhat-Tits building $\mathcal{B}({\rm G},k)$ can be seen as non-Archimedean analogue of a Riemannian symmetric space, it is not homogeneous under ${\rm G}(k)$; for example, if $k$ is discretely valued, the building carries a polysimplicial structure which is preserved by the action of ${\rm G}(k)$. There is a very simple way to remedy at this situation using field extensions, and this is where our functoriality assumption comes in.
Let us first of all recall that the notion of a special point was defined in Section 1, just before Definition \[defi - non-simplicial building\]. Its importance comes from the fact that, when ${\rm G}$ is split, the stabilizer of a special point is particularly nice (see the discussion after Theorem \[thm - subgroup\]). As simple consequences of the definition, one should notice the following two properties: if a point $x \in \mathcal{B}({\rm G},k)$ is special, then
- every point in the ${\rm G}(k)$-orbit of $x$ is again special;
- if moreover ${\rm G}$ is *split*, then $x$ remains special in $\mathcal{B}({\rm G},K)$ for any non-Archimedean field extension $K/k$ (indeed: the local Weyl group at $x$ over ${\rm K}$ contains the local Weyl group at $x$ over $k$, and the full Weyl group of ${\rm G}$ is the same over $k$ and over $K$).
We can now explain how field extensions allow to improve transitivity of the group action on the building.
\[prop - special point\] 1. Given any two points $x, y \in \mathcal{B}({\rm G},k)$, there exists a non-Archimedean field extension $K/k$ such that $x$ and $y$, identified with points of $\mathcal{B}({\rm G},K)$ via the canonical injection $\mathcal{B}({\rm G},k) \hookrightarrow \mathcal{B}({\rm G},K)$, belong to the same orbit under ${\rm G}(K)$.
2\. For every point $x \in \mathcal{B}({\mathrm{G}},k)$, there exists a non-Archimedean field extension $K/k$ such that the following conditions hold:
[(i)]{} The group ${\mathrm{G}}\otimes_{k} K$ is split; [(ii)]{} The canonical injection $\mathcal{B}({\mathrm{G}},k) \rightarrow \mathcal{B}({\mathrm{G}}, K)$ maps $x$ to a special point.
We give a proof of this Proposition since it is the a key result for the investigation of Bruhat-Tits buildings from Berkovich’s point of view. The second assertion follows easily from the first: just pick a finite separable field extension $k'/k$ splitting ${\rm G}$ and a special point $y$ in $\mathcal{B}({\rm G},k')$, then consider a non-Archimedean field extension $K/k'$ such that $x$ and $y$ belong to the same ${\rm G}(K)$-orbit. In order to prove the first assertion, we may and do assume that ${\rm G}$ is split. Let ${\rm S}$ denote a maximal split torus of ${\rm G}$ whose apartment ${\rm A}({\rm S},k)$ contains both $x$ and $y$. As recalled in Proposition 3.17, this apartment is an affine space under ${\rm X}_*({\rm S}) \otimes_{\mathbf{Z}} \mathbf{R}$, where ${\rm X}_*({\rm S})$ denotes the cocharacter space of ${\rm S}$, and ${\rm S}(k)$ acts on ${\rm A}({\rm S},k)$ by translation via a map $\nu : {\rm S}(k) \rightarrow {\rm X}_*({\rm S}) \otimes_{\mathbf{Z}} \mathbf{R}$. Using a basis of characters to identify ${\rm X}_*({\rm S})$ (resp. ${\rm S}$) with $\mathbf{Z}^n$ (resp. $\mathbf{G}_{\rm m}^n$), it turns out that $\nu$ is simply the map $$k^{\times} \longrightarrow \mathbf{R}^n, \ \ \ (t_1,\ldots, t_n) \mapsto (-\log |t_1|, \ldots, -\log |t_n|).$$ By combining finite field extensions and transcendental extensions as described in Example 4.1, we can construct a non-Archimedean field extension $K/k$ such that the vector $x-y \in \mathbf{R}^n$ belongs to the subgroup $\log |(K^\times)^n|$. This implies that $x$ and $y$, seen as points of ${\rm A}({\rm S},K)$, belong to the same orbit under ${\rm S}(K)$, hence under ${\rm G}(K)$.
If $|K^\times| = \mathbf{R}_{>0}$, then ${\rm G}(K)$ acts transitively on $\mathcal{B}({\rm G},K)$. However, it is more natural to work functorially than to fix arbitrarily an algebraically closed non-Archimedean extension $\Omega/k$ such that $|\Omega^\times| = \mathbf{R}_{>0}$.
### Affinoid subgroups {#sss - affinoid}
Let us now describe the key fact explaining the relationship between Bruhat-Tits theory and non-Archimedean analytic geometry. This result is crucial for all subsequent constructions.
\[thm - subgroup\] For every point $x \in \mathcal{B}({\mathrm{G}},k)$ there exists a unique $k$-affinoid subgroup ${\mathrm{G}}_x$ of ${\mathrm{G}}^{\rm an}$ satisfying the following condition: for every non-Archimedean field extension $K/k$, the group ${\mathrm{G}}_x(K)$ is the stabilizer in ${\mathrm{G}}(K)$ of the image of $x$ under the injection $\mathcal{B}({\mathrm{G}},k) \rightarrow \mathcal{B}({\mathrm{G}},K)$.
The idea of the proof is the following (see [@RTW1 Th. 2.1] for details). If ${\mathrm{G}}$ is split and $x$ is a special point in the building, then the integral model $\mathcal{G}_x$ of ${\rm G}$ described in (3.2.3) is a Chevalley group scheme, and we define ${\rm G}_x$ as the generic fibre $\widehat{\mathcal{G}_x}_\eta$ of the formal completion of $\mathcal{G}_x$ along its special fibre. This is a $k$-affinoid subgroup of ${\rm G}^{\rm an}$, and it is easy to check that it satisfies the universal property in our claim. Thanks to Proposition \[prop - special point\], we can achieve this situation after a suitable non-Archimedean extension $K/k$, and we apply faithfully flat descent to obtain the $k$-affinoid subgroup ${\rm G}_x$ [@RTW1 App. A]. Let us remark that, in order to perform this descent step, it is necessary to work with an extension which is not too big (technically, the field $K$ should be a $k$-affinoid algebra); since one can obtain $K$ by combining finite extensions with the transcendental ones described in Example \[ex - field\], this is fine.
### Closed embedding in the analytic group {#sss - closed embedding}
The $k$-affinoid subgroup ${\mathrm{G}}_x$ is the Berkovich spectrum of a $k$-affinoid algebra ${\rm A}_x$, i.e., ${\mathrm{G}}_x$ is the Gelfand spectrum $\mathcal{M}({\rm A}_x)$ of bounded multiplicative seminorms on ${\rm A}_x$. This is a compact and Hausdorff topological space over which elements of ${\rm A}_x$ define non-negative real valued functions. For any non-zero $k$-affinoid algebra ${\rm A}$, one can show that its Gelfand spectrum $\mathcal{M}({\rm A})$ contains a smallest non-empty subset, called its *Shilov boundary* and denoted $\Gamma({\rm A})$, such that each element $f$ of ${\rm A}$ reaches its maximum at some point in $\Gamma({\rm A})$.
\(i) If ${\rm A} = k\{{\rm T}\}$ is the Tate algebra of restricted power series in one variable, then $\mathcal{M}({\rm A})$ is Berkovich’s closed unit disc and its Shilov boundary is reduced to the point $o$ defined by the Gauss norm: for $f = \sum_{n \in \mathbf{N}} a_n{\rm T}^n$, one has $|f(o)| = \max_n |a_n|$.
\(ii) Let $a \in k$ with $0 < |a| < 1$. If ${\rm A} = k\{{\rm T}, {\rm S}\}/({\rm ST}-a)$, then $\mathcal{M}({\rm A})$ is an annulus of modulus $|a|$ and $\Gamma({\rm A})$ contains two points $o, o'$: for $f = \sum_{n \in \mathbf{Z}} a_n {\rm T}^n$, where ${\rm T}^{-1} = a^{-1}{\rm S}$, one has $|f(o)| = \max_n |a_n|$ and $|f(o')| = \max_n |a_n|.|a|^n$.
\(iii) For any non-zero $k$-affinoid algebra ${\rm A}$, its Shilov boundary $\Gamma({\rm A})$ is reduced to a point if and only if the seminorm $${\rm A} \rightarrow \mathbf{R}_{\geqslant 0}, \ \ \ f \mapsto \sup_{x \in \mathcal{M}({\rm A})} |f(x)|$$ is multiplicative.
For every point $x$ of $\mathcal{B}({\rm G}, k)$, it turns out that the Shilov boundary of ${\rm G}_x = \mathcal{M}({\rm A}_x)$ is reduced to a unique point, denoted $\vartheta(x)$. This is easily seen by combining the nice behavior of Shilov boundaries under non-Archimedean extensions, together with a natural bijection between the Shilov boundary of $\mathcal{V}_\eta$ and the set of irreducible components of $\mathcal{V} \otimes_{k^\circ} \widetilde{k}$ if $\mathcal{V}$ is a normal $k^\circ$-formal scheme; indeed, the smooth $k^\circ$-group scheme $\mathcal{G}_x$ has a connected special fibre when it is a Chevalley group scheme. Let us also note that the affinoid subgroup ${\rm G}_x$ is completely determined by the single point $\vartheta(x)$ via
$${\rm G}_x = \{z \in {\rm G}^{\rm an} \ ; \ \forall f \in \mathcal{O}({\rm G}), \ |f(z)| \leqslant |f(\vartheta(x))|\}.$$
In this way we define the desired map $$\vartheta: \mathcal{B}({\mathrm{G}}, k) \rightarrow {\mathrm{G}}^{\rm an},$$
and we show [@RTW1 Prop. 2.7] that it is injective, continuous and ${\mathrm{G}}(k)$-equivariant (where ${\mathrm{G}}(k)$ acts on ${\mathrm{G}}^{\rm an}$ by conjugation). If $k$ is a local field, $\vartheta$ induces a homeomorphism from $\mathcal{B}({\mathrm{G}},k)$ to a closed subspace of ${\mathrm{G}}^{\rm an}$ [@RTW1 Prop. 2.11].
Finally, the map $\vartheta $ is also compatible with non-Archimedean extensions $K/k$, i.e., the following diagram
$$\xymatrix{\mathcal{B}({\rm G},K) \ar@{->}[r]^{\vartheta_K} & ({\rm G} \otimes_k {\rm K})^{\rm an} \ar@{->}[d]^{p_{K/k}} \\ \mathcal{B}({\rm G},k) \ar@{->}[r]_\vartheta \ar@{->}[u]^{\iota_{K/k}} & {\rm G}^{\rm an}}$$ where $\iota_{K/k}$ (resp. $p_{K/k}$) is the canonical embedding (resp. projection) is commutative. In particular, we see that this defines a *section* of $p_{K/k}$ over the image of $\vartheta$. In fact, any point $z$ belonging to this subset of ${\rm G}^{\rm an}$ is universal (\[sss - n-A extensions\]) and $\vartheta_K(\iota_{K/k}(x))$ coincides with the canonical lift $\vartheta(x)_K$ of $\vartheta(x)$ to $({\rm G} \otimes_k K)^{\rm an}$ for any $x \in \mathcal{B}({\rm G},k)$.
Moreover, if $K/k$ is a Galois extension, then the upper arrow in the diagram is ${\rm Gal}(K/k)$-equivariant by [@RTW1 Prop. 2.7].
Compactifying buildings with analytic flag varieties {#ss-maps_to_flags}
----------------------------------------------------
Once the building has been realized in the analytic space ${\rm G}^{\rm an}$, it is easy to obtain compactifications. In order not to misuse the latter word, we assume from now one that $k$ is *locally compact*.
### Maps to flag varieties {#sss - maps to flags}
The embedding $\vartheta: \mathcal{B}({\mathrm{G}},k) \rightarrow {\mathrm{G}}^{\rm an}$ defined in \[sss - closed embedding\] can be used to compactify the Bruhat-Tits building $\mathcal{B}({\mathrm{G}},k)$. We choose a parabolic subgroup ${\mathrm{P}}$ of ${\mathrm{G}}$. Then the flag variety ${\mathrm{G}}/{\mathrm{P}}$ is complete, and therefore the associated Berkovich space $({\mathrm{G}}/{\mathrm{P}})^{\rm an}$ is compact. Hence we can map the building to a compact space by the composition $$\vartheta_P: \mathcal{B}({\mathrm{G}},k) \stackrel{\vartheta}{\longrightarrow} {\mathrm{G}}^{\rm an} \longrightarrow ({\mathrm{G}}/ {\mathrm{P}})^{\rm an}.$$ The map $\vartheta_P$ is by construction ${\mathrm{G}}(k)$-equivariant and it depends only on the ${\rm G}(k)$-conjugacy class of ${\rm P}$: we have $\vartheta_{g{\rm P}g^{-1}} = g\vartheta_{\rm P}g^{-1}$ for any $g \in {\rm G}(k)$.
However, $\vartheta_P$ may not be injective. By the structure theory of semisimple groups, there exists a finite family of normal reductive subgroups ${\mathrm{G}}_i$ of ${\mathrm{G}}$ (each of them quasi-simple), such that the product morphism $$\prod_i {\mathrm{G}}_i \longrightarrow {\mathrm{G}}$$ is a central isogeny. Then the building $\mathcal{B}({\mathrm{G}},k)$ can be identified with the product of all $\mathcal{B}({\mathrm{G}}_i,k)$. If one of the factors ${\mathrm{G}}_i$ is contained in ${\mathrm{P}}$, then the factor $\mathcal{B}({\mathrm{G}}_i,k)$ is squashed down to a point in the analytic flag variety $({\mathrm{G}}/ {\mathrm{P}})^{\rm an}$.
If we remove from $\mathcal{B}({\rm G},k)$ all factors $\mathcal{B}({\mathrm{G}}_i, k)$ such that ${\mathrm{G}}_i$ is contained in ${\rm P}$, then we obtain a building $\mathcal{B}_t({\mathrm{G}},k)$, where $t$ stands for the type of the parabolic subgroup ${\mathrm{P}}$, i.e., for its ${\rm G}(k)$-conjugacy class. The factor $\mathcal{B}_t({\mathrm{G}},k)$ is mapped injectively into $({\mathrm{G}}/{\mathrm{P}})^{\rm an}$ via $\vartheta_{\mathrm{P}}$.
\[rk - injectivity\] If ${\mathrm{G}}$ is almost simple, then $\vartheta_{\mathrm{P}}$ is injective whenever ${\mathrm{P}}$ is a proper parabolic subgroup in ${\mathrm{G}}$; hence in this case the map $\vartheta_{\mathrm{P}}$ provides an embedding of $\mathcal{B}({\mathrm{G}},k)$ into $({\mathrm{G}}/ {\mathrm{P}})^{\rm an}$.
### Berkovich compactifications {#sss - Berkovich compactifications}
Allowing compactifications of the building in which some factors are squashed down to a point, we introduce the following definition.
\[defi - Berkovich compactification\] Let $t$ be a ${\rm G}(k)$-conjugacy class of parabolic subgroups of ${\rm G}$. We define $\overline{\mathcal{B}}_t({\mathrm{G}},k)$ to be the closure of the image of $\mathcal{B}({\mathrm{G}},k)$ in $({\mathrm{G}}/{\mathrm{P}})^{\rm an}$ under $\vartheta_P$, where ${\rm P}$ belongs to $t$, and we endow this space with the induced topology. The compact space $\overline{\mathcal{B}}_t({\mathrm{G}},k)$ is called the [Berkovich compactification of type $t$]{} of the building $\mathcal{B}({\mathrm{G}},k)$.
Note that we obtain one compactification for each ${\rm G}(k)$-conjugacy class of parabolic subgroups.
\[rk - non locally compact\] If we drop the assumption that $k$ is locally compact, the map $\vartheta_{\mathrm{P}}$ is continuous but the image of $\mathcal{B}_t({\rm G},k)$ is not locally closed. In this case, the right way to proceed is to compactify each apartment ${\rm A}_t({\rm S},k)$ of $\mathcal{B}_t({\rm G},k)$ by closing it in ${\rm G}^{\rm an}/{\rm P}^{\rm an}$ and to define $\overline{\mathcal{B}}_t({\rm G},k)$ as the union of all compactified apartments. This set is a quotient of ${\rm G}(k) \times \overline{A}_t({\rm S},k)$ and we endow it with the quotient topology [@RTW1 3.4].
### The boundary {#sss - boundary}
Now we want to describe the boundary of the Berkovich compactifications. We fix a type $t$ (i.e., a ${\rm G}(k)$-conjugacy class) of parabolic subgroups.
\[defi - osculatory\] Two parabolic subgroups ${\mathrm{P}}$ and ${\rm Q}$ of ${\mathrm{G}}$ are called [osculatory]{} if their intersection ${\mathrm{P}}\cap {\rm Q}$ is also a parabolic subgroup.
Hence ${\mathrm{P}}$ and ${\rm Q}$ are osculatory if and only if they contain a common Borel group after a suitable field extension. We can generalize this definition to semisimple groups over arbitrary base schemes. Then for every parabolic subgroup ${\rm Q}$ there is a variety $\mathrm{Osc}_t({\rm Q})$ over $k$ representing the functor which associates to any base scheme ${\rm S}$ the set of all parabolics of type $t$ over ${\rm S}$ which are osculatory to ${\rm Q}$ [@RTW1 Prop. 3.2].
\[defi - t-relevant\] Let ${\rm Q}$ be a parabolic subgroup. We say that ${\rm Q}$ is [$t$-relevant]{} if there is no parabolic subgroup ${\rm Q}'$ strictly containing ${\rm Q}$ such that $\mathrm{Osc}_t({\rm Q}) = \mathrm{Osc}_t({\rm Q}')$.
Let us illustrate this definition with the following example.
\[example - relevant\] Let ${\mathrm{G}}$ be the group ${\rm SL}({\rm V})$, where ${\rm V}$ is a $k$-vector space of dimension $d +1$. The non-trivial parabolic subgroups of ${\mathrm{G}}$ are the stabilizers of flags $$(0 \subsetneq {\rm V}_1 \subsetneq \ldots \subsetneq {\rm V}_r \subsetneq {\rm V}).$$ Let ${\rm H}$ be a hyperplane in ${\rm V}$, and let ${\mathrm{P}}$ be the parabolic subgroup of ${\rm SL}({\rm V})$ stabilizing the flag $(0 \subset {\rm H} \subset {\rm V})$. We denote its type by $\delta$. Let ${\rm Q}$ be an arbitrary parabolic subgroup, stabilizing a flag $(0 \subsetneq {\rm V}_1 \subsetneq \ldots \subsetneq {\rm V}_r \subsetneq {\rm V})$. Then ${\rm Q}$ and ${\rm P}$ are osculatory if and only if ${\rm H}$ contains the linear subspace ${\rm V}_r$. This shows that all parabolic subgroups ${\rm Q}$ stabilizing flags contained in the subspace ${\rm V}_r$ give rise to the same variety $\mathrm{Osc}_\delta({\rm Q})$. Therefore, a non-trivial parabolic is $\delta$-relevant if and only if the corresponding flag has the form $0 \subsetneq {\rm W} \subsetneq {\rm V}$.
Having understood how to parametrize boundary strata, we can now give the general description of the Berkovich compactification $\overline{\mathcal{B}}_t({\mathrm{G}},k)$. The following result is Theorem 4.1 in [@RTW1].
\[thm - stratification\] For every $t$-relevant parabolic subgroup ${\rm Q}$, let ${\rm Q}_{\rm ss}$ be its semisimplification (i.e., ${\rm Q}_{\rm ss}$ is the quotient ${\rm Q}/\mathcal{R}({\rm Q})$ where $\mathcal{R}({\rm Q})$ denotes the radical of ${\rm Q}$). Then $\overline{\mathcal{B}}_t({\mathrm{G}},k)$ is the disjoint union of all the buildings $\mathcal{B}_t({\rm Q}_{ss},k)$, where ${\rm Q}$ runs over the $t$-relevant parabolic subgroups of ${\mathrm{G}}$.
The fact that the Berkovich compactifications of a given group are contained in the flag varieties of this group enables one to have natural maps between compactifications: they are the restrictions to the compactifications of (the analytic maps associated to) the natural fibrations between the flag varieties. The above combinatorics of $t$-relevancy is a useful tool to formulate which boundary components are shrunk when passing from a compactification to a smaller one [@RTW1 Section 4.2].
\[example - stratification\] Let us continue Example \[example - relevant\] by describing the stratification of $\overline{\mathcal{B}}_\delta({\rm SL}({\rm V}),k)$. Any $\delta$-relevant subgroup ${\rm Q}$ of ${\mathrm{G}}= {\rm SL}({\rm V})$ is either equal to ${\rm SL}({\rm V})$ or equal to the stabilizer of a linear subspace $0 \subsetneq {\rm W} \subsetneq {\rm V}$. In the latter case ${\rm Q}_{\rm ss}$ is isogeneous to ${\rm SL}({\rm W}) \times {\rm SL}({\rm V}/{\rm W})$. Now ${\rm SL}({\rm W})$ is contained in a parabolic of type $\delta$, hence $\mathcal{B}_\delta({\rm Q}_{\rm ss},k)$ coincides with $\mathcal{B}({\rm SL}({\rm V}/{\rm W}),k)$. Therefore $$\overline{\mathcal{B}}_\delta({\rm SL}({\rm V}),k) = \bigcup_{{\rm W} \subsetneq {\rm V}} \mathcal{B}\bigl({\rm SL}({\rm V}/{\rm W},k)\bigr),$$ where ${\rm W}$ runs over all linear subspaces ${\rm W} \subsetneq {\rm V}$.
Recall from \[SL-building\] that the Euclidean building $\mathcal{B}({\rm SL}({\rm V}),k)$ can be identified with the Goldman-Iwahori space $\mathcal{X}({\rm V},k)$ defined in \[defi - GI\]. Hence $\overline{\mathcal{B}}_\delta({\rm SL}({\rm V}),k)$ is the disjoint union of all $\mathcal{X}({\rm V}/{\rm W},k)$. Therefore we can identify the seminorm compactification $\overline{\mathcal{X}}(V,k)$ from \[ss - seminorm compactification\] with the Berkovich compactification of type $\delta$.
Invariant fans and other compactifications {#ss - fans}
------------------------------------------
Our next goal is to compare our approach to compactifying building with another one, developed in [@Wer07] without making use of Berkovich geometry. In this work, compactified buildings are defined by a gluing procedure, similar to the one defining the Bruhat-Tits building in Theorem \[th - gluing\]. In a first step, compactifications of apartments are obtained by a cone decomposition. Then these compactified apartments are glued together with the help of subgroups which turn out to be the stabilizers of points in the compactified building.
Let ${\mathrm{G}}$ be a (connected) semisimple group over $k$ and $\mathcal{B}({\mathrm{G}},k)$ the associated Bruhat-Tits building. We fix a maximal split torus ${\mathrm{T}}$ in ${\mathrm{G}}$, giving rise to the cocharacter space $\Sigma_{\rm vect} = {\rm X}_\ast({\mathrm{T}}) \otimes \mathbf{R}$. The starting point is a faithful, geometrically irreducible representation $\rho: {\mathrm{G}}\rightarrow {\rm GL}({\mathrm{V}})$ on some finite-dimensional $k$-vector space ${\mathrm{V}}$.
Let $R=R({\mathrm{T}},{\mathrm{G}}) \subset {\rm X}^\ast({\mathrm{T}})$ be the associated root system. We fix a basis $\Delta$ of $R$ and denote by $\lambda_0(\Delta)$ the highest weight of the representation $\rho$ with respect to $\Delta$. Then every other ($k$-rational) weight of $\rho$ is of the form $\lambda_0(\Delta) - \sum_{\alpha \in \Delta} n_\alpha \alpha$ with coefficients $n_\alpha \geqslant 0$. We write $[\lambda_0(\Delta) - \lambda] = \{ \alpha \in \Delta: n_\alpha >0\}$. We call every such subset $Y$ of $\Delta$ of the form $Y = [\lambda_0(\Delta) - \lambda]$ for some weight $\lambda$ *admissible*.
Let ${\rm Y} \subset \Delta$ be an admissible subset. We denote by ${\rm C}_{\rm Y}^\Delta$ the following cone in $\Sigma_{\rm vect}$: $${\rm C}_{\rm Y}^\Delta = \left\{ x \in \Sigma_{\rm vect} \ ; \ \begin{array}{ll} \alpha(x) = 0 & \mbox{ for all }\alpha \in {\rm Y}, \mbox{ and } \\
(\lambda_0(\Delta) - \lambda)(x) \geqslant 0 & \mbox{ for all weights }\lambda \mbox{ such that } [\lambda_0(\Delta) - \lambda] \not\subset {\rm Y}\end{array} \right\}$$
The collection of all cones ${\rm C}_{\rm Y}^\Delta$, where $\Delta$ runs over all basis of the root system and ${\rm Y}$ over all admissible subsets of $\Delta$, is a complete fan $\mathcal{F}_\rho$ in $\Sigma_{\rm vect}$. There is a natural compactification of $\Sigma_{\rm vect}$ associated to $\mathcal{F}_\rho$, which is defined as $\overline{\Sigma}_{\rm vect} = \bigcup_{{\rm C} \in \mathcal{F}_\rho} \Sigma_{\rm vect} / \langle {\rm C} \rangle$ endowed with a topology given by tubular neighborhoods around boundary points. For details see [@Wer07 Section 2] or [@RTW1 Appendix B].
We will describe this compactification in two examples.
\[regularweight\] If the highest weight of $\rho$ is regular, then every subset ${\rm Y}$ of $\Delta$ is admissible. In this case, the fan $\mathcal{F}_\rho$ is the full Weyl fan. In the case of a root system of type ${\rm A}_2$, the resulting compactification is shown on Figure 1. The shaded area is a compactified Weyl chamber, whose interior contains the corresponding highest weight of $\rho$.
\[identityrep\] Let ${\mathrm{G}}= {\rm SL}({\mathrm{V}})$ be the special linear group of a $(d+1)$-dimensional $k$-vector space ${\rm V}$, and let $\rho$ be the identical representation. We look at the torus ${\mathrm{T}}$ of diagonal matrices in ${\rm SL}({\mathrm{V}})$, which gives rise to the root system $R = \{\alpha_{i,j}\}$ of type ${\rm A}_{d}$ described in Example \[ex - RS of type A\]. Then $\Delta = \{\alpha_{0,1}, \alpha_{1,2}, \ldots, \alpha_{d-1, d}\}$ is a basis of $R$ and $\lambda_0(\Delta) = \varepsilon_0$ in the notation of Example \[ex - RS of type A\]. The other weights of the identical representation are $\varepsilon_1, \ldots, \varepsilon_d$. Hence the admissible subsets of $\Delta$ are precisely the sets ${\rm Y}_r = \{\alpha_{0,1}, \ldots, \alpha_{r-1 ,r}\}$ for $r = 1, \ldots, d$, and ${\rm Y}_{0} = \varnothing$. Let $\eta_0, \ldots, \eta_d$ be the dual basis of $\varepsilon_0, \ldots, \varepsilon_d$. Then $\Sigma_{\rm vect}$ can be identified with $\bigoplus_{i = 0}^d \mathbf{R} \eta_i / \mathbf{R} (\sum_{i} \eta_i)$, and we find $${\rm C}_{{\rm Y}_r}^\Delta = \{ x = \sum_i x_i \eta_i \in \Sigma_{\rm vect}: x_0 = \ldots = x_r \mbox{ and }x_0 \geqslant x_{r+1}, x_0 \geqslant x_{r+2}, \ldots, x_0 \geqslant x_d\} / \mathbf{R} (\sum_i \eta_i)$$ The associated compactification is shown in Figure 2. The shaded area is a compactified Weyl chamber and its codimension one face marked by an arrow contains the highest weight of $\rho$ (with respect to this Weyl chamber).
The compactification $\overline{\Sigma}_{\rm vect}$ induces a compactification $\overline{\Sigma}$ of the apartment $\Sigma = {\rm A}({\rm T},k)$, which is an affine space under $\Sigma_{\rm vect}$. Note that the fan $\mathcal{F}_\rho$ and hence the compactification $\overline{\Sigma}$ only depend on the Weyl chamber face containing the highest weight of $\rho$, see [@Wer07 Theorem 4.5].
Using a generalization of Bruhat-Tits theory one can define a subgroup ${\rm P}_x$ for all $x \in \overline{\Sigma}$ such that for $x \in \Sigma$ we retrieve the groups ${\rm P}_x$ defined in section 3.2, see [@Wer07 section 3]. Note that by continuity the action of ${\rm N}_{\rm G}({\rm T},k)$ on $\Sigma$ extends to an action on $\overline{\Sigma}$.
\[def - compact\] The compactification $\overline{\mathcal{B}}({\rm G},k)_\rho$ associated to the representation $\rho$ is defined as the quotient of the topological space ${\rm G}(k) \times \overline{\Sigma}$ by a similar equivalence relation as in Theorem \[th - gluing\]:
$(g,x) \sim (h,y)$ $\Longleftrightarrow$ there exists $n \in {\rm N}_{\rm G}({\rm T},k)$ such that $y=\nu(n).x$ and $g^{-1}hn \in {\rm P}_x$.
The compactification of $\mathcal{B}({\rm G},k)$ with respect to a representation with regular highest weight coincides with the polyhedral compactification defined by Erasmus Landvogt in [@La].
The connection to the compactifications defined with Berkovich spaces in section \[ss-maps\_to\_flags\] is given by the following result, which is proved in [@RTW2 Theorem 2.1].
\[comparison\] Let $\rho$ be a faithful, absolutely irreducible representation of ${\rm G}$ with highest weight $\lambda_0(\Delta)$. Define $${\rm Z} = \{\alpha \in \Delta : \langle \alpha, \lambda_0(\Delta) \rangle = 0\},$$ where $\langle\, , \, \rangle$ is a scalar product associated to the root system as in Definition \[def - root system\]. We denote by $\tau$ the type of the standard parabolic subgroup of ${\rm G}$ associated to ${\rm Z}$. Then there is a ${\rm G}(k)$-equivariant homeomorphism $$\overline{\mathcal{B}}({\rm G},k)_\rho \rightarrow
\overline{\mathcal{B}}_\tau({\rm G},k)$$ restricting to the identity map on the building.
In the situation of Example \[identityrep\] we have $\lambda_0(\Delta) = \varepsilon_0$ and ${\rm Z} = \{\alpha_{1,2},
\ldots, \alpha_{d-1, d}\}$. The associated standard parabolic is the stabilizer of a line. We denote its type by $\pi$. Hence the compactification of the building associated to ${\rm SL}({\mathrm{V}})$ given by the identity representation is the one associated to type $\pi$ by Theorem \[comparison\]. This compactification was studied in [@Wer01]. It is isomorphic to the seminorm compactification $\overline{\mathcal{X}}({\rm V}^\vee,k)$ of the building $\mathcal{X}({\rm V}^\vee,k)$.
Satake’s viewpoint {#ss - Satake}
------------------
If ${\rm G}$ is a non-compact real Lie group with maximal compact subgroup ${{\mathrm}K}$, Satake constructed in [@Satake2] a compactification of the Riemannian symmetric space ${\mathrm{S}}= {\mathrm{G}}/{\rm K}$ in the following way:
- [(i)]{} First consider the symmetric space ${\mathrm{H}}$ associated to the group ${\rm PSL}(n,\mathbf{C})$ which can be identified with the space of all positive definite hermitian $n \times n$-matrices with determinant $1$. Then ${\mathrm{H}}$ has a natural compactification $\overline{{\mathrm{H}}}$ defined as the set of the homothety classes of all hermitian $n \times n$-matrices.
- [(ii)]{} For an arbitrary symmetric space ${\mathrm{S}}= {\mathrm{G}}/{\rm K}$ use a faithful representation of ${\mathrm{G}}$ to embed ${\mathrm{S}}$ into ${\mathrm{H}}$ and consider the closure of ${\mathrm{S}}$ in $\overline{{\mathrm{H}}}$.
In the setting of Bruhat-Tits buildings we can imitate this strategy in two different ways.
[*Functoriality of buildings*]{} —- The first strategy is a generalization of functoriality results for buildings developed by Landvogt [@LandvogtCrelle]. Let $\rho: {\mathrm{G}}\rightarrow {\rm SL}({\mathrm{V}})$ be a representation of the semisimple group ${\mathrm{G}}$. Let ${\mathrm{S}}$ be a maximal split torus in ${\mathrm{G}}$ with normalizer ${\mathrm{N}}$, and let ${\rm A}({\mathrm{S}},k)$ denote the corresponding apartment in $\mathcal{B}({\mathrm{G}},k)$. Choose a special vertex $o$ in ${\rm
A}({\mathrm{S}},k)$. By [@LandvogtCrelle], there exists a maximal split torus ${\mathrm{T}}$ in ${\rm SL}({{\mathrm{V}}})$ containing $\rho({\mathrm{S}})$, and there exists a point $o'$ in the apartment ${\rm A}({\mathrm{T}},k)$ of ${\mathrm{T}}$ in $\mathcal{B}({\rm SL}(V),k)$ such that the following properties hold:
1. There is a unique affine map between apartments $i: {\rm
A}({\mathrm{S}},k) \rightarrow {\rm A}({\mathrm{T}},k)$ such that $i(o) = o'$. Its linear part is the map on cocharacter spaces ${\rm X}_*({\mathrm{S}})
\otimes_\mathbf{Z} \mathbf{R} \rightarrow {\rm X}_*({\mathrm{T}})
\otimes_{\mathbf{Z}} \mathbf{Z}$ induced by $\rho: {\rm S} \rightarrow {\mathrm{T}}$.
2. The map $i$ is such that $\rho({\mathrm{P}}_x) \subset {\mathrm{P}}'_{i(x)}$ for all $x \in {\rm A}({\mathrm{S}},k)$, where ${\mathrm{P}}_x$ denotes the stabilizer of the point $x$ with respect to the ${\mathrm{G}}(k)$-action on $\mathcal{B}({\mathrm{G}},k)$, and ${\mathrm{P}}'_{i(x)}$ denotes the stabilizer of the point $i(x)$ with respect to the ${\rm SL}({\mathrm{V}},k)$-action on $\mathcal{B}({\rm SL}({\mathrm{V}}),k)$.
3. The map $\rho_\ast: {\rm A}({\mathrm{S}},k) \rightarrow {\rm A}({\mathrm{T}},k)
\rightarrow \mathcal{B}({\rm SL}({\mathrm{V}}),k)$ defined by composing $i$ with the natural embedding of the apartment ${\rm A}({\mathrm{T}},k)$ in the building $\mathcal{B}({\rm SL}({{\mathrm{V}}}),k)$ is ${\mathrm{N}}(k)$-equivariant, i.e., for all $x \in {\rm A}({\mathrm{S}},k)$ and $n \in {\mathrm{N}}(k)$ we have $\rho_\ast (nx) = \rho(n) \rho_\ast(x)$.
These properties imply that $\rho_\ast: {\rm A}({\mathrm{S}},k) \rightarrow \mathcal{B}({\rm SL}({\mathrm{V}}), k)$ can be continued to a map $\rho_\ast: \mathcal{B}({\mathrm{G}},k) \rightarrow \mathcal{B}({\rm SL}({{\mathrm{V}}}),k)$, which is continuous and ${\mathrm{G}}(k)$-equivariant. By [@LandvogtCrelle 2.2.9], $\rho_\ast$ is injective.
Let $\mathcal{F}$ be the fan in ${\rm X}_*({\rm T}) \otimes_{\mathbf
Z} \mathbf{R}$ associated to the identity representation, which is described in Example \[identityrep\]. It turns out that the preimage of $\mathcal{F}$ under the map $\Sigma_{\rm vect} ({\mathrm{S}},k) \rightarrow \Sigma_{\rm
vect}({\mathrm{T}},k)$ induced by $\rho: {\mathrm{S}}\rightarrow {\mathrm{T}}$ is the fan $\mathcal{F}_\rho$, see [@RTW2 Lemma 5.1]. This implies that the map $i$ can be extended to a map of compactified apartments $\overline{\rm A}({\rm S},k) \rightarrow \overline{\rm A} ({\rm
T},k)$. An analysis of the stabilizers of boundary points shows moreover that $\rho({\rm P}_x) \subset {\rm P}'_{i(x)}$ for all $x \in \overline{\rm A}({\rm S},k)$, where ${\rm P}_x$ denotes the stabilizer of $x$ in ${\mathrm{G}}(k)$, and ${\rm P}'_{i(x)}$ denotes the stabilizer of $i(x)$ in ${\rm SL}({\mathrm{V}},k)$ [@RTW2 Lemma 5.2]. Then it follows from the definition of $\overline{\mathcal{B}}({\mathrm{G}},k)_\rho$ in \[def - compact\] that the embedding of buildings $\rho_\ast$ may be extended to a map $$\overline{\mathcal{B}}({\rm G},k)_\rho \longrightarrow \overline{\mathcal{B}}({\rm SL}({\mathrm{V}}),k)_{\rm id}.$$ It is shown in [@RTW2 Theorem 5.3] that this map is a ${\mathrm{G}}(k)$-equivariant homeomorphism of $\overline{\mathcal{B}}({\mathrm{G}},k)_\rho$ onto the closure of the image of $\mathcal{B}({\mathrm{G}},k)$ in the right hand side.
[*Complete flag variety*]{} —- Satake’s strategy of embedding the building in a fixed compactification of the building associated to ${\rm SL}({\mathrm{V}},k)$ can also be applied in the setting of Berkovich spaces. Recall from \[SL-building\] that the building $\mathcal{B}({\rm SL}({\mathrm{V}}),k)$ can be identified with the space $\mathcal{X}({\mathrm{V}},k)$ of (homothety classes of) non-Archimedean norms on ${\mathrm{V}}$. In section \[ss - seminorm compactification\], we constructed a compactification $\overline{\mathcal{X}}({\mathrm{V}},k)$ as the space of (homothety classes of) non-zero non-Archimedean seminorms on ${\mathrm{V}}$ and a retraction map $\tau: \mathbf{P}({\mathrm{V}})^{{\rm an}} \longrightarrow \overline{\mathcal{X}}({\mathrm{V}},k)$.
Now let ${\mathrm{G}}$ be a (connected) semisimple $k$-group together with an absolutely irreducible projective representation $\rho: {\mathrm{G}}\rightarrow {\rm PGL}({\mathrm{V}},k)$. Let ${{\rm Bor}}({\mathrm{G}})$ be the variety of all Borel groups of ${\mathrm{G}}$. We assume for simplicity that ${\mathrm{G}}$ is quasi-split, i.e., that there exists a Borel group $\rm B$ defined over $k$; this amounts to saying that ${\rm
Bor}({\rm G})(k)$ is non-empty. Then ${{\rm Bor}}({\mathrm{G}})$ is isomorphic to ${\mathrm{G}}/{\rm B}$. There is a natural morphism $${{\rm Bor}}({\mathrm{G}}) \longrightarrow \mathbf{P}({\mathrm{V}})$$ such that any Borel subgroup ${\rm B}$ in ${\mathrm{G}}\otimes K$ for some field extension $K$ of $k$ is mapped to the unique $K$-point in $\mathbf{P}({\mathrm{V}})$ invariant under ${\rm B} \otimes_k K$, see [@RTW2 Proposition 4.1]. Recall that in section \[sss - maps to flags\] we defined a map $$\vartheta_\varnothing: \mathcal{B}({\mathrm{G}},k) \rightarrow {\rm Bor(G)}^{{\rm an}}$$ ($\varnothing$ denotes the type of Borel subgroups). Now we consider the composition $$\mathcal{B}({\mathrm{G}},k) \stackrel{\vartheta_\varnothing}{\longrightarrow} {{\rm Bor}}({\mathrm{G}})^{{\rm an}} \rightarrow \mathbf{P}({\mathrm{V}})^{{\rm an}} \stackrel{\tau}{\longrightarrow} \overline{\mathcal{X}}({\mathrm{V}},k).$$ We can compactify the building $\mathcal{B}({\mathrm{G}},k)$ by taking the closure of the image. If $\rho^\vee$ denotes the contragredient representation of $\rho$, then it is shown in [@RTW2 4.8 and 5.3] that in this way we obtain the compactification $\overline{\mathcal{B}}({\mathrm{G}},k)_{\rho^\vee}$.
An intrinsic characterization of the building inside ${\rm G}^{\rm an}$
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In this last section, we complement [@RTW1] and [@RTW2] by establishing an intrinsic description of the building as a subspace of the analytic group ${\rm G}^{\rm an}$. The field $k$ is complete, discretely valued, with perfect residue field.
We described in 4.1 a canonical ${\rm G}(k)$-equivariant embedding $\vartheta :
\mathcal{B}({\rm G},k) \rightarrow {\rm G}^{\rm an}$, where ${\rm
G}(k)$ acts on ${\rm G}^{\rm an}$ by conjugation; in other words, this means that the building of ${\rm G}(k)$ has a natural realization as a space of multiplicative $k$-norms on the coordinate ring $\mathcal{O}({\rm G})$ of ${\rm G}$. It is very natural to ask for an intrinsic description of the image of $\vartheta$, i.e. a characterization of multiplicative norms on $\mathcal{O}({\rm G})$ which appear in Berkovich’s realization of $\mathcal{B}({\rm G},k)$. As we are going to see, one can answer this question in a very pleasant way: the image of $\vartheta$ is the set of points in ${\rm G}^{\rm an}$ satisfying a few simple conditions which we formulate below (Theorem \[Thm-image\] and Theorem \[characterization2\]).
Affinoid groups potentially of Chevalley type {#ss-Pot_Chevalley}
---------------------------------------------
Recall that we attached to any point $x$ of $\mathcal{B}({\rm G},k)$ a $k$-affinoid subgroup ${\rm G}_x$ of ${\rm
G}^{\rm an}$ satisfying the following condition: for any non-Archimedean extension $K/k$, the subgroup ${\rm G}_x(K)$ of ${\rm
G}(K)$ is the stabilizer of $x$ seen in the building $\mathcal{B}({\rm G},K)$. By definition, the point $\vartheta(x)$ is the unique element of the Shilov boundary of ${\rm
G}_x$, i.e., the only point of ${\rm G}_x$ such that $|f(y)| \leqslant |f(\vartheta(x))|$ for any $y \in {\rm G}_x$ and any $f \in \mathcal{O}({\rm G}_x)$. Conversely, one can recover ${\rm G}_x$ from $\vartheta(x)$ as its *holomorphic envelope* [@RTW1 Proposition 2.4,(ii)], which is to say: $${\rm G}_x = \{y \in {\rm G}^{\rm an} \ ; \ \forall f \in
\mathcal{O}({\rm G}), \ |f(y)| \leqslant |f(\vartheta(x))|\}.$$ This can be phrased equivalently in terms of multiplicative norms on $\mathcal{O}({\rm G})$ by saying that one recovers the affinoid algebra of ${\rm G}_x$ as the completion of the normed $k$-algebra $\left(\mathcal{O}({\rm G}), |.|(\vartheta(x) \right)$.
Let us say that a $k$-affinoid group ${\rm H}$ is of *Chevalley type* (or a *Chevalley $k$-affinoid group*) if it is the generic fibre of a $k^\circ$-formal group scheme $\mathcal{H}$ which is the formal completion of a $k^{\circ}$-Chevalley semisimple group along its special fibre. More generally, we will say that ${\rm H}$ is *potentially* of Chevalley type if there exists an affinoid extension $K/k$ such that ${\rm H}
\widehat{\otimes}_k K$ is a Chevalley affinoid group. By an *affinoid* extension, we simply mean that $K$ is a non-Archimedean field which is a $k$-affinoid algebra (see [@RTW1 Appendix A]; this restriction allows to recover $k$-affinoid algebras from $K$-affinoid algebras equipped with a descent datum). By construction, the $k$-affinoid group ${\rm G}_x$ attached to a point $x$ of $\mathcal{B}({\rm G},k)$ is always potentially of Chevalley type.
For a point $z$ of ${\rm G}^{\rm an}$, let us define its *holomorphic envelope* by $${\rm G}(z) = \{y \in
{\rm G}^{\rm an} \ ; \ \forall f \in \mathcal{O}({\rm G}),
\ |f(y)| \leqslant |f(z)|\}.$$ The above discussion brings out a first condition fulfilled by any point of ${\rm G}^{\rm an}$ belonging to the image of $\vartheta$.
<span style="font-variant:small-caps;">First condition</span> — *The holomorphic envelope of $z$ is a $k$-affinoid subgroup potentially of Chevalley type.*
It is easily checked that every point satisfying this condition does appear in the image of $\vartheta$ over some non-Archimedean extension of $k$.
\[Lemma-universal\] Let $z$ be a point of ${\rm G}^{\rm an}$ whose holomorphic envelope is a $k$-affinoid subgroup potentially of Chevalley type. Then $z$ is universal, and there exists a non-Archimedean extension $K/k$ such that the canonical lift $z_K$ of $z$ to ${\rm G}_K^{\rm an}$ belongs to the image of $\vartheta_K$.
We recall that the notion of a *universal point* was introduced in \[sss - n-A extensions\].
*Proof* — By assumption, there exists an affinoid field extension $K/k$ such that ${\rm G}(z) \widehat{\otimes}_k K$ is a $K$-affinoid subgroup of Chevalley type in ${\rm G}^{\rm an}
\widehat{\otimes}_k K$. Moreover, the Shilov boundary of ${\rm G}(z)^{\rm an} \widehat{\otimes}_k K$ is reduced to a unique universal point since this affinoid domain is the generic fibre of a formal scheme with geometrically integral special fibre. By faithfully flat descent, it follows that ${\rm G}(z)$ is a $k$-affinoid subgroup of ${\rm G}^{\rm an}$ whose Shilov boundary is reduced to the point $\{z\}$, which is universal [@RTW1 Appendix A].
Moreover, if $K/k$ is an affinoid extension as above, then the $K$-affinoid Chevalley subgroup ${\rm G}(z)
\widehat{\otimes}_k K$ is the stabilizer of a unique point $x$ of $\mathcal{B}({\rm G},K)$, hence $({\rm G}_K)_x = {\rm
G}(z) \widehat{\otimes}_k K$ and therefore $\vartheta_{K}(x)
= z_{K}$. We used the fact that any $K$-affinoid Chevalley subgroup ${\rm C}$ of $({\rm G} \otimes_k K)^{\rm an}$ occurs as the stabilizer of some point in the building. To see this, pick any special vertex $x$ in $\mathcal{B}({\rm G},K)$; its stabilizer is a $K$-affinoid subgroup of $({\rm G} \otimes_k K)^{\rm an}$ of Chevalley type, hence can be deduced from ${\rm C}$ by some $K$-automorphism $u$ of ${\rm G} \otimes_k K$. Since this automorphism $u$ acts on $\mathcal{B}({\rm G},K)$ by preserving special vertices [@TitsCorvallis 2.5], it follows that ${\rm C}$ coincides with the stabilizer of the special vertex $u^{-1}(z)$. $\square$
Galois-fixed points in buildings {#ss-Galois}
--------------------------------
The above condition does not suffice to characterize the image of $\vartheta$: it is (almost) always possible to construct a point $x \in {\rm G}^{\rm an}$ and a non-Archimedean extension $K/k$ such that :
- the canonical lifting $x_K$ of $x$ to ${\rm G}^{\rm an}_{\rm K}$ belongs to the building $\mathcal{B}({\rm G},K)$ ;
- the point $x$ does not belong to the building $\mathcal{B}({\rm G},k)$.
This is easy to realize if the residue characteristic of $k$ is positive, by considering a widely ramified Galois extension $K/k$. If the residue field is of characteristic zero, Galois theory cannot lead to such a situation, but we will see at the end of next section that it is enough to consider a transcendental non-Archimedean field extension $k_r/k$, where $r$ is a positive real number such that $r^\mathbb{Z} \cap |k^{\times}|=\{1\}$ and $$k_r = \left\{\sum_{n \in \mathbf{Z}} a_n t^n \ ; \ |a_n|r^n \rightarrow 0 \ {\rm when \ } |n| \rightarrow \infty\right\},$$ in order to construct an example (this works whatever the residue characteristic of $k$ is). In particular, even if the residue characteristic of $k$ is zero, the above condition fails to characterize the image of $\vartheta$.
Let us concentrate on the Galois-theoretic side in this section. We consider a finite Galois extension $k'/k$ and pick a point $z \in {\rm G}^{\rm an}$ such that ${\rm G}(z)$ is a $k$-affinoid subgroup potentially of Chevalley type. This point has a canonical lift $z_{k'}$ to ${\rm G}^{\rm an} \otimes_k k'$, which is fixed under the natural action of ${\rm Gal}(k'|k)$ on ${\rm G}^{\rm an} \otimes_k k'$; in particular, if $z_{k'}$ belongs to the building $\mathcal{B}({\rm G},k')$, then it lies in the subset of Galois-fixed points. Conversely, if we start with a point $x' \in {\rm G}^{\rm an} \otimes_k k'$ which belongs to $\mathcal{B}({\rm G},k')$ and is Galois-fixed, then the $k'$-affinoid subgroup $({\rm G}_{k'})_{x'}$ is equipped with a Galois descent datum. It follows that we can write ${\rm G}_{x'} =
{\rm G}(z) \otimes_k k'$ for some point $z \in {\rm G}^{\rm an}$, and that ${\rm G}(z)$ is a $k$-affinoid group potentially of Chevalley type.
By functoriality (Theorem \[th - functoriality of BT buildings\]), we identify $\mathcal{B}({\rm G},k)$ with a subset of $\mathcal{B}({\rm G},k')$ contained in the Galois-fixed locus. If the extension $k'/k$ is tamely ramified, then $\mathcal{B}({\rm G},k)$ coincides with the set of Galois-fixed points in $\mathcal{B}({\rm G},k')$ and therefore $z$ belongs to $\mathcal{B}({\rm G},k)$. If the extension $k'/k$ is widely ramified, there are in general more Galois-fixed points in $\mathcal{B}({\rm G},k')$, and any such point $x'$ leads to a point $z \in {\rm G}^{\rm an}$ whose holomorphic envelope is a $k$-affinoid subgroup potentially of Chevalley type but which does not belong to $\mathcal{B}({\rm G},k)$.
We want to illustrate this discussion by looking at an elementary example. Let us consider the group ${\rm G} = {\rm SL}_2$ over some discretely valued field $k$ and pick a finite Galois extension $k'$ of $k$. Via its canonical embedding in $\mathbf{P}^{1,{\rm an}}_{k'}$, the building $\mathcal{B}({\rm G},k')$ can be identified with the convex hull of $\mathbf{P}^1 (k')$ inside $\mathbf{P}^{1, {\rm an}}_{k'}$ with $\mathbf{P}^1 (k')$ omitted, i.e., with the subset $$\bigcup_{a \in k'} \eta_a\left(\mathbf{R}_{>0}\right),$$ where $\eta_a$ denotes the map from $\mathbf{R}_{>0}$ to $\mathbf{A}^{1,{\rm an}}_k$ sending $r$ to the maximal point of the ball of radius $r$ centered in $a$. The Galois action on $\mathcal{B}({\rm G},k')$ is induced by the Galois action on $\mathbf{P}^{1,{\rm an}}_{k'}$, and the sub-building $\mathcal{B}({\rm G},k)$ is the image of paths $\eta_a$ with $a \in k$. Since the field $k$ is discretely valued – hence spherically complete – there exists a well-defined Galois-equivariant retraction $$\tau : \mathbf{P}^{1, {\rm an}}_{k'} - \mathbf{P}^{1,{\rm an}}(k) \longrightarrow \mathcal{B}({\rm G},k)$$ defined by sending a point $x$ to the maximal point of the smallest ball with center in $k$ containing $x$.
Using this picture, one easily sees how a Galois-fixed point can appear in $\mathcal{B}({\rm G},k') -\mathcal{B}({\rm G},k)$. It suffices to find an element $\alpha$ of $k'$ such that all the paths $\eta_{\alpha^g}(\mathbf{R}_{>0})$ issued from conjugates $\alpha^g$ of $\alpha$ intersect at some point distinct from $\tau(\alpha)$; since the Galois action permutes these paths, their meeting point $x'$ will be fixed. Note that we have $$x' = \eta_{\alpha}(r) \ \ \ {\rm and} \ \ \ \tau(\alpha) = \eta_{\alpha}(r'),$$ where $r = \max \{|\alpha^g - \alpha| \ ; \ g \in {\rm Gal}(k'|k)\}$ is the diameter of the Galois orbit of $\alpha$ and $r' = \min \{|\alpha - a| \ ; \ a \in k\}$ is the distance from $\alpha$ to $k$.
Let $k'$ be any totally ramified finite Galois extension of $k$. It is well-known that $k'$ can be realized as the splitting field of some Eisenstein polynomial ${\rm P}({\rm T})={\rm T}^e + a_{e-1} {\rm T}^{e-1} + \ldots +
a_1 {\rm T} + a_0$, where $|a_i| \leqslant |a_0| < 1$ for all $i$ and $|k^{\times}| = |a_0|^{\mathbf{Z}}$. The group $|{k'}^{\times}|$ is generated by $|\alpha| =
|a_0|^{1/e}$ for any root $\alpha$ of ${\rm P}$.
We have ${\rm d}(\alpha,k) = |\alpha|$ and all conjugates of $\alpha$ are contained in the closed ball ${\rm E}(\alpha, |\alpha|) = {\rm E}(0,|\alpha|)$. The endomorphism of $\mathbf{A}^{1,{\rm an}}_{k'}$ defined by ${\rm P}({\rm
T})$ maps this ball onto the closed ball ${\rm E}(0,
|a_0|)$. In order to study the induced map ${\rm E}(0,|\alpha|) \rightarrow {\rm E}(0,|a_0|)$, set ${\rm U}
= {\rm T}/\alpha$ and write $${\rm Q}({\rm U}) = \frac{1}{a_0} {\rm
P}(\alpha{\rm U}) = \frac{\alpha^e}{a_0} {\rm U}^e + \frac{a_{e-1} \alpha^{e-1}}{a_0} {\rm U}^{e-1} +
\ldots + \frac{a_1}{a_0} \alpha {\rm U} + 1.$$ Since $|a_i|.|\alpha|^{i}< |a_i| \leqslant |a_0|$ for any $i \in \{1,
\ldots, e-1\}$, the polynomial ${\rm Q}$ reduces to $\widetilde{\frac{\alpha^e}{a_0}} {\rm U}^e+1 = 1 - {\rm U}^e$ in $\widetilde{k'}[{\rm U}]$. It follows that the following four conditions are equivalent:
- all paths $\eta_{\alpha^g}(\mathbf{R}_{>0})$, for $g \in {\rm Gal}(k'|k)$, intersect outside $\mathcal{B}({\rm G},k)$;
- all roots of ${\rm P}$ are contained in the *open* ball ${\rm D}(\alpha, |\alpha|)$;
- all roots of ${\rm Q}$ are contained in the *open* ball ${\rm D}(1,1)$;
- $e$ vanishes in $\widetilde{k}$.
In particular, for any totally ramified (finite) Galois extension $k'/k$, the building $\mathcal{B}({\rm G},k)$ is strictly smaller than the set of Galois-fixed points in $\mathcal{B}({\rm G},k')$ if and only if $[k':k]$ is divisible by the residue characteristic.
Let $k=\mathbf{Q}_2$ and $k' =
\mathbf{Q}_2(\alpha)$, where $\alpha^2=2$. The two paths $\eta_{\alpha}(\mathbf{R}_{>0})$ and $\eta_{-\alpha}(\mathbf{R}_{>0})$ intersect $\mathcal{B}({\rm G},k)$ along the image of $[2^{-1/2},\infty)$, whereas they meet along the image of $[2^{-3/2},\infty)$. The whole interval $\eta_\alpha\left([2^{-3/2},2^{-1/2})\right)$ consists of Galois-fixed points lying outside $\mathcal{B}({\rm G},k)$. In general, Rousseau gave an upper bound for the distance of a Galois-fixed point in $\mathcal{B}({\rm G},k')$ to $\mathcal{B}({\rm G},k)$ in terms of the ramification of $k'/k$ [@RousseauOrsay Prop. 5.2.7].
Apartments {#ss-Recovering apartments}
----------
The characterization of the building inside ${\rm G}^{\rm an}$ requires an additional condition involving maximal tori of ${\rm G}$. We will have to make use of the following fact.
\[tori\] Let ${\rm T}$ be a torus over $k$.
- Its analytification ${\rm T}^{\rm an}$ contains a largest bounded subgroup ${\rm T}^1$. This is an affinoid subgroup, which coincides with the affinoid domain cut out by the equations $|\chi|=1$, $\chi \in {\rm X}^*({\rm T})$, when ${\rm T}$ is split.
- The Shilov boundary of ${\rm T}^1$ is reduced to a point ${\rm o}_{\rm T}$, which is universal.
*Proof.* We consider first the case of a split torus. If $\chi_1, \ldots, \chi_n$ is a basis of characters of ${\rm T}$, then the equations $|\chi_1|=1,\ldots, |\chi_n|=1$ cut out a $k$-affinoid subgroup ${\rm T}^1$ of ${\rm T}^{\rm an}$ over which $|\chi|=1$ for any character $\chi \in {\rm X}^*({\rm T})$. Let $K/k$ be a non-Archimedean extension and $\Gamma$ a bounded subgroup of ${\rm T}(K)$. For any character $\chi$ of ${\rm T}$, both $\chi(\Gamma)$ and $(-\chi)(\Gamma) = \chi(\Gamma)^{-1}$ are bounded subgroups of $K^{\times}$, hence $|\chi(\Gamma)|=1$ and $\Gamma \subset {\rm T}^1({\rm K})$. The Shilov boundary of ${\rm T}^1$ is reduced to a universal point since the reduction of ${\rm T}^1$, a torus over $\widetilde{k}$, is geometrically irreducible.
In general, we pick a finite Galois extension $k'/k$ splitting ${\rm T}$ and set ${\rm T}_{k'} = {\rm T} \otimes_k k'$. The affinoid subgroup ${\rm T}_{k'}^1$ of ${\rm T}_{k'}^{\rm an}$ is stable under the natural Galois action on ${\rm T}_{k'}^{\rm an}$, hence its descends to a $k$-affinoid subgroup ${\rm T}^1$ of ${\rm T}$ such that ${\rm T}_{k'}^1 = ({\rm T}^1) \otimes_k k'$. Finally, since the Shilov boundary of ${\rm T}^1 \otimes_k k'$ is the preimage of the Shilov boundary of ${\rm T}^1$ under the canonical projection, we see that ${\rm T}^1$ contains an unique Shilov boundary point. Its universality follows from [@RTW1 Lemma A.10]. $\Box$
Let us now go back to our discussion toward a characterization of the building inside ${\rm G}^{\rm an}$. We assume temporarily that the group ${\rm G}$ is split. Given a point $z$ in ${\rm G}^{\rm an}$ whose holomorphic envelope ${\rm G}(z)$ is a $k$-affinoid group potentially of Chevalley type, let us consider a non-Archimedean extension $K/k$ such that the canonical lift $z_K$ of $z$ belongs to the image of $\vartheta_K$ and denote by $x$ its preimage: $z_K = \vartheta_K(x)$. Since the group ${\rm G}$ is split, the embedding $\mathcal{B}({\rm G},k) \hookrightarrow \mathcal{B}({\rm
G},K)$ identifies the left-hand side with the union of apartments of all maximal split tori in ${\rm G}_K$ which are defined over $k$. Therefore, in order to guarantee that the point $z$ itself belongs to the image of $\vartheta$, we should require that $x$ belongs to the apartment of a maximal split torus defined over $k$. The next proposition translates this additional condition in appropriate terms.
\[prop-bounded\_torus\] Let ${\rm S}$ be a maximal split torus and let $x$ be a point of $\mathcal{B}({\rm G},k)$. The following conditions are equivalent:
- $x$ belongs to the apartment ${\rm A}({\rm S},k)$;
- for every non-Archimedean extension $K/k$, the point $x$ is fixed by the action of ${\rm S}^1(K)$ on $\mathcal{B}({\rm
G},K)$;
- the affinoid subgroup ${\rm G}_x$ of ${\rm G}^{\rm an}$ contains ${\rm S}^1$.
[*Proof*]{} — Equivalence of points (ii) and (iii) follows immediately from the definition of the affinoid group ${\rm G}_x$, and it is obvious that (i) implies (ii). The converse implication is a direct consequence of Proposition \[characterization\_apartments\]. If the residue field of $k$ contains at least four elements, then $x$ belongs to ${\rm A}({\rm S},k)$ as soon as it is fixed by ${\rm S}^1(k)$. In general, we consider any non-Archimedean extension $K/k$ such that the residue field of $K$ contains at least four elements, so $x$ belongs to the apartment of ${\rm S} \otimes_k K$ in the building $\mathcal{B}({\rm G},K)$; since ${\rm S}$ is split, the embedding of $\mathcal{B}({\rm G},k)$ inside $\mathcal{B}({\rm G},K)$ identifies the apartments ${\rm A}({\rm S},k)$ and ${\rm A}({\rm S},K)$, hence $x$ belongs to ${\rm A}({\rm S},k)$. $\Box$
In the split case, the discussion above shows precisely which additional condition should be required in order to characterize the image of $\vartheta$ in ${\rm G}^{\rm an}$: *there exists a maximal split torus ${\rm S}$ in ${\rm G}$ such that ${\rm G}(z) \cap {\rm S}^{\rm an} = {\rm S}^1$*.
To deal with the general case, we recall the following definition from (3.2.2).
Let ${\rm T}$ be a maximal torus of ${\rm G}$ and let $k_{\rm T}$ the minimal Galois extension of $k$ which splits ${\rm T}$. We denote by $k_{\rm T}^{\rm ur}$ the maximal unramified extension of $k$ in $k_{\rm T}$.
The torus ${\rm T}$ is *well-adjusted* if the maximal split subtori of ${\rm T}$ and ${\rm T} \otimes_k k_{\rm T}^{\rm ur}$ are maximal split tori of ${\rm G}$ and ${\rm G} \otimes_k k_{\rm T}^{\rm ur}$.
<span style="font-variant:small-caps;">Second condition</span> — *There exists a well-adjusted maximal torus ${\rm T}$ such that ${\rm G}(z) \cap
{\rm T} = {\rm T}^1$.*
We now characterize the image of $\vartheta$ in the analytic space of ${\rm G}$.
\[Thm-image\] The image of the canonical embedding $\vartheta : \mathcal{B}({\rm
G},k) \hookrightarrow {\rm G}^{\rm an}$ is the subset of points $z$ satisfying the following two conditions:
- the holomorphic envelope ${\rm G}(z)$ of $z$ is a $k$-affinoid subgroup potentially of Chevalley type;
- there exists a well-adjusted maximal torus ${\rm T}$ of ${\rm G}$ such that ${\rm G}(z) \cap {\rm T}^{\rm an}$ is the maximal affinoid subgroup ${\rm T}^1$ of ${\rm T}^{\rm an}$.
*Proof.* We have already seen that the first condition is necessary. The same holds for the second one. Given a point $x \in \mathcal{B}
({\rm G}, k)$, Lemma \[functor-apartment\] guarantees the existence of a maximal split torus ${\rm S}$ and a well-adjusted maximal torus ${\rm T}$ containing ${\rm S}$ such that $x \in {\rm A}({\rm S}, k) \subset {\rm A}({\rm T},
k')$, where $k'$ is a finite Galois extension of $k$ which splits ${\rm T}$. It follows that ${\rm G}_x \otimes_k k'$ contains the bounded torus ${\rm T}^1 \otimes_k k'$, hence ${\rm T}^1 \subset {\rm G}_x$.
Now, let us show that the two conditions are sufficient. Let us consider a point $z \in {\rm G}^{\rm an}$ satisfying these conditions and pick a non-Archimedean extension $K / k$ and a point $x \in
\mathcal{B}({\rm G}, K)$ such that $z_K = \vartheta_K (x)$, where $z_K$ denotes the canonical lift of the universal point $z$ to ${\rm G}_K = ({\rm G} \otimes_k K)^{\rm an}$. This equality holds over any non-Archimedean extension of $K$. Since $z$ satisfies the second condition, we get a well-adjusted maximal torus ${\rm T}$ of ${\rm G}$ containing a maximal split torus ${\rm S}$ such that ${\rm G} (z) \cap {\rm T}^{\rm an} =
{\rm T}^1$. Enlarging $K$ if necessary, we assume that ${\rm T}$ splits over $K$. Since $${\rm T}_K^{\rm an} \cap \left( {\rm G}_K \right)_x =
{\rm T}_K^{\rm an} \cap {\rm G} (z)_K = \left(
{\rm T}^{\rm an} \cap {\rm G} (z) \right)_K = \left(
{\rm T}^1 \right)_K = \left( {\rm T}_K \right)^1,$$ it follows from Proposition \[prop-bounded\_torus\] that $x$ belongs to the apartment of ${\rm T}_K$. Once we know that $z_K$ belongs to the image of ${\rm A} ( {\rm T}, K)$ for some non-Archimedean extension $K / k$ splitting ${\rm T}$, this property holds for any such extension by compatibility of $\vartheta$ with field extension (see the end of 4.1.3). In particular, we can consider a finite Galois extension $k' / k$ which splits ${\rm T}$. It follows from the identity $( {\rm G}_{k'})_x = {\rm G}
(z)_{k'}$ that the point $x$ is fixed by ${\rm Gal} (k' |k)$. Since ${\rm A} ( {\rm T}, k')^{{\rm Gal}
(k' |k)}$ is the image of ${\rm A} ( {\rm S}, k)$ in $\mathcal{B} ( {\rm G}, k')$, we conclude that $x$ comes from a point $y$ of $\mathcal{B} ( {\rm G}, k)$ such that $z = \vartheta
(y)$. $\Box$
A reformulation in terms of norms {#ss-Reformulation norms}
---------------------------------
The above characterization of points of ${\rm G}^{\rm an}$ lying inside the building $\mathcal{B}({\rm G},k)$ (identified with its image by the canonical map $\vartheta$) can be conveniently rephrased in terms of (multiplicative) $k$-norms on the coordinate algebra $\mathcal{O}({\rm G})$.
This reformulation relies on a construction involving universal points, and we refer to (\[sss - n-A extensions\]) for a definition of this notion. Let ${\rm G}^{\rm an}_u$ denote the subset of universal points in ${\rm G}^{\rm an}$. Following Berkovich [@Ber1 5.2], there is a natural *monoid* structure on ${\rm G}^{\rm an}_u$ extending the group structure on ${\rm G}(k)$. Given any two points $g, h \in {\rm G}_u^{\rm an}$, the seminorm $g \otimes h$ on $\mathcal{O}({\rm G}) \otimes_k \mathcal{O}({\rm G})$ is multiplicative, and one defines $g \ast h$ as the induced multiplicative seminorm on $\mathcal{O}({\rm G})$ via the comultiplication map $\Delta : \mathcal{O}({\rm G}) \rightarrow \mathcal{O}({\rm G}) \otimes_k \mathcal{O}({\rm G})$. This binary operation is associative, with unit the element $1 \in {\rm G}(k)$; moreover, we recover the group law if $g$ and $h$ belong to ${\rm G}(k)$.
More generally, given an (analytic) action of ${\rm G}^{\rm an}$ on some $k$-analytic space ${\rm X}$, one can define in a similar way an action of the monoid ${\rm G}^{\rm an}_u$ on the topological space underlying ${\rm X}$, which extends the action of ${\rm G}(k)$.
\[universal\*universal\] Let ${\rm X}$ be a $k$-analytic space endowed with a ${\rm G}^{\rm an}$-action ${\rm G}^{\rm an} \times {\rm X} \rightarrow {\rm X}$. Let $g \in {\rm G}^{\rm an}$ and $x \in {\rm X}$. If both points $g$ and $x$ are universal, then so is $g \ast x$.
*Proof.* Let $K/k$ be any non-Archimedean field extension. We have to check that the tensor product of norms on $(\mathcal{H}(g) \otimes_k \mathcal{H}(x)) \otimes_k K$ is multiplicative. Since $x$ is universal, the tensor product norm on $\mathcal{H}(x) \otimes_k K$ is multiplicative. This implies that the underlying ring is a domain, and its quotient field $K'$ inherits an absolute value extending this norm. Since the canonical isomorphism $(\mathcal{H}(g) \otimes_k \mathcal{H}(x)) \otimes_k K \simeq \mathcal{H}(g) \otimes_k (\mathcal{H}(x) \otimes_k K)$ is an isometry, we thus get an isometric embedding $$(\mathcal{H}(g) \otimes_k (\mathcal{H}(x)) \otimes_k K \simeq \mathcal{H}(g) \otimes_k (\mathcal{H}(x) \otimes_k K) \hookrightarrow \mathcal{H}(g) \otimes_k \widehat{K'}.$$ The norm on the right-hand-side is multiplicative by universality of $g$ and the conclusion follows. $\Box$
Finally, we also recall that one can define a partial order on the set underlying ${\rm G}^{\rm an}$ as follows: $$x \preccurlyeq y \ \ \ {\rm if \ and \ only \ if} \ \ \forall a \in \mathcal{O}({\rm G}), \ |a(x)| \leqslant |a(y)|.$$
We can now give the following description of the building inside ${\rm G}^{\rm an}$.
\[characterization2\] A point $x \in {\rm G}^{\rm an}$ belongs to the image of the canonical embedding $\vartheta : \mathcal{B}({\rm G},k) \hookrightarrow {\rm G}^{\rm an}$ if and only if
- $x$ is universal;
- $x \ast x \preccurlyeq x$ and ${\rm inv}(x) \preccurlyeq x$ ;
- there exists a well-adjusted maximal torus ${\rm T}$ such that ${\rm o}_{\rm T} \preccurlyeq x$;
- $x$ is maximal with respect to the three conditions above.
\[Rk-characterization2\] 1. In particular, conditions (i)-(iv) imply that $x$ is a multiplicative $k$-*norm* on $\mathcal{O}({\rm G})$. 2. One way to understand condition (ii) is to say that $x$ defines a $k$-multiplicative norm on the commutative Hopf algebra $\mathcal{O}({\rm G})$, with respect to which commultiplication and the antipode are bounded. One should observe that (iii) obviously implies $1_{\rm G} \preccurlyeq x$, so that the counit is also bounded. 3. By Lemma \[\*-order\] below, the condition $1_{\rm G} \preccurlyeq x$ (resp. ${\rm inv}(x) \preccurlyeq x$) implies $x = 1_{\rm G} \ast x \preccurlyeq x \ast x$ (resp. $x \preccurlyeq {\rm inv}(x)$), hence condition (ii) could be replaced by $x \ast x = x$ and ${\rm inv}(x)=x$.
\[\*-order\] Let $x,y,x',y'$ be four points in ${\rm G}^{\rm an}$ such that both sets $\{x,y\}$ and $\{x',y'\}$ contain at least one universal point. If $x \preccurlyeq x'$ and $y \preccurlyeq y'$, then $x \ast y \preccurlyeq x' \ast y'$ and ${\rm inv}(x) \preccurlyeq {\rm inv}(x')$.
*Proof.* This follows directly from the formulas: $$\forall a \in \mathcal{O}({\rm G}), \ \ \ |a(x \ast y)| = \inf \max_i |b_i(x)| \cdot |c_i(y)| \ \ {\rm and } \ \ |a(x' \ast y')| = \inf \max_i |b_i(x')| \cdot |c_i(y')|,$$ where the infimum is taken over the set of all expressions $\sum_i b_i \otimes c_i$ representing $\Delta(a)$, and $$\forall a \in \mathcal{O}({\rm G}), \ \ \ |a({\rm inv}(x))| = |{\rm inv}(a)(x)| \leqslant |{\rm inv}(a)(x')| = |a({\rm inv}(x'))|.$$ $\Box$
\[\*-group\] Let $x$ be a universal point of ${\rm G}^{\rm an}$, with holomorphic envelope ${\rm G}(x) = \{z \in {\rm G}^{\rm an} \ ; \ z \preccurlyeq x\}$. The following conditions are equivalent:
- ${\rm G}(x)$ is a subgroup object of ${\rm G}^{\rm an}$, i.e., ${\rm G}(x)(K)$ is a subgroup of ${\rm G}(K)$ for any non-Archimedean extension $K/k$;
- $x$ satisfies $$1_{\rm G} \preccurlyeq x, \ \ \ {\rm inv}(x) \preccurlyeq x, \ \ {\rm and} \ \ \ x \ast x \preccurlyeq x.$$
Moreover, ${\rm G}(x)$ is bounded in ${\rm G}^{\rm an}$.
*Proof.* Assume that ${\rm G}(x)$ is a subgroup object of ${\rm G}^{\rm an}$. Since ${\rm G}(x)(k)$ is a subgroup of ${\rm G}(k)$, it contains the unit element $1_{\rm G}$ and therefore $1_{\rm G} \preccurlyeq x$. Let us now consider the canonical point $\underline{x} \in {\rm G}(\mathcal{H}(x))$ lying over $x$, i.e., the $\mathcal{H}(x)$-point defined by the canonical homomorphism $\mathcal{O}({\rm G}) \rightarrow \mathcal{H}(x)$. We have $|a(x)| = |a(\underline{x})|$ for any $a \in \mathcal{O}({\rm G})$, as well as $$|a(x \ast x)| = |a(\underline{x} \cdot \underline{x})| \ \ \ {\rm and} \ \ \ |a({\rm inv}(x))| = |a(\underline{x}^{-1})|.$$ Since $\underline{x} \cdot \underline{x}$ and $\underline{x}^{-1}$ belong to ${\rm G}(x)(\mathcal{H}(x))$, it follows that $$|a(x \ast x)| \leqslant |a(x)| \ \ \ {\rm and} \ \ \ |a({\rm inv}(x))| \leqslant |a(x)|$$ for all $a \in \mathcal{O}({\rm G})$, which exactly means that $x \ast x \preccurlyeq x$ and ${\rm inv}(x) \preccurlyeq x$.
We assume now that $x$ is a universal point of ${\rm G}^{\rm an}$ satisfying the conditions of (ii). Obviously, ${\rm G}(x)$ contains the $k$-rational point $1_{\rm G}$. Given a non-Archimedean extension $K/k$ and elements $g,h \in {\rm G}(x)(K)$, we have for all $a \in \mathcal{O}({\rm G})$: $$|a(g^{-1})| = |{\rm inv}(a)(g)| \leqslant |{\rm inv}(a)(x)| = |a({\rm inv}(x))|$$ and $$|a(gh)| = |\Delta(a)(g,h)| \leqslant \inf \max_i |b_i(g)|\cdot |c_i(h)| \leqslant \inf \max_i |b_i(x)|\cdot |c_i(x)| = |a(x \ast x)|,$$ where the infimum is taken over the set of all expressions $\sum_i b_i \otimes c_i$ representing $\Delta(a)$. Since ${\rm inv}(x) \preccurlyeq x$ and $x \ast x \preccurlyeq x$, we deduce $$|a(g^{-1})| \leqslant |a(x)| \ \ \ {\rm and} \ \ \ |a(gh)| \leqslant |a(x)|,$$ hence $g^{-1}, gh \in {\rm G}(x)({\rm K})$. This proves that ${\rm G}(x)$ is a subgroup object of ${\rm G}^{\rm an}$.
Boundedness is obvious: if $f_1, \ldots, f_n$ is a finite set generating $\mathcal{O}({\rm G})$ as a $k$-algebra, then $|f_i(y)| \leqslant \max_i |f_i(x)|$ for any point $y \in {\rm G}(x)$.$\Box$
*Proof of Theorem \[characterization2\].* We identify the building $\mathcal{B}({\rm G},k)$ with its image in ${\rm G}^{\rm an}$ by the embedding $\vartheta$.
If a point $x$ of ${\rm G}^{\rm an}$ belongs to $\mathcal{B}({\rm G},k)$, then $x$ is universal (Lemma \[Lemma-universal\]) and ${\rm G}(x)$ is a $k$-affinoid subgroup of ${\rm G}^{\rm an}$, hence $x \ast x \preccurlyeq$ and ${\rm inv}(x) \preccurlyeq x$ by Lemma \[\*-group\]. Moreover, there exists a well-adjusted maximal torus ${\rm T}$ such that ${\rm G}(x) \cap {\rm T}^{\rm an} = {\rm T}^1$, and this equality amounts to saying that $x$ dominates the distinguished point ${\rm o}_{\rm T}$ of ${\rm T}$. Finally, consider a universal point $z \in {\rm G}^{\rm an}$ satisfying condition (ii) and dominating $x$ (which implies that $z$ dominates $1_{\rm G}$). For any non-Archimedean extension $K/k$, Lemma \[\*-group\] implies that ${\rm G}(z)(K)$ is a bounded subgroup of ${\rm G}(K)$ containing ${\rm G}(x)(K)$; by maximality of the latter, we deduce ${\rm G}(z)(K) = {\rm G}(x)(K)$, hence ${\rm G}(x) = {\rm G}(z)$ and $z=x$. We have thus checked that $x$ satisfies conditions (i)-(iv).
Conversely, let $x$ be a point in ${\rm G}^{\rm an}$ satisfying conditions (i)-(iv). We observe that condition (iii) implies that $x$ dominates $1_{\rm G}$, hence it follows from Lemma \[\*-group\] that ${\rm G}(x)(K)$ is a bounded subgroup of ${\rm G}(K)$ for any non-Archimedean extension $K/k$. We are going to show that all these subgroups fix a common point in $\mathcal{B}({\rm G},k)$. By condition (iii), there exists a well-adjusted maximal torus ${\rm T}$ of ${\rm G}$ such that ${\rm T}^1(K) \subset {\rm G}(x)(K)$ for any $K$.
We first assume that ${\rm T}$ is split and that $\widetilde{k}$ contains at least four elements. For any non-Archimedean field extension $K/k$, we identify ${\rm A}({\rm T},k)$ with ${\rm A}({\rm T},K)$ in $\mathcal{B}({\rm G},K)$, and we let $\mathcal{F}(K)$ denote the fixed-point set of ${\rm G}(x)(K)$ in $\mathcal{B}({\rm G},K)$. This is a non-empty closed subset, which lies in the apartment ${\rm A}({\rm T},k)$ since ${\rm G}(x)(K)$ contains the group of units ${\rm T}^1(K)$ and there are at least four elements in $\widetilde{K}$ (Proposition \[characterization\_apartments\]). Considering in particular the extension $\mathcal{H}(x)/k$, we get a point $z \in {\rm A}({\rm T},k)$ fixed under the canonical element $\underline{x} \in {\rm G}(\mathcal{H}(x))$ over $x$. This means that $\underline{x}$ belongs to ${\rm G}(z)(\mathcal{H}(x))$, or equivalently that $x$ is contained in the $k$-affinoid subgroup ${\rm G}(z)$. This amounts to saying that $z$ dominates $x$, hence $x=z$ by maximality.
In general, we consider a finite Galois extension $k'/k$ which splits ${\rm T}$ and such that $\widetilde{k'}$ contains at least four elements. It follows from the previous argument that the canonical extension $x_{k'}$ of $x$ to ${\rm G}^{\rm an} \otimes_k k'$ belongs to the apartment of ${\rm T} \otimes_k k'$ in $\mathcal{B}({\rm G},k')$. Since $x_{k'}$ is invariant under the action of ${\rm Gal}(k'/k)$, this point belongs to the image of ${\rm A}({\rm S},k)$ in ${\rm A}({\rm T},k')$ by Remark \[Rk-universal\],2, and therefore $x$ belongs to ${\rm A}({\rm S},k)$. $\Box$
With this new characterization of the building $\mathcal{B}({\rm G},k)$ inside the analytic space ${\rm G}^{\rm an}$, we can complete the discussion at the beginning of section 5.2.
Let $r$ be a positive real number such that $r^\mathbb{Z} \cap |k^\times| = \{1\}$ and set ${\rm G}_r = {\rm G}^{\rm an} \widehat{\otimes}_k k_r$. The canonical projection $p_r$ of ${\rm G}_r$ onto ${\rm G}^{\rm an}$ restricts to an embedding of $\mathcal{B}({\rm G},k_r)$ into ${\rm G}^{\rm an}$.
*Proof*. We adapt an argument of Berkovich [@Ber2 Theorem 10.1]. Let ${\rm A}(r) = \mathcal{M}(k_r)$ denote the Gelfand spectrum of $k_r$; this is the annulus of width zero and radius $r$ in the affine line: ${\rm A}(r) = \{x \in \mathbf{A}^1 \ | \ |{\rm T}(x)| = r\}$. It comes with a simply transitive action of the affinoid torus $\mathbf{G}_{\rm m}^1 = {\rm A}(1)$ by multiplication: for any non-Archimedean field extension $K/k$, $$\mathbf{G}_{\rm m}^1(K) = \{t \in K^\times \ | \ |t|=1\}, \ \ \ {\rm A}(r)(K) = \{ \lambda \in K \ | \ |\lambda|=r\} \ \ \ {\rm and} \ \ \ t \cdot \lambda = t\lambda.$$ Since ${\rm G}_r = {\rm G}^{\rm an} \times_k {\rm A}(r)$, we get an action of $\mathbf{G}_{\rm m}^1$ on the $k$-analytic space ${\rm G}_r$, whose orbits coincide with the fibres of the canonical projection $p_r : {\rm G}_r \rightarrow {\rm G}^{\rm an}$.
If we consider the partial order $\preccurlyeq$ introduced above on ${\rm G}_r$, then each $\mathbf{G}_{\rm m}^1$-orbit ${\rm O} = p_r^{-1}(x)$ has a unique maximal point $\sigma(x)$. Indeed, the orbit of a point $z \in {\rm G}_r$ is the image of the natural map $$\mathcal{H}(z) \widehat{\otimes}_k \mathbf{G}_{\rm m}^1 \rightarrow {\rm G}_r \times_k \mathbf{G}_{\rm m}^1 \rightarrow {\rm G}_r,$$ which is compatible with partial orders on the source and target spaces; since $\mathbf{G}_{\rm m}^1$ has a unique maximal point ${\rm o}_{\mathbf{G}_{\rm m}}$, the conclusion follows. This maximal point can also be described as before in terms of “multiplication” by the universal point ${\rm o}_{\mathbf{G}_{\rm m}}$, namely $$\sigma(x) = z \ast {\rm o}_{\mathbf{G}_{\rm m}}$$ for any $z \in p^{-1}(x)$. The map $\sigma$ thus defined is a continuous section of the projection $p_r$.
In order to prove the Proposition, it is now enough to establish that the building $\mathcal{B}({\rm G},k_r)$ is contained in the image of $\sigma$. So let us consider a point $z \in \mathcal{B}({\rm G},k_r)$ and set $x = p_r(z)$. Since $\sigma(x)$ is the unique maximal point of the fibre $p_r^{-1}(x)$, one has $z \preccurlyeq \sigma(x)$. If we manage to check that $\sigma(x)$ satisfies conditions (i)-(iii) of Theorem \[characterization2\], then we are done by maximality of $z$.
\(i) Since both points $z \in {\rm G}_r$ and ${\rm o}_{\mathbf{G}_{\rm m}} \in \mathbf{G}_{\rm m}^1$ are universal (Theorem \[characterization2\] and Lemma \[tori\]), so is the point $\sigma(x) = z \ast {\rm o}_{\mathbf{G}_{\rm m}}$ by Lemma \[universal\*universal\].
\(ii) The diagonal action of $\mathbf{G}_{\rm m}^1$ on ${\rm G}_r \times_k {\rm G}_r$ induces an action of $\mathbf{G}_{\rm m}^1$ on the $k$-analytic space ${\rm G}_r \times_{k_r} {\rm G}_r$ with respect to which the multiplication map ${\rm G}_r \times_{k_r} {\rm G}_r \rightarrow {\rm G}_r$ is equivariant. This implies that $$\sigma(x) \ast \sigma(x) = (z \ast {\rm o}_{\mathbf{G}_{\rm m}}) \ast (z \ast {\rm o}_{\mathbf{G}_{\rm m}}) = (z \ast z) \ast {\rm o}_{\mathbf{G}_{\rm m}}.$$ From $z \ast z = z$ (Remark \[Rk-characterization2\], 3), we deduce that $\sigma(x) \ast \sigma(x) = \sigma(x)$. Similarly, the antipode ${\rm G}_r \rightarrow {\rm G}_r$ commutes with the action of $\mathbf{G}_{\rm m}^1$ and therefore $${\rm inv}(\sigma(x)) = {\rm inv}(z \ast {\rm o}_{\mathbf{G}_{\rm m}^1}) = {\rm inv}(z) \ast {\rm o}_{\mathbf{G}_{\rm m}^1}.$$ Since ${\rm inv}(z) = z$ (Remark \[Rk-characterization2\], 3), we deduce that ${\rm inv}(\sigma(x)) = \sigma(x)$.
\(iii) Finally, condition (iii) is obvious: we have $z \preccurlyeq \sigma(x)$ and ${\rm o}_{\rm T} \preccurlyeq z$ for some well-adjusted maximal torus ${\rm T}$, hence ${\rm o}_{\rm T} \preccurlyeq \sigma(x)$. $\Box$
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*Bertrand Rémy and Amaury Thuillier*\
Université de Lyon\
CNRS ; Université Lyon 1\
Institut Camille Jordan\
43 boulevard du 11 novembre 1918\
F-69622 Villeurbanne cedex\
$\{$remy; thuillier$\}$@math.univ-lyon1.fr
*Annette Werner*\
Institut für Mathematik\
Goethe-Universität Frankfurt\
Robert-Mayer-Str. 6-8\
D-60325 Frankfurt a. M.\
werner@math.uni-frankfurt.de
[^1]: Though the notion is taken from [@BT1a], the terminology we use here is not the exact translation of the French “donnée radicielle” as used in \[loc. cit.\]: this is because we have already used the terminology “root datum” in the combinatorial sense of [@SGA3]. Accordingly, we use the notation of [@SGA3] instead of that of [@BT1a], e.g. a root system is denoted by the letter $R$ instead of $\Phi$.
[^2]: This notion was introduced by Berkovich, who used the adjective *peaked* [@Ber1 5.2]. Its study was carried on by Poineau, who prefered the adjective *universal* [@Poineau].
|
---
abstract: 'A prevalent market structure in the Internet economy consists of buyers and sellers connected by a platform (such as Amazon or eBay) that acts as an intermediary and keeps a share of the revenue of each transaction. While the optimal mechanism that maximizes the intermediary’s profit in such a setting may be quite complicated, the mechanisms observed in reality are generally much simpler, e.g., applying an affine function to the price of the transaction as the intermediary’s fee. @LN07 [@LN13] initiated the study of such fee-setting mechanisms in two-sided markets, and we continue this investigation by addressing the question of when an affine fee schedule is approximately optimal for worst-case seller distribution. On one hand our work supplies non-trivial sufficient conditions on the buyer side (i.e. linearity of marginal revenue function, or MHR property of value and value minus cost distributions) under which an affine fee schedule can obtain a constant fraction of the intermediary’s optimal profit for all seller distributions. On the other hand we complement our result by showing that proper affine fee-setting mechanisms (e.g. those used in eBay and Amazon selling plans) are *unable* to extract a constant fraction of optimal profit in the worst-case seller distribution. As subsidiary results we also show there exists a constant gap between maximum surplus and maximum revenue under the aforementioned conditions. Most of the mechanisms that we propose are also prior-independent with respect to the seller, which signifies the practical implications of our result.'
author:
- 'Rad Niazadeh$^{\dag}$, Yang Yuan$^{\dag}$'
- 'Robert Kleinberg$^{\dag}$ [^1]'
bibliography:
- 'ebay.bib'
title: 'Simple and Near-Optimal Mechanisms For Market Intermediation'
---
Introduction {#sec:intro}
============
A prevalent market structure in the Internet economy consists of buyers and sellers connected by a platform (such as Amazon or eBay) that acts as an intermediary and keeps a share of the revenue each time a buyer makes a purchase from a seller. What mechanism should the intermediary use to maximize its profit? In cases the optimal mechanism is unacceptably complicated, can simpler mechanisms closely approximate the profit of the optimal mechanism? We approach these questions using the framework of Bayesian mechanism design and worst-case approximation guarantees.
To motivate our investigation it is instructive to consider the transaction fees that are commonly used by intermediaries in reality. For example, when an item is sold on eBay using a fixed price listing (as opposed to an auction), the seller is charged a fee of $0.3+0.1P$, where $P$ is the total amount of the sale in dollars[^2]. Amazon uses a similar pricing rule for individual sellers, which is $0.99+\alpha P$, where $\alpha$ is a real number determined by the category of the product, typically ranging from $8\%$ to $15\%$ [^3]. Generalizing these examples, we say that a *fee-setting mechanism* is one in which the intermediary names a function $w(\cdot)$, the seller names a price $P$, and the buyer chooses whether or not to take the item at price $P$. If the transaction takes place, then the intermediary keeps $w(P)$ and pays $P-w(P)$ to the seller. Otherwise, no money changes hands. We refer to $w$ as the *fee schedule* of the mechanism. We say that $w$ is *affine* if it can be represented in the form $w = (1-\alpha) P+\beta$ for some constants $\alpha,\beta$, and we say that an affine schedule $w(P)=(1-\alpha) P+\beta$ is *proper* if $\alpha\in [0,1], \beta\geq 0$. Note that the fee schedule used by eBay and Amazon (and many other intermediaries, for example real estate brokers) are affine and proper.
@LN07 [@LN13] initiated the study of fee-setting mechanisms in two-sided markets. They showed that if it is possible for the intermediary to choose a mechanism that implements a given allocation rule in Bayes-Nash equilibrium, then there is a fee-setting mechanism that does so. They also provided necessary and sufficient conditions for the intermediary’s optimal mechanism to be implemented by an affine fee-setting mechanism. The necessary and sufficient condition discovered by @LN07 [@LN13] requires the seller’s cost to be drawn from a generalized Pareto distribution (see Definition \[gpd\] below). Using results from extreme value theory, they show that in the limit as only the sellers with lowest cost and the buyers with highest value enter the market, the conditional distribution of the seller’s cost (conditional on entering the market) approaches a generalized Pareto distribution, thus providing a partial justification for the prevalence of affine fee-setting mechanisms in two-sided markets.
Our work draws inspiration from the aforementioned work of @LN07 [@LN13] and seeks a different type of justification for affine fee-setting mechanisms by asking the question, “When are affine fee-setting mechanisms approximately optimal?” Our results pertain to the case when the buyer’s virtual valuation function is affine, which is the characterization of generalized Pareto distributions, in ex-post IR setting. We first show that a specific choice of seller prior-independent affine fee schedule $w(P)=P-\phi_{\mathcal{B}}(P)$ is ex-post IR for every possible seller’s distribution, where prior-independent means the fee schedule only depends on the buyer’s value distribution but not on the seller’s cost distribution. Moreover, this affine fee schedule also achieves a constant-approximation to the maximum surplus — and hence, also, a constant-approximation to the optimal revenue. The approximation factor depends on the exponent of the buyer’s generalized power distribution but it is no more than $4$ comparing to optimal intermediary’s profit when the buyer’s PDF is monotone. Our results complement the results of @LN07 [@LN13] in the sense that combined with their results, we show that if either of the buyer side or the seller side has affine virtual valuation function, and the other side follows regular distributions, then the best affine fee schedule guarantees either optimal or near optimal revenue, which provides explanation for the phenomenon that affine fee schedule is widely used in the daily life.
Our second main result explores the setting that the difference between the values of the seller and the buyer follows MHR distribution, which indicates that the surplus and revenue are constant approximation to each other. Under this assumption, we may further extend the buyer’s distribution to MHR distributions, and still get constant approximation ratio with constant (and hence affine) fee schedule.
Intriguingly, without proper MHR assumptions the ex-post IR affine fee schedule in the aforementioned approximation result is *not* proper; in contrast to intermediaries in typical two-sided markets in practice, the intermediary in our approximation result may charge a transaction fee which is a decreasing function of the seller’s price. Our third main result shows that this reliance on improper affine fee schedules is unavoidable: even when the buyer’s value is assumed to be uniformly distributed on $[0,1]$, there exist seller cost distributions for which no proper affine fee-setting mechanism can achieve a constant-approximation to the optimal revenue.
In the special case that the buyer’s distribution is uniform $[0,1]$, we propose an improved mechanism, which gives $3$-approximation fee-setting mechanism to the optimal revenue. We also prove that if one needs a prior independent affine fee schedule when the buyer’s distribution is uniform $[0,1]$, then $\alpha-\beta=1$ is necessary. Moreover, among all the prior independent affine fee schedule, $w(P)=1-P$ gets the best approximation ratio $8$ comparing to maximum surplus. From this perspective, our proposed affine fee schedule is optimal. Finally, our proof techniques reveal the fact that there exists a constant gap between optimal revenue and maximum surplus when buyer’s distribution is generalized Pareto distribution as a side dish.
The primary source of difficulty in proving these results is that fee-setting mechanisms are not Bayes-Nash incentive compatible (BNIC). Thus, deriving a revenue guarantee for the intermediary requires first solving for the Bayes-Nash equilibrium of the mechanism. Our paper adopts the approach introduced by @LN07 [@LN13] for deriving the Bayes-Nash equilibrium. The technical heart of our paper lies in some surprising connections between the affine fee schedule, Bayes-Nash equilibrium payment function, and the cumulative hazard rate function. These connections are non-trivial, which make the proof succinct while the results are still general. Starting from that, we got expressions of the three quantities of interest — the maximum surplus, the optimal revenue, and the affine fee-setting mechanism’s revenue — in a closely related form. Then, leveraging our assumption that the buyer’s virtual value function is affine, we are able to choose an affine fee schedule to approximate the optimal revenue.
Related Work
------------
@MS83 showed that for one seller one buyer setting, if there is no intermediary between them, then no incentive-compatible individually rational mechanism can produce post efficient outcome, where post efficient outcome means the trade should take place whenever the buyer’s value is larger than the seller’s cost. Based on this impossibility result, they also considered the case that intermediary is allowed, and both the seller and the buyer can trade with the intermediary only.
@DGTZ12 studied the double auction, in which the intermediary designs mechanism for the buyers and the sellers to extract maximum revenue. In the paper, they provided optimal or near-optimal mechanisms for both single dimensional and multi-dimensional environments with continuous or discrete distributions. @JW12 studied the same problem with single unit-demand buyer and multiple sellers, and gave a characterization for the optimal solution in this setting. Since the optimal mechanism is generally hard to implement, they also proposed several approximation mechanisms, including picking the best item and sell, or using anonymous virtual reserve price combined with greedy algorithm.
Contract problem has a similar setting as the intermediary problem: the principle (intermediary) proposed a contract ($w(\cdot)$ function) to the agent (the seller), and the agent will choose his action and get a output ($P$ payment), and then give the principle $w(P)$, keep $P-c$ as its utility. Previously, researchers have found evidence showing that linear contract is powerful in this setting. @PBS07 studied linear contract problem, and found that linear contracts are common in practice not only because the simplicity, but also due to the fact that the optimal linear contract guarantees at least $90\%$ of the fully optimal contract in the canonical moral hazard setting. @GC13 proved that under mild assumptions, the optimal contract is actually linear.
Simple mechanisms and their approximation ratios to the corresponding optimal mechanisms have been an important research topic in the literature. For example, @BK94 showed that in the i.i.d., regular, single dimensional setting, second price auction with $n+1$ bidder will give more revenue than the optimal auction with $n$ bidders. @HR10 investigated the single dimensional setting where bidders have independent valuations, and showed that VCG with anonymous reserve price can achieve $4$-approximation to the optimal revenue. @DRY10 considered the auctions that are prior-independent, in the sense that the auction will achieve good approximation to the optimal revenue while the specific value distributions of the bidders are not used in the auction.
Preliminaries
=============
In this paper we consider the problem of single-item trade, in which a profit-maximizing broker mediates the exchange between a buyer and a seller. In particular, we follow the Bayesian mechanism design approach wherein a Bayesian designer looks to find the trade mechanism with the maximum possible revenue in expectation over the distributions from which the preferences of the buyer and seller are drawn. We assume the preferences of buyer and seller are private values drawn from product distributions, which are common knowledge.
Setting, notations, solution concepts, and basics
-------------------------------------------------
We assume the reader is familiar with the general model of single dimensional mechanism design for risk neutral agents, including the definitions of incentive compatibility and individual rationality, basics of Bayesian mechanism design, and adapting these concepts to the exchange setting (see Appendix \[mechbasic\]). Still, it is worth identifying a few aspects of our notations and terminology.
Suppose the seller $\mathcal{S}$ has a private cost $c$ and the buyer $\mathcal{B}$ has a private value $v$ for the item. We use $F$ (and $f$) to denote the CDF(and PDF) of $v$, and $G$ (and $g$) to denote the CDF (and PDF) of $c$. Unless stated otherwise, we assume the support of $f$ is $[0,\overline{v}]$ and the support of $g$ is $[0,\overline{c}]$. We define the marginal revenue functions (a.k.a. *virtual preferences*) of seller and buyer as follows. Let $\phi_{\mathcal{S}}(c)\triangleq c+\frac{G(c)}{g(c)}$ be defined as the virtual cost of the seller and $\phi_{\mathcal{B}}(v)\triangleq v-\frac{1-F(v)}{f(v)}$ be defined as the virtual value of the buyer. We also define buyer’s *hazard rate*, $h_B(v)\triangleq\frac{f(v)}{1-F(v)}$, and *cumulative hazard rate*, $H_B(v)\triangleq\int_{0}^{v}h_B(z)dz$. It can be easily shown that $1-F(v)=e^{-H_B(v)}$, which is a famous property of cumulative hazard rate.
We say a buyer (or a seller) is *buyer-regular* (or *seller-regular*) if $\phi_{\mathcal{B}}(v)$ (or $\phi_{\mathcal{S}}(c))$ is monotone non-decreasing. A buyer’s distribution is said to be *MHR (monotone hazard rate)* if $h_\mathcal{B}(v)$ is monotone non-decreasing (or equivalently $H_\mathcal{B}(v)$ is convex). For a regular buyer $v$, *monopoly price* is defined to be $\eta_v=\phi_{\mathcal{B}}^{-1}(0)$ (i.e. if $v\geq\eta_v$ virtual value is non-negative). Moreover, *monopoly revenue* $R_{\eta}^{v}\triangleq\eta_v(1-F(\eta_v))$ is the expected revenue one gets by posting $\eta_v$ to a buyer with value $v$.
Characterization of distributions with affine virtual value/cost {#paretosec}
----------------------------------------------------------------
A critical constraint throughout this paper, which is appearing in different forms in many of our results and background results on this subject, is when the buyer or seller has an affine virtual preference, i.e. when $\phi_\mathcal{S}(c)=xc+y$ or when $\phi_\mathcal{B}(v)=xv-y$ for $x,y\in\mathbb{R}$. We now characterize the buyer distributions and seller distributions with the above property as follows.
\[gpd\] A generalized Pareto distribution $F$ with parameters $\mu,\lambda,$ and $\xi$, where $\mu,\lambda,\xi\in\mathbb{R}$, $\lambda > 0$ and $\xi \geq 0$, is defined by the following cumulative density function. $$F(x)= \left\{ \begin{array}{rl}
1-(1-\xi \lambda (x-\mu))^{\frac{1}{\xi}} &\mbox{ if $\xi>0$} \\
1-e^{\lambda(x-\mu)} &\mbox{~if $\xi=0$}
\end{array} \right.\nonumber$$ and the support is bounded and equal to $[\mu,\mu+\frac{1}{\xi\lambda}]$ if $\xi >0$, and is unbounded and equal to $[\mu,+\infty)$ if $\xi=0$. When $\xi>0$ we refer to the distribution as *generalized power distribution* and when $\xi=0$ we refer to it as *generalized exponential distribution*.
It is worth mentioning that the family of Pareto distributions are skewed, heavy-tailed distributions that are sometimes used to model the distributions of incomes and other financial variables. For the cost of the seller, we define a similar distribution as follows.
\[reverspareto\] The seller with cost $c$ has a reverse-generalized Pareto distribution with parameters $\mu, \lambda,$ and $\xi$ if $-c$ is a random variable drawn from a generalized Pareto distribution with parameters $\mu,\lambda$ and $\xi$.
For generalized Pareto distribution family, one can easily prove the following corollary by definition.
\[cor:gpd\] If $v$ is drawn from a generalized Pareto distribution with parameters $\mu, \lambda,$ and $\xi$, then $\phi_B(v)=(1+\xi)v-(\frac{1}{\lambda}-\xi\mu)$. If $c$ is drawn from a reverse-generalized Pareto distribution with parameters $\mu, \lambda,$ and $\xi$, then $\phi_S(c)=(1+\xi)c+(\frac{1}{\lambda}+\xi\mu)$.
We can also prove that the inverse is true, i.e. affine virtual preferences implies the generalized Pareto distribution (To prove this lemma, simply solve the corresponding differential equations coming from the definitions, which we omit here)
\[lemma:gpd\] A buyer (or seller) has affine virtual value (or cost) only if its value (or cost) is drawn from a generalized Pareto distribution (or reverse-generalized Pareto distribution).
Background results {#backresult}
==================
In this section, we investigate a class of mechanisms known as *fee-setting*, introduced first by @LN07. In these mechanisms, the intermediary asks the seller to bid her preferred price. If a buyer is willing to buy the item with this price, the intermediary takes a share of the trade money and gives the rest to the seller. Fee-setting mechanisms are simple, intuitive, easy to implement and more robust compared with Myerson’s optimal mechanism.
Fee-setting exchange mechanisms {#indopt}
-------------------------------
We first define a *fee-setting mechanism* as follows.
A *a fee-setting mechanism* with common knowledge *fee schedule* $w(.):\mathbb{R}\rightarrow \mathbb{R}$ is an indirect mechanism for single buyer single seller exchange that runs the following steps subsequently:
- Trader asks the seller to bid its desired price $P$,
- Trader then posts the price $P$ for the buyer,
- If $v <P$ then the trade doesn’t happen and all the payments will be zero.
- If $v \geq P$ then the item will be traded, trader charges the buyer $P$, keeps its share of the trade $w(P)$, and pays $P-w(P)$ to the seller.
We now define *affine fee setting exchange mechanisms* formally below.
An *affine fee setting exchange mechanism* with parameters $\alpha$ and $\beta$ is a fee-setting mechanism with affine fee schedule $w(P)\triangleq(1-\alpha)P+\beta$, $\alpha, \beta \in \mathbb{R}$.
In this paper, we refer to an affine exchange fee mechanism with parameters $\alpha$ and $\beta$ as $\textrm{APX}(\alpha,\beta)$. We also define $\textrm{Rev-APX}(\alpha,\beta)$ to be the revenue of $\textrm{APX}(\alpha,\beta)$ when strategy profile of agents is a BNE (As we will discuss below, for affine exchange mechanisms there is a unique BNE under the regularity assumption). Moreover, $\textrm{OPT-Rev}$ is defined to be the revenue of optimal Myerson mechanism, and $\textrm{OPT-Surplus}$ to be the surplus of VCG mechanism.
Characterization of BNE strategy of the seller
-----------------------------------------------
By a standard argument similar to those used in the Bayes-Nash equilibrium characterization of single dimension mechanism [@RB78] one can characterize the BNE of the fee-setting mechanism. More formally, we have the following theorem, proved in [@LN07], that characterizes the BNE of the fee-setting mechanisms.
\[bnechar\] [@LN07] Consider a fee-setting mechanism with differentiable fee-setting $w(.)$, then $P:{[0,\overline{c}]}\rightarrow\mathbb{R}^{+}$ is a BNE strategy of the seller if and only if:
- $P(c)$ is monotone non-decreasing with respect to $c$.
- $P(c)$ satisfies $\phi_\mathcal{B}(P(c))=P(c)-\frac{P(c)-w(P(c))-c}{1-\frac{\partial w}{\partial p}(P(c))}$.
Although the characterization in Theorem \[bnechar\] is indirect, it has many nice implications in the special case of fee-settings mechanisms with affine fee schedule.
\[affine:bnechar\] Suppose in an exchange setting seller is regular. Then for an affine fee-setting mechanism with fee schedule $w(P)=(1-\alpha)P+\beta$, $P(c)=\phi_\mathcal{B}^{-1}(\frac{c+\beta}{\alpha})$ is the unique BNE strategy of seller.
From Theorem \[bnechar\] we know in any BNE, we have $$\phi_\mathcal{B}(P(c))=P(c)-\frac{P(c)-w(P(c))-c}{1-\frac{\partial w}{\partial p}(P(c))}=P(c)-\frac{\alpha P(c)+\beta+c}{1-(1-\alpha)}=\frac{c+\beta}{\alpha}\nonumber$$ and as buyer is regular, $\phi_\mathcal{B}$ is invertible, so in any BNE $P(c)=\phi_\mathcal{B}^{-1}(\frac{c+\beta}{\alpha})$.
Optimality of affine and non-affine fee-setting mechanisms
----------------------------------------------------------
Considering the class of fee-setting mechanisms, one important question is how well these mechanisms can perform comparing to Myerson’s optimal mechanism. @LN07 [@LN13] showed that with a proper choice of function $w(P)$ (not necessarily affine) one can design a fee-setting mechanism that extracts the same revenue in expectation as in Myerson’s optimal mechanism. While this result is surprising by itself, they also could show that optimal fee-setting mechanism will be affine when seller’s cost is drawn form a reverse-generalized Pareto distribution as in Definition \[reverspareto\] (in other words, when the seller’s virtual cost is affine). For more details on this result and a simple proof using revenue equivalence theorem [@RB78], see Appendix \[appsec3\].
Main results {#mainresult}
============
As can be seen from the discussion in the last section, @LN07 initiated the study of affine fee-setting mechanisms in two-sided markets and identified necessary and sufficient conditions for the intermediary’s optimal fee schedule to be affine for worst-case buyer distribution. In this section, we continue this investigation by addressing the question of when an affine fee schedule is optimal or approximately optimal for worst-case seller distribution. By simulation, one can show that there exists a pair of seller and buyer distributions for which the best affine mechanism is not optimal (for example see [@LN07]). However, in those cases, we may still be able to get constant approximations to maximum intermediary profit with affine fee-settings. We have three main results following this line of thought.
As our first result, intuitively when at least one side of the bilateral market has some linear behaviors it might be possible for the mechanism designer to extract optimal or approximately optimal revenue from the buyer and seller using affine fee-settings. Under this condition, we propose improper fee-setting mechanisms that can extract constant approximations to optimal revenue. More formally:
> **Main Result 1** *If the buyer has affine virtual value, under some mild assumptions, the affine fee-setting mechanism $w(P)=P-\phi_\mathcal{B}(P)$ extracts a constant approximation of optimal intermediary’s revenue in expectation for any seller-regular distributions. Moreover, optimal intermediary’s revenue and maximum surplus are in constant approximation of each other in expectation.*\
As the second result, when surplus and revenue are in constant approximation of each other (for example when the distributions involved in the trade are not heavy-tailed) posting a proper price for the buyer can always extract constant approximations to optimal surplus, and hence optimal revenue, and seller’s cost will not be an important issue. More formally:
> **Main Result 2** *If the random variables $v$ (buyer’s value) and $v-c$ (difference of buyer’s value and seller’s cost) are MHR, the constant fee-setting mechanism $w(P)=\eta_{v-c}$ extracts constant approximations to optimal intermediary’s revenue in expectation for any seller distributions, in which $\eta_{v-c}$ is the monopoly price for the random variable $v-c$ ($\eta_{v-c}=\phi_{v-c}^{-1}(0)$). Moreover, optimal intermediary’s revenue and maximum surplus are in constant approximation of each other in expectation.*
As the final result, we show that a mechanism designer who tries to get constant approximation to optimal revenue for all seller’s distribution (especially for heavy-tailed distributions), cannot avoid using the improper fee-setting mechanisms. Formally:
> **Main Result 3** *Even when the buyer’s value is drawn from $\textrm{unif}~[0,1]$, there exists seller cost distributions for which no proper affine fee-setting mechanism can achieve a constant-approximation to the optimal intermediary’s revenue.*
In the next Section, we first provide a proof sketch for our first main result, and then for the special case when buyer’s value is uniform we propose an improved fee-setting mechanism accompanied by a refined analysis, which gives us a better approximation ratio. Then in Section \[sec:MHR\] we sketch the proof of second main result. Finally in Section \[sec:inapprox\] we elaborate on our third inapproximability result.
Approximations for affine buyer’s virtual value
-----------------------------------------------
Suppose buyer’s virtual value is affine, i.e. $\phi_{\mathcal{B}}(v)=\alpha v -\beta$,[^4] and now look at the affine fee-setting mechanism $w(P)=P-\phi_\mathcal{B}=(1-\alpha)P+\beta$. We start by proving the following properties of this mechanism, which also show the mechanism is ex-post IR for seller, buyer and trader (and hence no party regrets attending the trade).
\[affine:property\]If $\phi_{\mathcal{B}}(v)=\alpha v -\beta$ and $P(c)$ is the BNE strategy of seller, then affine fee-setting mechanism $w(P)=P-\phi_\mathcal{B}(P)=(1-\alpha)P+\beta$ has the following properties:
(a) $\forall c:$ $w(\phi_\mathcal{B}(P))=\alpha w(P)$ and $\phi_\mathcal{B}(\phi_\mathcal{B}(P))=c$.
(b) Ex-post utilities of seller and trader are always non-negative.
(c) $\forall v: e^{-H_\mathcal{B}(v)}=(\frac{w(v)}{\beta})^{\frac{1}{\alpha-1}}$, when $\alpha\neq 1$.
To prove (a) we have $w(\phi_\mathcal{B}(P))=(1-\alpha)(\alpha P-\beta)+\beta=\alpha((1-\alpha)P+\beta)=\alpha w(P)$. Moreover, due to Corollary \[affine:bnechar\], $\phi_\mathcal{B}(P)=\frac{c+\beta}{\alpha}$ and hence $c=\alpha \phi_\mathcal{B}(P) -\beta=\phi_\mathcal{B}(\phi_\mathcal{B}(P))$. To prove (b), note that utility of trader is equal to $w(P)=P-\phi_\mathcal{B}(P)\geq 0$, due to properties of virtual value. Also, seller’s ex-post utility when trade happens is equal to $P-w(P)-c=\phi_\mathcal{B}(P)-c\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))-c=0$, due to property (a). To prove (c) we have $h_\mathcal{B}(v)=(v-\phi_\mathcal{B}(v))^{-1}=(w(v))^{-1}$. Now, the following calculation finds cumulative hazard rate $H_\mathcal{B}(v)$ which completes the proof of (c). $$H_\mathcal{B}(v)=\int_0^{v}h_\mathcal{B}(z)dz=\frac{\ln(w(v))}{1-\alpha}-\frac{\ln(w(0))}{1-\alpha}=\frac{\ln(\frac{w(v)}{\beta})}{1-\alpha}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\qed$$
Now, using the above properties we prove one can extract a constant portion of optimal revenue and optimal surplus by the above mechanism. The intuition behind the proof is as follows. Look at the special case when the buyer’s distribution is uniform on $[0,1]$. Then the fee schedule that we propose is $w(P)=1-P$. At the first glance this appears counterintuitive: as a seller, if you ask for a higher price then the broker gets less money from you. But the seller needs to take a trade-off when setting the price: if the seller picks $P=1$, which minimizes the broker’s fee as $w(1)=0$, then the chance of finding a buyer with this price will be zero, which produces zero utility to the seller. So the seller needs to find a balanced price, at which the chance of finding a buyer is large, and the fee paid to the intermediary is reasonable as well. In other words, the seller is buying “chance of trade” from the intermediary by paying $1-P$ to it. We formalize this argument by the following theorem. (Figure \[fig1\] presents a geometric proof sketch.)
=\[thick,font=,color=black!70!blue\]
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\[theorem:main1\] Suppose buyer’s virtual value is affine: $\phi_\mathcal{B}(v)=\alpha v- \beta$ for some $\alpha\geq 1$. Then the revenue of affine fee-setting mechanism $w(P)=P-\phi_\mathcal{B}(P)$ is $\alpha^{\frac{1}{\alpha-1}}-$approximation to optimal revenue and $\alpha^{\frac{\alpha+1}{\alpha-1}}-$approximation to optimal surplus in expectation.
Suppose $P(c)$ is seller’s BNE strategy. We first find an equivalent expression for the expectation of revenue of the fee-setting mechanism $w(P)=P-\phi_\mathcal{B}(P)=(1-\alpha)P+\beta$ (i.e. $\textrm{APX}(\alpha,\beta)$) conditioned on a fixed cost $c$ for the seller. For a fixed $c$, trade happens if buyer’s value is at least equal to the posted price, i.e. if $v\geq P(c)$. In this case, trader gets its share $w(P)$ and returns the rest to the seller. So, $$\mathbb{E}\{\textrm{Rev-APX}|c\}=\mathbb{E}\{w(P)\mathds{1}\{v\geq P\}|c\}=w(P)(1-F(P))=w(P)e^{-H_\mathcal{B}(P)}\overset{(i)}{=}\frac{w(P)^{\frac{\alpha}{\alpha-1}}}{\beta^{\frac{1}{\alpha-1}}},\nonumber$$ where equality (i) is due to property (c) in Lemma \[affine:property\]. Basically, the conditional revenue of $\textrm{APX}$ is the measured area under a rectangle with width equal to $w(P)$ and length equal to the interval $\{v: v\geq P\}$, when we use the distribution of buyer’s value as the measure function. For the special case of uniform distribution, this corresponds to normal area of this rectangle, as can be seen in Figure \[fig1\]. For the optimal revenue, we try to obtain a similar upper-bound. From Myerson’s theory of optimal mechanisms, we know $\mathbb{E}\{\textrm{OPT-Rev}|c\}=\mathbb{E}\{(\phi_\mathcal{B}(v)-\phi_{\mathcal{S}}(c))\mathds{1}\{\phi_\mathcal{B}(v)\geq \phi_{\mathcal{S}}(c)\}|c\}$. By plugging in buyer’s affine virtual function $\phi_\mathcal{B}(v)=\alpha v-\beta$ in this expression, we have: $$\begin{aligned}
&\mathbb{E}\{\textrm{OPT-Rev}|c\}=\mathbb{E}\left\{(\alpha v-\beta-\phi_{\mathcal{S}}(c))\mathds{1}\{\alpha v-\beta\geq \phi_{\mathcal{S}}(c)\}|c\right\}\nonumber\\
&=\alpha \mathbb{E}\left\{(v-\frac{\phi_{\mathcal{S}}(c)+\beta}{\alpha})\mathds{1}\{v\geq \frac{\phi_{\mathcal{S}}(c)+\beta}{\alpha}\}|c\right\}\overset{(i)}{=}\alpha\mathbb{E}\left\{\frac{1}{h_\mathcal{B}(v)}\mathds{1}\{v\geq \frac{\phi_{\mathcal{S}}(c)+\beta}{\alpha}\}|c\right\}\nonumber\\
&\overset{(ii)}{\leq} \alpha\mathbb{E}\left\{\frac{1}{h_\mathcal{B}(v)}\mathds{1}\{v\geq \phi_\mathcal{B}(P)\}|c\right\}\overset{(iii)}{=}\alpha\mathbb{E}\left\{w(v)\mathds{1}\{v\geq \phi_\mathcal{B}(P)\}|c\right\}\nonumber,\end{aligned}$$ where equality (i) is true because for any $x$, $\mathbb{E}\{(v-x)\mathds{1}\{v\geq x\}\}=\int_{t\geq x}(1-F(t)) dt=\mathbb{E}\{\frac{1}{h(v)}\mathds{1}\{v\geq x\}\}$, (ii) is true because $\frac{\phi_{\mathcal{S}}(c)+\beta}{\alpha}\geq \frac{c+\beta}{\alpha}=P(c)$, and (iii) is true because $w(v)=v-\phi_\mathcal{B}(v)=\frac{1}{h_\mathcal{B}(v)}$. The last upper-bound on the conditional revenue of optimal Myerson divided by $\alpha$ is the measured area under the curve $w(v)$ in the interval $\{v: v\geq \phi_\mathcal{B}(P)\}$, when we use the distribution of buyer’s value as the measure function. For the special case of uniform distribution, this corresponds to normal area under the curve, as can be seen in Figure \[fig1\]. One can calculate this term by taking the integral of an affine function and show this is equal to $\alpha^{\frac{1}{\alpha-1}}\mathbb{E}\{\textrm{Rev-APX}|c\}$, which gives us the desired approximation factor (see Appendix \[detail:proof\]). Here we take a different approach which results in a slightly weaker approximation factor, but is more intuitive. $w(v)$ is non-increasing (as $\alpha\geq 1$) and hence $w(v)\leq w(\phi_\mathcal{B}(P))$ in the region $\{v:v\geq \phi_\mathcal{B}(p)\}$. So, we can upper-bound the conditional expectation of optimal revenue further by $$\begin{aligned}
&\mathbb{E}\{\textrm{OPT-Rev}|c\}\leq \alpha w(\phi_\mathcal{B}(P))(1-F(\phi_\mathcal{B}(P)))=\alpha w(\phi_\mathcal{B}(P))e^{-H_\mathcal{B}(w(\phi_\mathcal{B}(P)))}\nonumber\\
&\overset{(i)}{=}\alpha\frac{w(\phi_\mathcal{B}(P))^{\frac{\alpha}{\alpha-1}}}{\beta^{\frac{1}{\alpha-1}}}\overset{(ii)}{=}\alpha^{\frac{\alpha}{\alpha-1}}\mathbb{E}\{\textrm{Rev-APX}|c\},\end{aligned}$$ in which (i) is true due to property (c) in Lemma \[affine:property\], and (ii) is true because $w(\phi_\mathcal{B}(P))=\alpha w(P)$ based on property (a) in Lemma \[affine:property\]. Taking expectation with respect to $c$ will prove the desired approximation factor with respect to optimal revenue.
To compare the revenue of our fee-setting mechanism with the surplus, we use the same machinery to find an expression for the expectation of maximum surplus for a fixed $c$. Similar to the calculations for optimal revenue we have $\mathbb{E}\{\textrm{OPT-Surplus}|c\}=\mathbb{E}\{(v-c)\mathds{1}\{v\geq c\}|c\}=\mathbb{E}\{\frac{1}{h_\mathcal{B}(v)}\mathds{1}\{v\geq c\}|c\}=\mathbb{E}\{w(v)\mathds{1}\{v\geq c\}|c\}=\mathbb{E}\{w(v)\mathds{1}\{v\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))\}|c\}$, where the last equality is true because $\phi_\mathcal{B}(\phi_\mathcal{B}(P))=c$ due to property (a) in Lemma \[affine:property\]. Again, the conditional maximum surplus is the measured area under the curve $w(v)$ in the interval $\{v: v\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))\}$, when we use the distribution of buyer’s value as the measure function. For the special case of uniform distribution, again this corresponds to normal area under the curve, as can be seen in Figure \[fig1\]. Now, again one can either calculate this term by taking integral of an affine function and show this is equal to $\alpha^{\frac{\alpha+1}{\alpha-1}}\mathbb{E}\{\textrm{Rev-APX}|c\}$ (which gives us the desired approximation factor, see Appendix \[detail:proof\] for the proof), or can use the following upper-bound for a slightly weaker factor (but more intuitive). $$\begin{aligned}
&\mathbb{E}\{\textrm{OPT-Surplus}|c\}\overset{(i)}{\leq}w(\phi_\mathcal{B}(\phi_\mathcal{B}(P)))(1-F(\phi_\mathcal{B}(\phi_\mathcal{B}(P))))= w(\phi_\mathcal{B}(\phi_\mathcal{B}(P)))e^{-H_\mathcal{B}(w(\phi_\mathcal{B}(\phi_\mathcal{B}(P))))}\nonumber\\
&=\frac{w(\phi_\mathcal{B}(\phi_\mathcal{B}(P)))^{\frac{\alpha}{\alpha-1}}}{\beta^{\frac{1}{\alpha-1}}}\overset{(ii)}{=}\alpha^{\frac{2\alpha}{\alpha-1}}\mathbb{E}\{\textrm{Rev-APX}|c\},\end{aligned}$$ where (i) is true because $w(v)$ is non-increasing, and (ii) is true because by using property (a) of Lemma \[affine:property\] twice we have $w(\phi_\mathcal{B}(\phi_\mathcal{B}(P)))=\alpha w(\phi_\mathcal{B}(P))=\alpha^2w(P)$. Taking expectation with respect to $c$ will complete the proof.
We are now ready to obtain approximation ratios for different cases of generalized Pareto distributions, namely general power distributions and exponential distributions. This is exactly the same class of distributions that @LN07 [@LN13] investigated.
Suppose $F(v)=1-e^{-\lambda v}$ over $[0,\infty)$ for $\lambda>0$. Then revenue of $\textrm{APX}(1,\frac{1}{\lambda})$ (i.e. fee-setting with $w(P)=\frac{1}{\lambda}$) is $e^2$-approximation to maximum surplus, and $e$-approximation to the optimal revenue in expectation.
This is the special case of Theorem \[theorem:main1\] when $\phi_\mathcal{B}(v)=v-\frac{1-F(v)}{f(v)}
=v-\frac{1}{\lambda}$. That gives us $\alpha=1, \beta=\frac{1}{\lambda}$. Following the fact that $\lim_{\alpha\rightarrow 1} \alpha^{\frac{1}{\alpha-1}}=e$ and $\lim_{\alpha\rightarrow 1} \alpha^{\frac{\alpha+1}{\alpha-1}}=e^2$, we prove the desired approximation factors.
\[coro:powerdistribution\] Suppose $F(v)=1-(1-\frac{v}{\bar{v}})^a$ over the support $[0,\bar{v}]$ for some $a\geq 1$. Then the revenue of $\textrm{APX}(\frac{a+1}{a},\frac{\bar{v}}{a})$ (i.e. fee-setting with $w(P)=\frac{-1}{a}P+\frac{\bar{v}}{a}$) is $8-$approximation to the maximum surplus, and $4-$approximation to the maximum revenue.
This is the special case of Theorem \[theorem:main1\] when $\phi_\mathcal{B}(v)=v-\frac{1-F(v)}{f(v)}=\frac{a+1}{a}v-\frac{\bar{v}}{a}$. So $\alpha=\frac{a+1}{a}$, $\beta=\frac{\bar v}{a}$. Note that as $a\geq 1$, we have $\alpha\leq 2$. Following the fact that for $\alpha\leq 2$, we have $\alpha^{\frac{1}{\alpha-1}}\leq 4$ and $\alpha^{\frac{\alpha+1}{\alpha-1}}\leq 8$, we prove the desired approximation factors.
Approximations for the uniform distribution
-------------------------------------------
Uniform distribution on $[0,1]$ is a special case of power distributions, so based on the results of the last section we can get approximation factors $4$ and $8$ with respect to optimal revenue and surplus respectively. However, we propose a different fee-setting mechanism that is $3-$approximation with respect to optimal revenue in expectation. Our technique is based on the “best of two" technique for designing approximation algorithms, which picks the best of two mechanisms each performs well on some class of input seller’s distribution. For the proof, see Appendix \[proof:unif\].
\[theorem:unif\]Suppose $F=\textrm{unif}~[0,1]$. Let $y\triangleq\min \{\phi_\mathcal{S}^{-1}(1),\overline{c}\}$[^5]. Then the mechanism which is best of $\textrm{APX}(2,1)$ and $\textrm{APX}(1,\frac{1-y}{2})$ in terms of revenue is $3$-approximation to optimal revenue in expectation.
The best affine fee-setting mechanism is at least a $3$-approximation to optimal revenue expectation when $F=\textrm{unif}~[0,1]$.
The best affine fee-setting mechanism has expected revenue at least as large as both $\textrm{APX}(2,1)$ and $\textrm{APX}(1,\frac{1-y}{2})$, and hence is a $3-$approximation to the maximum intermediary’s revenue.
Approximations for MHR distributions {#sec:MHR}
------------------------------------
In this section, we investigate the question of approximating surplus and revenue when neither buyer’s virtual value nor seller’s virtual cost is affine, but instead we have some proper distributional assumptions on the buyer and seller distributions. We look at the setting that the difference between the values of the seller and the buyer follows MHR distribution, which indicates that the surplus and revenue of an imaginary bidder with value $v-c$ are in constant approximation to each other. Moreover, we assume $v$ is coming from a MHR distribution and hence surplus approximation and revenue approximation are equivalent for this bidder. It is important to mention that many distributions in real economic exchange settings satisfy the following properties under independence assumption of seller and buyer (like uniform, normal, exponential, and etc.). Now, under these assumptions we get constant approximation ratio to both surplus and revenue in expectation with a constant fee schedule. Formally we have the following theorem.
\[theorem:MHR\] Suppose buyer’s value $v$ is MHR, and random variable $v-c$ is also MHR. Then a constant fee-schedule mechanism $w(P)=\eta_{v-c}$ is $e^2-$approximation to optimal surplus, and hence $e^2-$approximation to optimal revenue in expectation, where $\eta_{v-c}$ is monopoly price of random variable $v-c$.
Let $P(c)$ be the BNE strategy of seller. We know random variable $v-c$ is MHR, hence due to Lemma 4.18 in [@hartline2012approximation], monopoly revenue of $v-c$ is an $e-$approximation to maximum surplus of $v-c$ in expectation. In other words, $$R_\eta^{v-c}=\eta_{v-c}\textrm{Pr}\{v-c\geq \eta_{v-c}\}\geq \frac{1}{e}\mathbb{E}\{(v-c)_{+}\}=\frac{1}{e}\mathbb{E}\{\textrm{OPT-Surplus}\}.$$ Moreover, the expected revenue of fee-setting mechanism $w(P)$ is equal to $\textrm{APX-Rev}=\eta_{v-c}\textrm{Pr}\{v\geq P(c)\}$. We claim $\mathbb{E}\{\textrm{APX-Rev}\}\geq \frac{1}{e} R_{\eta_{v-c}}$, which implies the desired approximation bounds. In other words, $$\mathbb{E}\{\textrm{APX-Rev}\}\geq \frac{1}{e} R_{\eta_{v-c}}\geq \frac{1}{e^2} \mathbb{E}\{\textrm{OPT-Surplus}\}\geq \frac{1}{e^2}\mathbb{E}\{\textrm{OPT-Rev}\}.$$ To prove the claim, it is enough to show $\textrm{Pr}\{v\geq P(c)\}\geq\frac{1}{e} \textrm{Pr}\{v-c\geq \eta_{v-c}\}$. Note that from Corollary \[affine:bnechar\] we know $\phi_\mathcal{B}(P)=c+\eta_{v-c}$. Hence, conditioned on a fixed $c$ we have $$\begin{aligned}
\label{eq:3}
\textrm{Pr}\{v-c\geq \eta_{v-c}|c\}=\textrm{Pr}\{v\geq c+\eta_{v-c}|c\}=1-F(\phi_\mathcal{B}(P))=e^{-H_\mathcal{B}(\phi_\mathcal{B}(P))}\end{aligned}$$ Now, note that $H_{\mathcal{B}}(x)$ is convex (as $v$ is MHR), so $\forall x: H_{\mathcal{B}}(x)\geq H_{\mathcal{B}}(P)+h_{\mathcal{B}}(P)(x-P)$. Let $x=\phi_\mathcal{B}(P)$, and hence $$\label{eq:4}
H_\mathcal{B}(\phi_\mathcal{B}(P))\geq H_{\mathcal{B}}(P)+h_{\mathcal{B}}(P)(\phi_\mathcal{B}(P)-P)=H_{\mathcal{B}}(P)-1,$$ where the last equality is true because $\phi_\mathcal{B}(P)=P-\frac{1-F(P)}{f(P)}=P-\frac{1}{h_{\mathcal{B}}(P)}$. Combining (\[eq:3\]) and (\[eq:4\]) we have $$\label{eq:5}
\textrm{Pr}\{v-c\geq \eta_{v-c}|c\}\leq e e^{-H_{\mathcal{B}}(P)}=e (1-F(P))=e \textrm{Pr}\{v\geq P|c\}.$$ By taking expectation from both sides of (\[eq:5\]) with respect to $c$ we prove what we claimed, which completes the proof of theorem.
Inapproximability results {#sec:inapprox}
=========================
In this section, we give two inapproximability results. The first one shows that the proper fee schedules eBay and Amazon are currently using are not revenue-efficient, in the sense that for $\textrm{unif}[0,1]$ buyer distribution no *proper* fee schedule can get constant approximation to the optimal revenue for the worst case seller distribution. Meanwhile, as we showed before, there is an improper fee-setting mechanism that always gets $4$-approximation to the optimal revenue. The second result shows that for $\textrm{unif}[0,1]$ buyer distribution, $\textrm{APX}(\alpha,\beta)$ gives seller prior independent constant approximation to the maximum surplus for worst-case seller distribution *if and only if* $\alpha-\beta=1$ and $\alpha\neq 1$.
Inapproximability result for proper fee schedule
------------------------------------------------
First we investigate the question of how good proper fee schedule works. We define a proper fee schedule as the following.
A proper fee schedule is an affine fee schedule with parameters $\alpha$ and $\beta$ such that $0\leq \alpha \leq 1$ and $\beta\geq 0$.
Then we give definitions on the approximability of proper fee schedule.
Proper fee schedule revenue gap $RG_{F,G}$ under buyer distribution $F$, and seller distribution $G$ is the ratio of the optimal revenue to the approximation revenue using the best proper fee schedule.
Proper fee schedule surplus gap $SG_{F,G}$ under buyer distribution $F$, and seller distribution $G$ is the ratio of the maximum surplus to the approximation revenue using the best proper fee schedule.
As a direct consequence of Corollary \[coro:powerdistribution\], we can say optimal revenue is $8-$approximation to optimal surplus in expectation. Hence, for the special case of $\textrm{unif[0,1]}$ we have,
\[corollary:RGSG\] If $F$ is uniform distribution on $[0,1]$, then for any seller distribution $G$, $RG_{F,G}\geq \frac{1}{8}SG_{F,G}$.
We now show the following theorem (proved in Appendix \[inapprox:appendix\]), which shows that $RG_{F,G}$ could be arbitrarily large even if the buyer distribution is as simple as the uniform $[0,1]$ distribution. At the same time, $\textrm{APX}(2,1)$ is $4$-approximation to the optimal revenue, which means proper fee schedule can be arbitrarily worse than $\textrm{APX}(2,1)$.
\[inapprox:theorem\] When $F$ is uniform distribution on $[0,1]$, for every constant $d$, there exists a regular seller distribution $G$ with $RG_{F,G}\geq d$.
[ [ *Proof Sketch.* ]{}]{}Based on Corollary \[corollary:RGSG\], it suffices to show that for every constant $d$, there exists a regular seller distribution $G$ with $SG_{F,G}\geq d$. Assume $F$ is uniform distribution on $[0,1]$. Consider the following family of distributions with parameter $\delta$ (as in Figure\[fig2\]), defined on the interval $\left [0,1-\sqrt{\delta}\right ]$, $$\label{instance:bad}
g_\delta(x)=\frac{2\delta}{(1-\delta)(1-x)^3},
~~~G_\delta(x)=\frac{\delta}{1-\delta}\left (\frac{1}{(1-x)^2}-1\right ),
~~x\in \left [0,1-\sqrt{\delta}\right ].$$ the rest of the proof shows that for any $d>0$, $\exists \delta$ such that $RG_{F,G}\geq d$
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Inapproximability result for prior-independent approximation
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For the setting of seller prior-independent, one might still expect the existence of other constant approximations. However, we show our mechanism is the unique fee-setting mechanism that can get constant seller prior-independent approximations to surplus. More formally, we show that in the seller prior-independent setting when buyer’s value is drawn from uniform $[0,1]$ distribution, $w(P)=(1-\alpha)P +\beta$ gives constant approximation to the surplus if and only if $\alpha-\beta=1$. The proof is provided in Appendix \[inapprox:appendix\].
\[theorem:priorindinapprox\] If the buyer’s distribution is uniform $[0,1]$, $w(x)=(1-\alpha)x+\beta$, where $\alpha$ and $\beta$ are parameters independent form the seller distribution, then the revenue obtained using $w$ is a constant approximation to the surplus for every possible seller’s distribution if and only if $\alpha-\beta=1$ and $\alpha\neq 1$. Moreover, when $\alpha=2, \beta=1$, it achieves the best approximation ratio $8$.
Extension to multi-buyers case
==============================
Some of our results can extend to multi-buyers case, when the buyers are regular and i.i.d. In fact, if there are $n$ buyers with regular i.i.d. values $v_1,v_2,\ldots,v_n$ drawn from distribution $F$, one can replace the pool of buyers with one *effective* buyer $v=\underset{i}{\max}~v_i$ and still get the same revenue in expectation for any fee-setting mechanisms and optimal Myerson mechanism (because all buyers have the same non-decreasing virtual value function), and also the same surplus in expectation for VCG mechanism. Now, using the following lemma and the above reduction we can extend Theorem \[theorem:MHR\] to multi-buyers case, whose proof is found in Appendix \[lemmaproofs:appendix\].
\[iidMHR\] Suppose $v_1,v_2,\ldots,v_n$ are i.i.d. random variables drawn from MHR distribution $F$. Then $v=\underset{i}{\max}~v_i$ is also MHR.
Now, by combining Lemma \[iidMHR\] and Theorem \[theorem:MHR\] we have the following direct corollary.
Suppose $F$ is a MHR distribution and there are $n$ i.i.d. buyers whose values are drawn from $F$. Seller’s cost $c$ is drawn from $G$ and is independent from all buyers. Moreover, assume the random variable $\underset{i}{\max}~v_i-c$ is MHR. Then the revenue of constant fee-setting mechanism $w(P)=\eta_{\underset{i}{\max}~v_i -c}$, where $\eta_{\underset{i}{\max}~v_i-c}$ is the monopoly price for the distribution of $\underset{i}{\max}~v_i -c$, is $e^2-$approximation to optimal surplus and revenue in expectation.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to express their very great appreciations to Prof. Jason Hartline for his valuable and constructive suggestions during this research work, especially in developing Theorem \[theorem:MHR\]. His willingness to give his time so generously has been very much appreciated.
Basics of mechanism design for exchange
=======================================
\[mechbasic\] In this section, we provide details of solution concepts and definitions used in this paper. The provided details are following the normal trend of mechanism design literature, but they have been adapted for the exchange setting.
Similar to the definition of allocations and payments in single dimensional mechanism design framework, suppose $x_\mathcal{S}\in\{0,1\}$ and $p_\mathcal{S}\in[0,\infty)$ are seller’s allocation and payment, and $x_\mathcal{B}\in\{0,1\}$ and $p_\mathcal{B}\in[0,\infty)$ are buyer’s allocation and payment. In the context of exchange, feasible allocations are $(x_\mathcal{B},x_\mathcal{S})=\{(1,1),(0,0)\}$. We assume both seller and buyer are risk-neutral, i.e. $u_\mathcal{S}=p_\mathcal{S}-x_s c$ and $u_\mathcal{B}=x_B v-p_\mathcal{B}$ are seller’s and buyer’s utilities under allocations $\mathbf{x}=(x_\mathcal{B},x_\mathcal{S})$ and payments $\mathbf{p}=(p_\mathcal{B},p_\mathcal{S})$. We start by defining an *exchange mechanism* when we have one seller and one buyer as follows.
An *exchange mechanism* for 1-seller, 1-buyer is a tuple $\mathscr{M}=(A_{\mathcal{S}}\times A_{\mathcal{B}},\mathbf{x}(.),\mathbf{p}(.) )$, where $\mathbf{x}=(x_\mathcal{S}(.) ,x_\mathcal{B}(.))$, and $\mathbf{p}= (p_\mathcal{S}(.) ,p_\mathcal{B}(.))$. $A_\mathcal{S}$ and $A_\mathcal{B}$ are set of mechanism actions of seller and buyer respectively, $x_\mathcal{S}:A_{\mathcal{B}}\times A_{\mathcal{S}}\rightarrow \{0,1\}$ and $x_\mathcal{B}:A_{\mathcal{B}}\times A_{\mathcal{S}}\rightarrow \{0,1\}$ are *seller’s allocation* and *buyer’s allocation* respectively, and $p_\mathcal{S}:A_{\mathcal{S}}\rightarrow [0,\infty)$ and $p_\mathcal{B}:A_{\mathcal{B}}\rightarrow [0,\infty) $ are *seller’s payment* and *buyer’s payment* respectively. Moreover, mechanism $\mathscr{M}$ and strategy profile $\left(b_\mathcal{B}(.),b_{\mathcal{S}}(.)\right)$ where $b_\mathcal{B}:[0,\bar{v})\rightarrow \mathcal{A}_{\mathcal{B}}, b_\mathcal{S}:[0,\bar{c})\rightarrow \mathcal{A}_{\mathcal{S}}$ implement *allocation rules* $x_\mathcal{S}(c,v)\triangleq x_\mathcal{S}(b_{\mathcal{S}}(c),b_\mathcal{B}(v))$, $x_\mathcal{B}(c,v)\triangleq x_\mathcal{B}(b_{\mathcal{S}}(c),b_\mathcal{B}(v))$, and *payment rules* $p_\mathcal{S}(v,c)\triangleq p_\mathcal{S}(b_{\mathcal{S}}(c),b_\mathcal{B}(v))$, $p_\mathcal{B}(v,c)\triangleq p_\mathcal{B}(b_{\mathcal{S}}(c),b_\mathcal{B}(v))$.
Suppose we have an exchange mechanism $\mathscr{M}$ and a strategy profile $(b_\mathcal{S},b_\mathcal{B})$ that implement allocation/payment rules $(x_\mathcal{S}(v,c),x_{\mathcal{B}}(v,c))$ and $(p_\mathcal{S}(v,c),p_{\mathcal{B}}(v,c))$ respectively. Now, *interim allocation rule* (or *interim payment rule*) of an agent (either seller or buyer) is defined as the expectation of the allocation rule (or payment rule) of that agent conditioned on her private information (cost if seller, value if buyer). We denote interim allocation/payment rules by $(x_\mathcal{S}(c),x_\mathcal{B}(v))$ and $(p_\mathcal{S}(c),p_\mathcal{B}(v))$ by a bit of notation abuse (as allocation/payment rules use the same notation as interim allocation/payment rules but with different function inputs).
Bayesian mechanism design in general aims to define the rules of a game of incomplete information, a.k.a. the mechanism, played by the agents in the environment. Mechanism designer hopes that a solution of this game has desirable properties, in particular good objective functions such as revenue of the mechanism or surplus of the agents. To analyze the solution of the game, we need to look at the correct solution concept applicable to our application. To do so, we first formalize the game which is played by seller and buyer, and then we talk about solution concepts we use in this paper. As it can be seen from the above definition, a *strategy* for an agent (buyer or seller) is a mapping from its type space (i.e. value space of the buyer or cost space of the seller) to its corresponding mechanism’s action space ($A_\mathcal{S}$ for seller or $A_\mathcal{B}$ for buyer). We now define a *direct revelation exchange mechanism*.
A *direct revelation exchange mechanism* is a single-round, sealed bid exchange mechanism which has action spaces equal to the corresponding type spaces (i.e., the seller bids its cost for the item under trade and the buyer bids its value for that item).
We now can define a Bayes-Nash Equilibrium strategy profile of an exchange mechanism as follows.
A *Bayes-Nash Equilibrium* for an exchange mechanism $\mathscr{M}=(A_{\mathcal{S}}\times A_{\mathcal{B}},\mathbf{x}(.),\mathbf{p}(.) )$ under common prior $F\times G$ is a strategy profile $\left(b_\mathcal{B}(.),b_{\mathcal{S}}(.)\right)$ where $b_\mathcal{B}:[0,\bar{v})\rightarrow \mathcal{A}_{\mathcal{B}}, b_\mathcal{S}:[0,\bar{c})\rightarrow \mathcal{A}_{\mathcal{S}}$, and $$\begin{aligned}
&-\forall v\in [0,\bar{v}), \forall ~b'_{\mathcal{B}}: \\&~~\mathbb{E}_{c}\{u_{\mathcal{B}}[x_\mathcal{B}(b_{\mathcal{B}}(v),b_{\mathcal{S}}(c)) ,p_\mathcal{B}(b_{\mathcal{B}}(v),b_{\mathcal{S}}(c))]\}\geq \mathbb{E}_{c}\{u_{\mathcal{B}}[x_\mathcal{B}(b'_{\mathcal{B}}(v),b_{\mathcal{S}}(c)) ,p_\mathcal{B}(b'_{\mathcal{B}}(v),b_{\mathcal{S}}(c))]\}\end{aligned}$$ $$\begin{aligned}
&-\forall c\in [0,\bar{c}), \forall ~b'_{\mathcal{S}}: \\&~~\mathbb{E}_{v}\{u_{\mathcal{S}}[x_\mathcal{S}(b_{\mathcal{B}}(v),b_{\mathcal{S}}(c)) ,p_\mathcal{S}(b_{\mathcal{B}}(v),b_{\mathcal{S}}(c))]\}\geq \mathbb{E}_{v}\{u_{\mathcal{S}}[x_\mathcal{S}(b_{\mathcal{B}}(v),b'_{\mathcal{S}}(c)) ,p_\mathcal{S}(b_{\mathcal{B}}(v),b'_{\mathcal{S}}(c))]\}\end{aligned}$$
Similar to the definition of BNE, we adapt the solution concepts of *Bayesian Incentive Compatibility (BIC)*, *Dominant Strategy Incentive Compatibility (DSIC)*, *Interim Individual Rationality (Interim IR)*, and *Ex-post Individual Rationality (Ex-post IR)* to the setting of exchange as follows.
A direct revelation exchange mechanism $\mathscr{M}$ is *BIC* if truthful bidding (i.e. seller bids her cost, buyer bids her value) is a BNE.
An exchange mechanism $\mathscr{M}$ is *Interim IR* if neither buyer nor seller get a negative revenue in expectation at the interim stage of the game, i.e. when they know their private types.
An exchange mechanism $\mathscr{M}$ is *Ex-post* IR if neither buyer nor seller get a negative revenue at the ex-post stage of the game, i.e. when all the private types are revealed to all the players.
Below We look at the intermediary problem as a single dimensional mechanism design framework, and characterize the optimal Myerson’s mechanism for single item exchange problem. We implement this optimal revenue scheme using a more intuitive indirect mechanism in Section \[indopt\]. This second indirect mechanism will be the base-line of all of our proposed simple mechanisms in this paper.
Suppose both seller and buyer are regular. Then the following direct BIC and Interim IR mechanism[^6], which is maximizing *virtual surplus* ($\triangleq x_\mathcal{B}\phi_\mathcal{B}- x_\mathcal{S}\phi_\mathcal{S}$), is revenue optimal in expectation.
- Solicit seller’s and buyer’s bids for their cost and value respectively. Let these bids be ($b_\mathcal{S},b_{\mathcal{B}})$.
- If $\phi_\mathcal{B}(b_\mathcal{B})\geq \phi_\mathcal{S}(b_\mathcal{S})$ the trade happens, o.w. no item will be transferred.
- If trade happens, then charge the buyer its critical price, i.e. $\tau_{B}=\phi_\mathcal{B}^{-1}(\phi_\mathcal{S}(b_\mathcal{S}))$, and give the seller its critical price, i.e. $\tau_{S}=\phi_\mathcal{S}^{-1}(\phi_\mathcal{B}(b_\mathcal{B}))$. Otherwise, nobody will be charged.
Other than the objective of revenue, another important benchmark in mechanism design is *surplus*. Vickrey-Clarke-Groves (VCG) mechanism maximizes the surplus and satisfies the strongest incentive compatibility and individual rationality solution concepts. We adapt VCG mechanism to the setting of exchange as follows.
The following DSIC and ex-post IR mechanism is maximizing the surplus:
- Solicit seller’s and buyer’s bids for their cost and value respectively. Let these bids be ($b_\mathcal{S},b_{\mathcal{B}})$.
- If $b_\mathcal{B}\geq b_\mathcal{S}$ the trade happens, o.w. no item will be transferred.
- If trade happens, then pay the seller the amount of $b_\mathcal{B}$ and charge the buyer $b_\mathcal{S}$.
Finally, we need to define the notion of Prior-Independence with respect to seller market or buyer market for exchange mechanisms.
An exchange mechanism $M$ is known to be seller prior-independent with respect to the seller if no seller distributional information is needed by the mechanism.
Implantation of Myerson’s optimal by fee-setting {#appsec3}
================================================
One important question related to designing fee-setting mechanisms is whether they can be optimal or not. The following theorem, proved first in [@LN07], provides an answer to this question. It states with a proper choice of function $w(P)$ one can design a fee-setting mechanism that extracts the same revenue in expectation as in Myerson’s optimal mechanism. We provide a simple proof of this result using Revenue Equivalence theorem [@RB78].
[@LN07] Consider an exchange setting with regular buyer/seller. Define $\mathscr{P}(c)\triangleq\phi_{\mathcal{B}}^{-1}(\phi_{\mathcal{S}}(c))$. Consider a fee-setting exchange mechanism with fee schedule $w(P)=P-\mathbb{E}_v\{\mathscr{P}^{-1}(v)|v\geq P\}$. Then:
- $P(c)=\mathscr{P}(c)$ and $b(v)=v$ is a BNE of this mechanism.
- The interim allocation/payment rules are equal to those of Myerson’s optimal mechanism.
Suppose seller plays $p(c)=\mathscr{P}(c)=\phi_{\mathcal{B}}^{-1}(\phi_{\mathcal{S}}(c))$ and buyer plays $b(v)=v$. Now, let $(x_{\mathcal{S}}(v,c), x_{\mathcal{B}}(v,c)) $ be the interim allocation rule and $(p_{\mathcal{S}}(v,c), p_{\mathcal{B}}(v,c)) $ the interim payment rule of this mechanism under this strategy profile. Moreover, let $(x^M_{\mathcal{S}}(v,c), x^M_{\mathcal{B}}(v,c)) $ and $(p_{\mathcal{S}}(v,c), p_{\mathcal{B}}(v,c)) $ be the interim allocation rule and interim payment rule of Myerson’s optimal mechanism respectively (Note that we know Myerson’s is DSIC). In this mechanism, trade happens when $v \geq \phi_{\mathcal{B}}^{-1}(\phi_{\mathcal{S}}(c))$ which is equivalent to $\phi_{\mathcal{B}}(v)\geq\phi_{\mathcal{S}}(c)$. So both of our mechanism and Myerson’s optimal mechanism have the same allocation rule for both buyer and seller, i.e. $x_\mathcal{B}=x^M_\mathcal{B}=x_\mathcal{S}=x^M_\mathcal{S}=\mathds{1}\{\phi_{\mathcal{B}}(v)\geq\phi_{\mathcal{S}}(c)\}$. As in the Myerson’s mechanism the critical price of the buyer is $\tau_\mathcal{B}=\phi_{\mathcal{B}}^{-1}(\phi_{\mathcal{S}}(c))$ and we charge the buyer by $\mathscr{P}(c)$ if trade happens, the payment rule of buyers are the same in both mechanism. For the interim payment rule of the seller in our mechanism we have $p_{\mathcal{S}}(c)=\mathbb{E}_v\{\left(p(c)-w(p(c))\right)\mathds{1}\{v\geq p(c)\}\}=(p(c)-w(p(c)))(1-F(p(c))$. In the Myerson’s mechanism, we have $p^M_{\mathcal{S}}(c)=\mathbb{E}_v\{\tau_\mathcal{S}(v)\mathds{1}\{c\leq \tau_\mathcal{S}(v)\}\}=\mathbb{E}_v\{\mathscr{P}^{-1}(v)|v\geq p(c)\}\textrm{Pr}_v\{v\geq p(c)\}=(p(c)-w(p(c))(1-F(p(c)))=p_{\mathcal{S}}(c)$. Hence both mechanisms have the same interim allocation/payment rules. As Myerson’s mechanism is BIC, we conclude that $p(c)=\mathscr{P}(c)$ and $b(v)=v$ are also BNE of our mechanism due to revenue equivalence theorem [@RB78].
The indirect fee-setting exchange mechanism with fee schedule $w(P)=\mathbb{E}_v\{\mathscr{P}^{-1}(v)|v\geq P\}$ extracts the maximum revenue in expectation under BNE strategy profile $(\mathscr{P}(c), v)$ for seller and buyer.
For the special case when the seller’s virtual cost is affine, there is an interesting result due to @LN13 which shows the fee-setting mechanism that implements the optimal Myerson is also affine. More formally, we have the following theorem (modified a bit) due to @LN13.
[@LN13] Suppose the buyer is buyer-regular. Then the following are equivalent statements:
- Cost of the seller is drawn from a reverse-generalized Pareto distribution with parameters $\mu,\lambda$ and $\xi$.
- An affine fee mechanism, i.e. with fee schedule $w(P)=(1-\alpha)P+\beta$ where $\alpha=\frac{1}{1+\xi}$ and $\beta=-\frac{\frac{1}{\lambda}+\xi\mu}{1+\xi}$, is intermediary optimal for all buyer distributions.
In principle, one can run the dual of a fee-setting mechanism by swapping the roles of buyer and seller: mechanism asks the buyer for a price $P$ and post the price for the seller. If seller is willing to sell the item with price $P$, intermediary takes $w(P)$ as its share and charges the buyer by $w(P)+P$. Although guarantee bounds for these mechanisms are equivalent to those of ordinary fee-settings, these fee-setting mechanisms are often not used by online exchange platforms in reality, such as Amazon or eBay. Hence, they are out of focus of this paper.
Details of the proof of Theorem \[theorem:main1\] {#detail:proof}
=================================================
We showed the following relations while sketching the proof of Theorem \[theorem:main1\] in Section \[mainresult\]. $$\begin{aligned}
&\mathbb{E}\{\textrm{Rev-APX}|c\}=\frac{w(P)^{\frac{\alpha}{\alpha-1}}}{\beta^{\frac{1}{\alpha-1}}}\label{eq:first}\\
&\mathbb{E}\{\textrm{OPT-Rev}|c\}\leq \alpha\mathbb{E}\left\{w(v)\mathds{1}\{v\geq \phi_\mathcal{B}(P)\}|c\right\}\label{eq:sec}\\
&\mathbb{E}\{\textrm{OPT-Surplus}|c\}=\mathbb{E}\{w(v)\mathds{1}\{v\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))\}|c\}\label{eq:third}\end{aligned}$$ Now, we find equivalent expressions for upper-bounds in (\[eq:first\]) and (\[eq:sec\]) as follows. $$\begin{aligned}
&\alpha\mathbb{E}\left\{w(v)\mathds{1}\{v\geq \phi_\mathcal{B}(P)\}|c\right\}=\alpha \int_{t\geq \phi_\mathcal{B}(P)}(1-F(t))dt=\alpha \int_{t\geq \phi_\mathcal{B}(P)}\left (\frac{w(t)}{\beta}\right )^{\frac{1}{\alpha-1}}dt\nonumber\\
&=\frac{1}{\alpha\beta^{\frac{1}{\alpha-1}}}w(\phi_\mathcal{B}(P))^{\frac{\alpha}{\alpha-1}}=\alpha^{\frac{1}{\alpha-1}} \frac{w(P)^{\frac{\alpha}{\alpha-1}}}{\beta^{\frac{1}{\alpha-1}}},\end{aligned}$$ where in the last equality we use the fact that $w(\phi_\mathcal{B}(P))=\alpha w(P)$, due to property (a) in Lemma \[affine:property\]. Also, we have $$\begin{aligned}
&\alpha\mathbb{E}\left\{w(v)\mathds{1}\{v\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))|c\right\}=\alpha \int_{t\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))}(1-F(t))dt=\alpha \int_{t\geq \phi_\mathcal{B}(\phi_\mathcal{B}(P))}\left (\frac{w(t)}{\beta}\right )^{\frac{1}{\alpha-1}}dt\nonumber\\
&=\frac{1}{\alpha\beta^{\frac{1}{\alpha-1}}}w(\phi_\mathcal{B}\left (\phi_\mathcal{B}(P))\right )^{\frac{\alpha}{\alpha-1}}=\alpha^{\frac{\alpha+1}{\alpha-1}} \frac{w(P)^{\frac{\alpha}{\alpha-1}}}{\beta^{\frac{1}{\alpha-1}}},\end{aligned}$$ where in the last equality we use the fact that $w(\phi_\mathcal{B}(\phi_\mathcal{B}(P)))=\alpha^2 w(P)$, due to property (a) in Lemma \[affine:property\]. Comparing the above upper-bounds on optimal revenue and surplus with the revenue of the affine fee-setting mechanism given in completes the proof of the desired approximation factors.
Proof of Theorem \[theorem:unif\] {#proof:unif}
=================================
For uniform $[0,1]$ distribution, $F(x)=x$, $f(x)=1$, $\phi(x)=2x-1$. So $\alpha=2,\beta=1, P(c)=\frac{c+3}{4}$. We first derive an upperbound on $\textrm{OPT}$. We have $$\begin{aligned}
\textrm{OPT}
&=\int_{c=0}^{y}\left(\int_{0.5+0.5\phi_\mathcal{S}(c)}^{1}(2v-1-\phi_\mathcal{S}(c))dv\right) g(c)dc\nonumber\\
&=\int_{c=0}^{y} \Big( 1-(0.5+0.5\phi_\mathcal{S}(c))^2-0.5(1+\phi_\mathcal{S}(c))(0.5-0.5\phi_\mathcal{S}(c))\Big) g(c)dc\nonumber\\
&=\frac{1}{4}\int_{c=0}^{y}\Big(1-\phi_\mathcal{S}(c)\Big )^2 g(c)dc=\frac{1}{4}\int_{c=0}^{y}\left(1-c-\frac{G(c)}{g(c)}\right)^2 g(c)dc\nonumber\\
&=\frac{1}{4}\int_{c=0}^{y}\left(1+c^2+\frac{G^2(c)}{g^2(c)}-2c-2\frac{G(c)}{g(c)}+2c\frac{G(c)}{g(c)}\right)g(c)dc\nonumber\\
&=\frac{1}{4}\left( G(y)+y^2G(y)-2\int_{c=0}^y cG(c)dc -2yG(y)+2\int_{c=0}^{y}G(c)dc- 2\int_{c=0}^{y}cG(c)dc+2\int_{c=0}^{y}G(c)dc\right)
\nonumber\\
&+\frac{1}{4}\int_{c=0}^{y}\left(\frac{G(c)}{g(c)}\right)^2g(c)dc=\frac{(1-y)^2G(y)}{4}+\frac{1}{4}\int_{c=0}^{y}\left(\frac{G(c)}{g(c)}\right)G(c)dc\nonumber\\
&\leq \frac{(1-y)^2G(y)}{4}+\frac{1}{4}\int_{c=0}^{y}\left(1-c\right)G(c)dc,\end{aligned}$$
where the last inequality comes from the fact that due to seller regularity, for $c\in[0,y]:\phi_\mathcal{S}(c)\leq 1\Rightarrow \frac{G(c)}{g(c)}\leq (1-c)$. Now we have $$\begin{aligned}
\textrm{OPT}&\leq \frac{(1-y)^2G(y)}{4}+\frac{1}{4}\int_{c=0}^{y}\left(1-c\right)G(c)dc\nonumber\\
&=\frac{(1-y)^2G(y)}{4}-\frac{1}{8}(1-c)^2G(c)\bigr|_{c=0}^{c=y}+\frac{1}{8}\int_{c=0}^{y}(1-c)^2g(c)dc\nonumber\\
&=\frac{(1-y)^2G(y)}{8}+\frac{1}{8}\int_{c=0}^{y}(1-c)^2g(c)dc.\end{aligned}$$ Let $\textrm{OPT}_1\triangleq \frac{(1-y)^2G(y)}{8}$ and $\textrm{OPT}_2\triangleq \frac{1}{8}\int_{c=0}^{y}\left(1-c\right)^2g(c)$. We now show $\textrm{Rev-APX}(1,\frac{1-y}{2})\geq \textrm{OPT}_1 $ and $\textrm{Rev-APX}(2,1)\geq \frac{\textrm{OPT}_2}{2} $, and hence conclude the best of these two mechanisms is always a $3$-approximation to $\textrm{OPT}$. To show this, we look at the exact expression for $\textrm{Rev-APX}(\alpha,\beta)$. $$\begin{aligned}
\textrm{Rev-APX}(\alpha,\beta)&=\mathbb{E}_{c,v}\{w(P(c)\mathds{1}\{v\geq P(c)\}\}\nonumber\\
&=\int_{c=0}^{\alpha-\beta}\left(\frac{(1-\alpha)(c+\alpha)+\beta(1+\alpha)}{2\alpha}\right)\left( \frac{\alpha -c -\beta}{2\alpha}\right)g(c)dc\nonumber\\
&=\frac{1}{4\alpha^2}\int_{c=0}^{\alpha-\beta}\Big((1-\alpha)(c+\alpha)+\beta(1+\alpha)\Big) \left(\alpha-c-\beta\right)g(c)dc.
\label{eq3}\end{aligned}$$ Now, we first find a lower bound for $\textrm{Rev-APX}(1,\frac{1-y}{2})$. By applying integration by parts and using the fact that $G(.)$ is monotone non-decreasing we have $$\begin{aligned}
\textrm{Rev-APX}(1,\frac{1-y}{2})&=\frac{1}{4}\int_{c=0}^{\frac{1+y}{2}}(1-y)\left (\frac{1+y}{2}-c\right )g(c)dc\nonumber\\
&=\frac{1-y}{4}\left( \left (\frac{1+y}{2}-c\right )G(c)\bigr|_{0}^{\frac{1+y}{2}}\right)+\frac{1-y}{4}\int_{c=0}^{\frac{1+y}{2}}G(c)dc\nonumber\\
&=\frac{1-y}{4}\int_{c=0}^{\frac{1+y}{2}}G(c)dc\geq\frac{1-y}{4} \int_{c=y}^{\frac{1+y}{2}}G(c)dc\geq \frac{1-y}{4} \int_{c=y}^{\frac{1+y}{2}}G(y)dc\nonumber\\
&=\frac{(1-y)^2G(y)}{8}=\textrm{OPT}_1.\end{aligned}$$ Note that in the above calculation, $y=\min \{\phi_\mathcal{S}^{-1}(1),\overline{c}\}\leq 1$, as $\phi_\mathcal{S}(1)\geq 1$, and hence $\frac{1+y}{2}\geq y$. Based on the previous calculation, we know $\textrm{Rev-APX}(2,1)\geq \frac{1}{16}\int_{c=0}^{y}(1-c)^2g(c)dc=\frac{\textrm{OPT}_2}{2}$, which completes the proof.
Proof of inapproximibility results {#inapprox:appendix}
==================================
Consider the family of seller distributions proposed in (\[instance:bad\]). First of all, for this family of distributions we have $\phi_\mathcal{S}(c)=c+\frac{G_\delta(c)}{g_\delta(c)}=c+\frac{\frac{1}{(1-x)^2}-1}{\frac{2}{(1-x)^3}}=\frac{1+x}{2}-\frac{(1-x)^3}{2}$ is a non-decreasing function and hence seller is regular. Next step to prove the theorem is coming up with an expression for maximum social surplus in terms of parameter $\delta$. We have, $$\begin{aligned}
\textrm{Max-Surplus}_\delta&=\frac{1}{2}\mathbb{E}_c\left \{(1-c)^2\right \}\nonumber
=\frac{\delta}{1-\delta}\int_0^{1-\sqrt\delta}\frac{(1-c)^2}{(1-c)^3}dc=\frac{\delta}{1-\delta}\ln\frac{1}{\sqrt \delta}=\frac{\delta}{2(1-\delta)}\ln\frac{1}{\delta}.\end{aligned}$$ Let $\alpha-\beta=1-\epsilon$. Define $\xi\triangleq \max(\sqrt\delta,\epsilon)$. Based on (\[eq3\]), for every possible pair $(\alpha,\beta)$ under the distribution $G_\delta$, we have, $$\begin{aligned}
&\textrm{APX}_\delta(\alpha, \alpha-1+\epsilon)=\frac{1}{4\alpha^2}\mathbb{E}_c\left\{\Big((\alpha-1)(1-c)+\epsilon(\alpha+1)\Big)\left(1-c-\epsilon\right)_+\right\}
\\
=&\frac{1}{4\alpha^2}\int_{0}^{\min(1-\sqrt{\delta},1-\epsilon)}{\Big((\alpha-1)(1-c)+\epsilon(\alpha+1)\Big)\left(1-c-\epsilon\right)g_\delta(c)dc}\nonumber\\
=&\frac{\alpha-1}{4\alpha^2}\int_{0}^{1-\xi}(1-c)^2g_\delta(c)dc+\frac{\epsilon}{2\alpha^2}\int_{0}^{1-\xi}(1-c)g_\delta(c)dc-\frac{\epsilon^2(\alpha+1)}{4\alpha^2}\int_{0}^{1-\xi}g_\delta(c)dc.\nonumber\end{aligned}$$ By plugging $g_\delta(c)=\frac{2\delta}{(1-\delta)(1-c)^3}$ for $c\in [0,1-\xi]$ and computing integrals we have $$\begin{aligned}
\textrm{APX}_\delta(\alpha, \alpha-1+\epsilon)=\frac{(\alpha-1)\delta}{2\alpha^2(1-\delta)}\ln \frac{1}{\xi}+\frac{\delta\epsilon}{\alpha^2(1-\delta)}\frac{(1-\xi)}{\xi}-\frac{\epsilon^2(\alpha+1)\delta}{4\alpha^2(1-\delta)}\frac{(1-\xi^2)}{\xi^2}.\end{aligned}$$
Now we have the expressions for both $\textrm{Surplus}$ and $\textrm{APX}_\delta(\alpha, \beta)$. Below we want to show, if $\delta$ is small enough, $SG_{F,G}$ under $G_\delta$ can be arbitrarily large, which means for every constant $d$, we can find a $G_\delta$ with $SG_{F,G}\geq d$. In order to prove this, for fixed $\delta$ we discuss the possible values of $(\alpha,\epsilon)$, and then compute the ratio of $\textrm{APX}_\delta(\alpha, \beta)$ to $\textrm{Surplus}$. If the ratio goes to zero as $\delta$ goes to zero, the theorem is proved.
We now consider two cases:
- *Case 1* ($\epsilon\leq \sqrt{\delta}$): In this case we have $\xi=\sqrt{\delta}$, and hence $$\textrm{APX}_\delta(\alpha, \alpha-1+\epsilon)=\frac{(\alpha-1)\delta}{4\alpha^2(1-\delta)}\ln \frac{1}{\delta}+\Delta_\delta(\epsilon),$$ where $\Delta_\delta(\epsilon)\triangleq\frac{\sqrt\delta(1-\sqrt \delta)}{\alpha^2(1-\delta)}\epsilon-\frac{(\alpha+1)}{4\alpha^2}\epsilon^2$. This function is quadratic with respect to its argument and its maximum over the interval $[0,\sqrt\delta]$ happens at either $\epsilon^*=\frac{2\sqrt\delta(1-\sqrt \delta)}{(\alpha+1)(1-\delta)}$ or $\sqrt\delta$. Now, we develop an upperbound on the ratio of $\textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)$ and $\textrm{Max-Surplus}$ when $\epsilon\leq \sqrt\delta$ as follows: $$\label{ineq1}
\forall \epsilon\in[0,\sqrt \delta]: \frac{\textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)}{\textrm{Max-Surplus}_\delta}\leq \frac{\frac{\sqrt\delta(1-\sqrt \delta)}{\alpha^2(1-\delta)}\epsilon}{\frac{\delta}{2(1-\delta)}\ln\frac{1}{\delta}}\leq \frac{2(1-\sqrt \delta)}{\alpha^2\ln(1/\delta)} ,$$ where the first inequality comes from the fact that $\alpha\leq 1$( hence $\frac{(\alpha-1)\delta}{4\alpha^2(1-\delta)}\ln \frac{1}{\delta}\leq 0$), and second inequality is due to $\epsilon\leq \sqrt{\delta}$. Now, we have two cases:
- If $\alpha\geq 2\frac{1-\sqrt \delta}{1-\delta}-1$, then $\epsilon^*\leq \sqrt{\delta}$ and hence the maximum of $\Delta(\epsilon)$ happens at $\epsilon=\epsilon^*$. In this case, using inequality (\[ineq1\]) we have $$\forall \epsilon\in[0,\sqrt \delta]: \frac{\textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)}{\textrm{Max-Surplus}_\delta}\leq \frac{2(1-\sqrt \delta)}{\left (2\frac{1-\sqrt \delta}{1-\delta}-1\right )^2\ln(1/\delta)}\overset{\delta\rightarrow 0} {\longrightarrow }0,$$
- If $\alpha< 2\frac{1-\sqrt \delta}{1-\delta}-1$, then $\epsilon^*> \sqrt \delta$ and hence the maximum of $\Delta(\epsilon)$ happens at $\epsilon=\sqrt \delta$. In this case we have $$\begin{aligned}
\label{ineq2}
\forall \epsilon\in[0,\sqrt \delta]: ~ & \frac{\textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)}{\textrm{Max-Surplus}_\delta}\leq \frac{(\alpha-1)}{2\alpha^2}+\frac{\Delta(\sqrt \delta)}{\frac{\delta}{2(1-\delta)}\ln\frac{1}{\delta}}\nonumber\\&=\frac{(\alpha-1)}{2\alpha^2}+\frac{2(1-\sqrt \delta)}{\alpha^2\ln\frac{1}{\delta}}-\frac{(\alpha+1)(1-\delta)}{2\alpha^2\ln \frac{1}{\delta}}.\end{aligned}$$ Suppose $\alpha^*(\delta)$ be the $\alpha$ that maximizes the upperbound on revenue in (\[ineq2\]). If $\forall C>0, \exists \delta_C$ s.t. if $\delta\leq \delta_C$ then $\alpha^*(\delta)\geq \left (\frac{C}{\ln\frac{1}{\delta}}\right )^{1/2}$, then we would have $$\begin{aligned}
\label{ineq3}
\forall C>0, \forall \epsilon\in\left [0,\sqrt \delta\right ], \delta\in [0,\delta_C]:~&\frac{\textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)}{\textrm{Max-Surplus}_\delta}\leq
\frac{2(1-\sqrt \delta)}{\alpha^*(\delta)^2\ln\frac{1}{\delta}}\leq \frac{2}{C}.\end{aligned}$$ As the above uppderbound holds for all $C>0$, so the ratio goes to zero as $\delta$ goes to zero.
Now, suppose $\exists C_0$ s.t. $\forall \delta_0$, $\exists \delta<\delta_0$ s.t. $\alpha^*(\delta) < \left (\frac{C_0}{\ln\frac{1}{\delta}}\right )^{1/2}$. From (\[ineq2\]) we have: $$\begin{aligned}
\label{ineq4}
\forall \epsilon\in\left [0,\sqrt \delta\right ]: ~ & \frac{\textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)}{\textrm{Max-Surplus}_\delta}\leq
\frac{(\alpha^*(\delta)-1)}{2\alpha^*(\delta)^2}+\frac{2(1-\sqrt \delta)}{\alpha^*(\delta)^2\ln\frac{1}{\delta}}\nonumber\\
&\leq\frac{1}{2\alpha^*(\delta)^2}\left(\frac{4(1-
\sqrt\delta)}{\ln \frac{1}{\delta}}+\left (\frac{C_0}{\ln\frac{1}{\delta}}\right )^{1/2}-1\right).\end{aligned}$$ We can find arbitrarily small $\delta$ such that $\alpha^*(\delta) < \left (\frac{C_0}{\ln\frac{1}{\delta}}\right )^{1/2}$, in which case this upperbound is a negative number. Thus, we know the ratio can be arbitrarily small.
- *Case 2* ($\epsilon> \sqrt{\delta}$): In this case we have $\xi=\epsilon$, and hence $$\textrm{APX}_\delta(\alpha, \alpha-1+\epsilon)=\frac{\delta}{(1-\delta)4\alpha^2}\gamma(\epsilon,\alpha),$$ where $\gamma(\epsilon,\alpha)=2(\alpha-1)\ln \frac{1}{\epsilon}+4(1-\epsilon)-(\alpha+1)(1-\epsilon^2).$ We now investigate the choice of $\epsilon$ that maximizes $\gamma$ for a fixed $\alpha$. We have $$\frac{\partial\gamma}{\partial\epsilon}=\frac{2(1-\alpha)}{\epsilon}-4+2(\alpha+1)\epsilon=\frac{2}{\epsilon}(1-\alpha-2\epsilon+(\alpha+1)\epsilon^2).$$ Roots of $\frac{\partial\gamma}{\partial\epsilon}$, which are candidates for local extremum, are $\epsilon_1=1$ and $\epsilon_2=\frac{1-\alpha}{1+\alpha}$. We know if $\epsilon\geq 1$, then the mechanism cannot get any revenue as the interval $[1-\epsilon,0]$ is outside of the support of the seller’s distribution. So, maximum of $\gamma(\alpha,\epsilon)$ for any fixed $\alpha$ over $\epsilon\in [\sqrt \delta,\infty]$ either happens at $\epsilon= \sqrt \delta$ or $\epsilon=\epsilon_2$. If maximum happens at $\epsilon=\sqrt \delta$ then the analysis will be the same as Case 1 and we are done. Otherwise, assume maximum happens at $\epsilon=\frac{1-\alpha}{1+\alpha}$. We have $$\begin{aligned}
\label{ineq5}
&\frac{ \textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)\lvert_{\epsilon=\frac{1-\alpha}{1+\alpha}}}{\textrm{Max-Surplus}_\delta}\nonumber\\
&=\frac{\frac{\delta}{(1-\delta)4\alpha^2}\gamma(\alpha,\frac{1-\alpha}{1+\alpha})}{\frac{\delta}{2(1-\delta)}\ln\frac{1}{\delta}}=\frac{1}{2\alpha^2\ln\frac{1}{\delta}}\left( 2(\alpha-1)\ln\left (\frac{1+\alpha}{1-\alpha}\right )+\frac{4\alpha}{\alpha+1}\right)\nonumber\\
&\leq \frac{1}{\alpha^2\ln\frac{1}{\delta}}\left((\alpha-1)\ln\left (1+\frac{2\alpha}{1-\alpha}\right )+2\alpha\right)\leq \frac{2\alpha+(\alpha-1)\ln(1+2\alpha)}{\alpha^2\ln\frac{1}{\delta}}.\end{aligned}$$
Suppose $\alpha^*(\delta)$ be the $\alpha$ that maximizes the upperbound on revenue in (\[ineq5\]) for a particular $\delta$. If $\forall C>0, \exists \delta_C$ s.t. if $\delta\in [0,\delta_C]$ then $\alpha^*(\delta)\geq \frac{C}{\ln\frac{1}{\delta}}$, using (\[ineq5\]) we would have $$\begin{aligned}
\frac{ \textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)\lvert_{\epsilon=\frac{1-\alpha}{1+\alpha}}}{\textrm{Max-Surplus}_\delta}\leq \frac{1}{\alpha^*(\delta)\ln\frac{1}{\delta}}\leq \frac{1}{C}.\end{aligned}$$ Since $C$ is arbitrary, we know the ratio goes to zero as $\delta$ goes to zero.
Now suppose $\exists C_0$ s.t. $\forall\delta_0, \exists \delta<\delta_0$ such that $\alpha^*(\delta)<\frac{C_0}{\ln\frac{1}{\delta}}$. We now have $$\begin{aligned}
&\frac{ \textrm{APX}_\delta(\alpha,\alpha-1+\epsilon)\lvert_{\epsilon=\frac{1-\alpha}{1+\alpha}}}{\textrm{Max-Surplus}_\delta}\leq \frac{2\alpha^*(\delta)+(\alpha^*(\delta)-1)\ln(1+2\alpha^*(\delta))}{\alpha^*(\delta)^2\ln\frac{1}{\delta}}\\
\overset{\delta \rightarrow 0}{\approx}&
\frac{
2\alpha^*(\delta)+
(\alpha^*(\delta)-1)(2\alpha^*(\delta)-2\alpha^*(\delta)^2)
}{\alpha^*(\delta)^2\ln\frac{1}{\delta}}=
\frac{4}{\ln{\frac{1}{\delta}}}\overset{\delta \rightarrow 0}{\rightarrow} 0.\end{aligned}$$ The third expression is obtained using Taylor expansion for $\ln(1+x)$ at $x=0$. When $\delta$ goes to zero, we can always find a corresponding $\alpha^*(\delta)$ going to zero ( $\forall\delta_0,\exists\delta<\delta_0, \alpha^*(\delta)<\frac{C_0}{\ln\frac{1}{\delta}}$). So we may ignore the $o(\alpha^*(\delta)^2)$.
The “if" direction has been proved in Corollary \[coro:powerdistribution\]. The proof of “only if" direction is as follows. First, notice that when buyer’s distribution is uniform $[0,1]$, then $\phi_\mathcal{B}(v)=2v-1$, $P(c)=\frac{c+\beta}{2\alpha}+\frac{1}{2}$. Assume that $1-\epsilon=\alpha-\beta$. Since $P(c)\leq 1$, we have $1-\epsilon=\alpha-\beta\geq c$. First we compute $\textrm{Max-Surplus}$, and then $\textrm{Rev-APX}$, $$\begin{aligned}
\textrm{Max-Surplus}=
\int_0^1 \left (
\int_c^1
(1-v) dv\right ) g(c) dc
=
\int_0^1
\frac{(1-c)^2}{2}
d G(c)
=
\int_0^1
G(c) (1-c) dc,\end{aligned}$$ $$\begin{aligned}
\textrm{Rev-APX}(\alpha,\beta)&=\mathbb{E}_{v,c}\Big\{w(P(c))\mathds{1}\{v\geq P(c)\}\Big\}=\mathbb{E}_{c}\left \{\Big((1-\alpha)P(c)+\beta \Big)\Big(1-F(P(c))\Big)\right \}\\
&=
\mathbb{E}_{c}\left \{
\left ((1-\alpha)\left ( \frac{c+\beta}{2\alpha}
+\frac{\beta}{2}\right )+\beta \right )\Big(1-F(P(c))\Big)\right \}\\
&=
\mathbb{E}_{c}\left \{
\left (
\frac{(1-\alpha)(c+\alpha)+(1+\alpha)(\alpha-1+\epsilon)}{2\alpha}
\right )\left (\frac{\alpha-c-\beta}{2\alpha}\right )\right \}\\
&=
\frac{1}{4\alpha^2}
\int_0^{1-\epsilon}
\Big((1-\alpha)(c-1)+ (1+\alpha)\epsilon\Big)(1-\epsilon-c)
g(c) dc\\
&=
\frac{1}{4\alpha^2}
\Big((1-\alpha)(c-1)+ (1+\alpha)\epsilon\Big)(1-\epsilon-c)
G(c)|_0^{1-\epsilon}-\\
&~~\int_0^{1-\epsilon}
G(c) \Big ((1-\alpha)(1-\epsilon-c) -
(1-\alpha)(c-1)-(1+\alpha)\epsilon \Big )dc\\
&=
\frac{-1}{4\alpha^2}
\int_0^{1-\epsilon}
G(c) \Big((1-\alpha)(1-\epsilon-c) -
(1-\alpha)(c-1)-(1+\alpha)\epsilon)\Big)dc\\
&=
\frac{1}{2\alpha^2}
\int_0^{1-\epsilon}
G(c) \Big (
(\alpha-1)(1-c)+\epsilon
\Big )dc\nonumber\\
&=
\frac{\alpha-1}{2\alpha^2}
\int_0^{1-\epsilon}
G(c) (1-c)dc
+
\frac{\epsilon}{2\alpha^2}
\int_0^{1-\epsilon}
G(c) dc.\end{aligned}$$ When $\epsilon=0$, which means $\alpha-\beta=1$, $\textrm{Rev-APX}(\alpha,\beta)$ is $\frac{2\alpha^2}{\alpha-1}$ approximation to Surplus. When $\alpha=2$, it gets the maximum approximation ratio of $8$. If $\epsilon>0$, then we may consider a distribution $G$ which is supported at $(1-\epsilon,1]$, then we know $\textrm{Rev-APX}(\alpha,\beta)=0$, while Max-Surplus is positive. So it could not approximate Max-Surplus in this distribution. If $\epsilon<0$, we consider two cases. If $\alpha\geq 1$, then we consider a distribution $G$ which is uniform on $[1+\epsilon/2,1]$, so we know $\int_0^{1-\epsilon/2} G(c) (1-c) dc\leq 0$, so $\int_0^{1-\epsilon} G(c)(1-c) dc \leq 0$, which means $\textrm{Rev-APX}(\alpha,\beta)\leq 0$, but maximum surplus is positive. If $\alpha<1$, then since $\alpha-\beta=1-\epsilon>1$, so $\beta<0$. Consider the distribution $G$ which is uniform on $[0,-\beta/2]$, then we know $w(P(c))=(1-\alpha)(\frac{c+\beta}{2\alpha}+\frac{\beta}{2})+\beta<0$, so the intermediary could extract no revenue in this case.
Proof of Lemma \[iidMHR\] {#lemmaproofs:appendix}
=========================
Let $G(x)$ be the CDF of random variable $v=\max v_i$. We have $G(x)=(F(x))^n$ and $g(x)=nf(x)(F(x))^{(n-1)}$. Let $\tilde{h}$ be the hazard rate of $v$, hence $$\tilde{h}(x)=\frac{g(x)}{1-G(x)}=n\frac{nf(x)(F(x))^{(n-1)}}{1-(F(x))^n}=n\frac{f(x)}{1-F(x)}\frac{1}{\sum_{i=0}^{n-1} (1/F(x))^i}$$ which is non-decreasing as $h(x)=\frac{f(x)}{1-F(x)}$ is non-decreasing and $F(x)$ is non-decreasing.
Conclusions and open questions
==============================
In this paper we studied the problem of simple affine fee-setting mechanisms versus optimal intermediary mechanisms in the setting of 1-seller 1-buyer exchange. Our result complements the already existing result on optimality of affine fee-setting mechanisms when seller has an affine virtual cost function. In fact, we showed that under some technical assumptions, if the buyer has affine virtual value function there exist an affine fee-setting mechanism that extracts a constant approximation of optimal intermediary profit. Moreover, we showed if buyer’s value is MHR and the difference between buyer’s value and seller’s cost is MHR, then we get constant approximation to both surplus and revenue by a constant fee-schedule mechanism. Next, we provided inapproximability results by showing that proper affine fee-setting mechanisms (e.g. those used in eBay and Amazon selling plans) are *unable* to extract a constant fraction of optimal profit in the worst-case seller distribution. As subsidiary results we also show there exists a constant gap between maximum surplus and maximum revenue under the aforementioned conditions. Most of the mechanisms that we propose are also prior-independent with respect to the seller, which signifies the practical implications of our result.
There are many open questions left that might be interesting for future works on this topic:
- Can we extend the results to the case where there are multiple sellers?
- As has been conjectured in [@LN07], affine fee-setting mechanism seem to get a good fraction of optimal revenue even under worst-case distributions of both buyer and seller. Can the proof techniques provided in this paper be used to solve that problem?
- Can we generalize techniques provided in this paper to other exchange environments such as multi-item environments?
[^1]: All three authors were supported by NSF grant AF-0910940. Robert Kleinberg was also supported by a Microsoft Research New Faculty Fellowship and a Google Research Grant.
[^2]: See [http://pages.ebay.com/help/sell/fees.html]{}
[^3]: See [http://services.amazon.com/selling/pricing.htm]{}
[^4]: Note that due to Corollary \[cor:gpd\] and Lemma \[lemma:gpd\], $v$ is drawn from a generalized Pareto distribution, and hence $\alpha$ should be in $[1,\infty)$.
[^5]: We set $\phi_\mathcal{S}^{-1}(1)=+\infty$ when $\phi_\mathcal{S}(1)=1$ doesn’t have a solution.
[^6]: To be more precise, this mechanism is also Dominant Strategy Incentive Compatible (DSIC) and Ex-post individual rational under no-positive transfer assumption.
|
---
abstract: 'We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium solution may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behavior of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether or not the cusp catastrophe appears depends on the choice of the coupling function. These types of catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is a nonperturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.'
---
[**Neutron stars in scalar-tensor theories of gravity and catastrophe theory**]{}
[ Tomohiro Harada [^1]]{}\
[*Department of Physics, Kyoto University,*]{} [*Kyoto 606-01, Japan*]{}\
Introduction
============
Scalar-tensor theories [@will1993; @de1992] are among the generalized theories of gravitation. Brans-Dicke theory [@bd1961] is a member of the scalar-tensor theories. Scalar-tensor theories have recently attracted the attention of many researchers. One of the reasons is that the unified theories that contain gravity as well as other interactions, such as string theory [@gsw1987], naturally predict the existence of scalar fields that relate to gravity. In the hyperextended inflation model [@sa1990], scalar-tensor theories of gravity play an essential role. Moreover, projects of laser interferometric gravitational wave observations [@ligo; @virgo; @geo; @tama] will be soon in practical use, so that high-accuracy tests of the scalar-tensor theories may be expected [@wz1989; @will1994; @snn1994; @sst1995; @hcnn1997].
Scalar-tensor theories are viable theories of gravity for some choices of the coupling function which is contained in the theories. Predictions of these theories in a strong field may be drastically different from those of general relativity. Recently, Damour and Esposito-Farèse [@de1993; @de1996] discovered one example of such phenomena. They showed that, for some choices of the coupling function, the configuration of a massive neutron star deviates significantly from that in general relativity, even if the post-Newtonian limit of the theory is extremely close to or even agrees with that of general relativity. This deviation in a strong field may be easily tested from binary-pulsar timing observations, if it exists, because of the extra energy loss by scalar gravitational radiation [@de1996]. The deviation from general relativity can be no longer dealt with as a perturbative effect from general relativity. Damour and Esposito-Farèse referred to this nonperturbative strong-field effect as “spontaneous scalarization” in analogy to the spontaneous magnetization of the ferromagnets.
In this paper, we investigate spontaneous scalarization in detail with the technique of catastrophe theory. A many-parameter version of the turning point method [@katz1978; @katz1979; @sorkin1981; @sorkin1982] is used as a tool of a stability analysis of equilibrium solutions. The stability analysis of boson stars in scalar-tensor gravity via catastrophe theory in the case of one-dimensional control space was done in [@cs1997; @tls1997]. Catastrophe types of neutron star equilibrium solutions are classified as fold and cusp catastrophes. The occurrence of the cusp catastrophe depends on the choice of the coupling function. Spontaneous scalarization is classified as the cusp catastrophe. From this catastrophic feature, we conclude that the stable configuration of the neutron star may change discontinuously as the baryon number of the star or the asymptotic value of the scalar field changes continuously. The behavior of the scalar charge around the cusp point is explained by catastrophe theory. For the coupling function considered here, when the asymptotic value of the scalar field is such that the theory agrees with general relativity in the post-Newtonian limit, we find the sequence of the equilibrium solutions [*bifurcates*]{} to three branches at some critical central density. One branch consists of solutions that are identical to neutron stars in general relativity, and the other two consist of solutions that deviate significantly from neutron stars in general relativity. The general relativistic branch is secularly unstable in agreement with the result obtained by a perturbation study [@harada1997], while the non-general-relativistic branches are secularly stable.
This paper is organized as follows. In Sec. II, we summarize the field equations of scalar-tensor theory and the equations determining equilibrium solutions of neutron stars in this gravitational theory. In Sec. III, we present stability criteria on the grounds of the turning point method. In Sec. IV, the stability criteria are applied to equilibrium solutions of neutron stars in scalar-tensor theory and some consequences of catastrophe theory are discussed. Section V is devoted to conclusions. We use units in which $c=1$. The Greek indices run from 0 to 3. We follow the Misner-Thorne-Wheeler [@mtw1973] sign conventions for curvature quantities.
Basic equations
===============
Here we consider a class of scalar-tensor theories in which gravity is mediated by not only a metric tensor but also a massless scalar field. The action is given by [@de1992] $$\label{eq:action}
I=\frac{1}{16\pi G_*}\int\sqrt{-g_*}\left( R_* -
2g_*^{\mu\nu}\varphi_{,\mu}\varphi_{,\nu}
\right)d^4x
+I_m[\Psi_m,A^2(\varphi)g_{*\mu\nu}],$$ where $g_{*\mu\nu}$ is the “Einstein” frame metric tensor, $\Psi_m$ denotes matter fields collectively, and $G_*$ is some dimensionful constant. In this Einstein frame, the Einstein-Hilbert term is isolated from other sectors. The “Brans-Dicke” frame metric tensor $\tilde{g}_{\mu\nu}$ is related to the Einstein frame metric tensor by the following conformal transformation: $$\tilde{g}_{\mu\nu}=A^2(\varphi)g_{*\mu\nu}.
\label{eq:conformaltransformation}$$ Because of the “universal coupling,” which is the way of the coupling of the scalar field in the matter sector seen in Eq. (\[eq:action\]), a test particle moves on the geodesic of the Brans-Dicke frame metric $\tilde{g}_{\mu\nu}$. For this reason the Brans-Dicke frame is often called a “physical” frame. The tilde denotes the physical frame quantity.
In the Einstein frame, the field equations are given by $$\begin{aligned}
& & G_{*\mu\nu}=8\pi G_* T_{*\mu\nu}
+2\left(\varphi_{,\mu}\varphi_{,\nu}-\frac{1}{2}g_{*\mu\nu}
g_*^{\alpha\beta}\varphi_{,\alpha}\varphi_{,\beta}\right),
\label{eq:fieldeq1} \\
& & \Box_* \varphi = - 4\pi G_* \alpha(\varphi) T_*,
\label{eq:fieldeq2}\end{aligned}$$ while the equations of motion for matter are $$\label{eq:eom}
\nabla_{*\nu}T_{*\mu}^{\nu}=\alpha(\varphi)T_*\nabla_{*\mu}\varphi,$$ where the energy-momentum tensor of the matter, $T_*^{\mu\nu}$, is defined and related to the physical energy-momentum tensor $\tilde{T}^{\mu\nu}$ as $$\label{eq:defofem}
T_*^{\mu\nu} \equiv \frac{2}{\sqrt{-g_*}}
\frac{\delta I_m[\Psi_m,A^2(\varphi)g_{*\mu\nu}]}
{\delta g_{*\mu\nu}}
= A^6(\varphi) \tilde{T}^{\mu\nu}.$$ $G_{*\mu\nu}$ and $\Box_*$ are the Einstein tensor and d’Alembertian of $g_{*\mu\nu}$, respectively. $T_*$ and $\alpha(\varphi)$ are defined as $$\begin{aligned}
T_* &\equiv& T_{*\mu}^{~~\mu}
\equiv T^{\mu\nu}_* g_{*\mu\nu},\\
\alpha(\varphi) &\equiv& \frac{d\ln A(\varphi)}
{d\varphi}.\end{aligned}$$
The parameters in the parametrized post-Newtonian framework are given by [@will1993; @de1992] $$\begin{aligned}
& & 1-\gamma_{Edd} = \frac{2\alpha_0^2}{1+\alpha_0^2}, \\
& & \beta_{Edd}-1 = \frac{\beta_0\alpha_0^2}{2(1+\alpha_0^2)^2},\\
& & \xi = \alpha_1 = \alpha_2 = \alpha_3 = 0, \end{aligned}$$ where $\beta_{Edd}$ and $\gamma_{Edd}$ are the so-called Eddington parameters. We have defined $$\begin{aligned}
\label{eq:defofalpha0}
\alpha_0 &\equiv& \alpha(\varphi_0), \\
\label{eq:defofbeta0}
\beta_0 &\equiv& \frac{d\alpha}
{d\varphi}(\varphi_0),\end{aligned}$$ and $\varphi_0$ is the value of the scalar field $\varphi$ in the spatial asymptotic region. We assume that the cosmological evolution of the scalar field is sufficiently slow in comparison with the characteristic time scale of the local gravitational process of the isolated object considered here. From this assumption $\varphi_0$ is regarded as the cosmological value of the scalar field. On the other hand, we can identify the asymptotic value $\varphi_0$ to the value of the scalar field in the matching region in the matching approach to the $N$-compact-body problem (see Appendix A of [@de1992]). Then, the solar-system experimental constraints are [@lebach1995] $$\label{eq:constraint1}
\gamma_{Edd}= 0.9996\pm0.0017$$ and [@williams1995] $$\label{eq:constraint2}
4\beta_{Edd}-\gamma_{Edd}-3=-0.0007\pm0.0010.$$
We summarize equations for the structure of a relativistic star in scalar-tensor theory, following [@de1993]. We restrict ourselves to the static and spherically symmetric case. The metric is given in the following form: $$\label{eq:staticspherical}
ds_{*}^2 = - e^{\nu(r)}dt^2 + \left(1-\frac{2\mu(r)}{r}
\right)^{-1}dr^2
+r^2(d\theta^2+\sin^2 \theta d\phi^2).$$ The matter is described as a perfect fluid: i.e., $$\tilde{T}_{\mu\nu}=(\tilde{\rho}+\tilde{p})\tilde{u}_{\mu}
\tilde{u}_{\nu}+\tilde{p}\tilde{g}_{\mu\nu}.$$ Then, the following equations are obtained: $$\begin{aligned}
\label{eq:beq1}
\mu^{\prime}&=&4\pi G_* r^2 A^4 \tilde{\rho}
+\frac{1}{2}r(r-2\mu)\psi^2, \\
\label{eq:beq2}
\nu^{\prime}&=&8\pi G_* \frac{r^2 A^4 \tilde{p}}{r-2\mu}
+r\psi^2+\frac{2\mu}{r(r-2\mu)}, \\
\label{eq:beq3}
\varphi^{\prime}&=&\psi, \\
\label{eq:beq4}
\psi^{\prime}&=&4\pi G_* \frac{r A^4}{r-2\mu}
[\alpha(\tilde{\rho}-3\tilde{p})+r(\tilde{\rho}-\tilde{p})\psi]
-\frac{2(r-\mu)}{r(r-2\mu)}\psi, \\
\label{eq:beq5}
\tilde{p}^{\prime}&=&-(\tilde{\rho}+\tilde{p})
\left[4\pi G_*\frac{r^2 A^4 \tilde{p}}{r-2\mu}
+\frac{1}{2}r\psi^2+\frac{\mu}{r(r-2\mu)}
+\alpha(\varphi)\psi\right], \\
\label{eq:beq6}
\tilde{p}&=&\tilde{p}(\tilde{\rho}),\end{aligned}$$ where the prime denotes a derivative with respect to $r$. We use the polytropic equations of state: $$\begin{aligned}
\tilde{\rho} &=& \tilde{n}m_b+\frac{Kn_0m_b}{\Gamma-1}
\left(\frac{\tilde{n}}{n_0}\right)^{\Gamma}, \\
\tilde{p} &=& Kn_0m_b\left(\frac{\tilde{n}}{n_0}\right)^{\Gamma}, \\
m_b &=& 1.66\times10^{-24}~\mbox{g}, \\
n_0 &=& 0.1~\mbox{fm}^{-3},\end{aligned}$$ where $\tilde{n}$ is the baryon number density in the Brans-Dicke frame. We then take the parameters $\Gamma=2.34$ and $K=0.0195$ (EOS II of [@de1993]). Note that the total baryon number is given by $$N=\int 4\pi\tilde{n} A^3 r^2 \left(1-\frac{2\mu}{r}
\right)^{-1/2}dr.$$
Here we present the method of solving the above equations and obtaining the structure of a neutron star. First the initial values of the above set of ordinary differential equations are fixed as $$\mu (0)=0,~\nu(0)=0,~\varphi(0)=\varphi_c,
~\psi(0)=0,~\tilde{p}(0)=\tilde{p}_c,$$ and Eqs. (\[eq:beq1\])-(\[eq:beq6\]) are integrated numerically up to the stellar surface at which $\tilde{p}=0$. Thereafter the solution is matched with the static and spherically symmetric “vacuum” solution, where the term “vacuum” means only the absence of matter, i.e., $\tilde{T}_{\mu\nu}=0$. This solution is given in [@de1992]. The solutions are parametrized by three parameters $b$, $d$, and $\varphi_0$. From the matching conditions at the surface, we can obtain $\nu(r)$ including the constant term. In order to set $A(\varphi_0)$ to unity, we rescale the raw quantities to the renormalized ones as follows: $$\begin{aligned}
r_{ren}&=&A_0^2 r,~~ \mu_{ren}= A_0^2 \mu,~~ \nu_{ren}=\nu,
~~\varphi_{ren}=\varphi,~~ \psi_{ren}=A_0^{-2}\psi, \nonumber \\
\varphi_{0 ren} &=& \varphi_0,~~ a_{ren}=A_0^2 a,
~~b_{ren}=A_0^2 b, ~~d_{ren}=A_0^2 d, ~~N_{ren}=A_0^3 N.\end{aligned}$$ Then, from the asymptotic properties at spatial infinity for a static and isolated system $$\begin{aligned}
g_{*\mu\nu}&=&\eta_{\mu\nu}+\frac{2G_*m}{r_{ren}}\delta_{\mu\nu}
+O\left(\frac{G_*^2}{r_{ren}^2}\right), \\
\varphi&=&\varphi_0+\frac{G_*\omega}{r_{ren}}
+O\left(\frac{G_*^2}{r_{ren}^2}
\right), \end{aligned}$$ where $\eta_{\mu\nu}$ is the Minkowskian metric. We call $m$ the Arnowitt-Deser-Misner (ADM) energy and $\omega$ the scalar charge [@de1992]. $b_{ren}$ and $d_{ren}$ are related to $m$ and $\omega$ as $$\begin{aligned}
G_* m=\frac{b_{ren}}{2}, \\
G_*\omega= -d_{ren}.\end{aligned}$$ Hereafter the subscripts “[*ren*]{}” are omitted for simplicity. We use units in which $G_*=1$.
stability criteria
==================
In scalar-tensor theory, control parameters of the static, spherically symmetric, and isolated neutron star are not only the baryon number $N$ but also the “external field”, that is, the asymptotic value of the scalar field $\varphi_0$. For static systems, the partial derivative of $m$ in terms of $\varphi_0$ with $N$ constant is given by [@de1992] $$\label{eq:dermderphi0}
\left(\frac{\partial m}{\partial \varphi_0}\right)_{N}=-\omega.$$ The energy injection of the system by increasing baryons is described as $$\label{eq:injection}
\int 4\pi \tilde{u}_0 \delta\tilde{\rho}
A^3 r^2 \left(1-\frac{2\mu}{r}\right)^{-1/2}dr
=\int 4\pi e^{\nu/2} \tilde{\mu}
\delta \tilde{n}
A^4 r^2 \left(1-\frac{2\mu}{r}\right)^{-1/2}dr,$$ where $\tilde{\mu}\equiv d\tilde{\rho}/d\tilde{n}$ is the chemical potential. The first law of thermodynamics in an adiabatic process is $$\label{eq:1stlaw}
d\left(\frac{\tilde{\rho}}{\tilde{n}}\right)
=-\tilde{p}d\left(\frac{1}{\tilde{n}}\right).$$ From Eqs. (\[eq:beq2\]), (\[eq:beq3\]), (\[eq:beq5\]), and (\[eq:1stlaw\]), we find that the quantity $A e^{\nu/2}\tilde{\mu} $ is constant all over the star. Therefore this quantity can be estimated by its value at the stellar surface. Using this fact, the expression of the energy injection, Eq. (\[eq:injection\]), is rewritten as $$A_s e^{\nu_s/2}\tilde{\mu}_s\int 4\pi A^3 r^2
\left(1-\frac{2\mu}{r}\right)^{-1/2}\delta \tilde{n}dr
= A_s e^{\nu_s/2}\tilde{\mu}_s \delta N,$$ where the suffix “[*s*]{}” indicates that the quantity is evaluated at the stellar surface $r=r_s$. Therefore, the effective chemical potential, $\mu_{eff}$, is given by $$\mu_{eff}\equiv \left( \frac{\partial m}{\partial N}
\right)_{\varphi_0}
= A_s e^{\nu_{s}/2}\tilde{\mu}_{s}.$$ From the above discussions the variation of $m$ for static systems in a quasistatic process is written in the following form: $$\delta m = -\omega \delta\varphi_0 + \mu_{eff} \delta N,$$ where by “quasistatic process” we mean successive changes among the infinitesimally nearby equilibrium solutions.
Suppose that the isolated neutron star is perturbed slightly with $\varphi_0$ and $N$ constant for some reason other than incident waves, while spherical symmetry is preserved. Then the outgoing waves can carry out some positive energy to infinity and the system cannot keep its original state if there exists an energetically favorable configuration which is infinitesimally deformed from the system with the same $\varphi_0$ and $N$. Therefore, an equilibrium solution $X$ is secularly stable against spherically symmetric (infinitesimal) perturbations if and only if there is no momentarily static and spherically symmetric configuration $Y$ which is arbitrarily close to $X$ with the same $\varphi_0$ and $N$ but strictly smaller $m$.
In order to examine the stability of the equilibrium solution, we follow the turning point method [@katz1978; @katz1979; @sorkin1981; @sorkin1982]. In the present problem, $m$ is a potential function, since the equilibrium solution is a stationary point of $m$ ($\delta m=0$) and the stable equilibrium solution is a minimal point of $m$ ($\delta m=0$ and $\delta^2 m >0$). The asymptotic value of the scalar field $\varphi_0$ and baryon number $N$ form a two-dimensional control space. The equilibrium solutions are uniquely parametrized by two parameters, i.e., the central value of the scalar field, $\varphi_c$, and the central total baryonic density, $\tilde{\rho}_c$.
We adopt the following stability criteria [@sorkin1982]:
\(i) The stability of $X(\varphi_c,\tilde{\rho}_c)$ can change typically only at a “turning point.” Here the “turning point” $(\varphi_c^0,\tilde{\rho}_c^0)$ is a point where there exists a nontrivial vector $(\delta\varphi_c,\delta\tilde{\rho}_c)$ such that $$\begin{aligned}
\label{eq:tp1}
\delta \varphi_0 &=& \left(\frac{\partial \varphi_0}{\partial
\varphi_c}\right)_{\tilde{\rho}_c}
\delta \varphi_c + \left(\frac{\partial \varphi_0}{\partial
\tilde{\rho_c}}\right)_{\varphi_c}
\delta \tilde{\rho}_c=0,\\
\label{eq:tp2}
\delta N &=& \left(\frac{\partial N}{\partial \varphi_c}
\right)_{\tilde{\rho}_c}
\delta \varphi_c + \left(\frac{\partial N}{\partial \tilde{\rho_c}}
\right)_{\varphi_c}
\delta \tilde{\rho}_c=0.\end{aligned}$$ From Eqs. (\[eq:tp1\]) and (\[eq:tp2\]), $$\frac{\partial(\varphi_0,N)}{\partial(\varphi_c,\tilde{\rho}_c)}=0$$ at the turning point. Therefore the change of stability can be detected as envelopes of a family of curves $\tilde{\rho}_c=\mbox{const}$ in the $(\varphi_0, N)$ plane. Of course, this is also true for a family of curves $\varphi_c=\mbox{const}$.
\(ii) In order to specify an unstable branch at the turning point, we draw the sequence of equilibrium solutions in the $(\varphi_0,\omega)$ plane, maintaining $N$ constant. Then, as one proceeds along the curve in a counterclockwise direction, a branch beyond the turning point is unstable. This is also the case with the curve $(N,-\mu_{eff})$ with $\varphi_0$ constant. This is a direct consequence of theorem I of [@sorkin1982].
Here we describe the meaning of criterion (i) in the context of catastrophe theory. We regard the ADM energy $m$ as a function of three variables $\varphi_0$, $N$, and $\omega$. We take $\omega$ as a state variable. We consider an equilibrium space $$M_m=\left\{(\varphi_0,N,\omega)
\Biggm|\left(\frac{\partial m}{\partial \omega}
\right)_{\varphi_0,N}=0\right\}$$ and a control space $${\bf R}^2=\{(\varphi_0,N)\}.$$ We define a catastrophe map $$\begin{aligned}
\chi_m : M_m~ & & \longrightarrow~ {\bf R}^2, \nonumber \\
(\varphi_0, N, \omega)~ & & \longmapsto ~(\varphi_0,N). \end{aligned}$$ A point $P \in M_m$ is called a singular point of $\chi_m$ if the Jacobian of $\chi_m$ vanishes at $P$. A point $Q \in {\bf R}^2$ is called a singular value if there is at least one singular point in $\chi_m^{-1}(Q)$. A bifurcation set $B_m\subset {\bf R}^2$ is a set of singular values. At a singular point $P\in M_m$, a vector normal the tangent space of $M_m$, which is $$\left(\left(\frac{\partial^2 m}
{\partial\varphi_0\partial\omega}\right)
_{N,\omega},
\left(\frac{\partial^2 m}{\partial N\partial\omega}\right)
_{\omega,\varphi_0},
\left(\frac{\partial^2 m}{\partial\omega^2}\right)
_{\varphi_0,N}\right),$$ is parallel to the $\varphi_0 N$ plane. Therefore, the set of singular points, $\Sigma_m\subset M_m$, satisfies $$\Sigma_m=\left\{(\varphi_0,N,\omega)\Biggm|
\left( \frac{\partial m}{\partial \omega}
\right)_{\varphi_0,N}=\left(\frac{\partial^2 m}
{\partial \omega^2}
\right)_{\varphi_0,N}=0\right\},$$ and the bifurcation set $B_m \subset {\bf R}^2$ satisfies $$B_m = \left\{(\varphi_0,N)\Biggm|
\left( \frac{\partial m}{\partial \omega}
\right)_{\varphi_0,N}=\left(\frac{\partial^2 m}
{\partial \omega^2}
\right)_{\varphi_0,N}=0\right\}.$$ The envelopes of the family of the curves $\tilde{\rho}_c=\mbox{const} $ in the $(\varphi_0,N)$ plane form a bifurcation set $B_m$ of the catastrophe map $\chi_m$ because the Jacobian of $\chi_m $ vanishes at points on the envelopes. Criterion (i) says that a sequence of the equilibrium solutions can change its stability only at the points of the bifurcation set.
From criteria (i) and (ii), we examine the stability of the equilibrium solutions of neutron stars in scalar-tensor theory. From the turning point method alone, however, we cannot say that an equilibrium solution is [*stable*]{}. Therefore the stability of an equilibrium solution must be investigated by perturbation study [*once for all*]{}. For this purpose, we examine the case in which $\alpha(\varphi_0)=0$ and $m/r_s$ is sufficiently small ($\tilde{\rho}_c$ is sufficiently small). In this case, there is an equilibrium solution that is identical to that in general relativity. For this solution, the second-order variation of $m$ by regular, adiabatic, time-symmetric, and spherically symmetric perturbations with $\varphi_0$ and $N$ constant is $$\delta^2 m = \mbox{general relativistic terms} +
\frac{1}{2}
\int_{0}^{\infty} dr e^{-\nu/2}\left(1-\frac{2\mu}{r}\right)^{-1/2}
\zeta \left[-\frac{d^2}{dr_*^2}+V(r)\right]\zeta,$$ where $$\begin{aligned}
\zeta&\equiv& r\delta\varphi, \\
dr_* &\equiv& e^{\nu/2}\left(1-\frac{2\mu}{r}\right)^{1/2}dr, \\
V(r) &\equiv& \frac{1}{r}\left(1-\frac{2\mu}{r}\right)\left[
\frac{\nu^{\prime}}{2}-\frac{\mu^{\prime}r-\mu}{r(r-2\mu)}\right]
e^{\nu}-4\pi \beta_0 (-\tilde{\rho}+3\tilde{p})e^{\nu},\end{aligned}$$ and see Appendix B of [@htww1964] for the general relativistic terms. The general relativistic part is positive definite if $\Gamma>\Gamma_c$, and $\Gamma_c \to 4/3$ in the Newtonian limit [@chandrasekhar1964]. The second term is positive definite because the eigenvalues of the operator, $-d^2/dr_*^2+V$, are all positive for an arbitrary coupling function $A(\varphi)$ if $m/r_s$ is sufficiently small, which has been shown in [@harada1997]. Therefore, the general relativistic equilibrium solution in which the central density is sufficiently small is stable for the case $\alpha(\varphi_0)=0$ if $\Gamma > \Gamma_c \simeq 4/3$.
Results
=======
Hereafter we restrict our attention to the coupling function of the quadratic form $$\label{eq:quadratic}
A(\varphi)=\exp\left(\frac{1}{2}\beta\varphi^2\right).$$ For this model, the solar-system experiments constrain the present cosmological value of the scalar field through Eqs. (\[eq:constraint1\]) and (\[eq:constraint2\]) as $$|\varphi_0|\alt 0.032|\beta|^{-1}$$ and $$|\varphi_0|\alt\cases{
0.012(1+\beta)^{-1/2}|\beta|^{-1} & for $\beta>-1 $, \cr
0.029|1+\beta|^{-1/2}|\beta|^{-1} & for $\beta<-1 $, \cr
}$$ respectively. In particular, if $\varphi_0=0$, the post-Newtonian limit of this theory agrees completely with that of general relativity because $\alpha(\varphi_0)=0$.
$\beta\protect\agt-4.35$ case
-----------------------------
We present here the results of the case $\beta=-4$, but the features are basically common to the case $\beta\agt-4.35$. Figure 1 shows $\tilde{\rho}_c=\mbox{const}$ curves in the $(\varphi_0,N)$ plane, where the equilibrium solutions have been determined in the manner described in Sec. II. At a point on an envelope of the family of the curves seen in Fig. 1, the stability of the sequence of equilibrium solutions changes. Figure 2 shows the curves $(\varphi_0,\omega)$ with $N$ constant. In Fig. 2, the solid lines denote stable branches while the dotted lines denote unstable branches, where stability criteria (i) and (ii) are applied. Therefore, in region (A) in Fig. 1, only one stable equilibrium solution exists. For $\varphi_0=0$, this stable solution is identical to that of general relativity. In region (B) in Fig. 1, however, no stable solution exists. This is classified as the fold catastrophe in which the control space is two-dimensional. This catastrophe is elementary and structurally stable. Hence it is expected that this catastrophe structure is not changed by adding small higher-order terms to the exponent of the coupling function (\[eq:quadratic\]). The potential function $m$ is written locally around point $p(\varphi_{0p},N_{p})$ on the envelope (see Fig. 1) as, for $\varphi_{0p}>0$, $$\label{eq:fold}
m = \frac{A}{3}(\omega-\omega_{p})^3
+[B(\varphi_{0p}-\varphi_0)+B^{\prime}(N-N_{p})]
(\omega-\omega_{p})+m_p,$$ where $A$, $B$, and $B^{\prime}$ are some positive constants. Then the terms in the square brackets cancel out on the envelope, and are negative in region (A) and positive in region (B). For $\varphi_{0p}<0$, replace $\omega-\omega_p$ and $\varphi_{0p}-\varphi_0$ with $\omega_{p}-\omega$ and $\varphi_{0}-\varphi_{0p}$, respectively. For simplicity, we describe the behavior of the scalar charge for the case $\varphi_{0p}>0$. From Eq. (\[eq:fold\]), near point $p$, the scalar charge is given by the roots of the following quadratic equation: $$\left(\frac{\partial m}{\partial \omega}\right)_{\varphi_0,N}
=A (\omega-\omega_{p})^2 +
[B(\varphi_{0p}-\varphi_0)+B^{\prime}(N-N_{p})]=0.$$ The scalar charge is then given near point $p$ in region (A) by $$\omega = \omega_{p}\pm A^{-1/2}
[B(\varphi_{0}-\varphi_{0p})+B^{\prime}(N_{p}-N)]^{1/2},$$ where the upper sign denotes the stable branch and the lower denotes the unstable one. If a quadratic term in $(\omega-\omega_p)$ was involved in Eq. (\[eq:fold\]), the number of the roots of the equation $\partial m/\partial\omega=0$ did not change at point $p$. That is why Eq. (\[eq:fold\]) does not contain the quadratic term. The “scalar susceptibility” $\chi_{\varphi}$ is given near point $p$ by $$\chi_{\varphi}\equiv\left(\frac{\partial \omega}{\partial \varphi_0}
\right)_{N}
=\pm\frac{1}{2}A^{-1/2}B
[B(\varphi_{0}-\varphi_{0p})+B^{\prime}(N_{p}-N)]^{-1/2}.$$ The bifurcation set $B_m\subset {\bf R}^2$, which is the envelope seen in Fig. 1, is given by $$\begin{aligned}
\left(\frac{\partial m}{\partial \omega}\right)_{\varphi_0,N}
&=&A (\omega-\omega_{p})^2 +
[B(\varphi_{0p}-\varphi_0)+B^{\prime}(N-N_{p})]=0, \\
\left(\frac{\partial^2 m}{\partial \omega^2}\right)
_{\varphi_0,N}
&=& 2A(\omega-\omega_p)=0,\end{aligned}$$ i.e., $$\varphi_0=\frac{B^\prime}{B}(N-N_p)+\varphi_{0p},$$ near point $p$. This fold catastrophe appears also in general relativity in which $A(\varphi)=1$ identically. In general relativity, because of the absence of a gravitational scalar field, the control space is one-dimensional. For $\beta=-4$, the maximum ADM energy is greater than the general relativistic one for $\varphi_0\neq 0$. This is because, due to the presence of the scalar field, the effective gravitational constant becomes smaller and thereby gravity becomes weaker than in general relativity.
$\beta\protect\alt -4.35$ case
------------------------------
This case is more interesting than the above case. We present the results of the case $\beta=-6$. Figure 3 shows $\tilde{\rho}_c=\mbox{const}$ curves in the $(\varphi_0,N)$ plane. This figure is very different from Fig. 1. On the envelope of the family of curves, $e^{\prime}dcbab^{\prime}c^{\prime}de$, the sequence of equilibrium solutions changes its stability. Although there are other envelopes in region (B), they have nothing to do with the change of the number of stable equilibrium solutions. Figure 4 shows curves $(\varphi_0,\omega)$ with $N$ constant. From criteria (i) and (ii), the number of stable equilibrium solutions is as follows: In region (A), only one stable equilibrium solution exists. In region (B), two distinct stable equilibrium solutions exist. Surprisingly, these stable equilibrium solutions are different even for $\varphi_0=0$ from their counterparts in general relativity. For $\varphi_0=0$, the unstable solution agrees with the stable solution in general relativity. One of the two stable equilibrium solutions disappears on the envelope $dcbab^{\prime}c^{\prime}d$. In region (C), no stable equilibrium solution exists. Point $a$ is a bifurcation point for $\varphi_0=0$. This is seen in Fig. 5 which displays the curves $(\tilde{\rho_c},m)$ and $(\tilde{\rho_c},N)$ for $\varphi_0=0$, where the solid lines denote stable branches and the dotted lines denote unstable branches. In this figure, two stable branches are degenerate because, for $\varphi_0=0$, two stable equilibrium solutions are identical except for the sign of the scalar field. The equilibrium solution of the bifurcated stable branches is more compact for smaller mass but less compact for larger mass than the general relativistic sequence. Figure 6 shows the equilibrium space $M_m$ near point $a$. This type of the catastrophe at point $a$ is classified as the cusp catastrophe in which the control space is two-dimensional. The map $\chi_m$ is a cusp catastrophe map. This catastrophe is elementary and structurally stable, which suggests that this structure is stable against adding small higher order terms to the exponent of the coupling function (\[eq:quadratic\]). Point $a$ is called a cusp point.
We restrict our attention to cusp point $a(0,N_a)$. ($m_b N_a \simeq 1.24 M_{\odot}$ for $\beta=-6$.) The potential function $m$ is written around cusp point $a$ as $$\label{eq:cusp}
m=\frac{C}{4}\omega^4-\frac{D(N-N_a)}{2}\omega^2
-\varphi_0\omega+m_a,$$ where $C$ and $D$ are some positive constants. The reason why the coefficient of $\varphi_0 \omega$ is determined is that Eq. (\[eq:dermderphi0\]) holds. This form of expansion agrees with the usual Landau ansatz for a second-order phase transition, which has been used to explain spontaneous scalarization by Damour and Esposito-Farèse [@de1996]. The scalar charge is given by the roots of the following cubic equation: $$\label{eq:cube}
\left(\frac{\partial m}{\partial \omega}\right)
_{\varphi_0,N}=
C\omega^3-D(N-N_a)\omega-\varphi_0=0.$$ From Eq. (\[eq:cube\]), near cusp point $a$, the scalar charge is given by $$\omega=0,$$ for $N<N_a$ with $\varphi_0=0$. This is a stable branch. For $N>N_a$ with $\varphi_0=0$, $$\label{eq:critical}
\omega=\cases{
\pm\left(\frac{D}{C}\right)^{1/2}(N-N_a)^{1/2}
& for the stable branches, \cr
0 & for the unstable branch. \cr }$$ At point $a$ the stable equilibrium solution changes [*continuously*]{}, but its derivative with respect to $N$ is [*discontinuous*]{}. If Eq. (\[eq:cusp\]) involved a cubic term in $\omega$, the number of roots of the equation $\partial m/\partial \omega=0$ did change at point $a$. But this catastrophe was classified as the fold type and therefore not the case for point $a$ because of the shape of the bifurcation set seen in Fig. 3. That is why Eq. (\[eq:cusp\]) does not contain the cubic term. We also note that, for the case of two-dimensional control space, the structurally stable catastrophe is classified as either the fold or cusp type by Thom’s theorem. Therefore, at point $a$, a second-order phase transition occurs. If we fix $N$ to $N_a$, $$\omega= C^{-1/3}\varphi_0^{1/3}.$$ This is stable. From Eqs. (\[eq:cube\])-(\[eq:critical\]), with $\varphi_0=0$ near point $a$, it is derived that the scalar susceptibility $\chi_{\varphi}$ is given by $$\chi_{\varphi}=\cases{
D^{-1}(N_a-N)^{-1} & for $N<N_a$, \cr
\frac{1}{2}D^{-1}(N-N_a)^{-1} & for $N>N_a$. \cr}$$ Near point $a$ in region (A) in Fig. 3, only one real root of the cubic equation (\[eq:cube\]) corresponds to the stable equilibrium solution, while, in region (B), the smallest and largest roots of three real roots correspond to the stable equilibrium solutions and the intermediate root corresponds to the unstable one. If $\varphi_0>0$, the largest root corresponds to the globally stable one. If $\varphi_0<0$, the smallest root corresponds to the globally stable one. If $\varphi_0=0$, the two stable equilibrium solutions have identical ADM energies. The bifurcation set $B_m$, which is the envelope $b^{\prime}ab$, is given by $$\begin{aligned}
\left(\frac{\partial m}{\partial \omega}
\right)_{\varphi_0,N}&=&
C\omega^3-D(N-N_a)\omega-\varphi_0=0, \\
\left(\frac{\partial^2 m}{\partial \omega^2}
\right)_{\varphi_0,N}&=&
3C\omega^2-D(N-N_a)=0,\end{aligned}$$ i.e., $$\varphi_0=\pm\left(\frac{4D^3}{27C}\right)^{1/2}(N-N_a)^{3/2},$$ near point $a$. The cusp catastrophe has been named after this shape. On the envelope $dcbab^{\prime}c^{\prime}d$ except for points $a$ and $d$, one of the two distinct stable equilibrium solutions, the locally but not globally stable one, disappears, and hence a first-order phase transition occurs if the system obeys a perfect delay convention. On envelope $ede^{\prime}$, the stable equilibrium solution disappears. The catastrophic feature on the envelopes except for point $a$ is the fold catastrophe described in the last subsection. Point $d$ is not a cusp point but the intersection of two folds.
We should comment that, for the near critical case $-4.9\alt\beta\alt -4.35$, the behavior of the stable equilibrium solutions around the point of the maximum baryon number is somewhat complicated, although the structure of the cusp catastrophe at cusp point $a$ is not changed. Figures 7 and 8 show the curves $(\tilde{\rho}_c,N)$ with $\varphi_0=0$, for $\beta=-4.5$ and $-4.85$, respectively. For $-4.8\alt\beta\alt-4.35$, the number of stable equilibrium solutions changes as 1, 2 (degenerate in Fig. 7), 3, 1, 0 as the control parameter $N$ is increased continuously, as is seen in Fig. 7. For $-4.9\alt\beta\alt-4.8$, the number of stable equilibrium solutions changes as 1, 2 (degenerate in Fig. 8), 3, 2, 0 as $N$ is increased continuously, as is seen in Fig. 8. For $\beta\alt-4.6$ the maximum ADM energy with $\varphi_0=0$ is greater than that in general relativity, while, for $\beta\agt -4.6$, it is the same as that in general relativity.
The Kepler mass, which governs the Newtonian orbital motion of a test body, is not the ADM energy $m$ in general, but [@de1992] $$\tilde{\mu}=\frac{1+\alpha_0\alpha_A}{1+\alpha_0^2}m,$$ where $$\alpha_A\equiv\frac{\partial\ln m}{\partial\varphi}=
-\frac{\omega}{m}.$$ When we consider the case of $\alpha_0=\beta\varphi_0=0$, the Kepler mass is identical to the ADM energy. Therefore the argument above for $\varphi_0=0$ is also valid for the Kepler mass.
Here we present the physical interpretation as to why spontaneous scalarization occurs. In spite of the absence of the potential in the Lagrangian, the scalar field $\varphi$ obtains an effective potential term $W(\varphi)$ which satisfies $$\frac{\partial W}{\partial \varphi}=-4\pi \alpha(\varphi) T_*$$ because of the coupling with matter. Note that $T_*$ depends on $\varphi$. Then, if we consider $A(\varphi)$ of the form (\[eq:quadratic\]), $$\frac{\partial V}{\partial\varphi}=-4\pi \beta\varphi T_*,$$ and if $T_*= A^4(-\tilde{\rho}+3\tilde{p})$ is negative, $\varphi=0$ is an unstable stationary point of the effective potential, if $\beta<0$. On the other hand, the term from the spatial derivative in Eq. (\[eq:fieldeq2\]) has a contribution to stabilize the solution. By these two competing effects, the stability of the trivial configuration $\varphi=0$ against spontaneous scalarization is governed. For a detailed analysis of the stability of the trivial configuration, see [@harada1997].
If spontaneous scalarization occurs, the effective gravitational constant, which is $A^2(\varphi)=\exp(\beta\varphi^2)$ in the sense of the inverse of the Brans-Dicke scalar field, becomes considerably smaller than unity. Thereby the gravitation becomes weaker and a considerably larger mass than in general relativity can be supported by the lower matter pressure than in general relativity.
Summary and Discussions
=======================
The behavior of the equilibrium solutions of neutron stars in scalar-tensor theories of gravitation shows a catastrophic feature, which is characterized by a discontinuous change of the system. When we consider a function $A(\varphi)$ of the form $A(\varphi)=\exp(\frac{1}{2}\beta \varphi^2)$, the catastrophe types are classified as fold and cusp catastrophes. The appearance of the cusp catastrophe depends on whether $\beta\agt-4.35$ or $\beta\alt-4.35$. From the fact that those types of catastrophes are structurally stable, it is expected that they would be seen in a wide class of coupling functions.
For $\beta\agt-4.35$, the fold catastrophe on the two-dimensional control space does occur. The critical baryon number and critical ADM energy depend on $\varphi_0$. For a baryon number smaller than the critical one, one stable equilibrium solution exists, while, for a baryon number larger than the critical one, no stable equilibrium solution exists. In particular, for $\varphi_0=0$, the stable equilibrium solution is completely identical to that in general relativity. The behavior of the scalar charge and scalar susceptibility near the critical baryon number is explained by the form of the potential function of the fold catastrophe.
For $\beta\alt-4.35$, the cusp catastrophe does occur while the fold catastrophe also occurs. For $\beta\alt-4.9$, there is some critical value of the scalar field, $\varphi_0^{crit}>0$. If $|\varphi_0|>\varphi_0^{crit}$, there is only one critical number $N^{crit1}$ that depends on $\varphi_0$. For $N<N^{crit1}$, one stable equilibrium solution exists, while, for $N>N^{crit1}$, no stable equilibrium solution exists. If $0<|\varphi_0|<\varphi_0^{crit}$, there are three critical baryon numbers $N^{crit1}>N^{crit2}>N^{crit3}$. For $N<N^{crit3}$ or $N^{crit2}<N<N^{crit1}$, only one stable equilibrium solution exists. For $N^{crit3}<N<N^{crit2}$, two distinct stable equilibrium solutions exist and they do not agree with those in general relativity even for the limit $\varphi_0\to 0$. The almost general relativistic branch is unstable for $N>N^{crit2}$. For $N>N^{crit1}$, however, no stable equilibrium solution exists. If $\varphi_0=0$, the sequence of equilibrium solutions of neutron stars bifurcates at a point. Beyond this point, the general relativistic branch becomes unstable and another two (degenerate) sequences of equilibrium solutions far from the general relativistic one are stable. This bifurcation point is a cusp point, and the behavior of the scalar charge and scalar susceptibility near the cusp point is explained by the form of the potential function of the cusp catastrophe. At a point on the envelopes other than the cusp point, the fold catastrophe occurs. Since the critical baryon numbers $N^{crit1}$ and $N^{crit2}$ agree, the number of stable equilibrium solutions is 1 for $N<N^{crit3}$, 2 for $N^{crit3}<N<N^{crit2}=N^{crit1}$ and 0 for $N^{crit2}=N^{crit1}<N$. It should be noticed that, for the near critical case $-4.9\alt\beta\alt-4.35$, the structure of the cusp catastrophe does appear although the behavior becomes somewhat more complicated around the maximum baryon number for $\varphi_0\simeq 0$. This complicated feature agrees with the fact that the critical mass against zero-mode instability is not a monotonic function with respect to $\beta$, which is seen in Table I of [@harada1997].
Here we comment on the continuous change of the asymptotic value of the scalar field $\varphi_0$. If we identify $\varphi_0$ with the cosmological value of the scalar field, the evolution of $\varphi_0$ can be described by the equation of motion (\[eq:fieldeq2\]) in the Friedmann-Robertson-Walker universe. On the other hand, if we identify $\varphi_0$ with the value of the scalar field at the matching region in the $N$-compact-body problem, $\varphi_0$ should evolve due to the change of the density distribution around the neutron star. If the time scale of the variation of $\varphi_0$ is sufficiently longer than that of the local gravitational phenomena, such as the scalar gravitational wave radiation, the process due to the change of $\varphi_0$ can be regarded as quasistatic. Through the cosmological evolution of the scalar field $\varphi_0$, the neutron stars may collapse and radiate a scalar gravitational wave.
We also comment on the continuous change of $N$, which may be a result of a mass accretion onto the neutron star. If the baryon number of the neutron star exceeds the maximum value, the neutron star collapses and scalar gravitational waves are radiated and this is a candidate for the source of the scalar gravitational waves [@snn1994; @sst1995; @hcnn1997]. In a theory like the one of Fig. 7, there is a stable general relativistic neutron star that has the same baryon number and ADM energy within numerical accuracy as the maximum-mass non-general-relativistic neutron star has. Then, the transition of the non-general-relativistic neutron star to the general relativistic one due to a mass accretion occurs without any energy extraction.
Scalar-tensor theories of gravity naturally arise from the low-energy limit of string theory or other unified theories. For the moment, however, it is not clear how the scalar fields should couple to gravity (but see [@dp1994]). Experimental tests, such as binary pulsar timing observations, may constrain the way of coupling between the scalar fields and gravity. In particular, as for the case in which the single, massless scalar field couples to gravity with the coupling function $A(\varphi)=\exp[(1/2)\beta\varphi^2]$, Damour and Esposito-Farèse [@de1996] obtained the constraint on $\beta$ as $\beta\agt -5$, using the data of three binary pulsars. They showed that the occurrence of spontaneous scalarization makes it very difficult for the theory to maintain consistency with the results of binary pulsar timing experiments. The results obtained in this paper show that spontaneous scalarization is not an exceptional but robust phenomenon for the neutron star and common to a wide range of coupling functions. Gravitational experiments with high-precision and/or in a strong-field regime and gravitational wave observations may have the potential to constrain the way of coupling of the gravitational scalar fields and thereby we may catch a glimpse of string-scale physics.
I would like to thank T. Nakamura, M. Sasaki, Y. Eriguchi, N. Sugiyama, K. Nakao, M. Siino, T. Chiba, and M. Kaneko for useful discussions. I am also grateful to H. Sato for his continuous encouragement.
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0.3in
FIGURE CAPTION
0.05in
[Fig..]{}
A family of curves of $\tilde{\rho}_c=\mbox{const}$ in the $(\varphi_0,N)$ plane for the $\beta=-4$ case. The ordinate is $m_b N$ in place of the baryon number $N$. In region (A), only one stable equilibrium solution exists, while, in region (B), no stable equilibrium solution exists. At a point on the envelope of the family of the curves, the fold catastrophe occurs.
Curves $(\varphi_0,\omega)$ with $N$ constant for the $\beta=-4$ case. The number attached to each curve is $m_{b}N$ in the solar mass unit. The solid lines denote stable branches and the dotted lines denote unstable branches.
Same as Fig. 1, but for $\beta=-6$ case. In region (A), only one stable equilibrium solution exists. In region (B), two distinct stable equilibrium solutions exist. In region (C), no stable solution exists. Point $a$ is a cusp point. At point $a$, the cusp catastrophe occurs, while the fold catastrophe occurs at a point on the envelopes except for $a$.
Same as Fig. 2, but for the $\beta=-6$ case.
\(a) $(\tilde{\rho}_c,m)$ and (b) $(\tilde{\rho}_c,N)$ curves with $\varphi_0=0$ for $\beta=-6$. The solid lines denote stable branches, while the dotted lines denote unstable branches. The two distinct bifurcated branches are degenerate because they have identical ADM energies and baryon numbers but scalar fields of the opposite sign. The number of stable equilibrium solutions changes as 1, 2, 0 as $N$ increases.
Equilibrium space $M_m$ in the $(\varphi_0,N,\omega)$ space around cusp point $a$ for the $\beta=-6$ case. This structure of the equilibrium space is classified as the cusp catastrophe.
Same as Fig. 5(b), but for $\beta=-4.5$. The number of stable equilibrium solutions changes as 1, 2, 3, 1, 0 as $N$ increases. Two solutions are degenerate on the non-general-relativistic branches.
Same as Fig. 5(b), but for $\beta=-4.85$. The number of stable equilibrium solutions changes as 1, 2, 3, 2, 0 as $N$ increases. Two solutions are degenerate on the non-general-relativistic branches.
[^1]: Email address: harada@tap.scphys.kyoto-u.ac.jp
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abstract: |
This paper deals with probabilistic upper bounds for the error in functional estimation defined on some interpolation and extrapolation designs, when the function to estimate is supposed to be analytic. The error pertaining to the estimate may depend on various factors: the frequency of observations on the knots, the position and number of the knots, and also on the error committed when approximating the function through its Taylor expansion. When the number of observations is fixed, then all these parameters are determined by the choice of the design and by the choice estimator of the unknown function.
AMS (2010) Classification: 62K05, 41A5
author:
- 'Michel Broniatowski$^{1}$,Giorgio Celant$^{2}$, Marco Di Battista$^{2}$ , Samuela Leoni-Aubin$^{3}$'
- |
Michel Broniatowski$^{1}$,Giorgio Celant$^{2}$, Marco Di Battista$^{2}$ , Samuela Leoni-Aubin$^{3}$\
$^{1}$[Université Pierre et Marie Curie, LSTA,e-mail:michel.broniatowski@upmc.fr]{}\
$^{2}$[University of Padua, Department of Statistical Sciences,]{}\
$^{3}$[INSA Lyon, ICJ, e-mail: samuela.leoni@insa-lyon.fr]{}
title:
- Upper bounds for the error in some interpolation and extrapolation designs
- Upper bounds for the error in some interpolation and extrapolation designs
---
Introduction
=============
Consider a function $\varphi$ defined on some open set $D \subset\mathbb{R}$ and which can be observed on a compact subset $S$ included in $D$. The problem that we consider is the estimation of this function through some interpolation or extrapolation techniques. This turns out to define a finite set of points $s_{i}$ in a domain $\tilde{S}$ included in $S$ and the number of measurement of the function $\varphi$ at each of these points, that is to define a design $\mathcal{P}:=\left\{ \left( s_{i},n_{i}\right) \in S\times\mathbb{N}
, ~i=0,...,l,~\widetilde{S}\subsetneqq S\right\} $. The points $s_{i}$ are called the *knots*, $n_{i}$ is the frequency of observations at knot $s_{i}$ and $l+1$ is the number of knots. The choice of the design $\mathcal{P}$ is based on some optimality criterion. For example, we could choose an observation scheme that minimize the variance of the estimator of $\varphi$.
The choice of $\mathcal{P}$ has been investigated by many authors. Hoel and Levine and Hoel ([@10] and [@11]) considered the case of the extrapolation of a polynomial function with known degree in one and two variables. Spruill, in a number of papers (see [@17], [@18], [@19] and [@20]) proposed a technique for the (interpolation and extrapolation) estimation of a function and its derivatives, when the function is supposed to belong to a Sobolev space, Celant (in [@4] and [@5]) considered the extrapolation of quasi-analytic functions and Broniatowski-Celant in [@3] studied optimal designs for analytic functions through some control of the bias.
The main defect of any interpolation and extrapolation scheme is its extreme sensitivity to the uncertainties pertaining to the values of $\varphi$ on the knots. The largest the number $l+1$ of knots, the more unstable is the estimate. In fact, even when the function $\varphi$ is accurately estimated on the knots, the estimates of $\varphi$ or of one of its derivatives $\varphi^{(j)}$ at some point in $D$ may be quite unsatisfactory, due either to a wrong choice of the number of knots or to their location. The only case when the error committed while estimating the values $\varphi(s_{i})$ is not amplified in the interpolation procedure is the linear case. Therefore, for any more envolved case the choice of $l$ and $(s_{i}, n_{i} )$ must be handled carefully, which explains the wide literature devoted to this subject. For example, if we estimate $\varphi\left( v\right) ,v\in S\diagdown
\widetilde{S},$ by $\widehat{\varphi\left( s_{k}\right) }:=\varphi\left(
s_{k}\right) +\varepsilon\left( k\right) ,$ where $\varepsilon\left(
k\right) $ denotes the estimation error and $\widetilde{S}$ a Tchebycheff set of points $S $, we obtain $$\left\vert \varphi\left( v\right) -\widehat{\varphi\left( s_{k}\right)
}\right\vert \leq\left( \max_{k}\left\vert \varepsilon\left( k\right)
\right\vert \right) \Lambda_{l}\left( v,s_{k},0\right) ,$$ where $\Lambda_{l}\left( v,s_{i},j\right) $ is a function that depends on $\widetilde{S}$, the number of knots and on the order of the derivative that we aim to estimate (here $0$), and (see [@2] and [@14] ) $$\max_{k=0,...,l}\Lambda_{l}\left( v,s_{k},0\right) :=\frac{1}{l+1}\sum
_{k=0}^{l}ctg\left( \frac{2k+1}{4\left( l+1\right) }\pi\right) \sim
\frac{2}{\pi}\ln\left( l+1\right) \quad\text{ when }l\rightarrow\infty.$$ If equidistant knots are used, one gets (see [@16]) $$\max_{k=0,...,l}\Lambda_{l}\left( v,s_{k},0\right) \thicksim\frac{2^{l+1}}{el\left( \ln l+\gamma\right) }, \qquad\gamma
=0,577~~\text{(Euler-Mascheroni constant).}$$
When the bias in the interpolation is zero, as in the case when $\varphi$ is polynomial with known degree, the design is optimized with respect to the variance of the interpolated value (see [@10] ). In the other cases the criterion that is employed is the minimal MSE criterion. The minimal MSE criterion allows the estimator to be as accurate as possible but it does not yield any information on the interpolation/extrapolation error.
In this paper, we propose a probabilistic tool (based on the concentration of measure) in order to control the estimation error. In Section 2 we present the model, the design and the estimators. Section 3 deals with upper bouns for the error. Concluding remarks are given in Section 4.
The model, the design and the estimators
========================================
Consider an unknown real-valued analytic function $f$ defined on some interval $D$ : $$\begin{aligned}
f:~~ D:=\left( a,b\right) & \rightarrow\mathbb{R}\\
v & \mapsto f\left( v\right) .\end{aligned}$$
We assume that this function is observable on a compact subset $S$ included in $D$, $S:=\left[ \underline{s},\overline{s}\right] \subset D$, and that its derivatives are not observable at any point of $D$. Let $\widetilde{S}:=\left\{ s_{k\text{ }}\in\widetilde{S},k=0,...,l\right\} $ be a finite subset of $l+1$ elements in the set $S$. The points $s_{k}$ are called the *knots*.
Observations $Y_{i}$, $i=1, \ldots,n$ are generated from the following location-scale model $$\begin{aligned}
Y_{j}\left( s_{k}\right) = & f\left( s_{k}\right) +\sigma E\left(
Z_{j}\right) +\varepsilon_{j} ,\\
\varepsilon_{j} := & \sigma Z_{j}-\sigma E\left( Z_{j}\right) , \qquad
j=1,...,n_{k},~~k=0,...,l,\end{aligned}$$ where $Z$ is a completely specified continuous random variable, the location parameter $f\left( v\right) $ and the scale parameter $\sigma>0
$ are unknown parameters. $E\left( Z\right) ,\varsigma$ respectively denote the mean and the variance of $Z$, and $n_{k}$ is the *frequency of observations* at knot $s_{k}$.
We assume to observe $(l+1)$ i.i.d. samples, $\underline{Y}\left(
k\right) :=\left( Y_{1}\left( n_{k}\right) ,...,Y_{n_{k}}\left(
n_{k}\right) \right) ,k=0,...,l, $ and $Y_{i}\left( n_{k}\right) $ i.i.d. $Y_{1}\left( n_{k}\right) ,$ for all $i\neq k$, $i= 0, \ldots, l$.
The aim is to estimate a derivative of $f\left( v\right) $, $f^{(d)}(v)$, $d \in\mathbb{N}$, at a point $v \in\left( a,\overline
{s}\right) $.
Let $\varphi\left( v\right) :=f\left( v\right) +\sigma E\left(
Z\right) $, and consider the Lagrange polynomial $$L_{s_{k}}\left( v\right) :=\prod_{j\neq k,j=0}^{l}\frac{v-s_{j}}{s_{k}-s_{j}}.$$
We are interested in interpolating (or extrapolating) some derivatives of $\varphi$, $\varphi^{(d)}$, with $d \in\mathbb{N}$, $$\mathcal{L}_{l}\left( \varphi^{\left( d \right) }\right) \left( v\right)
:=\sum_{k=0}^{l}\varphi\left( s_{k}\right) L_{s_{k}}^{\left( d \right)
}\left( v\right) .$$
The domain of extrapolation is denoted $U := D\diagdown S$. It is convenient to define a generic point $v \in D$ stating that it is an *observed point* if it is a knot, an *interpolation point* if $v \in
S$ and an *extrapolation point* if $v \in U$.
For all $d \in\mathbb{N}$, for any $v \in S$, the Lagrange interpolation scheme converges for the function $\varphi^{(d)}$, that is, for $l \to\infty$, $$\mathcal{L}_{l}\left( \varphi^{\left( d \right) }\right) \left( s\right)
\rightarrow\varphi^{\left( d \right) }\left( s\right) , \quad\forall s\in
S.$$
Interpolating the derivative $\varphi^{\left( d+i\right) }\left(
s^{\ast}\right) $ at a point $s^{\ast}\in S$ opportunely chosen, a Taylor expansion with order $(m-1)$ of $\varphi^{\left( d \right) } (v)$ at point $v$ from $s^{\ast}$ gives $$T_{\varphi^{\left( d \right) },m,l}\left( v\right) :=\sum_{i=0}^{m-1}\frac{\left( v-s^{\ast}\right) }{i!}^{i}\mathcal{L}_{l}\left(
\varphi^{\left( d +i\right) }\right) \left( s^{\ast}\right) , \quad
s^{\ast}\in S,$$ and we have $$\lim_{m\rightarrow\infty}\lim_{l\rightarrow\infty}T_{\varphi^{\left( d
\right) },m,l}\left( v\right) =\varphi^{\left( d \right) }\left(
v\right) , \quad\forall v\in D.$$
When $\varphi^{(d)} \in\mathcal{C}^{\alpha}(D) $, $\forall\alpha$, $l\geq2\alpha-3$, the upper bound for the error of approximation is given in [@1], $$E_{t}:=\sup_{v \in D}\left\vert \varphi^{\left( d \right) }\left( v\right)
-T_{\varphi^{\left( d \right) },m,l}\left( v\right) \right\vert \leq
M(m,l, \alpha),$$
where $M(m,l, \alpha) =A\left( \alpha,l\right) +B\left( m\right)
$, $$A\left( \alpha,l\right) :=K\left( \alpha,l\right) \sum_{i=0}^{m-1}\left(
\sup_{s\in S}\left\vert \varphi^{\left( d+i+\alpha\right) }\left( s\right)
\right\vert \frac{1}{i!}\sup_{v\in U}\left\vert v-s^{\ast}\right\vert
^{i}\right) ,$$ $$K\left( \alpha,l\right) :=\left( \frac{\pi}{2\left( 1+l\right) }\left(
\overline{s}-\underline{s}\right) \right) ^{\alpha}\left( 9+\frac{4}{\pi
}\ln\left( 1+l\right) \right) ,$$ $$\text{and } \quad B\left( m\right) :=\sup_{v\in\left( a,\overline
{s}\right) }\left( \frac{\left\vert u-s^{\ast}\right\vert ^{m}\left\vert
\varphi^{\left( d+\alpha\right) }\left( v\right) \right\vert }{m!}\right)
.$$
The optimal design writes $\left\{ \left( n_{k},s_{k }\right)
\in\left( \mathbb{N}\setminus\{0\} \right) ^{l+1} \times\mathbb{R} ^{l+1}, ~
n:=\sum_{k=0}^{l}n_{k}, ~ n\text{ fixed}\right\} $, where $n$ is the total number of experiments and the $(l+1)$ knots are defined by $$s_{k}:=\frac{\overline{s}+\underline{s}}{2}-\frac{\overline{s}-\underline{s}}{2}\cos\frac{2k-1}{2l+2}\pi, \qquad k=0, \ldots, l,$$ with $n_{k}:=\left[ \frac{n\sqrt{{P}_{k}}}{\sum_{k=0}^{l}\sqrt{{P}_{k}}}\right] $, $\left[ .\right] $ denoting the integer part function, and (see [@3] for details) $$P_{k}:=\left\vert \sum_{\beta=0}^{m}\sum_{\alpha=0}^{m}\frac{\left(
u-s\right) ^{\alpha+\beta}}{\alpha!\beta!}L_{s_{k}}^{\left( \alpha\right)
}\left( s\right) L_{s_{k}}^{\left( \beta\right) }\left( s\right)
\right\vert , \qquad k=0,...,l.$$
The function $\varphi$ cannot be observed exactly at the knots. Let $\widehat{\varphi\left( s_{k}\right) }$ denote the least squares estimate of $\varphi\left( s_{k}\right) $ at the knot $s_{k}$ and $$\label{lagsch}\mathcal{L}_{l}\left( \widehat{\varphi^{\left( d+i\right) }}\right) \left( v\right) :=\sum_{k=0}^{l}\widehat{\varphi\left(
s_{k}\right) }L_{s_{k}}^{\left( d+i\right) }\left( v\right) .$$ We estimate the $d -$th derivative of $\varphi\left( v\right) $ at $v\in D$ as follows $$\widehat{T}_{\varphi^{\left( d \right) },m,l}\left( v\right) :=\sum
_{i=0}^{m-1}\frac{\left( v-s^{\ast}\right) }{i!}^{i}\mathcal{L}_{l}\left(
\widehat{\varphi^{\left( d +i\right) }}\right) \left( s^{\ast}\right) ,
\qquad s^{\ast}\in S.$$
The knots $s_{k}$ are chosen in order to minimize the variance of $\widehat{T}_{\varphi^{\left( d \right) },m,l}\left( v\right) $ and it holds $$\lim_{m\rightarrow\infty}\lim_{l\rightarrow\infty}{\lim}_{\min_{k=0, \ldots,
l} \left( n_{k}\right) \rightarrow\infty}\widehat{T}_{\varphi^{\left( d
\right) },m,l}\left( v\right) =\varphi^{\left( d \right) }\left(
v\right) , \qquad\forall v \in D.$$ $\widehat{T}_{\varphi^{\left( d \right) },m,l}\left( v\right) $ is an extrapolation estimator when $v\in U$ and an interpolation estimator when $v\in S.$
For a fixed degree $l$ of the Lagrange scheme (\[lagsch\]), the total error committed while substituting $\varphi^{\left( d \right) }\left(
v\right) $ by $\widehat{T}_{\varphi^{\left( d \right) },m,l}\left(
v\right) $ writes $$E_{Tot}\left( \varphi^{\left( d \right) }\left( v\right) \right)
:=\varphi^{\left( d \right) }\left( v \right) -\widehat{T}_{\varphi
^{\left( d \right) },m,l}\left( v\right) .$$
For the interpolation error concerning $\varphi^{\left( i+ d
\right) }$, we have the following result presented in [@6], p.293 : if $\varphi^{\left( i+ d \right) }\in\mathcal{C}^{ \alpha}\left( S\right) $, $\forall\alpha$, $l\geq2 \alpha-3$, then $$\sup_{s \in S}\left\vert \varphi^{\left( d +i\right) }\left( s\right)
-\mathcal{L}_{l}\left( \varphi^{\left( d +i\right) }\right) \left(
s\right) \right\vert \leq M_{1}:=K\left( \alpha,l\right) \sup_{s\in
S}\left\vert \varphi^{\left( d +i+ \alpha\right) }\left( s\right)
\right\vert .$$
This error depends on the very choice of the knots and is controlled through a tuning of $l$.
The error due to the Taylor expansion of order $\left( m-1\right)
$ $$\varphi^{\left( d \right) }\left( v\right) -\sum_{i=0}^{m-1}\frac{\left(
v-s^{\ast}\right) }{i!}^{i}\varphi^{\left( d +i\right) }\left( s^{\ast
}\right)$$ depends on $s^{\ast}$, it is a truncation error and it can be controlled through a tuning of $m$.
Let $\widehat{\varphi(s_{k})}$ be an estimate of $\varphi(s_{k})$ on the knot $s_{k}$ and $$\varepsilon\left( k\right) :=\varphi\left( s_{k}\right) -\widehat
{\varphi\left( s_{k}\right) }, \qquad k=0,...,l$$ denote the error pertaining to $\varphi(s_{k})$ due to this estimation. $\varepsilon\left( k\right) $ clearly depends on $n_{k}$, the frequency of observations at knot $s_{k}$.
Finally, when $n$ is fixed, the error committed while extrapolating depends on the design $\{ (n_{k} , s_{k}) \in\left( \mathbb{N}\setminus\{ 0
\} \right) ^{l+1} \times\mathbb{R}^{l+1}, ~ k=0, \ldots, l, ~n = \sum
_{k=0}^{l} n_{k} \}$, on $m$ and on $l$.
Without loss of generality, we will assume $\sigma=1$. In this case we have $\widehat{\varphi\left( s_{k}\right) }$ =$\overline{Y}\left(
s_{k}\right) :=\frac{\sum_{j=1}^{n_{k}}Y_{j}\left( k\right) }{n_{k}}$. The general case when $\sigma$ is unknown is described in [@3].
In the next Section we will provide upper bounds for the errors in order to control them.
Since $\varphi$ is supposed to be an analytic function, we can consider the extrapolation as an analytic continuation of the function out of the set $S$ obtained by a Taylor expansion from an opportunely chosen point $s^{\ast}$ in $S.$ So, the extrapolation error will depend on the order of the Taylor expansion and on the precision in the knowledge of the derivatives of the function at $s^{\ast}$. This precision is given by the interpolation error and by the estimation errors on the knots. The analyticity assumption also implies that the interpolation error will quickly converge to zero. Indeed, for all integer $r$, the following result holds: $$\lim_{l\rightarrow\infty}l^{r}\sup_{s\in S}\left\vert \varphi^{\left(
j\right) }\left( s\right) -\sum_{k=0}^{l}L_{s_{k}}^{\left( j\right)
}\left( s\right) \varphi\left( s_{k}\right) \right\vert =0.$$
We remark that the instability of the interpolation and extrapolation schemes discussed by Runge (1901) can be avoided if the chosen knots form a Tchebycheff set of points in $S$, or if they form a Feteke set of points in $S$, or by using splines.
Note that in all the works previously quoted the function is supposed to be polynomial with known degree (in [@10] and [@11]), to belongs to a Sobolev space (see [@17], [@18], [@19] and [@20]), or to be quasi analytic (in [@4] and [@5]), or analytic (in [@3]). Moreover, $\widetilde{S}$ is chosen as a Tchebycheff set of points in $S$ .
Bernstein in [@2] affirmed that polynomials of low degree are good approximations for analytic functions. In the case of the Broniatowski-Celant design ([@3]), the double approximation to approach $\varphi$ allows to choose any subset of $S$ as possible interpolation set. So, if the unknown function is supposed to be analytic, then we can choose a small interpolation set in order to obtain a small interpolation error.
Upper bounds and control of the error
=====================================
The extrapolation error depends on three kinds of errors: truncation error, interpolation error and error of estimation of the function on the knots. In order to control the extrapolation error, we split an upper bound for it in a sum of three terms, each term depending only on one of the three kinds of errors.
In the sequel, we will distinguish two cases: in the first case, we suppose that the observed random variable $Y$ is bounded, in the second case $Y$ is supposed to be a random variable with unbounded support. We suppose that the support is known.
Case 1: $Y$ is a bounded random variable
----------------------------------------
If $\tau_{1},\tau_{2}$ (assumed known) are such that $\Pr\left( \tau_{1}\leq
Y\leq\tau_{2}\right) =1$, it holds $\left\vert \varphi\left( v\right)
\right\vert \leq R$, where $R:=\max\left\{ \left\vert \tau_{1}\right\vert
,\left\vert \tau_{2}\right\vert \right\} .$ Indeed, $E\left( Y\right)
=\varphi\in\left[ -R,R\right] .$ Let $$\varepsilon\left( k\right) :=\frac{\sum_{j=1}^{n_{k}}Y_{j}\left( k\right)
}{n_{k}}-\varphi\left( s_{k}\right) .$$
The variables $Y_{j}\left( k\right) ,\forall j=1,...,n_{k},\forall
k=0,..,l,$ are i.i.d., with the same bounded support and for all $k$$,
E\left( Y_{j}\left( k\right) \right) =$ $\varphi\left( s_{k}\right) $, hence we can apply the Hoeffding’s inequality (in [@9]): $$\Pr\left\{ \left\vert \varepsilon\left( k\right) \right\vert \geq
\rho\right\} \leq2\exp\left( -\frac{2\rho^{2}n_{k}}{\left( \tau_{2}-\tau_{1}\right) ^{2}}\right) .$$
In Proposition 1, we give an upper bound for the extrapolation error denoted by $E_{ext}$. This bound is the sum of the three terms, $M_{Taylor}$, controlling the error associated to the truncation of the Taylor expansion which defines $\varphi^{\left( d\right) }$, $M_{interp}$, controlling the interpolation error and $M_{est}$, describing the estimation error on the knots.
For all $\alpha\in\mathbb{N} \setminus\{ 0 \} $, if $\varphi^{\left(
i+d\right) }\in\mathcal{C}^{\alpha} \left( a,b\right) $, $l\geq2\alpha-3$, then, $\forall u \in U$, $\left\vert E_{ext}\left( u\right) \right\vert \leq
M_{Taylor}+M_{interp}+M_{est}$, where $$M_{Taylor}:=R\frac{\left( d+m\right) !}{m!}\left( \frac{s^{\ast}-u}{b-a}\right) ^{m}\frac{1}{\left( b-a\right) ^{d}},$$ $$K\left( l,\alpha\right) :=\left( 9+\frac{4}{\pi}\ln\left( 1+l\right)
\right) \left( \frac{\pi}{2\left( 1+l\right) }\right) ^{\alpha},$$ $$M_{interp}:=K\left( l,\alpha\right) \frac{R}{\left( \overline{s}-\underline{s}\right) ^{d+\alpha}}\sum_{i=0}^{m-1}\left( \frac{s^{\ast}-u}{\overline{s}-\underline{s}}\right) ^{i}\frac{\left( d+i+\alpha\right)
!}{i!},$$ $$\Lambda\left( l,m\right) :=\sum_{i=0}^{m-1}\sum_{k=0}^{l}\frac{\left(
s^{\ast}-u\right) ^{i}}{i!}\left\vert L_{s_{k}}^{\left( d+i\right) }\left(
s^{\ast}\right) \right\vert ,$$ $$M_{est}:=\Lambda\left( l,m\right) \left( \max_{k=0,...,l}\left\vert
\varepsilon\left( k\right) \right\vert \right) .$$
By using the Cauchy’s Theorem on the derivatives of the analytic functions, we obtain $$\left\vert \varphi^{\left( d\right) }\left( u\right) -\widehat
{\varphi^{\left( d\right) }\left( u\right) }\right\vert =\left\vert
\varphi^{\left( d\right) }\left( u\right) +\sum_{i=0}^{m-1}\frac
{\varphi^{\left( d+i\right) }\left( s^{\ast}\right) }{i!}\left(
u-s^{\ast}\right) ^{i}-\sum_{i=0}^{m-1}\frac{\varphi^{\left( d+i\right)
}\left( s^{\ast}\right) }{i!}\left( u-s^{\ast}\right) ^{i}-\widehat
{\varphi^{\left( d\right) }\left( u\right) }\right\vert$$$$\leq\left\vert \varphi^{\left( d\right) }\left( u\right) -\sum_{i=0}^{m-1}\frac{\varphi^{\left( d+i\right) }\left( s^{\ast}\right) }{i!}\left( u-s^{\ast}\right) ^{i}\right\vert +\left\vert \sum_{i=0}^{m-1}\frac{\varphi^{\left( d+i\right) }\left( s^{\ast}\right) }{i!}\left( u-s^{\ast}\right) ^{i}-\widehat{\varphi^{\left( d\right)
}\left( u\right) }\right\vert$$$$\leq\frac{\sup_{v\in U}\left\vert \varphi^{\left( d+m\right) }\left(
v\right) \right\vert }{m!}\left( s^{\ast}-u\right) ^{m}+\left\vert
\sum_{i=0}^{m-1}\frac{\varphi^{\left( d+i\right) }\left( s^{\ast}\right)
}{i!}\left( u-s^{\ast}\right) ^{i}-\sum_{i=0}^{m-1}\frac{\widehat
{\varphi^{\left( d+i\right) }\left( s^{\ast}\right) }}{i!}\left(
u-s^{\ast}\right) ^{i}\right\vert$$$$\leq\frac{R\left( m+d\right) !}{\left( b-a\right) ^{d}m!}\left(
\frac{s^{\ast}-u}{b-a}\right) ^{m}+\left\vert \sum_{i=0}^{m-1}\frac{\left(
s^{\ast}-u\right) ^{i}}{i!}\left( \varphi^{\left( d+i\right) }\left(
s^{\ast}\right) -\widehat{\varphi^{\left( d+i\right) }\left( s^{\ast
}\right) }\right) \right\vert$$$$\leq\frac{R\left( m+d\right) !}{\left( b-a\right) ^{d}m!}\left(
\frac{s^{\ast}-u}{b-a}\right) ^{m}+\sum_{i=0}^{m-1}\frac{\left( s^{\ast
}-u\right) ^{i}}{i!}\left\vert \varphi^{\left( d+i\right) }\left( s^{\ast
}\right) -\widehat{\varphi^{\left( d+i\right) }\left( s^{\ast}\right)
}\right\vert$$$$\leq M_{Taylor}+\sum_{i=0}^{m-1}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}\left\vert
\begin{array}
[c]{c}\varphi^{\left( d+i\right) }\left( s^{\ast}\right) -\sum_{k=0}^{l}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right) \varphi\left(
s_{k}\right) \\
+\sum_{k=0}^{l}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right)
\varphi\left( s_{k}\right) -\widehat{\varphi^{\left( d+i\right) }\left(
s^{\ast}\right) }\end{array}
\right\vert$$$$\begin{aligned}
& \leq M_{Taylor}+\sum_{i=0}^{m-1}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}\left\vert \varphi^{\left( d+i\right) }\left( s^{\ast}\right)
-\sum_{k=0}^{l}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right)
\varphi\left( s_{k}\right) \right\vert +\\
& \sum_{i=0}^{m-1}\sum_{k=0}^{l}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right) \left\vert
\varphi\left( s_{k}\right) -\overline{Y}\left( k\right) \right\vert\end{aligned}$$$$\begin{aligned}
& \leq M_{Taylor}+\sum_{i=0}^{m-1}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}K\left( l,\alpha\right) \left( \sup_{s\in S}\left\vert \varphi^{\left(
d+i+\alpha\right) }\left( s\right) \right\vert \right) \\
& +\sum_{i=0}^{m-1}\sum_{k=0}^{l}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right) \left\vert
\varphi\left( s_{k}\right) -\overline{Y}\left( k\right) \right\vert\end{aligned}$$$$\begin{aligned}
& \leq M_{Taylor}+\frac{R}{\left( \overline{s}-\underline{s}\right)
^{d+\alpha}}\sum_{i=0}^{m-1}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}K\left(
l,\alpha\right) \frac{\left( d+i+\alpha\right) !}{\left( \overline
{s}-\underline{s}\right) ^{i}}\\
& +\sum_{i=0}^{m-1}\sum_{k=0}^{l}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right) \left\vert
\varphi\left( s_{k}\right) -\overline{Y}\left( k\right) \right\vert\end{aligned}$$$$\leq M_{Taylor}+M_{interp}+\sum_{i=0}^{m-1}\sum_{k=0}^{l}\frac{\left(
s^{\ast}-u\right) ^{i}}{i!}L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast
}\right) \left\vert \varphi\left( s_{k}\right) -\overline{Y}\left(
k\right) \right\vert$$$$\begin{aligned}
& \leq M_{Taylor}+M_{interp}+\left( \max_{k=0,...,l}\left\vert
\varepsilon\left( k\right) \right\vert \right) \sum_{i=0}^{m-1}\sum
_{k=0}^{l}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}\left\vert L_{s_{k}}^{\left( d+i\right) }\left( s^{\ast}\right) \right\vert \\
& =M_{Taylor}+M_{interp}+M_{est}.\end{aligned}$$
Proposition 2 yields the smallest integer such that the error of estimation is not greater than a chosen threshold with a fixed probability.
$\forall\eta\in\left[ 0,1\right] ,\forall\rho\in\mathbb{R} ^{+},\exists
n\in\mathbb{N} $ such that $$Pr \left( \max_{k=0,...,l}\left\vert \varepsilon\left( k\right) \right\vert
\geq\frac{\rho}{\Lambda\left( l,m\right) }\right) \leq\eta.$$
If, $\forall k$ $\left\vert \varepsilon\left( k\right) \right\vert \geq
\frac{\rho}{\Lambda\left( l,m\right) }$, then $\max_{k=0,...,l}\left\vert
\varepsilon\left( k\right) \right\vert \geq\frac{\rho}{\Lambda\left(
l,m\right) }.$ We have $$\ \Pr\left( \max_{k=0,...,l}\left\vert \varepsilon\left( k\right)
\right\vert \geq\frac{\rho}{\Lambda\left( l,m\right) }\right) \leq
\prod_{k=0}^{l}\Pr\left( \left\vert \varepsilon\left( k\right) \right\vert
\geq\frac{\rho}{\Lambda\left( l,m\right) }\right) \leq\prod_{k=0}^{l}2\exp\left( -\frac{2\rho^{2}}{\left( \Lambda\left( l,m\right) \right)
^{2}}n_{k}\right) .$$
So, we can choose $$n^{\ast}=\left[ \frac{\left( l+1\right) \ln2-\ln\eta}{2}\left(
\frac{\Lambda\left( l,m\right) \left( \tau_{2}-\tau_{1}\right) }{\rho
}\right) ^{2}\right] .$$
Proposition 3 gives an upper bound for the extrapolation error that depends on $\left( l,m,n\right) $. We recall that the number of knots $l+1$ controls the interpolation error, $m$ denotes the number of terms used in the Taylor expansion for $\varphi^{\left( d\right) }$ and $n$ is the total number of observations used to estimate $\varphi\left( s_{k}\right)
,k=0,..,l $. Hence $n$ controls the total estimation error.
With the same hypotheses and notations, we have that $$\forall\left( \rho_{m},\rho_{l},\rho_{n}\right) \in\mathbb{R}
( \mathbb{R} ^{+})^{3}, \quad\left\vert E_{ext}\left( u\right) \right\vert
\leq\rho_{m}+\rho_{l}+\rho_{n}$$ with probability $\eta$. $\eta$ depends on the choice of $\left(
\rho_{m},\rho_{l},\rho_{n}\right) $, which depends on $\left( m,l,n\right)
.$
When $\left( \rho_{m},\rho_{l}\right) $ is fixed , we can choose $\left(
m,l\right) $ as the solution of the system: $$\left( M_{Taylor},M_{interp}\right) =\left( \rho_{m},\rho_{l}\right) .$$ We end the proof by taking $\rho_{n}=\frac{\rho}{\Lambda\left(
l,m\right) }$ and $n=n^{\ast}$.
In the case of the estimation of $\varphi\left( u\right) $ (i.e., when $d=0$) we obtain for the couple $\left( m,n\right) $ the explicit solution $$m=\frac{\ln\rho_{m}-\ln R}{\ln\left( s^{\ast}-u\right) -\ln\left(
b-a\right) },$$ $$n=\left[ \frac{\left( l+1\right) \ln2-\ln\eta}{2}\left( \frac
{\Lambda\left( l\right) \left( \tau_{2}-\tau_{1}\right) }{\rho}\right)
^{2}\right] ,\Lambda\left( l\right) =\sum_{i=0}^{m-1}\sum_{k=0}^{l}\frac{\left( s^{\ast}-u\right) ^{i}}{i!}\left\vert L_{s_{k}}^{\left(
i\right) }\left( s^{\ast}\right) \right\vert .$$
When $l\geq2\alpha-3$, $l$ is the solution of the equation $$\text{ }\rho_{l}=\left( 9+\frac{4}{\pi}\ln\left( 1+l\right) \right)
\left( \frac{\pi}{2\left( 1+l\right) }\right) ^{\alpha}\frac{R}{\left(
\overline{s}-\underline{s}\right) ^{\alpha}}\sum_{i=0}^{m-1}\left(
\frac{s^{\ast}-u}{\overline{s}-\underline{s}}\right) ^{i}\frac{\left(
i+\alpha\right) !}{i!}.$$
Theorem 4, due to Markoff, provides an uniform bound for the derivatives of a Lagrange polynomial.
(Markoff) Let $P_{l}\left( s\right) :=\sum_{j}a_{j}s^{j}$ be a polynomial with real coefficients and degree $l$. If $\sup_{s\in S}\left\vert
P_{l}\left( s\right) \right\vert \leq W,$ then for all $s$ in $intS$ and for all $l$ in $\mathbb{N} ,$ it holds $$\left\vert P_{l}^{\left( j\right) }\left( s\right) \right\vert \leq
\frac{l^{2}\left( l^{2}-1\right) ...\left( l^{2}-\left( j-1\right)
^{2}\right) }{\left( 2j-1\right) !!}\left( \frac{2}{\left( \overline
{s}-\underline{s}\right) }\right) ^{j}W.$$
When applied to the elementary Lagrange polynomial, it is readily checked that $W=\pi$. Indeed, $$\left\vert L_{s_{k}}\left( s\right) \right\vert =\left\vert \frac{\left(
-1\right) ^{k}\sin\left( \frac{2k-1}{2l+2}\pi\right) }{l+1}\frac
{\cos\left( \left( l+1\right) \theta\right) }{\cos\theta-\cos\left(
\frac{2k-1}{2l+2}\pi\right) }\right\vert \leq$$ $$\leq\frac{\left\vert \sin\left( \frac{2k-1}{2l+2}\pi\right) \right\vert
}{l+1}\frac{\left\vert \cos\left( \left( l+1\right) \theta\right)
\right\vert }{\left\vert \cos\theta-\cos\left( \frac{2k-1}{2l+2}\pi\right)
\right\vert }\leq$$ $$\leq\frac{\left\vert \sin\left( \frac{2k-1}{2l+2}\pi\right) \right\vert
}{l+1}\frac{\left( l+1\right) \left\vert \theta-\frac{2k-1}{2l+2}\pi\right\vert }{\frac{1}{\pi}\sin\left( \frac{2k-1}{2l+2}\pi\right)
\left\vert \theta-\frac{2k-1}{2l+2}\pi\right\vert }=\pi.$$
We used $$\left\vert \cos\left( \left( l+1\right) \theta\right) \right\vert
=\left\vert \cos\left( \left( l+1\right) \theta\right) -\cos\left(
\left( l+1\right) \frac{2k-1}{2l+2}\pi\right) \right\vert \leq\left(
l+1\right) \left\vert \theta-\frac{2k-1}{2l+2}\pi\right\vert$$ and $\cos\left( \left( l+1\right) \frac{2k-1}{2l+2}\pi\right)
=0$. Moreover, $$\left\vert \cos\theta-\cos\left( \frac{2k-1}{2l+2}\pi\right) \right\vert
=2\sin\left( \frac{\theta+\frac{2k-1}{2l+2}\pi}{2}\right) \left\vert
\sin\left( \frac{\theta-\frac{2k-1}{2l+2}\pi}{2}\right) \right\vert .$$ The concavity of the sine function on $\left[ 0,\pi\right] $ implies $$\sin\left( \frac{\theta+\frac{2k-1}{2l+2}\pi}{2}\right) \geq\frac{1}{2}\left( \sin\theta+\sin\left( \frac{2k-1}{2l+2}\pi\right) \right)$$ $$\left\vert \sin\left( \frac{\theta-\frac{2k-1}{2l+2}\pi}{2}\right)
\right\vert \geq\frac{2}{\pi}\left\vert \theta-\frac{2k-1}{2l+2}\pi\right\vert
,\theta\in\left[ 0,\pi\right] .$$
The Cauchy theorem merely gives a rough upper bound. In order to obtain a sharper upper bound, we would assume some additional hypotheses on the derivatives of the function.
Case 2: $Y$ is an unbounded random variable
-------------------------------------------
If the support of the random variable $Y$ is not bounded and $\varphi$ is a polynomial of unknown degree $t$, $t \leq g-1$, with $g$ known, it’s still possible to give an upper bound for the estimation error. Since $$\varphi^{\left( d\right) }=\sum_{i=0}^{g-1}\frac{\varphi^{\left(
d+i\right) }\left( s^{\ast}\right) }{i!}\left( u-s^{\ast}\right) ^{i}=$$ $$=\sum_{i=0}^{g-1}\frac{\sum_{k=0}^{g-1}L_{s_{k}}^{\left( d+i\right) }\left(
s^{\ast}\right) \varphi\left( s_{k}\right) }{i!}\left( u-s^{\ast}\right)
^{i}=\sum_{k=0}^{g-1}L_{s_{k}}^{\left( d\right) }\left( u\right)
\varphi\left( s_{k}\right) ,$$
$\varphi^{\left( d\right) }$ can be estimated as follows $$\widehat{\varphi^{\left( d\right) }\left( u\right) }=\sum_{k=0}^{g-1}L_{s_{k}}^{\left( d\right) }\left( u\right) \overline{Y}\left(
s_{k}\right) .$$
We have in probability $\widehat{\varphi^{\left( d\right) }}\rightarrow\varphi^{\left( d\right) }$ for $\min\left( n_{k}\right)
\rightarrow\infty$. So, $$Var\left( \widehat{\varphi^{\left( d\right) }}\right) =\sum_{k=0}^{g-1}\left( L_{s_{k}}^{\left( d\right) }\left( u\right) \right)
^{2}\frac{\varsigma}{n_{k}}\rightarrow0,$$ where $\varsigma$ is the variance of $Z$. We use the Tchebycheff’s inequality in order to obtain an upper bound for the estimation error. For a given $\eta$, $$\Pr\left\{ \left\vert \widehat{\varphi^{\left( d\right) }}-\varphi^{\left(
d\right) }\right\vert \geq\eta\right\} \leq\frac{\sum_{k=0}^{g-1}\left(
L_{s_{k}}^{\left( d\right) }\left( u\right) \right) ^{2}\frac{\varsigma
}{n_{k}}}{\eta^{2}}.$$
If we aim to obtain, for all fixed $\omega$, $\Pr\left\{ \left\vert
\widehat{\varphi^{\left( d\right) }}-\varphi^{\left( d\right) }\right\vert
\geq\eta\right\} \leq\omega\text{,} $ we can choose $n^{\ast}$ as the solution of the equation $\frac{\sum_{k=0}^{g-1}\left( L_{s_{k}}^{\left(
d\right) }\left( u\right) \right) ^{2}\frac{\varsigma}{n_{k}}}{\eta^{2}}=\omega$, that is $$n^{\ast}=\frac{\sum_{k=0}^{g-1}\left( L_{s_{k}}^{\left( d\right) }\left(
u\right) \right) ^{2}\varsigma}{\omega\eta^{2}}.$$ The integer $\left[ n^{\ast}\right] $ is such that the inequality $\Pr\left\{ \left\vert \widehat{\varphi^{\left( d\right) }}-\varphi
^{\left( d\right) }\right\vert \geq\eta\right\} \leq\omega$ is satisfied.
We remark that if we know the degree $t$ of the polynomial, then it is sufficient to set $g-1=t$. When $\varphi\left( u\right) =\varphi
^{d}\left( u\right) $ (i.e., $d=0$), we have $\left[ n^{\ast}\right]
=\frac{\sum_{k=0}^{g-1}\left( L_{s_{k}}\left( u\right) \right)
^{2}\varsigma}{\omega\eta^{2}}.$
We underline that for $d=0$ and when $t$ is known $\widehat
{\varphi^{\left( d\right) }\left( u\right) }=\widehat{\varphi\left(
u\right) }$ coincides with Hoel’s estimator.
If the solely information on $\varphi$ is that $\varphi$ is analytic then we are constrained to give hypotheses on the derivatives of the function. More precisely, since $\operatorname{Im}\varphi\subseteq\mathbb{R}$, we can’t apply the Cauchy theorem on the analytic functions; we can only say that $\varphi\left( v\right) =E\left( Y\right) \in\mathbb{R}$. So, we are not able to find a constant $R$ such that $\left\vert \varphi\left( v\right)
\right\vert \leq R$. Moreover, since we can’t observe $\varphi\left(
v\right) $ for $v\notin S$, we don’t have any data to estimate $M_{Taylor}$.
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---
abstract: 'Circuit quantum electrodynamics allows spatially separated superconducting qubits to interact via a “quantum bus", enabling two-qubit entanglement and the implementation of simple quantum algorithms. We combine the circuit quantum electrodynamics architecture with spin qubits by coupling an InAs nanowire double quantum dot to a superconducting cavity. We drive single spin rotations using electric dipole spin resonance and demonstrate that photons trapped in the cavity are sensitive to single spin dynamics. The hybrid quantum system allows measurements of the spin lifetime and the observation of coherent spin rotations. Our results demonstrate that a spin-cavity coupling strength of 1 MHz is feasible.'
author:
- 'K. D. Petersson'
- 'L. W. McFaul'
- 'M. D. Schroer'
- 'M. Jung'
- 'J. M. Taylor'
- 'A. A. Houck'
- 'J. R. Petta'
title: Circuit Quantum Electrodynamics with a Spin Qubit
---
Electron spins trapped in quantum dots have been proposed as basic building blocks of a future quantum processor [@Loss1998; @Hanson2007]. With spin qubits, two qubit operations are typically based on exchange coupling between nearest neighbor spins, leading to a fast 180 ps entangling gate [@Petta2005]. However, a scalable spin-based quantum computing architecture will almost certainly require long-range qubit interactions. Unfortunately, the weak magnetic moment of the electron makes it difficult to couple spin qubits that are separated by a large distance. Approaches to transferring spin information by physically shuttling electrons or using exchange-coupled spin chains are experimentally challenging [@Hermelin2011; @McNeil2011; @Friesen2007]. In comparison, circuit quantum electrodynamics (cQED) has enabled long distance coupling of multiple superconducting qubits via a microwave cavity, providing a scalable architecture for quantum computation [@Wallraff2004; @Reed2012; @Sillanpaa2007]. Several proposals suggest coupling spatially separated spin qubits via a microwave cavity, but direct coupling between a single spin and the magnetic field of the cavity results in a spin-cavity vacuum Rabi frequency $g_{\rm S}$/2$\pi$ $\sim$ 10 Hz; far too weak to be useful for quantum information processing [@Imamoglu2009; @Childress2004].
Here we harness spin-orbit coupling in a hybrid quantum dot/cQED architecture to couple the electric field of a high quality factor superconducting cavity to a single “spin-orbit qubit" fabricated from an InAs nanowire double quantum dot (DQD) [@Wallraff2004; @Schroer2011; @NadjPerge2010; @Fasth2007]. The architecture allows us to achieve a charge-cavity vacuum Rabi frequency $g_{\rm C}$/2$\pi$ $\sim$ 30 MHz, consistent with coupling rates obtained in GaAs quantum dots and carbon nanotubes [@Frey2012; @Delbecq2011]. The strong spin-orbit interaction of InAs allows us to electrically drive spin rotations with a local gate electrode, while the 30 MHz cavity-charge interaction provides a measurement of the resulting spin dynamics. An alternative approach that has been recently explored consists of coupling ensembles of spins ($N$ $\sim$ 10$^{12}$) to superconducting resonators [@Schuster2010; @Kubo2010; @Amsuss2011; @Zhu2011].
Our hybrid spin-orbit qubit/superconducting device is shown in Fig. 1(a). We fabricate a $\lambda$/2 superconducting Nb resonator (the cavity) with a resonance frequency $f_{\rm 0}$ = $\omega_{\rm 0}$/2$\pi$ $\sim$ 6.2 GHz and quality factor, $Q$ $\sim$ 2000 [@SOM]. The amplitude and phase response of the cavity is detected using a homodyne measurement with a microwave probe frequency $f_R$ [@Wallraff2004]. We couple a single InAs nanowire spin-orbit qubit to the electric field generated by the cavity [@Trif2008]. The qubit consists of a DQD defined in an InAs nanowire [@Schroer2011; @NadjPerge2010]. A series of Ti/Au depletion gates create a simple double well confinement potential containing ($N_{\rm L}$, $N_{\rm R}$) electrons, where $N_{\rm L}$ ($N_{\rm R}$) is the number of electrons in the left (right) dot. We tune the tunnel coupling, $t_{\rm C}$, of the DQD by adjusting the voltage $V_{\rm M}$ on the middle barrier gate, labeled M in Fig. 1(c). A trapped electron in the DQD has an electric dipole moment $d$ $\sim$ 1000 $e$$a_{\rm o}$, where $a_{\rm o}$ is the Bohr radius and $e$ is the electronic charge. To maximize the electric field at the position of the DQD, the drain contact of the nanowire is connected to the ground plane of the resonator and the source contact is connected to an anti-node of the resonator \[see Fig. 1(b)\]. Standard dc transport measurements are made possible by applying a source-drain bias, $V_{\rm SD}$, to the DQD via a spiral inductor that is connected to the voltage node of the resonator [@Chen2011].
![\[fig1\] (Color online) (a) Circuit schematic and micrograph of a device similar to the one measured. Transmission through the $\lambda$/2 superconducting Nb resonator is measured using homodyne detection. Nanowire source-drain bias, $V_{\rm SD}$, is applied at the central voltage node of the cavity through a $\sim$ 4 nH inductor. (b) Expanded image of the region containing the DQD. One end of the nanowire is connected to the resonator ground plane and the other end is connected to the anti-node of the resonator. (c) Scanning electron micrograph image of the nanowire DQD. Seven gate electrodes are used to create a confinement potential along the length of the nanowire. (d) DQD energy levels (upper plot) and ac susceptibility, $\chi$, (lower plot) as a function of detuning, $\epsilon$. (e) The phase response of the resonator provides a direct measurement of the DQD charge stability diagram.](Figure1rev5.pdf){width="0.7\columnwidth"}
We focus on the cavity response near the ($M$, $N$+1) $\leftrightarrow$ ($M$+1, $N$) interdot charge transition. Neglecting spin for the moment, the DQD forms a two-level “artificial molecule" with an energy splitting $\Omega = \sqrt{\epsilon^2+4t_{\rm C}^2}$, where $\epsilon$ is the detuning \[upper diagram, Fig. 1(d)\]. Interdot tunnel coupling hybridizes the charge states around $\epsilon$ = 0 resulting in a tunnel splitting of 2$t_{\rm C}$. The detuning dependent dipole moment of the DQD has an admittance which loads the cavity. We characterize the strength of the interaction by the ac susceptibility $\chi$ \[lower panel, Fig. 1(d)\] [@Blais2004].
A qualitative understanding of the quantum dot/cavity coupling can be obtained considering the relevant energy scales in the system. The single dot charging energy, $E_{\rm C}$ $\sim$ 12 meV, is much larger than the relevant photon energies, $hf_{\rm R}$ $\sim$ 25 eV, and the cavity is largely unaffected by the DQD in Coulomb blockade. However, near interdot charge transitions (e.g. ($M$, $N$+1) $\leftrightarrow$ ($M$+1, $N$)), or transitions with the source and drain electrodes (e.g. ($M$, $N$)$\leftrightarrow$($M$, $N$+1)), the energy scales associated with the DQD are close to the cavity energy, and the cavity is damped, resulting in a negative phase shift in microwave transmission at the bare cavity frequency. The DQD charge stability diagram is measured in Fig. 1(e) by probing the phase response of the microwave cavity as a function of the gate voltages $V_{\rm R}$ and $V_{\rm L}$ [@Frey2012; @Delbecq2011].
Quantitative analysis of the cavity response requires a fully quantum mechanical model that accounts for photon exchange between the microwave field and the DQD [@Trif2008; @Chen2011]. In cavity QED, one considers interactions between an atom with transition frequency $\omega_{\rm a} = \Omega/\hbar$ and the photon field of the cavity, characterized by the resonance frequency $\omega_{\rm 0}$. The atom and cavity energy levels hybridize when the atom-cavity detuning $\Delta$ = $\omega_{\rm a}$ - $\omega_{\rm 0}$ $<$ $g_{\rm C}$, leading to the Jaynes-Cummings ladder of quantum states [@Jaynes1963]. When the atom and cavity are detuned in the dispersive limit ($\Delta$ $>$ $g_{\rm C}$), the cavity field exhibits a phase shift in microwave transmission at the bare cavity frequency that is given by $\phi$ = $\arctan[(2g_{\rm C}^2)/(\kappa \Delta)]$, where $\kappa$ is the cavity decay rate. The phase will therefore change sign as the atom-cavity detuning $\Delta$ is tuned from positive to negative values [@Wallraff2004].
We extract $g_{\rm C}$ by measuring the amplitude and phase response of the cavity for several values of the interdot tunnel coupling \[Figs. 2(e), (f)\]. For example, with $V_{\rm M}$ = -2.26 V, the qubit transition frequency is always greater than the cavity frequency ($\Omega/\hbar$ $>$ $\omega_{\rm 0}$), leading to a negative phase shift for all values of $\epsilon$. In contrast, for $V_{\rm M}$ = -2.32 V, the minimum qubit transition frequency 2$t_{\rm C}$/$\hbar$ $<$ $\omega_{\rm 0}$ and $\Delta$ changes sign with $\epsilon$, resulting in a phase shift that takes on both positive and negative values. We fit the data to a master equation model using a best fit value of $g_{\rm C}$/2$\pi$ = 30 MHz and a $V_{\rm M}$ dependent tunnel coupling that ranges from 2$t_{\rm C}$/$h$= 1.8 to 7.0 GHz [@SOM]. We assume a qubit lifetime of 15 ns and account for inhomogeneous broadening due to charge noise by convolving the phase and magnitude response with a Gaussian of width $\sigma_{\rm E}$ = 21 $\mu$eV [@Petersson2010b]. The vacuum Rabi frequency extracted here compares favorably to values obtained using Cooper pair box qubits, $g_{\rm C}$/2$\pi$ $\sim$ 6 MHz [@Wallraff2004], transmon qubits, $g_{\rm C}$/2$\pi$ $\sim$ 100 MHz [@Schuster2007], and many-electron GaAs quantum dots, $g_{\rm C}$/2$\pi$ $\sim$ 50 MHz [@Frey2012].
![\[fig2\] (Color online) (a) – (b) Phase and amplitude response of the cavity near the ($M$+1, $N$) $\leftrightarrow$ ($M$, $N$+1) charge transition measured using a fixed drive frequency, $f_{\rm R}$ = 6194.8 MHz. (c) – (d) Phase and normalized amplitude of the microwave field plotted as a function of $f_{\rm R}$ at the interdot charge transition (green curve) and in Coulomb blockade (blue curve). (e) – (f) Phase and amplitude response measured as a function of DQD detuning, $\epsilon$, for a range of tunnel couplings, as set by $V_{\rm M}$. Dashed lines are fits to the data, allowing the extraction of the cavity coupling strength, $g_{\rm C}$ $\sim$ 30 MHz (see main text). Inset: Cavity frequency relative to the qubit transition frequency, $\Omega/h$.](Figure2rev6.pdf){width="0.7\columnwidth"}
We access the spin-degree of freedom by operating the device as a spin-orbit qubit (Fig. 3). For simplicity, we label the charge states (1,1) and (0,2) [@Petta2005]. The ground state with two-electrons in the right quantum dot is the singlet S(0,2). At negative detuning, the four relevant spin-orbital states are $|$$\Uparrow\Uparrow >$, $|$$\Downarrow\Downarrow>$, $|$$\Uparrow\Downarrow>$, and $|$$\Downarrow\Uparrow>$ [@NadjPerge2010]. The level diagram is similar to a GaAs singlet-triplet spin qubit, with a key difference being that the g-factors for the two spins can vary significantly [@Petta2005]. Interdot tunnel coupling hybridizes the states with singlet character near $\epsilon$ = 0, and an external field results in Zeeman splitting $E_{\rm Z}$ = $\tilde{g}\mu_{\rm B} B$ of the spin states, where $\tilde{g}$ is the electronic g-factor, $\mu_{\rm B}$ is the Bohr magneton, $B$ is the magnetic field.
Spin selection rules result in Pauli blockade at the two-electron transition, a key ingredient for spin preparation and measurement \[see inset, Fig. 3(b)\] [@Petta2005; @NadjPerge2010; @Ono2002]. For example, the $|$$\Uparrow\Uparrow>$ state cannot tunnel to S(0,2) due to Pauli exclusion. Modulation of the confinement potential with a gate voltage results in spin-orbit-driven EDSR transitions that lift the Pauli blockade [@NadjPerge2010; @Golovach2006]. In Fig. 3(b) we plot the current, $I$, through the DQD with $V_{\rm SD}$ = 2.5 meV and the gates tuned in Pauli blockade \[blue dot, inset Fig. 3(b)\]. Hyperfine fields rapidly mix spin states when $E_{\rm Z}$ = $\tilde{g}\mu_{\rm B}B$ $<$ $B_{\rm N}$, where $B_{\rm N}$ $\sim$ 2 mT is the hyperfine field [@Schroer2011]. At finite fields, the leakage current is non-zero when the ac driving frequency on the gate, $f_{\rm G}$, satisfies the electron spin resonance condition $E_{\rm Z}$ = $h f_{\rm G}$. We observe two resonance conditions corresponding to single spin rotations in the left and right quantum dot, with g-factors of 8.2 and 10.6 [@NadjPerge2010].
![\[fig3\] (Color online) (a) EDSR transitions lift Pauli blockade, resulting in current flow through the device. (b) Leakage current measured in Pauli blockade at point $\epsilon$$'$ (inset) as a function of magnetic field, $B$, and microwave driving frequency, $f_{\rm G}$. Pauli blockade is lifted by EDSR driving when $E_{\rm Z}$ = $\tilde{g} \mu_{\rm B} B$ = $h f_{\rm G}$. Inset: Finite-bias triangles measured with $V_{\rm SD}$ = 2.5 mV indicate a suppression of current due to Pauli blockade. (c) Energy levels of the spin-orbit qubit plotted as a function of $\epsilon$. The data in (b) are acquired with $\epsilon$ = $\epsilon$$'$.](Figure3rev5.pdf){width="0.75\columnwidth"}
Around $\epsilon$ = 0, the DQD has a spin state dependent dipole moment that allows spin state readout via the superconducting cavity [@Petersson2010a]. We combine quantum control of the spins using EDSR and cavity detection of single spin dynamics using the pulse sequence shown in Figs. 4 (a),(b). Starting with the spin qubit in state $|$$\Uparrow\Uparrow>$, we pulse to negative detuning and apply a microwave burst of length $\tau_{\rm B}$ to drive EDSR transitions. For example, an EDSR $\pi$-pulse will drive a spin transition from $|$$\Uparrow\Uparrow>$ to $|$$\Uparrow\Downarrow>$. The resulting spin state is probed by pulsing back to $\epsilon$ = 0 for a time $T_{\rm M}$. The cavity is most sensitive to charge dynamics near $\epsilon$ = 0 due to the different ac susceptibility of the $|$$\Uparrow\Downarrow>$ and $|$$\Uparrow\Uparrow>$ spin states [@SOM]. In Fig. 4(c) we plot the cavity phase shift as a function of $f_{\rm G}$ and $B$. We again observe two features that follow the standard spin resonance condition, consistent with the dc transport data in Fig. 4(b). Varying the measurement time $T_{\rm M}$, we fit the measured phase response to an exponential decay and estimate a spin lifetime $T_{\rm 1}$ $\sim$ 1 $\mu$s \[Fig. 4(d)\].
We demonstrate time-resolved Rabi oscillations in the spin-orbit qubit and readout via the cavity by varying the EDSR microwave burst length $\tau_{\rm B}$. Figure 4(e) shows the measured phase as a function of $\tau_{\rm B}$ and gate drive power, $P_{\rm G}$. We observe Rabi oscillations with a minimum period of 17 ns, as shown in Fig. 4(f), consistent with an EDSR driving mechanism [@NadjPerge2010]. These results demonstrate that the microwave field of the cavity is sensitive to the spin state of a single electron.
![\[fig4\] (Color online) (a) Upper: Pulse sequence used for resonator readout is superimposed on the level diagram. Lower: The ac susceptibility, $\chi$, is dependent on the spin state of the DQD and allows for sensitive spin readout. (b) Pulse sequence used to drive EDSR transitions. (c) Phase response of the cavity measured as a function of EDSR drive frequency, $f_{\rm G}$, and external field, $B$, with $\tau_{\rm B}$ = 100 ns and $T_{\rm M}$ = 850 ns. EDSR transitions are observed in the phase response, in agreement with the dc transport data. (d) Measured phase shift as a function of $T_{\rm M}$ with $\tau_{\rm B}$ = 100 ns, $B$ = 90 mT and $f_{\rm G}$ = 13.1 GHz. The exponential decay yields a spin relaxation time of $T_{\rm 1}$ = 1 $\mu$s. (e) Phase response of the cavity as a function of EDSR burst length, $\tau_{\rm B}$, and driving power for fixed $B$ = 86 mT, $f_{\rm G}$ = 9.5 GHz, and $T_{\rm M}$ = 1.75 $\mu$s. (f) Rabi oscillations at different powers, indicated by the dashed lines (e). The solid curves are fits to a power law decay [@SOM].](Figure4rev6.pdf){width="0.75\columnwidth"}
In cQED, a large number of qubits can be connected via the electric field of the superconducting cavity. Trif *et al.* have proposed using the cQED approach to couple two spin-orbit qubits via a cavity mediated interaction [@Trif2008]. Based on our results, we can estimate the effective spin-cavity coupling strength using theory developed for single quantum dots, $g_{\rm S} \approx g_{\rm C} \dfrac{E_{\rm Z}}{\Delta E_{\rm 0}}\dfrac{R}{\lambda_{\rm SO}}$ [@Trif2008]. Taking $g_{\rm C}$/2$\pi$ = 100 MHz (which can be obtained by increasing the cavity frequency), $E_{\rm Z}$ = 70 $\mu$eV, an orbital level spacing $E_{\rm O}$ = 1.7 meV, dot radius $R$ = 25 nm, and spin-orbit length $\lambda_{\rm SO}$ = 100–200 nm, we find a spin coupling rate $g_{\rm S}$/2$\pi$ $\sim$ 1 MHz, which is five orders of magnitude larger than the coupling rate that would be obtained by coupling a single spin to the magnetic field of a microwave cavity.
In order to implement coherent state transfer between the qubit and the cavity the device must be in the strong coupling regime, where the spin-cavity coupling rate $g_{\rm S}$ is larger than the cavity decay rate $\kappa$ and the qubit decoherence rate $\gamma$. Optimization of the resonator design will reduce the cavity decay rate to well below 1 MHz [@SOM]. There are several options for decreasing the qubit decoherence rate. First, dynamical decoupling can be used to reduce the qubit decay rate to $\sim$ 1 MHz in the InAs system [@NadjPerge2010]. InAs could also be replaced by Ge/Si core/shell nanowires where hole spin-orbit coupling is predicted to be large [@Kloeffel2011]. Resonators can also be coupled to nuclear-spin-free Si/SiGe quantum dots by using micromagnets to create artificial spin-orbit fields [@PioroLadriere2008]. Based on our results we anticipate that the strong coupling regime for single spins can be reached, eventually allowing spin qubits to be interconnected in a quantum bus architecture.
Acknowledgements: Research at Princeton was supported by the Sloan and Packard Foundations, Army Research Office grant W911NF-08-1-0189, DARPA QuEST award HR0011-09-1-0007 and the National Science Foundation through the Princeton Center for Complex Materials, DMR-0819860, and CAREER award, DMR-0846341. JMT acknowledges support from ARO MURI award W911NF0910406.
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abstract: 'We derive the asymptotical dynamical law for Ginzburg-Landau vortices in an inhomogeneous background density under the Schrödinger dynamics, when the Ginzburg-Landau parameter goes to zero. New ingredients involve across the cores lower bound estimates and approximations.'
author:
- '[Robert L. Jerrard]{} & [Didier Smets]{}'
title: 'Vortex dynamics for the two dimensional non homogeneous Gross-Pitaevskii equation'
---
Introduction
============
We are interested in the two dimensional Gross-Pitaevskii equation
$$\label{eq:gpv}
i \partial_t u - \Delta u + \frac{1}{{\varepsilon}^2}\left( V(x) + |u|^2\right) u = 0$$
for $u:{\mathbb{R}}^2\times {\mathbb{R}}^+ \to {\mathbb{C}}$, where $0<{\varepsilon}\ll1$ and $V : {\mathbb{R}}^2 \to {\mathbb{R}}^+$ is a smooth potential such that
$$V(x) \to +\infty \text{ as } |x|\to +\infty.$$ The Gross-Pitaevskii equation is a widely used model to describe the dynamics of a Bose-Einstein condensate in a trapping potential $V$. The equation on ${\mathbb{R}}^2$ arises via dimension reduction from 3 dimensions; this has been justified for particular choices of $V$ in [@AMSW] for example. Equation is Hamiltonian, with Hamiltonian given by $$\mathcal{E}_{{\varepsilon},V}(u) = \int_{{\mathbb{R}}^2} \frac{|\nabla u|^2}{2} + \frac{1}{{\varepsilon}^2}\left(
V(x)\frac{|u|^2}{2} + \frac{|u|^4}{4}\right).$$ Another quantity which is preserved by the flow associated with is the total mass $M$, given by $$M(u) = \int_{{\mathbb{R}}^2} |u|^2.$$ For each $m>0$, there exists[^1] at least one positive ground state $\eta \equiv \eta_{{\varepsilon},m}:\ {\mathbb{R}}^2\to {\mathbb{R}}^+$ of total mass equal to $m$. By definition, a ground state $\eta$ realizes the infimum $$\mathcal{E}_{{\varepsilon},V}(\eta) = \inf\{ \mathcal{E}_{{\varepsilon},V}(g), \ g \in H^1({\mathbb{R}}^2,{\mathbb{C}}),\ M(g) = m\},$$ and satisfies the Euler-Lagrange equation $$-\Delta \eta + \frac{1}{{\varepsilon}^2}(V+\eta^2) \eta = \frac 1{{\varepsilon}^2}\lambda \eta,$$ where we write the Lagrange multiplier as $\frac{1}{{\varepsilon}^2}\lambda$ for some $\lambda\equiv \lambda_{{\varepsilon},m}.$
In the limit ${\varepsilon}\to 0,$ we have $$\eta^2 \to {\rho_{\scriptscriptstyle{TF}}}\qquad\text{in } L^2({\mathbb{R}}^2),$$ where the function ${\rho_{\scriptscriptstyle{TF}}}$, known as the [*Thomas-Fermi profile*]{} in the physics literature, is given by ${\rho_{\scriptscriptstyle{TF}}}(x) := (\lambda_0-V)^+(x)$ where the number $\lambda_0$ is uniquely determined by the mass condition $$\int_{{\mathbb{R}}^2}(\lambda_0-V(x))^+\, dx = m.$$ We will study the behaviour of solutions of which correspond, in a sense to be made precise later, to perturbations of the ground state $\eta$ by a finite number of quantized vortices, each carrying a single quantum of vorticity. Our goal is to prove that these vortices persist, and to describe their evolution in time.
We will show that to leading order the vortices do not interact, and that each one evolves (in a renormalized time scale) by the orthogonal gradient flow for the function $\log {\rho_{\scriptscriptstyle{TF}}}$, with a sign depending on the winding number of the given vortex. More precisely, let $${\Omega_{\scriptscriptstyle{TF}}}:= \{ x: {\rho_{\scriptscriptstyle{TF}}}(x)>0\}$$ be the interior of the limiting support[^2] of the ground state, let $\{b_i^0\}_{i=1}^l$ be distinct points in ${\Omega_{\scriptscriptstyle{TF}}}$, and let $d_1,\ldots, d_l \in \{-1,+1\}.$ For each $i\in \{1,\cdots,l\},$ we denote by $b_i(t)$ the solution of the ordinary differential equation $$\label{eq:limitdynamics0}
\dot b_i (t) = d_i \frac{\nabla^\perp{\rho_{\scriptscriptstyle{TF}}}}{{\rho_{\scriptscriptstyle{TF}}}}(b_i(t)),$$ where $\nabla^\perp = (-\partial_{x_2}, \partial_{x_1})$, with initial datum $b_i(0) = b_i^0.$
\[thm:limit\] Let $(u_{\varepsilon}^0)_{{\varepsilon}>0}$ be a family of initial data for such that $$M(u_{\varepsilon}^0) = m,$$ $$\mathcal{E}_{{\varepsilon},V}(u_{\varepsilon}^0) \le \mathcal{E}_{{\varepsilon},V}(\eta) + \pi \sum_{i=1}^l {\rho_{\scriptscriptstyle{TF}}}(b_i^0){|\!\log{\varepsilon}|}+ o({|\!\log{\varepsilon}|}),$$ and $${\rm curl} \big( \frac{j(u_{\varepsilon}^0)}{{\rho_{\scriptscriptstyle{TF}}}}\big) \longrightarrow 2\pi \sum_{i=1}^l d_i \delta_{b_i^0} \qquad \text{in } W^{-1,1}_{\rm loc}({\Omega_{\scriptscriptstyle{TF}}}),$$ as ${\varepsilon}\to 0.$ Then, as long as the points $\{b_i(t)\}_{i=1}^l$ remain distinct, $${\rm curl} \big( \frac{j(u_{\varepsilon}^t)}{{\rho_{\scriptscriptstyle{TF}}}}\big) \longrightarrow 2\pi \sum_{i=1}^l d_i \delta_{b_i(t)} \qquad \text{in } W^{-1,1}_{\rm loc}({\Omega_{\scriptscriptstyle{TF}}}),$$ as ${\varepsilon}\to 0,$ where $u_{\varepsilon}^t:= u_{\varepsilon}(\cdot,t{|\!\log{\varepsilon}|}).$
Here, $j(u_{\varepsilon}^t):=(iu_{\varepsilon}^t,\nabla u_{\varepsilon}^t)$ where $(z,w):={\rm Im}(z \bar{w}).$ Therefore, $$\frac12 {\rm curl} \big( \frac{j(u_{\varepsilon}^t)}{{\rho_{\scriptscriptstyle{TF}}}}\big) = \frac12 {\rm curl}\: j\big(\frac{u_{\varepsilon}^t}{\sqrt{{\rho_{\scriptscriptstyle{TF}}}}}\big) = J\big(\frac{u_{\varepsilon}^t}{\sqrt{{\rho_{\scriptscriptstyle{TF}}}}}\big)$$ is the Jacobian determinant of $u_{\varepsilon}^t/\sqrt{{\rho_{\scriptscriptstyle{TF}}}}.$ It is widely recognized, in the present regime for the Ginzburg-Landau energy, that the notion of a vortex of winding number $d_i$ located at the point $b_i(t)$ is appropriately described by the presence of the term $2\pi d_i \delta_{b_i(t)}$ in the limit of the vorticity field ${\rm curl}\: j\big(u_{\varepsilon}^t/\sqrt{{\rho_{\scriptscriptstyle{TF}}}}\big).$
[Note that the ordinary differential equations are decoupled. Also, since ${\rho_{\scriptscriptstyle{TF}}}(b_i(t)) = {\rho_{\scriptscriptstyle{TF}}}(b_i^0)$ for any $t\in {\mathbb{R}}$ the points $ \{b_i(t)\}_{i=1}^l$ remain distinct for all times unless two of them are located on the same level line of ${\rho_{\scriptscriptstyle{TF}}}$ and have opposite circulations. ]{}
Results of this sort in the homogeneous case $\eta\equiv 1$ were first proved in the late 1990s, see [@CoJe; @CoJe2], and have subsequently been developed by a number of authors, see for example [@LiXi; @BeJeSm; @JeSp]. The point of this paper is thus to understand the effect of the inhomogeneity on the dynamical law for the vortices.\
We remark that a number of authors have studied questions about vortex dynamics in inhomogeneous backgrounds for parabolic equations [@Li; @JianSong], or more recently [@SerfTice] for a quite general class of equations of mixed parabolic-Schrödinger type. The case of pure Schrödinger dynamics presents distinct difficulties and as far as we know has not been treated until now.
The sequel of this introduction is devoted to the presentation of the strategy leading to Theorem \[thm:limit\]. We notice that will actually prove a result (Theorem \[thm:main\] below) which is stronger in two respects than Theorem \[thm:limit\]: first it describes the dynamics of vortices at small but fixed value of ${\varepsilon}$, rather than asymptotically as ${\varepsilon}\to 0$ in Theorem \[thm:limit\], and second it applies to a broader class of inhomogeneous equations (see below) where $\eta$ need not necessarily be the profile of a ground state.
Perturbation equation and Theorem \[thm:main\]
----------------------------------------------
For the class of initial data which we consider in Theorem \[thm:limit\], it is convenient to rewrite the corresponding solutions of in the form $$\label{eq:change}
u(x,t) = \eta(x) w(x,t)$$ and to study the evolution equation for $w$. One easily checks that if $u$ is a solution to , then $w$ solves $$i\eta^2 \partial_t w - {\rm div}(\eta^2\nabla w) +\frac{1}{{\varepsilon}^2} \eta^4
(|w|^2-1)w = -\frac{ \lambda}{{\varepsilon}^2} \eta^2 w.$$ In particular, the change of phase and time scale $$v(x,t) =
\exp\left(i\frac{\lambda}{{\varepsilon}^2}\frac{t}{{|\!\log{\varepsilon}|}}\right)w\left(x,\frac{t}{{|\!\log{\varepsilon}|}}
\right)$$ leads to the equation
$$\label{eq:nhgp}
i{|\!\log{\varepsilon}|}\eta^2 \partial_t v - {\rm div}(\eta^2\nabla v) +\frac{1}{{\varepsilon}^2}
\eta^4 (|v|^2-1)v = 0$$
for $v$. Note that the change of time scale is related to the fact that the phenomenon which we wish to describe, namely vortex motion, arises in times of order one in that new time scale (see the definition of $u_{\varepsilon}^t$ in the statement of Theorem \[thm:limit\]).
Our analysis will henceforth focus on equation . Equation , like , is Hamiltonian, with Hamiltonian given by the weighted Ginzburg-Landau energy $$\label{Eepseta}
E_{{\varepsilon},\eta}(v) \equiv \int_{{\mathbb{R}}^2} e_{{\varepsilon},\eta}(v) = \int_{{\mathbb{R}}^2} \eta^2
\frac{|\nabla v|^2}{2} + \eta^4 \frac{(|v|^2-1)^2}{4{\varepsilon}^2}.$$ As a matter of fact, using the Euler-Lagrange equation for $\eta$, one realizes that $$\label{eq:parotide2}
\mathcal{E}_{{\varepsilon},V}(u) = \mathcal{E}_{{\varepsilon},V}(\eta) + E_{{\varepsilon},\eta}(v) + \frac{\lambda}{2{\varepsilon}^2}\Big( M(u)-M(\eta)\Big).$$ In the sequel, we enlarge our framework and consider equation where $\eta\ : \ {\mathbb{R}}^2 \to {\mathbb{R}}$ is any smooth positive function such that the corresponding Cauchy problem is globally well-posed for initial data in $H^1({\mathbb{R}}^2,\eta\,dx)$ and such that the corresponding solutions can be approximated by smooth solutions[^3]. In particular, under those assumptions the energy $E_{{\varepsilon},\eta}$ is preserved along the flow of .
Let ${\varepsilon}>0$ and let $\Omega \subset {\mathbb{R}}^2$ be a bounded open set. Let $\{a_i^0\}_{i=1}^l$ be distinct points in $\Omega$, and let $d_1,\ldots, d_l \in \{-1,+1\}.$ For each $i\in \{1,\cdots,l\},$ we denote by $a_i(t)$ the solution, as long as it does not reach $\partial\Omega,$ of the ordinary differential equation $$\label{eq:limitdynamics}
\dot a_i (t) = d_i \nabla^\perp \log\eta^2(a_i(t)),$$ where $\nabla^\perp = (-\partial_{x_2}, \partial_{x_1})$, with initial datum $a_i(0) = a_i^0.$\
We assume that $$\eta_{min} := \inf_{x\in \Omega} \eta(x) > 0,$$ and we fix a time $T_{\rm col}>0$ such that $$\rho_{min} := \min_{t\in [0,T_{\rm col}]}
\min\big\{\{ \frac 12 d(a_i(t),a_j(t))\}_{\ i\neq j}\cup \{d(a_i(t),\partial \Omega)\}_{i} \cup \{1\} \big\}
> 0.
\label{rhomin.def}$$ Finally, we consider a finite energy solution $v$ of , we set $v^t:=v(\cdot,t)$ and we define, for $t\in [0,T_{\rm col}],$ $$\label{eq:defrta}
r_a^t := \| J v^t - \pi \sum_{i=1}^l d_i
\delta_{a_i(t)}\|_{W^{-1,1}(\Omega)},$$ and $$\label{defSigmat}
\Sigma^t := \frac{{{E}_{\varepsilon,\eta}}(v^t)}{{|\!\log{\varepsilon}|}} - \pi \sum_{i=1}^l \eta^2(a_i(t)).$$
We will deduce Theorem \[thm:limit\] from
\[thm:main\] There exist positive constants ${\varepsilon}_0,$ $\gamma_0$ and $C_0$, depending only on $l$, $\eta_{min}$, $\rho_{min}$, and $\|\nabla \eta^2\|_{L^\infty(\Omega)}$, such that if $0<{\varepsilon}\leq {\varepsilon}_0$ and if $\Sigma^0 + r_a^0 \leq \gamma_0,$ then $$\label{eq:ineqmainalt}
r_a^t \leq r_a^0 + \Big( \Sigma^0 + r_a^0 + \frac{\log{|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}} \Big) \big( e^{C_0t}-1\big) + C_0{\varepsilon}^\frac12,$$ as long as $t\leq T_{\rm col}$ and $\Sigma^0 + r_a^t(t) \leq \gamma_0.$
\[rem:2\]
$i)$ As we shall discuss in Section \[sec:heuristics\] below, the quantity $r^t_a$, which is a sort of discrepancy measure, can be thought of as measuring the distances between the “actual vortex locations” and the desired vortex locations. The quantity $\Sigma^t$, multiplied by ${|\!\log{\varepsilon}|}$, corresponds to the excess of energy of the solution $v$ with respect to an energy minimizing field possessing the vortices at the points $a_i(t).$ Notice that since ${{E}_{\varepsilon,\eta}}$ is preserved by the flow for $v$ and $\eta^2$ is preserved by the flow for the $a_i's$, we have $$\Sigma^t \equiv \Sigma^0, \qquad \forall \ t\in [0,T].$$
$ii)$ Theorem \[thm:main\] is interesting for initial data such that $\Sigma^0 + r^0_a$ is small. The existence of such data is standard. For example, if we fix $f:[0,\infty)\to [0,1]$ such that $f'\ge 0$, $f(0)=0$, and $f(s)\to 1$ as $s\to \infty$, then for the initial data $$v^0(z) := \prod_{i=1}^l f(\frac{|z-a_i|}{\varepsilon}) \left( \frac{z- a_i^0}{|z-a_i^0|}\right)^{d_i},
\qquad\qquad\mbox{ $x := (x_1, x_2) \cong z = x_1+ix_2$},$$ one can check that $\Sigma^0 \le C{|\!\log{\varepsilon}|}^{-1}, r^0_a\le C {\varepsilon}$. In any case, contains the error term in $\log{|\!\log{\varepsilon}|}/{|\!\log{\varepsilon}|}$ which implies that only yield the inequality $\Sigma^0 +r_a^t \leq \gamma_0$ for times at most of order $\log{|\!\log{\varepsilon}|}.$
$iii)$ One could supplement the claims of Theorem \[thm:main\] with closeness estimate for $j(v)$ to a reference field $j_*$ of very simple form. This would follow from an application of Corollary \[cor:proche\] below; however at the level of approximation which we have adopted here it is only meaningful in a neighborhood of size $o(1/{|\!\log{\varepsilon}|})$ of the vortex core.
Elements in the proofs {#sec:heuristics}
----------------------
Under the conditions that will prevail throughout most of this paper, we will be able to identify points $\xi_1^t,\ldots \xi_l^t$ and a number $r_\xi^t$ such that $$\| Jv^t - \pi\sum_{i=1}^l d_i \delta_{\xi_i^t}\|_{W^{-1,1}(\Omega)}\ \le r^t_\xi \ \le \ {\varepsilon}^{1/2}\ \ll \ r^t_a.
\label{heuristics0}$$ This is expressed in Proposition \[prop:loca\] below, and entitles us to think of $\xi_i^t, i=1,\ldots, l$ as being the “actual locations of the vortices" in $v^t$, up to precision of order $\le r^t_\xi$. Admitting this interpretation, then basic facts about the $W^{-1,1}$ norm, recalled in Section \[sect:w11\], imply that $$r^t_a = \frac 1 \pi (1 + o(1))
\sum_{i=1}^l |\xi_i^t -a_i^t|
\label{heuristics1}$$ is essentially the aggregate distance between the actual vortex locations and the desired vortex locations, as remarked above.
Heuristic considerations also suggest that if $v^t$ is a function with vortices at points $\xi_1^t,\ldots, \xi_l^t$ (or more precisely, if holds), then $$\label{heuristics2}
E_{{\varepsilon}, \eta}(v^t) \gtrapprox \pi {|\!\log{\varepsilon}|}(1-o(1)) \sum_{i=1}^l \eta^2(\xi_i^t).$$ Hence $E_{{\varepsilon}, \eta}(v^t) - \pi {|\!\log{\varepsilon}|}\sum_{i=1}^l \eta^2(\xi_i^t)$ corresponds to energy that is not committed to the vortices, and this energy in principle can cause difficulties for our analysis. From , , we have $$\label{heuristics3}\begin{split}
E_{{\varepsilon}, \eta}(v^t) - \pi {|\!\log{\varepsilon}|}\sum_{i=1}^l \eta^2(\xi_i^t)
&\le
{|\!\log{\varepsilon}|}\left(\Sigma^t + \frac 1{\pi}(1+o(1))\| \nabla \eta^2\|_\infty r^t_a\right)\\
&\leq {|\!\log{\varepsilon}|}\left( \Sigma^0 +C r^t_a\right).
\end{split}$$ For our analysis, it suffices to use estimates in the spirit of that are a little weaker than those suggested in , these are established in Proposition \[prop:relate\]. We expect from and that control of $r^t_a$ should yield a good deal of information about $v^t$. This is expressed in Proposition \[prop:approx\], where we compare $j(v^t)$ to a reference field $j_*^t.$ An important feature of that approximation is that it holds across the vortex core.
In order to control the evolution in time of $r_a^t$, we rely on some evolution equations satisfied by smooth solutions of . Conservation of energy is a consequence of the identity $$\label{eq:conserve_e}
\partial_t e_{{\varepsilon},\eta}(v) = \ {\rm div}( \eta^2 (\nabla v, v_t)),$$ and the canonical equation for conservation of mass can be written $$\label{eq:continuity}
\frac {|\!\log{\varepsilon}|}2 \partial_t \big (\eta^2(|v|^2-1)\big) =
\nabla \cdot (\eta^2 j(v)) .$$ The vorticity $Jv$ satisfies an evolution equation that it is convenient to write in integral form: $$\frac{d}{dt} \int_\Omega \varphi Jv = \frac 1 {|\!\log{\varepsilon}|}\int_\Omega
\left( {\epsilon}_{lj} \varphi_{x_l}
\frac{\eta^2_{x_k}}{\eta^2} \left[ v_{x_j}\cdot v_{x_k} + \delta_{jk} \frac{\eta^2}{{\varepsilon}^2}
(|v|^2-1)^2\right] + {\epsilon}_{lj} \varphi_{x_kx_l} v_{x_j}\cdot v_{x_k}\right)
\label{eq:evoljac}$$ where $\varphi$ is any smooth, compactly supported test function and ${\varepsilon}_{lj}$ is the usual antisymmetric tensor. This follows from the fact that $Jv = \frac 12 {\rm curl}\: j(v)$ together with the equation for the evolution of $j(v)$, which is obtained from after multiplying by $\nabla v$ and rewriting the result.
Identity is central to our analysis of vortex dynamics, as in previous works [@CoJe; @CoJe2; @LiXi; @BeJeSm; @JeSp] on the homogeneous case (for which of course still holds, with $\eta \equiv 1$). Under the conditions that $Jv$ is approximately a measure of the form $\pi \sum_{i=1}^l d_i \delta_{\xi_i(t)}$, where $\xi_i(t)$ are the vortex locations and $d_i\in \{\pm 1\}$ their signs, one expects the left-hand side of to satisfy $$\frac{d}{dt} \int_\Omega \varphi Jv
\ \approx \
\frac{d}{dt} \int_\Omega \varphi (\pi \sum d_i \delta_{\xi_i(t)})
\ \approx \
\frac d{dt}( \pi \sum d_i \varphi(\xi_i(t)))
\ = \
\pi \sum d_i \nabla\varphi(\xi_i(t))\cdot \dot \xi_i(t).$$ Assuming that this holds, then to understand the vortex velocities $\dot \xi_i$, it only suffices to understand the right-hand side of . It turns out that it also suffices to consider test functions $\varphi$ that are linear near each vortex. For such test functions, in the homogeneous case $\nabla \eta^2\equiv 0$, the integrand on the right-hand side of is supported away from the vortex locations, and one is thus able to control vortex dynamics by controlling terms of the form $v_{x_i}\cdot v_{x_j}$ away from the vortex cores. This argument is a key feature of all existing work on vortex dynamics in the homogeneous case.
When $\nabla \eta^2\ne 0$, it becomes necessary to control terms like $v_{x_i}\cdot v_{x_j}$ across the vortex cores. Carrying this out, in particular relying on the approximation given by Proposition \[prop:approx\], is the main new point in our analysis. Once this is established, the whole argument is completed by using a Gronwall type argument on a quantity related to $r_a^t$, namely $\Sigma^0+g(r_a^t)$, where the function $g$b is defined in . This demonstrates in particular that the new information found in Proposition \[prop:approx\] is strong enough to close the estimates and conclude the proof.
This research was partially supported by the National Science and Engineering Research Council of Canada under operating Grant 261955, as well as by the projects “Around the dynamics of the Gross-Pitaevskii equation” (JC09-437086) and “Schrödinger equations and applications” (ANR-12-JS01-0005-01) of the Agence Nationale de la Recherche.
A useful lemma {#sect:w11}
==============
We frequently use the $W^{-1,1}$ norm. The specific convention we use is in our definition is $$\| \mu \|_{W^{-1,1}(\Omega)} := \sup \{ \langle \mu, \varphi \rangle \ : \ \varphi\in W^{1,\infty}_0(\Omega),
\max \{ \| \varphi\|_\infty, \|\nabla \varphi\|_\infty\} \le 1\} .$$ In this paper, we will only use this norm on measures or more regular objects, although of course it is well-defined for a somewhat larger class of distributions.
The following lemma, which we will use numerous times, is an easy special case of classical results (see [@BCL] for example).
Suppose that $\Omega$ is an open subset of ${\mathbb{R}}^n$, and that $\{ a_i\}_{i=1}^l$ are distinct points in $\Omega$. Define $\rho_a := \min \{ \{\frac 12 |a_i -a_j| \}_{i\ne j} \cup \{ d(a_i, \partial \Omega )\}_i \cup \{1\}\}$. Given any points $\{ \xi_i\}_{i=1}^l$ in $\Omega$ and $\{ d_i\}_{i=1}^l \in \{\pm 1\}^l$, if $$\label{eq:W11.1}
\| \sum_{i=1}^l d_i \delta(a_i - \xi_i) \|_{W^{-1,1}(\Omega)} \le \frac 14 \rho_a,$$ then (after possibly relabelling the points $\{ \xi_i\}_{i=1}^l$), $$\label{eq:W11.2}
\| \sum_{i=1}^l d_i \delta(a_i - \xi_i) \|_{W^{-1,1}(\Omega)} = \sum_{i=1}^l |a_i - \xi_i|.$$ \[lem:W-11\]
In the remainder of this paper, we will always tacitly assume that under the conditions of the lemma, the points $\xi_i$ are labelled so that the conclusion holds.
We give the short proof for the reader’s convenience.
For $i=1,\ldots, l$, define $\varphi_i(x) := d_i( \frac 12 \rho_a - |x-a_i|)^+$. Then $\max( \|\varphi_i\|_\infty, \|\nabla\varphi_i\|_\infty ) =\max( \frac 12 \rho_a, 1) =1$, for every $i$, so $$\| \sum_{i=1}^l d_i \delta(a_i - \xi_i) \|_{W^{-1,1}(\Omega)} \ge \langle \sum_{j=1}^l d_j (\delta_{a_j} - \delta_{\xi_j})
, \varphi_i\rangle = \frac {\rho_a} 2- \sum_j d_id_j (\frac {\rho_a} 2 -
|\xi_j-a_i|)^+.$$ Then implies that $\{ \xi_j \}_{j=1}^l \cap B(a_i,\rho_a/2)$ is nonempty for every $i$. Since $\{ B(a_i,\rho_a/
2) \}_{i=1}^l$ are pairwise disjoint, it follows (after possibly reindexing) that $\{ \xi_j \}_{j=1}^l \cap B(a_i,\rho_a/2) = \{\xi_i\}$ for all $i$. Now let $\varphi = \sum_i\varphi_i$. The functions $\{ \varphi_i\}$ have disjoint support, so $\max( \|\varphi\|_\infty, \|\nabla\varphi\|_\infty ) =1$, and thus $
\| \sum_{i=1}^l d_i \delta(a_i - \xi_i) \|_{W^{-1,1}(\Omega)} \ge \langle \sum_{i=1}^l d_i (\delta_{a_i} - \delta_{\xi_i})
, \varphi \rangle= \sum_{i=1}^l |a_i - \xi_i|$. On the other hand, if $\psi$ is any compactly supported function such that $\max( \|\psi\|_\infty, \|\nabla\psi\|_\infty ) \le 1$, then $$\langle \sum_{i=1}^l d_i (\delta_{a_i} - \delta_{\xi_i}), \psi \rangle\le
\sum_{i=1}^l |\psi(a_i) - \psi(\xi_i)| \le
\sum_{i=1}^l |a_i-\xi_i|.$$ Hence $\| \sum_{i=1}^l d_i \delta(a_i - \xi_i) \|_{W^{-1,1}(\Omega)} \le
\sum_{i=1}^l |a_i-\xi_i|$.
Relating weighted and unweighted energy {#sect:relating}
=======================================
In this section, we relate the weighted and unweighted energy under some localization assumptions on the Jacobian. For a measurable subset $A \subset {\mathbb{R}}^2$ and $v \in \dot H^1(A,{\mathbb{C}})$ we set $$E_{{\varepsilon},\eta}(v;A) := \int_A e_{{\varepsilon},\eta}(v) \quad\text{and}\quad E_{\varepsilon}(v;A):= E_{{\varepsilon},1}(v,A).$$ Define the function $g$ on ${\mathbb{R}}^+$ by $$\label{eq:defg}
g(x) = \left\{ \begin{array}{ll}
x + \frac{|\! \log x|}{{|\!\log{\varepsilon}|}} & \text{if } x>\frac{1}{{|\!\log{\varepsilon}|}}\\
\frac{1+ \log{|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}} & \text{otherwise.}
\end{array}
\right.$$ We have
\[prop:relate\] Let $\Omega\subset {\mathbb{R}}^2$ an open set, $\{a_i\}_{i=1}^l$ distinct points in $\Omega$, $\{d_i\}_{i=1}^l \in \{\pm 1\},$ and $\eta : \Omega \to {\mathbb{R}}$ a positive Lipschitz function such that $\inf_{\Omega}\eta =: \eta_{\min} >0$. Set $\rho_a :=
\min\{\{ \frac 12 d(a_i,a_j)\}_{\ i\neq j}\cup \{d(a_i,\partial \Omega)\}_{i} \cup \{1\}
\},$ and let ${\varepsilon}\leq \exp(-\frac{8}{\rho_a})$ and $v\in \dot
H^1(\Omega,{\mathbb{C}})$ be such that $$\label{eq:surplus}
\Sigma_a := \left( \frac{{{E}_{\varepsilon,\eta}}(v)}{{|\!\log{\varepsilon}|}} - \pi \sum_{i=1}^l \eta^2(a_i) \right)^+
< +\infty.$$ Assume also that $$\label{eq:localisee}
r_a := \| J v - \pi \sum_{i=1}^l d_i \delta_{a_i}\|_{W^{-1,1}(\Omega)} \leq
\frac{\rho_a}{8}.$$ Then there exists a constant $C$, depending only on $l$, $\|\nabla \eta^2\|_\infty$ and $\eta_{min}$, such that $$\label{eq:usuelle}
\begin{aligned}
\frac{E_{\tilde {\varepsilon}}(v; B(a_i, R))}{|\!\log\tilde{\varepsilon}|}
&\leq
\pi + C(\Sigma_a + g(r_a))\quad\quad\mbox{ for }i=1,\ldots, l\\
\frac{E_{\tilde {\varepsilon}}(v;\Omega\setminus \cup_{i=1}^l B(a_i, R))}{|\!\log\tilde{\varepsilon}|}
&\le
C(\Sigma_a + g(r_a))
\end{aligned}$$ where $R = 4\max( r_a, {|\!\log{\varepsilon}|}^{-1}) \le \frac{\rho_a}4$ and $\tilde{\varepsilon}:= \frac{{\varepsilon}}{\eta_{min}}$, and the function $g$ is defined in .
Let $r\in [r_a, \frac{\rho_{a}}{8}]$ be a number that will be fixed later. Then the balls $\{B(a_i,4r)\}_{i=1}^l$ are disjoint and contained in $\Omega.$ Let $i\in
\{1,\cdots,l\},$ by monotonicity of the $W^{-1,1}$ norms with respect to the domain, we deduce from that $$\| Jv - \pi d_i \delta_{a_i}\|_{W^{-1,1}(B(a_i,4r)} \leq r_a \le r.$$ It follows from the lower bounds estimates of Jerrard [@Je] or Sandier [@Sa] that $$\label{eq:lowerbound}
E_\delta(v,B(a_i,4r)) \geq \pi \log \frac{4r}{\delta} - K_1,$$ for every $\delta>0$, where $K_1$ is a universal constant. We next write $$\label{eq:borneinfeta}
\begin{split}
{{E}_{\varepsilon,\eta}}(v,B(a_i,4r)) &= \int_{B(a_i,4r)} \eta^2(x) \frac{|\nabla v|^2}{2}
+ \eta^4(x) \frac{(|v|^2-1)^2}{4{\varepsilon}^2}\\
&\geq \int_{B(a_i,4r)} \eta^2(x) \big[ \frac{|\nabla v|^2}{2} +
\frac{(|v|^2-1)^2}{4\left(\frac{{\varepsilon}}{\eta_{min}}\right)^2}\big]\\
&\geq \left( \inf_{x\in B(a_i,4r)} \eta^2(x)\right) E_{\tilde
{\varepsilon}}(v,B(a_i,4r)).
\end{split}$$ Therefore, from with the choice $\delta=\tilde{\varepsilon}$, and noting that $|\log r| \ge \log |\frac{\rho_a}{8}|
\ge \log 8 \ge 1$, we obtain $$\label{eq:borneinfetabis}
{{E}_{\varepsilon,\eta}}(v,B(a_i,4r)) \geq \eta^2(a_i) \pi {|\!\log{\varepsilon}|}- K_2 \left( r{|\!\log{\varepsilon}|}+ |\!\log r| \right),$$ where $K_2$ depends only on $\|\nabla\eta^2\|_\infty$ and $\eta_{min}.$
On the other hand, we deduce from and that $$\label{eq:bornesupeta}
\begin{split}
{{E}_{\varepsilon,\eta}}(v,B(a_i,4r)) &\leq {{E}_{\varepsilon,\eta}}(v,\Omega) - \sum_{j\neq i}
{{E}_{\varepsilon,\eta}}(v,B(a_j,4r))\\
&\leq \pi \eta^2(a_i){|\!\log{\varepsilon}|}+ \Sigma_a{|\!\log{\varepsilon}|}+ (l-1)K_2 \left( r{|\!\log{\varepsilon}|}+ |\!
\log r|\right).
\end{split}$$ Hence, going back to we obtain $$\label{eq:onballi}
\begin{split}
E_{\tilde {\varepsilon}}(v,B(a_i,4r)) &\leq \frac{1}{\left( \inf_{x\in B(a_i,4r)}
\eta^2(x)\right)} {{E}_{\varepsilon,\eta}}(v,B(a_i,4r))\\
&\leq \pi |\!\log \tilde{\varepsilon}| + K_3 (\Sigma_a {|\!\log{\varepsilon}|}+ r{|\!\log{\varepsilon}|}+ |\!\log r|),
\end{split}$$ where $K_3$ depends only on $l$, $\|\nabla\eta^2\|_\infty$ and $\eta_{min}.$
Concerning the energy outside the balls $B(a_i,4r)$, we have from and $$\label{eq:out}
\begin{split}
{{E}_{\varepsilon,\eta}}(v,\Omega \setminus \cup_iB(a_i,4r)) &= {{E}_{\varepsilon,\eta}}(v,\Omega) - \sum_{i}
{{E}_{\varepsilon,\eta}}(v,B(a_j,4r))\\
&\leq \Sigma_a{|\!\log{\varepsilon}|}+ lK_2 \left( r{|\!\log{\varepsilon}|}+ |\! \log r| \right).
\end{split}$$ Hence, $$\label{eq:outsideballs}
\begin{split}
E_{\tilde {\varepsilon}}(v,\Omega \setminus \cup_i B(a_i,4r)) &\leq \frac{1}{\inf
\eta^2} {{E}_{\varepsilon,\eta}}(v,\Omega\setminus \cup_i B(a_i,4r))\\
&\leq K_4 (\Sigma_a {|\!\log{\varepsilon}|}+ r{|\!\log{\varepsilon}|}+ |\!\log r|),
\end{split}$$ where $K_4$ depends only on $l$, $\|\nabla\eta^2\|_\infty$ and $\eta_{min}.$
The function $r\mapsto r + |\log r|/{|\!\log{\varepsilon}|}$ is minimized taking $r := \max(r_a,\frac{1}{{|\!\log{\varepsilon}|}})$, in which cas $r\leq \frac{\rho_{a}}{8}$ by assumption on $r_a$ and ${\varepsilon}.$ The conclusions follow with the choice $C := \max(K_3,K_4).$
If we define $\tilde \Sigma_a := \frac{{{E}_{\varepsilon,\eta}}(v)}{{|\!\log{\varepsilon}|}} - \pi \sum_{i=1}^l \eta^2(a_i)$ , then implies that $$\tilde \Sigma_a \ge \sum_i \left(\frac { E_{{\varepsilon},\eta}(v, B(a_j, 4r))}{|\!\log{\varepsilon}|}- \pi \eta^2(a_i)\right)
\ge - l K_2(r + \frac {|\log r|}{|\!\log{\varepsilon}|})$$ for every $r\in [r_a, \frac {\rho_a}{8}]$. Choosing $r = \max(r_a,\frac{1}{{|\!\log{\varepsilon}|}})$ as above, we find that $\tilde \Sigma_a \ge - l K_2 g(r_a)$. In particular, $\Sigma_a = (\tilde \Sigma_a)^+ \le \tilde \Sigma_a + 2 l K_2 g(r_a)$. So all our estimates remain true if we replace $C(\Sigma_a + g(r_a))$ by $C(\tilde \Sigma_a + (2lK_2+1) g(r_a)).$
Improved localization for Jacobians {#sect:imploc}
===================================
In this section, we prove that if the Jacobian of a function $v$ is known to be sufficiently localized, then, provided the excess energy of $v$ with respect to the points of localization is not to big, the localisation is actually potentially much stronger. A result in the same spirit was obtained by Jerrard and Spirn in [@JeSp0] for the Ginzburg-Landau functional without a weight. Our proof here below makes a direct use of Theorem $1.1$ and Theorem $1.2'$ in [@JeSp0] by relating the weighted and unweighted Ginzburg-Landau energies according to Section \[sect:relating\].
\[prop:loca\] Let $\Omega\subset {\mathbb{R}}^2$ be a bounded, open set, $\{a_i\}_{i=1}^l$ distinct points in $\Omega$, $\{d_i\}_{i=1}^l \in \{\pm 1\},$ and $\eta : \Omega \to {\mathbb{R}}$ a positive Lipschitz function such that $\inf_{\Omega}\eta =: \eta_{\min} >0$. Set $\rho_a =
\min\big\{ \{ \frac 12d(a_i,a_j)\}_{\ i\neq j}\cup \{d(a_i,\partial \Omega)\}_{i} \cup \{1\}
\big\}.$ Let ${\varepsilon}\leq \exp(-\frac{8}{\rho_a})$ and let $v\in \dot
H^1(\Omega,{\mathbb{C}})$ be such that $$\label{eq:surplus2}
\Sigma_a := \left( \frac{{{E}_{\varepsilon,\eta}}(v)}{{|\!\log{\varepsilon}|}} - \pi \sum_{i=1}^l \eta^2(a_i) \right)^+ < +\infty.$$ Also, assume that $$\label{eq:localmoins}
r_a = \| J v - \pi \sum_{i=1}^l d_i \delta_{a_i}\|_{W^{-1,1}(\Omega)} \leq
\frac{\rho_a}{16}.$$
Then there exists $C_1 \ge 1$, depending only on a lower bound for $\rho_a$ and $\eta_{min}$ and on an upper bound for $l$ and $\|\nabla\eta^2\|_\infty$, and for each $i\in\{1,\cdots,l\}$ there exists a point $\xi_i \in B(a_i,2r_a)$, such that $$\label{eq:localplus}
\| J v - \pi \sum_{i=1}^l d_i \delta_{\xi_i}\|_{W^{-1,1}(\Omega)}
\leq r_\xi \equiv r_\xi(\Sigma_a,r_a) \equiv {\varepsilon}\exp(C_1(\Sigma_a +
g(r_a)){|\!\log{\varepsilon}|}).$$ where $g$ is defined in Proposition \[prop:relate\].
Note that Lemma \[lem:W-11\] and , imply that $$\label{eq:aminusxi}
\sum_{i=1}^l |a_i - \xi_i| \le \frac 1\pi(r_a + r_\xi).$$
Since $g(r) \ge \frac {\log{|\!\log{\varepsilon}|}}{|\!\log{\varepsilon}|}$ for every $r$, our requirement that $C_1\ge 1$ implies that $$\label{C0gtr1}
r_\xi \ge {\varepsilon}{|\!\log{\varepsilon}|}.$$
As mentioned, the proof of Proposition \[prop:loca\] relies very heavily on estimates from [@JeSp0]. Following the proof, we discuss some small adjustments we have made in employing these estimates here. Also, from here upon in many places we will denote by $C$ constants whose actual value may change from line to line but which could eventually be given a common value depending only on $l$, $\rho_{min}$, $\eta_{min}$ and $\|\nabla \eta^2\|_\infty.$
Since ${\varepsilon}\leq \exp(-\frac{8}{\rho_a})$, our assumptions imply that the hypotheses of Proposition \[prop:relate\] are verified. Then, since $B(a_i, \frac {\rho_a}2) \subset B(a_i, R) \cup (\Omega\setminus \cup_{i=1}^l B(a_i,R))$ for any $R<\frac {\rho_a}2$, and recalling that $g(r) \ge \frac { \log{|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}}$ for all $r$, we deduce from that $$\label{eq:usualb}
\frac{E_{\tilde {\varepsilon}}(v; B(a_i, \frac{\rho_a}2 ))}{\!\log(\frac{\rho_a}{2\tilde{\varepsilon}})}
\leq
\frac{E_{\tilde {\varepsilon}}(v; B(a_i, \frac{\rho_a}2 ))}{|\!\log \tilde{\varepsilon}|}
(1+ 2 \frac{ |\log \frac{\rho_a}2|}{|\!\log\tilde{\varepsilon}|})
\le
\pi + C(\Sigma_a +g(r_a))$$ for $i=1,\ldots, l$, and similarly implies that $$\label{eq:usualc}
\frac{E_{\tilde {\varepsilon}}(v; \Omega \setminus \cup_{i=1}^l B(a_i, \frac{\rho_a}4 ))}{|\!\log\tilde{\varepsilon}|} \leq C(\Sigma_a + g(r_a)).$$ According to Theorem 1.2’ in [@JeSp0], it follows from and that for every $i\in \{1,\ldots,l\}$, there exists some $\xi_i\in B(a_i, 2 r_a)$ such that $$\label{eq:findxi}
\| Jv - \pi d_i \delta_{\xi_i} \|_{W^{-1,1}(B(a_i, \frac{\rho_a}2))} \le
C \,\tilde {\varepsilon}\exp[C (\Sigma_a + g(r_a)){|\!\log{\varepsilon}|}].$$ In addition, Theorem 1.1 in [@JeSp0] implies that if $V$ is any bounded, open subset of $\Omega$ then $$\label{eq:citeJeSp}
\|Jv\|_{W^{-1,1}(V)}
\ \le \ C \, \tilde{\varepsilon}\, E_{\tilde{\varepsilon}}(v, V) \exp\left( \frac{ E_{\tilde{\varepsilon}}(v, V)}{\pi}\right)
\ \le \
C \, \tilde{\varepsilon}\, \exp\left( E_{\tilde{\varepsilon}}(v, V) \right) .$$ In particular, this and imply that $$\label{eq:complement}
\| J v \|_{W^{-1,1}(\Omega\setminus \cup_{i=1}^l B(a_i \frac{ \rho_a}4))} \le
C \,\tilde {\varepsilon}\exp[C (\Sigma_a + g(r_a)){|\!\log{\varepsilon}|}].$$ Now for $i\in \{1,\ldots,l\}$, let $\chi_i\in C^\infty_c(B(a_i, \frac {\rho_a}2))$ be functions such that $\chi_i = 1$ on $B(a_i \frac{\rho_a}4)$, $0\le \chi_i\le 1$, and $\|\nabla \chi_i\|_\infty \le C\rho_a^{-1}$. Also, let $\chi_0 = 1-\sum_{i=1}^l\chi_i$. Then for any $\varphi\in C^\infty_0(\Omega)$, such that $\| \varphi\|_{W^{1,\infty}(\Omega)}\le 1$, $$\begin{aligned}
\langle \varphi, Jv - \pi \sum_{i=1}^l d_i \delta_{\xi_i} \rangle
&=
\sum_{j=0}^l
\langle \chi_j \varphi, Jv - \pi \sum_{i=1}^l d_i \delta_{\xi_i} \rangle
\\
&= \langle \chi_0 \varphi, Jv \rangle +
\sum_{i=1}^l \langle \chi_i \varphi, Jv - \pi d_i \delta_{\xi_i} \rangle
\\
&\le \sum_{i=1}^l \|\chi_i \varphi \|_{W^{1,\infty}} C {\varepsilon}\exp[ C (\Sigma_a+g(r_a)){|\!\log{\varepsilon}|})\end{aligned}$$ where we have used for $i=1,\ldots, l$ and for $i=0$. Thus $$\langle \varphi, Jv - \pi \sum_{i=1}^l d_i \delta_{\xi_i} \rangle
\le
\frac C{\rho_a} {\varepsilon}\exp[ C (\Sigma_a+g(r_a)){|\!\log{\varepsilon}|}]\\
\le
{\varepsilon}\exp[ C_1 (\Sigma_a+g(r_a)){|\!\log{\varepsilon}|}]$$ for a suitable $C_1$, depending on the lower bound $\rho_0$ for $\rho_a$ as well as $l, \eta_{min}, \|\nabla\eta^2\|_\infty$. This implies .
To facilitate comparison between some facts that we have used above and the precise statements in [@JeSp0], we make the following remarks.
First, we have used some estimates in cruder but simpler forms than they appear in [@JeSp0]. For example, on the right-hand side of , we have replaced an expressions of the form $(C + K_0)^2 \exp( \frac {K_0} \pi)$ from [@JeSp0], where here we take $K_0 = C(\Sigma_a+ g(r_a)){|\!\log{\varepsilon}|}$, by the simpler expression $C \exp( K_0)$. We have also used the fact that $K_0 = C(\Sigma_a+ g(r_a)){|\!\log{\varepsilon}|}\ge \log{|\!\log{\varepsilon}|}$ to allow us to absorb some lower-order terms from [@JeSp0].
Second, estimates in [@JeSp0] are stated in terms of a slightly different norm, $\| \mu \|_{\dot W^{-1.1}(V)} :=
\sup\{ \langle \mu, \phi \rangle : \phi\in C^\infty_c(V), \|\nabla \phi\|_\infty \le 1 \}$. This does not cause any problems for us, since clearly $\|\mu\|_{W^{-1,1}(V)} \le \|\mu \|_{\dot W^{-1,1}(V)}$.
Finally, the estimate corresponding to in [@JeSp0] is a special case of a more general result, and as stated there requires the additional assumption that $\frac{E_{\tilde{\varepsilon}}(v, V)}{|\!\log\tilde {\varepsilon}| }
<\pi$. However, since $\| J v\|_{W^{-1,1}(V)}\le \|Jv\|_{L^1(V)} \le 2 E_{\tilde {\varepsilon}}(v; V)$, it is clear that is still true if $\frac{E_{\tilde{\varepsilon}}(v, V)}{|\!\log\tilde {\varepsilon}| }
\ge \pi$.
If $\Omega$ is simply connected, then we can alternately argue by citing a result from [@JeSp] to obtain an estimate of the form with $C_1$ [*independent of $\rho_a$*]{}, at the rather small expense of having to replace $\frac {\rho_a}{16}$ on the right-hand side of by some smaller quantity depending on $l$ as well as $\rho_a$. This is in principle useful if one wants to consider large numbers of vortices. The relevant result (Theorem 3) from [@JeSp] is proved using facts from [@JeSp0], as in the proof above, but combining estimates on the balls and away from the balls in a more careful way, to avoid introducing the factors of $\rho_a^{-1}$ that arise from the cutoff functions that we have employed here.
The proof of Theorem 3 from [@JeSp] can surely be adapted to yield a similar result without the assumption that $\Omega$ be simply connected, but since the proof is slightly complicated, we prefer not to tinker with it here.
Across the core approximation by reference field {#sect:jstar}
================================================
In this section we prove
\[prop:approx\] Let $\Omega\subset {\mathbb{R}}^2$ be an open set, $\{\xi_i\}_{i=1}^l$ distinct points in $\Omega$, $\{d_i\}_{i=1}^l \in \{\pm 1\},$ and $\eta : \Omega \to {\mathbb{R}}$ a positive Lipschitz function such that $\inf_{\Omega}\eta =: \eta_{\min} >0.$ Set $\rho_\xi =
\min\big\{\{\frac 12 d(
\xi_i,\xi_j)\}_{i\neq j}\cup \{d(\xi_i,\partial \Omega)\}_{i} \cup \{1\}\big\}.$ Let ${\varepsilon}\leq \exp(-\frac{8}{\rho_\xi})$ and let $v\in \dot
H^1(\Omega,{\mathbb{C}})$ be such that $$\label{eq:surplusbis}
\Sigma_\xi = \left( \frac{{{E}_{\varepsilon,\eta}}(v)}{{|\!\log{\varepsilon}|}} - \pi \sum_{i=1}^l \eta^2(\xi_i) \right)^+ <
+\infty$$ and $$\label{eq:localiseebis}
\| J v - \pi \sum_{i=1}^l d_i \delta_{\xi_i}\|_{W^{-1,1}(\Omega)}
\leq r_\xi \equiv {\varepsilon}\exp(K{|\!\log{\varepsilon}|}) = {\varepsilon}^{1-K} $$ for some $K\leq \frac{1}{2}.$
Define $j_* = j_*(\{\xi_i\}, r_\xi)$ in $\Omega$ by $$j_*(x) =\left\{\begin{array}{ll}
d_i \frac{(x-\xi_i)^\perp}{\max (r_\xi, |x-\xi _i|)^2 } &\text{if } x \in B(\xi_i,\frac{1}{{|\!\log{\varepsilon}|}})\\
0 &\text{if } x \in \Omega \setminus \cup_{i=1}^l B(\xi_i,\frac{1}{{|\!\log{\varepsilon}|}})
\end{array}\right.$$ where $(y_1, y_2)^\perp := (-y_2, y_1)$. Then $$\label{eq:estimparsurplus}
E_{{\varepsilon},\eta}(|v|) + \frac{1}{2}\int_\Omega \eta^2 \left|
\frac{j(v)}{|v|}- j_*\right|^2 \leq
(C \Sigma_\xi + K){|\!\log{\varepsilon}|}+
C\log{|\!\log{\varepsilon}|},$$ and $$\label{eq:JminusJstar}
\| \nabla\times (j(v) - j_*) \|_{W^{-1,1}} \le C_3 r_\xi.$$ where the constant $C$ depends only on $l,$ $\|\nabla \eta^2\|_\infty$ and $\eta_{min}.$
Since $K\le \frac 12$, the assumption that ${\varepsilon}< \exp(- 8/\rho_\xi)$ implies that $r_\xi < \frac 1{|\!\log{\varepsilon}|}< \frac {\rho_\xi}8$. In particular, the balls $B(\xi_i, {|\!\log{\varepsilon}|}^{-1})$, $i=1,\ldots, l$ are pairwise disjoint and contained in $\Omega$.
We will use more than once the fact that $$\label{gradsplit}
|\nabla v|^2 = |\nabla |v|\,|^2 + \left| \frac {j(v)|}{|v|}\right|^2.$$
[**Step 1: verification of** ]{}. A direct calculation, using the definition of $j_*$, shows that for any smooth $\varphi$, $$\langle \varphi , \nabla\times j_* - 2 \pi \sum d_i \delta_{\xi_i} \rangle
= \sum_{i=1}^l \frac 2{r_\xi^2} \int_{B(\xi_i, r_\xi)} d_i(\varphi(x)- \varphi(\xi_i)) \ \le \,C\, l \|\nabla \varphi\|_\infty r_\xi.$$ Thus $\| \nabla\times j_* - 2 \pi \sum d_i \delta_{\xi_i} \|_{W^{-1,1}(\Omega)} \le C r_\xi$. Recalling that $Jv = \frac 12 \nabla \times j(v)$, we deduce from this estimate and our assumption .
It remains to prove .
[**Step 2: decomposing the energy.**]{} Note that our assumptions , about the points $\{ \xi_i\}_{i=1}^l$ are exactly the same as the hypotheses , about the points $\{ a_i\}_{i=1}^l$ in Proposition \[prop:relate\], except that here we impose an additional smallness condition on $r_\xi$. Thus estimates from Proposition \[prop:relate\] are all available here. In particular, recalling with the choice $r = \max(r_\xi, \frac 1{|\!\log{\varepsilon}|}) = \frac 1{|\!\log{\varepsilon}|}$, we see that $$E_{{\varepsilon},\eta}(v, \Omega\setminus \cup_i B(\xi_i, \frac 4 {|\!\log{\varepsilon}|}) ) \le
C(\Sigma_\xi {|\!\log{\varepsilon}|}+ \log{|\!\log{\varepsilon}|}).$$ In view of , and noting that $j_*$ is supported in $\cup_i B(\xi_i, \frac 1 {|\!\log{\varepsilon}|})$ to prove it therefore suffices to show that $$E_{{\varepsilon},\eta}\bigl(|v|, B(\xi_i, \frac 4{|\!\log{\varepsilon}|})\bigr)+
\frac 12 \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\eta^2 \left| \frac{j(v)}{|v|} - j_*\right|^2
\ \le \
(C \Sigma_\xi + K){|\!\log{\varepsilon}|}+
C\log{|\!\log{\varepsilon}|}\label{eq:jstar.reduce}$$ for $i=1,\ldots, l$. Toward this end, note that $$\begin{aligned}
{|\!\log{\varepsilon}|}\left[\pi \eta^2(\xi_i) +\Sigma_\xi + C \frac{ \log{|\!\log{\varepsilon}|}}{|\!\log{\varepsilon}|}\right]
&\overset{\eqref{eq:bornesupeta}}
\ge
E_{{\varepsilon},\eta}(v, B(\xi_i, \frac 4{|\!\log{\varepsilon}|}) )\\
&\overset{\eqref{gradsplit}}=
E_{{\varepsilon},\eta}\bigl(|v|, B(\xi_i, \frac 4{|\!\log{\varepsilon}|})\bigr)+
\frac 12 \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\!\!\!\!\eta^2 \left| \frac{j(v)}{|v|} - j_*\right|^2
\\&\quad\quad
+\frac 12 \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\eta^2 \left| j_*\right|^2
+\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\!\!\!\!\! \eta^2 (\frac {j(v)}{|v|} - j_*)\cdot j_*.
$$ Using the explicit form of $j_*$ and of $r_\xi$, $$\begin{aligned}
\frac 12 \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\eta^2 \left| j_*\right|^2
&\ge \frac 12 \ \min_{B(\xi_i, 4{|\!\log{\varepsilon}|}^{-1})}\eta^2 \
\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})} \left| j_*\right|^2 \\
&\ge
\bigl( \eta^2(\xi_i) - C\|\nabla\eta^2\|_\infty {|\!\log{\varepsilon}|}^{-1} \bigr)
\pi \log \frac {{|\!\log{\varepsilon}|}^{-1}}{r_\xi}\\
&=
\bigl(\pi \eta^2(\xi_i) - C\frac {\log{|\!\log{\varepsilon}|}}{|\!\log{\varepsilon}|}\bigr) {|\!\log{\varepsilon}|}(1-K).\end{aligned}$$ By combining the previous two inequalities and rearranging, we see that to prove , it suffices to check that $$\left|
\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})} \eta^2 (\frac {j(v)}{|v|} - j_*)\cdot j_*
\right| \le C \bigl( \Sigma_\xi {|\!\log{\varepsilon}|}+
\log{|\!\log{\varepsilon}|}\bigr).
\label{eq:jstar.reduce2}$$
[**Step 3: proof of .**]{}
First note that $$\begin{aligned}
\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})} \eta^2 (\frac {j(v)}{|v|} - j_*)\cdot j_*
&=
\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\bigl(\eta^2(x) - \eta^2(\xi_i)\bigr)(\frac {j(v)}{|v|} - j_*)\cdot j_*
\\
& \quad
+
\eta^2(\xi_i) \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}
\frac {j(v)}{|v|} \cdot j_*\
\left(1- |v| \right)
\\
& \quad
+
\eta^2(\xi_i) \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}
({j(v)} - j_*) \cdot j_*\ . \end{aligned}$$ We estimate the three terms on the right-hand side in turn. First, $$\begin{aligned}
|\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}\bigl(\eta^2(x) - \eta^2(\xi_i)\bigr)(\frac {j(v)}{|v|} - j_*)\cdot j_*| \
& \le \
\frac C {|\!\log{\varepsilon}|}\| \nabla \eta^2\|_\infty ( \| \nabla v\|_2^2 + \| j_*\|_2^2)
\\
& \le \ C\left[ \frac{E_{{\varepsilon},\eta}(v) }{|\!\log{\varepsilon}|}+ (1- K + {|\!\log{\varepsilon}|}^{-1}) \right]\\
\\
&\le
C(\Sigma_\xi + \log{|\!\log{\varepsilon}|}),\end{aligned}$$ where we have used the fact that ${|\!\log{\varepsilon}|}^{-1}E_{{\varepsilon},\eta}(v)
\overset{\eqref{eq:surplusbis}}\le C(\Sigma_\xi + l \pi \|\eta^2\|_\infty) \le C(\Sigma_\xi + \log{|\!\log{\varepsilon}|})$.
Next, $$\begin{aligned}
\eta^2(\xi_i) |\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}
\frac {j(v)}{|v|} \cdot j_*\ \left(1- |v| \right)|
&\le
C \|j_*\|_\infty \int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})} \frac {\varepsilon}2 |\nabla v|^2 + \frac 1 {2{\varepsilon}} (|v|^2-1)^2\\
&\le
C r_\xi^{-1}{\varepsilon}E_{{\varepsilon},\eta}(v)
\\
&\le
C(\Sigma_\xi + \log{|\!\log{\varepsilon}|}),\end{aligned}$$ using the lower bound for $r_\xi$ and arguing as above.
To estimate the final term, note that $j_* = \nabla^\perp h$, for $$h(x):= \begin{cases}
d_i \big[ \frac 1{2r_\xi^2}( |x-\xi_i|^2 - 1) +\log(r_\xi {|\!\log{\varepsilon}|})\big]
&\mbox{ if }|x-\xi_i|\le r_\xi\\
d_i \log (|x-\xi_i| \,{|\!\log{\varepsilon}|})&\mbox{ if }r_\xi \le |x-\xi_i| \le {|\!\log{\varepsilon}|}^{-1}\\[3pt]
0&\mbox{ if }x\not\in \cup_i B(\xi_i, {|\!\log{\varepsilon}|}^{-1}).
\end{cases}$$ Thus, we can integrate by parts to find that $$\begin{aligned}
|\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}
({j(v)} - j_*) \cdot j_*\ |
&=
|\int_{B(\xi_i, \frac 4{|\!\log{\varepsilon}|})}
h\ \nabla \times ({j(v)} - j_*) |
\\
&\le\max( \|h \|_{\infty} , \|\nabla h\|_\infty )\ \| {j(v)} - j_* \|_{W^{-1,1}} \\
&\le C,\end{aligned}$$ after using and noting that $\|\nabla h\|_\infty = \|j\|_\infty = r_\xi^{-1}$. This completes the proof.
It follows from the definitions and of $\Sigma_\xi$ and $\Sigma_a$, together with , that $$\Sigma_\xi \le \Sigma_a + C g(r_a)$$ for $C$ depending only on $\|\nabla \eta^2\|_\infty$. Combining this with Propositions \[prop:loca\] and \[prop:approx\], we immediately obtain
\[cor:proche\]Under the assumptions of Proposition \[prop:loca\] $$\label{eq:estimparsurplusbis}
E_{{\varepsilon},\eta}(|v|) + \frac{1}{2}\int_\Omega \eta^2 \left|
\frac{j(v)}{|v|}- j_*\right|^2 \leq C (\Sigma_a +g(r_a)){|\!\log{\varepsilon}|},$$ where $j_*=j_*(\{\xi_i\},r_\xi)$, the points $\{\xi_i\}_{i=1}^l$ are given by Proposition \[prop:loca\], and the constant $C$ depends only on $l,$ $\rho_a$, $\|\nabla \eta^2\|_\infty$ and $\eta_{\min}.$
Small time upper bound on the speed of vortices
===============================================
Let $C_1$ be the constant given by Proposition \[prop:loca\] corresponding to the lower bound $\rho_{min}$ (as defined in ) for $\rho_a.$. Let also ${\varepsilon}\leq \exp(-\frac{8}{\rho_{min}}).$ Then the conclusions of Proposition \[prop:loca\], applied to $v^t$ with this choice of constants, are available to us for all $0 \le t \le T_{col}$. Since the conclusions of Proposition \[prop:loca\] remain true if we increase $C_1$, we may assume that $$\label{eq:gullit}
\frac{1}{C_1}
\leq \frac{\rho_{min}}{8},
$$ which we do in the sequel. We define the stopping time $$T_{loc} = \sup \{t\leq T_{col}\ ;\ \Sigma^0+g(r_a^s) \leq \frac{1}{2C_1},\
\forall\: 0\leq s \leq t\}.$$ Since the function $g$ satisfies $g(r)\geq r$ on ${\mathbb{R}}^+$, for $t\leq T_{loc}$ we have $r_a^t \leq \frac{1}{2C_1}\leq \frac{\rho_{min}}{16}$. In particular, we may apply Proposition \[prop:loca\] to $v^t$, $\{a_i(t)\}_{i=1}^l$ and $\{d_i\}_{i=1}^l$, which yields points $\{\xi_i(t)\}$ such that $$\label{rxit.def}
\| J v^t - \pi \sum_{i=1}^l d_i \delta_{\xi_i(t)}\|_{W^{-1,1}(\Omega)}
\leq r_\xi^t \equiv r_\xi(\Sigma_a^t,r_a^t) \equiv {\varepsilon}\exp(C_1(\Sigma_a^t +
g(r_a^t)){|\!\log{\varepsilon}|}),$$ where[^4] $$\Sigma_a^t = \left(\frac{{{E}_{\varepsilon,\eta}}(v^t)}{{|\!\log{\varepsilon}|}} - \pi \sum_{i=1}^l \eta^2(a_i(t)) \right)^+.$$ Since $t\mapsto v^t|_\Omega$ is continuous in $H^1(\Omega)$, it is clear that $t\mapsto Jv^t$ is continuous as a function from ${\mathbb{R}}$ into $W^{-1,1}(\Omega)$, and hence we can choose $\{ \xi_i(t)\}$ to be piecewise constant, and in particular measurable, as functions of $t$. Since ${{E}_{\varepsilon,\eta}}$ is preserved by the flow for $v$ and $\eta^2(a_i)$ is preserved by the flow for the $a_i$’s, we have $$\Sigma_a^t \equiv \Sigma^0.$$ Note in particular that $r_\xi^t \le \sqrt{\varepsilon}$ for $t<T_{loc}$.
\[prop:jacspeed\] There exist positive constants $\tau_0, {\varepsilon}_0$ and $C$, depending only on $l$, $\rho_{min}$, $\eta_{min}$ and $\|\nabla \eta^2\|_\infty$, such that ${\varepsilon}_0\leq \exp(-\frac{8}{\rho_{min}})$ and if $0<{\varepsilon}<{\varepsilon}_0$ and $$\Sigma^0 + g(r_a^t) \leq \frac{1}{4C_1}$$ for some $t\leq T_{loc},$ then $T_{loc} \geq t + \tau_0$ and $$\begin{aligned}
&\|Jv^s- Jv^t\|_{W^{-1,1}(\Omega)}
\leq
C\bigl(|t-s| +
r_\xi^t\bigr), \label{small_t.1}
\\
&r_\xi^s \leq
r^t_\xi + C {|\!\log{\varepsilon}|}{\varepsilon}^{1/2} \bigl(|s-t| + r_\xi^t\bigr),
\label{small_t.2}\\
&\{ a_i(s), \xi_i(s) \} \subset B(a_i(t), \frac{\rho_{min}}{4}),\qquad i=1,\ldots,l
\label{small_t.3}\end{aligned}$$ for every $t\leq s\leq t+\tau_0$.
For the ease of notation in the present proof, $\|\cdot\|$ is understood to mean $W^{-1,1}(\Omega)$ while $|\cdot|$ denotes the Euclidean norm on ${\mathbb{R}}^2.$
[**Step 1**]{}. Let $t\leq s \leq \min\{ T_{loc}, t+ \tau_0\}$, for $\tau_0$ to be fixed below. We first use the fact that $Jv^s, Jv^t$ are well-approximated by sums of point masses to show that $\| Jv^s- Jv^t\|$ can be estimated by computing $\langle Jv^s - Jv^t, \varphi\rangle$ for a specific test function $\varphi$ with certain good properties (in particular, bounds on [*second*]{} derivatives of $\varphi$). Toward this end, note that $$\label{eq:decompjac}\begin{split}
\|Jv^s-Jv^t\| &\leq
\|Jv^s-\pi\sum_{i=1}^l d_i\delta_{\xi_i(s)}\|
+
\|Jv^t-\pi\sum_{i=1}^l d_i\delta_{\xi_i(t)}\|
+
\|\pi\sum_{i=1}^l d_i(\delta_{\xi_i(s)}-\delta_{\xi_i(t)})\|\\
&\leq r_\xi^s+r_\xi^t + \pi \sum_{i=1}^l
|\xi_i(s)-\xi_i(t)|.
\end{split}$$ We now fix $\tau_0$, depending only on $\|\nabla
\eta^2\|_{\infty}$, $\eta_{min}$ and $\rho_{min}$, such that if $t\leq s\leq t+\tau_0,$ we have $|a_i(s)-a_i(t)| \leq \frac{\rho_{min}}{8}$ for all $i\in \{1,\cdots
,l\}.$ By Proposition \[prop:loca\], the choice of $T_{loc}$, and Lemma \[lem:W-11\], for every $\tau\leq T_{loc}$ we have $|a_i(\tau)-\xi_i(\tau)|\leq 2 r_a^\tau \leq
\frac{\rho_{min}}{8}.$ By the triangle inequality, it follows that $\xi_i(s)\in B(a_i(t),
\frac{\rho_{min}}{4})$ for all $t\leq s \leq \min(t+\tau_0,T_{loc})$ and $i\in \{1,\cdots
,l\}.$ Let $$\label{eq:defvarphi}
\varphi(x) = \sum_{i=1}^l d_i
\frac{(x-a_i(t))\cdot(\xi_i(s)-\xi_i(t))}{|\xi_i(s)-\xi_i(t)|} \chi\Big(
|x-a_i(t)|\Big),$$ where $\chi\in\mathcal{C}^\infty({\mathbb{R}}^+,[0,1])$ is such that $\chi\equiv 1$ on $[0,\rho_{min}/4],$ $\chi\equiv 0$ on $[\frac{\rho_{min}}{2},+\infty).$ By construction and the definition of $\rho_{min}$, we have $\varphi \in
\mathcal{D}(\Omega)$ and it follows that $$\begin{aligned}
\pi \sum_{i=1}^l |\xi_i(s)-\xi_i(t)|
&= \langle \pi\sum_{i=1}^l
d_i(\delta_{\xi_i(s)}-\delta_{\xi_i(t)}), \varphi \rangle\\
&\le (r_\xi^t + r_\xi^s)\|\varphi\|_{W^{1,\infty}} +
\langle Jv^s - J v^t ,\varphi\rangle.
\end{aligned}$$ Combining this with , we conclude that $$\label{eq:forlan}
\mbox{ $\| Jv^s - Jv^t\| \le C( r^s_\xi + r^t_\xi) +\langle Jv^s - J v^t ,\varphi\rangle$,
\quad\quad\quad
with $\|\varphi\|_{W^{2,\infty}} \le C \rho_{min}^{-2}$.}$$
[**Step 2**]{}. We now deduce from , together with , the fact that $\Sigma^0 \le \frac 1{4C_1}$, and conservation of energy, that $$\label{eq:vanbasten}\begin{aligned}
\| Jv^s - Jv^t\|&\leq (r_\xi^t + r_\xi^s)\|\varphi\|_{W^{1,\infty}} + (s-t)
\sup_{\tau\in [t,s]}\|\frac{d}{d\tau}
Jv^\tau\|_{W^{-2,1}(\Omega)} \|\varphi\|_{W^{2,\infty}}\\
&\leq C(r_\xi^t + r_\xi^s + |t-s|),
\end{aligned}$$ for $t\leq s \leq \min(t+\tau_0,T_{loc})$, where $C$ depends only on $l$, $\rho_{min}$, $\eta_{min}$ and $\|\nabla \eta^2\|_\infty$.
[**Step 3**]{}. It remains to estimate $r_\xi^s$ and to show that $t+\tau_0\leq T_{loc}.$ For that purpose, since $r_\xi^s=r_\xi(\Sigma^0,r_a^s)$, we first use to compute $$\label{eq:rxis}\begin{split}
r_a^s &= \| J v^s - \pi \sum_{i=1}^l d_i\delta_{a_i(s)}\| \\
&\leq \|Jv^s - J v^t\| + \|Jv^t -\pi\sum_{i=1}^l
d_i\delta_{a_i(t)}\| + \|\pi\sum_{i=1}^l
d_i(\delta_{a_i(s)}-\delta_{a_i(t)})\|\\
&\leq r_a^t + C\bigl( |t-s| + r_\xi^t + r_\xi^s) \bigr).
\end{split}$$ Next, since $s\leq T_{loc}$, $$\label{eq:rxis2}\begin{split}
r_\xi^s &= {\varepsilon}\exp(C_1[\Sigma^0+g(r_a^s)]{|\!\log{\varepsilon}|})\\
&\leq r_\xi^t + C_1{|\!\log{\varepsilon}|}{\varepsilon}^{\frac{1}{2}}\|g'\|_\infty \ (r_a^s-r_a^t)^+\\
&\leq r_\xi^t + \frac{1}{2C}(r_a^s-r_a^t)^+,
\end{split}$$ provided we assume, and this is again no loss of generality, that $C|\!\log {\varepsilon}_0|{\varepsilon}_0^{\frac{1}{2}} \leq \frac{1}{2C}$ for the same constant $C$ as in . Combining with we obtain $$\label{eq:rxis3}
r_a^s - r_a^t \leq C\bigl( |t-s| + r_\xi^t\bigr).$$ Going back to , this yields the desired estimate of $r^s_\xi$ : $$r_\xi^s \leq r^t_\xi + C {|\!\log{\varepsilon}|}{\varepsilon}^{1/2} \bigl(|s-t| + r_\xi^t\bigr).$$ Then going back to , $$\|Jv^s-Jv^t\| \leq C(|t-s| + r_\xi^t)$$ for $t\leq s \leq \min(t+\tau_0,T_{loc}).$ Finally, by assumption we have $\Sigma^0+g(r_a^t) \leq 1/(4C_1)$ so that by and the fact that $g'\leq 1$, $$\Sigma^0 + g(r_a^s) \leq 1/(4C_1) + C\bigl( |t-s| + r_\xi^t\bigr) \leq 1/(4C_1)
+ C\bigl(\tau_0+{\varepsilon}^\frac{1}{2}\bigr) \leq 1/(3C_1),$$ provided we assume, and this is no loss of generality, that $C(\tau_0+{\varepsilon}_0^\frac{1}{2}) \leq 1/(12C_1).$ It follows that $\min(t+\tau_0,T_{loc}) = t+\tau_0$, and the proof is complete.
Control of the discrepancy
==========================
In this section, we prove a discrete differential inequality for the quantity $r_a^t.$ More precisely, we will prove
\[prop:discrineq\] There exist positive constants ${\varepsilon}_0$ and $C_0$, depending only on $l$, $\rho_{min}$, $\eta_{min}$ and $\|\nabla \eta^2\|_\infty$, such that ${\varepsilon}_0 \leq \exp(-\frac{8}{\rho_{min}})$ and if $0<{\varepsilon}<{\varepsilon}_0$ and $$\Sigma^0 + g(r_a^t) \leq \frac{1}{4C_1}
\label{rtasmall}$$ for some $t\leq T_{loc},$ then $$\frac{r_a^T -r_a^t}{T-t} \leq C_0(\Sigma^0 + g(r_a^t))$$ where $T = t+ \frac{(r^t_\xi)^2}{\varepsilon}\leq T_{loc}$.
This is the main estimate in the proof of Theorem \[thm:main\].
We first require the constant ${\varepsilon}_0$ to be smaller than the one appearing in the statement of Proposition \[prop:jacspeed\]. As in the proof of Proposition \[prop:jacspeed\], we will write simply $\| \cdot \|$ to denote the $W^{-1,1}(\Omega)$ norm. Note that the condition states exactly that $$r^\xi_t \le {\varepsilon}^{3/4}
\label{eps14}$$ and then the definition of $T$ and yield $$r^s_\xi \le 2 r^t_\xi \quad\quad\mbox{ for all }s\in [t,T]
\label{rsxi}$$ if $C$ is large enough and ${\varepsilon}_0$ small enough, which we henceforth take to be the case. Moreover, from , we see that $r_a^s \le r_a^t + C(T-t + r_\xi^t)$ for all $s\in [t,T]$, and then the choice of $T$ and the definition of $g$ imply that $$g(r_a^s) \le 2 g(r_a^t)\quad\quad\mbox{ for all }s\in [t,T].
\label{gras}$$
[**1**]{}. First note that $$\begin{aligned}
r^T_a - r^t_a
&=
\| Jv^T - \pi \sum_{i=1}^l d_i \delta_{a_i(T)}\|
-
\| Jv^t - \pi \sum_{i=1}^l d_i \delta_{a_i(t)}\|\nonumber\\
&\le
\pi \sum_{i=1}^l \left(|\xi_i(T)-a_i(T)| - |\xi_i(t) - a_i(t)| \right)\ + \ r^T_\xi + r^t_\xi
\nonumber\\
&\le
\pi \sum_{i=1}^l \nu_i \cdot \bigl(\xi_i(T)- \xi_i(t) + a_i(t)- a_i(T) \bigr)\ + \ r^T_\xi + r^t_\xi
\label{eq:p5.1}\end{aligned}$$ for $\nu_i = \frac {\xi_i(T)-a_i(T)}{|\xi_i(T)-a_i(T)|}$ (unless $\xi_i(T)- a_i(T)=0$, in which case $\nu_i$ can be any unit vector). We now define $$\varphi(x) = \sum_i d_i\nu_i \cdot(x - a_i(t)) \chi(|x - a_i(t)|)$$ for $\chi\in C^\infty({\mathbb{R}}^+, [0,1])$ such that $\chi \equiv 1$ on $[0,\frac 12 \rho_{min}]$ and $\chi\equiv 0$ on $(\rho_
{min},\infty)$. It follows from that (since $d_i{}^2=1$ for all $i$) $$\pi \sum_{i=1}^l \nu_i \cdot \bigl(\xi_i(T)- \xi_i(t) + a_i(t)- a_i(T) \bigr)\
=
\pi \sum_{i=1}^l d_i\Bigl[ \varphi(\xi_i(T))- \varphi(\xi_i(t)) - \varphi(a_i(T)) + \varphi(a_i(t)) \Bigr],$$ so that and the definition of $r^T_\xi$ imply that $$\label{eq:p5.2}
r^T_a-r^t_a \le \langle \varphi , Jv^T - Jv^t \rangle \ -
\ \pi\sum_{i=1}^l d_i \Bigl[\varphi(a_i(T)) - \varphi(a_i(t))\Bigr]
+\ C(r^T_\xi + r^t_\xi).$$
[**2**]{}. The remainder of the proof is devoted to an estimate of $ \langle \varphi , Jv^T - Jv^t \rangle $. First, using , $$\label{eq:p5.3}
\begin{aligned}
\langle \varphi , Jv^T - Jv^t \rangle
&=
\int_t^T \frac \partial{\partial s} \langle \varphi , Jv^s \rangle \, ds \ \\
&= \frac 1 {|\!\log{\varepsilon}|}\int_t^T \int_\Omega
\left( {\epsilon}_{lj} \varphi_{x_l}
{\eta^2_{x_j}}
\frac
{(|v_{{\varepsilon}}|^2-1)^2}{4{\varepsilon}^2}\ + {\epsilon}_{lj} \varphi_{x_kx_l} v_{{\varepsilon},x_j}\cdot v_{{\varepsilon},x_k}\right)
\\
&\quad\quad\quad+ \int_t^T\int_\Omega
{\epsilon}_{lj} \varphi_{x_l}
\frac{\eta^2_{x_k}}{\eta^2} \frac{ v_{{\varepsilon},x_j}\cdot v_{{\varepsilon},x_k} }{|\!\log{\varepsilon}|}.
\end{aligned}$$ We immediately see from that $$\label{eq:easyterm1}
\left| \frac 1 {|\!\log{\varepsilon}|}\int_\Omega
\left( {\epsilon}_{lj} \varphi_{x_l}
{\eta^2_{x_j}}
\frac {(|v|^2-1)^2}{4{\varepsilon}^2}\ \right)\right|
\le C(\Sigma^0+ g(r_a^s))$$ for every $s\in [t,T]$. Moreover, it follows from that $B(\xi_i(s);4 {|\!\log{\varepsilon}|}^{-1}) \subset B(a_i(t), \frac 12 \rho_{min})$ if ${\varepsilon}_0$ is small enough, and the definition of $\varphi$ implies that $\varphi_{x_i x_j} = 0$ in $\cup_{i=1}^l B(a_i(t), \frac 12 \rho_{min})$, so implies that $$\label{eq:easyterm2}
\begin{aligned}
\frac 1 {|\!\log{\varepsilon}|}\int_\Omega
\left|{\epsilon}_{lj} \varphi_{x_kx_l} v_{{\varepsilon},x_j}\cdot v_{{\varepsilon},x_k}\right|
&\le
C \frac 1{|\!\log{\varepsilon}|}E_{\varepsilon}(v; \Omega\setminus \cup B(\xi_i, 4{|\!\log{\varepsilon}|}^{-1}))\\
&\le C(\Sigma^0+ g(r_\xi^s)) \le C(\Sigma^0+ g(r_a^s))
\end{aligned}$$ for every $s\in [t,T]$.
[**3**]{}. We now decompose the remaining term in . For every $s\in [t,T]$, let[^5] $j_*^s = j_*( \{\xi_i(s)\}, r_\xi^t)$ be the approximation to $j(v^s)$ obtained in Proposition \[prop:approx\]. Note that $$v_{{\varepsilon},x_j}\cdot v_{{\varepsilon},x_k} =
|v_{{\varepsilon}}| _{x_j}\, |v_{{\varepsilon}}|_{x_k}
+
\frac {j(v)_j}{|v|} \ \frac {j(v)_k}{|v|}$$ where for example $j(v)_j = (iv, \partial_{x_j}v)$ denotes the $j$th component of $j(v)$. Thus, adding and subtracting $j_*^s$ in various places, and writing $\psi_{jk}$ as an abbreviation for ${\epsilon}_{lj} \varphi_{x_l}\frac{\eta^2_{x_k}}{\eta^2}$, we have for every $s\in [t,T]$, $$\label{eq:decompose}
\begin{aligned}
&\int_\Omega
\psi_{jk} \frac{ v_{{\varepsilon},x_j}\cdot v_{{\varepsilon},x_k} }{|\!\log{\varepsilon}|}=
\int_\Omega
\psi_{jk} \frac {( j_*^s)_j ( j_*^s)_k}{|\!\log{\varepsilon}|}\ \ \\
&\quad\quad\quad +
\int_\Omega
\psi_{jk} \frac 1 {|\!\log{\varepsilon}|}\left[
\left( \frac {j(v)}{|v|} - j_*^s\right)_j (j_*^s)_k +
\left( \frac {j(v)}{|v|} - j_*^s\right)_k (j^s_*)_j\right] \ \\
&\quad\quad\quad+
\int_\Omega
\psi_{jk} \frac 1 {|\!\log{\varepsilon}|}\left[ {|v_{{\varepsilon}}| _{x_j}\, |v_{{\varepsilon}}|_{x_k}}
+
\left( \frac {j(v)}{|v|} - j_*^s\right)_j
\left( \frac {j(v)}{|v|} - j_*^s\right)_k\right] \ .
\end{aligned}$$ We immediately dispense with the easiest terms by using to see that $$\label{eq:iniesta}
\int_\Omega
\psi_{jk} \frac 1 {|\!\log{\varepsilon}|}\left[ {|v_{{\varepsilon}}| _{x_j}\, |v_{{\varepsilon}}|_{x_k}}
+
\left( \frac {j(v)}{|v|} - j_*^s\right)_j
\left( \frac {j(v)}{|v|} - j_*^s\right)_k\right] \ \le C(\Sigma^0 + g(r_a^s))$$ for every $s\in [t,T]$.
[**4**]{}. We next consider the first term on the right-hand side of , which is the term that yields the dominant contribution. Since $j_*^s$ is supported in $\cup_i B(\xi_i(s), {|\!\log{\varepsilon}|}^{-1})$, clearly $$\int_\Omega
\psi_{jk}
\frac {( j_*^s)_j ( j_*^s)_k}{|\!\log{\varepsilon}|}\ \
=
\sum_{i=1}^l
\int_{B(\xi_i(s), {|\!\log{\varepsilon}|}^{-1})} \psi_{jk} \frac {( j_*^s)_j ( j_*^s)_k}{|\!\log{\varepsilon}|}\ .
$$ For each $i=1,\ldots, l$, if $x\in B(\xi_i(s), {|\!\log{\varepsilon}|}^{-1})$, then $|x-a_i(s)| \le {|\!\log{\varepsilon}|}^{-1} + r^s_a + r^s_\xi$, by , so for every $s\in [t,T]$, $$\begin{aligned}
\left|\int_{B(\xi_i(s), {|\!\log{\varepsilon}|}^{-1})}\Bigl( \psi_{jk}(x) - \psi_{jk}(a_i(s))\Bigr)\frac {( j_*^s)_j ( j_*^s)_k}{|\!\log{\varepsilon}|}\
\right|
\ &\le \ \| \nabla \psi_{jk}\|_\infty({|\!\log{\varepsilon}|}^{-1} + r^s_a + r^s_\xi) \frac{\| j_*^s\|_2^2}{|\!\log{\varepsilon}|}\\
\ &\le \ C ({|\!\log{\varepsilon}|}^{-1} + r^s_a + r^s_\xi) ,\end{aligned}$$ using the explicit form of $j_*^s$, which (together with the definition of $r_\xi^s$) also implies that $$\begin{aligned}
\int_{B(\xi_i(s), {|\!\log{\varepsilon}|}^{-1})}\frac {( j_*^s)_j ( j_*^s)_k}{|\!\log{\varepsilon}|}\
& = \ \frac{ \pi}{|\!\log{\varepsilon}|}\delta_{jk} (\log \frac 1 {r_\xi^s} - \log {|\!\log{\varepsilon}|}+ \frac 14) \ \\
&= \ \pi\, \delta_{jk}\ \bigl( 1 - C_1(\Sigma^0 + g(r^s_a)) \bigr) \ + \ O(\frac{\log {|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}}).\end{aligned}$$ Combining the above computations and recalling that $g(r)\ge \max(r, \frac{ \log {|\!\log{\varepsilon}|}}{|\!\log{\varepsilon}|})$ for all $r$ and that $g(r_a^s) \ge r_\xi^s$ for $s\le T_{loc}$, we conclude that $$\begin{aligned}
\int_\Omega
\psi_{jk}
\frac {( j_*^s)_j ( j_*^s)_k}{|\!\log{\varepsilon}|}\ \
&= \pi \sum_i \psi_{kk}(a_i(s)) + O(C_1(\Sigma^0 + g(r_a^s))\nonumber\\
&= \pi \frac d{ds}\left(\sum_i d_i \varphi(a_i(s)) \right) + O(C_1(\Sigma^0 + g(r_a^s)).
\label{eq:mainterm}\end{aligned}$$ In the last line we have used the definition $\psi_{kk} = {\epsilon}_{lk}\varphi_{x_l} \partial_{x_k}(\log \eta^2 )= \nabla\varphi \cdot \nabla^\perp(\log \eta^2)$ together with the ordinary differential equation satisfied by the points $a_i(\cdot)$.
[**5**]{}. Combining , , , , , , and , and recalling , , we find that $$\begin{aligned}
\label{eq:halfway}
r^T_a - r^t_a
&\le C(T-t) \bigl(\Sigma^0 + g(r_a^t) \bigr)\ \ + \ C r^t_\xi \ +
\ \int_t^T
\int_\Omega
\frac {\psi_{jk} }{|\!\log{\varepsilon}|}\left( \frac {j(v)}{|v|} - j_*^s\right)_j (j_*^s)_k \, dx\,ds\\
&
\quad\quad
\quad\quad
\quad\quad
\quad\quad
+
\int_t^T\int_\Omega
\frac {\psi_{jk} } {|\!\log{\varepsilon}|}\left( \frac {j(v)}{|v|} - j_*^s\right)_k (j^s_*)_j\, dx\,ds .\nonumber\end{aligned}$$ We now begin to control the integrals on the right-hand side above. We will consider only the first, since the estimate of the second is identical. First, $$\begin{aligned}
\int_t^T
\int_\Omega
\frac {\psi_{jk} }{|\!\log{\varepsilon}|}\left( \frac {j(v)}{|v|} - j_*^s\right)_j (j_*^s)_k \, dx\, ds
\ & = \
\int_\Omega
\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k
\int_t^T\left( \frac {j(v)}{|v|} - j_*^s\right)_j \ ds \, dx\label{hardterm1}\\
\ & + \
\int_\Omega
\int_t^T \frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^s - j_*^t)_k \left( \frac {j(v)}{|v|} - j_*^s\right)_j \ ds \, dx \, . \nonumber\end{aligned}$$ We claim that $$\int_\Omega
\int_t^T \frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^s - j_*^t)_k \left( \frac {j(v)}{|v|} - j_*^s\right)_j \ ds \, dx \,
\le
C (T-t) (\Sigma^0 + g(r_a^t)).
\label{hardterm1a}$$ Using the Cauchy-Schwarz inequality, , and , we see that it suffices to prove that $$\int_\Omega |j_*^s - j_*^t|^2 \ dx \ \le \
(\Sigma^0 + g(r_a^t)) {|\!\log{\varepsilon}|}\quad\quad\quad\mbox{ for every $s\in [t,T]$. }$$ Toward this end, we fix some such $s$, and we introduce the notation $$\bar \xi_i := \frac 12 (\xi_i^t + \xi_i^s),\quad\quad
\sigma :=
r_\xi^s + r_\xi^t +\sum_{i=1}^l |\xi_i(t) - \xi_i(s)|.
$$ Our choice of $T$ and imply that $\sigma \le
C\frac{(r_\xi^t)^2}{\varepsilon}$. Writing $B_i := B(a_i(t), \frac{\rho_{min}}{2})$, we deduce from and the support properties of $j_*$ that $$\int_\Omega |j_*^t - j_*^s|^2 \ dx
\
= \sum_{i=1}^l
\int_{B_i}|j^t_* - j^s_*|^2 \ dx.$$ For each $i$, $B(\bar \xi_i, \sigma) \subset B(\xi_i(t), 2\sigma)\cap B(\xi_i(s), 2\sigma)$, so by an explicit computation, and recalling and the definition of $r_\xi^t$, we find that $$\begin{aligned}
\int_{B(\bar \xi_i, \sigma) }\!\!\!\!|j^t_* - j^s_*|^2 \ dx
\le
2 \int_{B(\xi_i(t), 2 \sigma) }\!\!\!|j^t_* |^2 \ dx
+
2 \int_{B(\xi_i(s), 2 \sigma) }\!\!\!|j^s_* |^2 \ dx
&\le
2\log( \frac{2\sigma}{r_\xi^t}) +
2\log( \frac{2\sigma}{r_\xi^s}) + C
\\
&\le
C\log( \frac {r_\xi^t} {\varepsilon})\\
&\le
C (\Sigma^0 + g(r_a^t)) {|\!\log{\varepsilon}|}.\end{aligned}$$ Next, on $B(\bar \xi_i, \frac 1{2{|\!\log{\varepsilon}|}})$, the definitions imply that both $j^s$ and $j^t$ are nonzero, and in fact $$|j^t_*(x) - j^s_*(x)|^2 = \frac {|\xi_i(t)-\xi_i(s)|^2}{|x - \xi(t)|^2 |x - \xi_i(s)|^2}.$$ Since $|\xi_i(\tau) - \bar \xi_i| \le \frac \sigma 2$ for $\tau = t,s$, it follows that $$|j^t_*(x) - j^s_*(x)|^2 \ \ \le
\frac {4 \sigma^2}{|x - \bar \xi_i|^4} \quad\mbox{ on }B(\bar \xi_i, \frac 1{2{|\!\log{\varepsilon}|}})\setminus B(\bar \xi_i, \sigma)$$ and hence that $$\int_{B(\bar \xi_i, \frac 1{2{|\!\log{\varepsilon}|}})\setminus B(\bar \xi_i, \sigma) }|j^t_* - j^s_*|^2 \ dx \
\le C.$$ Finally, $$\int_{B_i \setminus B(\bar \xi_i, \frac 1{2{|\!\log{\varepsilon}|}}) }|j^t_* - j^s_*|^2 \ dx \
\le
2\int_{B_i \setminus B(\xi_i(t), \frac 1{4{|\!\log{\varepsilon}|}}) }|j^t_* |^2 \ dx \
+
2\int_{B_i \setminus B(\xi_i(s), \frac 1{4{|\!\log{\varepsilon}|}}) }|j^s_* |^2 \ dx \ \ \le C.$$ We deduce by combining the previous inequalities.
[**6**]{}. We now consider the first term on the right-hand side of . Clearly $$\begin{aligned}
\int_\Omega
\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k
\int_t^T\left( \frac {j(v)}{|v|} - j_*^s\right)_j \, ds \, dx
\ & = \
\sum_{i=1}^l
\int_{B_i}\int_t^T \frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k
\left( \frac {j(v)}{|v|} - j(v)\right)_j ds \, dx \nonumber \\
&\quad\quad+ \
\sum_{i=1}^l \int_{B_i}\int_t^T \frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k
\left( {j(v)} - j_*^s\right)_j ds \, dx.
\label{hardterm2}\end{aligned}$$ By elementary estimates, $$| \frac{j(v)}{|v|} - j(v)| = \frac{|j(v)|}{|v|} \, \Bigl| |v|-1\Bigr|
\le
\frac {\varepsilon}2 |\nabla v|^2 + \frac 1{2{\varepsilon}}(|v|^2 -1)^2 ,$$ and from the definitions, and recalling , we see that $\| j_*^t\|_\infty \le (r_\xi^t)^{-1} \le ({\varepsilon}{|\!\log{\varepsilon}|})^{-1}$. Thus for every $i$, $$\begin{aligned}
\left| \int_{B_i}\int_t^T \frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k \cdot
\left( \frac {j(v)}{|v|} - j(v)\right)
ds \, dx
\right|
&\ \le \
C\| j_*^t\|_\infty {\varepsilon}(T-t) \frac{E_{{\varepsilon},\eta}(v)}{{|\!\log{\varepsilon}|}}\\
&\le
\frac C {|\!\log{\varepsilon}|}(T-t))(\Sigma^0+C)\\
&\le
C(T-t)(\Sigma^0+ g(r_a^t)),
\end{aligned}
\label{hardterm3}$$ since $$\label{totalE}
\frac {E_{{\varepsilon},\eta}(v)}{{|\!\log{\varepsilon}|}} \le
\Sigma^0 + \pi \sum_{i=1}^l \eta^2(a_i(t))
\le
\Sigma^0 + C(l, \|\eta\|_\infty).$$
[**7**]{}. Now fix some $i\in \{1,\ldots, l\}$ and let $\tilde\chi^i\in C^\infty_c(B(a_i(t), \frac 34 \rho_{min}))$ be a function such that $\tilde\chi^i=1$ on $B_i$. Then for every $s\in [t,T]$, $$\label{eq:etaHodge}
\tilde \chi_i({j(v)} - j_*^s)
= \nabla f^s + \frac 1 {\eta^2}\nabla^\perp g^s \quad\quad\quad\mbox{ in } B(a_i(t), \frac 34 \rho_{min})$$ for $f^s$ and $g^s$, real-valued functions on $B(a_i(t), \frac 34 \rho_{min})$, solving $$\label{f.def}\begin{aligned}
\nabla\cdot(\eta^2\nabla f^s) &= \nabla\cdot\bigl(\tilde \chi^i \eta^2 ( j(v) - j_*^s)\bigr)
&\mbox{ in }B(a_i(t), \frac 34 \rho_{min}),\\
\nu \cdot\nabla f^s &= 0
&\mbox{ on }\partial B(a_i(t), \frac 34 \rho_{min}),
\end{aligned}$$ and $$\label{g.def}\begin{aligned}
-\nabla\cdot( \frac{ \nabla g^s}{\eta^2}) &= \nabla\times \bigl(\tilde \chi^i( j(v) - j_*^s)\bigr).
&\mbox{ in }B(a_i(t), \frac 34 \rho_{min}),\\
g^s &= 0
&\mbox{ on }\partial B(a_i(t), \frac 34 \rho_{min}).
\end{aligned}$$ Indeed, if we let $f^s$ be a solution of , then $\eta^2(\tilde \chi_i (j(v) - j_*^s) - \nabla f^s)$ is divergence-free and hence can be written as $\nabla^\perp g^s$ on $B(a_i(t), \frac 34 \rho_{min})$, so that holds. Then it follows from that $g^s$ satisfies the equation in , and that the boundary condition is satisfied after adding a constant to $g^s$.
Thus $$\int_{B_i}\int_t^T \frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k \,
\left( {j(v)} - j_*^s\right)_j ds \, dx
\ = \
\int_{B_i}\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k\, (\nabla F + \frac{\nabla^\perp G}{\eta^2})_j\, dx
\label{eq:moresplitting}$$ for $$F(x) = \int_t^T f^s(x) \ ds,
\quad\quad\quad
G(x) = \int_t^T g^s(x) \ ds.$$ We write $F = F_1+\cdots + F_4$, where $$\nabla\cdot(\eta^2 \nabla F_m) = A_m \ \ \mbox{ in }B(a_i(t), \frac 34 \rho_{min}),
\quad
\nu \cdot \nabla F_m = 0 \ \ \mbox{ in }\partial B(a_i(t), \frac 34 \rho_{min}),$$ for $$\begin{aligned}
A_1 &= \tilde \chi^i \int_t^T \nabla \cdot (\eta^2 j(v)) \ ds\\
A_2 &= - \tilde \chi^i \int_t^T \nabla \cdot (\eta^2j_*^s) \ ds\\
A_3 &= \int_t^T \eta^2 \nabla \tilde \chi^i\cdot \frac{j(v)}{|v|} (|v| - 1) \ ds\\
A_4 &= \int_t^T \eta^2 \nabla \tilde \chi^i\cdot( \frac {j(v)}{|v|} - j_*^s) \ ds.\end{aligned}$$ Using the continuity equation — this is a key point in our argument — and , we note that $$\begin{aligned}
\|A_1 \|_{L^2} & =
{|\!\log{\varepsilon}|}\left\| \tilde \chi^i \eta^2 \left. (|v|^2-1) \right|_t^T \right\|_{L^2}\\
&\le
C {\varepsilon}{|\!\log{\varepsilon}|}\left( E_{{\varepsilon},\eta}(v^T) + E_{{\varepsilon}, \eta}(v^t)\right)\\
&\le C {\varepsilon}{|\!\log{\varepsilon}|}^3 (\Sigma^0 + g(r_a^t))\end{aligned}$$ since $g(r) \ge \frac {1+\log{|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}}$ for all $r$. Next, the definition implies that $\nabla \cdot j_*^s = 0$ for every $s$ and that $\| j_*^s\|_{L^p} \le C_p {|\!\log{\varepsilon}|}^{1 - \frac 2p}$ for every $p<2$, so $$\begin{aligned}
\| A_2\|_{L^p} \le \left\| \tilde \chi_i \ \right\|_{L^\infty} (T-t)
\sup_{s\in [t,T]}
\| \nabla(\eta^2)\cdot j_*^s \|_{L^p} \le C (T-t) {|\!\log{\varepsilon}|}^{1-\frac 2p}\quad\mbox{for }p<2.\end{aligned}$$ Very much as in , we can check that $$\|A_3\|_{L^1} \ \le \ C (T-t) \sup_{s\in[t,T]} \| \frac {j(v)}{|v|}(|v|-1)\|_{L^1}
\le C (T-t) {\varepsilon}{|\!\log{\varepsilon}|}^2 (\Sigma^0 + g(r^t_a)),$$ and it follows from and that $$\|A_4\|_{L^2} \ \le \ C (T-t) \Big( {|\!\log{\varepsilon}|}(\Sigma^0 + g(r^t_a)) \Big)^{1/2}.$$ Clearly, for any $q_1,\ldots, q_4\in [1,\infty]$, $$\int_{B_i}\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k \cdot (\nabla F)_j\, dx
\le \frac C {|\!\log{\varepsilon}|}\sum_{m=1}^4 \| j_* \|_{q_m} \|\nabla F_m \|_{q_m'}$$ where $\frac 1{q_m} + \frac 1{q_m'}=1$. Using elliptic estimates and Sobolev embedding theorems, and taking $q_1=\frac 43$, $$\frac 1{|\!\log{\varepsilon}|}\| j_* \|_{\frac 4 3} \|\nabla F_1 \|_{4}
\le
\frac{C}{|\!\log{\varepsilon}|}\| j_* \|_{\frac 4 3} \| A_1 \|_{2}
\ \le
\ C
{\varepsilon}{|\!\log{\varepsilon}|}^{\frac 3 2} (\Sigma^0 + g(r_a^t)) \le C(T-t) (\Sigma^0 + g(r_a^t)).$$ The last inequality follows from the choice of $T$ and , which imply in particular that $T-t \ge {\varepsilon}{|\!\log{\varepsilon}|}^2$. Similarly, taking $q_4 = 4/3$, $$\frac 1{|\!\log{\varepsilon}|}\| j_* \|_{\frac 4 3} \|\nabla F_4 \|_{4}
\le
\frac C{{|\!\log{\varepsilon}|}^{\frac 3 2}} \| A_4 \|_{2}
\ \le
C\frac{(T-t)} {|\!\log{\varepsilon}|}(\Sigma^0+ g(r^t_a))^{1/2}\le C(T-t)(\Sigma^0 + g(r_a^t)),$$ since ${|\!\log{\varepsilon}|}^{-1} \le g(r^t_a)$. For any $q_2\in (1,2)$, taking $p_2<2$ such that $p_2^* = q_2'$, so that $\frac 1{p_2} = \frac 32 - \frac 1{q_2}$, we find from our estimate of $A_2$ that $$\frac 1{|\!\log{\varepsilon}|}\| j_* \|_{q_2} \|\nabla F_2 \|_{q_2'}
\le
\frac C{|\!\log{\varepsilon}|}\| j_* \|_{q_2} \| A_2 \|_{p_2}
\ \le
C(T-t) {|\!\log{\varepsilon}|}^{-2} \le C(T-t)g(r^t_a).$$ And, recalling by Stampacchia’s estimate that for any $p\in [1,2)$ there exists $C_p$ such that $\| \nabla F_3\|_p \le C_p \|A_3\|_1$, we compute, choosing $q_3=3$ for concreteness, $$\begin{aligned}
\frac 1{|\!\log{\varepsilon}|}\| j_* \|_{3} \|\nabla F_3 \|_{\frac 3 2}
\le
\| j_* \|_{3} \| A_3 \|_{1}
&\ \le C (T-t) (r_\xi^t)^{-\frac 13 } {\varepsilon}{|\!\log{\varepsilon}|}(\Sigma^0 + g(r^t_a))
\\
&\le (T-t) ( {\varepsilon}{|\!\log{\varepsilon}|})^{2/3} (\Sigma^0 + g(r^t_a))\end{aligned}$$ again using the fact that $r_\xi^t \ge {\varepsilon}{|\!\log{\varepsilon}|}$ for all $t$, see . Combining the above, we find that for every $i\in \{1,\ldots,l\}$ and $0<{\varepsilon}<{\varepsilon}_0$ with ${\varepsilon}_0$ sufficiently small, $$\begin{aligned}
\int_{B_i}\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k \cdot (\nabla F)_j\, dx
&\le
C(T-t) (\Sigma^0+ g(r_a^t)). \label{Fterms}\end{aligned}$$
[**8**]{}. Next, $$\int_{B_i}\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k \cdot \frac{(\nabla^\perp G)_j}{\eta^2}\, dx
=
\int_{B_i}\frac {\psi_{jk} }{\eta^2{|\!\log{\varepsilon}|}} (j_*^t)_k \cdot \nabla^\perp (G_1+G_2+G_3)_j \, dx$$ for $G_m$ solving $$-\nabla\cdot ( \frac{ \nabla G_m}{\eta^2}) = A_m'
\quad \mbox{ in }B(a_i(t), \frac 34 \rho_{min}),
\quad\quad\quad
g = 0
\mbox{ on }\partial B(a_i(t), \frac 34 \rho_{min}),$$ with $$\begin{aligned}
A_1'
&:=
\int_t^T \tilde \chi^i \nabla\times(j(v) - j_*^s) \ ds,\\
A_2'
& :=
\int_t^T \nabla^\perp \tilde \chi^i \cdot j(v)(1 - \frac 1{|v|}) \ ds,\\
A_3'
&:=
\int_t^T \nabla^\perp \tilde \chi^i \cdot (\frac{j(v)}{|v|} - j_*^s) \ ds.\end{aligned}$$ The terms containing $G_2$ and $G_3$ are estimated exactly as the terms containing $F_3$ and $F_4$ in Step 7 above, leading to $$\int_{B_i}\frac {\psi_{jk} }{\eta^2{|\!\log{\varepsilon}|}} (j_*^t)_k \ \nabla^\perp (G_2+G_3)_j\, dx
\
\le \ C(T-t)(\Sigma^0+ g(r_a^t)) . $$ For the remaining term, we invoke the interpolation inequality $$\| A_1'\|_{W^{-1,p}} \le C \| A_1'\|_{W^{-1,1}}^\theta \| A_1'\|_{L^1}^{1-\theta} \label{negative.interp}$$ for $p\in (1,2)$ and $\theta$ such that $\frac 1p = \frac \theta 1 + \frac {1-\theta}2$ (see e.g. [@Triebel] Theorem 2.4.1 combined with Sobolev embedding theorem). To estimate the $W^{-1,1}$ norm, we fix $\zeta \in C^\infty_c(\Omega)$, and we compute $$\begin{aligned}
\langle \zeta, A_1'\rangle
=
\int_t^T\langle \tilde \chi^i \zeta , \nabla\times (j(v) - j_*^s) \rangle
&\le
\int_t^T \| \tilde \chi^i \zeta \|_{W^{1,\infty}} \| \nabla\times (j(v) - j_*^s) \|_{W^{-1,1}} \ ds
\\
&\le C(T-t) r_\xi^t \|\zeta\|_{W^{1,\infty}} \ \end{aligned}$$ using and . Thus $$\label{A1primeflat}
\| A_1'\|_{W^{-1,1}} \le C(T-t) r_\xi^t .$$ Also, for every $s\in [t,T]$, $$\| \nabla \times (j(v)- j_*^s) \|_{L^1}
\le
\| 2 Jv \|_{L^1} + \| \nabla\times j_*^s\|_{L^1}
\le
C E_{{\varepsilon},\eta}(v) + 2\pi l.$$ Estimating $E_{{\varepsilon},\eta}$ as usual by $C{|\!\log{\varepsilon}|}(\Sigma^0 + g(r_a^t))$, integrating the last inequality from $t$ to $T$, and combining it with and , we obtain $$\|A_1'\|_{W^{-1,p}} \le C (T-t)(r_\xi^t)^\theta \left( C{|\!\log{\varepsilon}|}(\Sigma^0 + g(r_a^t))\right)^{1-\theta}.$$ Then using Hölder’s inequality and (again) the fact that $\| j_*^s\|_{p'} \le C (r_\xi^s)^{ \frac 2{p'}-1}$ for $p'>2$, $$\begin{aligned}
\int_{B_i}\frac {\psi_{jk} }{\eta^2 {|\!\log{\varepsilon}|}} (j_*^t)_k \cdot \nabla^\perp (G_1)_j\, dx
&\le
\frac C{|\!\log{\varepsilon}|}(T-t) (r_\xi^t)^{\theta + \frac 2 {p'}-1}\left( C{|\!\log{\varepsilon}|}(\Sigma^0 + g(r_a^t))\right)^{1-\theta}
\nonumber\\
&\le
C(T-t) {|\!\log{\varepsilon}|}^{-\theta} \left( \Sigma^0 + g(r_a^t)\right)^{1-\theta}
\nonumber\\
&\le
C(T-t) \left( \Sigma^0 + g(r_a^t)\right),
\label{G1est}\end{aligned}$$ since it turns out that $\theta+ \frac 2{p'}-1 = 0$, and noting that ${|\!\log{\varepsilon}|}^{-1} \le g(r_a^t)$ for all $t$. Assembling these estimates, we find that $$\int_{B_i}\frac {\psi_{jk} }{|\!\log{\varepsilon}|}(j_*^t)_k (\nabla^\perp G)_j\, dx
\le
C(T-t) \left( \Sigma^0 + g(r_a^t)\right). $$ Now by combining this with , , , , , we finally obtain $$\begin{aligned}
r^T_a - r^t_a \le
C(T-t) \left( \Sigma^0 + g(r_a^t)\right) .
\end{aligned}
\label{main.est}$$
Proof of Theorem \[thm:main\]
=============================
Our main result is a straightforward corollary of the discrepancy estimate proved in the previous section.
Let $Y$ denote the solution of the ordinary differential equation $$\dot Y(t) = C_0\Big(\Sigma^0 + g(Y(t))\Big),\qquad Y(0) = r_a^0,$$ where $g$ is the function defined in , and let $\{ Y_n\}_{n=0}^\infty$ be a discrete approximation to $Y(\cdot)$ obtained via an Euler approximation implicit in the statement of Proposition \[prop:discrineq\]. Thus, we define $$Y_0= r^0_a, \qquad Y_{n+1} = Y_n +(t_{n+1} - t_{n}) \,C_0 \left(\Sigma^0 + g(Y_n)\right),$$ $$t_{n+1} := t_{n} + \frac{(r^{n}_\xi)^2}{\varepsilon}$$ where $$r^n_\xi := r_\xi(\Sigma^0, Y_n) = {\varepsilon}\exp( C_0(\Sigma^0+g(Y^n))){|\!\log{\varepsilon}|}).$$ Since the function $f(Y):= C_0 \left(\Sigma^0 + g(Y)\right) $ is convex, a forward Euler approximation to the solution of the equation $Y' = f(Y)$ is always less than or equal to the actual solution, and it follows that $
Y_n \le Y(t_n)$ for all $t$. Then repeated application of Proposition \[prop:discrineq\] shows that $$r_a^{t_n} \le Y_n \le Y(t_n)
\mbox{ for every $n$ such that $t_n \le T_{col}$ and $\Sigma^0 + g(Y_n) \le \frac 1{4C_1}$.}$$ Given an arbitrary $t\in (0,T_{col}]$ such that $\Sigma^0+ g(Y(t))\le \frac 1{4C_1}$, there exists some $n$ such that $t\in [t_n, t_{n+1}]$ and $r_a^{t_n} \le Y(t_n)$. Then by Proposition \[prop:jacspeed\], see in particular , as well as , $$\label{eq:parotide0}
r_a^t \le r_a^{t_n} + C\Big( ( t_{n+1}-t_n) + r_\xi^{t_n}\Big) \le Y(t) + C{\varepsilon}^{1/2},$$ since the bound $\Sigma^0+ g(Y(t_n))\le \frac 1{4C_1}$ guarantees that $r_\xi^{t_n}\le {\varepsilon}^{3/4}$ and hence that $ t_{n+1}-t_n \le {\varepsilon}^{1/2}.$ It remains to bound the function $Y$ from above. For that purpose, we notice that since $g(y) \leq y + \log{|\!\log{\varepsilon}|}/{|\!\log{\varepsilon}|}$ for every $y\geq 0$, we have $Y(t) \leq \tilde Y(t)$ where $\tilde Y$ is the solution of the ordinary differential equation $$\dot{ \tilde Y}(t) = C_0\big( \Sigma^0 + \frac{\log{|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}} + \tilde Y(t)\big),\qquad \tilde Y(0) = r_a^0.$$ The solution of the latter is explicitly given by $$\tilde Y(t) = r_a^0 + \big( \Sigma^0 + r_a^0+ \frac{\log{|\!\log{\varepsilon}|}}{{|\!\log{\varepsilon}|}} \big)\big( e^{C_0t}-1\big),$$ and the conclusion therefore follows from , increasing the value of $C_0$ to the value of $C$ in if necessary.
Some properties of the ground state {#sect:groundstate}
===================================
In this section we briefly recall some facts about minimizers of the functional[^6] $$\label{eq:EepsV}
{\mathcal{E}}_{{\varepsilon},V}(u) = \int_{{\mathbb{R}}^N} \frac{ |\nabla u|^2}2 + \frac 1{2{\varepsilon}^2}\left( V(x) {|u|^2} + \frac12 {|u|^4}\right)dx$$ in the space $$\label{eq:constraint}
\mathcal{H}_m := \{ u\in H^1({\mathbb{R}}^N;{\mathbb{C}}) : \int_{{\mathbb{R}}^N} V |u|^2 < \infty, \int_{{\mathbb{R}}^N} |u|^2 = m \}$$ where $V:{\mathbb{R}}^N\to [0,\infty)$ is a smooth function such that $V(x)\to \infty$ as $|x|\to \infty$, and $m>0$ is a parameter.
For every positive ${\varepsilon},m$, the existence of a function $\eta_{{\varepsilon}, m}:{\mathbb{R}}^N\to (0,\infty)$ minimizing ${\mathcal{E}}_{{\varepsilon},V}$ in $\mathcal{H}_m$ is standard, and follows easily from the growth of $V$ (which implies that the $L^2$ constraint is preserved for weak limits of sequences with equi-bounded energy) together with the strong maximum principle and the fact that ${\mathcal{E}}_{{\varepsilon},V}(|u|) \le {\mathcal{E}}_{{\varepsilon},V}(u)$ for all $u$.
In the introduction, we already introduced the unique number $\lambda_0$ such that $$\int_{{\mathbb{R}}^N} (\lambda_0-V)^+ dx = m,$$ and we have denoted by ${\rho_{\scriptscriptstyle{TF}}}:= (\lambda_0 - V)^+$ the Thomas-Fermi profile associated to $V$ and $m.$ We also note $w := (\lambda_0-V)^-$. We will prove
\[prop:groundstate\] Let $\eta = \eta_{{\varepsilon}. m} \in \mathcal{H}_m$ be a positive minimizer of ${\mathcal{E}}_{{\varepsilon}, V}$ in $\mathcal{H}_m$.
Then $$\| \eta^2 - {\rho_{\scriptscriptstyle{TF}}}\|_{L^2({\mathbb{R}}^N)} \le C {\varepsilon}^{2/3}.
\label{eq:L2eta}$$ Moreover, for any $K\subset\subset {\Omega_{\scriptscriptstyle{TF}}}:= \{ x\in {\mathbb{R}}^N : {\rho_{\scriptscriptstyle{TF}}}(x)>0\}$, there exists a constant $C = C(m,V,K)$ such that $$\| \eta^2 - {\rho_{\scriptscriptstyle{TF}}}\|_{L^\infty(K)} \le C {\varepsilon}^{2/3},
\quad\quad
\|\nabla \eta^2 \|_{L^\infty(K)} \le C.
\label{eq:unifLipschitz}$$
This is quite standard, and is proved for particular potentials $V$ in [@IM1] for example. We include a complete proof, since the references we know all impose slightly more restrictive conditions than we consider here (for example, symmetry conditions, or the assumption that $\lambda_0$ is a regular value of $V$).
It suffices to prove the result for ${\varepsilon}\le{\varepsilon}_0$, for some ${\varepsilon}_0>0$.
[**1**]{}. First, as is standard, for $u\in \mathcal{H}_m$ we rewrite $$\begin{aligned}
{\mathcal{E}}_{{\varepsilon},V}(u)
&=
\int_{{\mathbb{R}}^N} \Big[\frac {|\nabla u|^2}2 + \frac 1{4{\varepsilon}^2}(|u|^2-{\rho_{\scriptscriptstyle{TF}}})^2 + \frac 1{2{\varepsilon}^2} w|u|^2\Big]\ dx
\ + \ \frac 1{{\varepsilon}^2}(\lambda_0 \frac m2 - \frac 14 \int_{{\mathbb{R}}^N} {\rho_{\scriptscriptstyle{TF}}}^2)\\
&=:
{\mathcal{E}}_{{\varepsilon}, {\rho_{\scriptscriptstyle{TF}}}}(u) + C_1({\varepsilon}, m).\end{aligned}$$ Thus, it is clear that a function minimizes ${\mathcal{E}}_{{\varepsilon},V}$ in $\mathcal{H}_m$ if and only if it minimizes ${\mathcal{E}}_{{\varepsilon},{\rho_{\scriptscriptstyle{TF}}}}$ in $\mathcal{H}_m$.
[**2**]{}. Next we claim that $$\inf_{\mathcal{H}_m}{\mathcal{E}}_{{\varepsilon}, {\rho_{\scriptscriptstyle{TF}}}} \le C {\varepsilon}^{-2/3}.
\label{eq:upper}$$ Note that this immediately implies . We verify by choosing $U_{\varepsilon}:= c_{\varepsilon}f_{\varepsilon}(\sqrt {\rho_{\scriptscriptstyle{TF}}})$, where $$f_{\varepsilon}(s) =
\begin{cases}{\varepsilon}^{-\alpha}s^2&\mbox{ if }s\le {\varepsilon}^\alpha\\
s&\mbox{ if }s\ge{\varepsilon}^\alpha,
\end{cases}$$ where $c_{\varepsilon}$ is chosen so that $U_{\varepsilon}\in \mathcal{H}_m$. Then straightforward estimates very much like those in [@IM1], for example, show that ${\mathcal{E}}_{{\varepsilon},\eta}(U_{\varepsilon}) \le C ({\varepsilon}^{-\alpha} + {\varepsilon}^{2\alpha-2})$, and follows by taking $\alpha = 2/3$. (This crude estimate has the advantage of holding for [*every*]{} $m >0$, so that we do not require $\lambda_0$ to be a regular value of $V$. If $\lambda_0$ is a regular value, then a variant of the same construction shows that $\inf_{\mathcal{H}_m} {\mathcal{E}}_{{\varepsilon}, {\rho_{\scriptscriptstyle{TF}}}} \le C {|\!\log{\varepsilon}|}$.)
[**3**]{}. Since $V- \lambda_0 = (V-\lambda_0)^+ - (V- \lambda_0 )^- = w-{\rho_{\scriptscriptstyle{TF}}}$, we may write the variational equation satisfied by $\eta$ in the form $$-\Delta \eta + \frac 1{{\varepsilon}^2}( \eta^2 - {\rho_{\scriptscriptstyle{TF}}}+ w)\eta = \frac 1{{\varepsilon}^2} (\lambda_{\varepsilon}-\lambda_0)\eta,$$ where $\frac 1{{\varepsilon}^2}\lambda_{{\varepsilon}}$ is a Lagrange multiplier. Multiplying by $\eta$ and integrating, and using the fact that $\eta\in \mathcal{H}_m$, we find that $$\frac m {{\varepsilon}^2}(\lambda_{\varepsilon}- \lambda_0) = \int_{{\mathbb{R}}^2} |\nabla \eta|^2 + \frac 1 {{\varepsilon}^2}\left[ w \eta^2 + (\eta^2-{\rho_{\scriptscriptstyle{TF}}})^2 + (\eta^2-{\rho_{\scriptscriptstyle{TF}}}){\rho_{\scriptscriptstyle{TF}}}\right].$$ It follows that $$\label{eq:lambdaeps0}
\frac m{{\varepsilon}^2}(\lambda_{\varepsilon}- \lambda_0) \le
4 {\mathcal{E}}_{{\varepsilon},{\rho_{\scriptscriptstyle{TF}}}}(\eta) +
\frac 1{{\varepsilon}^2}\|{\rho_{\scriptscriptstyle{TF}}}\|_{L^2({\mathbb{R}}^N)} \| \eta^2-{\rho_{\scriptscriptstyle{TF}}}\|_{L^2({\mathbb{R}}^N)}
\le C {\varepsilon}^{-4/3}$$ by and .
[**4**]{}. Now let $\rho_{{\scriptscriptstyle{TF}},{\varepsilon}} := (\lambda_{\varepsilon}- V)^+$. It follows from that $$\|\rho_{{\scriptscriptstyle{TF}},{\varepsilon}} - {\rho_{\scriptscriptstyle{TF}}}\|_{L^\infty({\mathbb{R}}^N)} = |\lambda_{\varepsilon}- \lambda_0| \le C {\varepsilon}^{2/3},
\label{eq:lambdaeps}$$ so that $K\subset\subset \Omega_{\varepsilon}:= \{ x\in {\mathbb{R}}^N : \rho_{{\scriptscriptstyle{TF}},{\varepsilon}}>0\}$ if ${\varepsilon}>0$ is sufficiently small, which we henceforth take to be the case. Note also that $$-\Delta \eta + \frac 1{{\varepsilon}^2}(\eta^2 - \rho_{{\scriptscriptstyle{TF}},{\varepsilon}})\eta = 0
\quad\quad\mbox{ in $\Omega_{\varepsilon}$.
}
\label{eq:etasol}$$ Now fix some $r \le \frac 12\operatorname{dist}(K,\partial {\Omega_{\scriptscriptstyle{TF}}})$. In view of , and since $V$ is $C^2$, there exists $a,k>0$ and ${\varepsilon}_0>0$ such that $$\label{ab}
\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}>a^2 \ \mbox{ and }\ \ |\Delta \sqrt \rho_{{\scriptscriptstyle{TF}},{\varepsilon}}| \le k
\mbox{ whenever }0<{\varepsilon}\le {\varepsilon}_0.$$ For any $x\in K$ and $b\in (0,a)$, define $$\zeta_{x,b}(y) = \zeta(y)= b( \frac{|y-x|^2}{r^2}-1)^2$$ in $B(x,r)$. Then for $b\in (0,\frac a2)$, $$\label{eq:zetasub}
-\Delta \zeta + \frac 1{{\varepsilon}^2}(\zeta^2-\rho_{{\scriptscriptstyle{TF}},{\varepsilon}})\zeta
\ \le \ -\Delta \zeta - \frac{3a^2}{4{\varepsilon}^2} \zeta \ < \ 0
\quad\quad\mbox{ in } B(x,r)
$$ whenever ${\varepsilon}$ is sufficiently small. It follows that $\eta\ge \zeta_{x,b}$ in $B(x,r)$ for every $b\in (0,\frac a2)$, as otherwise we could find some $b_0\in (0,\frac a2)$ such that $\min_{B(x,r)} (\eta-\zeta_{x,b_0}) = 0$. Since $\eta>0$, the minimum would have to be attained in the interior of $B(x,r)$, and this is impossible in view of and .
It follows that $$\eta(y)\ge \frac{9a}{32} =: \alpha \mbox{ in }B(x, r/2).
\label{eq:etalbd}$$ Note also that $\| \eta \|_{L^\infty({\mathbb{R}}^N)} \le \| \sqrt{\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}\|_{L^\infty({\mathbb{R}}^N)}$, since otherwise $\tilde\eta :=\min( \eta , \| \sqrt{ \rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}\|_\infty)$ would satisfy ${\mathcal{E}}_{{\varepsilon},{\rho_{\scriptscriptstyle{TF}}}}(\tilde \eta) < {\mathcal{E}}_{{\varepsilon},{\rho_{\scriptscriptstyle{TF}}}}(\eta)$, contradicting the minimality of $\eta$.
[**5**]{}. Now write $\theta :=\eta-\sqrt{\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}$. Then $$- \Delta \theta + a_{\varepsilon}(x) \theta = \Delta \sqrt {\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}
\quad\quad\mbox{ for }a_{\varepsilon}(x) = \frac 1{{\varepsilon}^2}(\theta + 2\sqrt{\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}})(\theta+\sqrt{\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}) \ \overset{\eqref{eq:etalbd}}\ge \ \frac {\alpha^2}{{\varepsilon}^2}$$ in $B(x, r/2)$, and $|\theta|\le 2 \|\sqrt\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}\|_{L^\infty({\mathbb{R}}^N)}$ on $B(x, r/2)$. Now for $y\in B(x,r/2)$ define $$\Theta_{\varepsilon}(y) :=
\frac k{\alpha^2} {\varepsilon}^2 + 2 \|\sqrt{\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}\|_{L^\infty({\mathbb{R}}^N)}
\exp\left[ \frac{\alpha}{r{\varepsilon}}( \frac{|y-x|^2}2- \frac {r^2} 8 )\right]$$ where $k$ is the bound for $\|\Delta \sqrt {\rho_{{\scriptscriptstyle{TF}},{\varepsilon}}}\|_\infty$ found in . Then $\Theta \ge \theta$ on $\partial B(x,r/2)$, and there exists ${\varepsilon}_0>0$ such that $$(-\Delta + a_{\varepsilon})\Theta \ \ge \ k \ \ge\ (-\Delta + a_{\varepsilon})\theta \mbox{\ \ in \, $B(x,r/2)$,\qquad
if $0<{\varepsilon}< {\varepsilon}_0$.}$$ It follows that $\Theta \ge \theta$ in $B(x, r/2)$, and similarly $-\Theta \ge -\theta$ in $B(x, r/2)$. Thus $$\label{eq:theta}
|\eta - \sqrt {\rho_{\varepsilon}}| \le C {\varepsilon}^2 \quad\mbox{ on }B(x, r/4).$$
[**6**]{}. Returning to , we see that $$-\Delta \eta + b_{\varepsilon}\eta = 0 \quad\mbox{ in }B(x, r/4), \mbox{ for }b_{\varepsilon}= \frac 1{{\varepsilon}^2}(\eta^2-\rho_{\varepsilon}),$$ and implies that $\|b_{\varepsilon}\|_{L^\infty(B(x, r/4)} \le C$ independent of ${\varepsilon}\in (0,{\varepsilon}_0)$ and $x\in K$. Since we already know that $\|\eta\|_{L^\infty({\mathbb{R}}^N)} \le C$, we conclude from standard elliptic regularity that $\| \nabla \eta\|_{L^\infty(B(x, r/8))} \le C $. Also, it follows from and that $\| \eta^2 - \rho \|_{L^\infty(K)} \le C {\varepsilon}^{2/3}$, so we have proved .
Proof of Theorem \[thm:limit\]
==============================
In view of , Theorem \[thm:limit\] is a direct consequence of Theorem \[thm:main\] combined with Proposition \[prop:groundstate\] and the continuity of the solution of an initial value problem with respect to the nonlinearity.
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[**Addresses and E-mails:** ]{}
Robert Jerrard. Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada. E-mail: [rjerrard@math.utoronto.ca]{}
Didier Smets. Laboratoire Jacques-Louis Lions, Université Pierre & Marie Curie, 4 place Jussieu BC 187, 75252 Paris Cedex 05, France. E-mail: [smets@ann.jussieu.fr]{}
[^1]: We refer to Section \[sect:groundstate\] for the details on a number of statements regarding the ground states which we state without justification in this introduction.
[^2]: Note that we do [*not*]{} assume that ${\Omega_{\scriptscriptstyle{TF}}}$ is simply connected or that its boundary is smooth.
[^3]: This can be verified for a wide variety of weight functions $\eta,$ but we wish not consider that discussion here since we already know by means of the change of unknown that it satisfied when $\eta$ is a ground state.
[^4]: Proposition \[prop:loca\] actually uses a version of surplus energy for which the weighted energy ${{E}_{\varepsilon,\eta}}$ is restricted to $\Omega.$ Since the energy density and the weight are non-negative, our definition of surplus $\Sigma_a^t$ here, integrating on the whole ${\mathbb{R}}^2$, yields a larger number, and is therefore compatible with the claim of the proposition.
[^5]: Note that the regularization scale $r_\xi^t$ is fixed for $s\in [t,T]$.
[^6]: Note that we make no restriction on the dimension $N$ here.
|
---
abstract: 'Stochastic growth of binary alloys on a weakly interacting substrate is studied by kinetic Monte Carlo simulation. The underlying lattice model relates to fcc alloys, and the kinetics are based on deposition, atomic migration with bond-breaking processes and exchange processes mediated by nearest neighbor hopping steps. We investigate the interrelation between surface processes and the emerging nonequilibrium structure at and below the growing surface under conditions where atoms in the bulk can be regarded as immobile. The parameters of the model are adapted to CoPt$_3$ alloys. Growing nanoclusters exhibit an anisotropic short range order, primarily caused by Pt segregation at the surface. The overall structural anisotropy depends on both Pt surface segregation and cluster shape, and can explain the perpendicular magnetic anisotropy (PMA) recently measured in CoPt$_3$ nanoclusters on a van der Waals substrate. The onset of L1$_2$ ordering in the cluster is induced by surface processes. The same kinetic model is applied also to continuous thin films, which in addition can exhibit a small bulk contribution to PMA.'
author:
- 'S. Heinrichs,$^1$ W. Dieterich,$^1$ and P. Maass$^2$'
date: 'July 11, 2006'
title: Epitaxial growth of binary alloy nanostructures
---
Introduction {#sec:introduction}
============
Molecular beam epitaxy (MBE) has become an important tool to prepare ultrathin films and nanostructures in nonequilibrium, frozen-in states that show novel properties distinctly different from the equilibrium bulk phases. Theoretically, an important question is how to relate the structural characteristics of the growing material to the incoming flux, substrate temperature, adatom diffusion coefficients and adatom interactions. Successive stages of growth, nucleation and island formation in the submonolayer regime,[@Venables+84] second layer nucleation,[@Rottler+99; @Heinrichs+00] growth of three-dimensional clusters and emerging film morphologies, are fairly well understood for the one-component case, i.e. for deposition of only one species of atoms.[@Brune98; @Michely+04] By contrast, nonequilibrium alloy nanostructures produced by codeposition of two or more atomic species can display a variety of additional new phenomena that are only partly understood up to now. From the viewpoint of theoretical modelling, this is true in particular for metallic nanoalloys, which recently became of great interest from the experimental and technological viewpoint.[@Albrecht+02]
In many MBE experiments the substrate temperature is chosen sufficiently low so that atomic configurations in the interior of a growing cluster or film can be considered as frozen. Structural relaxation then takes place only among lower-coordinated atoms within the growth zone. Specific to binary systems are the following questions:
- Suppose that one atomic species tends to segregate at the surface. What is the emerging bulk structure upon further deposition burying a segregated surface layer?
- How large is a possible anisotropy in the frozen short-range order, i.e. a difference in the alloy structure between the lateral and the perpendicular (growth) direction?
- To what extent can diffusion processes limited to the near-surface region give rise to long-range order in the bulk of an ordering alloy at temperatures below the equilibrium order-disorder transition point?
- In the case of magnetic alloys, what are the magnetic properties associated with the non-equilibrium short or long-range order? This question is central in the ongoing search for materials that display a stable perpendicular magnetic anisotropy (PMA), which is useful for the development of high-density magnetic storage media. Under this viewpoint fcc-alloy systems of Co-Pt,[@Albrecht+01; @Shapiro+99] Fe-Pt[@Zeng+02] or Fe-Co[@Andersson+06] draw great attention in current investigations.
Here we report on answers related to these problems, based on kinetic Monte Carlo (KMC) simulations of a statistical model for binary fcc alloys of composition AB$_3$, with emphasis on CoPt$_3$ alloys.
MBE-grown nanoclusters of CoPt$_3$ on a van der Waals WSe$_2$ substrate were found recently to exhibit PMA at room temperature and below the onset of L1$_2$-ordering.[@Albrecht+01] The origin of PMA is thought to lie in a structural anisotropy induced by the growth process, strong enough to overcome the dipolar form anisotropy favoring in-plane magnetization. The model we use is based on effective chemical interaction and kinetic parameters extracted, respectively, from known equilibrium properties of CoPt$_3$[@Sanchez+89; @Gauthier+92] and from separate diffusion measurements.[@Bott+96] The dependence of the magnetic anisotropy on the local structure is represented by a sum over bond contributions, with nearest neighbor Co-Pt bonds as the dominant part. The associated bond anisotropy parameter is deduced from magnetic measurements on Co-Pt multilayer systems.[@Johnson+96] Within this model it was shown that Pt surface segregation and cluster shape effects can explain the occurence of PMA in some temperature window, followed by the evolution of L1$_2$-order at higher temperatures.[@Heinrichs+06] In this paper we present further details on cluster structures and extend our previous investigations to homogeneous films.
In section \[sec:model\] we introduce our kinetic model, based on random deposition and subsequent hopping moves of A and B atoms to neighboring sites of an fcc lattice. Hopping rates are derived from the energetics in the specific environment of each atom. To our knowledge, most prior KMC simulations of three-dimensional growth of binary systems were in the framework of solid on solid (SOS) models,[@Barabasi+95] which are not well suitable to describe cluster shapes and facetting in a realistic way. In very recent work the growth of CoPt$_3$ films was simulated, using a more realistic tight binding, second moment approximation to the interatomic interaction potential.[@Maranville+06] Segregation of Co-atoms to step edges on the growing surface was predicted, which leads to in-plane Co-clustering.[@Cross+01] By this the authors were able to explain PMA in disordered films, known from previous measurements[@Shapiro+99] to occur at elevated temperatures ($T
\gtrsim 500$K).
Our model, based on effective nearest neighbor couplings, allows for fully three-dimensional kinetics, a feature which is important in simulations of cluster growth and L1$_2$ ordering. It yields cluster facets in agreement with those observed in experiments and consistently accounts for mass transport along the cluster side and top facets. Both height and lateral growth largely relies on interlayer diffusion that starts from adatoms on the substrate, reaching higher layers along side facets. The present model automatically entails step edge barriers, dissociation from the surface and the possibility of vacancy diffusion in the bulk. It is less comprehensive than recent simulations of island formation in compound semiconductors, which are based on extensive electronic density functional calculations for energy barriers of the processes involved. These calculations, however, were so far limited to the submonolayer regime.[@Kratzer+02] Section \[sec:magnetic\] also describes a phenomenological relationship between the local contributions to the magnetic anisotropy and atomic short range order within a bond picture.
The anisotropic short range order and associated magnetic properties of clusters up to 4000 atoms are presented in section \[sec:clusters\], and are discussed in detail in terms of surface segregation and cluster shape. The same algorithm is applied in section \[sec:films\] to thin films. In addition to surface -induced PMA a bulk contribution is identified which, however, is too small to account for the measured PMA in thick films. Further conclusions are drawn in section \[sec:conclusions\].
Simulation model {#sec:model}
================
Interactions and Kinetics {#sec:interactions}
-------------------------
Our model starts from a reference fcc lattice with nearest neighbor distance $a$ in a cubic simulation box $L\times L\times L_z$ where sites can be occupied by an atom of type A (Co), B (Pt), or by a vacancy V. In contrast to models with a small concentration of vacancies and fixed shape of the sample, most of the simulation box considered here consists of vacancies representing the free space in the vicinity of the cluster. This allows for an unconstrained evolution of the cluster morphology. Effective interactions between A and B atoms are restricted to nearest neighbor pairs. This simplified description already captures essential statistical properties of ordering fcc alloys, including the phase diagram which displays L1$_2$ and L1$_0$ ordered structures in the vicinity of the AB$_3$ and AB-composition, respectively.[@Sanchez+89; @Kessler+01_03]
Bond energies between the different atomic species are denoted by $V_{\rm AA}, V_{\rm AB}$ and $V_{\rm BB}$. The linear combinations $I
= \frac{1}{4}(V_{\rm AA} + V_{\rm BB} - 2 V_{\rm AB})$, $h = V_{\rm
BB} - V_{\rm AA}$ which acts as a surface field, and $V_0 =
\left(V_{\rm AB} + V_{\rm BB}\right)/2$ control the bulk order-disorder transition temperature, the degree of surface segregation of B-atoms, and the average bond energy in the L1$_2$ ordered state. Parameter values are adjusted to reproduce equilibrium properties of CoPt$_3$: (i) The transition from the disordered to the L1$_2$-structure occurs at $T_0 \simeq 958\,$K,[@Berg+72] which is related to $I$ by $k_B T_0 \simeq 1.83I$.[@Binder80] Setting $I =
1$, our energy unit becomes $k_B T_0/1.83 = 45\,$meV, corresponding to 523$\,$K. (ii) The observed Pt surface segregation of nearly 100%,[@Gauthier+92] caused by the larger size of Pt relative to Co, is compatible with $h \gtrsim 4$; here we choose $h = 4$ in most of our calculations. (iii) At temperatures of interest, the mobility of atoms is effectively restricted to the film or cluster surface so that their typical coordination is between 3 (for an adatom on top of a terrace) and 7 (for an atom attached to a step edge). Parameters for a variety of corresponding processes were calculated for Pt within electronic density functional theory (DFT).[@Feibelman99] Different processes were classified according to the number of broken bonds, and the average bond energy yielded $V_0 \simeq - 5$, which we use here. Clearly, these bond energies are to be understood as effective energies since real interactions can extend to several neighbors.[@Kentzinger+00]
Experiments for nanoclusters of CoPt$_3$ were performed on a WSe$_2$ (0001) substrate surface,[@Albrecht+01] which is of the van der Waals type. The interaction with the substrate is modelled by a weak attractive potential, represented by an additional energy $V_s$ for atoms in the first layer. In the cluster simulations we choose $V_s=-5$. Compared to an fcc Pt (111) surface with three bonds of typical strength $V_0$, this amounts to about 1/3 of the energy of a single Pt-Pt bond.
The total flux $F$ of incoming atoms is taken as $F = 3.5$ monolayers/s. Additional parameters needed to describe the kinetics of the system are the transition state energy ${U_{\rm t}}$ and the attempt frequency $\nu$ for atomic moves. These values are known from diffusion experiments[@Bott+96] of Pt on Pt (111) surfaces as $\nu
= 8.3 \cdot 10^{11}$ s$^{-1}$ and ${U_{\rm t}}\simeq 5$. The resulting diffusion coefficient is $D/a^2 = (\nu/4) e^{- {U_{\rm t}}/k_B T}$ for moves without change in energy.
The rate for an elementary hopping process with a final energy $E_f$ and initial energy $E_i$ is chosen to be $$\label{eq:hopping_rate}
w_{if} = \nu e^{- {U_{\rm t}}/k_B T} \min (1, e^{-(E_f - E_i)/k_B T})$$ which fulfills the condition of detailed balance. As long as the general kinetics are concerned, characterized by the tendency of the system to approach thermal equilibrium, the specific form (\[eq:hopping\_rate\]) of the hopping rates should not be of crucial importance.
In many cases,[@Gambardella+00; @Santis+02] including Co deposited on Pt(111), adatoms on the top of a terrace or at step edges can exchange positions with an atom underneath in one concerted move. In comparison to single atom moves considered so far, a larger binding energy has to be overcome in order to effect such a simultaneous move of two particles. However, at the same time the transition state energy can be reduced because of the higher coordination in the transition state configuration. Direct exchange processes are found for Co deposited on Pt(111) over a wide temperature range 250–520$\,$K.[@Gambardella+00; @Santis+02] They are especially frequent for low-coordinated atoms on top of terraces (with coordination 3) or at step edges (with coordinations 4 or 5). We therefore include direct exchange processes between unlike atoms in the simulation. The corresponding rate involves an exchange barrier ${U_{\rm x}}$ that adds to the migration barrier ${U_{\rm t}}$, while initial and final energies are taken into account as before. These exchange processes, however, are allowed only if one atom has a coordination in the range 3 – 5 and the other one in the range 8 – 10. Let us remark that the activation energy for exchange diffusion of Pt on Pt (100) is just 470$\,$mV,[@Kellog94] which is almost the same as the total barrier ${U_{\rm t}}+ {U_{\rm x}}$ when ${U_{\rm x}}= 5$, as used in most of our computations described below.
The time evolution in our simulation is determined by a rejectionless continuous time MC-algorithm that generates realizations of the master equation describing the growth kinetics. Each simulation step consists of: (i) incrementing the time by an interval $\Delta t$ drawn from an exponential distribution with mean $\langle \Delta t \rangle = (\sum_f
w_{if})^{-1} $, (ii) executing a process $i \rightarrow f$ with probability proportional to its rate $w_{if}$, (iii) updating of rates of processes affected by the moving atom.
![A typical cluster configuration with 2000 atoms for $h=4$, $T=1.2$ and ${U_{\rm x}}=0$. Pt atoms are marked in blue and Co atoms are marked in different colors depending on the number of nearest neighbor Pt atoms: if more (less) Pt atoms are found out of plane than in plane the atom is marked in green (yellow); otherwise it is marked in red. Facets with sixfold {111} and fourfold {100} symmetry can be clearly distinguished.[]{data-label="fig:cluster-shape"}](figures/CoPt-T1_2-config){width="7.3cm"}
The simulation box consists of 15 layers with 1440 lattice sites each. To generate an isolated cluster attached to the weakly binding substrate, we start from a seed with 5 atoms on the surface. For the deposition rate $F=3.5$ ML/s used in most simulations and temperatures $T>0.5$, the distance between clusters, which would be observed by self-organized nucleation, exceeds the box size. This could be effectively accounted for by increasing the deposition rate at the boundary of the box to match the extra flux of atoms deposited within the typical capture zone of an island. However, in order to reduce the number of parameters, we deliberately kept the deposition rate constant for all temperatures.
An example of a typical cluster with $N=2000$ atoms obtained in the KMC simulations is shown in Fig. \[fig:cluster-shape\]. The top facet has (111) orientation as expected for the (111) substrate, and side facets are of (111) and (100) type. The surface shows strong Pt segregation. Similar to the experiments,[@Albrecht+02] the aspect ratio of lateral to height dimension of the cluster is about 3:1. At the temperature $T = 1.2$, the ratio $D/F a^4 \simeq 2 \cdot 10^{10}$, and the realization of the growth process implied $3.3 \cdot 10^{10}$ elementary processes. A simple MC algorithm which generates equivalent dynamics, but is not rejection free, can be estimated to require about $N^2 D/(a^4 L^2 F)\simeq 6\cdot 10^{13}$ trials for elementary moves, i.e. about 2000 times more than the algorithm used here. This ratio increases further at lower temperatures, since bond breaking processes will acquire lower rates and consequently contribute to larger time increments $\Delta t$. Subsequent equilibration of the clusters under zero flux during a multiple of the deposition time did not induce significant changes in the cluster configurations and will therefore be ignored in the following.
To generate homogeneous films (where the width of the interfacial growth zone is much smaller than the average film thickness), discussed in section \[sec:films\], we choose a stronger surface potential $V_s = -15$ which guarantees complete wetting in the first layer in the temperature range considered.
$I$ $h$ $V_0$ $V_s$ ${U_{\rm t}}$ ${U_{\rm x}}$ $\nu$
----- ----- ------- ------- --------------- --------------- ---------------------------
1 4 -5 -5 5 5 $8.3\cdot 10^{11} s^{-1}$
: Parameters used in the simulation. All energies are given in units of $k_B T_0/1.83 = 45\,$meV.
Structure-induced magnetic properties {#sec:magnetic}
-------------------------------------
In CoPt$_3$-alloys magnetic moments are mostly due to the Co atoms, with $\mu^{\rm Co} \simeq 1.7 \mu_B$,[@Menzinger+66; @Shapiro+99] while Pt atoms carry a comparatively small induced moment $\mu^{\rm
Pt}$ of about $0.3\mu_B$, which has been shown to depend on the actual atomic environment.[@Sanchez+89] Hybridization of d-electrons between neighboring Co and Pt atoms and a strong spin-orbit interaction near the Pt sites lead to a magnetic anisotropy that tends to align the Co moment along the Co-Pt bond direction. In the following we adopt a bond picture as in earlier work on bulk systems,[@Neel54; @Victoria+93] where the structural part of the magnetic anisotropy, $H_A$, is expressed as a sum over bonds $\langle
i, \delta\rangle$ that connect a Co atom with moment ${\boldsymbol \mu}_i ^{\rm
Co}$ at site ${\mathbf R}_i$ with a species $\alpha$ ($\alpha$ = Co,Pt,V) at a site ${\mathbf R}_i + {\boldsymbol \delta}$, $$\label{Ha}
H_A = - \sum _{\langle i, \delta\rangle} \sum_\alpha A^{\rm Co
\, \alpha} ({\boldsymbol \mu}_i ^{\rm Co}\cdot {\boldsymbol \delta})^2/(|{\boldsymbol \mu}_i
^{\rm Co} | |{\boldsymbol \delta}|)^2$$ By ${\boldsymbol \delta}$ we have denoted the 12 possible nearest neighbor bond vectors in the fcc lattice. Note that this anisotropy term naturally entails both surface and bulk contributions by considering vacancies as a possible neighbor species.
The parameters $A^{\rm Co \, \alpha}$ are the anisotropy energies associated with a Co-$\alpha$ bond. These are deduced from experiment and the term $\alpha=$ Pt gives the dominant contribution. Note that for saturated magnetization a nearest neighbor contribution to the anisotropic part of dipole-dipole interactions has exactly the same form, so that this contribution is included already in the coefficients $A^{\rm Co\,\alpha}$. The isotropic part of the exchange interactions only leads to a small renormalization of chemical interactions.
Equation (\[Ha\]) provides a relationship between a given atomic structure, to be obtained from simulations, and the magnetic anisotropy energy. Magnetic anisotropies for disordered alloys, including Co-Pt alloys, have been obtained by microscopic calculations based on the KKR-CPA method.[@Staunton+98] Although less accurate in principle, the local expression (\[Ha\]) here allows us to relate effects in the atomic short range order to the anisotropy energy in a direct way. Note that 2nd order contributions of the anisotropy in the direction cosines as in eq. (\[Ha\]) do not yield a magneto-crystalline anisotropy in an fcc lattice occupied by a single species. Additional bulk contributions are of 4th order and generally much weaker than the magnetic anisotropy in lower symmetry configurations to be considered here.
Several observations support the bond picture underlying eq. (\[Ha\]) as a reasonable approximation. Magnetic torque and magneto-optical Kerr-effect measurements on Co-Pt multilayers show that the anisotropy energy in these systems is dominated by an interfacial contribution connected to Co-Pt bonds. Detailed measurements of multilayers yield $K^{\rm CoPt}=0.97$ mJ/m$^2$ for (111) orientation and $K^{\rm CoPt}=0.59$ mJ/m$^2$ for (100) orientation.[@Weller+93; @Johnson+96] Considering the different angles of bonds to the surface and the different packings, eq. (\[Ha\]) indeed can reproduce this difference with one consistent value for $A^{\rm CoPt} \approx 250\,\mu$eV per Co-Pt bond.[@comm-A_par] Hence, on a semiquantitative level, anisotropy energies of Co-moments in different chemical environments are consistent with a superposition of bond contributions and a bond energy $A^{\rm CoPt}$ as given above.
Similarly, from measurements on Co-vacuum interfaces[@Kohlhepp+95; @Beauvillain+94] and theoretical investigations of a freely standing Co-monolayer[@Wu+91] we estimate $A^{\rm CoV} \simeq -67\,\mu$eV. The remaining parameter is $A^{\rm CoCo}$ for which we retain only the nearest neighbor dipolar contribution as discussed above. Using $\mu^{\rm Co} = 1.7 \mu_{\rm
B}$ and $|{\boldsymbol \delta}| = 2.72$ Å in CoPt$_3$, this yields $A^{\rm
CoCo}=23\,\mu$eV.
It is now straightforward to express the part of the anisotropy energy caused by local chemical order, $$\label{anisotropy}
E_s = H_A\{{\boldsymbol \mu}{\rm{\, in\, plane} } \} - H_A \{{\boldsymbol \mu}{\rm {\, out\, of\, plane}}\}$$ in terms of the numbers of Co-$\alpha$ bonds in plane, $n_\parallel^{{\rm Co} \, \alpha}$, and out of plane, $n_\perp^{{\rm
Co} \, \alpha}$. Using a site occupation number representation and the condition that occupation numbers for the three species $\alpha$ add up to unity at each site, (\[anisotropy\]) can be brought into the form $$\label{Es}
E_s = \sum_{\alpha = {\rm Co, Pt}} E_s^{\rm Co\,\alpha}=\frac{N}{2}\sum_{\alpha = {\rm Co, Pt}}(A^{{\rm Co} \,
\alpha} - A^{\rm Co V} ) P^{{\rm Co }\, \alpha}$$ with structural anisotropy parameters $$\label{parameter}
P^{{\rm Co}\,\alpha} = \frac{1}{N}(n_\perp^{{\rm Co}\,\alpha} -
n_\parallel^{{\rm Co}\,\alpha}).$$ These parameters enter as the primary structural characteristics related to the magnetic anisotropy.
In addition one has to take into account the magnetic form anisotropy due to dipolar interactions. Dipolar sums for a given cluster with saturated magnetic moment ${\mathbf M}_s$ are carried out using moments for Co and Pt as given before. By $E_{\rm dip}$ we denote the difference in dipolar energies for ${\mathbf M}_s $ parallel and perpendicular to the substrate, respectively. The anisotropy within the substrate plane turns out to be negligibly small because of the approximate 3-fold symmetry of cluster shapes. Nearest neighbor dipole-dipole interactions were already incorporated in the quantity $E_s$, as described above. The total magnetic anisotropy energy of a cluster is then given by $$\label{Etot}
E_{\rm tot} = E_s + E_{\rm dip}$$ Dipolar interactions generally favor in plane magnetization of thin films and clusters, $E_{\rm dip} < 0$. For thin homogeneous films, $E_{\rm dip} = - (\mu_0/2) M_s ^2$.
As mentioned above and supported by the above estimates, the term $\alpha = $Pt dominates expression (\[Es\]). PMA hence requires $P^{\rm CoPt}$ to be sufficiently large that $E_s$ can overcome the negative dipolar energy.
Clusters {#sec:clusters}
========
![Structural anisotropy parameters $P_{\perp-\|}^{\rm
Co\!-\!Pt}$ (upper panel) and $P_{\perp-\|}^{\rm Co\!-\!Co}$ (lower panel) and corresponding structural anisotropy energies. The dashed lines correspond to data obtained with a deposition rate of $F=0.35$ ML/s, which is 10 times smaller than the rate used for the full lines.[]{data-label="fig:struct_anisotropy-T"}](figures/CoPt-T-h4-N1000-coord "fig:"){width="9cm"}\
![Structural anisotropy parameters $P_{\perp-\|}^{\rm
Co\!-\!Pt}$ (upper panel) and $P_{\perp-\|}^{\rm Co\!-\!Co}$ (lower panel) and corresponding structural anisotropy energies. The dashed lines correspond to data obtained with a deposition rate of $F=0.35$ ML/s, which is 10 times smaller than the rate used for the full lines.[]{data-label="fig:struct_anisotropy-T"}](figures/CoPt-T-h4-N1000-coord-AA "fig:"){width="9cm"}
The two anisotropy parameters $P^{{\rm CoPt}}$ and $P^{\rm CoCo}$ defined in (\[parameter\]) are shown in Fig. \[fig:struct\_anisotropy-T\] as a function of temperature for clusters consisting of 1000 atoms. The solid lines connect data points for different values of the extra barrier for exchange processes ${U_{\rm x}}$, and for a flux $F = 3.5$ ML/s. In the absence of exchange processes (${U_{\rm x}}= \infty)$ both anisotropy parameters are smaller than zero for all temperatures, leading to a preference of in-plane magnetization. The reason is that the clusters are relatively flat and that surface segregation of Pt is weak because of kinetic suppression. As a consequence, in the surface of a cluster there exist more Co-Pt bonds in-plane than out-of-plane, while atoms in the inner part of the cluster give essentially no contribution to $P$. The flat cluster shape, on the other hand, would favor PMA if strong Pt segregation as realized at equilibrium could build up.
Indeed, when including exchange processes, Pt segregation gets enhanced considerably and $P$ can become positive. Actually, the cluster displayed in Fig. \[fig:cluster-shape\], showing pronounced Pt segregation to the (111) and (100) facets, was computed with $U_{\rm x} = 0$. As seen from Fig. \[fig:struct\_anisotropy-T\]a, for $U_{\rm x} = 0$, $P^{\rm CoPt}$ changes sign near $ T = 0.58$ and reaches a maximum at an optimum temperature $T_{\rm max} \simeq 0.8$. Conversely, $P^{\rm CoCo} < 0$ for all parameters, which leads to a small reduction of the sum (\[Es\]) relative to its leading term $\alpha = $Pt, see Fig. \[fig:struct\_anisotropy-T\]b.
The third set of data in Fig. \[fig:struct\_anisotropy-T\] (triangles) refers to ${U_{\rm x}}= 5$. This means that the activation energy for direct exchange is twice the diffusion barrier. Yet exchange processes have an important influence as they still can render $P^{\rm
CoPt}$ positive. The onset of a positive $P^{\rm CoPt}$ and the temperature were $P^{\rm CoPt}$ takes its maximum are shifted to somewhat higher values than in the case ${U_{\rm x}}= 0$. Fig. \[fig:struct\_anisotropy-T\] also contains data for the flux $F
= 0.35$ ML/s. The reduction of $F$ by one order of magnitude apparently leads to a small increase of $P^{\rm CoPt}$.
Investigating the mechanism of PMA in more detail we find that the sign of $P^{\rm CoPt}$ is determined by two major factors which show opposing trends in their temperature dependence. These are the degree of Pt surface segregation and the cluster shape, which we now discuss in more detail.
![Concentration of Pt atoms in the outer shell of clusters for $h=E_{\rm PtPt}-E_{\rm CoCo}=4$ in the upper panel and $h=0$ in the lower panel. The uppermost lines result from the mean field equations (\[eq:MF-U\]) and (\[eq:MF-S\]).[]{data-label="fig:segregation"}](figures/CoPt-T-h4-N1000-surf_a "fig:"){width="9cm"}\
![Concentration of Pt atoms in the outer shell of clusters for $h=E_{\rm PtPt}-E_{\rm CoCo}=4$ in the upper panel and $h=0$ in the lower panel. The uppermost lines result from the mean field equations (\[eq:MF-U\]) and (\[eq:MF-S\]).[]{data-label="fig:segregation"}](figures/CoPt-T-h0-N1000-surf_a "fig:"){width="9cm"}
The concentration $C_{\rm Pt}$ of Pt-atoms in the outer shell of a 1000 atom cluster is plotted in Fig. \[fig:segregation\], again for a flux $F = 3.5$ ML/s and three values of ${U_{\rm x}}$. As discussed above, the observed degree of Pt surface segregation is generally smaller than in the equilibrium case because of kinetic hindrance: During growth, the time for attaining equilibrium through exchange and diffusion processes is limited due to continuous incorporation of newly deposited atoms. Fig. \[fig:segregation\] reveals that both direct exchange processes and an increasing temperature act towards restoring equilibrium, i.e. they facilitate Pt surface segregation.
Clearly, a large $C_{\rm Pt}$ in the topmost layer leads to an enrichment of Co-atoms in the layer underneath. Therefore, for oblate cluster shapes with more surface area oriented in the \[111\] direction than in directions perpendicular to \[111\], a sufficiently strong segregation of Pt to the surface will induce a positive sign of $P^{\rm CoPt}$. This favors PMA, even for a disordered structure within the cluster interior. Furthermore, the weakly binding van der Waals substrate also allows for segregation towards the substrate during growth, effectively doubling the available surface with favorable orientation.
It should be noted that segregation of the majority atoms (Pt) can be observed even when the parameter $h = V_{\rm PtPt} - V_{\rm CoCo}$ controlling segregation is zero. This can be understood from a mean-field argument that counts bonds in a slab with a free (111) surface: for a random fcc alloy structure, the exchange of a Co-atom at the surface with a Pt-atom in the bulk allows the system to lower its energy on average by 3$I$, so that the surface concentration $C_{\rm Pt}$ will exceed the stoichiometric concentration, see Fig \[fig:segregation\]. By contrast, for a fully L1$_2$ ordered alloy the stoichiometric composition at the surface yields the lowest energy.
In order to give a more quantitative estimate of segregation, we consider a mean field model that consists of three completely filled (111) layers of atoms. The first layer is the free surface and the third one has fixed stoichiometric composition. Exchange processes are allowed between the outermost two layers. This model takes into account the almost non-existent bulk diffusion within a cluster, which results in an increased concentration of Co in the second layer impeding further exchange. The internal energy term per atom for a given Co concentration in the first layer $C_{\rm Co,1}$ can be expressed in the case of 1:3 stoichiometry as $$U=6 V_0 - \frac{3}{16}h\,(1-4\,C_{\rm Co,1}) + \frac{3}{2} I\,(1-C_{\rm
Co,1}+4\,C_{\rm Co,1}^2).
\label{eq:MF-U}$$
The energy minimum at $T=0$ is attained for $C_{\rm
Co,1}=\frac{1}{8}(1-h/(2J))$ clearly predicting segregation for $h=0$.
The entropy is $$S=-\frac{{k_{\scriptscriptstyle\rm B}}}{2} \sum_{\rm \alpha=Co,Pt}\;\sum_{z=1}^2 C_{\alpha,z}\ln C_{\alpha,z}.
\label{eq:MF-S}$$ Minimizing the free energy $F=U-TS$ then yields the equilibrium concentration of Co and Pt in the first layer. The corresponding results for the Pt concentration are displayed in Fig. \[fig:segregation\]. Including the entropy term leads to a decrease in Pt concentration in the outer shell with increasing temperature and restores the stoichiometric concentrations in the limit of high temperatures. Such a negative slope with temperature is reproduced in the simulations for $h=0$ when exchange processes are included, but not in the simulations for $h=4$ and $h=0$ without exchange processes. The positive slope of simulated segregation values with temperature indicates that segregation differs from thermal equilibrium due to kinetic limitations. This model also does not account for the portions on the cluster surface which are not flat.
![Cluster shape represented by the ratio of gyration radii $l_\parallel/l_\perp$ (with line as guide to the eye). Decreasing values signify the transition from oblate to spherically shaped clusters.[]{data-label="fig:shape_parameter-T"}](figures/CoPt-gyration-radii){width="8cm"}
![Number of atoms per layer $N_z$ represented with a radius $R_z=\sqrt{N_z/\pi}$ and averaged over 20 realizations of the growth process. Layer one is the bottom layer on the (111) substrate.[]{data-label="fig:atoms-layers"}](figures/CoPt-h4-N1000-atoms_in_layers){width="9cm"}
The second factor important for PMA is the cluster shape. An increasing temperature drives the shape closer to the equilibrium shape which is less oblate. This is seen from the aspect ratio $l_\parallel/l_\perp$ of the gyration radii of cluster sizes in the direction parallel and perpendicular to the substrate, shown in Fig. \[fig:shape\_parameter-T\]. Cluster shapes are parameterized in more detail in Fig. \[fig:atoms-layers\]. The atomic layer $z$ contains $N_z$ atoms and has an effective radius $R_z =
\sqrt{N_z/\pi}$. The figure displays the connection between $z$ and $R_z$ for a series of temperatures. It clearly shows the evolution of cluster shapes from oblate to almost a half-sphere as temperature is increased. In this way temperature effects in the cluster shape counteract the temperature dependent, segregation induced PMA. The result is the existence of a certain temperature window where PMA prevails, in agreement with experiments.[@Albrecht+02; @Albrecht+01]
![Scaling plot of surface $E_{\rm s}$ and bulk $E_{\rm dip}$ contributions to the magnetic anisotropy $E_{\rm tot}=E_{\rm s}+E_{\rm dip}$ as a function of cluster size $N$. The lines fitting $E_{\rm s}$ and $E_{\rm dip}$ have slope 2/3 and 1 respectively. The line through $E_{\rm tot}$ is given by $E_{\rm tot}=K_{\rm s} N^{2/3}-K_{\rm dip}
N$, where the prefactors $K_{\rm s}$ and $K_{\rm tot}$ are obtained from the fits to $E_{\rm s}$ and $E_{\rm dip}$.[]{data-label="fig:magneticE-scaling"}](figures/scaling-N){width="8cm"}
![Magnetic anisotropy energies $E_{\rm tot}$ (solid lines) and $E_{\rm s}$ (dashed lines) as a function of temperature for different values of the additional barrier ${U_{\rm x}}$ for exchange processes (no exchange processes for ${U_{\rm x}}=\infty$). $E_{\rm tot}$ also includes the form anisotropy from dipolar interactions, while $E_{\rm s}$ only contains the bond contributions (with inclusion of the nearest neighbor part of the dipolar interactions).[]{data-label="fig:magnetic_anisotropy-T"}](figures/CoPt-T-h4-N1000-magneticE){width="8.5cm"}
The discussion so far makes it clear that PMA essentially emerges as a surface effect. This suggests that $n_\perp^{\rm CoPt} - n_\parallel
^{\rm Co Pt} \sim N^{2/3}$, and accordingly $P^{\rm CoPt} \simeq
N^{-1/3}$. As shown by the solid line in Fig. \[fig:magneticE-scaling\], this behavior is well obeyed by the simulated data (triangles).
Combination of our results for $P^{\rm Co \, \alpha }$ with (\[Es\]) yields the structural part of the anisotropy energy. Adding the dipolar energy (see section \[sec:model\]) we obtain the total anisotropy energy (\[Etot\]), which can be written as $$\label{Eaniso}
E_{\rm tot} = K_s N^{2/3} - K_{\rm dip} N$$ The temperature dependence of $E_{\rm tot}$ is shown in Fig. \[fig:magnetic\_anisotropy-T\] for different ${U_{\rm x}}$ and $N =
10^3$ atoms. An optimal temperature where $E_{\rm tot}$ is maximum can clearly be identified. For example, for ${U_{\rm x}}= 5$ we have $T_{\max}
\simeq 1$. Fig. \[fig:magneticE-scaling\] contains a double-logarithmic plot of $E_{\rm tot}$ as a function of $N$ at $T =
1, \, {U_{\rm x}}= 5$. From $E_{\rm s}$ and $E_{\rm dip}$ we obtain $K_{\rm
s} \simeq 285\,\mu$eV and $K_{\rm dip} \simeq 5.6\,\mu$eV. The line for $E_{\rm tot}$ in Fig. \[fig:magneticE-scaling\] represents the expression (\[Eaniso\]) and, upon extrapolation to larger $N$-values, predicts the existence of an optimal cluster size for PMA, which is $N_{\rm opt} \simeq 4\cdot 10^4$. PMA is expected to disappear when $N \simeq 1.3 \cdot 10^5$.
Experimental values for $E_{\rm tot}$ for two different cluster sizes can be inferred by measurements of the blocking temperature using SQUID devices.[@Albrecht+01] For clusters with $N = 300$ at room temperature $(T = 0.56)$, this leads to the estimate $E_{\rm tot}
\simeq 2.8\,$meV. Similarly, $E_{\rm tot} = 3.4$ meV for $N = 1200$ at 573 K ($ T = 1.1$). In a different set of experiments at $T = 0.56$, granular nanostructures (dense covering of surface, but not touching) with similar lateral size of 3 nm were obtained with considerably larger anisotropy constants $K \simeq 13\,\mu$eV per atom.[@Albrecht+01] In our simulation model values $E_{\rm tot}
\lesssim 12\,$meV for $N = 300$ and $E_{\rm tot} \lesssim 20\,$meV for $N = 1200$ are found for $T \gtrsim 0.9$. The comparison shows that the experimentally observed values of the anisotropy energy lie within the ranges predicted by the model. The degree of agreement with experiment must be regarded as satisfactory in view of the uncertainties in the analysis of experiments and the simplifying assumptions in our model. One should also note the slower deposition rate in the experiments with $F = 0.02$ ML/s, compared to $F = 3.5$ ML/s used in most simulations. An effect of slower deposition rates in the simulations can be seen in Fig. \[fig:struct\_anisotropy-T\].
Another effect to be mentioned is that larger spin-orbit couplings are known to occur for Co-atoms with low coordination on terraces or along steps on Pt-surfaces.[@Gambardella+05] Such an effect in principle can further enhance PMA in nanoclusters because of the flat cluster shape. However, because of strong Pt surface segregation in the nanoclusters most of the near-surface Co-atoms are buried in the second layer or have high coordination. Hence we expect that the few Co-atoms found in the outermost layer need not be treated separately.
![Dependence of the order parameter for L1$_2$ structure $\Psi$ on temperature for segregation controlling parameter $h=4$ (upper panel) and $h=0$ (lower panel). Values are averaged over 20 realizations for $h=4$ and 10 realizations for $h=0$.[]{data-label="fig:orderpar-T"}](figures/CoPt-T-h4-N1000-order "fig:"){width="8cm"}\
![Dependence of the order parameter for L1$_2$ structure $\Psi$ on temperature for segregation controlling parameter $h=4$ (upper panel) and $h=0$ (lower panel). Values are averaged over 20 realizations for $h=4$ and 10 realizations for $h=0$.[]{data-label="fig:orderpar-T"}](figures/CoPt-T-h0-N1000-order "fig:"){width="8cm"}
Next we turn to analyzing the L1$_2$-ordering of clusters. Analogous to experiments, we determine the L1$_2$ order parameter from the magnitudes of scattered intensities around the associated three superstructure peaks ${\mathbf K}_i$, $i = 1,2,3$ in reciprocal space. For that purpose, atoms of the cluster and the surrounding vacancies are represented by pseudo-spins $s_l \in \{1,0,-1\}$ at the lattice positions ${\mathbf R}_l$ with $s_l = 0$ for vacancies and $s_l = \pm 1$ for $A$ and $B$, respectively. The structure factor is then calculated from the amplitudes $F_{\mathbf k}= \sum_l s_l e^{-i{\mathbf k}\cdot {\mathbf R}_l}$ which account for both the atomic arrangement in the cluster and the cluster shape. The finite dimensions of the cluster lead to a significant broadening of peaks in Fourier space, especially in the vertical direction. In order to account for this broadening, an integration in ${\mathbf k}$-space was performed around each peak in form of a sphere with radius 0.1$/a$. The total scattering intensity is calculated as $ I =
\sum^3_{i=1} \sum _{|{\mathbf k}- {\mathbf K}_i| < 0.1/a} |F_{\mathbf k}|^2 $. As an order parameter we define $$\label{orderpara}
\Psi = \frac{I - I_{\rm random}}{I_{{\rm L1}_2}- I_{\rm random}}$$ Here $I_{L1_2} $ and $I_{\rm random}$ are the intensities of clusters with identical shape, but perfect L1$_2$ order and random occupation by A and B atoms, respectively.
Fig. \[fig:orderpar-T\] shows the temperature dependent order parameter $\Psi$ in cases of a strong surface field $h = 4$ (a) and $h
= 0$ (b). The inclusion of exchange processes (${U_{\rm x}}< \infty$) diminishes $\Psi$ due to reduced ordering through segregation effects. This can be understood from the large energy a Co adatom gains when it is incorporated by an exchange process into deeper layers of the cluster irrespective of its contribution to ordering.
In the presence of exchange processes we notice from the figures that ordering sets in at temperatures near $T \simeq 0.8$ to 1 (420 – 523$\,$K). In experiments, the onset of L1$_2$ ordering was found for temperatures near 423$\,$K for 3$\,$nm thick films consisting of a dense assembly of islands.[@Albrecht+02] This temperature range is somewhat lower but close to the corresponding range in the simulations.
Comparison with Fig. \[fig:magnetic\_anisotropy-T\] shows that PMA for ${U_{\rm x}}= 5$ starts to decrease for $T \gtrsim 1$, which coincides with the onset of L1$_2$ ordering. One should bear in mind, however, that PMA in clusters is a surface induced effect and its decrease is caused by the change in cluster shape rather than by the concomitant bulk ordering.
![Atoms per layer versus total number of atoms during the growth of one cluster. The curve on top shows the number of atoms in the first layer above the substrate and curves below represent successive layers on top. For the 4th to 8th layer nucleation events are visible as a strong initial increase in the number of atoms. []{data-label="fig:cluster-layer-growth"}](figures/CoPt-h0-T1-i5-cluster-layer-growth){width="9cm"}
To better visualize the statistics of growth, we present in Fig. \[fig:cluster-layer-growth\] the layer-resolved evolution of one cluster. One can see pronounced incubation periods, before a nucleation event on top of the last layer takes place. Such an event is accompanied by a rapid initial growth and completion of the newly formed layer, followed by a period of essentially lateral growth of the whole cluster without change in height. The slope in the atom numbers of the top layer immediately after nucleation is much higher than one. This means that most of the atoms incorporated in a new layer during its completion arrive through mass transport along the side facets, leading to a transient lateral shrinkage of all the layers below the growing top layer. This is seen in the figure by the small dips in all curves occurring simultaneously during the short time intervals when the top layer is filled.
The nucleation events on top can be described by a theory originally devised for second layer nucleation.[@Heinrichs+02] This theory describes nucleation in a confined geometry with an influx of particles through deposition and loss of particles over a step edge. An essential parameter is the Schwoebel barrier that atoms have to surmount when crossing the step edge. In our simulations, this extra energy barrier effectively is about ${\Delta E_{\scriptscriptstyle\rm S}}=V_0$ because the intermediate state for a transition from the top facet to a side facet has one bond less than adatoms with coordination three on the top surface. With the parameters typically used in the simulations, the fluctuation dominated regime III discussed in Ref. is the relevant one for critical nuclei of size $i=1,2$. However, with the extra influx of atoms from the side facets, it is likely that the mean number of adatoms on top of the facet satisfies $\bar n>i$, so that mean-field theory may become justified for top layer nucleation.[@Heinrichs+02]
Films {#sec:films}
=====
![Anisotropy parameter $P_{\rm bulk}$ for films as a function of temperature for different values of the exchange energy barrier $U_{\rm x}$ and surface field $h$. The error bars are calculated from the statistical error after averaging over 5 realizations of the growth process and the fit uncertainty ($P$ as a function of $1/N$, see text) with error propagation.[]{data-label="fig:film-anisotropy"}](figures/CoPt-films-anisotropy.eps){width="8cm"}
The same model is now applied to continuous films. We simulate the growth of films starting with an empty simulation box with 224 atoms per layer and simulate growth up to deposition of 29 layers. The surface binding energy of $V_s=3 V_0=-15$ induces film instead of cluster growth. As before, the surface binding is independent of the atom type so that segregation to the substrate similar to a free surface can be observed. Consequently, true bulk properties are only found a few layers above the substrate and below the top surface. In order to separate the bulk contribution from surface effects, we considered each realization of the growth process in the range of 2000 to 6500 atoms and made a linear fit of $P$ vs. $1/N$. The anisotropy parameter in the bulk $P_{\rm bulk}$ is then found by extrapolating $N\to \infty$.[@comm-P_bulk] Fig. \[fig:film-anisotropy\] shows the temperature dependence of the anisotropy parameter $P_{\rm bulk}$ for different values of ${U_{\rm x}}$ and $h$. For most sets of parameters, $P_{\rm bulk}<0$, i.e. Co-Pt bonds align preferentially in the film plane. However, for certain combinations of parameters, the simulations yield $P_{\rm bulk}>0$, supporting PMA.
Clearly, even if $P_{\rm bulk}>0$ it is too small to account for the measured PMA in thick CoPt$_3$ films at elevated temperatures. As shown recently, those measurements can be explained by a model with interatomic interaction potentials that depend on the coordination of the atoms involved.[@Maranville+06] One essential feature of this model is that an effective segregation parameter, related to $h$ in our model, changes sign for intermediate coordinations of Co-atoms and thus favors Co segregation towards step edges in the surface. This sign change of $h$, however, should not significantly affect our conclusions. There we focused on the temperature range $T \lesssim 1$, where Co-segregation according to Ref. ceases to be effective.
![Long range order parameter $\Psi$ for films as a function of temperature for different values of the exchange energy barrier $U_{\rm x}$ and surface field $h$. Values are averaged over 5 realizations of the growth process.[]{data-label="fig:film-order"}](figures/CoPt-films-order_par.eps){width="8cm"}
The degree of long-range order in films is shown in Fig. \[fig:film-order\] at temperatures below the transition temperature $T_0 \simeq 1.83$, where bulk kinetics are suppressed. With higher $T$, the order parameter $\Psi$ becomes larger. Similar to our findings for clusters, the order parameter $\Psi$ in the case $h=4$ is suppressed relative to $h=1$, as clearly seen from the data at $T=0.8$. Surface segregation therefore appears to impede the development of long range order.
Conclusions {#sec:conclusions}
===========
In order to study growth kinetics of binary alloys we developed a lattice model based on nearest neighbor bonds with bond energies chosen to match equilibrium properties of CoPt$_3$. The kinetic parameters were obtained from diffusion experiments,[@Bott+96] DFT calculations,[@Feibelman99_PRB] and in addition from experiments observing interatomic exchange processes in the Co/Pt system.[@Gambardella+00; @Lundgren+99]
Experiments on CoPt$_3$ nanoclusters on a weakly binding substrate[@Albrecht+01] have revealed PMA in a temperature window that is well reproduced by our simulations. This temperature range is bounded towards low temperatures by frozen-in surface kinetics. The disappearance of PMA at higher temperatures is explained by the interplay of Pt surface segregation facilitated by direct exchange processes, and a transition from oblate to spherical cluster shapes. Our analysis suggests that the transition is not caused by L1$_2$ ordering. Yet, in our simulations the onset of L1$_2$ ordering is detected in the same temperature range where PMA disappears, in qualitative agreement with the measurements. It should be noted that up to $T \simeq 1$ long range order is induced solely by surface processes, as the bulk kinetics remain still frozen.
The structural anisotropy responsible for the magnetic anisotropy is characterized in our model primarily by a difference in the numbers of Co-Pt bonds out-of-plane and in-plane. Within a bond picture each Co-Pt bond provides a local contribution to the magnetic anisotropy, which tends to align the Co-moment parallel to the bond. An associated magnetic bond energy, deduced from experiments on Co-Pt multilayers, can reasonably reproduce the magnitude of PMA measured for nanoclusters in the appropriate temperature range. PMA is predicted to be a surface effect, a feature which could be tested experimentally.
Dipolar interactions tend to turn the easy axis of magnetization into the plane. The two competing effects, bond anisotropy (surface term) and dipolar interactions (volume term), lead to an optimum cluster size where the anisotropy energy of a cluster is largest, and a second characteristic size where PMA disappears.
The importance of cluster shape effects revealed by our computations is also corroborated by experiment, where a rotation of the easy magnetization axis into the film plane and an associated change in the aspect ratio from oblate to prolate was found in separate measurements on cluster assemblies at higher coverages.[@Albrecht+02] In our simulations, we observed clusters to become more spherical in shape and to develop an excess of in-plane Co-Pt bonds with increasing temperature.
Application of the same methodology to growing continuous films yields a bulk structural anisotropy favoring in-plane magnetization for most of the parameters studied. However, it can yield a positive bulk contribution to $P^{\rm CoPt}$ in a certain temperature range provided direct exchange processes are sufficiently fast. The associated PMA, however, is quite small and disappears for $T \gtrsim 1$. At these higher temperatures a different mechanism has been proposed recently to describe PMA in thick films in terms of in-plane Co clustering.[@Maranville+06]
Finally, we like to draw attention to the question of growth of magnetic clusters or films in strong external magnetic fields that saturates the magnetization in the growth direction. The magnetic anisotropy energy should then induce an anisotropy in the probabilities for atomic hopping such that out-of-plane Co-Pt bonding and PMA become favored. In fact, simulation results for growth in a magnetic field suggest that this effect might become detectable in the Co-Pt system. In particular, for L1$_0$ ordered alloys estimates based on Landau theory suggest that this additional field-induced PMA becomes stronger.[@Einax+06]
We thank M. Albrecht, M. Einax, M. Kessler, A. Majhofer and G. Schatz for helpful discussions. This work was supported in part by the Deutsche Forschungsgemeinschaft (SFB 513).
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The value for $A^{\rm CoPt}$ can be afflicted with uncertainty arising from measurements of films with varying interface quality through different preparation methods. In a previous work[@Heinrichs+06] we estimated this parameter based on experiments with values $K^{\rm CoPt}$ falling in the range[@Harzer+92; @McGee+93] of 0.27–1.15 mJ/m$^2$. The model can also describe the major contribution of the anisotropy resulting from alternating Pt and Co atomic layers in the ordered L1$_0$-phase of CoPt, where the lattice distortion (tetragonalization) is only responsible for about 20% of the observed anisotropy.[@Razee+01_PRB]
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An alternative determination of $P_{\rm bulk}$ where the outermost two layers at the top and bottom are excluded yields comparable results. We defined the outermost two layers to consist of atoms with coordination less or equal to 11 together with their nearest neighbors.
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|
---
abstract: 'The world astronomical image archives represent huge opportunities to time-domain astronomy sciences and other hot topics such as space defense, and astronomical observatories should improve this wealth and make it more accessible in the big data era. In 2010 we introduced the [*Mega-Archive*]{} database and the [*Mega-Precovery*]{} server for data mining images containing Solar system bodies, with focus on near Earth asteroids (NEAs). This paper presents the improvements and introduces some new related data mining tools developed during the last five years. Currently, the [*Mega-Archive*]{} has indexed 15 million images available from six major collections (CADC, ESO, ING, LCOGT, NVO and SMOKA) and other instrument archives and surveys. This meta-data index collection is daily updated (since 2014) by a crawler which performs automated query of five major collections. Since 2016, these data mining tools run to the new dedicated EURONEAR server, and the database migrated to SQL engine which supports robust and fast queries. To constrain the area to search moving or fixed objects in images taken by large mosaic cameras, we built the graphical tools [*FindCCD*]{} and [*FindCCD for Fixed Objects*]{} which overlay the targets across one of seven mosaic cameras (Subaru-SuprimeCam, VST-OmegaCam, INT-WFC, VISTA-VIRCAM, CFHT-MegaCam, Blanco-DECam and Subaru-HSC), also plotting the uncertainty ellipse for poorly observed NEAs. In 2017 we improved [*Mega-Precovery*]{}, which offers now two options for calculus of the ephemerides and three options for the input (objects defined by designation, orbit or observations). Additionally, we developed [*Mega-Archive for Fixed Objects*]{} (MASFO) and [*Mega-Archive Search for Double Stars*]{} (MASDS). We believe that the huge potential of science imaging archives is still insufficiently exploited. In this sense, defining and making available a standard format for indexing meta-data needed to access the image archives could strongly enhance their use. We recommend to IAU to define such a standard and ask the astronomical observatories to adopt it for indexing their image archives in a homogeneous manner, and make these indexes available up to date, free of any proprietorship period.'
address:
- |
Isaac Newton Group (ING), Apt. de correos 321, E-38700, Santa Cruz de La Palma, Canary Islands, Spain\
Instituto de Astrofisica de Canarias (IAC), Via Lactea, 38205 La Laguna, Tenerife, Spain
- 'Amateur astronomer and computer programmer, Brasov, Romania'
- |
Instituto de Astrofisica de Canarias (IAC), Via Lactea, 38205 La Laguna, Tenerife, Spain\
Universidad de La Laguna, 38205 La Laguna, Tenerife, Spain
author:
- Ovidiu Vaduvescu
- Lucian Curelaru
- Marcel Popescu
bibliography:
- 'paper114.bib'
title: |
Mega-Archive and the EURONEAR Tools for Datamining\
World Astronomical Images
---
Data mining ,asteroids ,near Earth asteroids (NEAs) ,image archives ,Mega-Archive ,Mega-Precovery.
Introduction {#sec1}
============
The world astronomical image archives provide valuable means to improve the physical properties of Solar System bodies, and in particular of near Earth asteroids (NEAs) which remain observable for short period of times. NEAs represent laboratories for studying the formation and evolution of the minor planets and their physical interactions with Sun and the major planets. Part of NEAs, potentially hazardous asteroids (PHAs) and virtual impactors (VIs) could pose some risk due to their possibility of impact, but they also represent an opportunity for cheaper space missions and eventually future mining industries.\
Upon discovery, the recovery and follow-up of NEAs are essential for providing the initial orbital solution and for searching of possible linkage with previously known objects. In most cases, smaller NEAs fade rapidly and become invisible even for largest telescopes which are expensive to access and usually lack time for urgent reaction. However, the existing image archives represent a free opportunity to improve the orbital knowledge based on serendipitous encounters of targets searched using dedicated data mining tools.\
Searching for fixed objects (like stars or galaxies) in image archives is straightforward, because only the position of the target is needed to compare with the known telescope pointing and instrument field. The largest astronomical observatories or their collaborating institutions provide simple web searching tools or more sophisticated services which allow searches of fixed objects in their image archives. In 2009 P. Erwin released the TELARCHIVE Python code[^1] (which requires Linux installation) allowing searches of fixed objects by querying a few remote collections of image archives. Data mining of moving objects and especially those having less accurate orbits becomes more complex, requiring the intersection in space and time of the orbit with the searched archives.\
Thanks to their large field, few major photographic plate archives started to be used two decades ago in the first NEA data mining projects by D. Steel in Australia (AANEAS)[^2], A. Boattini in Italy (ANEOPP, [@Boattini2001]) and G. Hahn in Germany (DANEOPS)[^3]. Due to their initial tiny sizes, CCD cameras have been less appealing to the astronomical community for data mining. Only few major NEA surveys such as NEAT and Spacewatch (whose archives were integrated in SkyMorph[^4]) or the modern Pan-STARRS (not providing a public tool) allow precovery searches of known asteroids and NEAs in their own archives.\
Within the EURONEAR[^5] project, since 2007 we datamined some major image archives to improve known NEA orbits, involving many amateurs and students in a few public outreach and educational projects. In 2010 we published the [*Precovery*]{} tool which allows searches of all know NEAs in a few existing and any other given instrument archive indexed in a simple ASCII meta-data format. Using this tool with four powerful archives (CFHTLS-MegaCam, ESO/MPG-WFI, INT-WFC and Subaru-SuprimeCam) we improved the orbits of more than 400 NEAs searched through 800,000 images [@Vaduvescu2009; @Vaduvescu2011; @Vaduvescu2012; @Vaduvescu2013; @Vaduvescu2017].\
Similar work has been carried out recently for detection and data mining of asteroids in the Kilo-Degree Survey (KiDS) observed with the VST-OmegaCam [@Mahlke2018] which resulted in 20,221 candidate objects (about half known and unknown asteroids) detected in 346 sq.deg. Other authors proposed similar public asteroid data mining and citizen science projects. In 2012, S. Gwyn proposed the “MegaCam Archival Asteroid Search Verification” (MAASV) project focusing on the CFHT-MegaCam archive [@Gwyn2012c]. Following our call for collaboration in 2010, the former EURONEAR member E. Solano proposed the SVO-NEA [@Solano2014] citizen science tool to precover NEAs in the SDSS DR8, then later the similar SVO-ast project to measure NEAs and Mars-crossers in a collection of surveys (SDSS, UKIDSS, VISTA and VSS) [@Solano2018].\
Besides astrometry, image archives are valuable also for deriving photometry and physical properties of astronomical objects. Popescu et al. [@Popescu2016; @Popescu2018] retrieved the asteroids imaged by VISTA Hemisphere Survey, obtaining near-infrared colors for 53,447 solar system objects including 57 NEAs, 431 Mars Crossers, 612 Hungaria asteroids, 51382 main-belt asteroids, 218 Cybele asteroids, 267 Hilda asteroids, 434 Trojans and 29 Kuiper Belt objects.\
In 2010 we introduced the [*Mega-Precovery*]{} project [@Popescu2010], aiming to allow public searches of one or a few asteroids or NEAs in a large collection of image archives which aimed initially to include at least one million images. This project’s capabilities and databases have been strengthened during the last years [@Char2013; @Popescu2014; @Vaduvescu2013] and in this paper we present the latest version of [*Mega-Precovery*]{}. To our best knowledge, there is only another similar web-tool allowing searches of asteroids in a large collection of archive images, namely the “Solar System Object Image Search” (SSOIS)[^6] hosted at CADC in Canada. This project was published first in 2012 by S. D. J. Gwyn and colleagues [@Gwyn2012a; @Gwyn2012b], being focused first only on the CADC archive and enhanced later with many other archives [@Gwyn2014; @Gwyn2015a; @Gwyn2015b; @Gwyn2016]. Another similar project to search for moving objects in image archives was “The Planetary Archive”, which was announced in 2014 [@Penteado2014] but whose outcome is unknown.
Mega-Archive
============
The largest astronomical observatories and related data centres provide [*science archive collections*]{} taken by their instruments and telescopes. These archives are served by common user interfaces allowing anybody to search and download images, spectra, catalogs and other data products. Following our first EURONEAR [*Precovery*]{} projects [@Vaduvescu2009; @Vaduvescu2011; @Vaduvescu2013], in 2010 we joined the first three [*instrument archives*]{} (CFHTLS, ESO/MPG and ING/INT) to start building the [*Mega-Archive*]{} database [@Popescu2010], with the aim to index at least one million images and add more archives later.
Archive Format and Index
------------------------
To search for any fixed or moving objects, any archive collection must contain some basic information about the observed images. To index the instrument archives within our database, the following “meta-data” fields are essential to be included:\
[**Image ID**]{} - *the string needed for downloading the FITS image file*
[**Observing date and time**]{} - *start of exposure in JD format (at least 5 decimals)*
[**Telescope pointings**]{} - *J2000 equatorial coordinates RA (hours with decimals) and DEC (degrees with decimals)*
[**Exposure time**]{} - *in seconds*
[**Filter**]{} - *given as a string*
[**Targeted object or field**]{} - *the name of the actual object or the observed field (string)*
The actual FITS images are linked remotely to the servers which store the archive collections, and they come either in raw original state or sometimes processed (ex from surveys). To avoid any missing in the indexing process of archives, and to avoid any duplicates due to possible processed images, the [*Mega-Archive*]{} includes only the raw images from any given archive. In the future we plan to add in the output list links to the processed images, whenever these are available.\
The instrumental archives are summarized in the master ASCII index file `ArchiveLogs.txt`, which describes all instruments with the following information (one instrument in one line):\
[**Collection/Telescope-Instrument**]{} - *Three acronyms naming the instrument archive*
[**Web Address**]{} - *Root internet address serving a given current image ID*
[**FOV**]{} - *Field of view on sky (in sq.arcmin) of the instrument assumed a rectangle*
[**MPC**]{} - *Minor Planet Centre observatory code where the instrument is located*
[**Width**]{} - *Width of the field (in degrees) along RA*
[**Height**]{} - *Height of the field (in degrees) along DEC*
[**Mag**]{} - *Limiting visual magnitude (V-band) in one minute exposure, taking into account the telescope diameter[^7]*
[**JD\_START**]{} - *Julian date of the first image in the archive*
[**JD\_END**]{} - *Julian date of the last image in the archive*
[**Nr. imgs**]{} - *Number of images in the archive indexed in the given period*
Instrument Etendue and Archive Etendue
--------------------------------------
To characterize the efficiency of telescopes and instruments in survey work, the astronomers use the term [*etendue*]{} ($A\Omega$) defined as the product of the telescope collecting area $A$ (expressed in square meters) and the surface on sky of the imaging camera $\Omega$ (in square degrees). To characterize the data mining efficiency of entire instrument archives, we propose the term [*archive etendue*]{} ($A\Omega\,A$) defined as the product of the etendue and the number of science images included in the given instrument archive.
Archive Collections
-------------------
We briefly present the six image archive collections included in [*Mega-Archive*]{} by 23 Feb 2019.\
[**The Canadian Astronomical Data Centre (CADC)**]{} was established in 1986 to store Hubble Space Telescope (HST) and later Canada France Hawaii Telescope (CFHT) data. Today it manages data collections taken with many other North American telescopes. Since the beginning of our project we ingested some of the CFHT archives. Currently, the [*Mega-Archive*]{} includes about two million images from 29 instruments (visible and NIR) installed on 8 telescopes linked to the CADC collection. The CADC Advanced Search[^8] allows multiple archive selections using a CGI form which presents a few selection menus. The CADC archive also allows programmatic queries, and for each instrument we used the following options to retrieve the needed meta-data for all available optical and NIR images (all other search options being left as default):
[**The European Southern Observatory (ESO)**]{} started in 1994 to archive NTT images [@Albrecht1994], and provides today one of the largest collection taken with major ESO telescopes in La Silla and Paranal. The [*Mega-Archive*]{} includes more than 3.3 million images from 15 instruments installed on 9 ESO telescopes. In order to query the ESO archive, we used the [*ESO Science Archive Facility*]{}[^9]. It allows to search and retrieve Raw Data using a CGI form showing two main selection blocks about the target/program and observing information. For each instrument we used the following options to search all available images (all other existing search options being left as default):
[**Las Cumbres Observatory Global Telescope (LCOGT)**]{} provides since 2013 their Science Archive [@Lister2013], storing all the images taken with the instruments and telescopes of this network. In 2017 we collected 21 available imagers from all telescopes, treating each instrument mounted on any telescope as one distinct archive (due to the distinct locations and MPC codes of each telescope). Today [*Mega-Archive*]{} includes more than one million images from the LCOGT collection. The LCOGT Science Archive[^10] could be manually or programatically queried based on the following options:
[**The Isaac Newton Group (ING)**]{} built in 2014 the ING science data archive based on telescope observing logs. We used this facility to ingest in the [*Mega-Archive*]{} almost 1.5 million images taken with 8 instruments mounted at all 3 ING telescopes. Besides the night observing logs, the ING query form[^11] allows an “Advanced search and browse” section for retrieval of meta-data and links to the available ING images sorted based on a few constraints. We used the following parameters to search all available ING images (all other available search options on this form being left as default):
[**The U.S. National Optical Astronomical Observatory (NOAO)**]{} provides since 2007 the [*NOAO Science Archive*]{} (known also as the NOAO NVO Portal) [@Miller2007] which allows VO-compliant access to imaging data taken with major American telescopes in Arizona and Chile. In 2012 we manually selected the first NVO instrument archives to ingest in [*Mega-Archive*]{}. Until today we included almost 5 million NVO images from 19 imaging instruments mounted on 10 telescopes. The NVO collection is available for searches from the NOAO Science Archive website[^12]. Two options are possible, proprietary data access (requiring login) and general search for NOAO data (available for any other user). We used the second choice with the following options in the Simple Query Form:
[**The Subaru-Mitaka-Okayama-Kiso Archive (SMOKA)**]{} was built in 2001 to store the images taken with major Japanese telescopes [@baba2002]. It requires user registration and login before providing the FITS images, but the meta-data could be retrieved without registration. The [*Mega-Archive*]{} includes more than one million images from 17 instruments installed on 4 Japanese telescopes. The SMOKA Archive Advance Search[^13] provides meta-data selected based on the instruments and date intervals, allowing an output of maximum 20,000 rows in one query. We used the following parameters to search all available SMOKA images:
Other Instrument Archives
-------------------------
Additionally to major archive collections, we indexed two other instrument archives and surveys:\
[**The Wide Field Imager (WFI) of the Anglo-Australian Telescope (AAT)**]{} was ingested in the [*Mega-Archive*]{} in 2012. We considered important this instrument due to its large etendue, although this archive include only about 5,000 images observed in the period 2000-2006. To manually collect the meta-data we used the former AAT Data Archive form[^14] which was recently integrated in a modern portal serving more instrument archives[^15].
[**The Sloan Digital Sky Survey (SDSS-III DS9 release)**]{} was included in 2013, being queried via the DR9 Science Archive Server[^16]. The [*Mega-Archive*]{} includes almost one million images from the SDSS-III DR9 archive.
Migration to the SQL architecture
---------------------------------
The classic format of the [*Mega-Archive*]{} used between 2010 and 2017 was ASCII (txt files), each observation being stored in one line with columns defining data presented in Section 2.1 (separated by the “” character), and each instrument archive being stored in one file.\
Following the migration to the actual EURONEAR dedicated server, in July 2017 the old [*Mega-Archive*]{} ASCII database was imported to SQL. This change was recommended to keep the search time as short as possible, due to the increase in number of images in [*Mega-Archive*]{}. We presented the architecture of [*Mega-Archive*]{} and its connection to the new EURONEAR tools as a flowchart in another recent paper [@Curelaru2019]. Thanks to the new SQL architecture, the searches of [*Mega-Archive*]{} have become faster for Solar system objects (taking a few minutes necessary to build accurate ephemerides needed to search the entire archive of 15 million observations for one asteroid) and extremely fast for fixed objects (taking only few fractions of a second for one target).
Mega-Archive Daily Update Crawler
---------------------------------
Major observatories ingest in their archives observations and images on a daily basis, which makes these databases appealing for searches related to very rapid time-domain phenomena. In other cases, dedicated surveys become ingested in archives much later, following the image reduction, project completion or expiration of proprietorship periods (typically one year following the observing date). Nevertheless, some newly discovered objects could rapidly raise attention of the astronomical communities, society and mass-media, such as a potential impactor asteroid, the interstellar object ’Oumuamua or a close supernova. Sometimes such objects become very rapidly invisible even in largest telescopes, thus data mining remains the only possibility to prevent loss and obtain more information. In these cases, data mining of [**up to date**]{} existing archives for precovery observations (preferably closer in time to discovery) could bring crucial information regarding the objects’ nature, orbital classification and virtual or imminent impacts.\
A [*web crawler*]{} (known also as a spider), is an Internet robot tool which systematically browses the World Wide Web with the aim to create entries to build or update a search engine index (e.g. [@Olston2010]). For automated update of the [*Mega-Archive*]{}, in 2014 we designed and implemented a crawler to check major archive collections on a daily basis. This PHP tool includes some scripts to query the ESO, CADC, SMOKA, ING and LCO collections, crawling the programmable interfaces published by each server (which very rarely need some minor changes of format). The only collection not able for automate crawling is the NVO archive, which we update manually on a yearly basis. The daily update tool starts to run automatically in cron every midnight and typically takes less than one hour to update all instrument archives in the [*Mega-Archive*]{}. Every SQL databases holding the instrumental archive could be updated (if there is new information) and also the master `ArchiveLogs.txt` file is updated (changing the number of images and the observed interval, needed for comparison with next day crawling). Each instrument archive is searched at once, checking only the most recent data (ingested in the collection during the previous day). Most problems (possible mal-functions due to servers or internet connection) could be traced and corrected next day by the admin who checks the logs listing the operations with all instrumental archives crawled in all collections, but in practice such system errors very rarely happen.
Archive Comparisons
-------------------
As of March 2019, the [*Mega-Archive*]{} includes 111 instruments, listed in Table 1. The columns list several parameters, namely the [*archive etendue*]{} ($A\Omega\,A$), the field of view of the instrument (FOV), the archival date interval, and the total number of raw science images indexed to date. Using these data, we could compare the archives and assess their use for data mining.\
Figure 1 plots the histogram counting the total number of archives versus the number of images in each archive. The first three instruments are actually three surveys, namely CTIO1.3m (mostly 2MASS), VISTA and SDSS-DR9, followed by CTIO0.9m, INT-WFC and WHT-LIRIS.\
Based on the etendue ($A\Omega$), the most powerful survey facilities in the present [*Mega-Archive*]{} are Subaru-HSC, Blanco-DECam, VISTA-VIRCAM, Subaru-SuprimeCam, VLT-VIMOS and CFHT-MegaCam. These and the following are plotted in the histogram in Figure 2. The analysis is extended in Figure 3 which plots the histogram counting the number of archives versus the [*archive etendue*]{} ($A\Omega\,A$). Based on this factor, the most productive facilities to date are VISTA-VIRCAM, Blanco-DECam, VST-OmegaCam, CFHT-MegaCam, Subaru-SuprimeCam, VLT-VIMOS and INT-WFC. Quite many near infrared (NIR) instruments populate Figures 1 and 3 due to their fast cadence of images explained by short exposure times and very rapid readouts compared to the other instruments in visible.\
Most exposures are shorter than 80s, probably owing to the fast cadence of NIR instruments and other fast time-domain science, as one can observe the histogram in Figure 4. This makes [*Mega-Archive*]{} and data mining very appealing for asteroid science, as most targets should have stellar-like aspect (instead of longer trails) which results is easily measurable astrometry and photometry.\
The number of images as a function of the observing time when they were obtained is shown in Figure 5. The great majority of the images in the actual [*Mega-Archive*]{} were collected since 2005. The last two years show lower numbers due to later ingestion in the collections and proprietorship which make last images invisible to collections and [*Mega-Archive*]{}.
Data Mining Tools
=================
[*Mega-Archive*]{} was designed in 2010 for asteroid searches and it was improved later, mostly after 2017 following the migration of EURONEAR to the new private dedicated server. We will present next the applications accessing the [*Mega-Archive*]{}.
Mega-Precovery for Moving Objects
---------------------------------
In 2010 we introduced [*Mega-Precovery*]{} to search for Solar System Objects using the [*Mega-Archive*]{} collection [@Popescu2010]. Our project targeted mostly the precovery and recovery of observations for PHAs and VIs. The algorithm and a flowchart was presented in [@Vaduvescu2013]. During last years, several major improvements were added to the project, significantly extending its functionality. These are described bellow.\
The original version (2010-2016) resided on the old EURONEAR IMCCE server. It was embedded in the old wikiplugin PHP environment and used the old ASCII archives format. The code migrated to the new EURONEAR server where it is preserved (under the Older Tools section) as version v.1 [^17]. The second version (2016-2017) updated the code to pure PHP 7 environment installed in the new EURONEAR server, and converted the [*Mega-Archive*]{} to SQL database. The code has been preserved as version v.2[^18]. The actual version 3 (released in October 2017) has implemented two options for the calculus of the ephemerides of the searched object, allowing three options for the input which are offered as three different pages linked from the main EURONEAR Data Mining Tools[^19]. Moreover, additionally to the standard web form, the actual [*Mega-Precovery*]{} v.3 allows programmable queries via HTTP commands.
### Mega-Precovery from Designations
This is our first classic search for one or a few known asteroids or comets, based on the name, number or designation [^20]. The user can chose between two ephemerides generators, either using the classic Miriade server[^21] [@Berthier2009] (available in [*Mega-Precovery*]{} v.1, v.2 and v.3) or the [*OrbFit*]{}[^22] software [@OrbFit2011] installed only in [*Mega-Precovery*]{} v.3. The SsODNet service at IMCCE[^23] [@Berthier2007] is used to check object designations.
### Mega-Precovery from Orbit
If the object doesn’t have an official name, number or designation, but its orbital elements are know, then these can be input into Mega-Precovery[^24]. The required elements are the semimajor axis $a$, the eccentricity $e$, the inclination $i$, the ascending node $\Omega$, the argument of periapsis $\omega$, the mean anomaly $M$, and the epoch $MJD$.
### Mega-Precovery from Observations
If the object was discovered very recently or if the connection with the ephemerides server is broken, then the target can be searched using an input consisting in a block of observations in MPC format[^25]. In this case, [*OrbFit*]{} is run locally on the EURONEAR server, either using a single step ephemerides model (given at least 3 observations taken in one or few nights following discovery) or a three steps model (defined by blocks of discovery and follow-up data, first opposition data, and multi-opposition data).\
All three search options are provided with a few advanced options: All instruments or only some archives selected by the user could be searched at once in [*Mega-Precovery*]{}. The computational interval can be constrained in an interval around the observations (to speed up the search of the entire [*Mega-Archive*]{}) or left default to cover all archives. The safety search border allows some flexibility due to telescope pointings (sometimes insecure, affected by small dithering during multiple run exposures, or due to the pointing not matching the camera centre). The default value of the safety border is $0.02^\circ$, which means that the search could allow any target outside the image by up to $1.2^\prime$ to be included in the results. The running mode allows the user to chose between fast geocentric ephemerides (where all ephemerides and steps are calculated at geocentre) and slow topocentric mode (where each integration step is calculated for the topocentric observatory position (recorded as MPC code in `ArchiveLogs.txt` where the current searched instrument is hosted - this is important only for very close flyby observations of PHAs or VIs where topocentric correction could become important). Finally, three output options are provided: HTML, simple text (formatted as a table), and CSV (fields separated by coma).\
The [*Mega-Precovery*]{} output columns are: the archive name (in the archive/telescope-instrument format), the image ID (tentatively linked in HTML to the collection server to retrieve the FITS image), the observed time (YYYY/MM/DD HH:MM:SS UT format), exposure time (in seconds), the expected object magnitude ($V$ when available), the telescope limiting magnitude (rough limit given the aperture, based on conventions mentioned in Section 2.1), filter and targeted object, angular distance to the field pointing (in degrees, followed by a percentage compared with the circular FOV), and a link to a plot showing the camera overlay (only for available mosaic cameras) marking the expected position of the searched object.
MASFO for Fixed Objects
-----------------------
In July 2017 we introduced MASFO[^26] tool to allow [*Mega-Archive Search for Fixed Objects*]{} given in J2000 RA and DEC coordinates. One or a few objects (entered in successive lines) could be searched given the approximating $V$ magnitude of the object needed to compare with the limiting magnitude of the telescope (as given in Section 2). The output consists in the same columns as [*Mega-Precovery*]{}, including a link to the CCD overlay plot for most mosaic cameras included in [*Mega-Archive*]{}.
MASDS for Double Stars
----------------------
Following some data mining work in double stars using the OmegaCam archive [@Curelaru2017], in 2018 we implemented some tools for data mining the entire [*Mega-Archive*]{} for known double stars given by the Washington Double Stars Catalog (WDS) ID or the WDS Discoverer ID. This tool is named [*Mega Archive Search of Double Stars*]{} (acronym MASDS)[^27]. For observational planning we provide another tool named [*WDS Filter Datamining*]{}[^28] which allows as input the sky area (RA/DEC box limits), stars’ magnitudes, separation, existing observations and discoverer. Both these tools are presented fully in a recent paper [@Curelaru2019].
FindCCD for Moving and Fixed Objects
------------------------------------
Survey telescopes and larger field mosaic cameras are increasing in number, and [*Mega-Archive*]{} presently includes about 20 such powerful imaging instruments, with Subaru-HSC (104 CCDs) and Blanco-DECam (62 CCDs) leading the list. Ideally data mining tools need to point their users directly to the exact CCDs of such a mosaic camera possibly holding their fixed or mobile targets.\
In 2013 we provided [*Find CCD Subaru*]{}[^29] to search for known NEAs in the SuprimeCam archive [@Vaduvescu2017]. In 2016 we extended this work to other few cameras, namely VST-OmegaCam, INT-WFC, VISTA-VIRCAM, CFHT-MegaCam, Blanco-DECam and Subaru-HSC. This tool is named [*FindCCD*]{}[^30], being developed for NEA searches. It queries the SkyBoT server[^31] [@Berthier2006] to calculate positions of the known NEAs in the field and the NEODyS-2 server[^32] [@Chesley1999] to overlay the uncertainty ellipses for poorly observed NEAs. In Figure 6 we give an example of [*FindCCD*]{} plot for the very poorly observed NEA 2015 BS516 searched in a given Blanco-DECam image, where the uncertainty region (plotted in red) covers many CCDs.\
In March 2018 we released [*FindCCD for Double Stars*]{}[^33] to identify the particular CCD of few major mosaic cameras possible to hold known double stars from the Washington Star Catalog [@Curelaru2019].\
In June 2018 we deployed [*FindCCD for Fixed Objects*]{}[^34] to search for stars, galaxies or other fixed objects given as J2000 RA/DEC coordinates. Seven mosaic cameras are supported by March 2019, but we plan to extend the list soon. Figure 7 gives the result of the search of a galaxy in one VIRCAM image.
Conclusions and Recommendations
===============================
The EURONEAR [*Mega-Archive*]{} project started in 2010 by joining meta-data of the first three instrument archives (CFHTLS, ESO/MPG and ING/INT), becoming the first such server for data mining asteroids and NEAs. In the same year we introduced [*Mega-Precovery*]{} to search [*Mega-Archive*]{} images for one or a few known asteroids or comets given by designations.\
During past years we added other archives, mainly based on six archive collections (CADC, ESO, ING, SMOKA, NVO and LCOGT) and by 2018 [*Mega-Archive*]{} has indexed 15 million images. In 2014 we implemented a crawler for automate query of five collections (except for NVO which does not allow programmatic queries) for daily update of the [*Mega-Archive*]{}. In 2016 we released the second version of [*Mega-Archive*]{} and [*Mega-Precovery*]{} which migrated to the new EURONEAR server and adopted the new SQL database architecture. Since Oct 2017 the third actual version of [*Mega-Precovery*]{} introduced two options for calculus of the ephemerides and three input options (search by designation, based on orbit, and observations).\
During last few years we designed other tools aimed to take advantage of [*Mega-Archive*]{} for science other than NEAs. In 2017 we introduced MASFO tool to search the [*Mega-Archive for Fixed Objects*]{}. In 2018 we implemented the MASDS tool for [*Mega-Archive Search for Double Stars*]{}. To specify exactly the exact CCD of major mosaic cameras include in the [*Mega-Archive*]{}, in 2016 and 2018 we built the graphical [*FindCCD*]{} and [*FindCCD for Fixed Objects*]{} to overlay the moving or fixed targets over seven mosaic cameras, plotting also the uncertainty ellipse for poorly observed NEAs. In the near future we plan to grow the [*Mega-Archive*]{} and improve [*Mega-Precovery*]{} and other related data mining tools.\
We have entered already in the big data epoch, where present and future surveys will provide huge amount of imaging data valuable for data mining and time-domain astronomy. In this sense, we recommend to the IAU, specifically Division B (Facilities, Technologies and Data Science) and Commissions B2 (Data and Documentation) and B3 (Astroinformatics and Astrostatistics) to adopt a common format and recommend to all astronomical observatories to use it for indexing and storing their science images, and make them available for programmable queries.\
On a cosmic scale, NEAs pose real threat to mankind, and there are cases when faint or/and very fast moving virtual or imminent impactors could not be recovered even by the largest telescopes which typically are lacking immediate observing time. In such cases, tools able to datamine very recent archives could become crucial to precover such objects, improve their orbits, assess the risks and eventually eliminate the impact threats. In this sense, we recommend to astronomical observatories to index their entire available observations (completely up to to date) even though some images are considered the proprietorship of a given PI or project (who could decide to collaborate per request in making available such images whenever such scenario will raise).\
Acknowledgements
================
This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. This work is based on data obtained from the ESO Science Archive Facility. The SMOKA archive is operated by the Astronomy Data Center and the National Astronomical Observatory of Japan, being based on data collected at Subaru Telescope, Okayama Astrophysical Observatory, Kiso observatory (University of Tokyo) and Higashi-Hiroshima Observatory. This research uses services or data provided by the Science Data Archive at NOAO. NOAO is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under a cooperative agreement with the National Science Foundation. The first manual search of the archive was performed by the EURONEAR collaborator Farid Char (University of Antofagasta, Chile). This paper makes use of data obtained from the Isaac Newton Group of Telescopes Archive which is maintained as part of the CASU Astronomical Data Centre at the Institute of Astronomy, Cambridge. The ING archive was built from the observing logs during a summer project by the student Vlad Tudor (former ING student and EURONEAR collaborator). This work makes use of the Science archive and observations from the LCOGT network. The first archive search and retrieval was performed by the EURONEAR collaborators Ioana and Adrian Stelea (Bucharest Astroclub, Romania). MASDS for Double Stars tool has made use of the Washington Double Star catalogs maintained at the U.S. Naval Observatory. This research has made use of Miriade, SkyBoT and SsODNet VO servers developed at IMCCE, [*Observatoire de Paris*]{}. We thank to Jerome Berthier for his continuous support regarding the access to these services. MP acknowledges support from the AYA2015-67772-R (MINECO, Spain).
[lrrrrr]{}
\
Instrument Archive & $A\Omega\,A$ & FOV & Start Date & End Date & Imags\
\
Instrument Archive & $A\Omega\,A$ & FOV & Start Date & End Date & Imags\
AAT-WFI & 14831 & 1089.0 & 2000-08-21 & 2006-02-05 & 4453\
CADC/APASS & 42242 & 31329.0 & 2010-04-11 & 2014-06-02 & 121350\
CADC/CFHT-aobir & 184 & 1.4 & 1997-12-10 & 2011-11-23 & 45108\
CADC/CFHT-aobvis & 6 & 1.4 & 1996-05-04 & 1999-03-30 & 1573\
CADC/CFHT-CFHTIR & 1397 & 17.6 & 2001-01-10 & 2005-11-17 & 27996\
CADC/CFHT-Megacam & 1464861 & 3600.0 & 2003-02-22 & 2013-05-09 & 143896\
CADC/CFHT-MOCAM & 421 & 225.0 & 1995-05-28 & 1995-11-29 & 662\
CADC/CFHT-REDEYE & 241 & 4.4 & 1993-02-04 & 1998-09-03 & 19312\
CADC/CFHT-WIRCam & 437812 & 466.6 & 2005-11-18 & 2019-01-23 & 331845\
CADC/CTIO-CPAPIR & 19191 & 348.2 & 2005-02-13 & 2019-02-15 & 112290\
CADC/CTIO-CPAPIRVIS & 1616 & 348.2 & 2008-01-16 & 2018-10-27 & 9453\
CADC/DAO-E2VCCD & 9125 & 253.6 & 2008-02-11 & 2019-03-04 & 52018\
CADC/GeminiN-GNIRS & 190 & 1.4 & 2004-09-05 & 2015-03-15 & 9232\
CADC/GeminiN-GMOS & 12200 & 30.5 & 2001-08-14 & 2015-12-01 & 27973\
CADC/GeminiN-NIRINIR & 10436 & 3.2 & 2002-02-22 & 2015-12-02 & 225033\
CADC/GeminiN-NIRIVIS & 101 & 3.2 & 2004-06-28 & 2015-10-07 & 2181\
CADC/GeminiS-Flamingos2 & 6439 & 36.0 & 2013-10-10 & 2015-11-29 & 12496\
CADC/GeminiS-GMOS & 13299 & 30.5 & 2003-02-28 & 2015-11-29 & 30491\
CADC/HST-ACS & 1277 & 11.7 & 2002-04-02 & 2018-03-19 & 88290\
CADC/HST-NICMOS & 114 & 0.8 & 1997-03-21 & 2008-09-10 & 113867\
CADC/HST-NICMOS & 2 & 0.8 & 1997-07-23 & 2008-08-21 & 1730\
CADC/HST-WFC3/NIR & 186 & 4.9 & 2009-07-05 & 2012-11-15 & 30543\
CADC/HST-WFC3/Vis & 186 & 7.3 & 2009-07-13 & 2012-11-15 & 20626\
CADC/HST-WFPC/OPT & 41 & 9.0 & 1989-11-30 & 1993-12-03 & 3691\
CADC/HST-WFPC2/NIR & 15 & 9.0 & 1994-03-05 & 2008-04-06 & 1366\
CADC/HST-WFPC2/OPT & 1803 & 9.0 & 1993-12-20 & 2009-05-12 & 162072\
CADC/Mt.Stromlo-MACHO & 114404 & 1794.4 & 1992-07-21 & 2002-11-19 & 197868\
CADC/UKIRT-Michelle & 160 & 1.4 & 2001-10-04 & 2004-04-23 & 35337\
CADC/UKIRT-UFTINIR & 1871 & 3.2 & 1999-10-16 & 2011-07-18 & 183298\
ESO/3.6m-TIMMI2 & 305 & 1.9 & 2004-05-08 & 2006-06-28 & 63817\
ESO/MPG-WFI & 144865 & 1149.1 & 1999-04-15 & 2018-03-22 & 139642\
ESO/NTT-EMMI & 3834 & 83.2 & 2004-03-17 & 2008-04-01 & 17541\
ESO/NTT-SOFI & 17201 & 24.2 & 2006-03-30 & 2018-03-14 & 270421\
ESO/NTT-SUSI2 & 1366 & 30.5 & 2004-04-02 & 2008-12-29 & 17057\
ESO/VISTA-VIRCAM & 22270114 & 4726.6 & 2009-10-16 & 2018-03-23 & 1414673\
ESO/VLT-EFOSC2 & 24864 & 16.7 & 2004-07-03 & 2018-03-12 & 103169\
ESO/VLT-FORS1 & 23860 & 46.0 & 1999-01-23 & 2009-03-26 & 35852\
ESO/VLT-FORS2 & 129969 & 46.0 & 1999-10-30 & 2018-03-21 & 195289\
ESO/VLT-HAWKI & 105972 & 56.3 & 2007-08-01 & 2018-03-22 & 130127\
ESO/VLT-ISAAC & 19153 & 6.4 & 1999-03-01 & 2013-12-13 & 208325\
ESO/VLT-NACO & 4719 & 0.9 & 2001-12-02 & 2018-03-22 & 353652\
ESO/VLT-VIMOS & 1134094 & 841.8 & 2002-10-30 & 2018-02-15 & 93060\
ESO/VLT-VISIR & 137 & 0.1 & 2004-05-11 & 2015-11-22 & 73068\
ESO/VST-OMEGACAM & 1142071 & 3664.5 & 2011-04-01 & 2018-03-23 & 240767\
ING/JKT-JAG & 2158 & 100.4 & 2002-01-03 & 2003-08-01 & 98553\
ING/WHT-ACAM & 24856 & 63.7 & 2009-06-10 & 2018-03-05 & 101428\
ING/WHT-LDSS & 1276 & 100.4 & 1993-03-18 & 2000-03-24 & 3302\
ING/WHT-LIRIS & 39347 & 18.2 & 2004-03-03 & 2018-01-29 & 563405\
ING/WHT-PFIP & 20328 & 256.6 & 1993-12-03 & 2016-09-22 & 20582\
ING/WHT-WHIRCAM & 13 & 1.0 & 1995-02-10 & 1999-06-28 & 3174\
ING/INT-PFCCD & 221 & 121.0 & 1993-01-13 & 2014-05-17 & 1492\
ING/INT-WFC & 922193 & 1169.6 & 2002-01-02 & 2018-03-05 & 643626\
LCOGT/SSO-0M4-03 & 581 & 582.1 & 2015-10-15 & 2019-03-04 & 29931\
LCOGT/SSO-0M4-05 & 424 & 582.1 & 2015-10-02 & 2019-03-04 & 21830\
LCOGT/SSO-1M0-03 & 3482 & 169.5 & 2014-05-01 & 2019-03-04 & 94788\
LCOGT/SSO-1M0-11 & 2370 & 169.5 & 2014-05-01 & 2019-03-04 & 62911\
LCOGT/SSO-2M0-02 & 3016 & 100.4 & 2014-05-01 & 2019-03-04 & 34442\
LCOGT/HLK-0M4-04 & 206 & 582.1 & 2016-03-31 & 2019-03-04 & 10633\
LCOGT/HLK-0M4-06 & 1417 & 582.1 & 2016-03-31 & 2019-03-04 & 73010\
LCOGT/HLK-2M0-01 & 5386 & 100.4 & 2014-05-04 & 2019-03-04 & 61509\
LCOGT/CTIO-0M4-09 & 222 & 582.1 & 2017-12-04 & 2019-03-05 & 11432\
LCOGT/CTIO-1M0-04 & 1629 & 697.0 & 2014-05-12 & 2019-03-05 & 10790\
LCOGT/CTIO-1M0-05 & 15907 & 697.0 & 2014-06-01 & 2019-03-05 & 105336\
LCOGT/CTIO-1M0-09 & 793 & 697.0 & 2014-05-31 & 2019-03-05 & 5251\
LCOGT/OT-0M4-10 & 266 & 582.1 & 2015-06-03 & 2019-03-05 & 12645\
LCOGT/OT-0M4-14 & 1851 & 582.1 & 2015-02-13 & 2019-03-05 & 88046\
LCOGT/OT-0M4-XX & 54 & 582.1 & 2015-02-15 & 2015-06-02 & 2577\
LCOGT/SED-0M8-01 & 513 & 125.3 & 2014-05-02 & 2018-02-14 & 36883\
LCOGT/GOL-1M0-02 & 213 & 169.5 & 2014-05-02 & 2018-08-23 & 5795\
LCOGT/SAAO-1M0-10 & 17436 & 697.0 & 2014-05-01 & 2019-03-04 & 112576\
LCOGT/SAAO-1M0-12 & 10098 & 697.0 & 2014-05-01 & 2019-03-04 & 65198\
LCOGT/SAAO-1M0-13 & 15424 & 697.0 & 2014-05-01 & 2019-03-04 & 99589\
LCOGT/MDO-1M0-08 & 10158 & 697.0 & 2014-05-02 & 2019-03-04 & 67269\
NVO/CTIO0.9m-CCD & 24679 & 182.3 & 2000-04-04 & 2017-05-17 & 826261\
NVO/CTIO1m-Y4KCam & 22409 & 392.0 & 2006-12-14 & 2014-06-02 & 263817\
NVO/CTIO1.3m-ANDICAM & 4524 & 5.8 & 2000-04-04 & 2017-06-01 & 2141898\
NVO/Blanco-MOSAIC2 & 322956 & 1388.3 & 2004-08-11 & 2012-02-20 & 83665\
NVO/Blanco-NEWFIRM & 126046 & 785.1 & 2010-07-03 & 2011-05-19 & 57738\
NVO/Blanco-DECam & 15142160 & 17424.0 & 2012-09-12 & 2017-06-05 & 312542\
NVO/Blanco-ISPI & 30117 & 104.0 & 2006-06-05 & 2014-04-10 & 104106\
NVO/Bok-CCD & 197417 & 4844.2 & 2015-01-08 & 2017-06-04 & 35610\
NVO/KPNO2.1m-CCD & 859 & 29.2 & 2010-09-03 & 2014-07-02 & 35940\
NVO/Mayall-Misc & 64 & 0.4 & 2007-02-21 & 2015-06-29 & 70005\
NVO/Mayall-MOSAIC & 109052 & 1296.0 & 2004-09-01 & 2013-02-07 & 32998\
NVO/Mayall-NEWFIRM & 260610 & 785.1 & 2007-06-30 & 2013-02-04 & 130171\
NVO/SOAR-SAM & 581 & 9.0 & 2014-03-05 & 2017-05-23 & 18689\
NVO/SOAR-OptImg & 19150 & 27.3 & 2005-11-07 & 2017-06-14 & 203377\
NVO/SOAR-Spartan & 7929 & 25.4 & 2011-08-16 & 2017-05-06 & 90331\
NVO/WIYN0.9m-MOSAIC & 11534 & 3478.6 & 2004-11-25 & 2010-03-29 & 20232\
NVO/WIYN0.9m-S2KB & 11705 & 416.2 & 2004-09-09 & 2014-11-17 & 171614\
NVO/WIYN3.5m-MiniMosaic & 2887 & 100.4 & 2004-12-18 & 2013-04-20 & 12989\
NVO/WIYN3.5m-WHIRC & 6009 & 10.9 & 2008-04-15 & 2017-05-19 & 249238\
SDSS-DR9 & 131306 & 140.8 & 1998-09-19 & 2009-11-18 & 938046\
SMOKA/KANATA-HONIR & 2075 & 100.4 & 2014-03-05 & 2016-10-30 & 42098\
SMOKA/KANATA-HOWPol & 11803 & 225.0 & 2008-12-06 & 2016-10-24 & 106875\
SMOKA/Kiso-1kCCD & 1123 & 155.8 & 1993-02-26 & 2000-10-24 & 30004\
SMOKA/Kiso-2kCCD & 70609 & 2498.0 & 1998-09-08 & 2012-02-27 & 117640\
SMOKA/Kiso-KWFC & 373092 & 15620.0 & 2012-04-02 & 2017-04-30 & 99408\
SMOKA/Okayama-ISLE & 1146 & 17.1 & 2006-11-15 & 2016-04-24 & 86736\
SMOKA/Okayama-KOOLS & 64 & 21.8 & 2008-01-04 & 2016-01-10 & 3822\
SMOKA/Okayama-OASIS & 85 & 16.2 & 1998-08-14 & 1998-12-13 & 6843\
SMOKA/Subaru-CAC & 45 & 0.7 & 1999-01-06 & 2000-06-21 & 4318\
SMOKA/Subaru-COMICS & 433 & 0.5 & 1999-12-14 & 2016-07-26 & 56943\
SMOKA/Subaru-CISCO & 8046 & 3.7 & 1999-01-12 & 2007-04-09 & 148791\
SMOKA/Subaru-FOCAS & 6040 & 36.0 & 2000-02-02 & 2016-10-11 & 11437\
SMOKA/Subaru-HSC & 129398 & 8100.0 & 2014-03-26 & 2014-07-08 & 1089\
SMOKA/Subaru-ICRS & 2274 & 1.0 & 2000-09-22 & 2016-09-26 & 148973\
SMOKA/Subaru-Kyoto & 61 & 3.9 & 2004-04-08 & 2015-09-24 & 1065\
SMOKA/Subaru-MOIRCS & 30417 & 28.2 & 2004-06-11 & 2016-10-31 & 73474\
SMOKA/Subaru-SuprimeCam & 1133632 & 918.5 & 1999-01-05 & 2016-05-09 & 92447\
SMOKA/Subaru-SCExAO & 405 & 0.2 & 2000-01-22 & 2008-07-15 & 119955\
{width="14cm"}
[**Figure 1:** ]{} The number of archives versus the number of images in the [*Mega-Archive*]{}. Only the most populated instruments are labeled, and the step of the $X$ axis is changed above $X=1,000,000$ to be able to accommodate the most prolific two archives.
{width="14cm"}
[**Figure 2:** ]{} The number of archives versus the etendue ($A\Omega$) in the [*Mega-Archive*]{}. Only the most powerful instruments are labeled and the step on the $X$ axis is changed above $A\Omega=10$.
{width="14cm"}
[**Figure 3:** ]{} The number of archives versus the [*archive etendue*]{} ($A\Omega\,A$) in the [*Mega-Archive*]{}. Only the most powerful instrument archives are labeled across the variable step for the $X$ axis (changed at $X=100.000$ and $X=1,000,000$), to be able to accommodate the most prolific archives.
{width="14cm"}
[**Figure 4:** ]{} The number of images versus the exposure time (in seconds) in the [*Mega-Archive*]{}. The step on the $X$ axis is one unit larger for the last few cuts, to be able to accommodate all exposures.
{width="14cm"}
[**Figure 5:** ]{} The number of images indexed in the [*Mega-Archive*]{} versus the observing date (year).
[**Figure 6:** ]{} [*FindCCD*]{} plot showing the Blanco-DECam overlay of the image `c4d150213_085354_ooi_g_v1` and the uncertainty position of the poorly observed NEA 2015 BS516 (bordered in red) covering many CCDs.
[**Figure 7:** ]{} [*FindCCD for Fixed Objects*]{} plot showing the CCD holding a galaxy of coordinates 12:40:03.1 -11:40:04 overlaid on the VISTA-VIRCAM mosaic image `VCAM.2013-01-21T07:52:15.676`.
[^1]: http://www.mpe.mpg.de/ erwin/code
[^2]: http://users.tpg.com.au/users/tps-seti/spacegd4.html
[^3]: https://web.archive.org/web/20171221091544/http://earn.dlr.de/daneops
[^4]: https://skys.gsfc.nasa.gov/cgi-bin/skymorph/mobs.pl
[^5]: http://www.euronear.org
[^6]: http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ssois
[^7]: We adopt the following conventions: V=19 for 0.4m, V=21 for 1m, V=23 for 2m, V=25 for 4m, V=26 for 8m class telescopes and HST
[^8]: http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/search
[^9]: http://archive.eso.org/eso/eso\_archive\_main.html
[^10]: https://archive.lco.global
[^11]: http://www.ing.iac.es/astronomy/observing/inglogs.php
[^12]: http://archive.noao.edu
[^13]: https://smoka.nao.ac.jp/fssearch.jsp
[^14]: Old server http://site.aao.gov.au/arc-bin/wdb/aat\_database/observation\_log/make
[^15]: New server https://datacentral.org.au/archives
[^16]: https://dr9.sdss.org/fields
[^17]: http://www.euronear.org/tools/megaprecoveryV1.php
[^18]: http://www.euronear.org/tools/megaprecoveryV2.php
[^19]: http://www.euronear.org/tools.php
[^20]: http://www.euronear.org/tools/megaprecdes.php
[^21]: http://vo.imcce.fr/webservices/miriade
[^22]: http://adams.dm.unipi.it/orbfit
[^23]: http://vo.imcce.fr/webservices/ssodnet
[^24]: http://www.euronear.org/tools/megaprecorb.php
[^25]: http://www.euronear.org/tools/megaprecobs.php
[^26]: http://www.euronear.org/tools/masfo.php
[^27]: http://www.euronear.org/tools/dstars/masds.php
[^28]: http://www.euronear.org/tools/dstars/wdsfilter/wdsfilter.php
[^29]: http://www.euronear.org/tools/findsubaruccd.php
[^30]: http://www.euronear.org/tools/findccdaster.php
[^31]: http://vo.imcce.fr/webservices/skybot
[^32]: https://newton.spacedys.com/neodys
[^33]: http://www.euronear.org/tools/dstars/findccddstars.php
[^34]: http://www.euronear.org/tools/findccdfixed.php
|
---
abstract: 'We provide an overview of several non-linear activation functions in a neural network architecture that have proven successful in many machine learning applications. We conduct an empirical analysis on the effectiveness of using these function on the MNIST classification task, with the aim of clarifying which functions produce the best results overall. Based on this first set of results, we examine the effects of building deeper architectures with an increasing number of hidden layers. We also survey the impact of using, on the same task, different initialisation schemes for the weights of our neural network. Using these sets of experiments as a base, we conclude by providing a optimal neural network architecture that yields impressive results in accuracy on the MNIST classification task.'
bibliography:
- 'DNNactivation2017.bib'
---
Introduction {#sec:intro}
============
Building neural net architecture that work well in practice involves finding and tuning many optimal parameters and making several design decisions that make the issue not trivial. One of such decision is the type of layer that compose the the neural net, particularly involving the type of activation functions that will be implemented as hidden layers. The ability to model non-linearities in our data is at the foundation of using such activations function, which allow us to build more complex representations of our data.
Historically, non-linear activations functions like the logistic sigmoid functions or tanh functions, while successful with certain data distributions, have proven difficult to train, mostly due to their non-zero centered property and slope of the function [@DBLP:journals/corr/XuHL16]. Many activations functions have been introduced in machine learning literature, with some surfacing as working well with many practical applications.
In section \[sec:actfn\], we initially provide a theoretical overview of the activation functions used in our experiments.
We then provide, in section \[sec:actexpts\], an empirical analysis and comparison of the performance of five such activation functions on a fixed task: the MNIST task of classifying hand-written digits [@20001258711]. We use a dataset of 60,000 data points, each being a 28x28 B&W image of a handwritten representation of a digit. For each experiment, we split the dataset into a training set of 50,000 datapoints, with the remaining 10,000 used as validation set. We use a fixed number of epochs (100), regardless of weather our model converges earlier. We also use a batch size of 50.
Based on our results from this analysis, we follow in section \[sec:deepmodelexp\] with an exploration of the impact of the depth of our model for the same task, by ranging the number of hidden layers from 2 to 8.
Finally, in section \[sec:weightinitexp\] we analyse different initilisation schemas for our neural networks weights, and see how overall accuracies vary accordingly.
Activation functions {#sec:actfn}
====================
This section covers a brief theoretical background on the activation functions taken into consideration in this work. We will use two activation functions as baseline models and four other functions as variations with potential optimizations to achieve better accuracy. The baseline models are the sigmoid function and ReLU function (Rectified Linear Unit) [@DBLP:journals/corr/AroraBMM16]. We consider these two baseline models since they are commonly used in many machine learning applications, and particularly in the case of the ReLU function, have stemmed over the years, variations that account for different data distributions and have experimentally yielded better results. The other four activations functions are Leaky ReLU [@DBLP:journals/corr/XuWCL15], ELU or Exponential Linear Units [@DBLP:journals/corr/ClevertUH15], and SELU or Scaled Exponential Linear Units [@DBLP:journals/corr/KlambauerUMH17].
Following are the definitions of the considered activations functions with their respective gradients, followed, in figure \[fig:comparison\_all\], by a their visualisation on the x-y axis.
**Sigmoid** (baseline model): $$\operatorname{sigmoid}(x) = \frac{1}{1+\exp{-x}}$$
$$\frac{d}{dx} \operatorname{sigmoid}(x) = \operatorname{sigmoid}(x)(1-\operatorname{sigmoid}(x))$$
**ReLU** (baseline model): $$\operatorname{relu}(x) = \max(0, x)$$
$$\frac{d}{dx} \operatorname{relu}(x) =
\begin{cases}
0 & \quad \text{if } x \leq 0 \\
1 & \quad \text{if } x > 0 .
\end{cases}$$
**Leaky ReLU:** $$\operatorname{lrelu}(x) =
\begin{cases}
\alpha x & \quad \text{if } x \leq 0 \\
x & \quad \text{if } x > 0 .
\end{cases}$$
$$\frac{d}{dx} \operatorname{lrelu}(x) =
\begin{cases}
\alpha & \quad \text{if } x \leq 0 \\
1 & \quad \text{if } x > 0 .
\end{cases}$$
**ELU:** $$\operatorname{elu}(x) =
\begin{cases}
\alpha (\exp(x) - 1) & \quad \text{if } x \leq 0 \\
x & \quad \text{if } x > 0 .
\end{cases}$$
$$\frac{d}{dx} \operatorname{elu}(x) =
\begin{cases}
\alpha \exp(x) & \quad \text{if } x \leq 0 \\
1 & \quad \text{if } x > 0 .
\end{cases}$$
**SELU:** $$\operatorname{selu}(x) = \lambda
\begin{cases}
\alpha (\exp(x) - 1) & \quad \text{if } x \leq 0 \\
x & \quad \text{if } x > 0 .
\end{cases}$$
$$\frac{d}{dx} \operatorname{selu}(x) = \lambda
\begin{cases}
\alpha \exp(x) & \quad \text{if } x \leq 0 \\
1 & \quad \text{if } x > 0 .
\end{cases}$$
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![Plots of activation functions[]{data-label="fig:comparison_all"}](comparison_all){width="\columnwidth"}
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Experimental comparison of activation functions {#sec:actexpts}
===============================================
In this section we present the results and discussion of experiments comparing networks using the different activation functions on the MNIST task [@20001258711]. We use 2 hidden layers with 100 hidden units per layer for these experiments. We report the learning curves (error vs epoch) for validation, and the validation set accuracies.
We start the experimental comparison by analysing our baseline models, and how they perform with different learning rates. We use the following learning rate values in this section: 0.02, 0.05, 0.10. We fixed these values after experimentally defining lower and upper bound values that either make the model not converge during training, or that gets stuck in local minima.
Sigmoid
-------
Reported below in table \[tab:sigmoid-table\] are the values achieved for the Sigmoid activation function for the three learning rates. Figure \[fig:SigmoidLayer\_depth2\_learningrate002\_epochs100\], figure \[fig:SigmoidLayer\_depth2\_learningrate005\_epochs100\], and figure \[fig:SigmoidLayer\_depth2\_learningrate01\_epochs100\] show, in order of learning rates, the different values achieved for error (left) and accuracy (right) across all 100 epochs.
We notice that, as the training error decreases, the validation error always decreases, suggesting that the model does not contain unnecessary complexity that lead to overfitting.
We also notice that, as our learning rate increases, both our training error and our accuracy increase. This can be justified by the following two reasons:
- With smaller learning rates, gradient descent is more likely to get stuck in local minima.
- Smaller learning rates usually require more epochs before an optimal solution is found. With all three learning rates, however we use the same number of epochs (100).
LR Train error Valid error Train acc Final acc
------- ------------- ------------- ----------- -----------
0.020 9.36e-02 1.14e-01 0.97 0.968
0.050 2.70e-02 8.48e-02 0.99 0.975
0.100 5.77e-03 8.77e-02 1.00 0.977
: Sigmoid layers: Errors and accuracies for different learning rates[]{data-label="tab:sigmoid-table"}
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In figure \[fig:comparison\_sigmoid\] we visualise the performance, with regards to validation error and validation accuracy of our models with three different learning rates.
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![Sigmoid - Validation error and accuracy of different learning rates[]{data-label="fig:comparison_sigmoid"}](comparison_sigmoid){width="\columnwidth"}
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ReLu
----
Rectified Linear Units, our second activation function, is our other baseline model. From table \[tab:relu-table\] we see that it already seems to achieve better results than our sigmoid model. The sigmoid activation function model likely incurred incurred into the vanishing gradient problem [@DBLP:journals/corr/abs-1211-5063], where a big amount of possible inputs precisely $2^{(28*28)}$ are ’squashed’ into a relatively small range: (0, 1), so the computed gradients are small as a result.
LR Train error Valid error Train acc Final acc
------- ------------- ------------- ----------- -----------
0.020 1.99e-03 1.04e-01 1.00 0.977
0.050 3.99e-04 1.12e-01 1.00 0.978
0.100 1.46e-04 1.18e-01 1.00 0.979
: ReLu layers: Errors and accuracies for different learning rates[]{data-label="tab:relu-table"}
We also notice a stark contrast with the Sigmoid model during training. Our ReLu models tend to find an optimal solution much before our 100th epoch (around 15-25th epoch, more precisely), where our validation error starts increasing, while the training error decreases. This is a sign of overfitting, and could have been address with an early stop at our 15-25th epoch. Figure \[fig:ReluLayer\_depth2\_learningrate002\_epochs100\], figure \[fig:ReluLayer\_depth2\_learningrate005\_epochs100\], and figure \[fig:ReluLayer\_depth2\_learningrate005\_epochs100\] show this behaviour.
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![ReLu - Validation error and accuracy of different learning rates[]{data-label="fig:comparison_relu"}](comparison_relu){width="\columnwidth"}
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Other activation functions {#otherfn}
--------------------------
In this subsection we report the experimental results with our other three activation functions. As with our baseline models, we train three models for each activation function, corresponding to the same three learning rates. For the purpose of not cluttering the section with graphs, we add the detailed performance of each model’s learning rate to section \[appendixA\] (Appendix A).
Here, we will only report the statistics at the 100th epoch and a summary graph of the performance for each learning rate.
All three model seem to perform equivalently to the ReLu model, and better than the Sigmoid model. The final accuracies at the 100th epoch range from 0.977 to 0.980. The best accuracy is achieved by the Leaky ReLu model, with a final validation accuracy of 0.980, corresponding to the lowest training error of 0.000147.
It is important to note that all three models actually achieved the lowest validation error at a much earlier epoch, just like the ReLu model did. All three models, in fact, show signs of overfitting (validation error increases, while training error goes to 0). We can infer that, potentially, the validation accuracies could be much greater and adapt better to new data if early stopping was applied.
The final results are in the same range because all three models alleviate the vanishing gradient problem which the Sigmoid activation function tends to incur into. In fact, Leaky ReLu and SELU and ELU all have negative values that help achieve mean shifts toward zero. This justifies the
**Leaky ReLu:**
LR Train error Valid error Train acc Final acc
------- ------------- ------------- ----------- -----------
0.020 2.02e-03 1.03e-01 1.00 0.977
0.050 4.07e-04 1.13e-01 1.00 0.978
0.100 1.47e-04 1.13e-01 1.00 0.980
: Leaky ReLu layers: Errors and accuracies for different learning rates[]{data-label="tab:lrelu-table"}
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![Leaky ReLu - Validation error and accuracy of different learning rates[]{data-label="fig:comparison_lrelu"}](comparison_lrelu){width="\columnwidth"}
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**ELU:**
LR Train error Valid error Train acc Final acc
------- ------------- ------------- ----------- -----------
0.020 3.62e-03 1.03e-01 1.00 0.977
0.050 5.84e-04 1.18e-01 1.00 0.978
0.100 1.98e-04 1.22e-01 1.00 0.979
: ELU layers: Errors and accuracies for different learning rates[]{data-label="tab:elu-table"}
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![ELU - Validation error and accuracy of different learning rates[]{data-label="fig:comparison_elu"}](comparison_elu){width="\columnwidth"}
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**SELU:**
LR Train error Valid error Train acc Final acc
------- ------------- ------------- ----------- -----------
0.020 2.53e-03 1.09e-01 1.00 0.976
0.050 5.28e-04 1.19e-01 1.00 0.977
0.100 1.95e-04 1.26e-01 1.00 0.978
: SELU layers: Errors and accuracies for different learning rates[]{data-label="tab:selu-table"}
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![SELU - Validation error and accuracy of different learning rates[]{data-label="fig:comparison_selu"}](comparison_selu){width="\columnwidth"}
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Deep neural network experiments {#sec:dnnexpts}
===============================
This section reports on experiments on deeper networks for MNIST. Specifically, the first set of experiments explores the impact of the depth of the network, by varying the number of hidden layers from 2 to 8. A second set of experiments is aims at comparing different approaches to weight initialisation.
Experiments in these sections have been carried with a fixed model chosen from the first part of the report. We use a model with 4 hidden ELU layers, and a learning rate of 0.02.
This selection is not arbitrary. We selected ELU over other activation functions not only because it achieve slightly achieves better experimental results in the previous set of experiments, but also because we identified a faster training time for these layers, which came in useful when training deeper models. The faster training time is justified in [@DBLP:journals/corr/ClevertUH15].
Additionally, we chose a learning rate of 0.02 over 0.05 and 0.1 despite it having a slightly lower accuracy. This is to prevent overflows that resulted from our implementation of our softmax layer.
Deep models {#sec:deepmodelexp}
-----------
Table \[tab:deep-table\] reports the training/validation errors and accuracy for the models with ELU hidden layers, varying from a depth of 3 to 8. We use the one with 2 ELU hidden layers from the previous section as a base model.
The first thing that can be noticed is that we can strictly achieve a lower training error as we increase the depth, and thus the complexity of our layer. In fact, we can achieve, in our models with 7-8 hidden layers, a two-fold decrease of the training error, which is almost annulled after the 100th epoch.
This result does not always correspond to a better model–in fact as the training error decreases, the validation error or the accuracy do not improve. The baseline model is the one that achieves the lowest validation error, which is a sign that our data does not meaningfully benefit from the added complexity. Furthermore, we notice that the accuracy improves almost insignificantly from our baseline model, achieving at best a 0.003 increase.
Depth Train error Valid error Train acc Final acc
------- ------------- ------------- ----------- -----------
2 3.62e-03 1.03e-01 1.00 0.977
3 9.85e-04 1.19e-01 1.00 0.977
4 3.93e-04 1.29e-01 1.00 0.979
5 2.02e-04 1.44e-01 1.00 0.979
6 1.22e-04 1.48e-01 1.00 0.979
7 8.63e-05 1.52e-01 1.00 0.980
8 6.42e-05 1.43e-01 1.00 0.979
: Errors and accuracies for different depths of an model with ELU hidden layers after 100 epochs[]{data-label="tab:deep-table"}
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Figure \[fig:comparison\_depth\] shows how the validation error and accuracy vary over the 100 epochs as we change the model’s depth. This gives a better picture of how early stopping could have actually helped prevent overfitting and come out with a better model.
It is interesting to see how the lowest validation error is achieved by the model with 3 hidden ELU layers at around epoch 25. What happens is that after epoch 25, the validation error of this model increases with a steeper slope than the one with depth 2 (baseline), since we end up overfitting the data more due to the increase complexity. It results, at epoch 100, that its validation error is higher. We can still infer that a slightly more complex, and less overtrained model could achieve better results in terms of validation error.
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![Validation error and accuracy of models with different depths[]{data-label="fig:comparison_depth"}](comparison_depth){width="\columnwidth"}
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Weight initialisation experiments {#sec:weightinitexp}
---------------------------------
In all preceding experiments, our model’s weights have been initialised using the Glorot/Xavier initilisation [@pmlr-v9-glorot10a]. Following their convention, initial weights are sampled from a uniform distribution whose range depends on the incoming ($n_{in}$) and outcoming ($n_{out}$) connections for units of our hidden layers.
More specifically our weights are sampled in this way:
$$\label{faninout}
w_{i} \sim U\Bigg(-\sqrt{\frac{6}{n_{in}+n_{out}}}, \sqrt{\frac{6}{n_{in}+n_{out}}}\Bigg)$$
In this section, we report results from using two other initialisation schemes, more specifically weights sampled from equations \[fanin\] and \[fanout\].
$$\label{fanin}
w_{i} \sim U\Bigg(-\sqrt{\frac{3}{n_{in}}}, \sqrt{\frac{3}{n_{in}}}\Bigg)$$
$$\label{fanout}
w_{i} \sim U\Bigg(-\sqrt{\frac{3}{n_{out}}}, \sqrt{\frac{3}{n_{out}}}\Bigg)$$
In this experiment we once again fix the learning rate, the model’s depth and hidden layer types, to have comparable results. In an analogous way as the previous experiments, we use a learning rate of 0.02 and ELU hidden layer types. We also choose to use a deeper layer model (depth 4) to potentially magnify the effect of using different weights.
Table \[tab:fan-table\] shows that our baseline model with Glorot Uniform initilisation performs much better than using fan-in and fan-out techniques. Validation errors and accuracies are lower after 100 epochs.
This result was somewhat expected because sampling weights following Glorot/Xavier (equation \[faninout\]) is just a combination of equations \[fanin\] and \[fanout\], where both forward propagation and backward propagation are taken into consideration when we calibrate the variance of the input independent of the number of incoming and outcoming connections to a single unit.
Schema Train error Valid error Train acc Final acc
--------- ------------- ------------- ----------- -----------
Glorot 3.93e-04 1.29e-01 1.00 0.979
Fan-in 3.62e-03 1.78e-01 1.00 0.973
Fan-out 7.76e-04 3.46e-01 1.00 0.955
: Errors and accuracies for different depths of a model with ELU hidden layers after 100 epochs[]{data-label="tab:fan-table"}
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Figure \[fig:comparison\_fans\] shows how validation errors and accuracy vary across the 100 epochs. Given the same number of epochs, Glorot initilisation is always superior to both other methods. Also, fan-in initilisation performs better than fan-out, which is prone to overfitting the model in our case.
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![Validation error and accuracy of models with different initialisation schemes[]{data-label="fig:comparison_fans"}](comparison_fans){width="\columnwidth"}
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Conclusions {#sec:concl}
===========
As demonstrated with the first set of experiments, ReLu, Leaky ReLu, ELU and SELU activation functions all yield great results in terms of validation error and accuracy on the MNIST task, with the ELU layer overall performing better than all other models.
Given this results, we found that using an additional hidden layer (for a total of 3 hidden ELU layers) improves the overall performance. Applying early stopping has proven to yield lower validation error and higher accuracy, as the models, particularly the deeper ones, tend to overfit the training data.
With this model, we also establish that the Glorot/Xavier Uniform initialisation schema yields the best result, compared to techniques that only take into account the incoming or the outcoming connections to a single node in the network.
To conclude, the following model is the one we found to yield the best results:\
Activation function: ELU\
Learning rate: 0.02\
Number of epochs: 25\
Number of hidden layers: 3\
Weight initialisation: Glorot Uniform
In further work, we will generalise these results to more complex classification tasks, like CIFAR-10 or CIFAR-100 [@krizhevsky2009learning] as well as regression tasks.
Appendix A {#appendixA}
==========
In this section we append error and accuracy graphs for different learning rates, as companion data to section \[otherfn\].
**Leaky ReLu:**
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**ELU:**
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**SELU:**
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|
---
author:
- |
Chien-Sheng Yang, , Ramtin Pedarsani, ,\
and A. Salman Avestimehr, [^1] [^2][^3]
bibliography:
- 'references.bib'
title: 'Communication-Aware Scheduling of Serial Tasks for Dispersed Computing '
---
Dispersed Computing, Task Scheduling, Throughput Optimality, Max-Weight Policy.
Introduction {#sec:intro}
============
System Model {#sec:sys}
============
Capacity Region Characterization {#sec:cap}
================================
Queueing Network Model {#sec:queue}
======================
Throughput-Optimal Policy {#sec:optimal}
=========================
Complexity of throughput-optimal policy {#sec:complexity}
=======================================
Towards more general Computing Model {#sec:dag}
====================================
Conclusion
==========
[^1]: This material is based upon work supported by Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001117C0053, ARO award W911NF1810400, NSF grants CCF-1703575, CCF-1763673, NeTS-1419632, ONR Award No. N00014-16-1-2189, and the UC Office of President under grant No. LFR-18-548175. The views, opinions, and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. A part of this paper was presented in IEEE ISIT 2018 [@yang2018communication].
[^2]: C.-S. Yang and A. S. Avestimehr are with the Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089 USA (e-mail: chienshy@usc.edu; avestimehr@ee.usc.edu).
[^3]: R. Pedarsani is with the Department of Electrical and Computer Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA (e-mail: ramtin@ece.ucsb.edu).
|
---
abstract: 'The main result of this note is a **Giambelli formula** for the Peterson Schubert classes in the $S^1$-equivariant cohomology ring of a type $A$ Peterson variety. Our results depend on the Monk formula for the equivariant structure constants for the Peterson Schubert classes derived by Harada and Tymoczko. In addition, we give proofs of two facts observed by H. Naruse: firstly, that some constants which appear in the multiplicative structure of the $S^1$-equivariant cohomology of Peterson varieties are **Stirling numbers of the second kind**, and secondly, that the Peterson Schubert classes satisfy a **stability property** in a sense analogous to the stability of the classical equivariant Schubert classes in the $T$-equivariant cohomology of the flag variety.'
address:
- |
Department of Pure Mathematics and Mathematical Statistics\
Centre for Mathematical Sciences\
Wilberforce Road\
Cambridge CB3 0WA\
United Kingdom
- |
Department of Mathematics and Statistics\
McMaster University\
1280 Main Street West\
Hamilton, Ontario L8S4K1\
Canada
author:
- Darius Bayegan
- Megumi Harada
title: 'A Giambelli formula for the $S^1$-equivariant cohomology of type $A$ Peterson varieties'
---
[^1]
Introduction {#sec:intro}
============
The main result of this note is a **Giambelli formula** in the $S^1$-equivariant cohomology[^2] of type $A$ Peterson varieties. Specifically, we give an explicit formula which expresses an arbitrary **Peterson Schubert class** in terms of the degree-$2$ Peterson Schubert classes. We call this a “Giambelli formula” by analogy with the standard Giambelli formula in classical Schubert calculus [@Ful97] which expresses an arbitrary Schubert class in terms of degree-$2$ Schubert classes.
We briefly recall the setting of our results. **Peterson varieties** in type $A$ can be defined as the following subvariety $Y$ of $\mathcal{F}\ell ags({{\mathbb{C}}}^n)$: $$\label{eq:def intro}
Y := \{ V_\bullet = (0 \subseteq V_1 \subseteq V_2 \subseteq \cdots
\subseteq V_{n-1} \subseteq V_n = {{\mathbb{C}}}^n) {{\hspace{1mm}}}\mid {{\hspace{1mm}}}NV_i \subseteq
V_{i+1} \textup{ for all } i = 1, \ldots, n-1\}$$ where $N: {{\mathbb{C}}}^n \to {{\mathbb{C}}}^n$ denotes the principal nilpotent operator. These varieties have been much studied due to its relation to the quantum cohomology of the flag variety [@Kos96; @Rie03]. Thus it is natural to study their topology, e.g. the structure of their (equivariant) cohomology rings. We do so through Schubert calculus techniques. Our results extend techniques initiated and developed in [@HarTym09; @HarTym10], to which we refer the reader for further details and motivation.
There is a natural circle subgroup of $U(n,{{\mathbb{C}}})$ which acts on $Y$ (recalled in Section \[sec:background\]). The inclusion of $Y$ into ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ induces a natural ring homomorphism $$\label{eq:intro-proj}
H^*_T(\mathcal{F}\ell ags({{\mathbb{C}}}^n)) \to H^*_{S^1}(Y)$$ where $T$ is the subgroup of diagonal matrices of $U(n,{{\mathbb{C}}})$ acting in the usual way on ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$. One of the main results of [@HarTym09] is that a certain subset of the equivariant Schubert classes $\{\sigma_w\}_{w \in S_n}$ in $H^*_T({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))$ maps under the projection to a computationally convenient module basis of $H^*_{S^1}(Y)$. We refer to the images via of $\{\sigma_w\}_{w \in S_n}$ in $H^*_{S^1}(Y)$ as **Peterson Schubert classes**. Moreover, [@HarTym09 Theorem 6.12] gives a manifestly positive **Monk formula** for the product of a degree-$2$ Peterson Schubert class with an arbitrary Peterson Schubert class, expressed as a $H^*_{S^1}(\operatorname{pt})$-linear combination of Peterson Schubert classes. This is an example of equivariant Schubert calculus in the realm of Hessenberg varieties (of which Peterson varieties are a special case), and we view the Giambelli formula (Theorem \[theorem:giambelli\]) as a further development of this theory. The Giambelli formula for Peterson varieties was also independently observed by H. Naruse.
Our Giambelli formula also allows us to simplify the presentation of the ring $H^*_{S^1}(Y)$ given in [@HarTym09 Section 6]. This is because the previous presentation used as its generators all of the elements in the module basis given by Peterson Schubert classes, although the ring $H^*_{S^1}(Y)$ is multiplicatively generated by only the degree-$2$ Peterson Schubert classes. Details are explained in Section \[subsec:simplify\] below, where we also give a concrete example in $n=4$ to illustrate our results. We also formulate a conjecture (cf. Remark \[remark:quadratic\]), suggested to us by the referee of this manuscript, that the ideal of defining relations is in fact generated by the quadratic relations only. If true, this would be a significant further simplification of the presentation of this ring and would lead to interesting further questions (both combinatorial and geometric).
In Sections \[sec:stirling\] and \[sec:stability\], we present proofs of two facts concerning Peterson Schubert classes which we learned from H. Naruse. The results are due to Naruse but the proofs given here are our own. We chose to include these results because they do not appear elsewhere in the literature. The first fact is that **Stirling numbers of the second kind** (see Section \[sec:stirling\] for the definition) appear naturally in the product structure of $H^*_{S^1}(Y)$. The second is that the Peterson Schubert classes satisfy a **stability condition** with respect to the natural inclusions of Peterson varieties induced from the inclusions ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n) {\hookrightarrow}{{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+1})$.
**Acknowledgements.** We are grateful to Hiroshi Naruse for communicating to us his observations on the stability of Peterson Schubert classes and the presence of Stirling numbers of the second kind. We thank Julianna Tymoczko for support of and interest in this project and Alex Yong for useful conversations. We also thank an anonymous referee for a careful reading of the manuscript and many helpful suggestions; in particular, the referee suggested the conjecture regarding quadratic relations recorded in Remark \[remark:quadratic\], as well as the question which we record in Remark \[remark:normal crossings\].
Peterson varieties and $S^1$-fixed points {#sec:background}
=========================================
In this section we briefly recall the objects under study. For details we refer the reader to [@HarTym09]. Since we work exclusively in Lie type $A$ we henceforth omit it from our terminology.
By the **flag variety** we mean the homogeneous space $GL(n,{{\mathbb{C}}})/B$ where $B$ is the standard Borel subgroup of upper-triangular invertible matrices. The flag variety can also be identified with the space of nested subspaces in ${{\mathbb{C}}}^n$, i.e., $$\mathcal{F}\ell ags({{\mathbb{C}}}^n) :=
\{ V_{\bullet} = (\{0\} \subseteq V_1 \subseteq V_2 \subseteq \cdots \subseteq V_{n-1} \subseteq
V_n = {{\mathbb{C}}}^n) {{\hspace{1mm}}}\mid {{\hspace{1mm}}}\dim_{{{\mathbb{C}}}}(V_i) = i \} \cong GL(n,{{\mathbb{C}}})/B.$$ Let $N$ be the $n \times n$ principal nilpotent operator given with respect to the standard basis of ${{\mathbb{C}}}^n$ as the matrix with one $n \times n$ Jordan block of eigenvalue $0$, i.e., $$\label{eq:standard principal nilpotent}
N = \begin{bmatrix} 0 & 1 & 0 & & & \\
0 & 0 & 1 & & & \\
0 & 0 & 0 & & & \\
& & & \ddots & & \\
& & & & 0 & 1\\
& & & & 0 & 0 \\
\end{bmatrix}.$$ Fix $n$ a positive integer. The main geometric object under study, the **Peterson variety** $Y$, is the subvariety of $\mathcal{F}\ell
ags({{\mathbb{C}}}^n)$ defined in where $N$ is the standard principal nilpotent in . The variety $Y$ is a (singular) projective variety of complex dimension $n-1$.
We recall some facts from [@HarTym09]. The following circle subgroup of $U(n,{{\mathbb{C}}})$ preserves $Y$: $$\label{eq:def-circle}
S^1 = \left\{ \left. \begin{bmatrix} t^n & 0 & \cdots & 0 \\ 0 & t^{n-1} & &
0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & & t \end{bmatrix}
\; \right\rvert \; t \in {{\mathbb{C}}}, \; \|t\| = 1 \right\} \subseteq T^n
\subseteq U(n,{{\mathbb{C}}}).$$ Here $T^n$ is the standard maximal torus of $U(n,{{\mathbb{C}}})$ consisting of diagonal unitary matrices. The $S^1$-fixed points of $Y$ are isolated and are a subset of the $T^n$-fixed points of ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$. As is standard, we identify the $T^n$-fixed points in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ with the permutations $S_n$. In particular since $Y^{S^1}$ is a subset of ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)^{T^n}$, we think of the Peterson fixed points as permutations in $S_n$. There is a natural bijective correspondence between the Peterson fixed points $Y^{S^1}$ and subsets of $\{1,2,\ldots,n-1\}$ which we now briefly recall. It is explained in [@HarTym09 Section 2.3] that a permutation $w \in S_n$ is in $Y^{S^1}$ precisely when the one-line notation of $w^{-1}$ is of the form $$\label{eq:w-oneline}
w^{-1} = \underbrace{j_1 \, j_1 -1 \, \cdots \, 1}_{j_1 \textup{ entries}} \, \underbrace{j_2 \, j_2-1 \, \cdots \,
j_1+1}_{j_2-j_1 \textup{ entries}} \, \cdots \,
\underbrace{n \, n-1 \, \cdots \, j_m+1}_{n-j_m \textup{ entries}}$$ where $1 \leq j_1 < j_2 < \cdots < j_m < n$ is any sequence of strictly increasing integers. For example, for $n=9, m=2$ and $j_1 = 3, j_2 = 7$, then the permutation $w^{-1}$ in has one-line notation $321765498$. Thus for each permutation $w \in S_n$ satisfying we define $${\mathcal{A}} := \{ i: w^{-1}(i) = w^{-1}(i+1)+1 \textup{ for } 1 \leq i\leq n-1\} \subseteq
\{1,2,\ldots, n-1\}.$$ This gives a one-to-one correspondence between the power set of $\{1,2,\ldots, n-1\}$ and $Y^{S^1}$. We denote the Peterson fixed point corresponding to a subset $\mathcal{A} \subseteq \{1,2,\ldots, n-1\}$ by $w_{\mathcal{A}}$.
Let $n=5$ and suppose $\mathcal{A}=\{1,2,4\}$. Then the associated permutation is $w_{\mathcal{A}} = 32154$.
Indeed, for a fixed $n$, we can also easily enumerate all the Peterson fixed points by using this correspondence.
\[example:Peterson Variety\] Let $n=4$. Then $Y^{S^1}$ consists of $2^3 = 8$ elements in correspondence with the subsets of $\{1,2,3\}$, namely: $w_{\emptyset} = 1234, w_{\{1\}} = 2134, w_{\{2\}} = 1324, w_{\{3\}} =
1243, w_{\{1,2\}} = 3214, w_{\{2,3\}} = 1432, w_{\{1,3\}} = 2143,
w_{\{1,2,3\}} = 4321$.
Given a choice of subset $\mathcal{A} \subseteq \{1,2,\ldots,n-1\}$, there is a natural decomposition of $\mathcal{A}$ as follows. We say that a set of consecutive integers $$\{a, a+1, \ldots, a+k\} \subseteq {\mathcal{A}}$$ is a [**maximal consecutive (sub)string**]{} of ${\mathcal{A}}$ if $a$ and $k$ are such that neither $a-1$ nor $a+k+1$ is in ${\mathcal{A}}$. For $a_1 := a$ and $a_2 := a_1+k$, we denote the corresponding maximal consecutive substring by $[a_1,a_2]$. It is straightforward to see that any ${\mathcal{A}}$ uniquely decomposes into a disjoint union of maximal consecutive substrings $${\mathcal{A}} = [a_1, a_2] \cup [a_3,a_4] \cup \cdots \cup [a_{m-1},a_m].$$ For instance, if $\mathcal{A} = \{1,2,3, 5,6,8\}$, then its decomposition into maximal consecutive substrings is $\{1,2,3\} \cup \{5,6\} \cup \{8\} = [1,3]
\cup [5,6] \cup [8,8]$.
Suppose $\mathcal{A} = \{j_1 < j_2 < \cdots < j_m\}$. Finally we recall that we can associate to each $w_{\mathcal{A}}$ a permutation $v_{\mathcal{A}}$ by the recipe $$\label{eq:def vA}
w_{\mathcal{A}} \mapsto v_{\mathcal{A}} := s_{j_1} s_{j_2} \cdots s_{j_m}$$ where an $s_i$ denotes the simple transposition $(i,i+1)$ in $S_n$.
The Giambelli formula for Peterson varieties
============================================
The Giambelli formula
---------------------
In this section we prove the main result of this note, namely, a **Giambelli formula for Peterson varieties**.
As recalled above, the Peterson variety $Y$ is an $S^1$-space for a subtorus $S^1$ of $T^n$ and it can be checked that $Y^{S^1} = ({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))^{T^n} \cap Y.$ There is a forgetful map from $T^n$-equivariant cohomology to $S^1$-equivariant cohomology obtained by the inclusion $S^1 {\hookrightarrow}T$, so there is a commutative diagram $$\label{eq:comm-diag}
\xymatrix{
H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)) \ar[r] \ar[d] & H^*_{T^n}(({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))^{T^n}) \ar[d] \\
H^*_{S^1}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)) \ar[r] \ar[d] & H^*_{S^1}(({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))^{T^n}) \ar[d] \\
H^*_{S^1}(Y) \ar[r] & H^*_{S^1}(Y^{S^1}).
}$$ The **equivariant Schubert classes** $\{\sigma_w\}$ in $H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))$ are well-known to form a $H^*_{T^n}(\operatorname{pt})$-module basis for $H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))$. We call the image of $\sigma_w$ under the projection map $H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))
\to H^*_{S^1}(Y)$ the **Peterson Schubert class corresponding to $w$**. For the permutations $v_{\mathcal{A}}$ defined in , we denote by $p_{\mathcal{A}}$ the corresponding Peterson Schubert class, i.e. the image of $\sigma_{v_{\mathcal{A}}}$. (This is slightly different notation from that used in [@HarTym09].) We denote by $p_{\mathcal{A}}(w) \in H^*_{S^1}(\operatorname{pt}) \cong {{\mathbb{C}}}[t]$ the restriction of the Peterson Schubert class $p_{\mathcal{A}}$ to the fixed point $w \in Y^{S^1}$.
One of the main results of [@HarTym09] is that the set of $2^{n-1}$ Peterson Schubert classes $\{p_{\mathcal{A}}\}_{\mathcal{A} \subseteq
\{1,2,\ldots, n-1\}}$ form a $H^*_{S^1}(\operatorname{pt})$-module basis for $H^*_{S^1}(Y)$ where $v_{\mathcal{A}}$ is defined in . (The fact that $H^*_{S^1}(Y)$ is a free module of rank $2^{n-1}$ over $H^*_{S^1}(\operatorname{pt})$ fits nicely with the result [@SommersTymoczko Theorem 10.2] that the Poincaré polynomial of $Y$ is given by $(q^2+1)^{n-1}$.) It is also shown in [@HarTym09] that the $n-1$ degree-$2$ classes $\{p_i := p_{s_i}\}_{i=1}^{n-1}$ form a multiplicative set of generators for $H^*_{S^1}(Y)$. These classes $p_i$ are also (equivariant) Chern classes of certain line bundles over $Y$. Moreover, there is a **Monk formula** [@HarTym09 Theorem 6.12] which expresses a product $$p_i p_{\mathcal{A}}$$ for any $i \in \{1,2,\ldots,n-1\}$ and any $\mathcal{A} \subseteq
\{1,2,\ldots,n-1\}$ as a $H^*_{S^1}(\operatorname{pt})$-linear combination of the additive module basis $\{p_{\mathcal{A}}\}$. Since the $p_i$ multiplicatively generate the ring, this Monk formula completely determines the ring structure of $H^*_{S^1}(Y)$. Furthermore it is in principle possible to express any $p_{\mathcal{A}}$ in terms of the $p_i$. Our Giambelli formula is an explicit formula which achieves this (cf. for example [@Ful97] for the version in classical Schubert calculus).
We begin by recalling the Monk formula, for which we need some terminology. Fix ${\mathcal{A}} \subseteq \{1,2,\ldots,n-1\}$. We define $\mathcal{H}_{\mathcal{A}}: \mathcal{A}
\to \mathcal{A}$ by $${\mathcal{H}_{{\mathcal{A}}}(j)} = \textup{the maximal element in the maximal consecutive substring of ${\mathcal{A}}$ containing $j$}.$$ Similarly, we define $\mathcal{T}_{\mathcal{A}}: \mathcal{A}
\rightarrow \mathcal{A}$ by $${\mathcal{T}_{{\mathcal{A}}}(j)} =
\textup{the minimal element in the maximal consecutive substring of
${\mathcal{A}}$ containing } j.$$ We say that the maps $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{T}_{\mathcal{A}}$ give the “head" and “tail" of each maximal consecutive substring of $\mathcal{A}$. For an example see [@HarTym09 Example 5.6]. We recall the following.
\[theorem:Monk\] **“The Monk formula for Peterson varieties” ([@HarTym09 Theorem 6.12])** Fix $n$ a positive integer. Let $Y$ be the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ with the natural $S^1$-action defined by . For ${\mathcal A}
\subseteq \{1,2,\ldots, n-1\}$, let $v_{\mathcal{A}} \in S_n$ be the permutation in , and let $p_{\mathcal{A}}$ be the corresponding Peterson Schubert class in $H^*_{S^1}(Y)$. Then $$\label{eq:Monk-final}
p_i \cdot p_{\mathcal{A}} = p_i(w_{\mathcal{A}}) \cdot
p_{\mathcal{A}} + \sum_{{\mathcal{A}} \subsetneq {\mathcal{B}} \textup{ and } |{\mathcal{B}}|
= |{\mathcal{A}}|+1} c^{\mathcal{B}}_{i,{\mathcal{A}}} \cdot p_{\mathcal{B}},$$ where, for a subset $\mathcal{B} \subseteq \{1,2,\ldots,n-1\}$ which is a disjoint union $\mathcal{B} =
\mathcal{A} \cup \{k\},$
- if $i \not \in \mathcal{B}$ then $c^{\mathcal{B}}_{i,\mathcal{A}} = 0$,
- if $i \in {\mathcal{B}}$ and $i \not \in [{\mathcal{T}_{\mathcal{B}}(k)},
{\mathcal{H}_{\mathcal{B}}(k)}],$ then $c^{\mathcal{B}}_{i,{\mathcal{A}}} =
0$,
- if $k \leq i \leq {\mathcal{H}_{\mathcal{B}}(k)},$ then $$\label{eq:cBiA-formula-part1}
c^{\mathcal{B}}_{i,{\mathcal{A}}} = ({\mathcal{H}_{{\mathcal{B}}}(k)}-i+1) \cdot \left( \begin{array}{c} {\mathcal{H}_{{\mathcal{B}}}(k)} - {\mathcal{T}_{{\mathcal{B}}}(k)}+1 \\
k-{\mathcal{T}_{{\mathcal{B}}}(k)} \end{array} \right),$$
- if ${\mathcal{T}_{{\mathcal{B}}}(k)} \leq i \leq k-1,$ $$\label{eq:cBiA-formula-part2}
c^{\mathcal{B}}_{i,{\mathcal{A}}}
= (i-{\mathcal{T}_{{\mathcal{B}}}(k)}+1) \cdot \binom{{\mathcal{H}_{{\mathcal{B}}}(k)}-{\mathcal{T}_{{\mathcal{B}}}(k)}+1}{k-{\mathcal{T}_{{\mathcal{B}}}(k)}+1}.$$
We also recall that [@HarTym09 Lemma 6.7] implies that if $\mathcal{B}, \mathcal{B}'$ are two disjoint subsets of $\{1,2,\ldots,n-1\}$ such that there is no $i$ in $\mathcal{B}$ and $j$ in $\mathcal{B}'$ with $\lvert {i-j} \rvert=1$, then $p_{\mathcal{B}\cup\mathcal{B}'} =p_{\mathcal{B}}p_{\mathcal{B}'}$. It follows that for any $\mathcal{A}$ we have $$\label{eq:product pvA}
p_{\mathcal{A}} = p_{[a_1, a_2]} \cdot p_{[a_3,a_4]} \cdots p_{[a_{m-1},a_m]}$$ where ${\mathcal{A}} = [a_1, a_2] \cup [a_3,a_4] \cup \cdots \cup
[a_{m-1},a_m]$ is the decomposition of $\mathcal{A}$ into maximal consecutive substrings. In particular, in order to give an expression for $p_{\mathcal{A}}$ in terms of the elements $p_i$, from we see that it suffices to give a formula only for the special case in which $\mathcal{A}$ consists of a single maximal consecutive string.
We now state and prove our Giambelli formula.
\[theorem:giambelli\] Fix $n$ a positive integer. Let $Y$ be the Peterson variety in $\mathcal{F}\ell ags({{\mathbb{C}}}^n)$ with the $S^1$-action defined by . Suppose $\mathcal{A}=\{a,a+1,a+2, \ldots, a+k\}$ where $1 \leq a \leq n-1$ and $0 \leq k \leq n-1-a$. Let $v_{\mathcal{A}}$ be the permutation corresponding to $\mathcal{A}$ defined in and let $p_{\mathcal{A}}$ be the associated Peterson Schubert class. Then $$p_{\mathcal{A}} =\frac{1}{(k+1)!} \displaystyle\prod_{j \in
\mathcal{A}} p_{j}.$$
We use the following lemma.
\[lemma:simple monk if i not in A\] Suppose $i \in \{1,2,\ldots,n-1\}$ and $\mathcal{A} \subseteq
\{1,2,\ldots,n-1\}$. Suppose further that $i \not \in
\mathcal{A}$. Then the Monk relation $$p_{i} \cdot p_{\mathcal{A}} = p_{i}(w_{\mathcal{A}}) \cdot
p_{\mathcal{A}} + \displaystyle\sum_{\mathcal{A} \subset
{\mathcal{B}} \textup{ and }
|{\mathcal{B}}|=|\mathcal{A}|+1} c^{{\mathcal{B}}}_{i,\mathcal{A}}
\cdot p_{{\mathcal{B}}}$$ simplifies to $$\label{eq:final simple monk}
p_{i} \cdot p_{\mathcal{A}} = c^{{\mathcal{A} \cup \{i\}}}_{i,\mathcal{A}}
\cdot p_{{\mathcal{A} \cup \{i\}}}.$$
First observe that the Monk relation simplifies to $$\label{eq:simplified monk}
p_{i} \cdot p_{\mathcal{A}} = \displaystyle\sum_{\mathcal{A} \subset
{\mathcal{B}} \textup{ and }
|{\mathcal{B}}|=|\mathcal{A}|+1} c^{{\mathcal{B}}}_{i,\mathcal{A}}
\cdot p_{{\mathcal{B}}}$$ if $i \not \in \mathcal{A}$, since in that case $p_i(w_{\mathcal{A}}) = 0$ by [@HarTym09 Lemma 6.4]. Moreover, from Theorem \[theorem:Monk\] we also know that $c_{i,
\mathcal{A}}^{\mathcal{B}} = 0$ if $i \not \in \mathcal{B}$. Hence the summands appearing in correspond to $\mathcal{B}$ satisfying $\mathcal{A} \subseteq \mathcal{B}, \lvert \mathcal{B} \rvert = \lvert
\mathcal{A} \rvert +1$, and $i \in \mathcal{B}$. On the other hand, since $i \not \in \mathcal{A}$ by assumption, this means that there is only one non-zero summand in the right hand side of , namely, the term corresponding to $\mathcal{B} = \mathcal{A} \cup \{i\}$. The equation follows.
We now prove the main theorem.
We proceed by induction on $k$. First consider the base case where $k=0$. Then $A=\{a\}$, so $p_{v_\mathcal{A}}=p_a$. On the right hand side, we have $\frac{1}{(0+1)!}\prod_{j \in \mathcal{A}}p_{j}=p_a$. This verifies the base case.
By induction, suppose the claim holds for $k-1$. We now show that the claim holds for $k$. Consider $\mathcal{A}' := \{a,a+1,\ldots,a+k-1\}$ and consider the product $p_{a+k} \cdot p_{\mathcal{A}'}$. From the Monk formula in Theorem \[theorem:Monk\] we know that $$\label{eq:expansion}
p_{a+k} \cdot p_{\mathcal{A}'}
=
p_{a+k}(w_{\mathcal{A}'}) \cdot p_{\mathcal{A}'}+\displaystyle\sum_{{\mathcal{A}}'\subseteq
{\mathcal{B}} \textup{ and } |{\mathcal{B}}|=|{\mathcal{A}}'|+1}c^{{\mathcal{B}}}_{a+k,{\mathcal{A}}'} \cdot p_{\mathcal{B}}.$$ On the other hand since by definition $a+k \not \in \mathcal{A}'$, by Lemma \[lemma:simple monk if i not in A\] the equality further simplifies to $$p_{a+k} \cdot p_{\mathcal{A}'}=c^{{\mathcal{A}}}_{a+k,{\mathcal{A}}'} \cdot p_{\mathcal{A}}.$$ Moreover, since ${\mathcal{A}}={\mathcal{A}}'\cup \{a+k\}$, we have $\mathcal{H}_{{\mathcal{A}}}(a+k) = a+k$ and ${\mathcal{T}_{\mathcal{A}}(a+k)} =
a$. Hence by Theorem \[theorem:Monk\] $$\begin{split}
c^{{\mathcal{A}}}_{a+k,{\mathcal{A}}'} & =\left( \mathcal{H}_{{\mathcal{A}}}(a+k)-(a+k)+1 \right)
\binom{\mathcal{H}_{{\mathcal{A}}}(a+k)-\mathcal{T}_{{\mathcal{A}}}(a+k)+1}{(a+k)-\mathcal{T}_{\mathcal{A}}(a+k)} \\
& = ((a+k)-(a+k)+1) \binom{a+k-a+1}{(a+k)-a} \\
& = k+1.
\end{split}$$ Therefore $$p_{a+k} \cdot p_{\mathcal{A}'}=(k+1) \cdot p_{\mathcal{A}}.$$ By the inductive hypothesis we have for the set $\mathcal{A}' = \{a, a+1, \ldots, a+k-1\}$ $$p_{\mathcal{A}'} =\frac{1}{k!} \prod_{j \in
{\mathcal{A}'}}p_{j}.$$ Substituting into the above equation yields $$p_{\mathcal{A}} = \frac{1}{(k+1)!} \prod_{j \in {\mathcal{A}}}
p_{j}$$ as desired. This completes the proof.
\[remark:normal crossings\] We thank the referee for the following observation. The formula in Theorem \[theorem:giambelli\] suggests that the classes $p_i$ behave like a normal crossings divisor (up to quotient singularities), with all other classes arising (up to rational coefficients) as intersections of the components. It would certainly be of interest to understand more precisely the underlying geometry which gives rise not only to the Giambelli relation in Theorem \[theorem:giambelli\] but also to the original Monk formula [@HarTym09 Theorem 6.12].
From Theorem \[theorem:giambelli\] it immediately follows that for any subset $${\mathcal{A}} = [a_1, a_2] \cup [a_3,a_4] \cup \cdots \cup
[a_{m-1},a_m]$$ with its decomposition into maximal consecutive substrings, we have $$\label{eq:giambelli for general pA}
p_{\mathcal{A}} = \frac{1}{(a_2-a_1+1)!} \cdot
\frac{1}{(a_4-a_3+1)!} \cdots \frac{1}{(a_m-a_{m-1}+1)!} \prod_{j \in
\mathcal{A}} p_j.$$ For the purposes of the next section we introduce the notation $$\label{eq:def sigma}
\sigma(\mathcal{A}) := \frac{1}{(a_2-a_1+1)!} \cdot
\frac{1}{(a_4-a_3+1)!} \cdots \frac{1}{(a_m-a_{m-1}+1)!}$$ for the rational coefficient appearing in . The following is an immediate corollary of this discussion.
Let $${\mathcal{A}} = [a_1, a_2] \cup [a_3,a_4] \cup \cdots \cup
[a_{m-1},a_m].$$ Then $$p_{\mathcal{A}} = \sigma(\mathcal{A}) \prod_{j \in \mathcal{A}}
p_j.$$
Simplification of the Monk relations {#subsec:simplify}
------------------------------------
In this section we explain how to use the Giambelli formula to simplify the ring presentation of $H^*_{S^1}(Y)$ given in [@HarTym09 Section 6]. Recall that the Peterson Schubert classes $\{p_{\mathcal{A}}\}$ form an additive module basis for $H^*_{S^1}(Y)$ and the degree $2$ classes $\{p_i\}_{i=1}^{n-1}$ form a multiplicative basis, so the Monk relations give a presentation of the ring $H^*_{S^1}(Y)$ via generators and relations as follows.
\[theorem:presentation\] **([@HarTym09 Corollary 6.14])** Fix $n$ a positive integer. Let $Y$ be the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ with the $S^1$-action defined by . For ${\mathcal A} \subseteq
\{1,2,\ldots, n-1\}$, let $v_{\mathcal{A}} \in S_n$ be the permutation given in , and let $p_{\mathcal{A}}$ be the corresponding Peterson Schubert class in $H^*_{S^1}(Y)$. Let $t \in H^*_{S^1}(\operatorname{pt}) \cong {{\mathbb{C}}}[t]$ denote both the generator of $H^*_{S^1}(\operatorname{pt})$ and its image $t \in
H^*_{S^1}(Y).$ Then the $S^1$-equivariant cohomology $H^*_{S^1}(Y)$ is given by $$H^*_{S^1}(Y) \cong {{\mathbb{C}}}[t, \{p_{\mathcal{A}}\}_{\mathcal{A}
\subseteq \{1,2,\ldots, n-1\}}] \large/ \mathcal{J}$$ where $\mathcal{J}$ is the ideal generated by the relations .
In order to state the main result of this section we introduce some notation. For $i$ with $1 \leq i \leq n-1$ and $\mathcal{A} \subseteq
\{1,2,\ldots,n-1\}$ define $$m_{i,\mathcal{A}} :=
p_i \cdot p_{\mathcal{A}} - p_i(w_{\mathcal{A}}) \cdot
p_{\mathcal{A}} - \sum_{{\mathcal{A}} \subsetneq {\mathcal{B}} \textup{ and } |{\mathcal{B}}|
= |{\mathcal{A}}|+1} c^{\mathcal{B}}_{i,{\mathcal{A}}} \cdot
p_{\mathcal{B}}$$ thought of as an element in ${{\mathbb{C}}}[t, \{p_{\mathcal{A}}\}_{\mathcal{A}
\subseteq \{1,2,\ldots,n-1\}}]$ where the $c_{i,\mathcal{A}}^{\mathcal{B}} \in {{\mathbb{C}}}[t]$ are the coefficients computed in Theorem \[theorem:Monk\]. Motivated by the Giambelli formula we also define the following elements in ${{\mathbb{C}}}[t, p_1,
p_2,\ldots, p_{n-1}]$: $$q_{i,\mathcal{A}} :=
p_i \cdot \sigma(\mathcal{A}) \cdot \left( \prod_{j \in \mathcal{A}} p_j \right) -
p_i(w_{\mathcal{A}}) \cdot
\sigma(\mathcal{A}) \cdot \left( \prod_{j \in \mathcal{A}} p_j \right)
- \sum_{{\mathcal{A}} \subsetneq {\mathcal{B}} \textup{ and } |{\mathcal{B}}|
= |{\mathcal{A}}|+1} c^{\mathcal{B}}_{i,{\mathcal{A}}} \cdot
\sigma(\mathcal{B}) \left( \prod_{k \in \mathcal{B}} p_k \right)$$ where the $\sigma(\mathcal{A}) \in {{\mathbb{Q}}}$ is the constant defined in .
Let $n=4$ and $i=1$ and $\mathcal{A} = \{1,2\}$. Consider $$m_{1, \{1,2\}} = p_1 p_{v_{\{1,2\}}} - 2t \, p_{v_{\{1,2\}}} + p_{v_{\{1,2,3\}}}.$$ Expanding in terms of the Giambelli formula, we obtain $$q_{1, \{1,2\}} = \frac{1}{2} p_{1}^2p_{2} -
2t \cdot \left( \frac{1}{2} p_{1}p_{2} \right) +\frac{1}{6}
p_{1}p_{2}p_3 = t \, p_1 p_2 + \frac{1}{6} p_1 p_2 p_3.$$
The main theorem of this section gives a ring presentation of $H^*_{S^1}(Y)$ using fewer generators and fewer relations than that in Theorem \[theorem:presentation\]. More specifically let $\mathcal{K}$ denote the ideal in ${{\mathbb{C}}}[t,p_1, \ldots, p_{n-1}]$ generated by the $q_{i,\mathcal{A}}$ for which $i \not \in
\mathcal{A}$, i.e., $$\label{eq:def K}
\mathcal{K} := \bigg\langle q_{i,\mathcal{A}} \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots, n-1\}, i \not \in \mathcal{A}
\bigg\rangle \subseteq {{\mathbb{C}}}[t,p_1, \ldots, p_{n-1}].$$ We have the following.
\[theorem:simplify Monk\] Fix $n$ a positive integer. Let $Y$ be the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ equipped with the action of the $S^1$ in . Then the $S^1$-equivariant cohomology $H^*_{S^1}(Y)$ is isomorphic to the ring $${{\mathbb{C}}}[t, p_1, p_2, \ldots, p_{n-1}]/\mathcal{K}$$ where $\mathcal{K}$ is the ideal in .
To prove the theorem we need the following lemma.
\[lemma:some q are zero\] Let $i \in \{1,2,\ldots,n-1\}$ and $\mathcal{A}
\subseteq \{1,2,\ldots, n-1\}$. Suppose $i \not \in \mathcal{A}$. Then $q_{i, \mathcal{A}} = 0$ in ${{\mathbb{C}}}[t, p_1, p_2, \ldots, p_{n-1}]$.
Since $i \not \in \mathcal{A}$ by assumption, Lemma \[lemma:simple monk if i not in A\] implies that $$m_{i, \mathcal{A}} = p_{i} \cdot p_{\mathcal{A}} - p_{i}(w_{\mathcal{A}})\cdot
p_{\mathcal{A}} - \displaystyle\sum_{\mathcal{A} \subset
{\mathcal{B}} \textup{ and }
|{\mathcal{B}}|=|\mathcal{A}|+1} c^{{\mathcal{B}}}_{i,\mathcal{A}}
\cdot p_{{\mathcal{B}}}$$ simplifies to $$m_{i, \mathcal{A}} = p_{i} \cdot p_{\mathcal{A}} - c^{{\mathcal{A} \cup \{i\}}}_{i,\mathcal{A}}
\cdot p_{{\mathcal{A} \cup \{i\}}}.$$ Thus in order to compute the corresponding $q_{i,\mathcal{A}}$ it remains to compute $c^{{\mathcal{A} \cup \{i\}}}_{i,\mathcal{A}}$ and apply the Giambelli formula.
Let ${\mathcal{A}} = [a_1, a_2] \cup [a_3,a_4] \cup \cdots \cup
[a_{m-1},a_m]$ be the decomposition of $\mathcal{A}$ into maximal consecutive substrings. Consider the decomposition of $\mathcal{A} \cup \{i\}$ into maximal consecutive substrings. There are several cases to consider:
1. the singleton set $\{i\}$ is a maximal consecutive substring of $\mathcal{A} \cup \{i\}$, i.e. $i-1 \not \in \mathcal{A}$ and $i+1
\not \in \mathcal{A}$,
2. the inclusion of $i$ extends a maximal consecutive substring to its right by $1$ element, i.e., there exists a maximal consecutive string $[a_\ell, a_{\ell+1}]
\subseteq \mathcal{A}$ such that $i = a_{\ell+1}+1$ and that $[a_\ell, i]$ is a maximal consecutive substring of $\mathcal{A}
\cup \{i\}$,
3. the inclusion of $i$ extends a maximal consecutive substring to its left by $1$ element, i.e., there exists a maximal consecutive string $[a_\ell, a_{\ell+1}]
\subseteq \mathcal{A}$ such that $i = a_{\ell}-1$ and that $[i, a_{\ell+1}]$ is a maximal consecutive substring of $\mathcal{A}
\cup \{i\}$, or
4. the inclusion of $i$ glues together two maximal consecutive substrings of $\mathcal{A}$, i.e., there exist two maximal consecutive substrings $[a_{\ell}, a_{\ell+1}], [a_{\ell+2}, a_{\ell+3}]$ such that $i=a_{\ell+1}+1 = a_{\ell+2}-1$ and hence $[a_\ell, a_{\ell+3}] =
[a_{\ell}, a_{\ell+1}] \cup \{i\} \cup [a_{\ell+2}, a_{\ell+3}]$ is a maximal consecutive substring of $\mathcal{A} \cup \{i\}$.
We consider each case separately.
Case (1): Suppose $\{i\}$ is a maximal consecutive substring in $\mathcal{A} \cup \{i\}$. In this case, the coefficient $c^{{\mathcal{A} \cup
\{i\}}}_{i,\mathcal{A}}$ is $1$. Hence we have $$m_{i, \mathcal{A}} = p_{i} \, p_{\mathcal{A}} - p_{v_{\mathcal{A} \cup \{i\}}}.$$ Since $\{i\}$ is a maximal consecutive substring in $\mathcal{A} \cup \{i\}$, we have $\sigma(\mathcal{A}) = \sigma(\mathcal{A} \cup \{i\})$. We conclude $$q_{i, \mathcal{A}} = p_i \cdot \left( \sigma(\mathcal{A}) \cdot \left( \prod_{j \in \mathcal{A}} p_j \right) \right)
- \sigma(\mathcal{A} \cup \{i\}) \cdot \left( \prod_{j \in \mathcal{A} \cup \{i\}}
p_j \right) = 0$$ as desired.
Cases (2) and (3) are very similar, so we only present the argument for case (2). Suppose $i$ extends a maximal consecutive substring $[a_\ell, a_{\ell+1}]$ of $\mathcal{A}$ to its right. Then $$m_{i, \mathcal{A}} = p_i \cdot p_{\mathcal{A}} - (i - a_\ell + 1) p_{v_{\mathcal{A} \cup \{i\}}}$$ since $k=i={\mathcal{H}_{\mathcal{B}}(i)}$ and ${\mathcal{T}_{\mathcal{B}}(i)} =
a_\ell$ so $c_{i, \mathcal{A}}^{\mathcal{A} \cup \{i\}} =
i-a_\ell+1$. We compute $$\begin{split}
q_{i, \mathcal{A}} & = p_i \, \left( \left( \prod_{1 \leq s \leq m-1, \, s \textup{ odd}} \frac{1}{(a_{s+1}-a_s
+1)!} \right) \cdot \left( \prod_{j \in \mathcal{A}} p_j \right)
\right) \\
& -
(i - a_\ell+1) \cdot \left( \prod_{1 \leq s \leq m-1, \, s \textup{ odd and } s\neq \ell} \frac{1}{(a_{s+1}-a_s
+1)!} \right) \cdot \left(\frac{1}{(i - a_\ell + 1)!} \right)
\cdot \left( \prod_{j \in \mathcal{A} \cup \{i\}} p_j \right)
\end{split}$$ where one of the factors in the product in the second expression has changed because the maximal consecutive string $[a_\ell, a_{\ell+1}]$ has been extended in $\mathcal{A} \cup \{i\}$. Since $$(i - a_\ell+1) \left(\frac{1}{(i - a_\ell + 1)!} \right) =
\frac{1}{(a_{\ell+1}-a_\ell+1)!}$$ by assumption on $i$, we conclude $q_{i,\mathcal{A}} = 0$ as desired.
Finally, consider the case (4) in which the inclusion of $i$ glues together two maximal consecutive substrings $[a_\ell, a_{\ell+1}],
[a_{\ell+2}, a_{\ell+3}]$ in $\mathcal{A}$. In this case, $k=i, {\mathcal{H}_{\mathcal{B}}(i)} = a_{\ell+3},
{\mathcal{T}_{\mathcal{B}}(i)} = a_\ell.$ Hence the coefficient $c_{i,
\mathcal{A}}^{\mathcal{A} \cup \{i\}}$ is $$c_{i,
\mathcal{A}}^{\mathcal{A} \cup \{i\}} = (a_{\ell+3} - i+1)
\binom{a_{\ell+3}-a_\ell+1}{i-a_\ell} =
\frac{(a_{\ell+3}-a_\ell+1)!}{(i-a_\ell)! (a_{\ell+3}-i)!}.$$ The expansion of $p_i \cdot p_{\mathcal{A}}$ is the same as in the previous cases. The term corresponding to $c^{{\mathcal{A} \cup \{i\}}}_{i,\mathcal{A}}
\cdot p_{{\mathcal{A} \cup \{i\}}}$ is $$\frac{(a_{\ell+3}-a_\ell+1)!}{(i-a_\ell)! (a_{\ell+3}-i)!}
\cdot \left( \prod_{1 \leq s \leq m-1, \, s \textup{ odd
and }
s\neq \ell, \ell+2} \frac{1}{(a_{s+1}-a_s
+1)!} \right) \cdot \left(\frac{1}{(a_{\ell+3} - a_\ell + 1)!} \right)
\cdot \left( \prod_{j \in \mathcal{A} \cup \{i\}} p_j \right).$$ Since by assumption on $i$ we have $i=a_{\ell+1}+1 = a_{\ell+2}-1$, we obtain the simplification $$\begin{split}
\frac{(a_{\ell+3}-a_\ell+1)!}{(i-a_\ell)! (a_{\ell+3}-i)!}
\left(\frac{1}{(a_{\ell+3} - a_\ell + 1)!} \right) & =
\frac{1}{(i-a_\ell)! (a_{\ell+3}-i)!} \\
& = \frac{1}{(a_{\ell+1}-a_\ell+1)!} \cdot \frac{1}{(a_{\ell+3} -
a_{\ell+2} +1)!}
\end{split}$$ from which it follows that $q_{i,\mathcal{A}} = 0$ also in this case. The result follows.
Let $n=5$, $i=4$ and let $\mathcal{A} = \{1,2\}$. Consider $$m_{4, \{1,2\}} = p_4 \cdot p_{v_{\{1,2\}}} - c_{4,\{1,2\}}^{\{1,2,4\}} \cdot p_{v_{\{1,2,4\}}}.$$ From it follows that $c_{4,\{1,2\}}^{\{1,2,4\}} =1$. The corresponding $q_{4, \{1,2\}}$ can be computed to be $$q_{4, \{1,2\}} = p_4 \left( \frac{1}{2!} p_1 p_2 \right) - \left( \frac{1}{2!} p_1 p_2
\right) p_4 = 0.$$
We may now prove Theorem \[theorem:simplify Monk\].
By Theorem \[theorem:presentation\] we know $$H^*_{S^1}(Y) \cong {{\mathbb{C}}}[t, \{p_{\mathcal{A}}\}_{\mathcal{A}
\subseteq \{1,2,\ldots, n-1\}}] \large/ \mathcal{J}$$ where $\mathcal{J}$ is the ideal generated by the relations so we wish to prove $${{\mathbb{C}}}[t,p_1, \ldots, p_{n-1}]/\mathcal{K} \cong {{\mathbb{C}}}[t, \{p_{\mathcal{A}}\}_{\mathcal{A}
\subseteq \{1,2,\ldots, n-1\}}] \large/ \mathcal{J}.$$ The content of the Giambelli formula (Theorem \[theorem:giambelli\]) is that the expressions $$p_{\mathcal{A}} - \sigma(\mathcal{A}) \prod_{j \in \mathcal{A}}
p_j$$ are elements of $\mathcal{J}$. Hence $$\begin{split}
\mathcal{J} &= \bigg\langle m_{i,\mathcal{A}} \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \bigg\rangle + \bigg\langle p_{\mathcal{A}} - \sigma(\mathcal{A}) \prod_{j \in \mathcal{A}}
p_j \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \bigg\rangle \\
& = \bigg\langle q_{i, \mathcal{A}} \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \bigg\rangle + \bigg\langle p_{\mathcal{A}} - \sigma(\mathcal{A}) \prod_{j \in \mathcal{A}}
p_j \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \bigg\rangle.
\end{split}$$ We therefore have $$\frac{{{\mathbb{C}}}[t, \{p_{\mathcal{A}}\}_{\mathcal{A}
\subseteq \{1,2,\ldots, n-1\}}]}{\mathcal{J}} \cong \frac{{{\mathbb{C}}}[t,p_1,
\ldots, p_{n-1}]}{\bigg\langle q_{i, \mathcal{A}} \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \bigg\rangle}$$ but since $q_{i,\mathcal{A}} = 0$ if $i \not \in \mathcal{A}$ by Lemma \[lemma:some q are zero\] we conclude $$\bigg\langle q_{i, \mathcal{A}} \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \bigg\rangle =
\bigg\langle q_{i, \mathcal{A}} \, \bigg\vert \, 1 \leq i \leq
n-1, \mathcal{A} \subseteq \{1,2,\ldots,n-1\} \textup{ and } i
\not \in \mathcal{A} \bigg\rangle$$ from which the result follows.
We illustrate the theorem by an example.
Let $n=4$ and $Y$ the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^4)$. Then the degree-$2$ multiplicative generators are $p_1,
p_2$, and $p_3$. Then the statement of Theorem \[theorem:simplify Monk\] yields a presentation of the equivariant cohomology ring of $Y$ as $$H^*_{S^1}(Y) \cong {{\mathbb{C}}}[t, p_1, p_2, p_3]/\mathcal{K}$$ where $\mathcal{K}$ is the ideal generated by the following $12$ elements: $$\begin{gathered}
2\, p_1^2 - 2t \, p_1 - p_1 p_2, \\
2 \, p _2^2 - 2t \, p_2 - p_1 p_2 - p_2 p_3, \\
2 \, p_3^2 - 2t \, p_3 - p_2 p_3, \\
3 \, p_1^2 p_2 - 6t \, p_1 p_2 - p_1 p_2 p_3, \\
3 \, p_1 p_2^2 - 6t \, p_1 p_2 - 2 \, p_1 p_2 p_3, \\
2 \, p_1^2 p_3 - 2t \, p_1 p_3 - p_1 p_2 p_3, \\
2 \, p_1 p_3^2 - 2t \, p_1 p_3 - p_1 p_2 p_3, \\
3 \, p_2^2 p_3 - 6t \, p_2 p_3 - 2 \, p_1 p_2 p_3, \\
3 \, p_2 p_3^2 - 6t \, p_2 p_3 - p_1 p_2 p_3, \\
p_1^2 p_2 p_3 - 3t \, p_1 p_2 p_3, \\
p_1 p_2^2 p_3 - 4t \, p_1 p_2 p_3, \\
p_1 p_2 p_3^2 - 3t \, p_1 p_2 p_3.\end{gathered}$$ This list is not minimal: for instance, one can immediately see the sixth and seventh expressions in this list are multiples of the first and third ones, so evidently are unnecessary for defining the ideal $\mathcal{K}$. In fact, more is true: a Macaulay 2 computation shows that the ideal $\mathcal{K}$ is in fact generated by just the *quadratic* relations, i.e. the first three elements in the above list. (We thank the referee for pointing this out.) Note the original presentation given in Theorem \[theorem:presentation\] uses $8$ generators and $24$ relations, so this discussion shows that our presentation indeed gives a simplification of the description of the ring.
\[remark:quadratic\] We thank the referee for the following comment. Based on our Giambelli formula, Theorem \[theorem:simplify Monk\], and the example of $n=4$ discussed above, it seems natural to conjecture that for any value of $n$, the corresponding ideal $\mathcal{K}$ is generated by just the quadratic relations. Using Macaulay 2, we have verified that the conjecture holds for a range of small values of $n$, but we were unable to give a proof for the general case. If the conjecture is true, then it would be a very significant simplification of the presentation of this ring and would lead to many interesting geometric and combinatorial questions.
Stirling numbers of the second kind {#sec:stirling}
===================================
In this section we prove that Stirling numbers of the second kind appear in the multiplicative structure of the ring $H^*_{S^1}(Y)$. We learned this result from H. Naruse and do not claim originality, though the proof given is our own. The **Stirling number of the second kind**, which we denote $S(n,k)$, counts the number of ways to partition a set of $n$ elements into $k$ nonempty subsets (see e.g. [@Knu73 Section 1.2.6]). For example, $S(3,2)$ is the number of ways to put balls labelled $1$, $2$, and $3$ into two identical boxes such that each box contains at least one ball. It is then easily seen that $S(3,2)=3$. We have the following.
Fix $n$ a positive integer. Let $Y$ be the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ equipped with the action of the $S^1$ in . For ${\mathcal A} \subseteq \{1,2,\ldots,
n-1\}$, let $v_{\mathcal{A}}, p_{\mathcal{A}}$ be as in Theorem \[theorem:presentation\]. The following equality holds in $H^*_{S^1}(Y)$ for any $k$ with $1 \leq k \leq n-1$: $$\label{eq:stirling}
p_{1}^{k}=\sum_{j=1}^{k} S(k,j) t^{k-j} \, p_{v_{[1,j]}}.$$
We proceed by induction on $k$. Consider the base case $k=1$. Then becomes the equality $$p_{1}=S(1,1)p_{1}.$$ Here $S(1,1)$ is the number of ways to put $1$ ball into $1$ box, so $S(1,1)=1$ and the claim follows.
Now assume that holds for $k$. We need to show that it also holds for $k+1$, i.e., $$p_{1}^{k+1}=\sum_{j=1}^{k+1} S(k+1,j) t^{k+1-j} \, p_{v_{[1,j]}}.$$ By the inductive hypothesis this is equivalent to showing that $$\label{eq:inductive step}
\sum_{i=1}^{k}S(k,i)t^{k-i}p_{1}p_{v_{[1, i]}}=\sum_{j=1}^{k+1}S(k+1,j)t^{k+1-j}p_{v_{[1,j]}}.$$ We now expand the left hand side using the Monk formula. For each $i$ it can be computed that $$p_1 \, p_{v_{[1,i]}} = it \, p_{v_{[1,i]}} + p_{v_{[1,i+1]}}$$ where we have used [@HarTym09 Lemma 6.4] to compute $p_1(w_{[1,i]})$. Therefore $$\begin{split}
\sum_{i=1}^{k}S(k,i)t^{k-i}p_{1}p_{v_{[1, i]}} &=
\sum_{i=1}^{k}S(k,i)t^{k-i} (it \, p_{v_{[1,i]}} + p_{v_{[1,i+1]}}) \\
& = \sum_{i=1}^k i \, S(k,i) t^{k+1-i} p_{v_{[1,i]}} + \sum_{i=1}^k
S(k,i) t^{k-i} p_{v_{[1,i+1]}} \\
&= S(k,1) t^k p_1 + \sum_{i=2}^k i \, S(k,i) t^{k+1-i}
p_{v_{[1,i]}} + \sum_{i=1}^{k-1} S(k, i) t^{k-i} p_{v_{[1,i+1]}} + S(k,k)
p_{v_{[1,k+1]}} \\
& = S(k,1) t^k p_1 + \sum_{i=2}^k i \, S(k,i) t^{k+1-i}
p_{v_{[1,i]}} + \sum_{i=2}^k S(k, i-1) t^{k+1-i} p_{v_{[1,i]}} + S(k,k)
p_{v_{[1,k+1]}} \\
& = S(k+1,1) t^k p_1 + \sum_{i=2}^k (i \, S(k,i) + S(k,i-1))
t^{k+1-i} p_{v_{[1,i]}} + S(k+1,k+1) p_{v_{[1,k+1]}} \\
& = S(k+1,1) t^k p_1 + \sum_{i=2}^k S(k+1,j)
t^{k+1-i} p_{v_{[1,i]}} + S(k+1,k+1) p_{v_{[1,k+1]}} \\
& = \sum_{j=1}^{k+1} S(k+1,j) t^{k+1-j} p_{v_{[1,j]}}
\end{split}$$ where we have used the recurrence relation for Stirling numbers (see e.g. [@Knu73]) $$S(k+1, j) = j S(k, j) + S(k, j-1)$$ and the fact that $S(k,1)=S(k,k)=S(k+1,1)=S(k+1,k+1)=1$ for any $k$. The result follows.
Stability of Peterson Schubert classes {#sec:stability}
======================================
We now observe that the Peterson Schubert classes $\{p_{\mathcal{A}}\}$ for the Peterson varieties satisfy a stability property for varying $n$, similar to that satisfied by the classical equivariant Schubert classes. This is an observation we learned from H. Naruse; we do not claim originality. For this section only, for a fixed positive integer $n$ we denote by $Y_n$ the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$.
Let $X_{w,n} \subseteq {{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ denote the **Schubert variety** corresponding to $w \in S_n$ in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$. By the standard inclusion of groups $S_n {\hookrightarrow}S_{n+1}$, we may also consider $w$ to be an element in $S_{n+1}$. Furthermore there is a natural $T^n$-equivariant inclusion $\iota_n: {{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n) {\hookrightarrow}{{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+1})$ induced by the inclusion of the coordinate subspace ${{\mathbb{C}}}^n$ into ${{\mathbb{C}}}^{n+1}$. Then with respect to $\iota_n$ the Schubert variety $X_{w,n}$ maps isomorphically onto the corresponding Schubert variety $X_{w,n+1}$. Since the equivariant Schubert classes are cohomology classes corresponding to the Schubert varieties, this implies that for any $w \in S_n$ there exists an infinite sequence of Schubert classes $\{\sigma_{w,m}\}_{m=n}^{\infty}$ which lift the classes $\sigma_{w,n} \in H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))$, i.e., $$\label{eq:maps of flags}
\xymatrix{
\cdots \ar[r] & H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+2})) \ar[r] &
H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+1})) \ar[r] & H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)) \\
\cdots \ar@{|->}[r] & \sigma_{w,n+2} \ar@{|->}[r] & \sigma_{w, n+1}
\ar@{|->}[r] & \sigma_{w,n}
}$$ and furthermore for any $v \in S_n$ and any $m \geq n$, the restriction $\sigma_{w,m}(v)$ is equal to $\sigma_{w,n}(v)$. The theorem below asserts that a similar statement holds for Peterson Schubert classes. Observe that the inclusion $\iota_n: {{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)
{\hookrightarrow}{{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+1})$ mentioned above also induces a natural inclusion $j_n: Y_n {\hookrightarrow}Y_{n+1}$ since the principal nilpotent operator on ${{\mathbb{C}}}^{n+1}$ preserves the coordinate subspace ${{\mathbb{C}}}^n$. Moreover, since the central circle subgroup of $U(n,{{\mathbb{C}}})$ acts trivially on ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ for any $n$, the inclusion $j_n$ is equivariant with respect to the $S^1$-actions on $Y_n$ and $Y_{n+1}$ given by the two circle subgroups defined by in $U(n,{{\mathbb{C}}})$ and $U(n+1,{{\mathbb{C}}})$ respectively. Thus there is a pullback homomorphism $j_n^*:
H^*_{S^1}(Y_{n+1}) \to H^*_{S^1}(Y_n)$ analogous to the map $\iota_n:
H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+1})) \to H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))$ above. We have the following.
For a positive integer $n$ let $Y_n$ denote the Peterson variety in ${{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n)$ equipped with the natural $S^1$-action defined by . For $w \in
S_n$ let $p_{w,n} \in H^*_{S^1}(Y_n)$ denote the Peterson Schubert class corresponding to $w$. Then the natural inclusions $j_m: Y_m {\hookrightarrow}Y_{m+1}$ for $m \geq n$ induce a sequence of homomorphisms $j_m^*: H^*_{S^1}(Y_{m+1}) \to H^*_{S^1}(Y_m)$ such that $j_m^*(p_{w,m+1}) = p_{w,m}$, i.e., there exists a infinite sequence of Peterson Schubert classes $\{p_{w,m}\}_{m=n}^{\infty}$ which lift $p_{w,n} \in
H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))$ $$\xymatrix{
\cdots \ar[r] & H^*_{S^1}(Y_{n+2}) \ar[r] & H^*_{S^1}(Y_{n+1}) \ar[r]
& H^*_{S^1}(Y_n) \\
\cdots \ar@{|->}[r] & p_{w,n+2} \ar@{|->}[r] & p_{w, n+1}
\ar@{|->}[r] & p_{w,n}
}$$ and furthermore for any $v \in Y_n^{S^1}$ and any $m \geq n$, the restriction $p_{w,m}(v)$ is equal to $p_{w,n}(v)$.
By naturality and the definition of Peterson Schubert classes $p_{w,n}
\in H^*_{S^1}(Y_n)$ as the images of $\sigma_{w,n}$, it is immediate that can be expanded to a commutative diagram $$\xymatrix{
\cdots \ar[r] & H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+2})) \ar[r]^{\iota_{n+1}^*} \ar[d] &
H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^{n+1})) \ar[r]^{\iota_n^*} \ar[d] & H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^n))
\ar[d] \\
\cdots \ar[r] & H^*_{S^1}(Y_{n+2}) \ar[r]_{j_{n+1}^*} & H^*_{S^1}(Y_{n+1}) \ar[r]_{j_n^*}
& H^*_{S^1}(Y_n)
}$$ where the vertical arrows are the projection maps $H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^m)) \to H^*_{S^1}(Y_n)$ obtained by the composition of $H^*_{T^n}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^m)) \to H^*_{S^1}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^m))$ with $H^*_{S^1}({{\mathcal{F}\ell ags}}({{\mathbb{C}}}^m)) \to H^*_{S^1}(Y_n)$. In particular, for any $w \in S_n$ and $m \geq n$, the vertical maps send $\sigma_{w,m}$ to $p_{w,m}$. The result follows.
[10]{}
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[^1]: The second author is partially supported by an NSERC Discovery Grant, an NSERC University Faculty Award, and an Ontario Ministry of Research and Innovation Early Researcher Award.
[^2]: In this note, all cohomology rings are with coefficients in ${{\mathbb{C}}}$.
|
CU-TP-863
[A.H. Mueller]{} [Department of Physics, Columbia University, New York, N.Y. 10027]{}
[**Abstract**]{} A selected review of topics at the border of hard and soft physics is given. Particular emphasis is placed on diffraction dissociation at Fermilab and HERA. Recently, significant differences between diffraction dissociation at HERA and at Fermilab have become apparent. This may suggest that one already is reaching nonlinear (unitarity) effects which are extending from the soft physics region into the semihard regime of QCD.
Introduction
============
The focus in this paper is on the regime of hardness near the borderline between hard and soft high energy collisions with a special emphasis on searching for nonlinear QCD effects. This is an opportune time for such a discussion as there is now a significant body of complementary data from deep inelastic scattering and from hadron-hadron collisions. Perhaps the major object of this review is to compare and contrast hadronic and deep inelastic collisions, especially diffraction dissociation where there are major differences between the hadronic and virtual photon initiated processes.
Sec.2 is devoted to a brief review of some soft physics results on total cross sections and diffraction dissociation. Despite the fact that total cross sections grow\[1\] as $(s/s_0)^\epsilon,$ with $\epsilon
\approx 0.1,$ through the highest Fermilab energies it is argued that there already is strong evidence of unitarity corrections being important from the ISR to Fermilab energy regimes. In particular, we suggest the lack of growth of the single diffraction dissociation cross section\[2-5\] as due to the blackness of central proton-proton and proton-antiproton collisions.
In contrast to a very weak growth of the single diffraction dissociation cross section in hadronic collisions the energy dependence of virtual photon diffraction dissociation appears to be significantly stronge\[6-7\] than that expected from soft physics. In Sec.3, we argue that this may be due to blackness for the soft components of the virtual photon’s wavefunction and a subsequent dominance of the process by semihard components as suggested recently by Gotsman, Levin and Maor\[8\]. If this is indeed the case it means that for the first time one has evidence of unitarity (nonlinear) effects extending into the semihard regime of QCD.
In Sec.4, we remark that the new DØdata\[9,10\] on rapidity gaps between jets showing that the gap fraction of events decreases with energy may be due to the same physics which slows the growth of the single diffraction cross section. If this decrease is indeed due to an energy dependent (decreasing) survival probability a similar behavior would be expected for comparable photoproduction data involving resolved photons, but such a decrease would not be seen in deep inelastic scattering.
In Sec.5, we review BFKL searches performed at H1, ZEUS and DØ\[9,11-13\]. Each analysis, using two-jet inclusive measurements at Fermilab and a forward jet measurement at HERA, finds some evidence for BFKL behavior through an energy dependence which seems stronger than expected from leading and next-to-leading order perturbation theory. However, definitive results have not yet been achieved.
In Sec.6, progress in calculating the next-to-leading corrections in BFKL evolution is reviewed\[14\]. We may be near a rather complete understanding of these corrections. A preliminary estimate suggests a substantial reduction of the BFKL pomeron intercept.
Sec.7 is devoted to a brief discussion of some topics involving nuclear reactions. Parametrizations of diffraction dissociation at HERA have been successfully used to describe nuclear shadowing in fixed target deep inelastic lepton-nucleus scattering\[15\]. $J/\psi$ production in proton-nucleus and nucleus-nucleus scattering continues to be an important subject for research. New phenomenological success in describing all data except for Pb-Pb collisions by a simple absorption model\[16,17\], along with the suggestion that the Pb-Pb data may be qualitatively different\[18\] have made it even more important to connect $J/\psi$ production and scattering more firmly with QCD.
Clearly soft
============
Total cross sections
--------------------
Donnachie and Landshoff\[1\] have shown that all high energy total cross sections for hadron-hadron collisions can be written in the form
$$\sigma_{tot}=\sigma_0(s/s_0)^\epsilon + {\rm subleading\ terms}\eqno(2.1)$$
where $\sigma_0$ depends on the particular hadrons initiating the collision and the subleading terms go to zero roughly like $(s/s_0)^{-1/2}.$ $s_0$ is an arbitrary scale factor while $\epsilon$ appears to be universal and of size
$$\epsilon \approx 0.1.\eqno(2.2)$$
$1 + \epsilon = \alpha_p$ is the intercept of the soft pomeron in Regge-language. HERA data shows that (2.1) is also true for real photon-proton collisions. Of course a growth in energy as fast as that indicated in (2.1) cannot persist at arbitrarily high energies because of limitations required by the Froissart bound which does not permit total hadronic cross sections to rise faster than $\ell n^2 s/s_0$ at asymptotic energies. The fact that the behavior indicated in (2.1) persists up to the highest Fermilab energy region might seem to indicate that unitarity constraints, which are responsible for the Froissart bound, are not yet effective in the presently available energy region. However, as observed long ago\[19\], this is not the case. If one writes proton-proton total, inelastic and elastic cross sections in terms of the S-matrix at a given impact parameter of the collision, $S(b),$ as
$$\sigma_{in}=\int d^2b[1-S^2(b)]\eqno(2.3)$$
$$\sigma_{e\ell}=\int d^2b[1-S(b)]^2\eqno(2.4)$$
$$\sigma_{tot}=2\int d^2b[1-S(b)],\eqno(2.5)$$
then $S(b)$ depends on b roughly as indicated in Fig.1. (We take S(b) to be real for simplicity.)
For small values of impact parameter $S(b)$ is near zero for proton-proton collisions already in the ISR and Fermilab fixed target energy regime. For proton-antiproton collisions $S(b)$ is quite small for $b<1 fm$ in the Fermilab energy regime. $S(b)$ near zero is a signal that unitarity corrections are large though they are not so easy to see in the total cross section because the radius of interaction is expanding and the growth of $\sigma_{tot}$ is mainly coming from that expansion. Monte Carlo simulations\[20\] of the dipole formulation\[21,22\] of the Balitsky, Fadin, Kuraev, Lipatov\[23,24\] equation for the academic case of heavy onium-heavy onium scattering show a similar phenomenon. For rapidities less than about 15 the BFKL equation is reliable for the total onium-onium cross section, however, for small impact parameter collisions important unitarity corrections are visible for rapidities of 6 units.
Diffraction dissociation in hadron-hadron scattering
----------------------------------------------------
Single diffraction dissociation, illustrated in Fig.2, is given by
$$x_P{d\sigma_{SD}\over dx_Pdt}\ =\ x^{2(1-\alpha_P(t))}f(M_x^2)\eqno(2.6)$$
is terms of soft pomeron exchange. Integrating (2.6) over t and over $x_P\leq 0.05,$ but excluding the proton state $M_x = M_p,$ one gets a single diffractive cross section $\sigma_{SD}.$ From the Regge formalism one expects $\sigma_{SD}$ to grow with $s$ as $s^{2\epsilon},$ but this is not seen in the data as illustrated in Fig.3, which is a simplified version of the more complete plot in Ref.5 where detailed data points are shown. A factor of 2 is included in Fig.3 to account for single diffractive dissociation of either of the colliding protons (antiproton). At Fermilab collider energies there is a discrepancy of an order of magnitude between the Regge fit and the data.
It seems clear that this discrepancy and the slow growth of $\sigma_{SD}$ with energy signal a breakdown of the Regge analysis when ${\sqrt{s}}\geq 20
GeV.$ I think this breakdown can be expressed in various equivalent ways. (i) A low energies inelastic collisions induce, through unitarity, both elastic scattering and diffractive dissociation. However, as $S(b)$ goes to zero for small and moderate $b$ at high energies, these black regions of impact parameter space only induce elastic scattering and not diffraction dissociation. Thus as one increases energy the elastic cross section grows rapidly while the diffraction dissociation cross section, coming from those regions in impact parameter space where $S(b)$ is neither too close to zero or too close to one, grows very slowly. (ii) In the Regge language one must include multiple pomeron exchange in addition to the single pomeron exchange which is valid at lower energy. This multiple pomeron exchange gives absorptive (virtual) corrections which slow the growth coming from single pomeron exchange. (iii) The “gap survival” probability\[25,26\] decreases with energy compensating the growth due to single pomeron exchange. Although gap survival probability is a concept usually used for hard collisions I think the same idea applies to single diffraction dissociation, at least in a heuristic way, in hadron-hadron collisions. It is likely that (i), (ii) and (iii) are just different ways of saying the same thing.
Deep inelastic lepton-proton scattering analogs of the soft physics results
===========================================================================
An “elastic” scattering amplitude
---------------------------------
Recently, there has been an interesting suggestion as to how to test unitarity limits in deep inelastic scattering\[27\]. Of course there is no Froissart bound for virtual photon-proton scattering, nevertheless, we have become used to viewing the small-x structure function in terms of a high energy quark-antiquark pair (possibly accompanied by gluons) impinging on the target proton. Although the quark-antiquark pair is not on-shell the time evolution of the pair as it passes through the nucleon should be constrained by unitarity in much the same way that a quark-antiquark pair coming from, say, a pion state would be.
To be more specific, view deep inelastic scattering in the rest system of the proton and in the aligned jet (naive parton) model\[28,29\]. At small x the virtual photon, $\gamma^*(q),$ breaks up into a quark and antiquark pair long before reaching the proton. The relative transverse momentum of the quark and antiquark is small, of hadronic size $\mu\approx 350 MeV,$ while the longitudinal momenta are $q_z$ and $q_z\cdot
{\mu^2\over Q^2}$ respectively. Because the relative transverse momentum is small the transverse coordinate separation of the quark and antiquark can be expected to be on the order of a fermi, and the resulting cross section with the proton should be of hadronic size. The smallness of the overall cross section comes from the small probability, of order $\mu^2/Q^2,$ to find such an aligned jet configuration in the wavefunction of $\gamma^*.$ (More probable configurations in the $\gamma^*$ have smaller interaction probabilities. While the aligned jet model cannot be expected to be a precise model of deeply inelastic scattering it should reasonably characterize a significant portion of deep inelastic events.)
The inelastic reaction of this, longitudinally asymmetric, quark-antiquark pair with the proton should produce a shadow quark-antiquark pair in the final state. If the center of the proton is relatively black to the incoming quark-antiquark pair the shadow may be rather strong, as in the hadronic case discussed above, and unitarity limits may aleady be reached at present energies. The outgoing quark-antiquark pair should show up as a diffractively produced state, of mass $M_x \approx Q,$ following the direction of the $\gamma^*.$ Assuming that the scattering amplitude of the quark-antiquark pair with the proton is imaginary one may reconstruct this amplitude, dropping an i, as
$$F(x,\b{b}) = \int d^2p e^{i\b{p}\cdot \b{b}}{\sqrt{{d\sigma_{SD}\over
d^2p}}}\eqno(3.1)$$
with $\b{b}$ the impact parameter of the collision and $\b{p}$ the momentum transfer to the recoil proton. ${d\sigma_{SD}\over d^2p}$ is the single diffractive cross section for $M_x\approx Q.$
In the present circumstance we do not have good control of the magnitude of F near $\b{b}=0.$ However, if the proton is black for central collisions one can expect $F(x,\b{0})$ to show little x-dependence. (Here x plays the role that $s$ does for the hadronic collisions discussed above.) The authors of Ref.27 suggest looking at the b-dependence of
$$\Delta_{eff} = {d \ell n F(x,b)\over d \ell n 1/x}.\eqno(3.2)$$
Unitarity constraints can be expected to show up as smaller values of $\Delta_{eff}$ near $b=0.$ More quantitatively, unitarity limits at $b=0$ would give
$$\Delta_{eff}(b=0) < 2(\alpha_P-1).\eqno(3.3)$$
This is a clever idea, and it will be interesting to see what the data give.
Large mass ${\gamma}^*$ diffractive dissociation
------------------------------------------------
The traditional picture of large mass diffraction dissociation at small values of x is shown in Fig.4, where Dokshitzer, Gribov, Lipatov, Altarelli, Parisi (DGLAP)\[30-32\] evolution takes one from the hard scale Q to the soft scale $\mu$ where a soft diffractive scattering, represented by soft pomeron exchange, occurs. In Fig.4, one assumes the DGLAP ordering
$$\mu^2 \approx \b{k}^2 << \cdot\cdot\cdot << \b{k}_2^2 << \b{k}_1^2 <<
Q^2.\eqno(3.4)$$
However, this is a subtle process and it is worthwhile looking carefully at the argumentation that leads to the size of $\b{k}^2$ at the lower end of the DGLAP evolution\[8,33-35\]. It is convenient to view that evolution proceeding from the hard scale Q toward softer scales, a direction opposite to that which is usually taken. In Fig.5, we illustrate the process in two steps: The left-hand part of the figure shows the virtual photon wavefunction in terms of its quark and gluon components. As in Fig.4, k is supposed to be the softest gluon and $\Delta
x_\perp = 2/k_\perp$ gives the transverse size of the $\gamma^*$ state. The right-hand part of the figure gives the diffractive scattering part of the process proceeding by gluon exchange from the proton interacting with the octet dipole consisting of the gluon $k$ and the remainder of the $\gamma^*$ state. Schematically, one may write
$$d\sigma_{SD} = {\rm flux}\ dP_r(k_\perp)[1-S(\Delta x_\perp = 2/k_\perp,
\b{b}, Y=\ell n 1/x)]^2d^2bdx_P\eqno(3.5)$$
where
$$dP_r(k_\perp) = {dk_\perp^2\over Q^2}\eqno(3.6)$$
is the probability that the lowest transverse momentum gluon have momentum $k_\perp .$ Eq.(3.6) shows that gluons with small $k_\perp$ have a small probability in the $\gamma^*$ wavefunction, analogous to what we found earlier for low momentum quarks in the aligned jet model. $1-S$ represents the amplitude for a gluon having $k_\perp,$ along with the remainder of the $\gamma^*$ wavefunction, to scatter elastically on the proton. $b$ is the impact parameter of the overall collision while $Y=\ell n 1/x$ is the rapidity of the softest gluon with respect to the proton. In $lowest\ order,$ two-gluon exchange,
$$1-S \propto {x_PG(x_P,k_\perp^2)\over k_\perp^2}\eqno(3.7)$$
when $k_\perp$ is large and where an integration has been performed over impact parameter, $\b{b}.$ Using (3.6) and (3.7) in (3.5) one sees, dimensionally, that $k_\perp^2$ cannot be large and this is the logic that has led theorists to take $k_\perp \approx \mu$ and use soft pomeron exchange for the scattering amplitude, $1-S.$
However, if $S$ is near zero for $k_\perp = \mu$ and for $\b{b} =
0,$ and this is not unreasonable since the $S$ matrix is near zero for small impact parameter hadron-hadron collisions, then it is apparent from (3.5) and (3.6) that values of $k_\perp$ significantly larger than $\mu$ will be important. Indeed, the values of $k_\perp$ that will dominate large mass single diffractive production are those values where $S$ is near, but not too close to, one. This is the case since the probability in the $\gamma^*$ wavefunction is located in large $k_\perp-$ values. The situation here is quite different than for hadron-hadron collisions. In hadron-hadron collisions the wavefunction of the incoming hadron is, except for a very small part, in the soft physics region. If the $S$-matrix is near zero for central collisions then the inelastic reaction will feed into elastic scattering as a shadow. In deep inelastic scattering at small-x when $S$ becomes black there will certainly be a similar phenomenon which occurs, and which has been described in Sec.3.1, but, in addition, blackness in the small $k_\perp$ region will allow higher values of $k_\perp$ to become effective thus making the process semihard.
If central impact parameter collisions of $\gamma^*$-proton collisions are indeed black for $k_\perp \approx \mu$ then we would expect the x-dependence of the single diffractive cross section, $x_P{d\sigma\over d x_P},$ to vary more strongly with x than suggested by the soft pomeron. If one writes
$$x_P{d\sigma\over dx_P} \propto x^{-n}\eqno(3.8)$$
then both ZEUS\[6\] and H1\[7\] now suggest that $n \approx 0.4$ rather than the $n=2(\alpha_P-1) \approx 0.2$ predicted by the soft pomeron. If the ZEUS and H1 measurements hold up, and n really is near 0.4 in the small $\beta$ region, then I think it becomes clear that semihard physics is dominating the physics of large mass diffraction dissociation. In that case it is interesting to reexamine the “elastic” scattering analyses we described in Sec.3.1 to see if the proposed procedure to measure blackness is also destroyed by the dominance of gluons and quarks at higher $k_\perp$- values. Finally, it should be pointed out that there are already rather detailed calculations of the phenomenon, at least for $q\bar{q}$ and $q\bar{q}g$ components of the $\gamma^*$ wavefunction, which arrived at a value $n \approx
0.5,$ not too far from experiment\[8\]. Have we, for the first time, actually seen the long sought after evidence for nonlinearity (unitarity limits) in the semihard region of deep inelastic scattering?
Before leaving this section, it may be useful to again contrast hadron-hadron scattering with $\gamma^*$-proton scattering. In the purely hadronic case the energy dependence of the single diffraction cross section is much weaker than that predicted by the soft pomeron. We have interpreted this as due to blackness in central proton-proton collisions which enhances the elastic cross section but suppresses diffractive excitation. In $\gamma^*$-proton scattering, on the other hand, the energy dependence (x-dependence) is much stronger than that predicted by the soft pomeron. We have interpreted that also as due to blackness of the soft components of the $\gamma^*$ now leading to an enhanced role for the harder components of the $\gamma^*$-wavefunction and a resulting stronger energy dependence of the cross section.
Rapidity gaps between jets at Fermilab and HERA
===============================================
Suppose one measures two jets having comparable but opposite transverse momentum along with the requirement that there be a rapidity gap between two jets. One might hope that this would be a good process to measure the hard (BFKL) pomeron as illustated in Fig.6\[36\]. There are, however, at least two worries with using this process to measure the hard pomeron. (i) The pomeron contribution to the hard quark-antiquark scattering is\[37\].
$${d^{\sigma}\over dt} = (\alpha C_F)^4\ {\pi^3\over 4 t^2} {exp[2(\alpha_P-1)
\Delta Y]\over [{7\over 2} \alpha N_c\zeta(3) \Delta Y]^3}\eqno(4.1)$$
with $\Delta Y$ the rapidity between the two jets. Here $\alpha_P-1=
{4\alpha N_c\over \pi} \ell n 2$ is the BFKL pomeron intercept. The presence of the factor $(\Delta Y)^3$ in the denominator in (4.1) strongly reduces the effective growth of the cross section with $\Delta Y$ making the emergence of the hard pomeron more difficult at moderate values of $\Delta Y.$ (ii) Perhaps more serious yet is the fact that the cross section for producing two jets with a gap between them depends on the absence of a soft interaction between the spectator parts of the proton and antiproton, the so-called gap survival probability\[36\]. This lack of factorization makes it difficult to make a precise comparison between theory and experiment.
There is new data from DØ\[9,10\] and an interesting new analysis comparing the 1800 GeV data with that at 630 GeV. If $f_{gap}$ is the fraction of all two-jet events (separated by a given rapidity) with a gap between them then DØfinds that
$${f_{gap}(630)\over f_{gap}(1800)} = 2.6 \pm 0.6_{stat.}\eqno(4.2)$$
for $\Delta Y \geq 3.8$. Thus the gap fraction $decreases$ with increasing energy. While this number cannot be directly compared to BFKL dynamics because $\Delta Y$ has been taken to be the same at the two energies, while a BFKL test should have $\Delta Y(1800)-\Delta Y(630) = \ell n {1800\over
630},$ it does suggest that the survival probability has a rather strong energy dependence making BFKL tests more difficult in rapidity gap events. It will be interesting to see whether models of the gap survival probability can easily accomodate the energy dependence in Eq.(4.2)\[38\].
At Fermilab the gap fraction is typically 0.01 while at HERA more like 0.07. The gap survival probability is much larger at HERA as is natural for a point-like $\gamma^*.$ It would be interesting to have a HERA analysis similar to that of DØ to see if the energy dependence of the gap fraction is weaker, closer to x-independent, than at Fermilab. With respect to the DØdata the energy dependence of the gap fraction may be reflecting exactly the same phenomenon as observed in the energy dependence of the single diffraction cross section discussed in Sec.2.2. While the inclusive two-jet cross section increases at higher energies, because of the growth in the parton densities, the energy dependence of the gap cross section is likely to be much weaker because of the increasing blackness of central proton-antiproton collisions as already seen in the single diffractive cross section.
BFKL searches
=============
The hard (BFKL) pomeron or, equivalently, BFKL evolution shows up simply only in single transverse momentum hard scale processes. Thus in hadron-hadron collisions or in deep inelastic scattering where a soft scale, the size of the hadron (proton), is present a special class of events must be taken in order to isolate BFKL dynamics. Since this is generally very difficult to do experimentally it is perhaps useful to remind the reader why BFKL dynamics is so interesting for QCD and why it is worth the considerable effort necessary to uncover it.
There are at least two important reasons why hard single scale high energy scattering is interesting. (i) It is a high energy scattering problem that may be soluble, or nearly soluble. (ii) BFKL evolution leads to high parton densities and thus into a new domain of nonperturbative, but weak coupling, QCD. As parton distributions evolve from a momentum fraction $x_1$ to a smaller momentum fraction $x_2$, all at a fixed transverse momentum scale, BFKL dynamics gives the rate of increase of those (mainly gluon) densities. This evolution is illustrated in Fig.7. When gluon densities reach a density such that on the order of $1/\alpha$ gluons overlap, perturbation theory breaks down and one enters a new regime of strong field, $F_{\mu\nu} \sim 1/g,$ QCD. While it is unlikely that one can reach such densities at Fermilab or HERA at truly hard transverse momentum scales one should at least be able to see the approach to these high densities through BFKL evolution.
Inclusive two-jet cross sections at Fermilab and forward single jet inclusive cross sections at HERA can be used to measure the BFKL intercept\[39-43\]. These processes are illustrated in Figs.8 and 9 respectively where $k_1$ and $k_2$ represent measured jets. In proton-antiproton collisions one chooses $k_{1\perp}, k_{k_2\perp} > M, $ a fixed hard scale, while in deep inelastic scattering $k_{1\perp}$ is chosen to be on the order of Q, the photon virtuality. For the hadron-hadron case
$$\sigma_{2-jet}= f(x_1, x_2, M^2) {e^{(\alpha_P-1)\Delta Y}\over \sqrt{\Delta
Y}}\eqno(5.1)$$
while for deep inelastic scattering
$$\sigma_{jet} = f(x_1,Q^2) {e^{(\alpha_P-1)\ell n x_1/x}\over \sqrt{\ell n
\ x_1/x}}\eqno(5.2)$$
with $x_1$ and $x_2$ being the longitudinal momentum fractions of the measured jets. $\alpha_P-1= {4\alpha N_c\over \pi} \ell n 2$ and the $f$’s in (5.1) and (5.2) are known in terms of the quark and gluon distributions of the proton and antiproton. In (5.1) $\Delta Y$ is the rapidity difference between the two measured jets. One can get a measurement of $\alpha_P-1$ in (5.1) by varying $\Delta Y$ with $x_1, x_2$ and $M^2$ fixed, and this can be done at Fermilab by comparing the inclusive two-jet cross section at different incident energies. In (5.2) one can measure $\alpha_P-1$ by varying $x$ for fixed $x_1$ and $Q^2.$
Sometime ago H1\[11,12\] presented an analysis showing $\sigma_{jet}$ increasing by about a factor of four as $x$ goes from about $3x10^{-3}$ to about $7x10^{-4}$ for $k_{1\perp} > 3.5 GeV.$ This is a growth quite a bit faster than given in conventional Monte Carlos and much faster than the growth from single gluon exchange between the measured jet and the quark-antiquark pair coming from the virtual photon. The growth is comparable to that given in (5.2), for $\alpha_P-1 \approx 1/2,$ however, the comparison is not completely convincing because a comparison of partonic energy dependences, from (5.2), with hadron final states is not very reliable when $k_{1\perp}$ is as small as in the $H1$ analyses.
Recently ZEUS\[13\] has completed an analysis of this process. Since the ARIADNE Monte Carlo gives a good fit to the ZEUS data this Monte Carlo is used to unfold the hadronization and thus get a better comparison with BFKL evolution. The data agree much better with BFKL evolution than with the Born term or with next-to-leading order QCD calculations. A definitive comparison with BFKL dynamics is hindered by the lack of ability to include hadronization corrections along with the BFKL evolution. One can hope that the situation will soon improve in this regard.
A new DØ\[9\] comparing 1800 GeV and 630 GeV data for $k_{1\perp}, k_{2\perp} \geq 20 GeV$ gives $\alpha_P=1.35 \pm 0.04 (stat) \pm
0.22$ (syst) when (5.1) is used to fit the data. The strength of the DØanalysis is that $k_\perp > 20 GeV$ which makes uncertainties due to jet definition minimal. Weaknesses of the analysis are the large systematic error and the smallness of $\Delta Y,$ equal to 2, at the lower energy. We can hope that the systematic errors will come down in the near future.
Overall, I think the BFKL searches are encouraging but not yet definitive. The fact that all three analyses suggest a strong increase with energy of reliable quantities for isolating BFKL effects is certainly positive. An attempt will also be made to measure $\alpha_P-1$ at LEP\[44\] in the next year by measuring the $\gamma^*-\gamma^*$ total cross section. This is a very clean process, although the cross section is rather small.
Higher order corrections to BFKL evolution
==========================================
In general in QCD next-to-leading corrections are very important. It is only after next-to-leading corrections have been calculated that scales have a real meaning and normalizations can be trusted. In the case of BFKL evolution the next-to-leading corrections are also important to show that, in principle, corrections to the BFKL answer can be calculated, thus making single scale high energy scattering systematically calculable in QCD.
There has been a long program\[14,45-47\], led by the work of V. Fadin and L. Lipatov, to calculate the next-to-leading corrections to BFKL evolution and it now appears that program may be coming to completion. When the work is finished one should get the next correction to $\alpha_P$ as well as next-to-leading resummations for anomalous dimension and coefficient functions. If one writes
$$\alpha_P = {4 N_c\over \pi} \ell n 2 \alpha(Q)[1-c \alpha(Q)]\eqno(6.1)$$
then there is the suggestion $c$ may be near 3, a very large correction, although there is some work yet to be done before one can accept this number with confidence\[14\].
For the anomalous dimension matrix one writes
$$\gamma_n = \sum_{i=1}^\infty \gamma_{n i}^{(0)}\left[{\alpha N_c\over
\pi(n-1)}\right]^i + \alpha \sum_{i=1}^\infty \gamma_{n i}^{(1)}\left[{\alpha
N_c\over \pi(n-1)}\right]^i + \cdot \cdot \cdot\eqno(6.2)$$
where the first series represents the leading order (BFKL) answer. We should soon know the second series, the constants $\gamma_{n i}^{(1)}$ along with similar terms for the coefficient functions. When the BFKL corrections are known at next-to-leading order we should reap several benefits. (i) A better understanding of the importance of BFKL (resummation) effects in $\nu
W_2$ should be possible. Recall, that as a two-scale process BFKL dynamics does not directly govern the small-x behavior of $\nu W_2.$ However, BFKL effects are certainly present and can be systematically included through resummations such as the one indicated in (6.2). When the next-to-leading corrections are known we should begin to get a reliable indication of the importance of these resummation effects in the HERA regime. (ii) The next-to-leading resummations should help us to better understand where the operator expansion is valid in small-x physics, that is at what $x$ and $Q^2$ are coefficient and anomalous dimension functions sufficiently safe from diffusion effects to be reliably calculated perturbatively\[47,48\].
Nuclear reactions
=================
Nuclear shadowing in deep inelastic scattering
----------------------------------------------
Nuclear shadowing in deep inelastic scattering is known to be a leading twist phenomenon\[29\] and thus dominated by soft physics, at least at current x-values. In the DGLAP formalism shadowing effects are put into initial parton distributions. However, when shadowing is not too strong it can be calculated from diffractive deep inelastic scattering using the Gribov-Glauber formalism as illustrated in Fig.10. In that figure the left-hand part represents the imaginary part of the forward $\gamma^*$-scattering amplitude which, by the optical theorem, is equal to the $\gamma^*$-A total cross section or, equivalently, $\nu W_2.$ The first term on the right-hand side of Fig.10 represents the incoherent scattering off the A nucleons in the nucleus, the nucleons being labeled by $N_i.$ The second term on the right-hand side of the figure represents the double scattering term which is dominated by diffractive scattering, off nucleon $N_i$ in the amplitude and $N_j$ in the complex conjugate amplitude. So long as shadowing correction are not too large it should not be necessary to go beyond the double scattering term. Indeed, the double scattering term can be obtained from diffractive data at HERA while triple and higher scattering terms would involve the scattering of partonic systems in the nucleus, terms which cannot be reliably determined. Using a parametrization which fits the HERA diffractive data, a pretty good description of fixed target deep inelastic scattering of nuclei is obtained with the double scattering term giving a shadowing correction of about the right size\[15\].
$J/\psi$ production in proton-nucleus and nucleus-nucleus collisions
--------------------------------------------------------------------
Recently there has been much interest, and excitement, about the NA50 data\[18\]. In a nutshell one can summarize the experimental situation as follows: (i) All P-A and A-A collisions, except for Pb-Pb, [*look like*]{} $J/\psi$ production in proton-proton collisions with absorptive final state interactions corresponding to a $``J/\psi''$ cross section with nucleons of 7mb\[16,17\]. (ii) Pb-Pb central collisions have a $J/\psi$ cross section which is significantly suppressed with respect to (i)\[18\] .
Kharzeev and Satz\[49\] have suggested a picture that considers the system moving though the nuclei, after the hard collision which produces the $c\bar{c}$ pair, to be a $(c\bar{c}g)$ color singlet system which, if it suffers no reaction with the nuclear medium, turns into a $J/\psi$ after the $(c\bar{c}g)$ system has passed through the material. The $(c\bar{c}g)$ system is supposed to have a size comparable to that of the $J/\psi,$ but, because of the fact that it looks like an octet dipole formed from $(c\bar{c})_8,$ and a gluon a cross section of 7mb is natural. This is an interesting picture, however, there are a lot of unanswered questions. (i) Where does the gluon in the $(c\bar{c}g)$ system come from? Does it come from the hard scattering or is it part of the gluon distribution of the incident hadron or nucleus? If it is the latter does this $enhance\ J/\psi$ production in nuclear collisions because there are so many more spectator gluons at the impact parameter of the collision. (ii) How does the gluon know what size system to form with the $c\bar{c}$ since the $(c\bar{c}g)$ does not interact, due to a slowing down of the rate of interaction for high velocity states, while traversing the material? Why should the relevant size for the $\psi$ and $\psi^\prime$ be the same? While a very interesting, and successful, phenomenology has developed around this picture it is important to determine whether the whole picture is reasonable from a QCD point of view.
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---
abstract: 'Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the Appendix we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.'
address:
- 'Universidad de Granada, Departamento de Álgebra and IEMath-Granada. Facultad de Educación, Econonía y Tecnología de Ceuta. Cortadura del Valle, s/n. E-51001 Ceuta, Spain'
- 'University of Turin, Department of Mathematics “Giuseppe Peano”, via Carlo Alberto 10, I-10123 Torino, Italy'
author:
- Laiachi El Kaoutit
- Paolo Saracco
title: 'Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras'
---
[^1]
Introduction
============
Motivation and overviews {#ssec:muchocaldo}
------------------------
Let ${{\mathcal M}}$ be a smooth connected real manifold and denote by $A=C^{\infty}({{\mathcal M}})$ its smooth ${\mathbb{R}}$-algebra [@nestruev §4.1]. All vector bundles considered below are over ${{\mathcal M}}$ and by definition they are locally trivial with constant rank [@nestruev §11.2]. For a given Lie algebroid ${{\mathcal L}}$ with anchor map $\omega:{{\mathcal L}}\,\to\,T{{\mathcal M}}$ (see Example \[exam:VayaCon\]), we consider the category $\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}({{\mathcal L}})$ consisting of those vector bundles ${{\mathcal E}}$ with a (right) ${{\mathcal L}}$-action. That is, an $A$-module morphism $\varrho_{-}: \Gamma({{\mathcal L}}) \to {\mathrm{End}_{{\scriptscriptstyle{{\mathbb{R}}}}}(\Gamma({{\mathcal E}}))}$ which is a Lie algebra map satisfying $\varrho_{X}(f s) =f \varrho_{X}(s) +\Gamma(\omega)_{X}(f) s$, for any section $s \in \Gamma({{\mathcal E}})$ and any function $f \in A$. Morphisms in the category $\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}({{\mathcal L}})$ are morphisms of vector bundles $\varphi: {{\mathcal E}}\to {{\mathcal F}}$ which commute with the actions, that is, such that $\Gamma(\varphi) \circ \varrho_{X} \, =\, \varrho_{X}' \circ \Gamma(\varphi)$, for every section $X \in \Gamma({{\mathcal L}})$. Thanks to the properties of the smooth global sections functor $\Gamma$, expounded in [@nestruev Theorems 11.29, 11.32 and 11.39], the category $\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}({{\mathcal L}})$ is a (not necessarily abelian) symmetric rigid monoidal category, which is endowed with a fiber functor $\boldsymbol{\omegaup}: \mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}({{\mathcal L}}) \to {\mathsf{proj}(A)}$ to the category of finitely generated and projective $A$-modules. The identity object in this monoidal structure is the line bundle ${{\mathcal M}}\times {\mathbb{R}}$ with action $\Gamma(\omega)$ (composed with the canonical injection $\mathrm{Der}_{{\scriptscriptstyle{{\mathbb{R}}}}}(A) \subset {\mathrm{End}_{{\scriptscriptstyle{{\mathbb{R}}}}}(A)}$), and the action on the tensor product of two objects ${{\mathcal E}}$ and ${{\mathcal F}}$ in ${\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}(\mathcal{L})}$, is given by $$(\varrho {\otimes_{\scriptscriptstyle{A}}} \varrho')_{X}: \Gamma({{\mathcal E}}){\otimes_{\scriptscriptstyle{A}}}\Gamma({{\mathcal F}}) \longmapsto \Gamma({{\mathcal E}}){\otimes_{\scriptscriptstyle{A}}}\Gamma({{\mathcal F}}), \quad \Big( s{\otimes_{\scriptscriptstyle{A}}}s' \longmapsto \varrho_{X}(s){\otimes_{\scriptscriptstyle{A}}} s' + s{\otimes_{\scriptscriptstyle{A}}}\varrho_{X}'(s')\Big),$$ for every $X \in \Gamma({{\mathcal L}})$. The symmetry is the one given by the tensor product over $A$. Up to the natural $A$-linear isomorphism $\Gamma({{\mathcal E}}^{*}) = \Gamma\big(\mathsf{Hom}({{\mathcal E}},{{\mathcal M}}\times {\mathbb{R}})\big) \cong \mathsf{Hom}_{\mathcal{C}^{\infty}({{\mathcal M}})}\big(\Gamma\left({{\mathcal E}}\right),\mathcal{C}^{\infty}({{\mathcal M}})\big) = \Gamma({{\mathcal E}})^{*}$ (see [e.g. ]{}[@nestruev Theorem 11.39]), the action on the dual vector bundle ${{\mathcal E}}^{*}$ is provided by the following $A$-module and Lie algebra map $\varrho^{*}: \Gamma({{\mathcal L}}) \to {\mathrm{End}_{{\scriptscriptstyle{{\mathbb{R}}}}}(\Gamma({{\mathcal E}})^{*})}$, sending any section $X \in \Gamma({{\mathcal L}}) $ to the ${\mathbb{R}}$-linear map $$\varrho^{*}_{X} : \Gamma({{\mathcal E}})^{*} \longrightarrow \Gamma({{\mathcal E}})^{*}, \quad \Big( s^{*} \longmapsto \Gamma(\omega)_{X} \circ s^{*}- s^{*} \circ \varrho_{X} \Big)$$ (see also, for example, [@Crainic §1.4], [@ModularClasses page 731]).
The Tannaka reconstruction process shows then that the pair $({\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}(\mathcal{L})}, \boldsymbol{\omegaup})$ leads to a (universal) commutative Hopf algebroid which we denote by $(A, {{\mathcal U}}^{\circ})$ and then to a complete commutative (or topological) Hopf algebroid $(A, {\widehat{{{\mathcal U}}^{\circ}}})$, where $A$ is considered as a discrete topological ring. It turns out that $(A, {{\mathcal U}}^{\circ})$ is the finite dual Hopf algebroid, in the sense of [@LaiachiGomez], of the (co-commutative) universal enveloping Hopf algebroid $(A, {{\mathcal U}}:={{\mathcal V}}_{{\scriptscriptstyle{A}}}(\Gamma({{\mathcal L}}))$ of the Lie-Rinehart algebra $(A,\Gamma({{\mathcal L}}))$, because in this case (${{\mathcal M}}$ connected) the category ${\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}(\mathcal{L})}$ can be identified with the category of (right) ${{\mathcal U}}$-modules with finitely generated and projective underlying $A$-module structure. In this way, we end up with a continuous $(A{\otimes_{\scriptscriptstyle{{\mathbb{R}}}}}A)$-algebra map $\zeta: {{\mathcal U}}^{\circ} \to {{\mathcal U}}^{*}$, where the latter is the convolution algebra of ${{\mathcal U}}$. This algebra admits a topological Hopf algebroid structure, rather than just a topological bialgebroid one as it is known in the literature, see [@Kapranov:2007]. It is worthy to point out that, even if the Tannaka reconstruction process may be applied in this context, the pair $({\mathrm{Rep}_{{\scriptscriptstyle{{{\mathcal M}}}}}(\mathcal{L})}, \boldsymbol{\omegaup})$ does not necessarily form a Tannakian category in the sense of [@Deligne:1990].
The main motivation of this paper is to set up, in a self-contained and comprehensive way, the basic notions and tools behind the theory of complete commutative (or topological) Hopf algebroids and the connection of this theory with Lie algebroids as above. Our aim is to provide explicitly the topological Hopf algebroid structure on ${{\mathcal U}}^{*}$ mentioned previously and to show that the completion ${\widehat{\zeta}}$ of the map $\zeta$ leads not only to a continuous map, but also to a continuous morphism of topological Hopf algebroids. Besides, we will provide some conditions under which ${\widehat{\zeta}}$ becomes an homeomorphism, as well.
Now, knowing that the map ${\widehat{\zeta}}$ is always a morphism of topological Hopf algebroids, one may analyse the distinguished case when ${{\mathcal L}}=T {{\mathcal M}}$ is the tangent bundle with its obvious Lie algebroid structure. It can be shown, by using for instance [@MR1811901 Corollary 15.5.6], that in this case the universal enveloping algebroid ${{\mathcal U}}$ can be identified with the algebra $\mathrm{Diff}(A)$ of all differential operators on $A$. On the other hand, one can extend the duality between $l$-differential operators and $l$-jets of the bundle of $l$-jets of functions on ${{\mathcal M}}$ (see e.g. [@nestruev Theorem 11.64]) to an homeomorphism between the convolution algebra ${{\mathcal U}}^{*}$ and the algebra of infinite jets ${{\mathcal J}}(A):={\widehat{A{\otimes_{\scriptscriptstyle{{\mathbb{R}}}}}A}}$ (see Example \[exam:Sacarrelli\] below). These are isomorphic not only as complete algebras, but also as topological Hopf algebroids. Summing up, we have a commutative diagram $$\begin{gathered}\label{eq:maindiagram}
\xymatrix@R=15pt@C=30pt{ {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \ar@{->}^-{{\widehat{\zeta}}}[rr] & & \mathrm{Diff}(A)^{*} \\ & {{\mathcal J}}(A) \ar@{->}^-{\cong}_-{{\widehat{\vartheta}}}[ru] \ar@{->}^-{{\widehat{\etaup}}}[lu] & }
\end{gathered}$$ of topological Hopf algebroids, where the algebra maps $\vartheta: A{\otimes_{\scriptscriptstyle{{\mathbb{R}}}}}A \to {{\mathcal U}}^{*}$ and $\etaup:A{\otimes_{\scriptscriptstyle{{\mathbb{R}}}}}A \to {{\mathcal U}}^{\circ}$ define the source and the target of ${{\mathcal U}}^{*}$ and ${{\mathcal U}}^{\circ}$ respectively. Furthermore, in view of [@Kapranov:2007 Proposition A.5.10], if the map ${\widehat{\zeta}}$ is an homeomorphism, then both ${\rm Spf}({\widehat{{{\mathcal U}}^{\circ}}})$ and ${\rm Spf}(\mathrm{Diff}(A)^{*})$ can be seen as formal groupoids that integrate the Lie algebroid ${{\mathcal L}}$. Nevertheless, it is not always guaranteed that ${\widehat{\zeta}}$ is an homeomorphism, even for the simplest case, see [@LaiachiPaolo] for a counterexample and Remark \[rem:laventanadelfrente\] for more details.
Description of the main results
-------------------------------
Recall that a complete Hopf algebroid in the sense of §\[ssec:CCHAlgds\] is a co-groupoid object in the category of complete commutative algebras, where the latter is endowed with a suitable topological tensor product (see Appendix \[sec:CBCF\]).
Let $(A,L)$ be a Lie-Rinehart algebra whose underlying module $L_{{\scriptscriptstyle{A}}}$ is finitely generated and projective and consider ${{\mathcal U}}:={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$, its universal enveloping Hopf algebroid (see §\[ssec:ULA\] for details). This is a co-commutative (right) Hopf algebroid to which one can associate two complete commutative (or topological) Hopf algebroids, where the base algebra $A$ is trivially filtered. The first one of these is the completion $(A, {\widehat{{{\mathcal U}}^{\circ}}})$ of the finite dual $(A,{{\mathcal U}}^{\circ})$, the commutative Hopf algebroid constructed as in [@LaiachiGomez], see §\[ssec:Fdual\] for more details on this construction. The second one is $(A, {{\mathcal U}}^{*})$, where ${{\mathcal U}}^{*}$ is the commutative convolution algebra of ${{\mathcal U}}$. The topological Hopf structure of this algebroid is explicitly retrieved in §\[sec:Ustar\]. It is noteworthy to mention that, in general, even if a given co-commutative Hopf algebroid possesses an admissible filtration (see §\[ssec:FUstra\] and §\[ssec:AFiltration\]), it is not clear, at least to us, how to endow its convolution algebra with a topological antipode. Namely, one needs the translation map to be an homeomorphism, or at least a filtered algebra map. A condition which we show that is always fulfilled for ${{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ with $L$ as above.
As we already mentioned, at the algebraic level we have a canonical $(A{\otimes_{\scriptscriptstyle{}}}A)$-algebra map $\zeta: {{\mathcal U}}^{\circ} \to {{\mathcal U}}^{*}$. The following result, which concerns the properties of its completion ${\widehat{\zeta}}$, is our main theorem (stated below as Theorem \[thm:triangle\] and its Corollary \[coro:Equivalence\]).
\[thm:A\] Let $(A,{{\mathcal U}})$ be a co-commutative (right) Hopf algebroid with an admissible filtration (see §\[ssec:FUstra\]) and assume that the translation map $\delta$ of ${{\mathcal U}}$, defined in , is a filtered algebra map. Then the algebra map $\zeta: {{\mathcal U}}^{{{\boldsymbol{\circ}}}} \to {{\mathcal U}}^*$ factors through a continuous morphism ${\widehat{\zeta}}: {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \to {{\mathcal U}}^*$ of complete Hopf algebroids.
In particular, this applies to ${{\mathcal U}}={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ for any Lie-Rinehart algebra $(A,L)$ with $L_{{\scriptscriptstyle{A}}}$ a finitely generated and projective module. Moreover, if ${{\mathcal U}}^{\circ}$ is an Hausdorff topological space with respect to its canonical adic topology and ${\widehat{\zeta}}$ is an homeomorphism, then $\zeta$ is injective and therefore there is an equivalence of symmetric rigid monoidal categories between the category of right $L$-modules and the category of right ${{\mathcal U}}^{\circ}$-comodules, with finitely generated and projective underlying $A$-module structure.
As a consequence, we also discuss conditions under which the map ${\widehat{\zeta}}$ is an homeomorphism, for instance when ${\widehat{\zeta}}$ is injective and ${\widehat{\vartheta}}$ is a filtered isomorphism (as in the example at the end of §\[ssec:muchocaldo\]). See Proposition \[prop:zeroneveropen\] for further conditions.
Notation and basic notions
--------------------------
Throughout this paper and if not stated otherwise, ${\Bbbk}$ will denote a fixed field (eventually a commutative ring in the appendices), all algebras, coalgebras and Lie algebras will be assumed to be over ${\Bbbk}$ (eventually, all modules will have an underlying structure of central ${\Bbbk}$-modules. That is, the left and right actions satisfy $k\cdot m=m\cdot k$ for every $m\in M$ and $k\in {\Bbbk}$. Such a category of central ${\Bbbk}$-modules will be denoted by ${{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}$). Given two algebras $R$ and $S$, for an *$(S, R)$-bimodule* $M$ we denote the underlying module structures by ${}_{{\scriptscriptstyle{S}}}M$, $M_{{\scriptscriptstyle{R}}}$ and ${}_{{\scriptscriptstyle{S}}}M_{{\scriptscriptstyle{R}}}$. The *right and left duals* stand for the bimodules $M^*:= {\mathrm{Hom}_{{\scriptscriptstyle{-R}}}\left(M,\,R\right)}$ and ${}^*M:={\mathrm{Hom}_{{\scriptscriptstyle{S-}}}\left(M,\,S\right)}$, which we consider canonically as $(R, S)$-bimodule and $(S, R)$-bimodule, respectively. The unadorned tensor product $\otimes$ is understood to be over ${\Bbbk}$. For an algebra $A$, an *$A$-(co)ring* is a (co)monoid inside the monoidal category of $A$-bimodules.
We denote by ${\mathrm{CAlg}_{{\Bbbk}}}$ the category of commutative algebras. Given an algebra $A$ in ${\mathrm{CAlg}_{{\Bbbk}}}$, and two bimodules ${}_{{\scriptscriptstyle{A}}}M_{{\scriptscriptstyle{A}}}$ and ${}_{{\scriptscriptstyle{A}}}N_{{\scriptscriptstyle{A}}}$. The notations like $M_{{\scriptscriptstyle{A}}} ~{\otimes_{\scriptscriptstyle{A}}}~ {}_{{\scriptscriptstyle{A}}}N$, ${}_{{\scriptscriptstyle{A}}}M ~{\otimes_{\scriptscriptstyle{A}}}~ N_{{\scriptscriptstyle{A}}}$ or ${}_{{\scriptscriptstyle{A}}}M ~{\otimes_{\scriptscriptstyle{A}}}~ {}_{{\scriptscriptstyle{A}}}N$, are used for different tensor products over $A$, in order to specify the sides on which we are making such a tensor product. The subscript joining bimodules will be omitted in that notation whenever the sides are clear from the context. For more details on the tensor product construction, see for instance [@MR A II.50, §3].
Recall finally that algebras and bimodules over algebras form a bicategory ${{\mathcal B}}im_{{\Bbbk}}$. For the notion of bicategories and 2-functors (i.e., morphisms of bicategories) we refer the reader to [@Benabou].
Complete commutative Hopf algebroids. {#sec:CHA}
=====================================
In this section we recall explicitly the structure maps involved in the definition of complete commutative Hopf algebroids. All algebras are assume to be commutative, in particular, all Hopf algebroids are understood to be so. We will freely use the notations and notions expounded in Appendix \[sec:CBCF\]. Following the standard literature on the subject, we will assume ${\Bbbk}$ to be a field, even if the results presented here can be stated for a commutative ring in general.
Commutative Hopf algebroids: Definition and examples {#ssec:CHAlgds}
----------------------------------------------------
We recall here from [@ravenel Appendix A1] the definition of commutative Hopf algebroid. We also expound some examples which will be needed in the forthcoming sections.
A *Hopf algebroid* is a *cogroupoid* object in the category $\mathrm{CAlg}_{{\Bbbk}}$ (equivalently a groupoid in the category of affine schemes). Thus, a Hopf algebroid consists of a pair of algebras $\left( A,\mathcal{H}\right) $ together with a diagram of algebra maps: $$\label{Eq:loop}
\xymatrix@C=45pt{ A \ar@<1ex>@{->}|(.4){\scriptstyle{s}}[r] \ar@<-1ex>@{->}|(.4){\scriptstyle{t}}[r] & \ar@{->}|(.4){ \scriptstyle{\varepsilon}}[l] \ar@(ul,ur)^{\scriptstyle{{{\mathcal S}}}} {{\mathcal H}}\ar@{->}^-{\scriptstyle{\Delta}}[r] & {{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}, }$$ where to perform the tensor product, ${{\mathcal H}}$ is considered as an $A$-bimodule of the form ${}_{\scriptstyle{s}}{{\mathcal H}}_{\scriptstyle{t}}$, i.e., $A$ acts on the left through $s$ while it acts on the right through $t$. The maps $s,t:A\rightarrow \mathcal{H}$ are called the *source* and the *target* respectively, $\varepsilon :\mathcal{H}\rightarrow A$ the *counit*, $\Delta :\mathcal{H}\rightarrow \mathcal{H}{\otimes_{\scriptscriptstyle{A}}} \mathcal{H}$ the *comultiplication* and ${{\mathcal S}}:\mathcal{H}\rightarrow \mathcal{H}$ the *antipode*. These have to satisfy the following compatibility conditions.
- The datum $({{\mathcal H}}, \Delta, \varepsilon)$ has to be a coassociative and counital comonoid in ${{}_{{\scriptscriptstyle{A}}}\mathsf{Bim}{}_{{\scriptscriptstyle{A}}}}$, i.e., an $A$-coring. At the level of groupoids, this encodes a unitary and associative composition law between arrows.
- The antipode has to satisfy ${{\mathcal S}}\circ s=t$, ${{\mathcal S}}\circ t = s$ and ${{\mathcal S}}^2={\mathrm{Id}}_{{{\mathcal H}}}$, which encode the fact that the inverse of an arrow interchanges source and target and that the inverse of the inverse is the original arrow.
- The antipode has to satisfy also ${{\mathcal S}}(h_1)h_2=(t\circ \varepsilon)(h)$ and $h_1{{\mathcal S}}(h_2)=(s\circ \varepsilon)(h)$, which encode the fact that the composition of a morphism with its inverse on either side gives an identity morphism (the notation $h_1\otimes h_2$ is a variation of the Sweedler’s Sigma notation, with the summation symbol understood, and it stands for $\Delta(h)$).
Note that there is no need to require that $\varepsilon \circ s= {\mathrm{Id}}_A= \varepsilon \circ t$, as it is implied by the first condition.
A *morphism of Hopf algebroids* is a pair of algebra maps $\left( \phi _{{\scriptscriptstyle{0}}},\phi _{{\scriptscriptstyle{1}}}\right) :\left( A,\mathcal{H}\right)
\rightarrow \left( B,\mathcal{K}\right)$ such that $$\begin{aligned}
\phi _{1}\circ s =s\circ \phi _{0},&\qquad& \phi _{1}\circ t=t\circ \phi
_{0}, \\
\Delta \circ \phi _{1} =\chi \circ \left( \phi _{1}\otimes _{A}\phi
_{1}\right) \circ \Delta ,&\qquad& \varepsilon \circ \phi _{1}=\phi _{0}\circ
\varepsilon , \\
{{\mathcal S}}\circ \phi _{1} =\phi _{1}\circ {{\mathcal S}}&&\end{aligned}$$ where $\chi :\mathcal{K}{\otimes_{\scriptscriptstyle{A}}}\mathcal{K\rightarrow }\mathcal{K} {\otimes_{\scriptscriptstyle{B}}}\mathcal{K}$ is the obvious map induced by $\phi_0$, that is $\chi \left(h{\otimes_{\scriptscriptstyle{A}}}k\right) =h{\otimes_{\scriptscriptstyle{B}}}k$.
\[exm:Halgd\] Here are some common examples of Hopf algebroids (see [@ElKaoutit:2015] for more):
1. Let $A$ be an algebra. Then the pair $(A, A\otimes_{}A)$ admits a Hopf algebroid structure given by $s(a)=a\otimes_{}1$, $t(a)=1\otimes_{}a$, $S(a\otimes_{}a')=a'\otimes_{}a$, $\varepsilon(a{\otimes_{\scriptscriptstyle{}}}a')=aa'$ and $\Delta(a\otimes_{}a')= (a\otimes_{}1) {\otimes_{\scriptscriptstyle{A}}} (1\otimes_{}a')$, for any $a, a' \in A$.
2. Let $(B,\Delta, \varepsilon, {\mathscr{S}})$ be a Hopf algebra and $A$ a right $B$-comodule algebra with coaction $A \to A\otimes_{}B$, $a \mapsto a_{{\scriptscriptstyle{(0)}}} \otimes_{} a_{{\scriptscriptstyle{(1)}}}$. This means that $A$ is a right $B$-comodule and the coaction is an algebra map (see e.g. [@Montgomery:1993 §4]). Consider the algebra ${{\mathcal H}}= A\otimes_{}B$ with algebra extension $ \eta: A\otimes_{}A \to {{\mathcal H}}$, $a'\otimes_{}a \mapsto a'a_{{\scriptscriptstyle{(0)}}}\otimes_{}a_{{\scriptscriptstyle{(1)}}}$. Then $(A,{{\mathcal H}})$ has a structure of Hopf algebroid, known as a *split Hopf algebroid*: $$\Delta(a\otimes_{}b) = (a\otimes_{}b_{{\scriptscriptstyle{1}}}) \otimes_{A} (1_{{\scriptscriptstyle{A}}}\otimes_{}b_{{\scriptscriptstyle{2}}}), \;\;\varepsilon(a\otimes_{}b)=a\varepsilon(b),\;\; {{\mathcal S}}(a\otimes_{}b)= a_{{\scriptscriptstyle{(0)}}}\otimes_{} a_{{\scriptscriptstyle{(1)}}}{\mathscr{S}}(b).$$
3. Let $B$ be as in part $(2)$ and $A$ any algebra. Then $(A, A\otimes_{}B\otimes_{}A)$ admits in a canonical way a structure of Hopf algebroid. For $a,a'\in A$ and $b\in B$, its structure maps are given as follows $$\begin{gathered}
s(a)=a\otimes 1_{{\scriptscriptstyle{B}}}\otimes 1_{{\scriptscriptstyle{A}}}, \quad t(a)=1_{{\scriptscriptstyle{A}}}\otimes 1_{{\scriptscriptstyle{B}}}\otimes a, \quad \varepsilon(a\otimes b\otimes a')=aa'\varepsilon(b), \\
\Delta(a\otimes b\otimes a')= \big(a\otimes b_1\otimes 1_{{\scriptscriptstyle{A}}}\big) {\otimes_{\scriptscriptstyle{A}}} \big(1_{{\scriptscriptstyle{A}}}\otimes b_2\otimes a'\big), \quad {{\mathcal S}}(a\otimes b\otimes a')=a'\otimes {\mathscr{S}}(b)\otimes a.\end{gathered}$$
Complete commutative Hopf algebroids: Definition and examples {#ssec:CCHAlgds}
-------------------------------------------------------------
We keep the notation from Appendix \[sec:CBCF\] and we refer to the same section for all definitions and results as well.
In the spirit of [@quillen Appendix A], our next aim is to show that the completion functor from Appendix \[sec:CBCF\] now induces a functor from the category of Hopf algebroids ${\mathsf{CHAlgd}_{{\Bbbk}}}$ to that of complete Hopf algebroids ${\mathsf{CHAlgd}_{{\Bbbk}}^{\text{c}}}$ in the sense of [@MR1320989 §1], for example. To our knowledge this is not immediately clear. To perform such a construction one needs tools from the $I$-adic topology [@MR0163908 chap 0 § 7] as well as from linear topology in bimodules and their topological tensor product as retrieved in the Appendices.
Assume we are given a diagram $S \leftarrow A \rightarrow R$ of filtered algebras and consider the filtered $(R,S)$-bimodule $R{\otimes_{\scriptscriptstyle{A}}}S$ with filtration given as in Equation :
$${{\mathcal F}}_{n}(R{\otimes_{\scriptscriptstyle{A}}}S)= \sum_{p+q=n} {\mathsf{Im}\left({F_pR {\otimes_{\scriptscriptstyle{A}}} F_qS}\right)}.$$ This tensor product becomes then a filtered algebra over $A$, i.e., $R{\otimes_{\scriptscriptstyle{A}}}S$ is a filtered $A$-algebra.
Using this notion of filtered tensor product, the definition of Hopf algebroid as given in §\[ssec:CHAlgds\] can be canonically adapted to the filtered context. Thus, a pair $(A,{{\mathcal H}})$ of filtered algebras is said to be a *filtered Hopf algebroid* provided that there is a diagram as in equation , where the involved maps satisfy the compatibility conditions in the filtered sense. Morphisms between filtered Hopf algebroids are easily understood and the category so obtained will be denoted by ${\mathsf{CHAlgd}_{{\Bbbk}}}^{{\mathsf{flt}}}$. The following example shows that there is a canonical functor ${\mathsf{CHAlgd}_{{\Bbbk}}}\to {\mathsf{CHAlgd}_{{\Bbbk}}}^{{\mathsf{flt}}}$.
\[exam:Hflt\] Consider a Hopf algebroid $(A,{{\mathcal H}})$. Assume that $A$ is trivially filtered and endow $\mathcal{H}$ with the augmented filtration $F_{0}\mathcal{H=H}$ and $F_{n}\mathcal{H}:=K^{n}$ for every $n\geq 1$, where $K:={\mathrm{Ker}\left({ \varepsilon :\mathcal{H}
\rightarrow A}\right)} $ and $\mathcal{H}{\otimes_{\scriptscriptstyle{A}}}\mathcal{H}$ with the usual tensor product filtration $${{\mathcal F}}_{0}\left( \mathcal{H}{\otimes_{\scriptscriptstyle{A}}}\mathcal{H}\right) =\mathcal{H
}{\otimes_{\scriptscriptstyle{A}}}\mathcal{H},\quad {{\mathcal F}}_{n}\left( \mathcal{H}{\otimes_{\scriptscriptstyle{A}}}\mathcal{H}
\right) =\sum_{p+q=n}{\mathsf{Im}\left({K^{p}{\otimes_{\scriptscriptstyle{A}}}K^{q}}\right)}$$ for every $n\geq 1$. Then $(A,{{\mathcal H}})$ is a filtered Hopf algebroid.
As it is already known, the usual (algebraic) tensor product of complete bimodules is not necessarily a complete bimodule. Thus, in order to introduce complete Hopf algebroids, one needs to use the topological (or complete) tensor product (see Definition \[def:CTP\] and Theorem \[th:adjunction\] for more details). In this way, a *complete Hopf algebroid* is a pair $(A,{{\mathcal H}})$ of complete algebras together with a diagram of filtered algebra maps similar to the one of equation , with the exception that $\Delta: {{\mathcal H}}\to{{\mathcal H}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal H}}$, and where all the maps satisfy compatibility conditions analogous to that of a Hopf algebroid. In different but equivalent words, a complete Hopf algebroid is a cogroupoid object in the category of complete algebras. Henceforth, by complete Hopf algebroid we will always mean a *commutative* complete Hopf algebroid.
\[rem:cmptensalgebra\] A detail has to be highlighted here. Assume that $A$ and ${{\mathcal H}}$ are complete algebras and that ${{\mathcal H}}$ is also an $A$-bimodule via two structure maps $s,t:A\to{{\mathcal H}}$ as above. Then ${{\mathcal H}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal H}}$ is a complete algebra. Indeed, it can be checked that the obvious multiplication $${{\mu_{{\scriptscriptstyle{{{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}}}}} \colon {\left({{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}\right)\otimes\left({{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}\right)} \rightarrow {{{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}};\, \Big({(x{\otimes_{\scriptscriptstyle{A}}} y)\otimes (x'{\otimes_{\scriptscriptstyle{A}}}y')} \mapsto {xx'{\otimes_{\scriptscriptstyle{A}}} yy'}\Big)}$$ is well-defined and makes of ${{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}$ a filtered algebra. Since ${\widehat{(-)}}$ is monoidal (cf. Corollary \[prop:moncatcomplbimod\]), we have that ${\widehat{{{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}}}={{\mathcal H}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal H}}$ is a complete algebra with ${\widehat{\mu}}$ as a multiplication.
The following result can be seen as a consequence of Theorem \[thm:Athm\]. Nevertheless, for the convenience of reader, we outline here the proof.
\[coro:CHc\] The completion functor induces a functor $$\xymatrix{ {\mathsf{CHAlgd}_{{\Bbbk}}}^{{\mathsf{flt}}} \ar@{->}^-{{\widehat{(-)}}}[rr] & & {\mathsf{CHAlgd}_{{\Bbbk}}^{\text{c}}}}$$ to the category of complete Hopf algebroids with continuous morphisms of Hopf algebroids. In particular, we have the following composition of functors $$\xymatrix@R=15pt{ {\mathsf{CHAlgd}_{{\Bbbk}}}^{{\mathsf{flt}}} \ar@{->}^-{{\widehat{(-)}}}[rr] & & {\mathsf{CHAlgd}_{{\Bbbk}}^{\text{c}}}\\ {\mathsf{CHAlgd}_{{\Bbbk}}}\ar@{->}[u] \ar@/_1ex/@{-->}[urr] & & }$$
Let $(A,{{\mathcal H}})$ be a filtered Hopf algebroid with filtered algebra maps $s,t,\varepsilon,\Delta,{{\mathcal S}}$. In particular, ${{\mathcal H}}$ is an object in ${{}^{}_{{\scriptscriptstyle{A}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{A}}}}$. Consider ${\widehat{A}}$ and ${\widehat{\mathcal{H}}}$, which are complete modules as well as complete algebras (see Theorem \[thm:Athm\]). We have that ${\widehat{{{\mathcal H}}}}$ is an object in ${{}^{}_{{\scriptscriptstyle{{\widehat{A}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{A}}}}}}$ and we have complete algebra maps $$\begin{gathered}
{{{\widehat{s}}~,~{\widehat{t}}} \colon {{\widehat{A}}} \rightarrow {{\widehat{{{\mathcal H}}}}}},\qquad {{{\widehat{\varepsilon }}} \colon {{\widehat{{{\mathcal H}}}}} \rightarrow {
{\widehat{A}}}}, \\
{{{\widehat{\Delta }}} \colon {{\widehat{{{\mathcal H}}}}} \rightarrow {{\widehat{\left(
{{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}\right) }}\stackrel{\eqref{Eq:Psi}}{\cong}{\widehat{{{\mathcal H}}}}~{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\scriptscriptstyle{{\widehat{A}}}}}}}~}~{\widehat{{{\mathcal H}}}}}},\quad \text{and} \quad {{{\widehat{{{\mathcal S}}}}} \colon {{\widehat{{{\mathcal H}}}}} \rightarrow {{\widehat{{{\mathcal H}}}}}}.\end{gathered}$$ These maps satisfy the same axioms as the original ones, because ${\widehat{\left( -\right) }}$ is a monoidal functor by Corollary \[prop:moncatcomplbimod\]. The unique detail that needs perhaps a few words more is the antipode condition. Consider the maps ${c_l}:{{{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}}\to{{{\mathcal H}}}$ and ${c_r}:{{{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal H}}}\to{{{\mathcal H}}}$ such that $c_l\left(x{\otimes_{\scriptscriptstyle{A}}} y\right)={{\mathcal S}}(x)y$ and $c_r\left({x{\otimes_{\scriptscriptstyle{A}}} y}\right)={x{{\mathcal S}}(y)}$ respectively, for all $x,y\in{{\mathcal H}}$. These allow us to write the antipode conditions as the commutativity of the diagram $$\xymatrix @C=55pt @R=15pt {
{{\mathcal H}}& {{\mathcal H}}\otimes {{\mathcal H}}\ar[d]^-{p} \ar[r]^-{\mu~\circ~\left({{\mathcal H}}~\otimes~ {{\mathcal S}}\right)} \ar[l]_-{\mu~\circ~\left({{\mathcal S}}~\otimes~{{\mathcal H}}\right)} & {{\mathcal H}}\\
& {{\mathcal H}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal H}}\ar[ul]^-{c_l} \ar[ur]_-{c_r} & \\
A \ar[uu]^-{t} & {{\mathcal H}}\ar[l]_-{\varepsilon} \ar[r]^-{\varepsilon} \ar[u]^-{\Delta} & A \ar[uu]_-{s}
}$$ One can check that $p$, $c_l$ and $c_r$ are all filtered. Indeed, for $p$ it is immediate and for $c_l$ and $c_r$ it follows from the fact that ${{\mathcal S}}$ is filtered. We can now apply the functor ${\widehat{(-)}}$ to get a commutative diagram $$\label{Eq:TopAntip}
\xymatrix @C=55pt @R=15pt {
{\widehat{{{\mathcal H}}}} & {\widehat{{{\mathcal H}}}}~{\widehat{ \otimes}}~ {\widehat{{{\mathcal H}}}} \ar[d]^-{{\widehat{p}}} \ar[r]^-{{\widehat{\mu}}~\circ~\left({\widehat{{{\mathcal H}}}}~{\widehat{\otimes}}~ {\widehat{{{\mathcal S}}}}\right)} \ar[l]_-{{\widehat{\mu}}~\circ~\left({\widehat{{{\mathcal S}}}}~{\widehat{\otimes}}~{\widehat{{{\mathcal H}}}}\right)} & {\widehat{{{\mathcal H}}}} \\
& {\widehat{{{\mathcal H}}}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{A}}}}}~} {\widehat{{{\mathcal H}}}} \ar[ul]^-{{\widehat{c_l}}} \ar[ur]_-{{\widehat{c_r}}} & \\
{\widehat{A}} \ar[uu]^-{{\widehat{t}}} & {\widehat{{{\mathcal H}}}} \ar[l]_-{{\widehat{\varepsilon}}} \ar[r]^-{{\widehat{\varepsilon}}} \ar[u]^-{{\widehat{\Delta}}} & {\widehat{A}} \ar[uu]_-{{\widehat{s}}}
}$$ which shows that ${\widehat{{{\mathcal S}}}}$ is the antipode of ${\widehat{{{\mathcal H}}}}$. Therefore, $\left( {\widehat{A}},{\widehat{
\mathcal{H}}}\right) $ is a complete Hopf algebroid.
Let $\left( \phi _{{\scriptscriptstyle{0}}},\phi _{{\scriptscriptstyle{1}}}\right) :\left( A,\mathcal{H}\right)
\rightarrow \left( B,\mathcal{K}\right)$ be a morphism of filtered Hopf algebroids. Hence we can consider ${{{\widehat{\phi_0}}} \colon {{\widehat{A}}} \rightarrow {{\widehat{B}}}}$ and ${{{\widehat{\phi_1}}} \colon {{\widehat{\mathcal{H}}}} \rightarrow {{\widehat{\mathcal{K}}}}}$ and these are morphisms of complete algebras. Since $\chi:{{\mathcal K}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal K}}\to {{\mathcal K}}{\otimes_{\scriptscriptstyle{B}}}{{\mathcal K}}$ is filtered, $\left({\widehat{\phi _{0}}},{\widehat{\phi _{1}}}\right) $ becomes a morphism of complete Hopf algebroids by the functoriality of ${\widehat{(-)}}$. In light of Example \[exam:Hflt\], ${\widehat{(-)}}$ restricts to a functor $${{{\widehat{(-)}}} \colon {{\mathsf{CHAlgd}_{{\Bbbk}}}} \rightarrow {{\mathsf{CHAlgd}_{{\Bbbk}}^{\text{c}}}}},$$ and this finishes the proof.
\[exam:Sacarrelli\] Let $A$ be an algebra and consider the pair $(A,A{\otimes_{\scriptscriptstyle{}}}A)$ as an Hopf algebroid in a canonical way. Then the pair $(A, {\widehat{A{\otimes_{\scriptscriptstyle{}}}A}})$ is a complete Hopf algebroid by Proposition \[coro:CHc\], where $A$ is trivially filtered and $A{\otimes_{\scriptscriptstyle{}}}A$ is given the $K$-adic topology where $K:={\mathrm{Ker}\left({\mu_{A}:A{\otimes_{\scriptscriptstyle{}}} A\to A}\right)}$ is the kernel of the multiplication. The complete algebra ${\widehat{A{\otimes_{\scriptscriptstyle{}}}A}}$ is known under the name *algebra of infinite jets*. This terminology is justified by looking at the case when $A=C^{\infty}({{\mathcal M}})$. Namely, in this case, ${\widehat{A{\otimes_{\scriptscriptstyle{}}}A}}$ coincides with the inverse limit of the smooth global sections $\Gamma({{\mathcal J}}^{l}({{\mathcal M}}))$ of the bundle of $l$-jets of functions on ${{\mathcal M}}$, see [@nestruev §11.46], [@Krasilshchik:1997 Proposition 9.4 (iv)].
The complete commutative Hopf algebroid structure of the convolution algebra {#sec:Ustar}
============================================================================
The main task of this section is to show that the convolution algebra of a given (right) co-commutative Hopf algebroid, which is endowed with an admissible filtration (see subsection \[ssec:AFiltration\]) and whose translation map is a filtered algebra map, is a complete Hopf algebroid in the sense of subsection \[ssec:CCHAlgds\]. At the level of topological (right) bialgebroids this was mentioned in [@Kapranov:2007 A.5]. However, it seems that the literature is lacking a precise construction for a topological antipode. Here we will supply this by providing the explicit description of all the involved maps in the complete Hopf algebroid structure of the convolution algebra. Such a description will be crucial in proving the results of the forthcoming section. The prototype example, which we have in mind and which fulfils the above assumptions, is the convolution algebra of the universal Hopf algebroid of a finitely generated and projective Lie-Rinehart algebra.
Co-commutative Hopf algebroids: Definition and examples {#ssec:CoHAlg}
-------------------------------------------------------
Next, we recall the definition of a (right) co-commutative Hopf algebroid (see for instance [@Kapranov:2007 A.3.6], compare also with [@Kowalzig Definition 2.5.1]). A *(right) co-commutative Hopf algebroid* over a commutative algebra is the datum of a commutative algebra $A$, a possibly noncommutative algebra ${{\mathcal U}}$ and an algebra map $s=t: A\to {{\mathcal U}}$ landing not necessarily in the center of ${{\mathcal U}}$, with the following additional structure maps:
- A morphism of right $A$-modules $\varepsilon: {{\mathcal U}}\to A$ which satisfies $\varepsilon(uv)=\varepsilon(\varepsilon(u)v)$, for all $u, v \in {{\mathcal U}}$;
- An $A$-ring map $\Delta: {{\mathcal U}}\to {{\mathcal U}}\times_{{\scriptscriptstyle{A}}} {{\mathcal U}}$, where the module $${{\mathcal U}}\times_{{\scriptscriptstyle{A}}} {{\mathcal U}}:=\left\{ \sum_{{\scriptscriptstyle{i}}} u_{{\scriptscriptstyle{i}}}{\otimes_{\scriptscriptstyle{A}}} v_{{\scriptscriptstyle{i}}} \in {{\mathcal U}}_{{\scriptscriptstyle{A}}}~ {\otimes_{\scriptscriptstyle{A}}} ~{{\mathcal U}}_{{\scriptscriptstyle{A}}} \mid~ \sum_{{\scriptscriptstyle{i}}} au_{{\scriptscriptstyle{i}}}{\otimes_{\scriptscriptstyle{A}}} v_{{\scriptscriptstyle{i}}}= \sum_{{\scriptscriptstyle{i}}} u_{{\scriptscriptstyle{i}}}{\otimes_{\scriptscriptstyle{A}}} av_{{\scriptscriptstyle{i}}} \right\}$$ is endowed with the algebra structure $$\sum_{{\scriptscriptstyle{i}}} u_{{\scriptscriptstyle{i}}}\times_{{\scriptscriptstyle{A}}} v_{{\scriptscriptstyle{i}}} ~ .~ \sum_{{\scriptscriptstyle{j}}} u'_{{\scriptscriptstyle{j}}}\times_{{\scriptscriptstyle{A}}}v'_{{\scriptscriptstyle{j}}} = \sum_{{\scriptscriptstyle{i,j}}} u_{{\scriptscriptstyle{i}}}u'_{{\scriptscriptstyle{j}}} \times_{{\scriptscriptstyle{A}}} v_{{\scriptscriptstyle{i}}}v'_{{\scriptscriptstyle{j}}}, \qquad 1_{{{\mathcal U}}\times_{{\scriptscriptstyle{A}}} {{\mathcal U}}}=1_{{{\mathcal U}}}{\otimes_{\scriptscriptstyle{A}}}1_{{{\mathcal U}}}$$ and the $A$-ring structure given by the algebra map $1: A \rightarrow {{\mathcal U}}\times_{{\scriptscriptstyle{A}}}{{\mathcal U}},~ \Big( a \mapsto a \times_{{\scriptscriptstyle{A}}} 1_{{\scriptscriptstyle{{{\mathcal U}}}}}= 1_{{\scriptscriptstyle{{{\mathcal U}}}}} \times_{{\scriptscriptstyle{A}}} a \Big)$;
subject to the conditions
- $\Delta$ is coassociative, co-commutative in a suitable sense and has $\varepsilon$ as a right and left counit;
- the canonical map $$\label{Eq:beta}
{{\beta} \colon {{{\mathcal U}}_{{\scriptscriptstyle{A}}} ~{\otimes_{\scriptscriptstyle{A}}}~ {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}} \longrightarrow {{{\mathcal U}}_{{\scriptscriptstyle{A}}}~{\otimes_{\scriptscriptstyle{A}}}~ {{\mathcal U}}_{{\scriptscriptstyle{A}}}};\quad \Big({u{\otimes_{\scriptscriptstyle{A}}}v} \longmapsto {uv_{{\scriptscriptstyle{1}}}{\otimes_{\scriptscriptstyle{A}}}v_{{\scriptscriptstyle{2}}}}\Big)}$$ is bijective, where we denoted $\Delta(v)=v_{{\scriptscriptstyle{1}}}\otimes_{A}v_{{\scriptscriptstyle{2}}}$ (summations understood). As a matter of terminology, the map $\beta^{-1}(1{\otimes_{\scriptscriptstyle{A}}}-):{{\mathcal U}}\to {{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}{_{{\scriptscriptstyle{A}}}{{\mathcal U}}}$ is the so-called *translation map*. The following is a standard notation: $$\label{eq:amore}
\delta(u):=\beta^{-1}(1{\otimes_{\scriptscriptstyle{A}}}u)=u_-{\otimes_{\scriptscriptstyle{A}}}u_+ \quad\textrm{(summation understood)}.$$
Right $A$-linear maps ${{\mathcal U}}_{{\scriptscriptstyle{A}}} \to A$ form the module ${{\mathcal U}}^{*}$ with a structure of algebra given by the *convolution product* $$\label{Eq:convolution}
f *g: {{\mathcal U}}\longrightarrow A, \quad \Big( u \longmapsto f(u_{{\scriptscriptstyle{1}}}) g(u_{{\scriptscriptstyle{2}}}) \Big).$$ It comes endowed with an algebra map $$\label{Eq:vartheta}
\vartheta=s_*\otimes_{}t_*: A\otimes_{}A \longrightarrow {{\mathcal U}}^{*}, \quad \Big( a'\otimes_{}a \longmapsto \left[ u \mapsto \varepsilon(a u)a' \right] \Big).$$
\[exam:URSO\] Let $A=\Bbbk[X]$ be the polynomial algebra with one variable. Consider its associated first Weyl algebra ${{\mathcal U}}:=A[Y, \partial/\partial X]$, that is, its differential operators algebra. Then the pair $(A,{{\mathcal U}})$ is a (right) co-commutative Hopf algebroid with structure maps: $$\Delta(Y)\,=\, 1{\otimes_{\scriptscriptstyle{A}}}Y+ Y{\otimes_{\scriptscriptstyle{A}}}1,\quad \varepsilon(Y)\,=\, 0, \quad Y_{{\scriptscriptstyle{-}}}{\otimes_{\scriptscriptstyle{A}}} Y_{{\scriptscriptstyle{+}}}\,=\, 1{\otimes_{\scriptscriptstyle{A}}}Y - Y{\otimes_{\scriptscriptstyle{A}}}1.$$
The universal enveloping Hopf algebroid of a Lie-Rinehart algebra {#ssec:ULA}
-----------------------------------------------------------------
Let $A$ be a commutative algebra over a field ${\Bbbk}$ of characteristic $0$ and denote by $\mathrm{Der}_{{\scriptscriptstyle{{\Bbbk}}}}(A)$ the Lie algebra of all linear derivations of $A$. Consider a Lie algebra $L$ which is also an $A$-module, and let $\omega: L \to \mathrm{Der}_{{\scriptscriptstyle{{\Bbbk}}}}(A)$ be an $A$-linear morphism of Lie algebras. Following [@Rin:DFOGCA], the pair $(A,L)$ is called a *Lie-Rinehart algebra* with *anchor* map $\omega$ provided that $${[ X, aY ]} = a{[X,Y]}+X(a)Y,$$ for all $X, Y \in L$ and $a, b \in A$, where $X(a)$ stands for $\omega(X)(a)$.
Apart from the natural examples $(A,\mathrm{Der}_{{\scriptscriptstyle{{\Bbbk}}}}(A))$ (with anchor the identity map), another basic source of examples are the smooth global sections of a given Lie algebroid over a smooth manifold.
\[exam:VayaCon\] A *Lie algebroid* is a vector bundle $\mathcal{L} \to \mathcal{M}$ over a smooth manifold, together with a map $\omega: \mathcal{L} \to T\mathcal{M}$ of vector bundles and a Lie structure $[-,-]$ on the vector space $\Gamma({{\mathcal L}})$ of global smooth sections of $\mathcal{L}$, such that the induced map $\Gamma(\omega): \Gamma({{\mathcal L}}) \to \Gamma(T\mathcal{M})$ is a Lie algebra homomorphism, and for all $X, Y \in \Gamma({{\mathcal L}})$ and any $f \in \mathcal{C}^{\infty}(\mathcal{M})$ one has $$\label{eq:LieAlgd}
[X,fY]\,=\, f[X,Y]+ \Gamma(\omega)(X)(f)Y.$$ Then the pair $(\mathcal{C}^{\infty}(\mathcal{M}), \Gamma({{\mathcal L}}))$ is obviously a Lie-Rinehart algebra.
As it has been pointed out by the referee, the fact that the map $\Gamma(\omega): \Gamma({{\mathcal L}}) \to \Gamma(T\mathcal{M})$ in Example \[exam:VayaCon\] is a Lie algebra homomorphism is a consequence of the Jacobi identity and of Relation (see [e.g. ]{}[@Grabowski; @Herz; @Magri]). Therefore, it should be omitted from the definition of a Lie algebroid. Nevertheless, we decided to keep the somewhat redundant definition above to make it easier for the unaccustomed reader to see the parallel with Lie-Rinehart algebras.
The *universal enveloping Hopf algebroid* of a Lie-Rinehart algebra is an algebra $ {{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right) $ endowed with a morphism $\iota _{{\scriptscriptstyle{A}}}:A\rightarrow {{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right) $ of algebras and an $A$-linear Lie algebra morphism $\iota _{{\scriptscriptstyle{L}}}:L\rightarrow {{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right) $ such that $$\label{eq:compLRalg}
\iota _{{\scriptscriptstyle{L}}}\left(X\right) \iota _{{\scriptscriptstyle{A}}}\left( a\right) - \iota _{{\scriptscriptstyle{A}}}\left( a\right) \iota _{{\scriptscriptstyle{L}}}\left( X\right)=\iota _{{\scriptscriptstyle{A}}}\left( X
\left( a\right) \right)$$ for all $a\in A$ and $X\in L$, which is universal with respect to this property. In details, this means that if $\left( W,\phi _{{\scriptscriptstyle{A}}},\phi _{{\scriptscriptstyle{L}}}\right) $ is another algebra with a morphism $\phi _{{\scriptscriptstyle{A}}}:A\rightarrow W$ of algebras and a morphism $\phi _{{\scriptscriptstyle{L}}}:L\rightarrow W$ of Lie algebras and $A$-modules such that $$\phi _{{\scriptscriptstyle{L}}}\left( X\right)\phi _{{\scriptscriptstyle{A}}}\left( a\right) -\phi _{{\scriptscriptstyle{A}}}\left( a\right) \phi _{{\scriptscriptstyle{L}}}\left( X\right)=\phi _{{\scriptscriptstyle{A}}}\left( X \left(a\right) \right) ,$$ then there exists a unique algebra morphism $\Phi : {{\mathcal V}}_{{\scriptscriptstyle{A}}}
\left( L\right) \rightarrow W$ such that $\Phi \, \iota _{{\scriptscriptstyle{A}}}=\phi _{{\scriptscriptstyle{A}}}$ and $
\Phi \, \iota _{{\scriptscriptstyle{L}}}=\phi _{{\scriptscriptstyle{L}}}$.
Apart from the well-known constructions of [@Rin:DFOGCA] and [@MoerdijkLie], the universal enveloping Hopf algebroid of a Lie-Rinehart algebra $(A,L)$ admits several other equivalent realizations. In this paper we opted for the following. Set $\eta :L\rightarrow A\otimes L;\, X\longmapsto 1_{{\scriptscriptstyle{A}}}\otimes X$ and consider the tensor $A$-ring $T_{{\scriptscriptstyle{A}}}\left( A\otimes L\right) $ of the $A$-bimodule $A\otimes L$. It can be shown that $${{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right) \cong \frac{T_{{\scriptscriptstyle{A}}}\left( A\otimes L\right) }{{{\mathcal J}}}$$ where the two sided ideal ${{\mathcal J}}$ is generated by the set
$${{\mathcal J}}:=\left\langle \left.
\begin{array}{c}
\eta \left( X\right) \otimes _{{\scriptscriptstyle{A}}}\eta \left( Y\right) -\eta \left( Y\right) \otimes _{{\scriptscriptstyle{A}}}\eta \left( X\right) -\eta \left( \left[ X,Y\right] \right) , \\
\eta \left( X\right) \cdot a- a\cdot \eta \left( X\right) -\omega \left( X\right)\left( a\right)
\end{array}
\right| X,Y\in L, \; a\in A\right\rangle$$
We have the algebra morphism $\iota _{{\scriptscriptstyle{A}}}:A\rightarrow
{{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right);\, a\longmapsto a+{{\mathcal J}}$ and the right $A$-linear Lie algebra map $\iota _{{\scriptscriptstyle{L}}}:L\rightarrow {{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right);\, X\longmapsto \eta \left( X\right) +{{\mathcal J}}$ that satisfy the compatibility condition . It turns out that ${{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ is a co-commutative right $A$-Hopf algebroid with structure maps induced by the assignment $$\begin{gathered}
\varepsilon \left( \iota _{{\scriptscriptstyle{A}}}\left( a\right) \right) =a, \qquad \varepsilon \left( \iota _{{\scriptscriptstyle{L}}}\left( X\right) \right) =0, \\
\Delta \left( \iota _{{\scriptscriptstyle{A}}}\left( a\right) \right) =\iota _{{\scriptscriptstyle{A}}}\left( a\right) \times _{{\scriptscriptstyle{A}}}1_{{\scriptscriptstyle{ {{\mathcal V}}_{{\scriptscriptstyle{A}}}\left( L\right)}}
}=1_{ {\scriptscriptstyle{{{\mathcal V}}_A( L)}} }\times _{{\scriptscriptstyle{A}}}\iota _{{\scriptscriptstyle{A}}}\left( a\right) , \\
\Delta \left( \iota _{{\scriptscriptstyle{L}}}\left( X\right) \right) =\iota _{{\scriptscriptstyle{L}}}\left( X\right) \times _{{\scriptscriptstyle{A}}}1_{ {\scriptscriptstyle{{{\mathcal V}}_A\left( L\right)}}
}+1_{{\scriptscriptstyle{{{\mathcal V}}_A\left( L\right)}} }\times _{{\scriptscriptstyle{A}}}\iota _{{\scriptscriptstyle{L}}}\left( X\right) , \\
\beta^{-1}\left( 1_{{\scriptscriptstyle{ {{\mathcal V}}_A\left( L\right) }}} {\otimes_{\scriptscriptstyle{A }}} \iota_{{\scriptscriptstyle{A}} }\left( a \right)\right) = \iota _{{\scriptscriptstyle{A}}}\left( a\right) {\otimes_{\scriptscriptstyle{A}}}1_{{\scriptscriptstyle{ {{\mathcal V}}_A\left( L\right)}} }=1_{{\scriptscriptstyle{ {{\mathcal V}}_A\left( L\right) }}}{\otimes_{\scriptscriptstyle{A}}}\iota _{{\scriptscriptstyle{A}}}\left( a\right) , \\
\beta^{-1}\left( 1_{ {\scriptscriptstyle{{{\mathcal V}}_A\left( L\right) }}} {\otimes_{\scriptscriptstyle{A }}} \iota_{{\scriptscriptstyle{L }}}\left( X \right)\right) = 1_{ {\scriptscriptstyle{{{\mathcal V}}_A\left( L\right) }}}{\otimes_{\scriptscriptstyle{A}}}\iota _{{\scriptscriptstyle{L}}}\left( X\right) -\iota
_{{\scriptscriptstyle{L}}}\left( X\right) {\otimes_{\scriptscriptstyle{A}}}1_{{\scriptscriptstyle{ {{\mathcal V}}_A\left( L\right)}} }.\end{gathered}$$ Another realization for ${{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ can be obtained as a quotient of the smash product (right) $A$-bialgebroid $A\#U_{{\Bbbk}}(L)$, as introduced firstly by Sweedler in [@sweedler], by the ideal ${{\mathcal I}}:=\langle a\#X- 1\#aX \mid a\in A, X\in L\rangle$.
\[exam:laventana\] The first Weyl algebra considered in Example \[exam:URSO\], is in fact the universal enveloping Hopf algebroid of the Lie-Rinehart algebra $(A, \mathrm{Der}_{{\scriptscriptstyle{{\Bbbk}}}}(A))$, where $A={\Bbbk}[X]$ is the polynomial algebra.
Admissible filtrations on general rings. {#ssec:AFiltration}
----------------------------------------
Let ${{\mathcal U}}$ be an $A$-ring or, equivalently, let $A \to {{\mathcal U}}$ be a ring extension[^2]. Throughout this subsection we assume $A$ to be trivially filtered (i.e., $F^{n}A=A$ for all $ n\geq 0$, zero otherwise) and we refer to Appendix \[sec:TGFr\] for a more detailed treatment in the framework of filtered bimodules. Following [@Kapranov:2007 Definition A.5.1], we say that ${{\mathcal U}}$ has a *right admissible filtration*, if $A \subset {{\mathcal U}}$ (as a ring) and there is an increasing filtration ${{\mathcal U}}=\bigcup_{{\scriptscriptstyle{n \, \in \, {\mathbb{N}}}}} F^{n}{{\mathcal U}}$ of $A$-subbimodules such that $F^0{{\mathcal U}}=A$, $F^n{{\mathcal U}}\,.\, F^m{{\mathcal U}}\subseteq F^{n+m}{{\mathcal U}}$ and each one of the quotient modules in $\big\{F^n{{\mathcal U}}/F^{n-1}{{\mathcal U}}\big\}_{{\scriptscriptstyle{n \geq 0}}}$ is a finitely generated and projective right $A$-module. We will denote by ${\tau_{{\scriptscriptstyle{n}}}}: F^n{{\mathcal U}}\to {{\mathcal U}}$ and by ${{\tau}_{{\scriptscriptstyle{n,\,m}}}^{{\scriptscriptstyle{}}}}:F^n{{\mathcal U}}\to F^{m}{{\mathcal U}}$ the canonical inclusions for $m\geq n\geq 0$. Notice that ${{\mathcal U}}$ can be identified with the direct limit of the system $\{F^n{{\mathcal U}},\tau{{\scriptscriptstyle{n,m}}}\}$, that is: ${{\mathcal U}}={\varinjlim_{n}\left({F^n{{\mathcal U}}}\right)}$. Thus, the underlying $A$-bimodule ${}_{{\scriptscriptstyle{A}}}{{\mathcal U}}_{{\scriptscriptstyle{A}}}$ is an increasingly filtered $A$-bimodule which is locally finitely generated and projective as a right $A$-module by definition (see §\[ssec:LFGr\]). The following Proposition summarizes the properties of rings with an admissible filtration.
\[prop:FnUfgp\] Let ${{\mathcal U}}$ be an $A$-ring endowed with a right admissible filtration $\left\{F^n{{\mathcal U}}\mid n\geq 0\right\}$. The following properties hold true
1. Each of the structural maps $\tau_{{\scriptscriptstyle{n,n+1}}}: F^n{{\mathcal U}}\to F^{n+1}{{\mathcal U}}$ is a split monomorphism of right $A$-modules. In particular, each of the submodules $F^n{{\mathcal U}}$ is a finitely generated and projective right $A$-module and each of the monomorphisms ${\tau_{{\scriptscriptstyle{n}}}}: F^n{{\mathcal U}}\to {{\mathcal U}}$ splits in right $A$-modules with retraction ${\theta_{{\scriptscriptstyle{n}}}}$.
2. As a filtered right $A$-module, ${{\mathcal U}}$ satisfies $${{\mathcal U}}\cong gr({{\mathcal U}})\,=\, A\oplus \frac{F_1{{\mathcal U}}}{A}\oplus \frac{F_2{{\mathcal U}}}{F_1{{\mathcal U}}}\oplus \cdots \oplus \frac{F_{n}{{\mathcal U}}}{F_{n-1}{{\mathcal U}}}\oplus \cdots$$ In particular ${{\mathcal U}}$ is a faithfully flat right $A$-module.
Follows from Lemma \[lemma:fgpquotients\], Corollary \[coro:FnL\] and Remark \[rem:vartheta\] of the Appendices. The faithfully flatness is a consequence of the fact that ${{\mathcal U}}$ is the direct sum of the faithfully flat right $A$-module $A$ and the flat right $A$-module $\bigoplus_{n\in{\mathbb{N}}}F_{n+1}{{\mathcal U}}/F_n{{\mathcal U}}$ (see e.g. [@Bou:AC12 Proposition 9, I §3]).
\[rem:dualbasis\] Given a right admissible filtration $\left\{F^n{{\mathcal U}}\mid n\geq 0\right\}$ on an $A$-ring ${{\mathcal U}}$, it follows from Lemma \[lemma:fgpquotients\] that we have right $A$-linear isomorphisms $\psi_n:F^n{{\mathcal U}}\cong\bigoplus_{k=0}^n F^k{{\mathcal U}}/F^{k-1}{{\mathcal U}}$ for all $n\geq 0$. Let us fix a dual basis $\left\{\left(u_i^n,\gamma_i^n\right)\mid i=1,\ldots,d'_n\right\}$ for every $F^n{{\mathcal U}}/F^{n-1}{{\mathcal U}}$, $n\geq 0$. These induce a distinguished dual basis $\left\{\left(e_i^n,\lambda_i^n\right)\mid i=1,\ldots,d_n:=\sum_{j=0}^nd'_j\right\}$ on $F^n{{\mathcal U}}$ for all $n\geq 0$, which is given as follows. The generating set $\left\{e_i^n\mid i=1,\ldots,d_n\right\}$ is given by $\left\{\psi_n^{-1}\left(u_i^k\right)\mid k=0,\ldots,n, i=1,\ldots,d'_k\right\}$, that is $\psi_n^{-1}\left(u_i^k\right)=e_{i+d_{k-1}}^n$ for all $0\leq k\leq n$ and all $1\leq i\leq d_k'$ ($d_{-1}=0$ by convention). The dual elements are given by extending $\gamma_i^k:F^k{{\mathcal U}}/F^{k-1}{{\mathcal U}}\to A$ to $\left(\gamma'\right)_i^k:\bigoplus_{k=0}^nF^k{{\mathcal U}}/F^{k-1}{{\mathcal U}}\to A$, letting $\left(\gamma'\right)_i^k\mid_{F^h{{\mathcal U}}/F^{h-1}{{\mathcal U}}}=0$ for $h\neq k$, and then composing with $\psi_n$, i.e., $\left(\gamma'\right)_i^k\circ\psi_n=\lambda^n_{i+d_{k-1}}$ for all $k,i$ as above.
This distinguished dual basis has the following useful property which will be helpful in the sequel. Let us denote by $\tau_{m,n}\colon F^{m}{{\mathcal U}}\to F^n{{\mathcal U}}$ and $j_{m,n}\colon \bigoplus_{h=0}^m F^h{{\mathcal U}}/F^{h-1}{{\mathcal U}}\to \bigoplus_{k=0}^n F^k{{\mathcal U}}/F^{k-1}{{\mathcal U}}$ the inclusion morphisms for $m\leq n$. Then $\psi_n\circ \tau_{m,n}=j_{m,n}\circ \psi_m$, whence $\tau_{m,n}\left(e^m_i\right)=e^n_i$ for all $i=1,\ldots,d_m$ and $\lambda^n_i\circ\tau_{m,n}=\lambda^m_i$ if $i=1,\ldots,d_m$, $\lambda^n_i\circ\tau_{m,n}=0$ otherwise.
A filtration on the convolution algebra ${{\mathcal U}}^*$ of a co-commutative Hopf algebroid {#ssec:FUstra}
---------------------------------------------------------------------------------------------
The idea for this example came to us from [@Kapranov:2007 A.5.8]. Let $(A,{{\mathcal U}})$ be a co-commutative (right) Hopf algebroid. We say that $(A,{{\mathcal U}})$ is endowed with a *(right) admissible filtration* if the $A$-ring ${{\mathcal U}}$ has a right admissible filtration ${{\mathcal U}}=\bigcup_{{\scriptscriptstyle{n \, \in \, {\mathbb{N}}}}} F^{n}{{\mathcal U}}$ as in §\[ssec:AFiltration\], which is also compatible with the comultiplication in the sense that it satisfies $$\Delta(F^n{{\mathcal U}}) \subseteq \sum_{p+q=n}{\mathsf{Im}\left({F^p{{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} F^q{{\mathcal U}}_{{\scriptscriptstyle{A}}}}\right)}=\sum_{p+q=n}F^p{{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} F^q{{\mathcal U}}_{{\scriptscriptstyle{A}}}=F^n\left({{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}\right)$$ (the counit is automatically filtered). The inclusion ${{\tau_0} \colon {A} \rightarrow {{{\mathcal U}}}}$ plays the role of the morphisms $s=t$.
\[exm:ULAdmiss\] As in §\[ssec:ULA\], consider the universal enveloping Hopf algebroid ${{\mathcal U}}:={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ of a given Lie-Rinehart algebra $(A,L)$. Since the tensor $A$-ring $T_{{\scriptscriptstyle{A}}}\left( A\otimes _{{\Bbbk}}L\right) $ is endowed with an increasing filtration: $$F^{n}\left( T_{{\scriptscriptstyle{A}}}\left( A{\otimes_{\scriptscriptstyle{{\Bbbk}}}}L\right) \right)
:=\bigoplus_{k=0}^{n}T_{{\scriptscriptstyle{A}}}\left( A{\otimes_{\scriptscriptstyle{{\Bbbk}}}}L\right) ^{k},$$ where $T_{{\scriptscriptstyle{A}}}\left( A{\otimes_{\scriptscriptstyle{{\Bbbk}}}}L\right) ^{k}:=\left( A{\otimes_{\scriptscriptstyle{{\Bbbk}}}}L\right) {\otimes_{\scriptscriptstyle{A}}}\left( A{\otimes_{\scriptscriptstyle{{\Bbbk}}}}L\right) {\otimes_{\scriptscriptstyle{A}}}\cdots
{\otimes_{\scriptscriptstyle{A}}}\left( A{\otimes_{\scriptscriptstyle{{\Bbbk}}}}L\right)$ for $k$ times, this induces a filtration on ${{\mathcal U}}$ via the canonical projection. Thus, the $n$-th term of the filtration $F^n{{\mathcal U}}$ is the right $A$-module generated by the products of the images of elements of $L$ in ${{\mathcal U}}$ of length at most $n$. If we assume as usual that $A$ is trivially filtered then both $\iota _{A}:A\rightarrow \mathcal{U}$ and $\varepsilon :\mathcal{U} \rightarrow A$ are filtered. Moreover, from $$\Delta \left( \iota _{L}\left( X\right) \right) =\iota _{L}\left( X\right)
{\otimes_{\scriptscriptstyle{A}}}1_{\mathcal{U} }+1_{\mathcal{U}
}{\otimes_{\scriptscriptstyle{A}}}\iota _{L}\left( X\right)
\, \in \, {\mathsf{Im}\left({F^{{\scriptscriptstyle{1}}}\mathcal{U}
{\otimes_{\scriptscriptstyle{A}}}F^{{\scriptscriptstyle{0}}}\mathcal{U}}\right)} +{\mathsf{Im}\left({F^{{\scriptscriptstyle{0}}}\mathcal{U}
{\otimes_{\scriptscriptstyle{A}}}F^{{\scriptscriptstyle{1}}}\mathcal{U}}\right)} =F^{{\scriptscriptstyle{1}}}\left( \mathcal{U} {\otimes_{\scriptscriptstyle{A}}}\mathcal{U} \right)$$ it follows that $$\Delta \left( F^{n}\mathcal{U} \right) \subseteq
\sum_{k=0}^{n}\Delta \left( \iota _{L}\left( L\right) ^{k}\right) \subseteq
\sum_{k=0}^{n}\Delta \left( \iota _{L}\left( L\right) \right) ^{k}\subseteq
\sum_{k=0}^{n}F^{k}\left( \mathcal{U} {\otimes_{\scriptscriptstyle{A}}}\mathcal{U} \right) \subseteq F^{n}\left( \mathcal{U}
{\otimes_{\scriptscriptstyle{A}}}\mathcal{U} \right)$$ (notice that this makes sense since $\mathrm{Im}\left( \Delta \right)
\subseteq \left. \mathcal{U} \right. \times
_{A}\left. \mathcal{U} \right. $, which is a filtered algebra with filtration induced by the one of $\mathcal{U} {\otimes_{\scriptscriptstyle{A}}}\mathcal{U}$). Summing up, $\mathcal{U} $ is a filtered co-commutative Hopf algebroid. Furthermore, it is well-known that if $L$ is a projective right $A$-module, then we have a graded isomorphism of $A$-algebras $\mathrm{gr}\mathcal{U} \cong S_{{\scriptscriptstyle{A}}}\left( L\right)$, the symmetric algebra of $L_{{\scriptscriptstyle{A}}}$ (see e.g. [@Rin:DFOGCA Theorem 3.1]). From this, one deduces that if $L$ is also finitely generated as right $A$-module, then the quotient modules $F^{n}\mathcal{U} /F^{n-1}
\mathcal{U} $ are finitely generated and projective as right $
A$-modules as well. Therefore, under these additional hypotheses, $\mathcal{U}$ turns out to be a co-commutative right Hopf algebroid endowed with an admissible filtration.
We are going to show in the forthcoming subsections that the convolution algebra ${{\mathcal U}}^*$ of a co-commutative right Hopf algebroid $(A,{{\mathcal U}})$ with an admissible filtration is a projective limit of algebras and a complete Hopf algebroid, where the comultiplication $\Delta_*: {{\mathcal U}}^{*} \to {{\mathcal U}}^{*}_{{\scriptscriptstyle{A}}}~ {~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}~{^{}_{{\scriptscriptstyle{A}}}{{\mathcal U}}^*}$ is induced by the multiplication of ${{\mathcal U}}$, $A$ is trivially filtered and ${{\mathcal U}}^{*}$ is considered as an $A$-bimodule via the source and the target induced by the algebra map $\vartheta: A{\otimes_{\scriptscriptstyle{}}}A \to {{\mathcal U}}^*$ of equation . The counit will be the map ${{\varepsilon_*} \colon {{{\mathcal U}}^*} \rightarrow {A}}$ such that $f \mapsto f(1)$. The construction of the antipode for ${{\mathcal U}}^*$ will require an additional hypothesis and it will be treated separately in §\[ssec:AUstra\].
Notice that ${{\mathcal U}}^*\cong {\varprojlim_{n}\left({F^n{{\mathcal U}}^*}\right)}$ as $A$-bimodules via the isomorphism $$\label{eq:iso*}
{{\Phi} \colon {{{\mathcal U}}^*} \rightarrow {{\varprojlim_{n}\left({(F^n{{\mathcal U}})^*}\right)}};\, \Big({f} \mapsto {\left(\tau_n^*(f)\right)_{n\geq 0}}\Big)};\;\; {\varprojlim_{n}\left({(F^n{{\mathcal U}})^*}\right)} \rightarrow {{\mathcal U}}^*; \; \left( (g_n)_{{\scriptscriptstyle{n \geq 0}}} \mapsto g:={\varinjlim_{n}\left({g_n}\right)} \right),$$ and it can be endowed with a natural decreasing filtration $$\label{Eq:FUstar}
F_0{{\mathcal U}}^*:={{\mathcal U}}^* \quad \text{ and } \quad F_{n+1}{{\mathcal U}}^*:={\mathrm{Ker}\left({{\tau_{{\scriptscriptstyle{n}}}}^*}\right)}=\mathsf{Ann}(F^n{{\mathcal U}})$$ (see §\[ssec:DLF\] for the general case). Besides, ${{\mathcal U}}^*$ is also a projective limit of $(A\otimes_{}A)$-algebras, where the projective system is endowed with the algebra maps $\vartheta_n=(\tau_n)^* \circ \vartheta: A\otimes_{}A \to (F^n{{\mathcal U}})^*$ and where $(F^n{{\mathcal U}})^*$ is the convolution algebra of the $A$-coalgebra $(F^n{{\mathcal U}})_{{\scriptscriptstyle{A}}}$.
\[lemma:U\*complete\] The pair $\left({{\mathcal U}}^*,F_n{{\mathcal U}}^*\right)$ gives a complete $A$-bimodule as well as a complete algebra.
In §\[ssec:DLF\] it is proved that it is a complete $A$-bimodule and so a complete module over ${\Bbbk}$ as well. In light of Remark \[rem:calg\], it is enough for us to prove that the filtration $F_n{{\mathcal U}}^*$ is compatible with the convolution product to conclude the proof. Notice that the ${\mathsf{Ann}\left({F^n{{\mathcal U}}}\right)}$’s are ideals, whence we have that $F_n{{\mathcal U}}^**F_m{{\mathcal U}}^*\subseteq F_{n+m}{{\mathcal U}}^*$ whenever $n$ or $m$ is $0$. If $mn>0$ then, given $f\in F_n{{\mathcal U}}^*$ and $g\in F_m{{\mathcal U}}^*$, we have that $(f*g)\left(F^{m+n-1}{{\mathcal U}}\right)\subseteq \sum_{p+q=n+m-1}f\left(F^p{{\mathcal U}}\right)g\left(F^q{{\mathcal U}}\right)=0$, because when $p\geq n$ it happens that $q=m+n-1-p\leq m-1$. Therefore, $f*g\in {\mathsf{Ann}\left({F^{n+m-1}{{\mathcal U}}}\right)}=F_{n+m}{{\mathcal U}}^*$ and hence $F_n{{\mathcal U}}^**F_m{{\mathcal U}}^*\subseteq F_{n+m}{{\mathcal U}}^*$ for all $m,n\geq 0$. Recalling that we consider $A$ trivially filtered, we have that $F_n{{\mathcal U}}^*$ induces on ${{\mathcal U}}^*$ a filtration as a algebra and as an $A$-bimodule at the same time.
\[rem:Kstar\] We already know that the convolution algebra ${{\mathcal U}}^*$ is an augmented one and the augmentation is given by the algebra map (which is going to be the counit) $\varepsilon_*: {{\mathcal U}}^* \to A$, $f \mapsto f(1)$. Therefore, one can consider the ${{\mathcal I}}$-adic topology on ${{\mathcal U}}^*$ with respect to the two-sided ideal ${{\mathcal I}}:={\mathrm{Ker}\left({\varepsilon_*}\right)}$. However, in the filtration of Lemma \[lemma:U\*complete\] we have ${{\mathcal I}}=F_1{{\mathcal U}}^*$, and so ${{\mathcal I}}^n \subseteq F_n{{\mathcal U}}^*$, for every $n \geq 0$. Thus the ${{\mathcal I}}$-adic topology is finer than the linear topology obtained from the filtration $\{F_n{{\mathcal U}}^*\mid n \geq 0\}$ of Equation .
The topological comultiplication and counit of ${{\mathcal U}}^*$ {#ssec:CUUstar}
-----------------------------------------------------------------
Next we want to show that the multiplication ${{\mu} \colon {{{\mathcal U}}\otimes{{\mathcal U}}} \rightarrow {{{\mathcal U}}}}$ induces a comultiplication ${{\Delta_*} \colon {{{\mathcal U}}^*} \rightarrow {{{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*}}$ which endows ${{\mathcal U}}^*$ with a structure of comonoid in the monoidal category of complete bimodules $\left({{}^{}_{{\scriptscriptstyle{A}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{A}}}},{\widehat{\otimes}}_{{\scriptscriptstyle{A}}},A,\alpha\right)$ where $A$ is trivially filtered. We will often make use of the notation and conventions of §\[Not:infty\] and Remark \[rem:limits\]. The unaccustomed reader is invited to go through them before proceeding.
Let us perform the tensor product ${{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}{_{{\scriptscriptstyle{A}}}{{\mathcal U}}}$. The multiplication $\mu:{{\mathcal U}}\otimes {{\mathcal U}}\to{{\mathcal U}}$ which gives the algebra structure to ${{\mathcal U}}$ factors through the tensor product over $A$: $$\mu\left(x\tau_0(a)\otimes y\right)=x\tau_0(a)y=\mu(x\otimes \tau_0(a)y),$$ and it is $A$-linear with respect to both $A$-actions on $ {{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}{_{{\scriptscriptstyle{A}}}{{\mathcal U}}}$, namely $$a(x{\otimes_{\scriptscriptstyle{A}}} y)=(\tau_0(a)x){\otimes_{\scriptscriptstyle{A}}} y \quad \text{and} \quad (x{\otimes_{\scriptscriptstyle{A}}} y)a=x{\otimes_{\scriptscriptstyle{A}}} (y\tau_0(a)).$$ Therefore, it induces a filtered $A$-bilinear morphism $\mu^*:{{\mathcal U}}^*\to ({{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}})^*$ and maps $\mu_{{\scriptscriptstyle{n,m}}}:F^n{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}\, {{ }_{{\scriptscriptstyle{A}}}F^m{{\mathcal U}}}\to F^{n+m}{{\mathcal U}}$, which dually give rise to a family of morphisms of $A$-bimodules $$\xymatrix{ \Delta_{n,m}: {{\mathcal U}}^* \ar@{->}^-{\tau_{{\scriptscriptstyle{m+n}}}^*}[r] & \left(F^{m+n}{{\mathcal U}}\right)^* \ar@{->}^-{{\mu_{{\scriptscriptstyle{n,m}}}}^*}[r] & \left(F^n{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}{_{{\scriptscriptstyle{A}}}F^m{{\mathcal U}}}\right)^*, }$$ for every $n,m \geq 0$ such that $\Delta_{n,m}(f)\left(x{\otimes_{\scriptscriptstyle{A}}} y\right)=f(xy)$ for all $f\in{{\mathcal U}}^*$, $x\in F^n{{\mathcal U}}$ and $y\in F^m{{\mathcal U}}$. Given $f \in {{\mathcal U}}^*$, for each element $u \in {{\mathcal U}}$ we define $f\leftharpoonup u: {{\mathcal U}}_{{\scriptscriptstyle{A}}} \to A_{{\scriptscriptstyle{A}}}$ to be the linear map which acts as $v \mapsto f(uv)$.
\[lema:Deltamn\] For any $f \in {{\mathcal U}}^*$ and for all $n,m\in{\mathbb{N}}$, we have $$\label{eq:delta}
\big(\phi^{-1}_{m,n} \circ \Delta_{n,m}\big)(f)=\sum_{k=1}^{d_n}\tau_m^*\big(f\leftharpoonup \tau_n\big(e^n_k\big)\big){\otimes_{\scriptscriptstyle{A}}}\lambda^n_k \, \in \left(F^m{{\mathcal U}}\right)^*{}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}{}_{{\scriptscriptstyle{A}}}\left(F^n{{\mathcal U}}\right)^*$$ where $\left\{e_k^n,\lambda_k^n\mid k=1,\cdots,d_n\right\}$ is the dual basis of $(F^n{{\mathcal U}})_{{\scriptscriptstyle{A}}}$ given in Remark \[rem:dualbasis\] and $\phi_{m,n}:\left(F^mM^*\right)^{}_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}} {^{}_{{\scriptscriptstyle{R}}}\left(F^nM^*\right)} \cong \left(F^nM_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}} {{}_{{\scriptscriptstyle{R}}}F^mM}\right)^*$ are the canonical isomorphisms (see Corollary \[coro:FnL\]).
As a consequence of Lemma \[lema:Deltamn\] and of the fact that $({{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}})^*\cong {{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*$ as filtered bimodules via the completion of canonical map $\phi_{{\scriptscriptstyle{{{\mathcal U}},\,{{\mathcal U}}}}}$ (see Proposition \[prop:MRN\] and diagram below), we have an $A$-bilinear comultiplication $${{\Delta_{*}:={\psiup_{{\scriptscriptstyle{{{\mathcal U}},{{\mathcal U}}}}} } \circ \mu^*} \colon {{{\mathcal U}}^*} \rightarrow {({{\mathcal U}}^*)_{{\scriptscriptstyle{A}}} ~ {~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~} ~ {_{{\scriptscriptstyle{A}}}({{\mathcal U}}^*)}}}$$ which makes the following diagram to commute $$\label{diag:Delta*}
\xymatrix @C=40pt @R=20pt{
{{\mathcal U}}^* \ar@{-->}[rr]^-{\Delta_*} \ar[dr]_{\mu^*} \ar[dd]_{\tau_{m+n}^*} & & {{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^* \ar[dd]^-{\Pi_{m,n}} \ar@<-2pt>[dl]_-{{\widehat{\phi_{{{\mathcal U}},{{\mathcal U}}}}}}\\
& ({{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}})^* \ar@<-2pt>[ur]_-{\psiup_{{{\mathcal U}},{{\mathcal U}}}} \ar[d]^-{(\tau_n{\otimes_{\scriptscriptstyle{A}}}\tau_m)^*} & \\
F^{n+m}{{\mathcal U}}^* \ar[r]^-{\mu_{n,m}^*} & (F^n{{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}F^m{{\mathcal U}})^* & F^m{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}F^n{{\mathcal U}}^* \ar[l]_-{\phi_{m,n}}}$$ for all $m,n\geq0$ where the projections $\Pi_{{\scriptscriptstyle{m,n}}}$ make of ${{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^* $ the projective limit of the projective system $\left(F^m{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}F^n{{\mathcal U}}^*, \tau_{{\scriptscriptstyle{p,\,m}}}^{*}{\otimes_{\scriptscriptstyle{A}}}\tau_{{{\scriptscriptstyle{q,\,n}}}}^{*}\right)$, see §\[ssec:TTAC\]. Thanks to relations of the Appendices and and by resorting to the notations introduced in Remark \[rem:dualbasis\], one may write explicitly $$\label{eq:deltastar}
\Delta_{*}(f) \,=\,{\underset{n\to\infty}{\lim}}\left( \sum_{i=1}^{d_n} \big(f\leftharpoonup \tau_n\big(e_i^n\big)\big) {\otimes_{\scriptscriptstyle{A}}}E_{\scriptstyle{\lambda_i^n}} \right),$$ where we set $E_{\scriptstyle{\lambda_i^n}}:={\theta_{{\scriptscriptstyle{n}}}}^*(\lambda^n_i)$ (recall that for $g\in {{\mathcal U}}^*$ we have that $g-{\theta_{{\scriptscriptstyle{n}}}}^*{\tau_{{\scriptscriptstyle{n}}}}^*(g)\in{\mathrm{Ker}\left({{\tau_{{\scriptscriptstyle{n}}}}^*}\right)}=F_{n+1}{{\mathcal U}}^*$). Furthermore, the comultiplication $\Delta_{*}$ is uniquely determined by the following rule. For every $f \in {{\mathcal U}}^*$, $$\label{Eq:fuv}
\Delta_*(f) ={\underset{n\to\infty}{\lim}}\left( \sum_{(f)} f_{{\scriptscriptstyle{(1), \,n}}} {\otimes_{\scriptscriptstyle{A}}} f_{{\scriptscriptstyle{(2),\, n}}}\right) \, \Longleftrightarrow \left[ f(uv) \,=\, {\underset{n\to\infty}{\lim}}\left(\sum_{(f)} f_{{\scriptscriptstyle{(1),\, n}}}\big( f_{{\scriptscriptstyle{(2),\, n}}}(u) v \big)\right), \; \text{for every } u, v \in {{\mathcal U}}\right].$$
Now we can state the subsequent lemma.
\[lemma:deltaepsialg\] The comultiplication ${{\Delta_*} \colon {{{\mathcal U}}^*} \rightarrow {{{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*}}$ is a morphism of filtered $A$-bimodules when ${{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*$ is endowed with its canonical filtration. Both the comultiplication ${{\Delta_*} \colon {{{\mathcal U}}^*} \rightarrow {{{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*}}$ and the counit ${{\varepsilon_*} \colon {{{\mathcal U}}^*} \rightarrow {A}}$ are morphisms of complete algebras.
By definition of $\Delta_*$, the first claim follows from the filtered isomorphism $({{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}})^*\cong {{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*$, once observed that the transpose of a filtered morphism of increasingly filtered modules is filtered with respect to the induced decreasing filtrations on the duals. For the second claim, we only give the proof for the comultiplication, since the counit is clearly a morphism of complete algebras. To show that $\Delta_*$ is unital, recall first that the unit of ${{\mathcal U}}^*$ is the counit $\varepsilon=1_{{\scriptscriptstyle{{{\mathcal U}}^*}}}$ of ${{\mathcal U}}$ and the unit of ${{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*$ is $\varepsilon {\otimes_{\scriptscriptstyle{A}}}\varepsilon=1_{{\scriptscriptstyle{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*}}}$, so $1_{{\scriptscriptstyle{{\widehat{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*}}}}}={\widehat{\varepsilon{\otimes_{\scriptscriptstyle{A}}}\varepsilon}}$ (the notation is that of Remark \[rem:limits\]). Since $${\widehat{\phi_{{\scriptscriptstyle{{{\mathcal U}},{{\mathcal U}}}}}}}\left({\widehat{\varepsilon{\otimes_{\scriptscriptstyle{A}}}\varepsilon}}\right)(u{\otimes_{\scriptscriptstyle{A}}}v) = {\widehat{\phi_{{\scriptscriptstyle{{{\mathcal U}},{{\mathcal U}}}}}}} \left(\gamma_{{\scriptscriptstyle{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*}}}(\varepsilon{\otimes_{\scriptscriptstyle{A}}}\varepsilon)\right)(u{\otimes_{\scriptscriptstyle{A}}}v) = \phi_{{\scriptscriptstyle{{{\mathcal U}},{{\mathcal U}}}}}(\varepsilon{\otimes_{\scriptscriptstyle{A}}}\varepsilon)(u{\otimes_{\scriptscriptstyle{A}}}v) = \varepsilon(\varepsilon(u)v)=\varepsilon(uv)$$ it follows from that $\Delta_*(\varepsilon)={\widehat{\varepsilon{\otimes_{\scriptscriptstyle{A}}}\varepsilon}}$.
In view of Remark \[rem:counit\], to prove that $\Delta_{*}$ is multiplicative it is enough to show that $\Delta_*(f*g)=\boldsymbol{\mu}\left(\Delta_*(f){~{\widehat{\otimes}}_{{\scriptscriptstyle{}}}~}\Delta_*(g)\right)$, for every $f,g \in {{\mathcal U}}^*$, where $\boldsymbol{\mu}$ is the multiplication of the complete algebra ${{\mathcal U}}^* {~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*$ as in Remark \[rem:cmptensalgebra\]. By employing the notation of , we know that $$\begin{gathered}
\Delta_*(f*g)={{\underset{n\to\infty}{\lim}}}\left( \sum_{k=1}^{d_n} ((f*g)\leftharpoonup \tau_n\big(e_k^n\big)) {\otimes_{\scriptscriptstyle{A}}} E_{\scriptstyle{\lambda_k^n}} \right), \label{Eq:fstarg}\\
\boldsymbol{\mu}\left(\Delta_*(f) {~{\widehat{\otimes}}_{{\scriptscriptstyle{}}}~} \Delta_*(g)\right) = {{\underset{n\to\infty}{\lim}}}\left( \sum_{i,\, j, \, =1}^{d_n, \,d_n} (f\leftharpoonup \tau_n\big(e_i^n\big))*(g\leftharpoonup \tau_n\big(e_j^n\big)) {\otimes_{\scriptscriptstyle{A}}} (E_{\scriptstyle{\lambda_i^n}} *E_{\scriptstyle{\lambda_j^n}} ) \right).\end{gathered}$$ To show that the last two equations are equal, we need to show that the involved Cauchy sequences converge to the same limit. In view of , this amounts to show that $\Pi_{p,q}\left(\Delta_*(f*g)\right) = \Pi_{p,q}\left(\boldsymbol{\mu}\left(\Delta_*(f) {~{\widehat{\otimes}}_{{\scriptscriptstyle{}}}~} \Delta_*(g)\right) \right)$ for all $p,q\in{\mathbb{N}}$. Let $x\in F^q{{\mathcal U}}$, $y\in F^p{{\mathcal U}}$ for some $p+q=n$ and set $k=n+1=p+q+1$. We compute $$\begin{aligned}
& \phi_{p,q} \left( \Pi_{p,q} \Big( \boldsymbol{\mu}\left(\Delta_*(f){~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}\Delta_*(g)\right)\Big)\right)(x{\otimes_{\scriptscriptstyle{A}}}y) \overset{ \eqref{eq:phiPi}}{=} \sum_{i,\, j}^{d_{k},\, d_{k}}\left(\left(f\leftharpoonup \tau_{k}\big(e_j^{k}\big)\right)*\left(g\leftharpoonup \tau_{k}\big(e_j^{k}\big)\right)\right)\left( \left(E_{{\scriptscriptstyle{\lambda_i^k}}}*E_{{\scriptscriptstyle{\lambda_j^k}}}\right)(\tau_q(x)) \tau_p(y)\right) \\
& \stackrel{(**)}{=}\sum_{ (x)}\sum_{i,\, j}^{d_{k},\, d_{k}} \left(\left(f\leftharpoonup e_i^k\right)*\left(g\leftharpoonup e_j^k\right)\right)\left(E_{{\scriptscriptstyle{\lambda_i^k}}}(x_1)E_{{\scriptscriptstyle{\lambda_j^k}}}(x_2) y\right)\stackrel{(*)}{=} \sum_{(x),\, (y)}\sum_{i,\, j}^{d_k,\, d_k} f\left(\big(e_i^k E_{{\scriptscriptstyle{\lambda_i^k}}}(x_1)\big)E_{{\scriptscriptstyle{\lambda_j^k}}}(x_2) y_1\right)g(e_j^ky_2) \\
&\stackrel{(\triangle)}{=} \sum_{ (x),\, (y)}\sum_{j}^{d_k} f\left(x_1E_{{\scriptscriptstyle{\lambda_j^k}}}(x_2) y_1\right)g(e_j^ky_2) \stackrel{(\blacktriangle)}{=} \sum_{(x),\, (y)} f(x_1y_1) g \left( \sum_{j}^{d_k} (e_j^kE_{{\scriptscriptstyle{\lambda_j^k}}}(x_2))y_2\right)
\\
& \stackrel{(\triangle)}{=} \sum_{(x),\, (y)} f(x_1y_1)g(x_2y_2) \,=\, \sum_{(xy)} f((xy)_1)g((xy)_2) = (f*g)(xy) \stackrel{\eqref{diag:Delta*}}{=} \phi_{p,q} \left( \Pi_{p,q}\left( \Delta_{*}(f*g) \right) \right)(x{\otimes_{\scriptscriptstyle{A}}}y)\end{aligned}$$ where in $(*)$ we used the left $A$-linearity of $\Delta_{{{\mathcal U}}}$ and in $(\blacktriangle)$ the fact that ${\mathsf{Im}\left({\Delta_{{{\mathcal U}}}}\right)}\subseteq {{\mathcal U}}\times_{{\scriptscriptstyle{A}}}{{\mathcal U}}$. The equalities $(\triangle)$ follow from the fact that $\Delta_{{{\mathcal U}}}$ is compatible with the filtration and from the observation after equation . From $(**)$ up to the end of the computation, we omitted the inclusions $\tau_h$’s. In conclusion, we have $$\Pi_{p,q} \Big( \boldsymbol{\mu}\left(\Delta_*(f){~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}\Delta_*(g)\right)\Big)\,=\, \Pi_{p,q} \left(\Delta_{*}(f*g)\right), \quad \text{ for every } p,q \geq 0,$$ whence $\Delta_*$ is multiplicative as well.
\[prop:coalgebra\] Let $\left(A,{{\mathcal U}}\right)$ be a co-commutative Hopf algebroid with ${{\mathcal U}}$ endowed with an admissible filtration $\left\{F^n{{\mathcal U}}\mid n\geq 0\right\}$. Then $\left({{\mathcal U}}^*,\Delta_*,\varepsilon_*\right)$ is a coalgebra in the monoidal category $\left({{}^{}_{{\scriptscriptstyle{A}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{A}}}},{\widehat{\otimes}}_A,A,\alpha\right)$ of complete $A$-bimodules, where ${{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*$ is filtered with its canonical filtration.
We already know from Lemma \[lemma:U\*complete\] that ${{\mathcal U}}^*\cong {\varprojlim_{n}\left({{{\mathcal U}}^*/\mathsf{Ann}\left(F^n{{\mathcal U}}\right)}\right)}$ is a complete $A$-bimodule. The map $\Delta_*$ is $A$-bilinear by construction and we know from Lemma \[lemma:deltaepsialg\] that it is also filtered. The $A$-bilinear map ${{\varepsilon_*} \colon {{{\mathcal U}}^*} \rightarrow {A}}$ is filtered as well, in view of Lemma \[lemma:deltaepsialg\].
Let us prove then that $\Delta_*$ is coassociative and counital, with counit $\varepsilon_*$. Let us begin with counitality. Since $\varepsilon_*$ is filtered, $\varepsilon_*\otimes_A{{\mathcal U}}^*$ is filtered and hence we have ${{\varepsilon_*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*} \colon {{{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*} \rightarrow {{{\mathcal U}}^*}}$ which acts as $$\left(\varepsilon_*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*\right) \left( {\underset{n\to\infty}{\lim}}\left( \sum_{i=1}^{r_n}f_i^{(n)}{\otimes_{\scriptscriptstyle{A}}} g_i^{(n)} \right) \right) \,=\, {\underset{n\to\infty}{\lim}}\left(\sum_{i=1}^{r_n} f_i^{(n)}\left(1_{{\scriptscriptstyle{{{\mathcal U}}}}}\right) \cdot g_i^{(n)} \right).$$ Applying this formula to $\Delta_{*}(f)$ for any $f \in {{\mathcal U}}^*$, we get $$\left(\varepsilon_*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*\right) \left(\Delta_{*}(f)\right) = {\underset{n\to\infty}{\lim}}\left(\sum_{i=1}^{d_n} f(\tau_n\big(e_i^n\big))\cdot E_{{\scriptscriptstyle{\lambda_i^{n}}}} \right) = {\underset{n\to\infty}{\lim}}\left(\sum_{i=1}^{d_n} f(\tau_n\big(e_i^n\big))\cdot \lambda_i^{n}\theta_n \right) = {\underset{n\to\infty}{\lim}}\left(f\tau_n\theta_n\right) = f,$$ since $\{e_i^n,\lambda_i^n\mid i=1\cdots d_n\}$ is the dual basis of $F^n{{\mathcal U}}$ given in Remark \[rem:dualbasis\] and $f\tau_n\theta_n-f\in \mathsf{Ann}(F^{n-1}{{\mathcal U}})=F_n{{\mathcal U}}^*$. This shows that $\left(\varepsilon_*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*\right)\circ \Delta_*={\mathrm{Id}}_{{{\mathcal U}}^*}$. Analogously, we obtain $\left({{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}\varepsilon_*\right)\circ \Delta_*={\mathrm{Id}}_{{{\mathcal U}}^*}$.
Finally, we have to check the coassociativity of the comultiplication. To this end we will use the characterization of $\Delta_{*}$ given in . For a given $f \in {{\mathcal U}}^*$, we have $$\begin{split}
(\Delta_{*}{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*) \left(\Delta_{*}(f)\right)\,=\, {\underset{n\to\infty}{\lim}}\sum_{(f)}\left( \left({\underset{k\to\infty}{\lim}}\sum_{\left(f_{(1)}\right)} (f_{{\scriptscriptstyle{(11),\,n,\,k}}} {\otimes_{\scriptscriptstyle{A}}}f_{{\scriptscriptstyle{(12),\,n,\,k}}}) \right) {\otimes_{\scriptscriptstyle{A}}} f_{{\scriptscriptstyle{(2),n}}} \right), \\
({{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}\Delta_{*}) \left( \Delta_{*}(f)\right)\,=\, {\underset{n\to\infty}{\lim}}\sum_{(f)}\left( f_{{\scriptscriptstyle{(1),\,n}}} {\otimes_{\scriptscriptstyle{A}}} \left({\underset{k\to\infty}{\lim}}\sum_{\left(f_{(2)}\right)}(f_{{\scriptscriptstyle{(21),\,n,\,k}}} {\otimes_{\scriptscriptstyle{A}}}f_{{\scriptscriptstyle{(22),\,n,\,k}}}) \right) \right) .
\end{split}$$ For simplicity we will drop the sum $\sum_{(f)}$ accompanying the algebraic tensor product as all the involved topologies are linear. In light of , the coassociativity of $\Delta_{*}$ will follow once it will be shown that $$\label{eq:coass}
{\underset{n\to\infty}{\lim}}\left( f_{{\scriptscriptstyle{(11),\,n,\,n}}} {\otimes_{\scriptscriptstyle{A}}}f_{{\scriptscriptstyle{(12),\,n,\,n}}} {\otimes_{\scriptscriptstyle{A}}} f_{{\scriptscriptstyle{(2),\,n}}} \right) = {\underset{n\to\infty}{\lim}}\left( f_{{\scriptscriptstyle{(1),n}}} {\otimes_{\scriptscriptstyle{A}}} f_{{\scriptscriptstyle{(21),\,n,\,n}}} {\otimes_{\scriptscriptstyle{A}}}f_{{\scriptscriptstyle{(22),\,n,\,n}}} \right)$$ as a limit point in the complete space ${\widehat{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*}}$ (the completion of ${{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*$) . By omitting both the associativity constraint $a$ and its transpose $a^*$ observe that, for all given $u,v, w \in {{\mathcal U}}$, we have $$\begin{split}
{\widehat{\phi_{{\scriptscriptstyle{{{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}, {{\mathcal U}}}}}}} \left( {\widehat{\phi_{{\scriptscriptstyle{{{\mathcal U}},{{\mathcal U}}}}} {\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}^*}} \right) \left( {\underset{n\to\infty}{\lim}}\left( f_{{\scriptscriptstyle{(11),n,n}}} {\otimes_{\scriptscriptstyle{A}}}f_{{\scriptscriptstyle{(12),n,n}}} {\otimes_{\scriptscriptstyle{A}}} f_{{\scriptscriptstyle{(2),n}}} \right) \right) (u{\otimes_{\scriptscriptstyle{A}}}v{\otimes_{\scriptscriptstyle{A}}}w) = {\underset{n\to\infty}{\lim}}f_{{\scriptscriptstyle{(11),n,n}}}\Big(f_{{\scriptscriptstyle{(12),n,n}}}(f_{{\scriptscriptstyle{(2),n}}}(u)v) w\Big) \overset{\eqref{Eq:fuv}}{=} f(u(vw)), \\
{\widehat{\phi_{{\scriptscriptstyle{{{\mathcal U}}, {{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}}}}}} \left( {\widehat{{{\mathcal U}}^* {\otimes_{\scriptscriptstyle{A}}} \phi_{{\scriptscriptstyle{{{\mathcal U}},{{\mathcal U}}}}}}} \right) \left( {\underset{n\to\infty}{\lim}}\left( f_{{\scriptscriptstyle{(1),n}}} {\otimes_{\scriptscriptstyle{A}}} f_{{\scriptscriptstyle{(21),n,n}}} {\otimes_{\scriptscriptstyle{A}}}f_{{\scriptscriptstyle{(22),n,n}}} \right) \right) (u{\otimes_{\scriptscriptstyle{A}}}v{\otimes_{\scriptscriptstyle{A}}}w) = {\underset{n\to\infty}{\lim}}f_{{\scriptscriptstyle{(1),n}}}\Big( f_{{\scriptscriptstyle{(21),n,n}}}(f_{{\scriptscriptstyle{(22),n,n}}}(u)v)w \Big) \overset{\eqref{Eq:fuv}}{=} f((uv)w) .
\end{split}$$ Comparing this last equations leads to equality and then to the coassociativity of $\Delta_{*}$.
The topological antipode of ${{\mathcal U}}^*$ {#ssec:AUstra}
----------------------------------------------
Now we proceed to construct the topological antipode for ${{\mathcal U}}^*$, under the further hypothesis that the translation map of ${{\mathcal U}}$ is a filtered morphism of algebras. Such an assumption is always fulfilled in the case of the universal enveloping Hopf algebroid of a Lie-Rinehart algebra with finitely generated and projective module $L_{{\scriptscriptstyle{A}}}$, as we will see in Example \[exam:SUL\].
At the level of the algebra structure, the antipode is provided by the following map (compare with [@KowalzigExtBVA Theorem 3.1] and [@CGK:2016 Theorem 5.1.1] for the case when ${{\mathcal U}}$ is finitely generated and projective right $A$-module): $$\label{Eq:tantip}
{{\mathcal S}}_{*}: {{\mathcal U}}^* \longrightarrow {{\mathcal U}}^*,\quad \Big( f \longmapsto \left[ u \mapsto \varepsilon\big( f(u_{{\scriptscriptstyle{-}}}) u_{{\scriptscriptstyle{+}}}\big) \right] \Big),$$ where $u \mapsto \beta^{-1}(1{\otimes_{\scriptscriptstyle{A}}}u) = u_{{\scriptscriptstyle{-}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+}}} $ is the translation map obtained from the inverse of the map $\beta: {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {\,}_{{\scriptscriptstyle{A}}}{{\mathcal U}}\to {{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}$, which sends $u{\otimes_{\scriptscriptstyle{A}}}v \mapsto uv_{{\scriptscriptstyle{1}}}{\otimes_{\scriptscriptstyle{A}}}v_{{\scriptscriptstyle{2}}}$. Consider $\delta: {{\mathcal U}}\to {{\mathcal U}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}:= {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}$, $ u \mapsto u_{{\scriptscriptstyle{-}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+}}}$ the map of Equation . As it was shown in [@Schau:DADOQGHA Proposition 3.7], the map $\delta$ enjoys a series of properties. Here we recall few of them which will be needed in the sequel. First notice that $\beta^{-1} (v{\otimes_{\scriptscriptstyle{A}}} u)=v{u_{{\scriptscriptstyle{-}}}}{\otimes_{\scriptscriptstyle{A}}}{u_{{\scriptscriptstyle{+}}}}$, so for all $u,v\in{{\mathcal U}}$ and $a\in A$ we have $$\begin{aligned}
1{\otimes_{\scriptscriptstyle{A}}} u & =& {u_{{\scriptscriptstyle{-}}}}u_{{\scriptscriptstyle{+,\,1}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+,\,2}}} \,\, \in \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} \, {\otimes_{\scriptscriptstyle{A}}} \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} \label{Eq:B4} \\ u_{{\scriptscriptstyle{1,\, -}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{1,\,+}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{2}}} &=& {u_{{\scriptscriptstyle{-}}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+,\, 1}}}{\otimes_{\scriptscriptstyle{A}}}u_{{\scriptscriptstyle{+,\, 2}}} \,\, \in \, ({{\mathcal U}}_{{\scriptscriptstyle{A}}} \,{\otimes_{\scriptscriptstyle{A}}} \, {}_{{\scriptscriptstyle{A}}}{{\mathcal U}})\, {\otimes_{\scriptscriptstyle{A}}} \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} \label{Eq:B5} \\
u_{{\scriptscriptstyle{+,\,-}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+,\,+}}} &=& u_{{\scriptscriptstyle{-,\,1}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-,\,2}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+}}} \,\, \in \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} \,{\otimes_{\scriptscriptstyle{A}}} \, {{\mathcal U}}_{{\scriptscriptstyle{A}}}\, {\otimes_{\scriptscriptstyle{A}}} \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} \label{Eq:B55} \\
{u_{{\scriptscriptstyle{-}}}}\, {u_{{\scriptscriptstyle{+}}}}&=& \tau_{{\scriptscriptstyle{0}}}(\varepsilon(u) ) \,\, \in F^0{{\mathcal U}}= A \label{Eq:B555}
\\
(uv)_{{\scriptscriptstyle{-}}} {\otimes_{\scriptscriptstyle{A}}} (uv)_{{\scriptscriptstyle{+}}} & =& v_{{\scriptscriptstyle{-}}} u_{{\scriptscriptstyle{-}}} {\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+}}}v_{{\scriptscriptstyle{+}}} \,\, \in {{\mathcal U}}_{{\scriptscriptstyle{A}}} \, {\otimes_{\scriptscriptstyle{A}}} \, {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}\label{Eq:B66}
\\
a{\otimes_{\scriptscriptstyle{A}}} 1\,\,=\,\, 1{\otimes_{\scriptscriptstyle{A}}} a &=& a_{{\scriptscriptstyle{-}}} {\otimes_{\scriptscriptstyle{A}}} a_{{\scriptscriptstyle{+}}} \,\, \in \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} \,{\otimes_{\scriptscriptstyle{A}}} \, {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}. \label{Eq:B6}\end{aligned}$$ In particular, by equation we have that $\delta$ is an algebra map, viewed as a map from ${{\mathcal U}}$ to ${{\mathcal U}}^{{\scriptscriptstyle{\mathrm{op}}}} \times_{{\scriptscriptstyle{A}}} {{\mathcal U}}$. The subsequent lemma is crucial for showing that ${{\mathcal S}}_{*}$ is multiplicative.
\[lema:Sstar\] Let $f,g, h \in {{\mathcal U}}^*$ and $u \in {{\mathcal U}}$. Then we have $$\begin{split}
{{\mathcal S}}_{*}(f*g) (u) \,=\, {{\mathcal S}}_{*}(f)\Big( g(u_{{\scriptscriptstyle{-}}})u_{{\scriptscriptstyle{+}}} \Big) \hspace{2cm} \label{Eq:S1} \\
\big({{\mathcal S}}_{*}(f) * h\big) (u) \,=\, \big(h\leftharpoonup f(u_{{\scriptscriptstyle{-}}})\big) (u_{{\scriptscriptstyle{+}}}) \,=\, \Big(\big( \varepsilon\leftharpoonup f(u_{{\scriptscriptstyle{-}}}) \big) * h\Big) (u_{{\scriptscriptstyle{+}}})
\end{split}$$
We will implicitly use the co-commutativity of the comultiplication of ${{\mathcal U}}$ as well as the $A$-linearity of $\delta$. Computing the left hand side of the first equality gives $${{\mathcal S}}_{*}(f*g) (u) = \varepsilon\Big( (f*g)(u_{{\scriptscriptstyle{-}}})\, u_{{\scriptscriptstyle{+}}} \Big) = \varepsilon\Big( f(u_{{\scriptscriptstyle{-,1}}})\,g(u_{{\scriptscriptstyle{-,2}}})\, u_{{\scriptscriptstyle{+}}} \Big) \overset{\eqref{Eq:B55}}{=} \varepsilon\Big( f(u_{{\scriptscriptstyle{+,-}}})\,(g(u_{{\scriptscriptstyle{-}}})\, u_{{\scriptscriptstyle{+,+}}}) \Big) = {{\mathcal S}}_{*}(f)\Big( g(u_{{\scriptscriptstyle{-}}})u_{{\scriptscriptstyle{+}}} \Big),$$ where in the last equality we used and . This leads to the stated first equality.
As for the second one, we have $$({{\mathcal S}}_{*}(f) *h)(u) = {{\mathcal S}}_{*}(f)(u_{{\scriptscriptstyle{1}}}) h(u_{{\scriptscriptstyle{2}}}) = \varepsilon\Big( f(u_{{\scriptscriptstyle{1,-}}})\,u_{{\scriptscriptstyle{1,+}}}\Big)\, h(u_{{\scriptscriptstyle{2}}}) \overset{\eqref{Eq:B5}}{=}
\varepsilon\Big( f(u_{{\scriptscriptstyle{-}}})\,u_{{\scriptscriptstyle{+,1}}}\Big)\, h(u_{{\scriptscriptstyle{+,2}}})$$ from which one deduces on the one hand that $({{\mathcal S}}_{*}(f) *h)(u) = \varepsilon\Big( f(u_{{\scriptscriptstyle{-}}})\,u_{{\scriptscriptstyle{+,1}}}\Big)\, h(u_{{\scriptscriptstyle{+,2}}}) = \Big( (\varepsilon \leftharpoonup f(u_{{\scriptscriptstyle{-}}})) * h\Big) (u_{{\scriptscriptstyle{+}}})$ and on the other hand that $({{\mathcal S}}_{*}(f) *h)(u) = \varepsilon\Big( f(u_{{\scriptscriptstyle{-}}})\,u_{{\scriptscriptstyle{+,1}}}\Big)\, h(u_{{\scriptscriptstyle{+,2}}}) = \varepsilon( u_{{\scriptscriptstyle{+,1}}})\, h(f(u_{{\scriptscriptstyle{-}}})\,u_{{\scriptscriptstyle{+,2}}})=h(f(u_{{\scriptscriptstyle{-}}})\,u_{{\scriptscriptstyle{+}}})$.
The map $\beta$ is compatible with the increasing filtration on both ${{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {\,}_{{\scriptscriptstyle{A}}}{{\mathcal U}}$ and ${{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}$. Namely, $${{\mathcal F}}^{{\scriptscriptstyle{n}}}\big( {{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}}\big)\,=\, \sum_{p+q=n} {F^{p}{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}F^{q}{{\mathcal U}}_{{\scriptscriptstyle{A}}}} \subseteq {{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}}\quad \text{and} \quad {{\mathcal F}}^{{\scriptscriptstyle{n}}}\left({{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {\,}_{{\scriptscriptstyle{A}}}{{\mathcal U}}\right) = \sum_{p+q=n}{\mathsf{Im}\left({F^{{\scriptscriptstyle{p}}}{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}{_{{\scriptscriptstyle{A}}}F^{{\scriptscriptstyle{q}}}{{\mathcal U}}}}\right)}.$$ The left-most inclusion is clear since the ${\tau_{{\scriptscriptstyle{n}}}}: F^n{{\mathcal U}}_{{\scriptscriptstyle{A}}} \to {{\mathcal U}}_{{\scriptscriptstyle{A}}}$’s are split monomorphisms of right $A$-modules. The structure of $A$-bimodule on ${{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}}$ is given by $a(u{\otimes_{\scriptscriptstyle{A}}}v)a'=au{\otimes_{\scriptscriptstyle{A}}}a'v$. Applying $\beta$ to each term of the canonical filtration of ${{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}}$, we have $$\begin{split}
\beta\Big( {{\mathcal F}}^n\big( {{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}}\big) \Big) & = \sum_{p+q=n}F^p{{\mathcal U}}\cdot \Delta(F^q{{\mathcal U}}) \subseteq \sum_{p+l+k=n}F^p{{\mathcal U}}\cdot F^k{{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} F^l{{\mathcal U}}_{{\scriptscriptstyle{A}}} \subseteq \sum_{p+l+k=n}{F^{p+k}{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}F^{l}{{\mathcal U}}_{{\scriptscriptstyle{A}}}} \\ & \subseteq \sum_{i+j=n} {F^{i}{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}}F^{j}{{\mathcal U}}_{{\scriptscriptstyle{A}}}} = {{\mathcal F}}^n({{{\mathcal U}}_{{\scriptscriptstyle{A}}}{\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}}).
\end{split}$$ This means that $\beta$ is a filtered morphism of $A$-bimodules. On the other hand, $\beta^{-1}$ is a filtered bilinear map if and only if $\delta$ is a filtered algebra map. We point out that, in general, none of the equivalent conditions may be true, although this is the case for universal enveloping Hopf algebroids, as the next example shows.
\[exam:SUL\] Take $(A,L)$ and ${{\mathcal U}}={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ as in Example \[exm:ULAdmiss\]. Then, the following computation $$\delta \left( F^{n}\mathcal{U} \right) \,\subseteq\,
\sum_{k=0}^{n}\delta \left( \iota _{L}\left( L\right) ^{k}\right) \,\subseteq\,
\sum_{k=0}^{n}\delta \left( \iota _{L}\left( L\right) \right) ^{k} \,\subseteq\,
\sum_{k=0}^{n}F^{k}\left( \mathcal{U} {\otimes_{\scriptscriptstyle{A}}}\mathcal{U}
\right) \subseteq F^{n}\left( \mathcal{U}
{\otimes_{\scriptscriptstyle{A}}}\mathcal{U} \right)$$ shows that $\delta$ is a filtered algebra map, which implies that $\beta ^{-1}$ is also filtered, as we have $$\begin{aligned}
\beta ^{-1}\left( F^{n}\left( \mathcal{U} {\otimes_{\scriptscriptstyle{A}}}
\mathcal{U} \right) \right) & \subseteq &\sum_{p+q=n}\beta
^{-1}\left( F^{p}\mathcal{U} {\otimes_{\scriptscriptstyle{A}}}F^{q}\mathcal{U}
\right) \subseteq \sum_{p+q=n}F^{p}\mathcal{U} \centerdot \delta\left( F^{q}\mathcal{U}\right) \\
&\subseteq &\sum_{p+k+h=n}F^{p}\mathcal{U} \cdot {\mathsf{Im}\left({F^{k}
\mathcal{U} {\otimes_{\scriptscriptstyle{A}}}F^{h}\mathcal{U} }\right)}
\subseteq F^{n}\left( \mathcal{U} {\otimes_{\scriptscriptstyle{A}}}\mathcal{U}
\right).\end{aligned}$$
\[rem:delta\] In the case of the universal enveloping Hopf algebroids of Lie-Rinehart algebras, as we have seen in Example \[exam:SUL\], the translation map is always continuous and then the associated map ${{\mathcal S}}_{*}$ of equation is a filtered map. Indeed, if we assume that $\delta$ is filtered, then $${\tau_{{\scriptscriptstyle{n}}}}(u)_{{\scriptscriptstyle{-}}}{\otimes_{\scriptscriptstyle{A}}} {\tau_{{\scriptscriptstyle{n}}}}(u)_{{\scriptscriptstyle{+}}} \in \sum_{p+q=n} \mathsf{Im}\big( {\tau_{{\scriptscriptstyle{p}}}}{\otimes_{\scriptscriptstyle{A}}}{\tau_{{\scriptscriptstyle{q}}}}\big),$$ for every $u \in F^n{{\mathcal U}}$ and $n \geq 0$. Therefore, ${{\mathcal S}}_{*}(F_n{{\mathcal U}}^*) \subseteq F_n{{\mathcal U}}^*$, for every $n \geq 0$.
\[prop:tantip\] Let $(A,{{\mathcal U}})$ be a co-commutative (right) Hopf algebroid endowed with an admissible filtration and assume $\delta$ is a filtered algebra map. Then the map ${{\mathcal S}}_{*}$ of equation is a morphism of complete algebras such that ${{\mathcal S}}_*\circ s_* = t_*$ and ${{\mathcal S}}_*\circ t_* = s_*$.
We need to check that ${{\mathcal S}}_{*}$ is multiplicative and that it exchanges the source with the target, as we already know that it preserves the filtration, in view of Remark \[rem:delta\]. Recall that the unit of ${{\mathcal U}}^*$ is given by $\vartheta: A{\otimes_{\scriptscriptstyle{}}}A \to {{\mathcal U}}^*$ sending $a{\otimes_{\scriptscriptstyle{}}}a' \mapsto \left[ u \mapsto a\varepsilon(a'u)\right]$. Given $a, a' \in A$ we have $$\begin{split}
{{\mathcal S}}_{*}(\vartheta(a{\otimes_{\scriptscriptstyle{}}}a'))(u) & = \varepsilon\Big( \vartheta(a{\otimes_{\scriptscriptstyle{}}}a')(u_{{\scriptscriptstyle{-}}}) {u_{{\scriptscriptstyle{+}}}}\Big) = \varepsilon\Big( \varepsilon(a'u_{{\scriptscriptstyle{-}}})a {u_{{\scriptscriptstyle{+}}}}\Big) = \varepsilon\Big( a'u_{{\scriptscriptstyle{-}}}a {u_{{\scriptscriptstyle{+}}}}\Big) = \varepsilon\Big( u_{{\scriptscriptstyle{-}}}a {u_{{\scriptscriptstyle{+}}}}a' \Big) \\
& = \varepsilon\Big( u_{{\scriptscriptstyle{-}}}a {u_{{\scriptscriptstyle{+}}}}\Big)a' = \varepsilon\Big( (au)_{{\scriptscriptstyle{-}}} (au)_{{\scriptscriptstyle{+}}} \Big)a' \overset{\eqref{Eq:B555}}{=} \varepsilon(au)a',
\end{split}$$ whence ${{\mathcal S}}_{*} (\vartheta (a{\otimes_{\scriptscriptstyle{}}}a')) = \vartheta(a'{\otimes_{\scriptscriptstyle{}}}a)$. Therefore, ${{\mathcal S}}_{*} \circ s_* = t_*$ and ${{\mathcal S}}_{*} \circ t_* = s_*$, where $s_*, t_*$ are as in equation . Let us check that ${{\mathcal S}}_{*}$ is multiplicative. If we consider $f,g \in {{\mathcal U}}^*$ and $u \in{{\mathcal U}}$, we have $$\Big({{\mathcal S}}_{*}(f) *{{\mathcal S}}_{*}(g)\Big)(u) \overset{\eqref{Eq:S1}}{=} \Big({{\mathcal S}}_{*}(g) \leftharpoonup f(u_{{\scriptscriptstyle{-}}})\Big) (u_{{\scriptscriptstyle{+}}}) = {{\mathcal S}}_{*}(g) \Big(f(u_{{\scriptscriptstyle{-}}}) \, u_{{\scriptscriptstyle{+}}} \Big)
\overset{\eqref{Eq:S1}}{=} {{\mathcal S}}_{*}(g*f) (u)= {{\mathcal S}}_{*}(f*g)(u).$$ This shows that ${{\mathcal S}}_{*}(f*g)={{\mathcal S}}_{*}(f) * {{\mathcal S}}_{*}(g)$, which finishes the proof.
The following is the main result of this subsection.
\[prop:Bosco\] Let $(A,{{\mathcal U}})$ and $\delta$ be as in Proposition \[prop:tantip\]. Then $(A,{{\mathcal U}}^*)$ is a complete Hopf algebroid with structure maps $s_*$, $t_*$, $\Delta_*$, $\varepsilon_*$ and ${{\mathcal S}}_*$. In particular, this is the case for the universal enveloping Hopf algebroid ${{\mathcal U}}={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ of a Lie-Rinehart algebra $(A,L)$, where $L_{{\scriptscriptstyle{A}}}$ is finitely generated and projective.
We only need to check that the algebra map ${{\mathcal S}}_{*}$ enjoys the properties for being an algebraic antipode since it is already continuous. Let $f \in {{\mathcal U}}^*$ and take an arbitrary element $u \in {{\mathcal U}}$. Then $$\begin{split}
{{\underset{n\to\infty}{\lim}}}\sum_{(f)} \Big( f_{{\scriptscriptstyle{1,\,n}}}*{{\mathcal S}}_*(f_{{\scriptscriptstyle{2,\,n}}})\Big) (u) & \,=\, {{\underset{n\to\infty}{\lim}}}\sum_{(f)} \Big( {{\mathcal S}}_*(f_{{\scriptscriptstyle{2,n}}}) *f_{{\scriptscriptstyle{1,\,n}}}\Big) (u) \,\overset{\eqref{Eq:S1}}{=}\, {{\underset{n\to\infty}{\lim}}} \sum_{(f)} f_{{\scriptscriptstyle{1,\,n}}} \Big(f_{{\scriptscriptstyle{2,\,n}}}({u_{{\scriptscriptstyle{-}}}})\,{u_{{\scriptscriptstyle{+}}}}\Big) \\
& \overset{\eqref{Eq:fuv}}{=}\, f({u_{{\scriptscriptstyle{-}}}}{u_{{\scriptscriptstyle{+}}}}) \,=\, f(\varepsilon(u))\,=\, f(1)\varepsilon(u)\,=\, s_{*}(\varepsilon_{*}(f))(u).
\end{split}$$ Therefore, for every $f \in {{\mathcal U}}^*$, we have $${{\underset{n\to\infty}{\lim}}}\sum_{(f)} \Big( f_{{\scriptscriptstyle{1,\,n}}} * {{\mathcal S}}_{*}(f_{{\scriptscriptstyle{2,\,n}}} )\Big) \,=\, s_{*}(\varepsilon_{*}(f)).$$ Now let us check that ${{\mathcal S}}_*^2=id_{{\scriptscriptstyle{{{\mathcal U}}^*}}}$ which will be sufficient to claim that ${{\mathcal S}}_*$ is an antipode for the complete bialgebroid ${{\mathcal U}}^*$. Recall that $\delta: {{\mathcal U}}_{{\scriptscriptstyle{A}}} \to {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} { }_{{\scriptscriptstyle{A}}}{{\mathcal U}}$ is right $A$-linear, so that we can consider the map $$(\delta{\otimes_{\scriptscriptstyle{A}}}{}_{{\scriptscriptstyle{A}}}{{\mathcal U}}) \circ \delta: {{\mathcal U}}_{{\scriptscriptstyle{A}}} \longrightarrow {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}},\; \; \Big( u \longmapsto u_{{\scriptscriptstyle{-,\,-}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-,\,+}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{+}}}\Big)$$ Let us compute the image of the element $u_{{\scriptscriptstyle{-,\,-}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-,\,+}}} u_{{\scriptscriptstyle{+}}} \in {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} {}_{{\scriptscriptstyle{A}}}{{\mathcal U}}$ by the map $\beta$: $$\begin{aligned}
\beta\big( u_{{\scriptscriptstyle{-,\,-}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-,\,+}}} u_{{\scriptscriptstyle{+}}} \big) & =& \big( u_{{\scriptscriptstyle{-,-}}}{\otimes_{\scriptscriptstyle{A}}} 1\big) . (u_{{\scriptscriptstyle{-,\,+}}} u_{{\scriptscriptstyle{+}}}) \quad \in\; {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{}}} {{\mathcal U}}_{{\scriptscriptstyle{A}}}\\
&=& \Big( u_{{\scriptscriptstyle{-,\,-}}} u_{{\scriptscriptstyle{-,\,+,\,1}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-,\,+,\,2}}} \Big). {u_{{\scriptscriptstyle{+}}}}\\
&\overset{\eqref{Eq:B4} }{=}& \Big( 1{\otimes_{\scriptscriptstyle{A}}} {u_{{\scriptscriptstyle{-}}}}\Big) . {u_{{\scriptscriptstyle{+}}}}\,\,=\,\, u_{{\scriptscriptstyle{+,\,1}}}{\otimes_{\scriptscriptstyle{A}}}{u_{{\scriptscriptstyle{-}}}}u_{{\scriptscriptstyle{+,2}}} \\ &=& u_{{\scriptscriptstyle{+,\,2}}}{\otimes_{\scriptscriptstyle{A}}}{u_{{\scriptscriptstyle{-}}}}u_{{\scriptscriptstyle{+,\,1}}} \,\, \overset{\eqref{Eq:B4} }{=}\,\, u{\otimes_{\scriptscriptstyle{A}}} 1\,\,=\,\, \beta(u{\otimes_{\scriptscriptstyle{A}}} 1)\end{aligned}$$ Therefore, for every $u \in {{\mathcal U}}$, we have $$\label{Eq:uminus}
u_{{\scriptscriptstyle{-,\,-}}}{\otimes_{\scriptscriptstyle{A}}} u_{{\scriptscriptstyle{-,\,+}}} \, u_{{\scriptscriptstyle{+}}} \,\,=\,\, u{\otimes_{\scriptscriptstyle{A}}} 1 \, \in \, {{\mathcal U}}_{{\scriptscriptstyle{A}}} {\otimes_{\scriptscriptstyle{A}}} \, {}_{{\scriptscriptstyle{A}}} {{\mathcal U}}.$$ In this way, taking a function $f \in {{\mathcal U}}^*$ and an element $u \in {{\mathcal U}}$, we then get $${{\mathcal S}}_*^2(f)(u) ={{\mathcal S}}_*\big( {{\mathcal S}}_*(f) \big) (u) = \varepsilon\Big( {{\mathcal S}}_*(f)({u_{{\scriptscriptstyle{-}}}}) \, {u_{{\scriptscriptstyle{+}}}}\Big) = \varepsilon\Big( f(u_{{\scriptscriptstyle{-,\,-}}}) \, u_{{\scriptscriptstyle{-,\,+}}} \, {u_{{\scriptscriptstyle{+}}}}\Big) \overset{\eqref{Eq:uminus}}{=} \varepsilon\big( f(u) \, 1\big) = f(u),$$ whence ${{\mathcal S}}_*^2=id_{{\scriptscriptstyle{{{\mathcal U}}^*}}}$ and this finishes the proof of the fact that $({{\mathcal U}}^*, \Delta_{*}, \varepsilon_{*}, {{\mathcal S}}_{*})$ is a complete Hopf algebroid. The particular case follows immediately from Remark \[rem:delta\].
\[rem:deltaii\] As we have seen in this subsection, the fact that $\delta$ is a continuous map seems to be essential in carrying out the construction of the topological antipode for the convolution algebra ${{\mathcal U}}^*$ of an admissible co-commutative (right) Hopf algebroid ${{\mathcal U}}$. However, notice that no restrictive condition was imposed on the $A$-module $L$ in order to obtain the continuity of the translation map $\delta$ for the Hopf algebroid ${{\mathcal U}}={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$. In particular, the fact that the filtration on ${{\mathcal U}}$ is admissible is not used.
Besides, both conditions (i.e., admissibility of the filtration of ${{\mathcal U}}$ and continuity of $\delta$) are satisfied for any universal enveloping algebra of a finitely generated and projective Lie-Rinehart algebra, that is to say, our results apply to the classical geometric context of Lie algebroids that was of our interest.
Summing up, we have been able to show that the convolution algebra ${{\mathcal U}}^*$ of a filtered co-commutative (right) Hopf algebroid ${{\mathcal U}}$ is a complete Hopf algebroid if ${{\mathcal U}}$ is admissibly filtered and $\delta$ is continuous. A question which we consider worthy to be addressed is if the converse is true as well. If this is not the case, we would be very glad to see a counterexample. In this sense, providing examples of admissibly filtered co-commutative Hopf algebroid whose translation map is not filtered could be of great interest. Concluding, we think that both questions deserve further attention but also that they are out of the purposes of the present paper. Therefore, we will not go into details here and we will leave them for future investigation.
The main morphism of complete commutative Hopf algebroids {#sec:MCH}
=========================================================
In this section we give our main result. It is concerned with the universal (right) Hopf algebroid of a Lie-Rinehart algebra with an admissible filtration, as in §\[ssec:FUstra\]. This in particular encompasses the situation of a Lie algebroid over a smooth (connected) manifold, by using the global sections functor as in §\[ssec:muchocaldo\].
The finite dual of a co-commutative Hopf algebroid. {#ssec:Fdual}
---------------------------------------------------
We recall from [@LaiachiGomez] the construction of what is known as *the finite dual Hopf algebroid*. This construction is one of the main tools used in building up our application in the forthcoming subsections, so it is convenient to recall it in some detail.
Following [@LaiachiGomez], given a (right) co-commutative Hopf algebroid $(A,{{\mathcal U}})$, we consider the category ${\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}}$ of those right ${{\mathcal U}}$-modules whose underlying right $A$-module structure is finitely generated and projective. This category is a symmetric rigid monoidal linear category with identity object $A$, whose right ${{\mathcal U}}$-action is given by $a \centerdot u =\varepsilon(au)$. Furthermore, the forgetful functor $\boldsymbol{\omegaup}: {\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}} \to {\mathsf{proj}(A)}$ to the category of finitely generated and projective $A$-modules, plays the role of *the fibre functor* which is a non trivial symmetric strict monoidal faithful functor (as we are assuming that ${\rm Spec}(A) \neq \emptyset$).
The tensor product of two right ${{\mathcal U}}$-modules $M$ and $N$ is the $A$-module $M{\otimes_{\scriptscriptstyle{A}}}N$ endowed with the following right ${{\mathcal U}}$-action: $$(m{\otimes_{\scriptscriptstyle{A}}}n) \centerdot u = (m \centerdot u_{{\scriptscriptstyle{1}}}) {\otimes_{\scriptscriptstyle{A}}} (n \centerdot u_{{\scriptscriptstyle{2}}}).$$ The dual object of a right ${{\mathcal U}}$-module $M$ belonging to ${\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}}$ is the $A$-module $M^{*}={\mathrm{Hom}_{{\scriptscriptstyle{-A}}}\left(M,\,A\right)}$ with the right ${{\mathcal U}}$-action $$\label{Eq:star}
\varphi \centerdot u : M \longrightarrow A, \quad \Big( m \longmapsto \varphi(m\centerdot u_{{\scriptscriptstyle{-}}}) \centerdot u_{{\scriptscriptstyle{+}}} \Big),$$ where $u_{{\scriptscriptstyle{-}}}{\otimes_{\scriptscriptstyle{A}}}u_{{\scriptscriptstyle{+}}} = \beta^{-1}(1{\otimes_{\scriptscriptstyle{A}}}u)$. We will often omit the symbol $\centerdot$ in what follows, when the action will be clear from the context.
The commutative Hopf algebroid constructed from the data $\left({\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}}, \boldsymbol{\omegaup}\right)$, will be denoted by $(A,{{\mathcal U}}^{{{\boldsymbol{\circ}}}})$ and refereed to as *the finite dual Hopf algebroid* of $(A,{{\mathcal U}})$. From its own definition, ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ is the quotient algebra $$\label{Eq:Uo}
{{\mathcal U}}^{{{\boldsymbol{\circ}}}}=\frac{\underset{M\,\in\, \mathrm{Ob}
( {\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}}) }{\bigoplus} M^{\ast }{\otimes_{\scriptscriptstyle{T_{M}}}}M
}{\mathfrak{J}_{{\scriptscriptstyle{{\mathcal{A}}_{{{\mathcal U}}}}}}}$$ by the two sided ideal $\mathfrak{J}_{{\scriptscriptstyle{{\mathcal{A}}_{{{\mathcal U}}}}}}$ generated by the set $$\Big\{ \big(\varphi {\otimes_{\scriptscriptstyle{T_{N}}}}f\left( m\right)\big) -\big(\varphi \circ f{\otimes_{\scriptscriptstyle{T_{M}}}}m\big) \mid \varphi \in \left. N^{\ast }\right. ,m\in M,f\in T_{MN},M,N\in \mathrm{Ob}\left( {{\mathcal{A}}_{{{\mathcal U}}}} \right) \Big\},$$ where we denoted by $M$ and $\left. M^{\ast }\right. $ the objects $\boldsymbol{\omegaup}
\left( M\right) $ and $\left. \boldsymbol{\omegaup} \left( M\right)^{\ast } \right. $ and where we used the following short forms: $T_{MN}:={\mathrm{Hom}_{{\scriptscriptstyle{{\scriptscriptstyle{{\mathcal{A}}_{{{\mathcal U}}}}}}}}\left(M,\,N\right)}$, $T_{M}:={\mathrm{Hom}_{{\scriptscriptstyle{{\scriptscriptstyle{{\mathcal{A}}_{{{\mathcal U}}}}}}}}\left(M,\,M\right)}$. Therein, we also identify each element of the form $\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}m \in M^{\ast }{\otimes_{\scriptscriptstyle{T_{M}}}}M$ with its image in the direct sum $\underset{{\scriptscriptstyle{M\,\in\, \mathrm{Ob}({\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}})}}}{\bigoplus} M^{\ast }{\otimes_{\scriptscriptstyle{T_{M}}}}M$.
The structure maps of the finite dual Hopf algebroid $(A,{{\mathcal U}}^{{{\boldsymbol{\circ}}}})$ are given as follows. Write $\overline{\varphi \otimes _{T_{M}}m}$ for the equivalence class of the image (in the above direct sum) of a generic element for the form $\varphi {\otimes_{\scriptscriptstyle{T_M}}}m \in M^*{\otimes_{\scriptscriptstyle{T_M}}}M$, for some object $M \in {\mathcal{A}}_{{\scriptscriptstyle{{{\mathcal U}}}}}$. Since all involved maps are linear, we will be dealing most of all just with generic elements of the form $\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}m}$, bypassing the more general summation notation. Thus the structure maps on ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ are given by $$\begin{aligned}
&&{{\texttt{u}} \colon {{\Bbbk}} \rightarrow {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}};\, \Big({1_{{\Bbbk}}} \mapsto {\overline{{\mathrm{Id}}_{A}{\otimes_{\scriptscriptstyle{{\Bbbk}}}}1_{A}}}\Big)}, \\
&&{{\texttt{m}} \colon {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}{\otimes_{\scriptscriptstyle{}}}{{\mathcal U}}^{{{\boldsymbol{\circ}}}}} \rightarrow {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}};\, \Big({\overline{\psi {\otimes_{\scriptscriptstyle{ T_{N}}}}n}{\otimes_{\scriptscriptstyle{}}}\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}m}} \mapsto {\overline{\left( \psi \star \varphi\right) {\otimes_{\scriptscriptstyle{T_{M{\otimes_{\scriptscriptstyle{A}}}N}}}}\left( m{\otimes_{\scriptscriptstyle{A}}}n\right) }}\Big)}, \\
&&{{\etaup} \colon {A{\otimes_{\scriptscriptstyle{}}}A} \rightarrow {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}};\, \Big({a{\otimes_{\scriptscriptstyle{}}}b} \mapsto {\overline{l_{a}{\otimes_{\scriptscriptstyle{{\Bbbk}}}}b}}\Big)},\;\; \text{where } l_a: A \to A, \big( 1 \mapsto a \big)\text{ is the left multiplication by }\, a,\\
&&{{\varepsilon_{\circ}} \colon {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}} \rightarrow {A};\, \Big({\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}m}} \mapsto {\varphi \left( m\right)}\Big)} ,\\
&&{{\Delta_{\circ}} \colon {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}} \rightarrow {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^{{{\boldsymbol{\circ}}}}};\, \Big({\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}m}} \mapsto {\sum_{i=1}^{r}\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}e_{i}}{\otimes_{\scriptscriptstyle{A}}}\overline{e_{i}^{\ast }{\otimes_{\scriptscriptstyle{T_{M}}}}m}}\Big)}, \;\; \text{where } \{e_i,e_i^*\}_i \text{ is a dual basis for } M_{{\scriptscriptstyle{A}}}\\
&&{{{{\mathcal S}}_{\circ}} \colon {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}} \rightarrow {{{\mathcal U}}^{{{\boldsymbol{\circ}}}}};\, \Big({\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}}m}} \mapsto {\overline{\mathrm{ev}_{m}{\otimes_{\scriptscriptstyle{T_{\left. M^{\ast }\right. }}}}\varphi }}\Big)}, \;\; \text{where } {\rm ev}_m: M^* \to A\; \text{ is the evaluation at } m \text{ map}.\end{aligned}$$ For every $\psi \in \left. N^{\ast }\right. $ and $\varphi \in \left. M^{\ast }\right.$, the map $\psi \star \varphi :M{\otimes_{\scriptscriptstyle{A}}}N\rightarrow A$ acts as $m{\otimes_{\scriptscriptstyle{A}}}n\mapsto \varphi \left( m\right) \psi \left( n\right) $.
Notice that there is a linear map $$\label{Eq:zeta}
\zeta: {{\mathcal U}}^{{{\boldsymbol{\circ}}}} \longrightarrow {{\mathcal U}}^{*},\quad \Big( {\overline{\varphi{\otimes_{\scriptscriptstyle{T_M}}}m}} \longmapsto \left[ u \mapsto \varphi(mu)\right] \Big).$$
The following lemma is a straightforward computation, see [@LaiachiGomez].
\[lema:zeta\] The linear map $\zeta$ is an homomorphism of $(A{\otimes_{\scriptscriptstyle{}}}A)$-algebras.
It is noteworthy to mention that the algebra map $\zeta$, in contrast with the case of algebras over a field, is not known to be injective. However, if the base algebra $A$ is a Dedekind domain for example, then it is guaranteed that $\zeta$ is injective for every ${{\mathcal U}}$, see [@LaiachiGomez] for more details.
The completion of the finite dual and the convolution algebra {#ssec:zeta}
-------------------------------------------------------------
Let $(A,{{\mathcal U}})$ be a co-commutative (right) Hopf algebroid and consider its finite dual $(A,{{\mathcal U}}^{{{\boldsymbol{\circ}}}})$ as a commutative Hopf algebroid with structure maps given as in §\[ssec:Fdual\]. Here we assume that ${{\mathcal U}}$ is endowed with an admissible (increasing) filtration $\{F^n{{\mathcal U}}\}_{n\,\in \, \mathbb{N}}$ as in §\[ssec:FUstra\]. The admissible filtration on the Hopf algebroid ${{\mathcal U}}$ induces a filtration on the convolution algebra ${{\mathcal U}}^*$ given as in of §\[ssec:FUstra\]. It turns out that $(A,{{\mathcal U}}^*)$ with this filtration is a complete Hopf algebroid with structure maps explicitly given in §\[sec:Ustar\].
\[prop:App\] Let $(A,{{\mathcal U}})$ be a co-commutative (right) Hopf algebroid with an admissible filtration and consider its finite dual $(A,{{\mathcal U}}^{{{\boldsymbol{\circ}}}})$. Then the canonical map $\zeta:{{\mathcal U}}^{{{\boldsymbol{\circ}}}}\to{{\mathcal U}}^*$ of equation is filtered with respect to the filtrations $F_n{{\mathcal U}}^{{{\boldsymbol{\circ}}}}={{\mathcal K}}^n$ and $F_{n+1}{{\mathcal U}}^*={\mathsf{Ann}\left({F^{n}{{\mathcal U}}}\right)}$ for all $n\geq 0$ as in , where ${{\mathcal K}}={\mathrm{Ker}\left({\varepsilon_{\circ}:{{\mathcal U}}^{{{\boldsymbol{\circ}}}}\to A}\right)}$ is the kernel of the counit of ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$.
It can be easily checked that $\varepsilon_*\circ \zeta=\varepsilon_\circ$, where $\varepsilon_*:{{\mathcal U}}^*\to A$ and $\varepsilon_\circ:{{\mathcal U}}^{{{\boldsymbol{\circ}}}} \to A$ are the counits. In particular this implies that $\zeta\left({{\mathcal K}}\right)\subseteq {\mathrm{Ker}\left({\varepsilon_*}\right)}$. Hence the claim will be proved if we will be able to show that ${\mathrm{Ker}\left({\varepsilon_*}\right)}\subseteq F_1{{\mathcal U}}^*={\mathsf{Ann}\left({F^0{{\mathcal U}}}\right)}={\mathsf{Ann}\left({A}\right)}$, because in this case multiplicativity of $\zeta$ will imply that $$\zeta\left(F_n{{\mathcal U}}^{{\boldsymbol{\circ}}}\right)=\zeta({{\mathcal K}}^n)\subseteq \zeta({{\mathcal K}})^n\subseteq \left(F_1{{\mathcal U}}^*\right)^n\subseteq F_n{{\mathcal U}}^*.$$ However, if $f\in{\mathrm{Ker}\left({\varepsilon_*}\right)}$ then $f(1_{{\mathcal U}})=0$, whence $f(\tau_0(a))=f(1_{{\mathcal U}}\blacktriangleleft a)=f(1_{{\mathcal U}})a=0$. Consequently, $F^0{{\mathcal U}}=A\subseteq {\mathrm{Ker}\left({f}\right)}$, from which it follows that ${\mathrm{Ker}\left({\varepsilon_*}\right)}\subseteq {\mathsf{Ann}\left({F^0{{\mathcal U}}}\right)}$ as desired.
In light of Proposition \[coro:CHc\], $(A,{\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}})$ is a complete Hopf algebroid. On the other hand, we know from Proposition \[prop:Bosco\] that $(A,{{\mathcal U}}^{*})$ admits a structure of complete Hopf algebroid whenever the translation map of ${{\mathcal U}}$ is a filtered algebra map. Combining all this allows us to improve the content of Lemma \[lema:zeta\] and claim our main result as follows.
\[thm:triangle\] Let $(A,{{\mathcal U}})$ be a co-commutative (right) Hopf algebroid with an admissible filtration and assume that the translation map $\delta$ of ${{\mathcal U}}$ is a filtered algebra map. Then the $(A\otimes_{}A)$-algebra map $\zeta: {{\mathcal U}}^{{{\boldsymbol{\circ}}}} \to {{\mathcal U}}^*$ of equation factors through a continuous morphism ${\widehat{\zeta}}: {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \to {{\mathcal U}}^*$ of complete Hopf algebroids. Thus we have a commutative diagram: $$\xymatrix@R=15pt@C=30pt{ {{\mathcal U}}^{{{\boldsymbol{\circ}}}} \ar@{->}^-{\zeta}[rr] \ar@{->}_-{\gamma}[rd] & & {{\mathcal U}}^* \\ & {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \ar@{->}_-{{\widehat{\zeta}}}[ru] & }$$ In particular, this applies to ${{\mathcal U}}={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$, the universal enveloping Hopf algebroid of any Lie-Rinehart algebra $(A,L)$ such that $L_{{\scriptscriptstyle{A}}}$ is a finitely generated and projective module.
In Proposition \[prop:App\] we showed that $\zeta$ is a filtered algebra map. Thus, by applying the completion 2-functor of Theorem \[thm:Athm\] to $\zeta$ ($A$ is trivially filtered), we obtain that ${\widehat{\zeta}}$ is a continuous morphism of complete algebras. Now, since we already know that $\varepsilon_*\circ\zeta =\varepsilon_\circ$ and in view of Lemma \[lema:zeta\], we are left to show that ${\widehat{\zeta}}$ is compatible with the comultiplications and the antipodes. That is, the following relations hold $$\left(~{\widehat{\zeta}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{\widehat{\zeta}}~\right) \circ {\widehat{\Delta_{\circ}}}=\Delta_* \circ {\widehat{\zeta}}\quad \text{and} \quad {\widehat{\zeta}} \circ {\widehat{{{\mathcal S}}_\circ}} = {{\mathcal S}}_*\circ {\widehat{\zeta}}.$$ However, notice that to this aim it will be enough to show the following ones $$\gamma_{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*} \circ (\zeta{\otimes_{\scriptscriptstyle{A}}}\zeta) \circ \Delta_\circ=\Delta_* \circ \zeta\quad \text{and} \quad \zeta \circ {{\mathcal S}}_\circ = {{\mathcal S}}_*\circ \zeta.$$ Hence, let us consider an element of the form $\overline{\varphi {\otimes_{\scriptscriptstyle{T_M}}} m}\in {{\mathcal U}}^{{\boldsymbol{\circ}}}$. So we obtain an element in ${\widehat{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*}}={{\mathcal U}}^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{A}}}~}{{\mathcal U}}^*$ given by $$(\gamma_{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*}(\zeta{\otimes_{\scriptscriptstyle{A}}}\zeta)\Delta_\circ)\left(\overline{\varphi {\otimes_{\scriptscriptstyle{T_M}}} m}\right)={\widehat{\left(\sum_{i} \zeta \left( \overline{\varphi {\otimes_{\scriptscriptstyle{T_M}}} e_i} \right) {\otimes_{\scriptscriptstyle{A}}} \zeta \left( \overline{e_i^* {\otimes_{\scriptscriptstyle{T_M}}} m} \right) \right)}}={\underset{n\to\infty}{\lim}}\left(\sum_{i} \zeta \left( \overline{\varphi {\otimes_{\scriptscriptstyle{T_M}}} e_i} \right) {\otimes_{\scriptscriptstyle{A}}} \zeta \left( \overline{e_i^* {\otimes_{\scriptscriptstyle{T_M}}} m} \right) \right).$$ For every $u,v\in{{\mathcal U}}$, it satisfies $${\underset{n\to\infty}{\lim}}\left(\sum_{i} \zeta \left( \overline{\varphi {\otimes_{\scriptscriptstyle{T_M}}} e_i} \right) \left( \zeta \left( \overline{e_i^* {\otimes_{\scriptscriptstyle{T_M}}} m} \right)(u)v \right)\right) = {\underset{n\to\infty}{\lim}}\left(\sum_{i} \varphi\left( e_ie_i^*(mu)v \right)\right)=\varphi(m(uv)) = \zeta\left(\overline{\varphi {\otimes_{\scriptscriptstyle{T_M}}} m}\right)(uv)$$ whence, by the criterion of equation , we have that $\gamma_{{{\mathcal U}}^*{\otimes_{\scriptscriptstyle{A}}}{{\mathcal U}}^*} \circ (\zeta{\otimes_{\scriptscriptstyle{A}}}\zeta) \circ \Delta_\circ=\Delta_* \circ \zeta$. Moreover, $$\left(\zeta{{\mathcal S}}_\circ\left(\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}} m}\right)\right)(u)=\zeta\left(\overline{\mathrm{ev}_m {\otimes_{\scriptscriptstyle{T_{M^*}}}} \varphi}\right)(u)=\left(\varphi \centerdot u \right)(m)\stackrel{\eqref{Eq:star}}{=}\varepsilon_*\left(\varphi \left(m u_-\right)u_+\right)\stackrel{\eqref{Eq:tantip}}{=}{{\mathcal S}}_*\left(\zeta\left(\overline{\varphi {\otimes_{\scriptscriptstyle{T_{M}}}} m}\right)\right)(u),$$ for every $u \in {{\mathcal U}}$, and the proof is complete.
As in Example \[exam:Sacarrelli\], we are going to consider the $A$-bimodule $A{\otimes_{\scriptscriptstyle{}}} A$ to be endowed with the $K$-adic filtration given by $K:={\mathrm{Ker}\left({\mu_{A}:A{\otimes_{\scriptscriptstyle{}}} A\to A}\right)}$, even if $A$ itself is trivially filtered.
\[prop:zeroneveropen\] Let $(A,{{\mathcal U}})$ and $(A,{{\mathcal U}}^{{{\boldsymbol{\circ}}}})$ be as in Proposition \[prop:App\] and assume that $\zeta:{{\mathcal U}}^{{{\boldsymbol{\circ}}}}\to{{\mathcal U}}^{*}$ is injective. Then the following assertions are equivalent
1. the morphism ${\widehat{\zeta}}: {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \to {{\mathcal U}}^*$ is a filtered isomorphism,\[list:1\]
2. the morphism ${\mathrm{gr}}\left(\,{\widehat{\zeta}}\,\right):{\mathrm{gr}}{\left({\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}}\right)}\to {\mathrm{gr}}{\left({{\mathcal U}}^{*}\right)}$ is a graded isomorphism,\[list:2\]
3. the morphism ${\widehat{\zeta}}$ is surjective and the ${{\mathcal K}}$-adic filtration on ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ coincides with the one induced from ${{\mathcal U}}^{*}$ via $\zeta$,\[list:3\]
4. the graded morphism ${\mathrm{gr}}\left(\,{\widehat{\zeta}}\,\right):{\mathrm{gr}}{\left({\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}}\right)} \to {\mathrm{gr}}{\left({{\mathcal U}}^{*}\right)}$ is surjective and the ${{\mathcal K}}$-adic filtration on ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ coincides with the one induced from ${{\mathcal U}}^{*}$ via $\zeta$,\[list:4\]
5. the graded morphism ${\mathrm{gr}}\left({\zeta}\right):{\mathrm{gr}}{\left({{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}\right)} \to {\mathrm{gr}}{\left({{\mathcal U}}^{*}\right)}$ is surjective and the ${{\mathcal K}}$-adic filtration on ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ coincides with the one induced from ${{\mathcal U}}^{*}$ via $\zeta$,\[list:5\]
Moreover, the following assertions are equivalent as well
1. the morphism ${\widehat{\zeta}}: {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \to {{\mathcal U}}^*$ is an homeomorphism, \[list:6\]
2. the morphism ${\widehat{\zeta}}: {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \to {{\mathcal U}}^*$ is open and injective and ${{\mathcal U}}^{{\boldsymbol{\circ}}}$ is dense in ${{\mathcal U}}^*$, \[list:7\]
3. the ${{\mathcal K}}$-adic topology on ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ is equivalent to the one induced from ${{\mathcal U}}^{*}$ via $\zeta$ and ${{\mathcal U}}^{{\boldsymbol{\circ}}}$ is dense in ${{\mathcal U}}^*$.\[list:8\]
If in addition the morphism ${\widehat{\vartheta}}$ induced by the algebra map $\vartheta:A\otimes A\to{{\mathcal U}}^*$ of equation is a filtered isomorphism (as in the example mentioned in the introduction), then all the assertions from \[list:1\] to \[list:8\] are equivalent.
Before proceeding with the proof, there are some facts that have to be highlighted or recalled. First of all, notice that injectivity of $\zeta$ implies that the filtration on ${{\mathcal U}}^{{\boldsymbol{\circ}}}$ is separated. Secondly, recall that a morphism of filtered bimodules $f:M\to N$ is said to be strict if $f(F_kM)=f(M)\cap F_kN$ for all $k\geq 0$. In particular, $\zeta$ is strict if and only if the ${{\mathcal K}}$-adic filtration on ${{\mathcal U}}^{{\boldsymbol{\circ}}}$ coincides with the one induced from ${{\mathcal U}}^*$ via $\zeta$ itself. Thirdly, a filtered morphism (as ${\widehat{\zeta}}$ for example) is a filtered isomorphism if and only if it is bijective and strict. Finally, we have that ${\mathrm{gr}}\left(\gamma_{{\scriptscriptstyle{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}}\right):{\mathrm{gr}}\left({{\mathcal U}}^{{\boldsymbol{\circ}}}\right)\to{\mathrm{gr}}\left({\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}\right)$ is always an isomorphism (see e.g. [@NasOys Proposition D.3.1]), so that ${\mathrm{gr}}\left(\,{\widehat{\zeta}}\,\right)$ is injective (resp. surjective, bijective) if and only if ${\mathrm{gr}}(\zeta)$ is. Now, by applying [@NasOys Cor. D.III.5, D.III.6 and D.III.7] one proves that \[list:3\] $\Leftrightarrow$ \[list:1\] $\Leftrightarrow$ \[list:2\] $\Leftrightarrow$ \[list:4\] $\Leftrightarrow$ \[list:5\].
For the remaining equivalent facts, notice that ${\widehat{\zeta}}$ is surjective if and only if for every $x\in{{\mathcal U}}^*$ and for all $k\geq 0$, there exists $m_k\in {{\mathcal U}}^{{\boldsymbol{\circ}}}$ such that $x-m_k\in F_k{{\mathcal U}}^*$ or, equivalently, if and only if ${{\mathcal U}}^{{\boldsymbol{\circ}}}$ is dense in ${{\mathcal U}}^*$. This proves the equivalence between \[list:6\] and \[list:7\], so that we may focus on \[list:7\] $\Leftrightarrow$ \[list:8\]. Assume initially that ${\widehat{\zeta}}$ is an open and injective map. From this it follows that for all $h\geq 0$, $F_h{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}$ is open in ${{\mathcal U}}^*$. In particular, there exists $k\geq 0$ such that $F_k{{\mathcal U}}^*\subseteq F_h{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}$. Thus, $M\cap F_k{{\mathcal U}}^*\subseteq M\cap F_h{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}=F_h{{\mathcal U}}^{{\boldsymbol{\circ}}}$, which expresses the fact that the ${{\mathcal K}}$-adic topology is equivalent to the induced one. Conversely, assume that these two topologies are equivalent and that ${{\mathcal U}}^{{\boldsymbol{\circ}}}$ is dense in ${{\mathcal U}}^*$ (that is, that ${\widehat{\zeta}}$ is surjective). We plan to prove first that every $F_t{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}$ is open in ${{\mathcal U}}^*$ (which implies that ${\widehat{\zeta}}$ is open) and then that ${\widehat{\zeta}}$ is injective. To this aim, pick $t\geq 0$ and consider $k$ (which we may assume greater or equal than $t$) such that ${{\mathcal U}}^{{\boldsymbol{\circ}}}\cap F_k{{\mathcal U}}^*\subseteq F_t{{{\mathcal U}}^{{\boldsymbol{\circ}}}}$. Then every $y\in F_k{{\mathcal U}}^*$ is of the form $y={\widehat{\zeta}}\left(\left(m_i+F_i{{\mathcal U}}^{{\boldsymbol{\circ}}}\right)_{{\scriptscriptstyle{i\geq 0}}}\right)=\left(m_i+F_i{{\mathcal U}}^*\right)_{{\scriptscriptstyle{i\geq 0}}}$ for some $\left(m_i+F_i{{\mathcal U}}^{{\boldsymbol{\circ}}}\right)_{{\scriptscriptstyle{i\geq 0}}}\in{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}$ such that $m_k\in F_k{{\mathcal U}}^*\cap {{\mathcal U}}^{{\boldsymbol{\circ}}}\subseteq F_t{{\mathcal U}}^{{\boldsymbol{\circ}}}$, whence $$m_t+F_t{{\mathcal U}}^{{\boldsymbol{\circ}}}= m_k+F_t{{\mathcal U}}^{{\boldsymbol{\circ}}}= 0$$ in the quotient ${{\mathcal U}}^{{\boldsymbol{\circ}}}/F_t{{\mathcal U}}^{{\boldsymbol{\circ}}}$ and so $\left(m_i+F_i{{\mathcal U}}^{{\boldsymbol{\circ}}}\right)_{{\scriptscriptstyle{i\geq 0}}}\in F_t{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}$. Summing up, we showed that for every $t\geq 0$, there exists a $k\geq t$ such that $F_k{{\mathcal U}}^*\subseteq F_t{\widehat{{{\mathcal U}}^{{\boldsymbol{\circ}}}}}$ and hence that ${\widehat{\zeta}}$ is an open map. Let us show now that it is injective as well. To this aim, let $\left(m_i+F_i{{\mathcal U}}^{{\boldsymbol{\circ}}}\right)_{{\scriptscriptstyle{i\geq 0}}}$ be an element in ${\mathrm{Ker}\left({\,{\widehat{\zeta}}\,}\right)}$. This implies that $m_k\in F_k{{\mathcal U}}^*\cap {{{\mathcal U}}^{{\boldsymbol{\circ}}}}$ for all $k\geq 0$ and that, since the two topologies are equivalent, for every $i\geq 0$ there exists $j_i\geq i$ such that $F_j{{\mathcal U}}^*\cap {{\mathcal U}}^{{\boldsymbol{\circ}}}\subseteq F_i{{\mathcal U}}^{{\boldsymbol{\circ}}}$, whence $$m_k+F_k{{\mathcal U}}^{{\boldsymbol{\circ}}}= m_{j_k}+F_k{{\mathcal U}}^{{\boldsymbol{\circ}}}\in \left(F_{j_k}{{\mathcal U}}^*\cap {{\mathcal U}}^{{\boldsymbol{\circ}}}\right)+F_k{{\mathcal U}}^{{\boldsymbol{\circ}}}=F_k{{\mathcal U}}^{{\boldsymbol{\circ}}},$$ so that $\left(m_i+F_i{{\mathcal U}}^{{\boldsymbol{\circ}}}\right)_{{\scriptscriptstyle{i\geq 0}}}=0$. With this we conclude the proof that \[list:6\] $\Leftrightarrow$ \[list:7\] $\Leftrightarrow$ \[list:8\].
Finally, \[list:1\] clearly implies \[list:6\] and since $\zeta\circ \eta=\vartheta$, if ${\widehat{\vartheta}}$ is a filtered isomorphism then ${\widehat{\zeta}}$ admits the filtered section ${\widehat{\etaup}}\circ {\widehat{\vartheta}}^{-1}$. Therefore, if in such a case ${\widehat{\zeta}}$ is also injective, then it is a filtered isomorphism.
The subsequent Corollary gives another condition for the injectivity of the map $\zeta$, and so another application of the result [@LaiachiGomez Theorem 4.2.2].
\[coro:Equivalence\] Let $(A,L)$ be a Lie-Rinehart algebra and consider ${{\mathcal U}}={{\mathcal V}}_{{\scriptscriptstyle{A}}}(L)$ its universal enveloping Hopf algebroid. Assume that ${{\mathcal U}}^{\circ}$ is an Hausdorff topological space with respect to the ${{\mathcal K}}$-adic topology and that ${\widehat{\zeta}}$ is an homeomorphism. Then $\zeta$ is injective, and therefore, there is an equivalence of symmetric rigid monoidal categories between the category of right $L$-modules and the category of right ${{\mathcal U}}^{\circ}$-comodules, with finitely generated and projective underlying $A$-module structure.
\[rem:laventanadelfrente\] As a final remark, we point out that the completion of $\zeta$ might fail to be an homeomorphism, even if $\zeta$ is injective and $A$ is the base field, as it is shown in [@LaiachiPaolo] for an apparently trivial example: namely the enveloping Hopf algebra ${{\mathcal U}}=U(L)$ of the one dimensional complex Lie algebra $L$. Nevertheless, we believe that in the Hopf algebroid framework some unexpected result may show up. For instance, we just mention that the classical Sweedler dual coalgebra ${{\mathcal U}}^{o}$ of the first Weyl algebra ${{\mathcal U}}$ as in Example \[exam:URSO\] is zero, while the finite dual Hopf algebroid ${{\mathcal U}}^{{{\boldsymbol{\circ}}}}$ is not. This, in our opinion, suggests that the problem of ${\widehat{\zeta}}$ being an homeomorphism or not for universal enveloping Hopf algebroids is still worthy to be studied.
In fact, we believe that the presence of a non-trivial algebra of infinite jets ${{\mathcal J}}(A)={\widehat{A\otimes A}}$ (see Example \[exam:Sacarrelli\] for the definition) could make the difference. Let us consider again the diagram for $(A,{{\mathcal U}})$ a co-commutative Hopf algebroid with an admissible filtration such that the translation map $\delta$ is filtered, $$\xymatrix@R=15pt@C=30pt{ {\widehat{{{\mathcal U}}^{{{\boldsymbol{\circ}}}}}} \ar@{->}^-{{\widehat{\zeta}}}[rr] & & {{{\mathcal U}}}^{*} \\ & {{\mathcal J}}(A) \ar@{->}_-{{\widehat{\vartheta}}}[ru] \ar@{->}^-{{\widehat{\etaup}}}[lu] & }$$ In case ${\widehat{\vartheta}}$ turns out to be an isomorphism of complete Hopf algebroids (as for example when $A={\mathbb{C}}[X]$ and ${{\mathcal U}}={\mathsf{Diff}}(A)$), then one may reasonably conjecture that ${\widehat{\zeta}}$ could become an isomorphism as well.
If we look instead at the aforemenioned case, that is to say, $A={\mathbb{C}}$ and $L={\mathbb{C}}X$, then ${{\mathcal J}}({\mathbb{C}})={\widehat{{\mathbb{C}}\otimes{\mathbb{C}}}}\cong {\mathbb{C}}\otimes{\mathbb{C}}\cong {\mathbb{C}}$ and hence the completion ${\widehat{\vartheta}}$ of the algebra map $\vartheta\colon{\mathbb{C}}\otimes{\mathbb{C}}\to U(L)$ corresponds to the unit ${\mathbb{C}}\to {\mathbb{C}}[[X]]$, which is far away from being an isomorphism.
Complete bimodules and the completion 2-functor {#sec:CBCF}
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In this section we revise some notions on linear topology of rings and modules which are well-known or folklore, apart perhaps from the adjunction between the topological tensor product and the continuous hom functor. For a more exhaustive treatment of the material of this section, we refer to [@MR0163908; @MR0358652; @MR1420862; @NasOys]. The reason that pushed us to put this material in a comprehensive way was the apparent lack of a single reference in the literature which could clarify in an exhaustive way the constructions performed for complete Hopf algebroids in Subsection \[sec:CHA\]. We decided then to include a detailed exposition, especially for readers who are not familiar with this context.
Filtered bimodules over filtered algebras {#ssec:CBFA}
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Here we retrieve some basic notions and results in order to make explicit our assumptions and fix some notations. For further details on this subsection, we refer to [@MR1420862 Chapter I, §§1-3] and [@MR0358652 Chapter I-III].
As far as we will be concerned with this, a *linear topology* on an algebraic structure is a topology on the underlying set with respect to which all structure maps are continuous. An algebra $R$ is said to be *filtered* if there exists a decreasing chain of two-sided ideals $$R=F_{0}R\supseteq F_{1}R\supseteq \cdots$$ that satisfies $F_{n}R\cdot F_{m}R\subseteq F_{n+m}R$ for every $m,n\in {\mathbb{N}}$. We will denote it as a pair $\left(
R,F_{n}R\right) $ or we will just say that $R$ is filtered. Given a filtration on $R$, this induces a linear topology on it such that $\left\{ F_{n}R\mid n\in {\mathbb{N}}\right\} $ is a fundamental system of neighborhoods of $0$ and $\left\{x+F_{n}R\mid n\in {\mathbb{N}}\right\} $ is a fundamental system of neighborhoods of $x\in R$. A subset $U$ is open in $R$ if and only if for every $x\in U$ there exists an $n\in {\mathbb{N}}$ such that $x+F_{n}R\subseteq U$, while a subset $V$ is closed if and only if $V=\cap_{n \geq 0}\big( V+ F_{n}R\big)$. This, in particular, implies that $\left\{ x+F_{n}R\mid x\in R,n\in {\mathbb{N}}\right\} $ is a basis for this topology. Furthermore, by [@MR0358652 III.49, § 6.3], this topology is compatible with the ring structure on $R$ (cf. also Example 3 in the same page).
\[rem:K\] We will always endow the base ring ${\Bbbk}$ with the trivial filtration $F_0({\Bbbk})={\Bbbk}$ and $F_n({\Bbbk})=0$ for every $n\geq 0$. This filtration induces on ${\Bbbk}$ the discrete topology because $\left\{0\right\}$ is open by definition, whence every point is open. This topology is always compatible with all algebraic structures on ${\Bbbk}$ (even $${{(-)^{-1}} \colon {{\Bbbk}^*} \rightarrow {{\Bbbk}^*};\, \Big({k} \mapsto {k^{-1}}\Big)}$$ in case ${\Bbbk}$ is a field, cf. e.g. [@MR0358652 III.55, § 6.7, Example 1]).
In view of Remark \[rem:K\] and of [@MR0358652 III.53, § 6.6, Remarque], the linear topology on $R$ is compatible with the module structure, too. In other words, $R$ is a topological algebra.
Let $R$, $S$ be filtered algebras. An $\left( S,R\right) $-bimodule $M$ is said to be *filtered* (as a bimodule) if there exists a decreasing chain of submodules $$M=F_{0}^{S,R}M\supseteq F_{1}^{S,R}M\supseteq \cdots$$ that satisfies $$F_{n}S\cdot F_{m}^{S,R}M\subseteq F_{n+m}^{S,R}M\text{\qquad and\qquad }F_{n}^{S,R}M\cdot F_{m}R\subseteq F_{n+m}^{S,R}M$$ for every $m,n\in {\mathbb{N}}$. We will denote it as a pair $\left(M,F_{n}^{S,R}M\right) $ or we will just say that $M$ is filtered. Note that each $F_{n}^{S,R}M$ is in particular an $\left( S,R\right) $-subbimodule. If $M$ is a filtered $\left( S,R\right) $-bimodule then it can be endowed with a linear topology such that the given filtration forms a fundamental system of neighborhoods of 0. As above, a basis for this topology is given by the open sets $\left\{m+F_{n}^{S,R}M\mid m\in M,n\in {\mathbb{N}}\right\} $. For the sake of simplicity, the filtration on a $(S,R)$-bimodule $M$ will be denoted by $\left\{F_nM\mid n\in {\mathbb{N}}\right\}$.
A filtration $\left\{F_nM\mid n\in{\mathbb{N}}\right\}$ on an $(S,R)$-bimodule $M$ is said to be *finer* than another filtration $\left\{G_nM\right\}$ on $M$ if and only if for every $n\in{\mathbb{N}}$ there exists an $m\in{\mathbb{N}}$ such that $F_mM\subseteq G_nM$ (cf. [@MR0358652 I.38, § 6.3, Proposition 4]). As a consequence, the linear topology induced by the filtration $\left\{F_nM\right\}$ is finer than the one induced by the filtration $\left\{G_nM\right\}$. Two filtrations $\left\{F_nM\mid n\in{\mathbb{N}}\right\}$ and $\left\{G_nM\mid n\in{\mathbb{N}}\right\}$ on an $(S,R)$-bimodule $M$ are said to be *equivalent* if and only if each one is finer than the other one. In particular, the linear topologies induced on $M$ are equivalent.
The category of filtered $\left( S,R\right) $-bimodules, denoted by ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$, is defined as follows. The objects are filtered $\left(S,R\right) $-bimodules $M$. The arrows are $\left(S,R\right) $-bimodule maps $f:M\rightarrow N$ that satisfies $f\left(F_{n}M\right)\subseteq F_{n}N$, for every $n\in {\mathbb{N}}$.[^3] It is a (co)complete additive category with kernels and cokernels. If $(M_\lambda,f_{\lambda,\,\mu})$ is a projective system of filtered $(S,R)$-bimodules then its projective limit ${\varprojlim_{\lambda}\left({M_\lambda}\right)}$ is filtered with filtration $$\label{eq:projFiltr}
F_k\left({\varprojlim_{\lambda}\left({M_\lambda}\right)}\right)={\varprojlim_{\lambda}\left({F_k\left(M_\lambda\right)}\right)}.$$ We have a functor ${{D} \colon {{{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}_{{\scriptscriptstyle{R}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}}}$ which associates to $M$ in ${{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}_{{\scriptscriptstyle{R}}}}$ the $(S,R)$-bimodule $M$ itself with filtration $$F_nM:=\sum_{h+k=n}F_kS\cdot M\cdot F_hR$$ for all $n\in{\mathbb{N}}$. This filtration is called the *induced* filtration. If $(M,F_nM)$ is a filtered $(S,R)$-bimodule, for $k\in{\mathbb{N}}$ the *$k$-shifted module $M[k]$* is the same $(S,R)$-bimodule as $M$, but filtered with a different filtration given by $F_nM[k]=F_{n+k}M$.[^4]
\[rem:continuity\] As every function from a discrete topological space to any topological space is continuous, every morphism from a trivially filtered bimodule to any bimodule is automatically filtered.
Moreover, independently from being filtered or not, an $(S,R)$-bimodule homomorphism ${{f} \colon {M} \rightarrow {N}}$ is continuous with respect to the linear topologies induced by the filtrations if and only if for every $n\in {\mathbb{N}}$ there exists $m\left(n\right) \in {\mathbb{N}}$ such that $f\left( F_{m\left( n\right) }M\right) \subseteq F_{n}N$. In particular, any morphism of filtered $(S,R)$-bimodules is continuous (cf. also [@MR1320989]). On the other hand, given $M,N$ two filtered $(S,R)$-bimodules, one can prove that an $(S,R)$-bimodule homomorphism ${{f} \colon {M} \rightarrow {N}}$ is continuous with respect to the linear topologies induced by the given filtrations if and only if there exists a sub-filtration on $M$ equivalent to the former one and with respect to which $f$ is filtered.
In light of Remark \[rem:continuity\], we will distinguish *homeomorphism* as topological spaces from *filtered isomorphism* as filtered bimodules: the second terminology will be used for isomorphism of filtered bimodules whose inverse is also filtered. Note that every filtered isomorphism is in fact an homeomorphism.
Now, if $M$ and $N$ are filtered $(S,R)$ and $(R,T)$-bimodules respectively, then there is a natural filtration on their tensor product $M{\otimes_{\scriptscriptstyle{R}}}N$ given by $$\label{eq:filtrations}
{{\mathcal F}}_n\left(M{\otimes_{\scriptscriptstyle{R}}}N\right):=\sum_{p+q=n}{\mathsf{Im}\left({F_{p}M{\otimes_{\scriptscriptstyle{R}}}F_{q}N}\right)}$$ for all $n\in{\mathbb{N}}$, where the notation in the right hand side is the obvious one. We will consider this one as the standard filtration on the tensor product of filtered $(S,R)$ and $(R,T)$-bimodules, for all algebras $S$, $R$, $T$.
If ${{f} \colon {M} \rightarrow {M^{\prime}}}$ and ${{g} \colon {N} \rightarrow {N^{\prime}}}$ are morphisms of filtered $(S,R)$ and $(R,T)$-bimodules respectively, then $f{\otimes_{\scriptscriptstyle{R}}}g : M{\otimes_{\scriptscriptstyle{R}}}N \rightarrow M'{\otimes_{\scriptscriptstyle{R}}}N'$ is a morphism of filtered $(S,T)$-bimodules. In particular, we have a bicategory ${{\mathcal B}}im_{\Bbbk}^{\mathsf{flt}}$ which has filtered algebras as $0$-cells and whose categories of $\{1,2\}$-cells are the categories of filtered bimodules over filtered algebras with vertical and horizontal compositions given by the composition of bilinear morphisms and the usual tensor product, filtered as in , respectively.
Notice that the category ${{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{flt}}$ of filtered modules is monoidal with tensor product $\otimes$ and unit ${\Bbbk}$. Filtered algebras are monoids in ${{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{flt}}$ and filtered $(S,R)$-bimodules are objects in ${}_{{\scriptscriptstyle{S}}}^{}({{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{flt}})^{}_{{\scriptscriptstyle{R}}}$. This in particular, means that the categories of $\{1,2\}$-cells ${}_{{\scriptscriptstyle{S}}}({{\mathcal B}}im_{\Bbbk}^{\mathsf{flt}}){}_{{\scriptscriptstyle{R}}}$ of ${{\mathcal B}}im_{\Bbbk}^{\mathsf{flt}}$ are exactly $ {}_{{\scriptscriptstyle{S}}}^{}({{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{flt}})^{}_{{\scriptscriptstyle{R}}}$.
Similar to the discrete case (i.e., usual bimodules), we have an isomorphism between the category ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$ of filtered $(S,R)$-bimodules and the category ${_{S\otimes R^{{\scriptscriptstyle{{\mathsf{op}}}}}}{\mathsf{Mod}}^{{\mathsf{flt}}}}$ of filtered $S\otimes R^{{\scriptscriptstyle{\text{op}}}}$-modules, where $R^{{\scriptscriptstyle{{\mathsf{op}}}}}$ denotes the opposite algebra of $R$ and $S\otimes R^{{\scriptscriptstyle{\text{op}}}}$ is a filtered algebra with filtration as in .
\[ex:HomFiltr\] Let $P\in {{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{T}}}}$ and $N\in {{}^{}_{{\scriptscriptstyle{R}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{T}}}}$ and let us denote by ${\mathsf{Hom}^{\mathsf{flt}}_{{\scriptscriptstyle{-,T}}}\left({N},{P}\right)}$ the abelian group of filtered morphisms ${{f} \colon {N} \rightarrow {P}}$ which are $T$-linear. As one can expect it is an object in ${{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}_{{\scriptscriptstyle{R}}}}$. It is also filtered with filtration given by $$\label{eq:FiltrHom}
F_n{\mathsf{Hom}^{\mathsf{flt}}_{{\scriptscriptstyle{-,T}}}\left({N},{P}\right)}={\mathsf{Hom}^{\mathsf{flt}}_{{\scriptscriptstyle{-,T}}}\left({N},{P[n]}\right)}=\Big\{f\in {\mathrm{Hom}_{{\scriptscriptstyle{-,T}}}\left(N,\,P\right)}\mid f\left(F_kN\right)\subseteq F_{n+k}P\text{ for all }k\geq0\Big\}.$$ Observe that this is the filtration induced by the filtered bimodule of all homomorphisms of finite degree $\mathsf{HOM}_{-,T}\left(N,P\right)$ onto its subgroup $F_0\mathsf{HOM}_{-,T}\left(N,P\right)={\mathsf{Hom}^{\mathsf{flt}}_{{\scriptscriptstyle{-,T}}}\left({N},{P}\right)}$ (see e.g. [@MR1420862 I.2.5]).
It is worthy to mention that the linear topology makes of $M$ a linearly topologized $\left( S,R\right) $-bimodule in the sense of [@MR1608699 Definition 1.1], whence it endows ${\mathsf{End}^{\mathsf{cnt}}_{-R}\left({M}\right)}$ with the topology of uniform convergence on $M$.
The completion 2-functor {#ssec:CF}
------------------------
In this subsection we will recall the construction of the completion functor from the category of filtered bimodules to the one of complete bimodules. As a main reference for the material presented here, we suggest [@NasOys Chap. D, §§ I-II] and [@MR1420862 Chap. I, §3].
Let $S$, $R$ be filtered algebras and let $\left( M, F_{n}M\right) $ be a filtered $(S,R)$-bimodule. We recall that $M$ is *Hausdorff* (or *separable*) if and only if for every pair of elements $x,y\in M$ there exist two open sets $U,V\subseteq M$ such that $x\in U$, $y\in V$ and $U\cap V=\emptyset$. However, by definition of the linear topology on $M$, this is equivalent to say that $\bigcap_{n\in{\mathbb{N}}}F_nM=0$. Moreover, a sequence $\left\{m_k \mid k\geq0\right\}$ in a Hausdorff filtered $(S,R)$-bimodule $M$ is a *Cauchy sequence* if and only if for every $p\in {\mathbb{N}}$, there exists $q\in {\mathbb{N}}$ such that for all $k,h\geq q$ we have that $m_k-m_h\in F_p(M)$. It is *convergent* to an element $m\in M$ if and only if for every $p\in {\mathbb{N}}$, there exists $q\in {\mathbb{N}}$ such that for all $k\geq q$ we have that $m-m_k\in F_p(M)$. The bimodule $M$ is said to be *complete* with respect to the linear topology induced by the filtration if and only if every Cauchy sequence is convergent.
Now, the filtration on $M$ gives rise to a projective system of $(S,R)$-bimodules given by $$\label{eq:filtprojsyst}
{{\pi _{m,n}^{M}} \colon {\frac{M}{F_{m}M}} \longrightarrow {\frac{M}{F_{n}M}};\quad \Big({x+F_{m}M} \longmapsto { x+F_{n}M}\Big)}$$ for all $m\geq n$ and this allows us to give an effective characterization of when a filtered bimodule is Hausdorff and complete, as well as a universal construction of its Hausdorff completion. To this aim, set $${\widehat{M}}:={\varprojlim_{n}\left({ \frac{M}{F_{n}M}}\right)}$$ and consider the canonical morphism ${{\gamma_M} \colon {M} \rightarrow {{\widehat{M}}}}$ rendering commutative the diagram $$\label{Eq:gamma}
\xymatrix @R=15pt @C=15pt{ M \ar@{-->}^-{\gamma_M}[rr] \ar@{->}_-{\pi_n}[dr] & & {\widehat{M}} \ar@{->}^-{p_n}[dl] \\& \frac{M}{F_nM} &}$$ for all $n\in{\mathbb{N}}$, where ${{p_{n}} \colon {{\widehat{M}}} \rightarrow {{M}/{F_{n}M}}}$ are the natural projections. The subsequent result can be proven directly (see also [@NasOys Proposition D.II.3]).
\[prop:completion\] An object $\left( M, F_{n}M\right) $ in ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$ is complete and Hausdorff as a topological space if and only if the map $\gamma_{{\scriptscriptstyle{M}}}$ of diagram is an isomorphism.
This justifies the following definition.
\[def:completion\] \[Def:T2compl\]For a filtered $\left( S,R\right) $-bimodule $M$, we define its *Hausdorff completion* to be the inverse limit ${\widehat{M}}$ over the natural projective system as in .
As a matter of terminology, henceforth we will understood that a complete bimodule is Hausdorff as well, whence we will just refer to *complete* bimodules and *completions* of filtered bimodules.
The fact that Definition \[def:completion\] is consistent (i.e., that the completion ${\widehat{M}}$ of a filtered $(S,R)$-bimodule $M$ is a complete $(S,R)$-bimodule) follows from the subsequent Lemma \[lemma:Mhatcomplete\] (see also [@NasOys Proposition D.II.3]).
\[lemma:Mhatcomplete\] Let $\left(M,F_nM\right)$ be a filtered $(S,R)$-bimodule. We have an isomorphism ${\widehat{M}}/F_{n}{\widehat{M}}\cong {M}/{F_{n}M}$ in ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$ for all $n\geq 0$ which is compatible with the morphisms of the projective system $\eqref{eq:filtprojsyst}$. In particular, ${\widehat{M}}={\varprojlim_{n}\left({M/F_nM}\right)}$ is a complete $(S,R)$-bimodule.
The proof is omitted, but we point out that every quotient module $M/F_nM$ is filtered with the discrete filtration $F_k\left(M/F_nM\right):=F_kM/F_nM$ for all $k\geq 0$ and that ${\widehat{M}}$ is filtered with the filtration given in , which satisfies $$\label{eq:Mhatfiltration}
F_{m}{\widehat{M}}={\mathrm{Ker}\left({ p_{m}:{\widehat{M}}\rightarrow \frac{M}{F_{m}M}}\right)}.$$ In particular, the canonical $(S,R)$-bilinear morphism ${{\gamma_{{\scriptscriptstyle{M}}}} \colon {M} \rightarrow {{\widehat{M}}}}$ is always filtered and every $M/F_nM$ is a complete $(S,R)$-bimodule. Moreover, the canonical projections $\pi_n^{{\scriptscriptstyle{M}}}:M\to M/F_nM$ are filtered for all $n\geq 0$, whence continuous, and every $F_nM$ is closed in $M$, as preimage of the closed set $0$ in $M/F_nM$.
\[rem:gamma\] Assume that $M$ is a complete $(S,R)$-bimodule. Then the inverse morphism of ${{\gamma_{{\scriptscriptstyle{M}}}} \colon {M} \rightarrow {{\widehat{M}}}}$ is given by the assignment $$\label{eq:invCan}
{{\sigma_{{\scriptscriptstyle{M}}}} \colon {{\widehat{M}}} \rightarrow {M};\, \Big({\left(x_k+F_kM\right)_{k\geq 0}} \mapsto {\lim_{k\to\infty}(x_k)}\Big)},$$ which is well-defined because the limit $\underset{k\to\infty}{\lim}(x_k)$ is independent of the representatives chosen for the equivalence classes $x_k+F_kM\in {M}/{F_kM}$, $k\geq0$.
The following conventions turn out to be very useful in dealing with completions, whence we opted for introduce them in this general context.
\[Not:infty\] Given a filtered $(S,R)$-bimodule $M$, by a slight abuse of notation we are going to denote the elements of its completion ${\widehat{M}}$ by ${\widehat{x}}_\infty$, meaning by that an ${\mathbb{N}}$-tuple $\big( x_n + F_nM \big)_{n \geq 0} \in \prod_{n \geq 0} {M}/{F_nM}$ satisfying $x_{n+1}-x_{n} \in F_{n}M$ for all $n \geq 0$. Observe that this condition is in fact equivalent to claim that $\big( x_n + F_nM \big)_{n \geq 0} \in {\varprojlim_{n}\left({ {M}/{F_{n}M}}\right)}$. If $x \in M$, then its image via $\gamma_{{\scriptscriptstyle{M}}}$ in ${\widehat{M}}$ will be denoted by ${\widehat{x}}$, which corresponds to the ${\mathbb{N}}$-tuple $\big( x + F_nM \big)_{n \geq 0}$. When $M$ is complete, and so $\gamma_{{\scriptscriptstyle{M}}}$ and $\sigma_{{\scriptscriptstyle{M}}}$ are mutually inverse functions, the element $x_\infty:=\sigma_{{\scriptscriptstyle{M}}}\left({\widehat{x}}_\infty\right)$ belongs to $M$ and the condition $$\big( x_n + F_nM \big)_{n \geq 0}={\widehat{x}}_{\infty}=\gamma_{{\scriptscriptstyle{M}}}\sigma_{{\scriptscriptstyle{M}}}\left({\widehat{x}}_\infty\right) = \gamma_{{\scriptscriptstyle{M}}}\left(x_\infty\right)=\big( x_\infty + F_nM \big)_{n \geq 0}$$ implies the following: for all $n \geq 0$, there exists $k(n) \geq 0$ such that for every $p \geq k(n)$ we have $x_{\infty} - x_p \in F_nM$, i.e., $x_\infty={\underset{n\to\infty}{\lim}(x_n)}$ in $M$.
\[rem:limits\] Observe that a sequence $\left\{x_n\mid n\in{\mathbb{N}}\right\}$ in $M$ is Cauchy if and only if the sequence $\left\{{\widehat{x_n}}\mid n\in{\mathbb{N}}\right\}$ in ${\widehat{M}}$ is Cauchy. Moreover, every element ${\widehat{x}}_\infty\in{\widehat{M}}$ can be seen as a Cauchy sequence in $M$, in the sense that if ${\widehat{x}}_\infty=\left(x_n+F_nM\right)_{n\geq0}\in {\widehat{M}} $ then it follows that $\left\{x_n\mid n\in{\mathbb{N}}\right\}$ is Cauchy in $M$. It turns out, with the conventions introduced, that ${\widehat{x}}_\infty={\underset{n\to\infty}{\lim}}\left({\widehat{x_n}}\right)$ in ${\widehat{M}}$ where $\left\{x_n\mid n\in{\mathbb{N}}\right\}$ is the Cauchy sequence defining ${\widehat{x}}_\infty$.
Again, by a slightly but consistent abuse of notation, we are going to write ${\widehat{x}}_\infty={\underset{n\to\infty}{\lim}}\left(x_n\right)$ whenever ${\widehat{x}}_\infty=\left(x_n+F_nM\right)_{n\geq0}$. This proves to be very useful when one will have to compute, for example, ${\widehat{f}}\,\left({\widehat{x}}_\infty\right)$ for a given $f:M\to N$ (the meaning of ${\widehat{f}}$ is the expected one). Indeed $$\label{eq:fhat}
{\widehat{f}}\,\left({\underset{n\to\infty}{\lim}}(x_n)\right)={\widehat{f}}\,\left({\widehat{x}}_\infty\right)=\left(f(x_n)+F_nM\right)_{n\in{\mathbb{N}}}={\underset{n\to\infty}{\lim}}(f(x_n))=:{\widehat{f(x)}}_{\infty}.$$
Notice further that for two given Cauchy sequences $\{x_n\mid n\in{\mathbb{N}}\}$ and $\{y_n\mid n\in{\mathbb{N}}\}$ in $M$, we have that ${\underset{n\to\infty}{\lim}}(x_n)={\underset{n\to\infty}{\lim}}(y_n)$ in ${\widehat{M}}$ if and only if ${\widehat{x}}_\infty={\widehat{y}}_\infty$, if and only if $x_n-y_n\in F_nM$ for all $n\in{\mathbb{N}}$.
\[rem:gammafiltiso\] With this new notations, it is easy to show that if $M$ is complete, then the morphism $\sigma_{{\scriptscriptstyle{M}}}$ is filtered as well, so that $\gamma_{{\scriptscriptstyle{M}}}$ is a filtered isomorphism. Indeed, if we have ${\widehat{x}}_\infty \in {\mathrm{Ker}\left({ p_{n}:{\widehat{M}}\rightarrow {M}/{F_{n}M}}\right)}$ and if $x_\infty=\sigma_{{\scriptscriptstyle{M}}}\left({\widehat{x}}_\infty\right)$ is the limit of the sequence $\left\{x_k\mid k\geq 0\right\}$ as in Remark \[rem:gamma\], then $0=p_k\left({\widehat{x_\infty}}\right)=x_\infty + F_kM$ for all $0\leq k\leq n$. In particular $x_\infty\in F_nM$.
For the sake of completeness, recall that the inverse limit topology on ${\widehat{M}}$ is the coarsest topology for which all the canonical projections $p_n$’s are continuous. It can be proven that the inverse limit topology is equivalent to the linear topology induced by the filtration .
Consistently with our definition of a complete bimodule over filtered algebras, we say that a filtered algebra $R$ is a *complete* algebra if it is also complete as a module. Given a filtered algebra $R$, its completion ${\widehat{R}}$ as a filtered module inherits a structure of filtered algebra itself, which is complete as a module and such that the natural map $\gamma_{{\scriptscriptstyle{R}}} : R \to {\widehat{R}}$ is a map of filtered algebras. Explicitly, the multiplication ${{\widetilde{\mu}} \colon {{\widehat{R}}\times {\widehat{R}}} \rightarrow {{\widehat{R}}}}$ is given by $\widetilde{\mu}\left({\widehat{x}}_\infty, {\widehat{y}}_\infty\right)={\widehat{xy}}_\infty$ and the unit is $1_{{\scriptscriptstyle{{\widehat{R}}}}}={\widehat{1_{{\scriptscriptstyle{R}}}}}$. Therefore, the completion of a filtered algebra is a complete algebra, as expected. We point out in advance that ${\widehat{R}}$ (in fact, any complete algebra) with a slightly different multiplication will turn out to be a monoid inside the monoidal category of complete modules with a suitable tensor product (see the Lemma \[rem:calg\] below): in general, indeed, the ordinary tensor product $\otimes$ does not endow ${{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}}$ with a monoidal structure. In this way, we will be able to recover the definition of a complete algebra as a monoid in a monoidal category.
\[rem:reflimit\] It is well-known that the forgetful functor from a category of modules to the category of abelian groups creates, preserves and reflects limits (for the terminology see e.g. [@MR1712872 §V.1] and [@adamek §13]). As a consequence, for a given filtered $(S,R)$-bimodule $M$, if we consider $\gamma_{{\scriptscriptstyle{S}}}\otimes \gamma_{{\scriptscriptstyle{R^{\text{op}}}}}:S\otimes R^{\text{op}}\to {\widehat{S}}\otimes {\widehat{R}}^{\text{op}}$ and if ${{\mathcal R}}:{{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}_{{\scriptscriptstyle{{\widehat{R}}}}}}\to {{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}_{{\scriptscriptstyle{R}}}}$ is the restriction of scalars functor, then a projective cone $\tau:N\to{{\mathcal D}}_{{\scriptscriptstyle{M}}}$ in ${{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}_{{\scriptscriptstyle{{\widehat{R}}}}}}$ on the functor ${{\mathcal D}}_{{\scriptscriptstyle{M}}}:{\mathbb{N}}\to {{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}_{{\scriptscriptstyle{{\widehat{R}}}}}}$ mapping $n$ to $M/F_nM$ is a limiting cone of ${{\mathcal D}}_{{\scriptscriptstyle{M}}}$ if and only if ${{\mathcal R}}(\tau):{{\mathcal R}}(M)\to {{\mathcal R}}{{\mathcal D}}_{{\scriptscriptstyle{M}}}$ is a limiting cone of ${{\mathcal R}}{{\mathcal D}}_{{\scriptscriptstyle{M}}}$.
\[rem:invCan\] Given a filtered $(S,R)$-bimodule $M$, its completion ${\widehat{M}}$ is a complete $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bimodule.
The completion ${\widehat{M}}$ of a filtered $(S,R)$-bimodule $M$ can be endowed with an $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bimodule structure as follows. If $\left(M,F_nM\right)$ is a filtered $(S,R)$-bimodule, then for every $n\geq 0$ we have that $F_nS\cdot M\subseteq F_nM$ and $M\cdot F_nR\subseteq F_nM$, whence $M/F_nM$ is an $\left(S/F_nS,R/F_nR\right)$-bimodule and a filtered $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bimodule via restriction of scalars through the canonical projections $p_n^S:{\widehat{S}}\to S/F_nS$ and $p_n^R:{\widehat{R}}\to R/F_nR$ respectively. In this way, the morphisms ${{\pi^M_{n,m}} \colon {M/F_nM} \rightarrow {M/F_mM}}$ turn out to be $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bilinear as well. It follows that an $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bimodule structure is induced on the $(S,R)$-bimodule ${\widehat{M}}$ and it is explicitly given as follows: if ${\widehat{r}}_\infty\in{\widehat{R}}$, ${\widehat{s}}_\infty\in{\widehat{S}}$ and ${\widehat{x}}_\infty\in{\widehat{M}}$ then ${\widehat{(s x r)}}_{\infty}={\widehat{s}}_\infty \cdot {\widehat{x}}_\infty \cdot {\widehat{r}}_\infty\in{\widehat{M}}$ corresponds to the ${\mathbb{N}}$-tuple $\left(s_n\cdot x_n\cdot r_n+ F_nM\right)_{n\geq 0}$.
In this way, the canonical projections $p_{m}:{\widehat{M}}\rightarrow {M}/{F_{m}M}$ are $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bilinear, too. In particular, $\left({\widehat{M}},F_n{\widehat{M}}\right)$ is a filtered $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bimodule and the family of canonical projections ${\widehat{M}}\rightarrow {{\widehat{M}}}/{F_{\ast}{\widehat{M}}}$ is a projective cone in ${{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}_{{\scriptscriptstyle{{\widehat{R}}}}}}$. Hence ${\widehat{M}} \cong {\varprojlim_{n}\left({{{\widehat{M}}}/{F_n{\widehat{M}}}}\right)}$ as $\left(\,{\widehat{S}},{\widehat{R}}\,\right)$-bimodules as well in view of Remark \[rem:reflimit\] and Lemma \[lemma:Mhatcomplete\]. This concludes the proof of the statement.
In principle, we may consider on the one hand the full subcategory ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}$ of ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$ given by complete $(S,R)$-bimodules. Objects are filtered $(S,R)$-bimodules $\left(M,F_nM\right)$ such that $M\cong {\varprojlim_{n}\left({{M}/{F_nM}}\right)}$ as bimodules and arrows are filtered morphisms of complete bimodules $${\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{N}\right)}={\mathsf{Hom}^{\mathsf{flt}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{N}\right)}.$$ On the other hand, analogously, we may consider the full subcategory ${{}^{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{R}}}}}}$ of ${{}^{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{{\widehat{R}}}}}}$ given by complete $\left(\, {\widehat{S}},{\widehat{R}}\,\right)$-bimodules and filtered morphisms of complete bimodules.
\[prop:catequiv\] We have an equivalence of categories between ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}$ and ${{}^{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{R}}}}}}$.
The proof is a consequence of Proposition \[prop:completion\] and Remark \[rem:reflimit\] together with Lemma \[rem:invCan\].
The key role played by Proposition \[prop:catequiv\] will be that of allowing us to work in both categories ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}$ and ${{}^{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{R}}}}}}$ indifferently, depending on our needs or on what we would like to stress, even if the algebras $S$ and $R$ are not themselves complete.
Denote by $\mathscr{U}:{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}} \rightarrow {{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$ the functor that forgets the completeness, i.e., that associates to every complete $\left( S,R\right) $-bimodule its underlying filtered $\left( S,R\right)$-bimodule structure. What we showed in Remark \[rem:gammafiltiso\] can be restated now by saying that if $M$ is complete, then $M\cong {\widehat{\mathscr{U}(M)}}$ is a filtered isomorphism.
The other way around, we have a functor $${{{\widehat{\left( -\right) }}} \colon {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}}}$$ that associates every filtered bimodule with its completion and every morphism of filtered bimodules ${{f} \colon {M} \rightarrow {N}}$ with the morphism ${\widehat{f}}:={\varprojlim_{n}\left({\widetilde{f_n}}\right)}$, where ${{\widetilde{f_n}} \colon {{M}/{F_nM}} \rightarrow {{N}/{F_nN}}}$ is the map induced on the quotients (see e.g. [@MR1420862 Chapter I, §3]).
\[rem:counit\] Using Proposition \[prop:completion\] and Remark \[rem:gammafiltiso\] one can check that ${\widehat{(-)}}$ is left adjoint to the forgetful functor $\mathscr{U}$, i.e., that we have a natural isomorphism $$\label{eq:rightadj}
{\mathsf{Hom}^{\mathsf{flt}}_{{\scriptscriptstyle{S,\,R}}}\left({N},{\mathscr{U}(M)}\right)} \cong {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({ {\widehat{N}}},{M}\right)}.$$ The unit of this adjunction is the canonical map ${{\gamma_{{\scriptscriptstyle{N}}}} \colon {N} \rightarrow {\mathscr{U}\left({\widehat{N}}\right)}}$ for all $N\in {{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}$; the counit is “its inverse” ${{\sigma_{{\scriptscriptstyle{M}}}} \colon {{\widehat{\mathscr{U}(M)}}} \rightarrow {M}}$ for all $M\in {{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}$. [^5]
Furthermore, we point out that the bijection in equation encodes the universal property of the completion: every filtered morphism ${{g} \colon {N} \rightarrow {M}}$ from a filtered $(S,R)$-bimodule $N$ to a complete $(S,R)$-bimodule $M$ factors through the completion of $N$, i.e., we have a commutative diagram of filtered morphisms $$\label{eq:univprophat}
\xymatrix@R=15pt@C=30pt{N \ar[rr]^{g} \ar[dr]_{\gamma_N} & & M \\
& {\widehat{N}} \ar[ur]_{{\widehat{g}}} & }$$
Summing up, we obtained a commutative diagram $$\label{Eq:dash}
\xymatrix @R=5pt @C=30pt{ & {{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}} \ar@{<->}[dd] \\
{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}} \ar@{->}^-{{\widehat{(-)}}}[ru] \ar[rd]_-{{\widehat{(-)}}} & \\
& {{}^{}_{{\scriptscriptstyle{{\widehat{S}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{R}}}}}} }$$ where the vertical arrow is the equivalence of Proposition \[prop:catequiv\]. We end this subsection by recalling the following fact, which will be used in the forthcoming constructions.
Given filtered algebras $S$, $R$ and $T$, for every complete $(R,T)$-bimodule $N$ the assignment $$\label{eq:homfunct}
{{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,T}}}\left({N},{-}\right)}} \colon {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{T}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}}}$$ gives a well-defined functor.
The topological tensor product of filtered bimodules {#ssc:TTPFB}
----------------------------------------------------
The main objective of this subsection is to show (or rather to recall) briefly that the functor $${{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{-}\right)}}\right)}} \colon {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{T}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}}}$$ is representable for every pair of complete bimodules ${}_{{\scriptscriptstyle{S}}}M_{{\scriptscriptstyle{R}}}$ and ${}_{{\scriptscriptstyle{R}}}N_{{\scriptscriptstyle{T}}}$. The representing object will be called *the complete (or topological) tensor product* of $M$ and $N$. For a more exhaustive treatment of the construction of this tensor product over a single commutative ring we refer to [@MR0163908].
\[th:adjunction\] Let $R$, $S$ and $T$ be filtered algebras. For every complete bimodules ${}_SM_R$, ${}_RN_T$, ${}_SP_T$ we have a filtered isomorphism, whence an homeomorphism as linear topological spaces, $${\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}\cong {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}$$ which is natural in $M$ and $P$ and where ${\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}$ is the completion of the filtered tensor product $M{\otimes_{\scriptscriptstyle{R}}}N$.
The usual Hom-tensor adjunction for bimodules tells us that we have a pair of natural isomorphisms of abelian groups: $$\begin{gathered}
{{\psi} \colon {{\mathrm{Hom}_{{\scriptscriptstyle{S,\,R}}}\left(M,\,{\mathrm{Hom}_{{\scriptscriptstyle{-,\,T}}}\left(N,\,P\right)}\right)}} \rightarrow {{\mathrm{Hom}_{{\scriptscriptstyle{S,\,T}}}\left(M{\otimes_{\scriptscriptstyle{R}}}N,\,P\right)}}}, \quad \Big( f \longmapsto \big[ x{\otimes_{\scriptscriptstyle{R}}}y \mapsto f(x)(y) \big] \Big) \\
{{\phi} \colon {{\mathrm{Hom}_{{\scriptscriptstyle{S,\,T}}}\left(M{\otimes_{\scriptscriptstyle{R}}}N,\,P\right)}} \rightarrow {{\mathrm{Hom}_{{\scriptscriptstyle{S,R}}}\left(M,\,{\mathrm{Hom}_{{\scriptscriptstyle{-,\,T}}}\left(N,\,P\right)}\right)}}}, \quad \Big( g \longmapsto \big[ x \mapsto [ y \mapsto g(x{\otimes_{\scriptscriptstyle{R}}}y)] \big] \Big).\end{gathered}$$ It is easy to see that if $f\in {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}$, then $\psi(f)$ is filtered. Therefore we can further associate to $\psi(f)$ the (unique) morphism $\sigma_P\circ{\widehat{\psi(f)}}$ and the assignment $$f \longmapsto \Big[{{\big(\sigma_P\circ{\widehat{\psi(f)}}\big)} \colon {{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}} \rightarrow { P}}\Big]$$ leads to a well-defined map $${{\psiup} \colon {{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}} \rightarrow {{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}}}.$$ Explicitly, for all $f\in {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}$ and all ${\widehat{(x{\otimes_{\scriptscriptstyle{R}}}y)}}_{\infty} \in {\widehat{M{\otimes_{\scriptscriptstyle{R}}} N}}$ $$\psiup(f)\Big( {\widehat{(x{\otimes_{\scriptscriptstyle{R}}}y)}}_{\infty} \Big) \,=\, {\widehat{f(x)(y)}}_{\infty}\,=\,
\underset{k\to \infty}{\lim}\Big(f(x_{k})(y_{k})\Big),$$ (finite summations in the tensor product are understood and notation is used).
To check that it is filtered, let $f\in F_n{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}={\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}[n]}\right)}$. For all $l\geq 0$, if ${\widehat{(x{\otimes_{\scriptscriptstyle{R}}}y)}}_{\infty} \in F_l\left({\widehat{M{\otimes_{\scriptscriptstyle{R}}} N}}\right)$, then we may assume that $x_k{\otimes_{\scriptscriptstyle{R}}}y_k=0$ for every $k\leq l$ and that $x_k{\otimes_{\scriptscriptstyle{R}}}y_k\in {{\mathcal F}}_l\left(M{\otimes_{\scriptscriptstyle{R}}}N\right)$ for every $k>l$. Thus, the Cauchy sequence $\{ f(x_k)(y_k)\}_{ k \geq 0}$ lies in $F_{l+n}P$ as well as its limit, since this is a closed subset in $P$. Therefore, for all $n\geq 0$ we have $$\psiup\left(F_n{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}\right)\subseteq F_n{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}.$$ The other way around, let us check that $${{\phiup} \colon {{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}} \rightarrow {{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}}\right)}}}$$ which assigns to every $g\in {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}$ the morphism $${{\phiup(g)} \colon {M} \rightarrow {{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}};\, \Big({x} \mapsto {\left[y\mapsto g\left(\gamma_{M{\otimes_{\scriptscriptstyle{R}}} N}\left(x{\otimes_{\scriptscriptstyle{R}}} y\right)\right)\right]}\Big)},$$ is a well-defined filtered map. To this end, set $\gamma=\gamma_{M{\otimes_{\scriptscriptstyle{R}}} N}$ and observe that for every $h,k,n\in{\mathbb{N}}$ and for all $g\in F_n{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}={\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P[n]}\right)}$ we have that $$\left(\phiup(g)(F_hM)\right)(F_kN)\subseteq g\left(\gamma\left({\mathsf{Im}\left({F_hM{\otimes_{\scriptscriptstyle{R}}} F_kN}\right)}\right)\right)\subseteq g\left(\gamma\left({{\mathcal F}}_{h+k}\left(M{\otimes_{\scriptscriptstyle{R}}}N\right)\right)\right)\subseteq g\left(F_{h+k}\left({\widehat{M{\otimes_{\scriptscriptstyle{R}}} N}}\right)\right) \subseteq F_{h+k+n}P.$$ For all $g\in {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,\,T}}}\left({{\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}},{P}\right)}$, this proves at once that $\left(\phiup(g)(M)\right)(F_kN)\subseteq F_kP$ (take $n=0=h$), whence $\phiup(g)$ lands into ${\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P}\right)}$, that $\phiup(g)(F_hM)\subseteq {\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,\,T}}}\left({N},{P[h]}\right)}$ (take $n=0$), whence $\phiup(g)$ is filtered, and also that $\phiup$ itself is filtered as well. Since $\phiup$ and $\psiup$ are mutually inverse functions, they establish a filtered isomorphism, as claimed.
\[def:CTP\] In view of Theorem \[th:adjunction\], we say that ${\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}$ is the *complete* (or *topological*) tensor product over the filtered algebra ${R}$ of the complete bimodules ${_SM_R}$ and ${_RN_T}$ as indicated. It is coherent with the notion of completeness introduced in §\[ssec:CF\]. As a matter of notation, we will write $M~{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N:={\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}$.
Furthermore, Theorem \[th:adjunction\] can be restated by saying that, for $_RN_T$ complete, the functor $${{-{~{\widehat{\otimes}}_{{\scriptscriptstyle{{R}}}}~}N} \colon {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{T}}}}};\, \Big({M} \mapsto {M{~{\widehat{\otimes}}_{{\scriptscriptstyle{{R}}}}~}N}\Big)}$$ is left adjoint to the functor $${{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,T}}}\left({N},{-}\right)}} \colon {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{T}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}};\, \Big({P} \mapsto {{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,T}}}\left({N},{P}\right)}}\Big)}.$$
Following Theorem \[th:adjunction\], it is reasonable to call this complete tensor product a *topological* tensor product as it is the left adjoint to the *continuous* $\mathsf{Hom}$ functor between complete bimodules. We point out however that our definition of a topological tensor product satisfies a different universal property with respect to, e.g., [@Seal Definition 2.1] or [@Semadeni Theorem 20.1.2]. Namely, assume that ${_SM_R},{_RN_T}, {_SP_T}$ are complete bimodules over filtered algebras as indicated. Endow $M\times N$ with the filtration $F_k(M\times N)=F_kM\times F_kN$. The induced linear topology coincides with the product linear topology, i.e., the coarsest linear topology for which the canonical projections are continuous, and $M\times N$ is a complete $(S,T)$-bimodule with respect to this filtration. The canonical morphism $M\times N\to M\otimes_{{\scriptscriptstyle{R}}} N$ maps $F_k(M\times N)$ into ${\mathsf{Im}\left({F_kM {\otimes_{\scriptscriptstyle{R}}} F_kN}\right)}\subseteq {{\mathcal F}}_{2k}(M{\otimes_{\scriptscriptstyle{R}}} N)\subseteq {{\mathcal F}}_k(M{\otimes_{\scriptscriptstyle{R}}}N)$, whence it is filtered (and continuous) and the same hold for the composition $\tau:=\left(M\times N\to M{\otimes_{\scriptscriptstyle{R}}} N\to M{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N\right)$. Endow $M\times N$ with the bi-filtration $F_{h,k}(M\times N)=F_hM\times F_kN$ (for the definition of a bi-filtration see, e.g., [@Borel §X.2]). We observe that $\tau$ is bi-filtered[^6] as well. The bijective correspondence between $R$-balanced $(S,T)$-bilinear morphisms ${_SM_R}\times{_RN_T}\to {_SP_T}$ and morphisms in ${\mathsf{Hom}_{{\scriptscriptstyle{S,R}}}\left({M},{{\mathsf{Hom}_{{\scriptscriptstyle{-,T}}}\left({N},{P}\right)}}\right)}$ restricts to a bijective correspondence between $R$-balanced $(S,T)$-bilinear bi-filtered morphisms $M\times N\to P$ and elements in ${\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{S,R}}}\left({M},{{\mathsf{Hom}{}^{\mathsf{c}}_{{\scriptscriptstyle{-,T}}}\left({N},{P}\right)}}\right)}$. From this it follows that the complete tensor product could be considered as a topological tensor product in the sense that it satisfies the following universal property: there exists a complete $(S,T)$-bimodule and a bi-filtered $R$-balanced $(S,T)$-bilinear morphism $\tau:M\times N\to M{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N$ such that for every other complete $(S,T)$-bimodule $P$ and every bi-filtered $R$-balanced $(S,T)$-bilinear morphism $f:M\times N\to P$ there exists a unique filtered $(S,T)$-bilinear morphism $\widetilde{f}:M{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N\to P$ such that $f=\widetilde{f}\circ\tau$.
Given $M,N$ two filtered $R$-bimodules over a filtered algebra $R$, we have three (in principle, different) ways to obtain a complete ${\widehat{R}}$-bimodule from $M{\otimes_{\scriptscriptstyle{R}}}N$. The first and more natural one is ${\widehat{M{\otimes_{\scriptscriptstyle{R}}} N}}$: since $M{\otimes_{\scriptscriptstyle{R}}} N$ is a filtered $R$-bimodule, Lemma \[rem:invCan\] ensures that ${\widehat{M{\otimes_{\scriptscriptstyle{R}}} N}}$ is a complete ${\widehat{R}}$-bimodule.[^7] The other two come from the construction we performed in this subsection. Namely, they are ${\widehat{M}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}{\widehat{N}}$ and ${\widehat{M}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{N}}$, i.e., the complete tensor product of the complete ${\widehat{R}}$-bimodules ${\widehat{M}}$, ${\widehat{N}}$ over the filtered algebras $R$ and ${\widehat{R}}$. It turns out, however, that the three constructions give rise to the same complete bimodule up to the isomorphism of the following proposition (cf. also Remark \[rem:superiso\]).
\[prop:coherentcompl\] Assume that ${_{{\scriptscriptstyle{S}}}M_{{\scriptscriptstyle{R}}}}$ and ${_{{\scriptscriptstyle{R}}}N_{{\scriptscriptstyle{T}}}}$ are two filtered bimodules over filtered algebras as denoted. Then we have a filtered isomorphism of $(S,T)$-bimodules ${\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}\cong {\widehat{M}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{N}}$, natural in both variables, explicitly given by $$\begin{gathered}
\varphi_{M,N}: {\widehat{M}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{N}} \longrightarrow {\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}, \quad \left[\, {\underset{n\to\infty}{\lim}}\Big( {\underset{k\to\infty}{\lim}}(x_{k,n} ){\otimes_{\scriptscriptstyle{{\widehat{R}}}}}{\underset{l\to\infty}{\lim}}(y_{l,n})\Big) \longmapsto {\underset{n\to\infty}{\lim}}\big( x_{n,n}{\otimes_{\scriptscriptstyle{R}}}y_{n,n} \big) \,\right], \label{Eq:Phi} \\
\psi_{M,N}: {\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}} \longrightarrow {\widehat{M}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{N}} , \quad \left[\, {\underset{n\to\infty}{\lim}}\Big( x_{n} {\otimes_{\scriptscriptstyle{R}}}y_{n}\Big) \longmapsto {\underset{n\to\infty}{\lim}}\big( {\widehat{x_{n}}}{\otimes_{\scriptscriptstyle{{\widehat{R}}}}}{\widehat{y_{n}}} \big) \,\right]. \label{Eq:Psi}\end{gathered}$$
\[rem:superiso\] Keeping assumptions and notations from Proposition \[prop:coherentcompl\], the algebra morphism $\gamma_R:R\to{\widehat{R}}$ induces a filtered $(S,T)$-bilinear morphism ${\widehat{M}}{\otimes_{\scriptscriptstyle{R}}}{\widehat{N}}\to {\widehat{M}}{\otimes_{\scriptscriptstyle{{\widehat{R}}}}}{\widehat{N}}$. Moreover, we can consider the filtered $(S,T)$-bilinear morphism $\gamma_M{\otimes_{\scriptscriptstyle{R}}}\gamma_N:M{\otimes_{\scriptscriptstyle{R}}}N\to {\widehat{M}}{\otimes_{\scriptscriptstyle{R}}}{\widehat{N}}$. These induce the following composition $$\xymatrix @C=18pt{
\frac{M{\otimes_{\scriptscriptstyle{R}}}N}{{{\mathcal F}}_n\left(M{\otimes_{\scriptscriptstyle{R}}}N\right)} \ar[rr]^-{\widetilde{\gamma_M{\otimes_{\scriptscriptstyle{R}}}\gamma_N}} & & \frac{{\widehat{M}}{\otimes_{\scriptscriptstyle{R}}}{\widehat{N}}}{{{\mathcal F}}_n\left({\widehat{M}}{\otimes_{\scriptscriptstyle{R}}}{\widehat{N}}\right)} \ar[r] & \frac{{\widehat{M}}{\otimes_{\scriptscriptstyle{{\widehat{R}}}}}{\widehat{N}}}{{{\mathcal F}}_n\left({\widehat{M}}{\otimes_{\scriptscriptstyle{{\widehat{R}}}}}{\widehat{N}}\right)}
}$$ for all $n\geq 0$, which in turn induces exactly the morphism $\psi_{M,N}$ of the statement of Proposition \[prop:coherentcompl\].
Nevertheless, in what follows we will be concerned mainly with the complete tensor product over complete algebras. Thus, we decided to focus on the bare minimum to introduce the isomorphisms and . Furthermore, we will often omit these in the computations and we will identify ${\widehat{M{\otimes_{\scriptscriptstyle{R}}}N}}$ with ${\widehat{M}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{N}}$ as well, in order to simplify the exposition.
As it happened for filtered algebras and bimodules, complete algebras and bimodules form a bicategory.
We have a bicategory ${{\mathcal B}}im_{\Bbbk}^{\mathsf{c}}$ which has complete algebras as $0$-cells and whose categories of $\{1,2\}$-cells are the categories of complete bimodules over complete algebras. The vertical compositions are given by the ordinary compositions of morphisms. The horizontal compositions are given by the composition functors $-{~{\widehat{\otimes}}_{{\scriptscriptstyle{B}}}~}-:={\widehat{(-)}}\circ (-{\otimes_{\scriptscriptstyle{B}}}-)$ $$-{~{\widehat{\otimes}}_{{\scriptscriptstyle{B}}}~}-:{{}^{}_{{\scriptscriptstyle{A}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{B}}}}\times {{}^{}_{{\scriptscriptstyle{B}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{C}}}}\to {{}^{}_{{\scriptscriptstyle{A}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{C}}}}$$ for all complete algebras $A,B,C$. The constraints are induced by those of the bicategory ${{\mathcal B}}im_{\Bbbk}^{\mathsf{flt}}$.
In view of Proposition \[prop:catequiv\], the natural isomorphisms $\varphi_{-,-}$ and $\psi_{-,-}$ described in equations and can be regarded as isomorphisms of complete $\Big(\,{\widehat{S}}, {\widehat{T}}\,\Big)$-bimodules and for this reason we are going to denote them in the same way. The left and right unit constraints (or identities, as they are called in [@Benabou]) are deduced by using the natural isomorphisms $\varphi_{A,-}$ and $\varphi_{-,B}$ in conjunction with the natural isomorphism ${{\sigma_{{\scriptscriptstyle{X}}}} \colon {{\widehat{\mathscr{U}(X)}}} \rightarrow {X}}$ of Remark \[rem:gamma\] for $X$ an object in ${{}^{}_{{\scriptscriptstyle{{A}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{B}}}}}$ and $A,B$ complete algebras (cf. also Remark \[rem:counit\]). The associativity constraint ${{\alpha_{{\scriptscriptstyle{M,\,N,\,P}}}} \colon {\left(M{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}N\right){~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}P} \rightarrow {M{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}\left(N{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}P\right)}}$ is obtained by observing that for every ${}_{{\scriptscriptstyle{S}}}X_{{\scriptscriptstyle{R}}}$, ${}_{{\scriptscriptstyle{R}}}Y_{{\scriptscriptstyle{T}}}$ and ${}_{{\scriptscriptstyle{T}}}Z_{{\scriptscriptstyle{K}}}$ filtered bimodules, there is a unique $\Big(\,{\widehat{S}}, {\widehat{K}}\,\Big)$-bilinear map $\alpha_{{\scriptscriptstyle{{\widehat{X}},\,{\widehat{Y}},\,{\widehat{Z}}}}}$ making commutative the following diagram
$$\label{eq:defalpha}
\xymatrix @C=60pt @R=15pt{
\left({\widehat{X}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{Y}}\right){~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{T}}}}}~}{\widehat{Z}} \ar@{.>}[d]_-{\alpha_{{\widehat{X}},{\widehat{Y}},{\widehat{Z}}}} & {\widehat{X{\otimes_{\scriptscriptstyle{R}}} Y}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{T}}}}}~}{\widehat{Z}} \ar[l]_-{\psi_{X,\,Y}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{T}}}}}~}{\widehat{Z}}} & {\widehat{\left(X{\otimes_{\scriptscriptstyle{R}}}Y\right){\otimes_{\scriptscriptstyle{T}}}Z}} \ar[d]^-{{\widehat{a_{X,\,Y,\,Z}}}} \ar[l]_-{\psi_{X{\otimes_{\scriptscriptstyle{R}}}Y,\,Z}} \\
{\widehat{X}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}\left({\widehat{Y}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{T}}}}}~}{\widehat{Z}}\right) & {\widehat{X}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}{\widehat{Y{\otimes_{\scriptscriptstyle{T}}}Z}} \ar[l]_-{{\widehat{X}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}\psi_{Y,\,Z}} & {\widehat{X{\otimes_{\scriptscriptstyle{R}}}\left(Y{\otimes_{\scriptscriptstyle{T}}}Z\right)}} \ar[l]_-{\psi_{X,\,Y{\otimes_{\scriptscriptstyle{T}}}Z}}
}$$
where $a_{X,\,Y,\,Z}$ is the usual associativity constraint.
It turns out then that the completion functor fits properly in the wider framework of bicategories.
\[thm:Athm\] Let ${\Bbbk}$ be a commutative ground ring which we consider trivially filtered. Then the completion construction developed in this section induces a 2-functor $$\xymatrix@R=0pt{ & {{\mathcal B}}im_{{\scriptscriptstyle{{\Bbbk}}}}^\mathsf{flt} \ar@{->}[rr] & & {{\mathcal B}}im_{{\scriptscriptstyle{{\Bbbk}}}}^\mathsf{c} \\ \rm{0\text{-}cells} & R \ar@{|->}[rr] & & {\widehat{R}} \\ \rm{1\text{-}cells} & {}_{{\scriptscriptstyle{R}}}M_{{\scriptscriptstyle{S}}} \ar@{|->}[rr] & & {}_{{\scriptscriptstyle{{\widehat{R}}}}}{\widehat{M}}_{{\scriptscriptstyle{{\widehat{S}}}}} \\ \rm{2\text{-}cells} & \Big[f: M \to N \Big] \ar@{|->}[rr] & & \Big[{\widehat{f}}: {\widehat{M}} \to {\widehat{N}} \Big] }$$ from the bicategory ${{\mathcal B}}im_{{\scriptscriptstyle{{\Bbbk}}}}^\mathsf{flt}$ of filtered algebras and filtered bimodules to the bicategory ${{\mathcal B}}im_{{\scriptscriptstyle{{\Bbbk}}}}^\mathsf{c}$ of complete algebras and complete bimodules.
The construction of the stated 2-functor at the level of 0-cells is clear. At the level of $\{\text{1,2}\}$-cells, the needed family of functors is given by the functors exhibited in diagram , precisely by the lower diagonal one. The required natural transformations for ${\widehat{(-)}}$ are given in Proposition \[prop:coherentcompl\]. Finally, the coherence axioms (i.e., the hexagons and the squares in [@Benabou Definition 4.1]) are fulfilled by construction.
\[prop:moncatcomplbimod\] Let $R$ be a filtered algebra. Then the category of complete ${\widehat{R}}$-bimodules ${{}^{}_{{\scriptscriptstyle{{\widehat{R}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{R}}}}}}$ is monoidal with tensor product the topological tensor product $-{~{\widehat{\otimes}}_{{\scriptscriptstyle{{\widehat{R}}}}}~}-$ and with unit the completion algebra ${\widehat{R}}$ of $R$. Moreover, the completion functor ${{{\widehat{\left(-\right)}}} \colon {{{}^{}_{{\scriptscriptstyle{R}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}} \rightarrow {{{}^{}_{{\scriptscriptstyle{{\widehat{R}}}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{{\widehat{R}}}}}}}}$ is a monoidal functor.
\[rem:calg\] It follows from Corollary \[prop:moncatcomplbimod\] that $\left({{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}},{\widehat{\otimes}},{\Bbbk}\right)$ is a monoidal category. It can be checked that for a filtered algebra $(R,\mu,\eta)$, $R$ is a complete module (i.e., a complete algebra) if and only if $\left(R,\sigma_{{\scriptscriptstyle{R}}}\,{\widehat{\mu}},\eta\right)$ is a monoid in the monoidal category ${{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}}$. A similar thing happens for complete bimodules over complete algebras. In fact, up to an equivalence of categories, we may regard complete algebras as monoids in the monoidal category ${{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}}$ and complete bimodules over complete algebras $A$ and $B$ as objects in ${{}_{{\scriptscriptstyle{A}}}\left({{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}}\right){}_{{\scriptscriptstyle{B}}}}$ and conversely (in accordance with [@MR1320989 §A.1], for example). Furthermore, we point out that it could possible to deduce Theorem \[thm:Athm\] from a more general framework as claimed in [@shulman Examples 2.2 and 6.2], once proven that $\left({{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{flt}},\otimes,{\Bbbk}\right)$ and $\left({{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}},{\widehat{\otimes}},{\Bbbk}\right)$ are monoidal categories and that ${\widehat{(-)}}:{{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{flt}}\to {{\mathsf{Mod}}_{{\scriptscriptstyle{{\Bbbk}}}}}^{\mathsf{c}}$ is a monoidal functor.
Topological tensor product of linear duals of locally finitely generated and projective filtered bimodules {#sec:TGFr}
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In this appendix we plan to study the linear dual of the tensor product of two locally finitely generated and projective filtered modules (for instance, rings with an admissible filtration as in §\[ssec:FUstra\]). In particular, we will show that this bimodule is homeomorphic to the topological tensor product of the duals.
Locally finitely generated and projective filtered modules {#ssec:LFGr}
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Let $R$ be a ring and $M$ a right $R$-module endowed with an *ascending filtration* $\left\{F^nM\mid n\in{\mathbb{N}}\right\}$. This is said to be *exhaustive* if $\bigcup_{n\geq 0}F^nM=M$. In view of our aims, we assume $R$ trivially filtered. We denote by ${\mathrm{gr}}^n\left(M\right)$ the quotient module $F^nM/F^{n-1}M$ for all $n\geq 0$ ($F^{-1}M=0$ by convention), and by ${\mathrm{gr}}\left(M\right)$ the associated graded module ${\mathrm{gr}}\left(M\right)=\bigoplus_{n\geq 0}{\mathrm{gr}}^n\left(M\right)$. Henceforth and in line with Appendix \[sec:CBCF\], we denote increasing filtrations with upper indices and decreasing ones with lower indices. Moreover, $\tau_{{\scriptscriptstyle{m,\,n}}}:F^nM\to F^mM$ and $\tau_{{\scriptscriptstyle{n}}}:F^nM\to M$ for all $m\geq n\geq 0$ will denote the canonical inclusions.
\[lemma:fgpquotients\] Let $R$ be any ring, $M$ a right $R$-module endowed with an ascending filtration $\left\{F^kM\mid k\in{\mathbb{N}}\right\}$ and let $n\in{\mathbb{N}}$. If the quotient modules $F^kM/F^{k-1}M$ are projective right $R$-modules for all $0\leq k\leq n$, then $F^nM\cong {\mathrm{gr}}\left(F^nM\right)$ as filtered modules. In particular, $F^nM$ is projective. If moreover the quotient modules $F^kM/F^{k-1}M$ are finitely generated for $0\leq k\leq n$, then $F^nM$ is finitely generated as well. Finally, if the filtration is exhaustive and the quotient modules $F^nM/F^{n-1}M$ are projective for all $n\in{\mathbb{N}}$, then there exists an isomorphism of filtered modules $M\cong {\mathrm{gr}}(M)$ and $M_{{\scriptscriptstyle{R}}}$ itself is projective.
Since every quotient module $F^kM/F^{k-1}M$ is projective as right $R$-module, for all $0\leq k\leq n$, we have a split exact sequence of right $R$-modules $$\xymatrix{
0 \ar[r] & F^{n-1}M \ar@<+0.5ex>[r]^-{\tau_{{\scriptscriptstyle{n-1,\,n}}}} & F^{n}M \ar@<+0.5ex>[r]^-{} \ar@{.>}@<+0.5ex>[l]^-{} & \left(F^nM/F^{n-1}M\right) \ar[r]\ar@{.>}@<+0.5ex>[l]^-{} & 0
}$$ from which it follows that, as right $R$-modules, $$F^nM\cong F^{n-1}M\oplus \left(F^nM/F^{n-1}M\right).$$ Proceeding inductively, we have that $$\label{Eq:isoFn}
F^nM\cong \bigoplus_{k=0}^n \frac{F^kM}{F^{k-1}M}={\mathrm{gr}}\left(F^nM\right).$$ Observing that for all $m\leq n$, $F^m{\mathrm{gr}}\left(F^nM\right)=\bigoplus_{k=0}^m F^kM/F^{k-1}M={\mathrm{gr}}\left(F^mM\right)$ and $F^mF^nM=F^mM$, it is clear that the isomorphism preserves the filtrations as claimed. Moreover, as direct sum of projective right $R$-modules, $F^nM$ is projective as well.
The second claim is clear, as the direct sum is finite. About the last claim in the statement, saying that the filtration is exhaustive means that $M\cong {\varinjlim_{n}\left({F^nM}\right)}$ as filtered modules. Since $F^nM\cong{\mathrm{gr}}\left(F^nM\right)\cong F^n\left({\mathrm{gr}}(M)\right)$ as filtered modules, we have that $M\cong{\varinjlim_{n}\left({F^nM}\right)}\cong {\varinjlim_{n}\left({F^n\left({\mathrm{gr}}(M)\right)}\right)}\cong{\mathrm{gr}}(M)$ as claimed. As direct sum of projective right $R$-modules, $M$ is itself projective.
Henceforth, all ascending filtrations will be exhaustive. In analogy with [@Cartan:1958 §4], we will say that an increasingly filtered right $R$-module $M$ such that the quotient modules $F^nM/F^{n-1}M$ are finitely generated and projective is a *locally finitely generated and projective* (filtered) module.
The topology on the linear dual of a locally finitely generated and projective filtered bimodule {#ssec:DLF}
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Assume that we are given an increasingly filtered $R$-bimodule $M$ which is locally finitely generated and projective as a filtered right $R$-module (the definition of an increasingly filtered bimodule can be easily obtained by dualizing that for decreasingly filtered bimodules in Appendix \[sec:CBCF\]). In particular, this means that each member of the increasing filtration $\{F^nM\mid n\in{\mathbb{N}}\}$ is actually an $R$-subbimodule with a monomorphism ${\tau_{{\scriptscriptstyle{n}}}}:F^nM \to M$ and that the factors $F^nM/F^{n-1}M$ are finitely generated and projective right $R$-modules.
Since the filtration $\left\{F^nM\mid n\in{\mathbb{N}}\right\}$ is exhaustive, we may identify the right $R$-module ${M_{{\scriptscriptstyle{R}}}}$ with the inductive limit $M = {\varinjlim_{n}\left({F^nM}\right)}$ of the system $\left\{F^nM,\tau_{{\scriptscriptstyle{n,\,n+1}}}\right\}_{n\in{\mathbb{N}}}$. Therefore, $M^*={\mathsf{Hom}_{{\scriptscriptstyle{-,A}}}\left({M},{A}\right)}\cong {\varprojlim_{n}\left({F^nM^*}\right)}$ as a left $R$-module via the left $R$-linear isomorphism $$\label{Eq:D}
M^* \rightarrow {\varprojlim_{n}\left({F^nM^*}\right)}, \quad \Big( f \mapsto ({\tau_{{\scriptscriptstyle{n}}}}^*(f))_{{\scriptscriptstyle{n \geq 0}}}\Big); \qquad {\varprojlim_{n}\left({F^nM^*}\right)} \rightarrow M^*, \quad \Big( (g_{{\scriptscriptstyle{n}}})_{{\scriptscriptstyle{n \geq 0}}} \mapsto g:={\varinjlim_{n}\left({g_{{\scriptscriptstyle{n}}}}\right)} \Big)$$ where $(r\cdot f)(x)=rf(x)$ for all $f\in M^*$, $r\in R$ and $x\in M$. However, $M^*$ is also a right $R$-module with $\left(f\leftharpoonup x\right)(m)=f(x\cdot m)$ for all $f\in M^*$, $m\in M$ and $x\in R$, and it turns out that the isomorphism is right $R$-linear as well. Therefore, $M^*\cong {\varprojlim_{n}\left({F^nM^*}\right)}$ as $R$-bimodules. Notice that $g:M\to R$ is the unique right $R$-linear map that extends all the $g_{{\scriptscriptstyle{n}}}$’s at the same time, that is $g \, \tau_{{\scriptscriptstyle{n}}}= g_{{\scriptscriptstyle{n}}}$ for all $n\geq 0$.
\[coro:FnL\] Let $M$ be an increasingly filtered $R$-bimodule which is locally finitely generated and projective as right $R$-module. The following properties hold true.
1. Each of the subbimodules $F^nM$ is a finitely generated and projective right $R$-module and each of the structural maps $\tau_{{\scriptscriptstyle{n,\,n+1}}}: F^nM \to F^{n+1}M$ is a split monomorphism of right $R$-modules. Moreover, the transposes $\tau_{{\scriptscriptstyle{n}}}^*:M^*\to F^nM^*$ are surjective, so that for all $n\in{\mathbb{N}}$, $\tau_{{\scriptscriptstyle{n}}}$ is a split monomorphism too.
2. For every $m,n \geq 0$, we have an isomorphism of $R$-bimodules $$\phi_{{\scriptscriptstyle{m,\, n}}}: \,\left(F^mM^*\right)^{}_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}} {^{}_{{\scriptscriptstyle{R}}}\left(F^nM^*\right)} \cong \left(F^nM_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}} {{}_{{\scriptscriptstyle{R}}}F^mM}\right)^*$$ such that $\phi_{{\scriptscriptstyle{m,\, n}}}\left(f{\otimes_{\scriptscriptstyle{R}}}g\right)\left(x{\otimes_{\scriptscriptstyle{R}}}y\right)=f\left(g(x)y\right)$ for all $x \in F^nM$, $y\in F^mM$, $f\in F^mM^*$ and $g \in F^nM^*$.
The first claim of $(i)$ follows directly from Lemma \[lemma:fgpquotients\]. To prove the second one, we proceed as follows. If $\left\{S_i,\varphi_{j,i}\mid i,j\in\mathbb{N},j\geq i\right\}$ is an inverse system in the category of left $R$-modules with surjective transition maps $\varphi_{j,i}:S_j\to S_i$, $j\geq i$, then every projection $\varphi_i:{\varprojlim_{n}\left({S_n}\right)}\to S_i$ is surjective as well (cf. e.g. [@Gh Remark 2.14]). Since the transition maps $\tau_{n,n+1}$ are split monomorphisms, their transposes $\tau_{n,n+1}^*$ are surjective, whence the canonical maps $\varphi_i:{\varprojlim_{n}\left({(F^nM)^*}\right)}\to (F^iM)^*$ are surjective as well. If we denote by $\Phi$ the isomorphism of , then it satisfies $\varphi_n\circ\Phi=\tau_n^*$, whence $\tau_n^*$ is surjective.
Finally, in view of [@BSZ Lemma 11.3] and the hom-tensor adjunction respectively, we have the chain of isomorphisms of $R$-bimodules $$\left(F^mM\right)^*{}_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}} {_{{\scriptscriptstyle{R}}}\left(F^nM\right)}^* \cong {\mathrm{Hom}_{\scriptscriptstyle{\text{-}R}}(F^nM_{{\scriptscriptstyle{R}}},\left(F^mM\right)_{{\scriptscriptstyle{R}}}^*)}\cong {\mathrm{Hom}_{\scriptscriptstyle{\text{-}R}}(F^nM_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}}{}_{{\scriptscriptstyle{R}}}F^mM,R)}$$ which proves $(ii)$.
\[rem:vartheta\] Since ${\tau_{{\scriptscriptstyle{n}}}}^*$ is surjective and $F^nM^*$ is a finitely generated and projective left $R$-module, there is a left $R$-linear section $F^nM^*\to M^*$ of ${\tau_{{\scriptscriptstyle{n}}}}^*$ which induces a right $R$-linear retraction ${\theta_{{\scriptscriptstyle{n}}}}: M \to F^nM$ of ${\tau_{{\scriptscriptstyle{n}}}}$. In particular, each of the maps ${\tau_{{\scriptscriptstyle{n}}}}^*: M^* \to F^nM^*$ is a split epimorphism of left $R$-modules with section ${\theta_{{\scriptscriptstyle{n}}}}^*: F^nM^* \to M^*$ as well. Denote temporarily by $\pi_{{\scriptscriptstyle{n}}}:M^*\to M^*/{\mathrm{Ker}\left({{\tau_{{\scriptscriptstyle{n}}}}^*}\right)}$ the canonical projection. Even if ${\theta_{{\scriptscriptstyle{n}}}}^*$ is just left $R$-linear, the composition $\pi_{{\scriptscriptstyle{n}}}\circ {\theta_{{\scriptscriptstyle{n}}}}^* : F^nM^*\to M^*/{\mathrm{Ker}\left({{\tau_{{\scriptscriptstyle{n}}}}^*}\right)}$ is $R$-bilinear as it is the inverse of the $R$-bilinear isomorphism $\widetilde{{\tau_{{\scriptscriptstyle{n}}}}^*}:M^*/{\mathrm{Ker}\left({{\tau_{{\scriptscriptstyle{n}}}}^*}\right)}\to F^nM^*$.
Now, the right linear dual $M^*$ inherits naturally a decreasing filtration which converts it into a complete $R$-bimodule. Namely, mimicking [@MSS Appendix A.2], let us consider the filtration $$\label{eq:filtdual}
F_0M^*=M^* \quad \text{and} \quad F_{n+1}M^*={\mathrm{Ker}\left({\tau_{{\scriptscriptstyle{n}}}^*}\right)}, \quad \text{for} \,\, n \geq 0. \,\footnote{Observe that ${\mathrm{Ker}\left({\tau_{{\scriptscriptstyle{n}}}^*}\right)}=\left\{f\in M^*\mid F^nM\subseteq {\mathrm{Ker}\left({f}\right)}\right\}$, whence we will often use the notation ${\mathsf{Ann}\left({F^nM}\right)}$ to refer to it.}$$ Notice that no confusion may arise in the notation, as the upper or lower indices help in distinguishing between $F_nM^*$, the $n$-th term of the decreasing filtration on $M^*$, and $F^nM^*:=(F^{n}M)^{*}$, the dual of the $n$-th term of the increasing filtration on $M$. In view of (i) of Corollary \[coro:FnL\], we have an isomorphism of $R$-bimodules $F^nM^*\cong M^*/F_nM^*$. From this together with the isomorphism and Proposition \[prop:completion\] we deduce that the filtration $\{F_nM^*\mid n \in{\mathbb{N}}\}$ induces a linear topology over $M^*$ for which it is a complete $R$-bimodule.
In order to be able to evaluate limits of Cauchy sequences in $M^*$ on an element of $M$ it is useful to notice the following. Let $\{f_n\}_{{\scriptscriptstyle{n \geq 0}}}$ be a Cauchy sequence of right $R$-linear maps in $M^*$ and let $f = {\underset{n\to\infty}{\lim}}(f_n)$ denote its limit in $M^*$. Therefore we have that $f-f_n\in F_nM^*={\mathrm{Ker}\left({\tau_{n-1}^*}\right)}$ for every $n\geq 1$. For all $x\in M$, there exists an $l\geq0$ such that $x\in F^lM$ and hence for every $k\geq l+1$ we have that $$f_{k}(x)=f_{k}(\tau_l(x))=f(\tau_l(x))=f(x).$$ This means that the sequence of elements $\left\{f_n(x)\right\}_{n\geq0}$ eventually becomes constant in $A$ and equal to the value of $f$ on $x$. Thus, it is meaningful to set $f(x)=\left({\underset{n\to\infty}{\lim}}(f_n)\right)(x):={\underset{n\to\infty}{\lim}}(f_n(x))$.
On the other hand, notice that we may consider the inductive limit function of the inductive cone $\{\tau_n^*(f_{n+1})\}_{n \, \in \, \mathbb{N}}$. However, ${\varinjlim_{n}\left({\tau_n^*(f_{n+1})}\right)} = {\varinjlim_{n}\left({\tau_n^*(f)}\right)}=f={\underset{n\to\infty}{\lim}}(f_n)$.
The topological tensor product and the associativity constraint {#ssec:TTAC}
---------------------------------------------------------------
It is useful to recall that the full subcategory of $R$-bimodules which are locally finitely generated and projective on the right is closed under taking tensor products (compare with [@Majewski Theorem C.24, page 93]). Indeed, let $M,N$ be filtered $R$-bimodules which are locally finitely generated and projective on the right. Then we have an $R$-bilinear isomorphism $$\begin{gathered}
\bigoplus_{p+q=n}\frac{F^pM}{F^{p-1}M}{\otimes_{\scriptscriptstyle{R}}}\frac{F^qN}{F^{q-1}N} \, \longrightarrow \,\frac{{{\mathcal F}}^n(M{\otimes_{\scriptscriptstyle{R}}}N)}{{{\mathcal F}}^{n-1}(M{\otimes_{\scriptscriptstyle{R}}}N)}, \quad \\ \bigg((x_p+F^{p-1}M){\otimes_{\scriptscriptstyle{R}}}(y_q+F^{q-1}N)\mapsto (x_p{\otimes_{\scriptscriptstyle{R}}}y_q)+{{\mathcal F}}^{n-1}(M{\otimes_{\scriptscriptstyle{R}}}N)\bigg)\end{gathered}$$ where ${{\mathcal F}}^n(M{\otimes_{\scriptscriptstyle{R}}}N)=\sum_{p+q=n}F^pM{\otimes_{\scriptscriptstyle{R}}}F^qN$. Thus the factors ${{\mathcal F}}^n(M{\otimes_{\scriptscriptstyle{R}}}N)/{{\mathcal F}}^{n-1}(M{\otimes_{\scriptscriptstyle{R}}}N)$ are finitely generated and projective as right $R$-modules. Therefore, $M{\otimes_{\scriptscriptstyle{R}}}N$ is locally finitely generated and projective as claimed.
Next, we want to compare the topology that the linear dual $(N{\otimes_{\scriptscriptstyle{R}}}M)^*$ inherits from the structure of locally finitely generated and projective module, with that of ${\widehat{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}}=M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N^*$, the topological tensor product of the complete $R$-bimodules $M^*$ and $N^*$. At the algebraic level, we have a canonical $R$-bilinear map $$\label{Eq:eva}
\xymatrix@R=0pt{ (M^*)_{{\scriptscriptstyle{R}}} {\otimes_{\scriptscriptstyle{R}}} {_{{\scriptscriptstyle{R}}}(N^*)} \ar@{->}^-{\phi_{{\scriptscriptstyle{M,N}}}}[rr] & & (N_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}}{_{{\scriptscriptstyle{R}}}M})^* \\ f{\otimes_{\scriptscriptstyle{R}}}g \ar@{|->}^-{}[rr] & & \left[ y{\otimes_{\scriptscriptstyle{R}}}x \longmapsto f(g(y)x) \right] }$$ which makes the following diagram to commute $$\label{eq:phifilt}
\xymatrix@R=25pt @C=30pt{ M^*{\otimes_{\scriptscriptstyle{R}}} N^* \ar@{->}^-{\phi_{{\scriptscriptstyle{M,N}}}}[r] \ar@{->}_-{ \left({\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}} \right)^*{\otimes_{\scriptscriptstyle{R}}} \left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}} \right)^*}[d] & \left(N{\otimes_{\scriptscriptstyle{R}}}M\right)^* \ar@{->}^-{ \left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}} \right)^*}[d] \\ F^mM^*{\otimes_{\scriptscriptstyle{R}}}F^nN^* \ar@{->}^-{\phi_{{\scriptscriptstyle{m,n}}}}[r] & \left( F^nN{\otimes_{\scriptscriptstyle{R}}}F^mM \right)^*. }$$
In view of the technical subsequent Lemma \[lemma:filtrintersection\], it turns out that the natural transformation $\phi_{{\scriptscriptstyle{M,\, N}}}$ is a continuous map (in fact a morphism of filtered $R$-bimodules) where ${{\mathcal F}}_n\left(M^*{\otimes_{\scriptscriptstyle{R}}} N^*\right)=\sum_{p+q=n}{\mathsf{Im}\left({F_pM^*{\otimes_{\scriptscriptstyle{R}}}F_qN^*}\right)}$ and the decreasing filtration on $(N{\otimes_{\scriptscriptstyle{R}}}M)^*$ is given as in , that is, $F_0(N{\otimes_{\scriptscriptstyle{R}}}M)^*=(N{\otimes_{\scriptscriptstyle{R}}}M)^*$ and $F_n(N{\otimes_{\scriptscriptstyle{R}}}M)^*={\mathrm{Ker}\left({\boldsymbol{\tau}_{{\scriptscriptstyle{n-1}}}^*}\right)}$, $n \geq 1$, where $\boldsymbol{\tau}_{{\scriptscriptstyle{n}}}^*: (N{\otimes_{\scriptscriptstyle{R}}}M)^* \to {{\mathcal F}}^n(N{\otimes_{\scriptscriptstyle{R}}}M)^*$ are the canonical projections.
\[lemma:filtrintersection\] Let $R$ be a ring and $V,W$ be decreasingly filtered $R$-bimodules such that $W/F_nW$ are finitely generated and projective as left $R$-modules for all $n\in{\mathbb{N}}$. Then $${{\mathcal F}}_{n}\left(V{\otimes_{\scriptscriptstyle{R}}} W\right) :\,=\, \sum_{p+q=n}F_{p}V {\otimes_{\scriptscriptstyle{R}}} F_{q}W\,=\, \bigcap_{p+q=n+1}{\mathrm{Ker}\left({\pi_{{\scriptscriptstyle{p}}}^{{\scriptscriptstyle{V}}}{\otimes_{\scriptscriptstyle{R}}} \pi_{{\scriptscriptstyle{q}}}^{{\scriptscriptstyle{W}}}}\right)}$$ where $\pi_{{\scriptscriptstyle{p}}}^{{\scriptscriptstyle{V}}}\colon V\to V/F_pV$ and $\pi_{{\scriptscriptstyle{q}}}^{{\scriptscriptstyle{W}}}\colon W\to W/F_qW$ are the canonical projections. In particular, for $M$ and $N$ $R$-bimodules such that $N$ is locally finitely generated and projective on the right, we have $${{\mathcal F}}_n(M^*{\otimes_{\scriptscriptstyle{R}}}N^*)=\bigcap_{p+q=n-1}{\mathrm{Ker}\left({\tau_p^*{\otimes_{\scriptscriptstyle{R}}}\tau_q^*}\right)}.$$
The following proposition gives the desired comparison between the linear topologies on the filtered bimodules ${\widehat{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}}={M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N^*}$ and $(N{\otimes_{\scriptscriptstyle{R}}}M)^*$.
\[prop:MRN\] Let $M$ and $N$ be two $R$-bimodules, locally finitely generated and projective as right $R$-modules. Then the natural transformation $\phi_{{\scriptscriptstyle{M,N}}}$ of equation induces an homeomorphism (in fact, a filtered isomorphism) ${M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N^*} \cong (N{\otimes_{\scriptscriptstyle{R}}}M)^*$ such that the following diagram is commutative: $$\xymatrix@R=20pt@C=45pt{ \left(M^*\right)^{}_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}}{^{}_{{\scriptscriptstyle{R}}}\left(N^*\right)} \ar@{->}^-{\phi_{{\scriptscriptstyle{M,N}}}}[rr] \ar@{->}_-{\gamma_{{\scriptscriptstyle{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}}}}[rd] & & (N_{{\scriptscriptstyle{R}}}{\otimes_{\scriptscriptstyle{R}}}{_{{\scriptscriptstyle{R}}}M})^* \\ & {\left(M^*\right)^{}_{{\scriptscriptstyle{R}}}{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}{^{}_{{\scriptscriptstyle{R}}}\left(N^*\right)}} \ar@{-->}_-{{\widehat{\phi_{{\scriptscriptstyle{M,N}}}}}}^-{\cong}[ru] & }$$
We know that $N{\otimes_{\scriptscriptstyle{R}}}M$ is a locally finitely generated and projective right $R$-module, whence $(N{\otimes_{\scriptscriptstyle{R}}}M)^*$ is a complete $R$-bimodule with respect to the filtration $\mathsf{Ann}(F^k(N{\otimes_{\scriptscriptstyle{R}}}M))=F_{k+1}(N{\otimes_{\scriptscriptstyle{R}}}M)^*$ (see §\[ssec:DLF\]). In view of , for all $m+n=k$ we have that $$\left({\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}\right)^*\left(\phi_{M,N}\left({{\mathcal F}}_{k+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)\right)\right)=\phi_{m,n}\left(\left(\left({\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}\right)^*{\otimes_{\scriptscriptstyle{R}}}\left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}\right)^*\right)\left({{\mathcal F}}_{k+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)\right)\right)=0.$$ In particular, there exists a unique $R$-bilinear morphism $\sigma_{{\scriptscriptstyle{m,n}}}:{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}/{{{\mathcal F}}_{k+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)} \to ( F^nN{\otimes_{\scriptscriptstyle{R}}}F^mM )^*$ such that $\sigma_{{\scriptscriptstyle{m,n}}}\,{{\pi}_{{\scriptscriptstyle{k+1}}}^{{\scriptscriptstyle{M^*{\otimes_{\scriptscriptstyle{R}}} N^*}}}}=\left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}\right)^*\,\phi_{{\scriptscriptstyle{M,N}}}$. Notice that the completion ${\widehat{\phi_{{\scriptscriptstyle{M,N}}}}}$ of the filtered morphism $\phi_{M,N}$ fits into the following commutative diagram $$\label{eq:diagrams}
\xymatrix@R=20pt@C=30pt{ M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~} N^* \ar@{->}^-{{\widehat{\phi_{{\scriptscriptstyle{M,N}}}}}}[r] \ar@{->}_-{\mathfrak{p}_{k+1}}[d] & (N{\otimes_{\scriptscriptstyle{R}}}M)^* \ar@{->}^-{\left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}\right)^*}[d] \\ \frac{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}{{{\mathcal F}}_{k+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)} \ar^-{\sigma_{{\scriptscriptstyle{m,n}}}}[r] & \Big( F^nN{\otimes_{\scriptscriptstyle{R}}}F^mM \Big)^* }$$
Our next aim is to construct explicitly a filtered inverse for ${\widehat{\phi_{{\scriptscriptstyle{M,N}}}}}$. Set $\gamma:=\gamma_{{\scriptscriptstyle{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}}}: M^{*}{\otimes_{\scriptscriptstyle{R}}}N^{*} \to {\widehat{M^{*}{\otimes_{\scriptscriptstyle{R}}}N^{*}}}$. For all $m+n=k$ consider the composition $\Pi_{m,n}:=\phi^{-1}_{m,n}\circ \sigma_{m,n}\circ \mathfrak{p}_{k+1}$ which gives an $R$-bilinear morphism $\Pi_{m,n}:M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N^*\to F^mM^*{\otimes_{\scriptscriptstyle{R}}}F^nN^*$. It satisfies $\Pi_{m,n}\circ\gamma=\left({\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}\right)^*{\otimes_{\scriptscriptstyle{R}}}\left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}\right)^*$ and $$F_{k+1}\left(M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N^*\right):={\mathrm{Ker}\left({\mathfrak{p}_{k+1}}\right)}=\bigcap_{m+n=k}{\mathrm{Ker}\left({\Pi_{m,n}}\right)}.$$ Now, by considering the $R$-bilinear maps $\phi^{-1}_{{\scriptscriptstyle{m,n}}}\, \sigma_{{\scriptscriptstyle{m,n}}}:M^*{\otimes_{\scriptscriptstyle{R}}}N^*/{{\mathcal F}}_{k+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right) \to F^mM^*{\otimes_{\scriptscriptstyle{R}}}F^nN^* $ and $$\label{eq:}
\xi_{m,n}:F^mM^*{\otimes_{\scriptscriptstyle{R}}}F^nN^*\to \frac{M^*{\otimes_{\scriptscriptstyle{R}}}N^*}{{{\mathcal F}}_{h+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)}; \qquad \left(f{\otimes_{\scriptscriptstyle{R}}}g\mapsto \theta_{{\scriptscriptstyle{m}}}^*(f){\otimes_{\scriptscriptstyle{R}}}{\theta_{{\scriptscriptstyle{n}}}}^*(g)+{{\mathcal F}}_{h+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)\right), \, \footnote{Notice that we cannot perform the tensor product $\theta_{{\scriptscriptstyle{m}}}^*{\otimes_{\scriptscriptstyle{R}}}{\theta_{{\scriptscriptstyle{n}}}}^*$ as the maps ${\theta_{{\scriptscriptstyle{n}}}}^*$ are just left $R$-linear. Nevertheless, the stated morphism is well-defined. It is a consequence of Remark \ref{rem:vartheta} and of the fact that ${\mathrm{Ker}\left({\left({\tau_{{\scriptscriptstyle{p}}}^{{\scriptscriptstyle{M}}}}\right)^*{\otimes_{\scriptscriptstyle{R}}}\left({\tau_{{\scriptscriptstyle{q}}}^{{\scriptscriptstyle{N}}}}\right)^*}\right)} \subseteq {{\mathcal F}}_{m+1}\left(M^*{\otimes_{\scriptscriptstyle{R}}}N^*\right)$, $m=\min(p,q)$.}$$ for all $m+n=k$ and $h=\min(m,n)$, one can show that $M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~} N^*$ together with the family of morphisms $\left\{\Pi_{{\scriptscriptstyle{m,n}}}\mid n,m\geq0\right\}$ is isomorphic to the inverse limit of the projective system $F^mM^*{\otimes_{\scriptscriptstyle{R}}}F^nN^*$ with structure maps $\left({\tau_{{\scriptscriptstyle{p,m}}}^{{\scriptscriptstyle{M}}}}\right)^* {\otimes_{\scriptscriptstyle{R}}}\left({\tau_{{\scriptscriptstyle{q,n}}}^{{\scriptscriptstyle{N}}}}\right)^*$ for all $p\leq m$ and $q\leq n$. Now, by definition of $\Pi_{{\scriptscriptstyle{m,n}}}$ we have that $$\label{eq:phiPi}
\phi_{{\scriptscriptstyle{m,n}}}\circ \Pi_{{\scriptscriptstyle{m,n}}}\stackrel{(\text{def})}{=}\sigma_{{\scriptscriptstyle{m,n}}}\circ \mathfrak{p}_{k+1}\stackrel{\eqref{eq:diagrams}}{=}\left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}\right)^*\circ {\widehat{\phi_{{\scriptscriptstyle{M,N}}}}},$$ whence ${\widehat{\phi_{{\scriptscriptstyle{M,N}}}}}$ is also the unique morphism induced by the map of projective systems $\phi_{{\scriptscriptstyle{m,n}}}$. By considering $\phi_{{\scriptscriptstyle{m,n}}}^{-1}$ instead, one deduces that there exists a unique morphism $\psi_{{\scriptscriptstyle{M,N}}}:\left(N{\otimes_{\scriptscriptstyle{R}}} M\right)^*\to M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~} N^*$ such that $\Pi_{{\scriptscriptstyle{m,n}}}\circ \psi_{{\scriptscriptstyle{M,N}}}=\phi_{{\scriptscriptstyle{m,n}}}^{-1} \circ \left({\tau_{{\scriptscriptstyle{n}}}^{{\scriptscriptstyle{N}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{m}}}^{{\scriptscriptstyle{M}}}}\right)^*$. It is not difficult now to see that ${\widehat{\phi_{{\scriptscriptstyle{M,N}}}}}$ and $\psi_{{\scriptscriptstyle{M,N}}}$ are filtered morphisms which are mutually inverses.
Given $z\in M^*{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~} N^*$, we already know that $z={\underset{n\to\infty}{\lim}}{\left(\mathfrak{p}_n(z)\right)}$, up to a choice of a representative in $M^*{\otimes_{\scriptscriptstyle{R}}} N^*$ for each element $\mathfrak{p}_n(z)$. Fix $n\geq 0$, for all $h,k\geq n$ such that $n=\min(h,k)$, it turns out from the previous proof that $$\begin{aligned}
\left(\xi_{h,k}\circ \Pi_{h,k}\right)(z) & \stackrel{(\text{def})}{=} \left( \xi_{h,k} \circ \phi^{-1}_{h,k} \circ \sigma_{h,k}\circ \mathfrak{p}_{h+k+1} \right)(z) = \left( \xi_{h,k} \circ \phi^{-1}_{h,k} \circ \sigma_{h,k}\circ \pi_{h+k+1} \right)(x) \\
& = \left( \xi_{h,k} \circ \phi^{-1}_{h,k} \circ \left({\tau_{{\scriptscriptstyle{k}}}^{{\scriptscriptstyle{N}}}}{\otimes_{\scriptscriptstyle{R}}}{\tau_{{\scriptscriptstyle{h}}}^{{\scriptscriptstyle{M}}}}\right)^*\circ \phi_{{\scriptscriptstyle{M,N}}} \right)(x) \stackrel{\eqref{eq:phifilt}}{=} \left( \xi_{h,k} \circ \left( \left({\tau_{{\scriptscriptstyle{h}}}^{{\scriptscriptstyle{M}}}}\right)^*{\otimes_{\scriptscriptstyle{R}}}\left({\tau_{{\scriptscriptstyle{k}}}^{{\scriptscriptstyle{N}}}}\right)^* \right) \right)(x) = \pi_{n+1}(x)\end{aligned}$$ for some $x\in M^*{\otimes_{\scriptscriptstyle{R}}} N^*$ and where $\pi_i:M^*{\otimes_{\scriptscriptstyle{R}}} N^*\to M^*{\otimes_{\scriptscriptstyle{R}}} N^*/{{\mathcal F}}_i\left(M^*{\otimes_{\scriptscriptstyle{R}}} N^*\right)$ is the canonical projection, for all $i\geq 0$. Now, $\mathfrak{p}_{h+k+1}(z)=\pi_{h+k+1}(x)=\mathfrak{p}_{h+k+1}(\gamma(x))$ implies that $z-\gamma(x)\in{\mathrm{Ker}\left({\mathfrak{p}_{h+k+1}}\right)}\subseteq {\mathrm{Ker}\left({\mathfrak{p}_{n+1}}\right)}$, since $n\leq h+k$. Thus $\pi_{n+1}(x)=\mathfrak{p}_{n+1}(\gamma(x))=\mathfrak{p}_{n+1}(z)$ as well and hence $\xi_{h,k}\circ \Pi_{h,k} = \mathfrak{p}_{\min(h,k)+1}$. In particular, $$\label{eq:zlim}
z={\underset{n\to\infty}{\lim}}{\left(\mathfrak{p}_{n+1}(z)\right)}={\underset{n\to\infty}{\lim}}{\left(\left(\xi_{n,n}\circ \Pi_{n,n}\right)(z)\right)}.$$
**Acknowledgement.** Both authors would like to thank the referee for the careful reading and the helpful comments on a previous version of the paper. Moreover, the second author would like to thank the members of the campus in Ceuta for their friendship and the warm hospitality during his stays there.
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[^1]: This paper was written while P. Saracco was member of the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INdAM). His stay as visiting researcher at the campus of Ceuta of the University of Granada was financially supported by IEMath-GR. Research supported by the Spanish Ministerio de Economía y Competitividad and the European Union FEDER, grant MTM2016-77033-P
[^2]: Even if we plan to apply the results of this subsection to a co-commutative Hopf algebroid $(A,{{\mathcal U}})$, these are interesting on their own and this justifies the choice of presenting them in the present form.
[^3]: Such homomorphisms are called *of degree $0$* in the literature. Cf. e.g. [@NasOys Definition D.I.5].
[^4]: In [@MR1420862 I.2.7] the shifted module is denoted by $T(n)M$. Notice that here we consider only positively filtered modules with decreasing filtration.
[^5]: One should notice that here some confusion may arise, as we denoted by ${{\gamma_{{\scriptscriptstyle{M}}}} \colon {M} \rightarrow {{\widehat{M}}={\widehat{\mathscr{U}(M)}}}}$ also the canonical isomorphism in the category of complete bimodules whose actual inverse is $\sigma_M$. Cf. Remark \[rem:gamma\]. Nevertheless, as we may embed ${{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{c}}_{{\scriptscriptstyle{R}}}}\left(M,{\widehat{\mathscr{U}(M)}}\right)\subseteq {{}^{}_{{\scriptscriptstyle{S}}}\mathsf{Bim}{}^{\mathsf{flt}}_{{\scriptscriptstyle{R}}}}\left(\mathscr{U}(M),\mathscr{U}\left({\widehat{\mathscr{U}(M)}}\right)\right)$ and $\mathscr{U}(\gamma_{{\scriptscriptstyle{M}}})=\gamma_{\mathscr{U}(M)}$, we can identify the two morphism and it will be clear from the context which one we are referring to.
[^6]: By a bi-filtered morphism we mean a morphism $f:M\times N\to P$ such that $f(F_hM\times F_kN)\subseteq F_{h+k}P$. In particular, if $f$ is bi-filtered then $f(-,n):M\to P$ and $f(m,-):N\to P$ are filtered for all $m\in M$ and $n\in N$. Bi-filtered morphisms can be seen as a counterpart of *separately continuous* functions (for an account on the subject we refer the reader to [@Piotrowski]).
[^7]: Note that the writing $M{~{\widehat{\otimes}}_{{\scriptscriptstyle{R}}}~}N$ doesn’t make sense in this context, unless both $M$ and $N$ are complete, whereas ${\widehat{M{\otimes_{\scriptscriptstyle{R}}} N}}$ does.
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---
abstract: 'We propose a method for programmable shaping of the amplitude and phase of the XUV and x-ray attosecond pulses produced by high-order harmonic generation. It overcomes the bandwidth limitations of existing spectral filters and enables removal of the intrinsic attosecond chirp as well as the synthesis of pulse sequences. It is based on partial phase matching, such as quasi-phase matching, using a longitudinally addressable modulation.'
author:
- 'Dane R. Austin'
- Jens Biegert
title: Attosecond pulse shaping using partial phase matching
---
=1
Coherent extreme ultraviolet (XUV) and soft x-ray radiation, produced by high-order harmonic generation (HHG) in an intense laser field, is central to attoscience [@Nisoli-2009-New; @*Krausz-2009-Attosecond]. Spectra spanning 200-1600eV, with the potential to support temporal features of 2.5as duration, have been generated, providing the raw material for coherent excitation of atomic scale electron dynamics down to the inner shell. At present, there is no general means of controlling the spectral and temporal profile of radiation produced via HHG on the attosecond timescale. Bandpass and dispersive filtering, the latter being necessary to eliminate the attosecond chirp that is intrinsic to HHG, have been demonstrated using metal films, gases, and multilayer mirrors, but these lack tunability and have been demonstrated below 150eV for bandwidths of $\approx 50$eV [@Goulielmakis-2008-Single-Cycle; @*Hofstetter-2011-Attosecond; @*Ko-2010-Attosecond]. The ability to arbitrarily shape HHG over its entire bandwidth would improve existing experiments and enable others, analogous to the role of dispersion control [@Walmsley-2001-role] and pulse shaping [@Weiner-2000-Femtosecond] in femtosecond science and technology. Using harmonic spectra from standard IR sources, pulses as short at 2.5as could be generated [@Popmintchev-2012-Bright]. Current XUV-pump IR-probe studies of single-photon ionization [@Schultze-2010-Delay; @*Cavalieri-2007-Attosecond] could be performed over a range of photon energies, accessing a wider range of initial states and potentially disentangling the roles of the Coulomb potential and the IR probe. Coherently controlled wavepackets [@Remetter-2006-Attosecond] could be launched and probed. With sufficient intensity, XUV-pump XUV-probe [@Hu-2006-Attosecond] spectroscopy and coherent control could be achieved.
Macroscopic effects — the coherent sum of the dipole response of all the target atoms — play a crucial role in attosecond pulse generation via HHG [@Gaarde-2008-Macroscopic]. Of primary importance is the wave vector mismatch $\Delta k$ between the laser-driven dipole excitation and the propagating harmonics. Whilst the latter propagate very close to $c$, the former is influenced by diffraction in the focus or waveguide, dispersion of the neutral gas and free-electron plasma, and the intensity dependence of the electron in the continuum. Partial phase matching [@OKeeffe-2012-Quasi; @*Zhang-2007-Quasi-phase-matching], achieved with a longitudinal modulation in the dipole excitation of wavevector $K = \Delta k$, can be used to overcome a phase mismatch. Partial phase matching is inherently dependent on the harmonic frequency and hence offers a degree of control over the spectrum. Here, we show that using partial phase matching with a longitudinally addressable modulation, one may compensate the attosecond chirp and also synthesize an arbitrary in-situ amplitude and phase filter for HHG. The method is applicable over the whole spectrum up to kiloelectron-volt photon energies.
The essence of our method is that in HHG, a phase velocity mismatch is almost always accompanied by a group velocity mismatch i.e. $\Delta k(\omega)=\Delta n \omega / c$, where $\Delta n=n(\omega_1)-n(\omega)$ is the difference between the refractive indices at the fundamental frequency $\omega_1$ and the harmonics $\omega$. This is because in HHG, the group delay of the constituent attosecond bursts is dictated by the laser field oscillations (rather than their envelope as with perturbative harmonic generation). The phase velocity of the laser is therefore imparted on the group velocity of the dipole response. Partial phase matching with wavenumber $K$ occurs at a single frequency $\omega=Kc/\Delta n$. If $K$ varies along the propagation axis, then multiple frequencies are phase matched, but because of the group-velocity mismatch, their group delays will differ, producing a relative chirp between the dipole excitation and the macroscopically generated field. A linear variation $K_1 = {\text{d}}K / {\text{d}}z$ leads to a quadratic spectral phase of $(\Delta n/c)^2/K_1$, tunable in both magnitude and sign through $K_1$. As we will show, this effect can compensate the attosecond chirp leading to transform limited pulses, or be generalized to enable arbitrary pulse shaping.
Figure \[fig:diagram\] is a cartoon illustration of the concept, depicting compensation of the positive chirp of the short trajectories assuming a subluminal laser phase velocity. For simplicity, we assume $n(\omega)=1$. At two points along the propagation axis $z$, the laser field $E$ (red) and the kinetic energy KE (purple) of the recombining electron are plotted versus time in the retarded frame $t=\bar{t}-z/c$, where $\bar{t}$ is time in the lab frame. The laser field and the electron motion that it drives are delayed upon propagation, as shown by the sloped grey lines through the laser field peaks (red dots) and classical cutoffs (violet dots). A negatively chirped modulation (blue) achieves partial phase matching at decreasing harmonic frequencies, indicated by the horizontal dashed lines. The negative chirp is chosen such that the recombination time of the phase matched harmonics is constant (vertical dashed line), resulting in unchirped emission from the short trajectory.
![\[fig:diagram\]Attosecond chirp removal through negatively chirped partial phase matching; longitudinal modulation (blue), laser field (red) and electron kinetic energy (purple) versus retarded time, frequency of partial phase matching (horizontal dashed lines), and recombination time of partially phase matched harmonics (vertical dashed line).](attochirp-comp-diag_2D_subluminal)
We now present a simulation of a concrete implementation. Our laser propagation code includes dispersion, diffraction, self-phase modulation, and plasma dephasing and absorption. The single-atom response HHG code is an augmented Lewenstein model, with ADK ionization rates and photorecombination cross sections [@Austin-2012-Strong]. The propagation of the harmonics includes diffraction, and absorption and dispersion by the neutral gas. All quantities (including those pertaining to quasi-phase matching) have full spatio-temporal dependence with cylindrical symmetry. A temporally and spatially Gaussian driving pulse with 9fs full-width at half maximum (FWHM) duration, 1.8m center wavelength and 220J energy is focused to a $e^{-2}$ radius of 50m a distance 1.4mm before a jet of helium with peak pressure 5bar and FWHM thickness 1.4mm. The retardation of the fundamental field by the neutral gas causes a positive phase mismatch. A counter-propagating pulse train (CPT) of wavelength $\lambda{_{\text{c}}}=800$nm and total energy 15J is focused to a $e^{-2}$ radius of 50m in the middle of the gas jet. The temporal intensity profile $I{_{\text{c}}}=|E{_{\text{c}}}^2|/(2Z_0)$ of the CPT is plotted against its own comoving time $t{_{\text{c}}}=\bar{t}+z/c$ in Fig. \[fig:TLP\](a). The relative phase between a peak of the drive field and the CPT field varies as $\gamma=4\pi z/\lambda{_{\text{c}}}$. To lowest order, the CPT perturbs the action accumulated by the electron in the continuum, resulting in a phase shift $\alpha=\alpha_0 |E{_{\text{c}}}| \cos \gamma$ [@Cohen-2007-Optimizing], where $\alpha_0$ is given by first-order perturbation of the SFA action integral [@Dudovich-2006-Measuring]. Averaging the induced phase modulation $e^{i \alpha}$ over the longitudinally rapid variation of $\gamma$, the CPT is found to modulate the harmonic emission by a factor $J_0(\alpha_0 |E{_{\text{c}}}|)$ where $J_0$ is the zeroth-order Bessel function [@Cohen-2007-Optimizing]. The slow variations of $|E{_{\text{c}}}|$ are experienced by the drive field as a longitudinal modulation of frequency $K=4\pi/(cT{_{\text{c}}})$, overlaid in Fig. \[fig:TLP\](a). Here, $T{_{\text{c}}}$ is the (longitudinally varying) period of the CPT. The decrease of $K$ with $z$ induces quasi-phase matching at a decreasing harmonic frequency in the manner of Fig. \[fig:diagram\]. The generated macroscopic field is passed through a 100nm silver spectral filter and a 0.5mrad radius far-field spatial filter to eliminate the long trajectories. The resulting temporal profile is shown in Fig. \[fig:TLP\](b). A 31as pulse is produced, quite close to the transform limited duration of 20as. Note that all the temporal and spectral profiles in this Letter are radially integrated to infinity (corresponding to the experimental observable in e.g. photoelectron spectroscopy), proving the absence of significant spatio-temporal distortion. Temporal gating of the emission to a single half-cycle is achieved by the short pulse duration, following Goulielmakis et al. [@Goulielmakis-2008-Single-Cycle]. However, because of the spectral selectivity of the quasi-phase matching, a high-pass spectral filter is not needed. To verify that an isolated pulse is produced, Fig. \[fig:TLP\](c) shows a zoomed out temporal profile on a logarithmic scale. Satellite pulses are at the 1% intensity level, below the detection threshold of current experiments, and the main pulse contains $>90$% of the energy. The spectral density and phase, the latter being an intensity-weighted radial average, are shown in Fig. \[fig:TLP\](d). A maximum phase deviation of 0.6rad across the full width at 10% bandwidth 204–310eV shows that the attochirp has been completely compensated. Note that in Fig. \[fig:TLP\](d), as with the other spectra in this Letter, a numerical window, shown in grey in Fig. \[fig:TLP\](c), has been applied to the temporal profile to prevent the weak satellite pulses from causing fine interference fringes which obscure the key result.
![\[fig:TLP\]Production of a transform limited 32as pulse with a few-cycle infrared drive field and chirped quasi-phase matching.(a) Temporal intensity of the counterpropagating pulse train (blue solid, left $y$-axis) and corresponding longitudinal spatial frequency (red dashed, right $y$-axis). (b) Instantaneous power of the generated pulse (red solid), with indicated FWHM duration and Fourier-transform limited pulse (blue dashed). (c) Same as (b) but zoomed out and on logarithmic scale. The temporal window used for computing spectra is shown in grey. (d) Spectral density (blue, left $y$-axis) and phase (red, $y$-axis).](TLP)
We now examine the design of the CPT, Fig. \[fig:TLP\](a), in more detail. We focus on the emission from the short trajectory by the dominant laser half-cycle. Figure \[fig:TLP\_Deltak\](a) shows the phase mismatch $\Delta k$ extracted from the simulation in the $(\omega,z)$ domain. It shows that the phase mismatch is not perfectly proportional to the harmonic frequency, and varies with $z$. These departures from the simple picture leading to Fig. \[fig:diagram\] are due to the intensity-dependent dipole phase, the Gouy phase, plasma-induced dephasing of the fundamental and (weakly) the dispersion of the neutral gas at the XUV frequencies. They necessitate a slight refinement. Our method is to extract the group delay $\tau(\omega,z)$ of the harmonics, including propagation to the end of the gas. This is shown in Fig. \[fig:TLP\_Deltak\](b). Starting at $(\omega_0,0)$, where $\omega_0=259$eV is a chosen center frequency, we trace out a contour $(\omega {_{\text{P}}}(z),z)$ of $\tau$. The required longitudinal spatial frequency of the CPT is the phase mismatch evaluated along this contour i.e. $K(z)=-\Delta k(\omega{_{\text{P}}}(z),z)$. The red lines superimposed on Fig. \[fig:TLP\_Deltak\](a) and (b) illustrate this process. The amplitude $|E{_{\text{c}}}(t{_{\text{c}}})|$ of the CPT is then $$\left| E{_{\text{c}}}\left(\frac{2z}{c}\right)\right|=f[\Phi(z),\epsilon]/\alpha_0(\omega{_{\text{P}}}(z),z) \label{eq:Ec}$$ where $\Phi(z)=\int K(z) {\text{d}}z$ is the phase of the CPT, $\alpha_0(\omega{_{\text{P}}},z)$ the frequency-dependent phase perturbation coefficient defined above, and $f(\Phi,\epsilon)$ is chosen to map the full range of the zeroth order Bessel function onto a sinusoid of phase $\Phi$ and amplitude $\epsilon$: $$f(\Phi,\epsilon)=J_0^{-1} \left[ \epsilon (0.701\cos \Phi + 0.299) \right].$$ The parameter $\epsilon$ is used below to extend the method to amplitude modulation; for now, we take $\epsilon=1$. The particular form (\[eq:Ec\]) gives optimal QPM efficiency and minimizes unwanted spatial overtones, but is not essential. In an experiment, $\tau(\omega,z)$ and $\Delta k(\omega,z)$ could be obtained in a calibration step, with trial CPT sequences generated by a programmable femtosecond pulse shaper [@Weiner-2000-Femtosecond].
 Phase mismatch and (b) group delay of short trajectories versus harmonic frequency and propagation distance. The reference for the group delay is defined at $(z,\omega)=$(0mm,259eV). The red lines show the group delay contour $\tau=0$. The hatched area is above the classical cutoff, with no meaningful phase.](TLP_Deltak)
The preceding method may be generalized to produce an arbitrary smoothly varying group delay $\tau{_{\text{S}}}(\omega)$ by tracing a contour of $\Delta \tau = \tau(\omega,z)-\tau{_{\text{S}}}(\omega)$. All the subsequent steps are identical. Figure \[fig:chirp\](a) shows the CPT spatial frequencies, relative to the transform limited case, required to produce a quadratic spectral phase of $\pm1170\,$as$^2$ ($\pm 2$atomic units) and a cubic spectral phase of -14154as$^3$ (-1 atomic unit). Qualitatively, the curves correspond to the imparted group delay: approximately linear (around $z=0$) for the quadratic spectral phase, and quadratic for the cubic spectral phase. The phases of the resulting attosecond pulses, along with the target phases are shown in Fig. \[fig:chirp\](d),(e) and (f). There is excellent agreement between the actual and target spectral phases. However, the bandwidth of the chirped pulses is different to the transform limited case, shown by the dashed blue lines. This coupling of chirp to bandwidth is caused by clipping of the $z$-dependent phase matched frequency and is inherent to the method. Figure \[fig:chirp\](b) compares the phase-matched frequency versus $z$ of the chirped and transform limited cases. For a positively chirped pulse, closer to the intrinsic attosecond chirp of the dipole response, the phase-matched frequency sweep is faster, and a larger bandwidth is generated within the interaction region. The opposite applies for a negatively chirped pulse. The trend is illustrated in Fig. \[fig:chirp\](c), which shows the root-mean square bandwidth versus quadratic spectral phase. In general, the interaction length cannot be arbitrarily increased; plasma-induced defocusing reduces the intensity of the fundamental and the neutral gas absorbs harmonics generated at the start. This sets the ultimate limit to the pulse shaping capability of the method.
 QPM spatial frequency relative to transform-limited case for generation of labelled spectral phases. (b) Phase-matched harmonic frequency for generation of the labelled quadratic spectral phases and the transform limited case (dashed). (c) Root-mean square bandwidth of the generated pulse versus generated quadratic spectral phase coefficient. (d)-(f) Spectral density (blue, left $y$-axis) and phase (red, right $y$-axis) of the resulting attosecond pulses corresponding to (a). The spectral density for the transform-limited case is shown for reference by the dotted blue lines. The spectral phases are blanked out at the 1% intensity level. The target spectral phases, with zeroth- and first-order terms adjusted for best fit, are shown by dashed red lines.](chirp)
A further generalization is to exploit the one-to-one correspondence between $z$ and $\omega$ by directly modulating the phase and amplitude of the CPT, producing an arbitrary transfer function $H(\omega)$. In (\[eq:Ec\]), one sets $\Phi(z) \rightarrow \Phi(z) - \arg H(\omega{_{\text{P}}}(z))$ and $\epsilon=|H(\omega)|$. In this sense, the method is analogous to programmable acousto-optic filters used for femtosecond pulse shaping [@Verluise-2000-Amplitude]. Figure \[fig:delay\] shows some results. The CPT in Fig. \[fig:delay\](a) leads to the double attosecond pulse in Fig. \[fig:delay\](b) and features a slow modulation corresponding to the pulse separation of 97as. This separation can be continuously tuned. Subfigures (c) and (d) show the same for a pulse separation of 194as. Additionally, the relative phase of the subpulses can be adjusted, shown by subfigures (e) and (f).
![\[fig:delay\] (a),(c),(e) Counterpropagating pulse trains for the generation of isolated double attosecond pulses (b),(d),(f). (a),(b): 97 as separation. (c),(d): 194 as separation. (e),(f) 97 as separation with $\pi/2$ phase shift on the second pulse.](delay)
In summary, we have shown that quasi-phase matching HHG with a shaped counterpropagating pulse train enables control over the spectral amplitude and phase of the harmonics, including elimination of the attosecond chirp. The concept may be applied to any implementation of partial phase matching that permits longitudinal addressing of the modulation frequency, including grating-assisted phase matching [@Cohen-2007-Grating] which has the potential for high efficiency extension to keV photon energies.
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abstract: 'UM 625, previously identified as a narrow-line active galactic nucleus (AGN), actually exhibits broad [H$\alpha$]{} and [H$\beta$]{} lines whose width and luminosity indicate a low black hole mass of $1.6 \times 10^6$ [$M_{\odot}$]{}. We present a detailed multiwavelength study of the nuclear and host galaxy properties of UM 625. Analysis of [*Chandra*]{} and [*XMM-Newton*]{} observations suggests that this system contains a heavily absorbed and intrinsically X-ray weak (${\ensuremath{\alpha_{\rm{ox}}}}=-1.72$) nucleus. Although not strong enough to qualify as radio-loud, UM 625 does belong to a minority of low-mass AGNs detected in the radio. The broad-band spectral energy distribution constrains the bolometric luminosity to ${\ensuremath{L\mathrm{_{bol}}}}\approx(0.5-3)\times10^{43}$ [erg s$^{-1}$]{} and ${{\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}}\approx0.02-0.15$. A comprehensive analysis of Sloan Digital Sky Survey and [*Hubble Space Telescope*]{} images shows that UM 625 is a nearly face-on S0 galaxy with a prominent, relatively blue pseudobulge ([Sérsic]{} index $n = 1.60$) that accounts for $\sim$60% of the total light in the $R$ band. The extended disk is featureless, but the central $\sim150-400$ pc contains a conspicuous semi-ring of bright, blue star-forming knots, whose integrated ultraviolet luminosity suggests a star formation rate of $\sim$0.3 [$M_{\odot}$]{} yr$^{-1}$. The mass of the central black hole roughly agrees with the value predicted from its bulge velocity dispersion but is significantly lower than that expected from its bulge luminosity.'
author:
- '[Ning Jiang (蒋凝)]{}, Luis C. Ho, Xiao-Bo Dong (董小波), Huan Yang (杨欢) and Junxian Wang (王俊贤)'
title: 'UM 625 Revisited: Multiwavelength Study of A Seyfert 1 Galaxy with a Low-mass Black Hole'
---
[UTF8]{}[gbsn]{}
Introduction
============
Supermassive black holes (BHs), with masses in the range of $10^{6}-10^{10}$ [$M_{\odot}$]{} as measured via stellar and gas kinematics, have been convincingly inferred to be present in the centers of nearby inactive massive galaxies and are generally believed to reside in all galaxies with a spheroidal stellar component (see Kormendy & Ho 2013 for a review). Moreover, there are tight relations between the BH mass ([$M_\mathrm{BH}$]{}) and the properties of the spheroidal component (namely, ellipticals and the bulges of disk galaxies), including stellar velocity dispersion ($\sigma_\star$; Ferrarese & Merritt 2000, Gebhardt et al. 2000a), luminosity ($L_{\rm bulge}$; Kormendy & Richstone 1995; Magorrian et al. 1998; Marconi & Hunt 2003), and mass ($M_{\rm bulge}$; Häring & Rix 2004).
However, the situation is far from clear in the low-mass regime (${\ensuremath{M_\mathrm{BH}}}\lesssim 10^6$ [$M_{\odot}$]{}) because such BHs are largely beyond the reach of current capabilities for direct dynamical measurement. A practical approach is to search for them in low-luminosity type 1 active galactic nuclei (AGNs) with mass estimated from their broad-line width and luminosity using empirical virial relationships (e.g., Gebhardt et al. 2000b; Kaspi et al. 2000). This technique has yielded a sample of $\sim 200-300$ candidate BHs with masses between $10^5$ to $10^6$ [$M_{\odot}$]{}(Greene & Ho 2004, 2007b; Dong et al. 2012b), including a couple below $10^5$ [$M_{\odot}$]{}, in the regime of so-called intermediate-mass BHs (IMBH, $10^{3-6}$ [$M_{\odot}$]{}; Filippenko & Ho 2003; Barth et al. 2004; Dong et al. 2007; see reviews in Ho 2008; Kormendy & Ho 2013). The host galaxies of low-mass BHs thus found appear very different from their supermassive counterparts. The best nearby example, NGC 4395, is a dwarf Sdm galaxy without a bulge at all (Filippenko & Ho 2003), while the second prototype, POX 52, is instead a spheroidal or dwarf elliptical galaxy (Barth et al. 2004; Thornton et al. 2008). According to the studies to date, the [$M_\mathrm{BH}$]{}–$\sigma_{\star}$ relation of local inactive massive galaxies appears to roughly extend to the low-mass end (Barth et al. 2005; Greene & Ho 2006; Xiao et al. 2011), but the [$M_\mathrm{BH}$]{}–$L_{\rm bulge}$ relation does not. Photometric decomposition by Greene et al. (2008) and Jiang et al. (2011a) indicates that the majority of the host galaxies with disks are likely to contain pseudobulges; the rest resemble spheroidals according to their position on the fundamental plane. Very few live in classical bulges. Moreover, the [$M_\mathrm{BH}$]{}–$L_{\rm bulge}$ relation flattens out at the low-mass end, and on average the bulge luminosity is larger by 1–2 index at fixed [$M_\mathrm{BH}$]{} (Greene et al. 2008; Jiang et al. 2011b).
The low BH masses and high accretion rates that generally typify the low-mass AGN sample also provide a unique opportunity to probe accretion processes in an under-explored regime of parameter space. Especially interesting is the broad-band spectral energy distribution (SEDs) of these systems. Unlike quasars, which have luminosities that far exceed that of their host galaxies, the SEDs of Seyferts, particularly those with low-mass BHs, can be strongly contaminated by emission from their hosts. Observations with high angular resolution are absolutely essential to decouple the AGN from the host to construct proper nuclear SEDs. In spite of these challenges, studies so far already suggest that low-mass AGNs may possess unusual multiband properties, including for the tendency to be very radio-quiet (Greene et al. 2006; Greene & Ho 2007b) and relatively X-ray bright (Greene & Ho 2007a; Miniutti et al. 2009; Desroches et al. 2009; Dong et al. 2012a).
The most extensive multiwavelength studies on low-mass BHs thus far have concentrated, unsurprisingly, on the two prototypes, NGC 4395 and POX 52. Interestingly, the SED of NGC 4395 differs markedly from those of both quasars and typical low-luminosity AGNs (Moran et al. 1999). Specifically, the big blue bump, which dominates the SED of luminous
\[specfit\]
Seyferts and quasars at optical and UV bands, is absent, leading to an optical-to-X-ray spectral index of ${\ensuremath{\alpha_{\rm{ox}}}}=-0.97$ (Dewangan et al. 2008). NGC 4395 is a rare example of a low-mass AGN with a relatively low Eddington ratio (${{\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}}$) of $1.2\times10^{-3}$. Most likely because of strong selection effects, optically selected samples of low-mass BHs tend to have higher ${{\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}}$, ranging from $\lesssim0.01$ to $\sim1$ with a median of 0.2 (Dong et al. 2012b). POX 52, with $L_{\rm bol}=1.3\times10^{43}$ [erg s$^{-1}$]{} and ${{\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}}=0.2
-0.5$, exhibits a much more normal SED that is broadly similar to that of a scaled-down version of radio-quiet quasars (Thornton et al. 2008). The broad-band SED of NGC 4051, a narrow-line Seyfert 1 galaxy with ${\ensuremath{M_\mathrm{BH}}}=(1.7\pm0.5)\times10^6{\ensuremath{M_{\odot}}}$ (Denney et al. 2009) accreting at $1\%-5\%$ of [$L\mathrm{_{Edd}}$]{}, is best fit by a relativistically outflowing jet model (Maitra et al. 2011).
This paper reports a detailed multiwavelength data analysis of a low-mass BH hosted in the pseudobulge of UM 625. First noted as a “neutral compact spherical disc galaxy” by Zwicky et al. (1975), the blue color of UM 625 ($B-V=0.67$ mag; Salzer et al. 1989) placed it in early catalogs of blue compact galaxies (e.g., Campos-Aguilar et al. 1993). A high spatial resolution optical image was obtained with [*HST*]{} as part of the nearby AGN survey of Malkan et al. (1998), in which UM 625 was noted as an S0 galaxy with a partial nuclear ring structure. Later analysis of a near-infrared (NIR) [*HST*]{}/NICMOS image revealed a point-like nucleus embedded in an exponential surface brightness profile (Quillen et al. 2001; see also Hunt & Malkan 2004). An [*HST*]{} ACS/HRC image taken in the near-ultraviolet (NUV) shows a very bright, compact, yet partially resolved nucleus (Mu[ñ]{}oz Mar[í]{}n et al. 2007).
An optical spectrum of UM 625 was first acquired as part of the University of Michigan objective-prism survey for emission-line galaxies (MacAlpine & Williams 1981), from which its common name originates. Since then UM 625 has been classified as a Seyfert 2 galaxy (or occasionally as an H II galaxy) in light of its extremely strong [\[O[III]{}\]]{} $\lambda\lambda 4959,5007$ lines and the apparent absence of broad emission lines (e.g., Salzer et al. 1989; Terlevich et al. 1991). The spectral classification was re-examined critically by Dessauges-Zavadsky et al. (2000); using emission-line measurements from the literature, they classified it as a Seyfert 2 according to the three standard optical line-ratio diagnostic diagrams of Veilleux & Osterbrock (1987). Using a high-quality Sloan Digital Sky Survey (SDSS; York et al. 2000) spectrum with a resolution $R \approx 2000$, we clearly detect both broad [H$\alpha$]{} and [H$\beta$]{} and revise the spectral classification of UM 625 to a Seyfert 1. Combining the broad line width and luminosity, we estimate a BH mass of $1.6 \times 10^{6}$ [$M_{\odot}$]{}, placing UM 625 in our sample of AGNs with low-mass BHs (Dong et al. 2012b).
We assume a cosmology with $H_{0} =70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{m} =
0.3$, and $\Omega_{\Lambda} = 0.7$. At a redshift of $z=0.0250$, UM 625 has a luminosity distance of 109.1 Mpc.
Analysis of the Optical Spectrum
================================
UM 625 was spectroscopically observed by SDSS on 18 June 2002 UT with an exposure time of 4803 s exposure. It was classified as a galaxy by the spectroscopic pipeline of the SDSS Fourth Data Release (Adelman-McCarthy et al. 2006). In the course of our systematic spectral fitting of all extragalactic objects in the SDSS, we first noted UM 625 because it showed evident broad [H$\alpha$]{} and [H$\beta$]{} emission; its inferred virial BH mass placed it in the sample of type 1 AGNs with BH masses below $2 \times 10^6$ [$M_{\odot}$]{} described by Dong et al. (2012b). The details of the spectral analysis are given in Dong et al. (2012b). Here we present a brief description of the continuum modeling and emission-line profile fitting, which are based on the MPFIT package (Markwardt 2009) that performs [$\chi^2$]{}-minimization by the Levenberg–Marquardt technique.
UM 625 has a redshift of $z=0.0250$, and its SDSS spectrum is dominated by host galaxy starlight. The median signal-to-noise (S/N) ratio in the [H$\beta$]{}–[\[O[III]{}\]]{} and [H$\alpha$]{}–[\[N[II]{}\]]{}–[\[S[II]{}\]]{} regions are 45 and 60 pixel$^{-1}$, respectively, high enough to fit accurately the continuum and emission lines. We begin by correcting the spectrum for Galactic extinction using the extinction map of Schlegel et al. (1998) and the reddening curve of Fitzpatrick (1999). We model the starlight component with the stellar templates of Lu et al. (2006), which were built from the simple stellar population spectra (Bruzual & Charlot 2003). The AGN continuum is modeled as a power law. The stellar absorption lines must be subtracted well to ensure reliable measurement of weak emission lines (e.g., Ho et al. 1993, 1997). This is achieved by broadening and shifting the starlight templates to match the stellar velocity dispersion of the galaxy. As shown in Figure 1 (left panel), the fit is good; the absorption features are well matched, and the residuals in the emission line-free regions are consistent with the noise level.
Next, we fit the emission lines with Gaussians, using the code described in detail in Dong et al. (2005). The spectrum of UM 625 is dominated by strong, narrow emission lines such as [\[O[III]{}\]]{}$\lambda\lambda4959,5007$, [H$\beta$]{}, [H$\alpha$]{}, and [\[N[II]{}\]]{}$\lambda\lambda6548,6583$. But even a cursory inspection of the spectrum reveals that [H$\alpha$]{} has a clear broad component, even before continuum subtraction; after continuum subtraction, a broad component to [H$\beta$]{} also emerges. This is reminiscent of the situation of the two prototypal broad-line AGNs with intermediate-mass BHs, namely NGC4395 (Filippenko & Sargent 1989) and POX52 (Barth et al. 2004). Because of their narrowness as well as the high S/N ratio of the SDSS spectrum, the doublet lines of both [\[S[II]{}\]]{} and [\[N[II]{}\]]{} are well isolated. [\[S[II]{}\]]{}, [\[N[II]{}\]]{}, and the narrow components of the Balmer lines have very similar full widths at half-maximum (FWHMs). We fit [\[S[II]{}\]]{} first, assuming that the doublet lines have the same width; a good fit is achieved with reduced ${\ensuremath{\chi^2}}=1.1$ (29 degrees of freedom) by using two Gaussians for each line. Then we fit the [H$\alpha$]{}–[\[N[II]{}\]]{} region, taking the best-fit model of [\[S[II]{}\]]{} as a template to model [\[N[II]{}\]]{} and narrow [H$\alpha$]{}. The line ratio of [\[N[II]{}\]]{} $\lambda$6583/$\lambda$6548 is set to the theoretical value 2.96, and their separation is fixed to the laboratory value. We use additional Gaussians to model the broad component of [H$\alpha$]{}, starting with one Gaussian and adding in more if the fit can be improved significantly according to the $F-$test. A good fit is achieved for broad [H$\alpha$]{} with just two Gaussians (Figure 1, right panel). As the broad [H$\beta$]{} component is weak, we fit the total [H$\beta$]{} profile by assuming that its narrow component has the same profile as [\[S[II]{}\]]{} and that its broad component has the same profile as broad [H$\alpha$]{}. With the exception of [\[O[III]{}\]]{}$\lambda4959,5007$, 0.1cm we fit the other narrow emission lines simply with a single Gaussian. For [\[O[III]{}\]]{}, we modeled each of its doublet lines with two Gaussians, one accounting for the bulk component (line core) and the other for a weak, yet apparent, blue wing. All the line parameters are listed in Table 1. We make available online the data and the detailed fitting parameters.[^1]
The ratios of the prominent narrow lines, [\[O[III]{}\]]{}$\lambda5007$/[H$\beta$]{} $= 5.5$ and [\[N[II]{}\]]{}$\lambda6583$/[H$\alpha$]{} $= 0.4$, place UM 625 within the regime of AGNs (Baldwin et al. 1981). With a total [H$\beta$]{} to [\[O[III]{}\]]{}$\lambda5007$ ratio of 0.2, UM 625, like NGC 4395 and POX 52, would be considered a Seyfert 1.8 in the classification scheme of Osterbrock (1981). From the best-fit model of broad [H$\alpha$]{} yields FWHM = $1801 \pm 68$ [$\mathrm{km~s^{-1}}$]{} and a line dispersion ($\sigma_{\rm line}$, the second moment of the line profile) of $1583 \pm 59$ [$\mathrm{km~s^{-1}}$]{} (both corrected for the SDSS instrumental resolution of 139 [$\mathrm{km~s^{-1}}$]{} FWHM). The luminosity of broad [H$\alpha$]{} is $2.3 \times 10^{40}$ [erg s$^{-1}$]{}. The observed (not corrected for internal extinction) luminosity of the narrow [H$\alpha$]{} component is 3 times higher ($7.6 \times 10^{40}$ [erg s$^{-1}$]{}), and the luminosity of [\[O[III]{}\]]{}$\lambda5007$ is $1.1 \times 10^{41}$ [erg s$^{-1}$]{}.
To test the reliability of the broad components of Balmer lines, we refit the [H$\alpha$]{}–[\[N[II]{}\]]{} complex and [H$\beta$]{} using as a template for each line the [\[O[III]{}\]]{}$\lambda 5007$ profile derived from its double-Gaussian model. The results are totally unacceptable; the reduced ${\ensuremath{\chi^2}}= 12$ and 37, respectively, for the [H$\beta$]{} and the [H$\alpha$]{}–[\[N[II]{}\]]{} complex, with obvious large residuals. This is evident by directly comparing the profiles of [H$\beta$]{} and [\[O[III]{}\]]{}$\lambda5007$, as shown in the inset of Figure 1 (with [\[O[III]{}\]]{}$\lambda 5007$ scaled to have the same peak flux density as [H$\beta$]{}); [H$\beta$]{} has an additional much broader, albeit low-contrast ($f_{\lambda}
\lesssim 8 \times 10^{-17}$ [erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$]{}), component that is not present in [\[O[III]{}\]]{}$\lambda 5007$. Also evident in the inset is that [\[O[III]{}\]]{} is slightly broader than [H$\beta$]{} \[by $\sim120$ [$\mathrm{km~s^{-1}}$]{}, in height range $f_{\lambda} \approx (10-20) \times
10^{-17}$ [erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$]{}\]. We also tried another fitting scheme in which we model the narrow Balmer lines with the total profile of [\[O[III]{}\]]{}$\lambda 5007$ and the broad component of the Balmer lines with two Gaussians; the fitting results are also unacceptable, with reduced ${\ensuremath{\chi^2}}= 7$ and 23, respectively, for the [H$\beta$]{} and [H$\alpha$]{}–[\[N[II]{}\]]{} regions.
The extinction of the broad-line region (BLR) can be derived
\[galfit\]
from the observed Balmer decrement [H$\alpha$]{}/[H$\beta$]{} for normal AGNs (Dong et al. 2008). Assuming the extinction curve of the Small Magellanic Cloud (Hopkins et al. 2004; Wang et al. 2005) and an intrinsic broad-line [H$\alpha$]{}/[H$\beta$]{} = 3.1 (Dong et al. 2008), we get $E(B-V)=0.57$ mag for the BLR. Likewise, the observed narrow [H$\alpha$]{} and [H$\beta$]{} gives $E(B-V)=0.25$ mag for the NLR.
Analysis of the Images
======================
There are archival [*HST*]{} images in three bands for UM 625: F330W (roughly the $U$ band of the Johnson system), F606W ($R$), and F160W ($H$), observed through ACS, WFPC2, and NICMOS, respectively. From these images, we can see clearly that there is a bright point-like source present in the center of UM 625, and that the galaxy is almost round in shape and has no spiral arms, indicating an elliptical/spheroidal or a face-on S0 galaxy. Closer inspection reveals that there is, in addition, a nuclear semi-ring on scales of $\sim
150-400$ pc; this feature, most conspicuous in the F330W image, is resolved into several knots, presumably sites of intense star formation. The ring region is indicated in Figure 2 with the green polygon. These high-resolution images give us a distinct opportunity to study not only the multiwavelength properties of the AGN, but also its host galaxy. The [*HST*]{} images, however, suffer from one major limitation: their field of view is too small to properly sample the disk component or to reliably measure the sky background.
\[sdssimg\]
Fortunately, UM 625 was imaged with SDSS. Thus, before we discuss the [*HST*]{} images in detail, we first turn to the SDSS images. Our strategy is to use the SDSS $r$-band image, which closely approximates the [*HST*]{} F606W band, to constrain the photometric parameters of the disk component, which is otherwise difficult to determine from the [*HST*]{} data alone. With the disk thus constrained, we will use the [*HST*]{} F606W image to measure the parameters of the AGN point source and the bulge, which are central to our scientific analysis. The F330W and F160W images provide further photometric points for the nuclear SED, as well as color information to diagnose the stellar population of the bulge.
SDSS Imaging
------------
UM 625 was observed by SDSS in $ugriz$ on 24 May 2001 UT (Figure 3). Although the standard SDSS images have a relatively short exposure time of only 54 s per filter, the drift-scan mode in which the imaging survey was conducted (Gunn et al. 1998) ensures very accurate flat-fielding. Moreover, for most galaxies the field of view is quite large. Both factors are crucial for obtaining an accurate measurement of the sky background and its associated error, which determines the limiting surface brightness sensitivity. For most galaxies the azimuthally averaged surface brightness profile can be measured reliably down to $\mu_r \approx 27$ magarcsec$^{-2}$, which is deep enough to study the outer structure of galaxies (Pohlen & Trujillo 2006; Erwin et al. 2008; Li et al. 2011).
We chose to work with the $r$ band, which is closest to the [*HST*]{} F606W filter. We first mask all photometric objects in the field identified either by Sextractor (Bertin & Arnouts 1996) or by the SDSS photometric pipeline and then run the IRAF[^2] task [ellipse]{} to fit isophotes with a linear step of 2 pixels between successive steps, allowing the center, position angle, and ellipticity of each ellipse to vary. The sky background level is determined from the best-fit ellipses that have constant surface brightness with respect to radius, and the uncertainty on the background is estimated from the root mean square fluctuations ($\sigma$) about the mean value. The limiting surface brightness, defined as 3 $\sigma$, is 26.57 $\rm mag~arcsec^{-2}$. After sky subtraction, the azimuthally averaged one-dimensional (1-D) surface brightness profile is extracted in logarithmic steps, to increase the S/N in the noisier outer regions. We can clearly see that the profile (right panel of Figure 3) at radii larger than $\sim 10\arcsec$ behaves like an exponential disk, the scale length of which is 421$\pm$017 ($2.12\pm0.09$ kpc). We do not decompose the inner regions of the galaxy with this data set, as the SDSS image lacks sufficient resolution; for that, we turn to the high-resolution [*HST*]{} images below.
[*HST*]{}: Optical Image
------------------------
The F606W image is our best choice for probing the host galaxy structure because (1) the bandpass is relatively red and thus less sensitive to dust and young stars, (2) the field of view of WFPC2/PC1 is larger than that of ACS/HRC and NICMOS, and (3) its point-spread function (PSF) has less extended wings that NICMOS. A single 500 s exposure was obtained on 21 July 1994 (Proposal ID: 5479), with UM 625 placed near the center of the Planetary Camera detector (PC1), which has a plate scale of 0046 pixel$^{-1}$. We remove cosmic rays using LA Cosmic[^3] (van Dokkum 2001), which can detect cosmic ray hits of arbitrary shape and size. Two of the pixels in the center of the galaxy are saturated; we masked them out in the analysis below.
Precise sky subtraction is of great importance to obtain accurate photometric measurements and structural decomposition. From the surface brightness profile of the SDSS $r$-band image, \[f606w-sbp\] 0.3cm we know that UM 625 extends close to a radius of $20\arcsec$, which means that it fills nearly the entire field of PC1 (radius $\sim 18\arcsec$). We estimate the background level and its uncertainty from the outermost edges of the PC1 chip and from the flanking WF2 chips; the limiting surface brightness of the F606W image is 25.31 $\rm mag\,arcsec^{-2}$. To examine the effect of the star-forming ring on the underlying galaxy structure, we also plot the surface brightness profile with the ring region (the region enclosed by the green polygon as indicated in Figure 2) masked out. Figure 4 shows that the circumnuclear star-forming region mainly affects the profile on scales of 02$\lesssim r \lesssim$08; the outer profile ($r \gtrsim 2\arcsec$) agrees well with the SDSS $r$-band profile.
Separating the central AGN light from the host galaxy starlight requires knowing the PSF to a high accuracy. Unfortunately there is no bright star in the field. According to the image simulations of Kim et al. (2008), a synthetic PSF generated by the TinyTim software (Krist 1995) works reasonably well. We have also searched for empirical stellar PSFs from the WFPC2 PSF Library[^4] and chosen the one closest to the location of UM 625 on PC1. As shown in Figure 4, the stellar PSF agrees well with the TinyTim PSF, except in the outer regions where the empirical PSF is much noisier. For the rest of the analysis, we simply adopt the TinyTim PSF.
After background subtraction, we perform a two-dimensional (2-D) decomposition of UM 625 using GALFIT (Peng et al. 2002, 2010). The AGN is represented by a point source modeled with the TinyTim PSF, and the galaxy is modeled by bulge, fit with a [Sérsic]{} (1968) $r^{1/n}$ function, and a disk, fit with an exponential function (equivalent to $n = 1$). A single-component model for the host leaves unacceptably large
\[galfit\_1d\_f606w\] 0.3cm residuals. The best-fitting two-component model (Figure 5) gives a central component with [Sérsic]{} index $n = 1.60$ and effective radius $r_e =$ 137 (693 pc), and a disk with scale length of 386 (1.95 kpc), consistent with the results from the 1-D decomposition of the SDSS $r$-band image. The bulge-like component is mildly disky, with $c = -0.11$. The best-fit parameters are summarized in Table 2. We identify the central $n = 1.60$ component with a pseudobulge, because pseudobulges generally have $n \lesssim 2$ (Kormendy & Kennicutt 2004). The residual image (top right panel of Figure 2) reveals a faint ring-shaped structure on the eastern side of the galaxy, opposite to the masked ring region.
[*HST*]{}: NUV Image
--------------------
The F330W image was observed on 21 March 2003 using ACS/HRC as part of a study on the starburst–AGN connection (Proposal ID: 9379). ACS is located away from the optical axis of [*HST*]{}, and so it suffers from significant geometric distortion that is not corrected by the internal optics. For ease of rejecting cosmic rays, the total exposure was divided into two equal exposures of 10 min each. These images are then combined using [astrodrizzle]{}[^5], a new software replacing [multidrizzle]{}, to remove cosmic ray hits and to correct for the geometric distortion. The pixel scale of the combined image is 0025. A narrow ring-like star-forming region, ornamented with several bright knots, encircles nearly half the nucleus. We begin with creating two synthetic PSFs for each exposure using TinyTim. They are centered in the same position as the nucleus
in the HRC image to properly reproduce the geometric changes involved in the processing. During the modeling, we have also considered the dofocus offset ($\sim6~\mu$m) given by web-based model[^6]. The offset is mostly due to spacecraft breathing effects. The two TinyTim PSFs, still distorted, are almost identical except for slightly differences due to defocus offsets. We combined them with [astrodrizzle]{} in the same manner as the science images, producing a distortion-corrected PSF image.
We decompose the image using GALFIT with the region containing the star-forming ring masked out. Due to the small field of view of the ACS/HRC (26$\times$29), we cannot independently measure the sky background; we set it as a free parameter in the fit. Nor can be obtain any meaningful, independent constraint on the parameters of the disk. Our initial attempt to fit the image with a PSF + [Sérsic]{} + disk model failed to converge. As our primary goal for the F330W image is to measure the brightness of the nucleus, we simply set the disk scale length to be identical to that obtained from the F606W fit. The resulting fit yields a bulge with $n=2.64$ and $r_e =$ 102 (515 pc), and a point source flux that is very insensitive to the assumptions of the disk component. The AGN magnitude changes by only $\pm 0.01$ mag depending on whether the disk is included or not. If we further force the bulge to have the same $n$ and $r_e$ as derived from the F606W fit (i.e. allowing only the luminosity to adjust), the resulting AGN magnitude changes by 0.06 mag.
[*HST*]{}: NIR Image
--------------------
The NICMOS/NIC1 F160W image was observed on 31 July 1997 (Proposal ID: 7328), in three 256 s exposures dithered in an “L”-shaped pattern. Each exposure was processed with the standard pipeline [calnica]{} within IRAF/STSDAS. This task corrects for the nonlinearity of the detector and removes the bias value, dark current, amplifier glow, and shading; however, it does not remove the ghost “pedestal” effect produced by the variable quadrant bias (see, e.g., Hunt & Malkan, 2004). To correct for this effect, we apply the [pedsub]{} task to the calibrated image, quadrant by quadrant, determine the shifts of the dithered images with [xregister]{}, register them with [imshift]{}, and finally combine them with [imcombine]{}.
UM 625 extends far beyond the $\sim11\arcsec\times11\arcsec$ field of view of NIC1. In spite of this, a GALFIT model consisting of a PSF + [Sérsic]{} bulge can still give a fairly robust measurement of the nuclear magnitude. During the fitting, we set the background free and masked the ring region as before. It proved to be impossible to independently measure the disk component at all. Since we know from the optical images that a disk [*is*]{} present, we constrain it by fixing its structural parameters to those derived from the F606W image and normalizing its luminosity such that $R-H=2$ mag, a value typical of disks (e.g., MacArthur et al. 2004). However, we note that none of the parameters for the bulge or the AGN point source are significantly affected by our assumptions for the disk component. The best-fit model
\[xray\]
yields $n=1.54$ and $r_e =$ 098 (491 pc) for the bulge. If we fix the bulge $n$ and $r_e$ to the values derived from the F606W fit, the AGN magnitude changes by 0.21 mag.
Other Multiwavelength Data
==========================
X-ray Observations
------------------
UM 625 was observed by [*XMM-Newton*]{} on three separate occasions between 2004 and 2008 (ObsID: 0200430901, 0505930101, 0505930401) and once by [*Chandra*]{} in 2008 (ObsID: 9557). We used [*XMM-Newton*]{} SAS version 12.0.1 and the calibration file of June 2012 to reduce the EPIC PN (Strüder et al. 2001) archival data. After filtering background flares, the net exposure times are 6.4 ks, 13 ks, and 13 ks, respectively, for the three PN observations. The source was detected at a large offset angle ($\sim 10\arcsec -11\arcsec$) from the field center for the later two PN observations. To extract the source and background spectra, we defined the source region as a circle with radius $\sim 30\arcsec
-40\arcsec$ centered at the source position and the background regions as three or four circles, far removed from the CCD edges and other sources, with radii $\sim 30\arcsec -40\arcsec$ around the source region. The [*Chandra*]{} ACIS-S (Garmire et al. 2003) data were reduced with CIAO 4.4.1 (Fruscione et al. 2006) and the CALDB 4.4.10 database, adopting standard procedures. The net exposure time was 49 ks. Because of the large off-axis angle (106), we extracted the spectra using a 20-radius circle for the source and four circles around the source region for the background. All [*XMM-Newton*]{} and [*Chandra*]{} spectra were rebinned to have a minimum of 5 counts in each energy bin after background subtraction, and they were fit simultaneously with the same models using [Xspec]{} (Arnaud 1996). In view of the low source counts, we fit the spectra using the Cash-statistic instead of $\chi^2$.
We first fit the spectra over the energy range 0.2–10 keV using a single absorbed power-law model with the minimum absorption set to the Galactic value of ${\ensuremath{N_\mathrm{H}}}= 4.04\times10^{20} \, \rm cm^{-2}$ (Kalberla et al. 2005). The fit yields a column density equal to the Galactic value, a photon index $\Gamma_{1} = 2.65_{-0.15}^{+0.15}$, and C-statistic = 145 (125 degrees of freedom). The residuals, however, clearly show a flat, hard excess above 3 keV, which implies that an obscured component exists. We then added an absorbed power-law component with a photon index fixed to $\Gamma_{2} = 1.90$, typical of AGNs (Shemmer et al. 2008; Constantin et al. 2009). As shown in Figure 6, the fit is improved significantly, resulting in C-statistic = 116 (123 degrees of freedom). The soft-band power law, with $\Gamma_1 =
2.86_{-0.16}^{+0.17}$, is slightly steeper than the previous fit. The absorption column density of the absorbed power-law component is ${\ensuremath{N_\mathrm{H}}}=
9.7_{-5.8}^{+18}\times10^{22} \, \rm cm^{-2}$. The observed 0.5–2 keV and 2–10 keV fluxes are $2.5_{-0.2}^{+0.2}\times10^{-14}$ and $4.6_{-0.4}^{+0.4}\times10^{-14}$ [erg s$^{-1}$ cm$^{-2}$]{}, corresponding to $4.0_{-0.3}^{+0.4}\times10^{40}$ and $6.5_{-0.6}^{+0.6}\times10^{40}$ [erg s$^{-1}$]{}, respectively. Assuming that the obscured power-law component is the intrinsic emission from the corona, the unabsorbed 2–10 keV flux is $6.7\times10^{-14}$ [erg s$^{-1}$ cm$^{-2}$]{}, corresponding to $9.5\times10^{40}$ [erg s$^{-1}$]{}. The monochromatic flux at 2 keV is $7.8\times10^{-32}$ . As the spectra are cut off above 10 keV and no variations are detected in each single exposure or among the four exposures, we cannot exclude the possibility that the absorption is Compton-thick.
If we link the spectral slope of the hard component to the soft one, we obtain $\Gamma = 2.84\pm0.16$, ${\ensuremath{N_\mathrm{H}}}= 14.6_{-7.8}^{+23}\times10^{42} \,\rm cm^{-2}$, and a scattering fraction of $10\%$ if we attribute the soft component to scattered nuclear emission. We note that this fraction is consistent with those commonly reported in other obscured AGNs ($0.1\% - 10\%$; Noguchi et al . 2010).
Ultraviolet Observations
------------------------
The Optical Monitor (OM) on [*XMM-Newton*]{} took data for UM 625 in the UVW1 (2910 Å) and UVW2 (2120 Å) filters, each for 1 ks, during the 2004 X-ray observations. Because of the coarse resolution of the OM (PSF $\approx$ 18), the nucleus is not resolved from the host galaxy, and we consider these global (AGN plus host) flux measurements. We adopt an aperture radius of 10 to compute the source flux after estimating the sky background from the region between a radius of 15 and 20. The flux is corrected for Galactic extinction using the maps of Schlegel et al. (1998) and the reddening curve of Fitzpatrick (1999), resulting in $f_{\nu}$(2910 Å)=$(4.5\pm0.3)
\times10^{-27}$ and $f_{\nu}$(2120 Å)=$(2.7\pm0.2)\times10^{-27}$ .
Apart from the [*XMM-Newton*]{} OM data, UM 625 was also imaged simultaneously in the near-UV (NUV; 2316 Å) and far-UV (FUV; 1539 Å) bands of [*GALEX*]{} during its All-sky Imaging Survey (AIS), for a total exposure time of 112 s on 6 April 2004. UM 625 was later reobserved during the Medium Imaging Survey (MIS) on 18 May 2009, for a total of 1578 s; we adopt the MIS data (Bianchi et al. 2012). After correcting for Galactic extinction, we obtain an NUV flux density of $f_{\nu}$(2316 Å)=$(3.17\pm0.04)\times10^{-27}$ and an FUV flux density of $f_{\nu}$(1539 Å)=$(1.42\pm0.04)\times10^{-27}$ . The NUV flux density is slightly larger than that of the OM UVW2 band, whose effective wavelength is only $\sim200$ Å shorter.
Radio Observations
------------------
UM 625 was detected by Faint Images of the Radio Sky at Twenty-cm (FIRST; Becker et al. 1995) using the Very Large Array in its B configuration. The peak and integrated 20 cm flux density from the FIRST catalog[^7] (White et al. 1997) is 1.89 and 2.69 mJy, corresponding to a monochromatic radio luminosity at 20 cm of $2.66\times10^{28}$ and $3.79\times10^{28}~\rm erg~s^{-1}~Hz^{-1}$. These measurements are derived by fitting a 2-D Gaussian function to the source, using a map generated from twelve coadded images adjacent to the pointing center; the map has 18 pixel$^{-1}$, a resolution of FWHM = 5, and an rms noise of 0.152 mJy beam$^{-1}$. Customarily, the radio-loudness parameter $R$ is defined as the ratio of flux densities between 6 cm and 4400 Å. Assuming a radio spectral index $\alpha_r=-0.46$ (Lal & Ho 2010), $f_{\nu}(6~\rm cm)=1.55\times10^{-26}$ , which, in combination with $f_{\nu}(4400$ Å) determined from the SDSS spectral fitting, gives $R=5.0$. Although UM 625 is formally radio-quiet according to the widely used division of $R=10$ for radio-loud and radio-quiet AGNs (Kellermann et al. 1989), its level of radio activity is still somewhat unusual for low-mass AGNs (e.g., Greene et al. 2006).
Infrared Observations
---------------------
UM 625 is also contained in the *Infrared Astronomical Satellite* ([*IRAS*]{}) faint source catalog (Moshir et al. 1990). The faint source catalog contains data for point sources in unconfused regions with flux densities typically above 0.2 Jy at 12, 25, and $60\,\mu$m, and above 1.0 Jy at $100\,\mu$m. UM 625 is reliably detected at $60\,\mu$m with a flux density of $0.32\pm0.08$ Jy, but only upper limits are given for the other three bands (0.13, 0.12, and 0.77 Jy for 12, 25, and $60\,\mu$m, respectively).
*Wide-field Infrared Survey Explorer* ([*WISE*]{}; Wright et al. 2010), the mission most comparable to [*IRAS*]{} yet with a sensitivity more than 100 times higher at $12\,\mu$m, has mapped the entire sky in four bands centered at 3.4, 4.6, 12, and $22\,\mu$m. UM 625 is detected with high S/N in all four bands, with a flux density, converted from the magnitudes in the [*WISE*]{} All-sky Source Catalog, of 4.1, 4.2, 24, and 95 mJy, respectively.
For completeness, we take the following NIR magnitudes from the Two Micron All Sky Survey point source catalog (Skrutskie et al. 2006): $J=14.25\pm0.05, \, H=13.60\pm0.06$ and $K_{s}=13.24\pm0.05$ mag, which correspond to a flux density of 3.3, 3.8, and 3.4 mJy, respectively.
Results and Discussion
======================
Black Hole Mass
---------------
With the detection of broad emission lines, we can estimate the mass of the central BH using commonly used virial mass estimators for broad-line AGNs. We calculate the virial BH mass using the [H$\alpha$]{} formalism given in Xiao et al. (2011; their Equation 6), which is based on Greene & Ho (2005b, 2007b) but updated with the more recent relationship between BLR size and luminosity of Bentz et al. (2009). For FWHM([H$\alpha$]{}) = 1801 [$\mathrm{km~s^{-1}}$]{} and a broad [H$\alpha$]{} luminosity of $2.4 \times 10^{40}$ [erg s$^{-1}$]{}, $M_{\rm BH} = 1.6 \times 10^{6}$ [$M_{\odot}$]{}, which justifies inclusion in the low-mass BH sample of Dong et al. (2012b).
The uncertainty of the above BH mass estimate is not well understood. As pointed out by Vestergaard & Peterson (2006), the statistical accuracy of the masses from single-epoch virial mass estimators is a factor of $\sim 4$ ($1\sigma$), and, for individual mass estimates, the uncertainty can be as large as an order of magnitude. We note that recently Wang et al. (2009) recalibrated the BH mass formulas based on single-epoch spectra, stressing the nonlinear relation between the virial velocity of the BLR clouds and the FWHM of single-epoch broad emission lines. This nonlinearity probably arises from several kinds of nonvirial components incorporated into the total profile of broad emission lines in single-epoch spectra (see §4.2 of Wang et al. \[2009\] for a detailed discussion, as well as Collin et al. \[2006\] and Sulentic et al. \[2006\]). The formulas of Wang et al. (2009), calibrated using reverberation mapping data available to date, which span the mass range from $M_{\rm BH}
\approx 10^7$ to $10^9$ [$M_{\odot}$]{}, does not cover the low-mass regime of interest in this paper. Furthermore, we would like to point out that nonvirial components, if any, may be less significant in UM 625 than in other AGNs because its broad [H$\alpha$]{} profile is roughly symmetrical and close to a Gaussian.
Host Galaxy
-----------
The abundant archival [*HST*]{} images covering a wide wavelength baseline provide us an excellent opportunity to study the host galaxy of UM 625 with the central AGN point source removed. Our 2-D decomposition of the optical (F606W, $R$ band) WFPC2/PC1 shows that the stellar distribution consists of two main components: (1) a dominant compact, bulge-like component with [Sérsic]{} $n=1.60$, $r_e=693~\rm pc$, and $M_R=-19.62$ mag; (2) an extended disk ($n \approx 1$) component with a scale length of $\sim2$ kpc. The disk is more evident in the SDSS $r$-band image owing to its deeper limiting surface brightness and larger field of view, but its photometric parameters are well recovered with the WFPC2/PC1 image alone. The very limited field of view of the ACS/HRC and NICMOS/NIC1 images prevent us from placing any meaningful constraints on the disk in the UV and NIR. We suspect that even the structural parameters of the bulge may be partly compromised in these bands as a result of the small field size.
Taking the F606W decomposition as reference, the bulge-to-total light ratio ($B/T$) in the $R$ band is 0.66. If we use a total host luminosity calculated from the SDSS $r$-band image (after subtracting the AGN point source derived from the [*HST*]{} decomposition), which may be more reliable because of its larger field of view, the bulge-to-total ratio reduces slightly to $B/T =
0.60$. Either of these values lies within the range of $B/T$ for S0 galaxies (e.g., Simien & de Vaucouleurs 1986), in agreement with the absence of spiral arms in the main disk of the galaxy. The star-forming ring, which resembles a tightly wound spiral, in UM 625 is located in the circumnuclear region ($\sim 0.5$ kpc) resolved only by [*HST*]{} observations. Interestingly, the UV-optical and optical-NIR colors of the bulge of UM 625 are much bluer than those of typical S0 galaxies; they more closely resemble those of an Sb spiral. This is apparent from comparing the observed F330W$-$F606W and F606W$-$F160W colors with synthetic colors calculated using [calcphot]{} in IRAF/SYNPHOT package for various galaxy templates (Kinney et al. 1996; Polletta et al. 2007).
In view of its blue color, low [Sérsic]{} index ($n<2$), and disky isophote shape ($c<0$), the bulge of UM 625 can be convincingly categorized as a pseudobulge, commonly present in low-luminosity disk galaxies, including some S0s (see Kormendy & Kennicutt 2004 for a review). The presence of an ongoing central star formation, manifested through the nuclear star-forming ring (Section 5.3), further supports the pseudobulge interpretation. This is consistent with previous studies, which find that the majority of the host galaxies of low-mass BHs with disks are likely to contain pseudobulges rather than classical bulges (Greene et al. 2008; Jiang et al. 2011b). The only possible anomaly is that UM 625 has a much higher $B/T$ than usual; for example, the objects in the sample of Jiang et al. (2011b) with a detected disk component have an average $B/T = 0.23$.
Nuclear Star-forming Ring
-------------------------
As mentioned above, the circumnuclear region of UM 625 contains a blue ($\rm F330W-\rm F606W=-0.73$ mag, corrected for Galactic extinction) semi-ring on scales of $\sim150-400$ pc. It is most prominent in the NUV (F330W) band, which shows a number of bright, compact knots, reminiscent of star-forming galaxies with nuclear hotspots (e.g., Barth et al. 1995). Assuming that the UV light mainly arises from young stars, we can use the integrated luminosity to estimate the star formation rate. Summing the flux of the residuals above the smooth galaxy model within the ring region (green polygon in Figure 2) yields a luminosity of $L_{\nu}=2.0\times10^{27}$ erg s$^{-1}$ Hz$^{-1}$ after correcting for Galactic extinction, which corresponds to a star formation rate of 0.28 [$M_{\odot}$]{} yr$^{-1}$ (Kennicutt 1998; his Equation 1). As a check, aperture photometry between radius 015 and $1\arcsec$ in the residual image yields nearly the same flux, only 0.02 magnitude higher.
Given that the central region of UM 625 experiences significant ongoing star formation, as evidenced by the UV light, it is interesting to note that some of the narrow emission lines, especially the Balmer lines, are inevitably contaminated by stellar photoionization. From Equation (2) of Kennicutt (1998), a star formation rate of $\sim 0.3$ [$M_{\odot}$]{} yr$^{-1}$ produces an [H$\alpha$]{} luminosity of $3.5 \times 10^{40}$ [erg s$^{-1}$]{}, which is roughly half of the total observed narrow [H$\alpha$]{} emission. This is nonnegligible.
The star-forming region can potentially contaminate the emission in the radio and X-ray bands as well, due to the contribution from high-mass X-ray binaries, young supernova remnants, and hot interstellar plasma. According to the empirically calibrated linear relation between star formation rate and X-ray luminosity of Ranalli et al. (2003), the ring region produces $L_{\rm 0.5-2~keV}=1.3\times10^{39}$ [erg s$^{-1}$]{}; this is only a few percent of the observed X-ray luminosity, and so the AGN totally dominates the X-ray emission. By contrast, the Ranalli et al.’s relation between star formation rate and radio emission predicts $L_{\rm 1.4~GHz}=1.1\times10^{28}$ [erg s$^{-1}$]{} Hz$^{-1}$, which is nearly 30% of the integrated monochromatic radio power detected by FIRST.
X-ray Spectral Properties
-------------------------
Although the detection of broad [H$\alpha$]{} and [H$\beta$]{} qualifies UM 625 as a type 1 AGN, its X-ray spectrum indicates that it contains a significant intrinsic absorbing column density of ${\ensuremath{N_\mathrm{H}}}=9.7_{-5.8}^{+18}\times10^{22}\,\rm cm^{-2}$. This apparent disagreement between optical and X-ray classification is not uncommon (e.g., Garcet et al. 2007), as it can arise if the gas responsible for the X-ray absorption is highly ionized, instead of neutral, so that the accompanying dust would sublimate to yield a much smaller dust-to-gas ratio. Indeed, such absorbed type 1 AGNs are usually characterized by either complex or warm/ionized absorption (Malizia et al. 2012) arising from ionized gas possibly associated with a disk wind (Murray et al. 1995) or ionization cones as seen in some objects.
The ratio $T$ between 2–10 keV luminosity to extinction-corrected [\[O[III]{}\]]{} luminosity is also a powerful diagnostic of nuclear X-ray obscuration. Previous studies have suggested that objects with $T\leqslant0.1$ are invariably Compton-thick, whereas objects with $T\geqslant1$ are almost exclusively Compton-thin or unobscured (Guainazzi et al. 2005). With observed $L_{2-10\,\rm keV} = 4.6 \times 10^{40}$ [erg s$^{-1}$]{} and an extinction-corrected $L_{\rm [O~III]} = 2.3 \times 10^{41}$ [erg s$^{-1}$]{}, $T=0.2$, placing UM 625 intermediate between Compton-thick and Compton-thin.
The power-law component for the soft X-ray band is likely different from the soft X-ray excess commonly seen in type 1 AGNs since it should have been largely obscured, if present. It could arise from a scattered component from the nucleus or contamination from the host galaxy; the contribution from the star-forming ring is negligible (Section 5.3). For AGNs with low-mass BHs, thermal emission from the accretion disk can also contribute significantly to the soft X-ray band (e.g., Thornton et al. 2008; Miniutti et al. 2009). We have tried to fit our X-ray data using a (disk) blackbody model, but the data quality is insufficient to reach meaningful conclusions regarding the nature of the soft X-rays.
Spectral Energy Distribution and Bolometric Luminosity
------------------------------------------------------
\[sed\]
We combine all of the photometric data of UM 625 described in Sections 3 and 4 to construct its broad-band SED (Figure 7); apart from NGC 4395 and POX 52, this is one of the most complete SEDs available for low-mass AGNs. The median SEDs of radio-quiet and radio-loud quasars (Elvis et al. 1994), scaled to UM 625 in the optical band, are overplotted for comparison.
Even after correcting for absorption, the X-ray emission of UM 625 is still somewhat weak compared with the median SED of quasars. The optical-to-X-ray slope [$\alpha_{\rm{ox}}$]{} is $-1.72$, where we adopt the standard definition $\alpha_{\rm ox}\equiv
-0.384{\rm log}[f_{\nu}(2500$ Å)/$f_{\nu}({\rm 2\,keV})]$ (Tananbaum et al. 1979) and estimate $f_{\nu}(2500$ Å) from $f_{\nu}(5100$ Å) assuming an optical-UV continuum spectral index of $-0.44$ (Vanden Berk et al. 2001)[^8]. By contrast, previous X-ray studies of low-mass AGNs indicate that [$\alpha_{\rm{ox}}$]{} is, on average, larger than in high-mass AGNs (e.g., Greene & Ho 2007a; Miniutti et al. 2009; Desroches et al. 2009; Dong et al. 2012a), falling systematically below the low-luminosity extension of the [$\alpha_{\rm{ox}}$]{}-$L_\nu(2500$Å) relation of Steffen et al. (2006). The previous samples span a wide range in [$\alpha_{\rm{ox}}$]{}, from $\approx -1.7$ to $-1$. Thus, UM 625 lies among the weakest X-ray sources with the lowest [$\alpha_{\rm{ox}}$]{}. The origin of the X-ray weakness is not known. Some may be simply highly absorbed, but others may be intrinsically X-ray weak (Dong et al. 2012a). UM 625 may belong to the latter category.
Low-mass AGNs appear to be exceptionally radio-quiet, if not radio-silent. In a Very Large Array 6 cm survey of the 19 low-mass AGNs from Greene & Ho (2004), Greene et al. (2006) detected radio emission from only one ($\sim
5$%), which has $R$=2.8. This detection rate is approximately the same as that found in the larger sample of Greene & Ho (2007b), whose detected sources have $R \approx 1-80$. UM 625 is clearly detected at 20 cm, at $R =
5.0$. Even after correcting for possible contamination from star formation (Section 5.3), the source still has $R \approx 3.3$. While UM 625 is, strictly speaking, not radio-loud, it is still somewhat unusual in that it belongs to the minority of low-mass AGNs that show any radio emission at all.
Despite the fact that the SED of UM 625 still contains many gaps in wavelength coverage, it should still offer a more reliable measurement of the bolometric luminosity of the system than any estimate based on a single band. Integrating the median radio-quiet quasar SED of Elvis et al. (1994) after scaling it to the nuclear optical point derived from [*HST*]{} for UM 625, we obtain [$L\mathrm{_{bol}}$]{} = $2.4\times10^{43}$ [erg s$^{-1}$]{}, which corresponds to [[$L\mathrm{_{bol}}$]{}/[$L\mathrm{_{Edd}}$]{}]{} = 0.11 for $M_{\rm BH} = 1.6 \times 10^6$ [$M_{\odot}$]{}; had we chosen the radio-loud SED template instead of the radio-quiet one, these values would be $\sim$8% higher. Since the SED of UM 625 does not, in fact, exactly match the standard shape of the quasar templates, an alternative approach is to simply perform a piecewise power-law integration of the observed points (from radio to X-ray). This yields a bolometric luminosity that is lower by a factor $\sim$5, reducing [[$L\mathrm{_{bol}}$]{}/[$L\mathrm{_{Edd}}$]{}]{} to 0.02. Yet a third estimate can be obtained from the strength of the [H$\alpha$]{} emission, which scales with the optical continuum emission (e.g., Greene & Ho 2005b), and hence with [$L\mathrm{_{bol}}$]{} assuming some canonical bolometric correction for the optical band. Following the formalism of Greene & Ho (2007b), the total (broad plus narrow) [H$\alpha$]{} luminosity of $1.0 \times 10^{41}$ [erg s$^{-1}$]{} leads to [$L\mathrm{_{bol}}$]{} = $3.2\times10^{43}$ [erg s$^{-1}$]{} and [[$L\mathrm{_{bol}}$]{}/[$L\mathrm{_{Edd}}$]{}]{} = 0.14. [^9] To summarize, UM 625 is estimated to have ${\ensuremath{L\mathrm{_{bol}}}}\approx(0.5-3)\times10^{43}$ [erg s$^{-1}$]{}and ${{\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}}\approx0.02-0.15$, which is between the values for NGC 4395 and POX 52.
Black Hole–Bulge Connection
---------------------------
We end with a short discussion concerning the connection between the mass of the central BH in UM 625 and the properties of the bulge of its host galaxy. While the SDSS spectrum (Figure 1) does have detected stellar absorption features, we are not confident that we have resolved them well enough to trust the stellar velocity dispersion derived from the continuum fitting (Section 2). In lieu of the stars, we use the velocity dispersion of the narrow-line gas as traced by the low-ionization lines [\[S[II]{}\]]{} $\lambda\lambda$6716, 6731, which are resolved, to estimate the stellar velocity dispersion (Greene & Ho 2005a; Ho 2009). For $\sigma_{\star} \simeq \sigma_{\rm [S~II]} = 68.1$ [$\mathrm{km~s^{-1}}$]{}, the [$M_\mathrm{BH}$]{}–$\sigma_{\star}$ relation of G[ü]{}ltekin et al. (2009; their Equation 7) implies [$M_\mathrm{BH}$]{} = $1.7 \times 10^{6}$ [$M_{\odot}$]{}. The more recent extension of the [$M_\mathrm{BH}$]{}–$\sigma_{\star}$ relation to the low-mass end by Xiao et al. (2011) yields [$M_\mathrm{BH}$]{} = $1.3\times10^6\, {\ensuremath{M_{\odot}}}$. Both of these estimates are in surprisingly good agreement with our virial mass estimate, [$M_\mathrm{BH}$]{} = $1.6\times10^{6}$ [$M_{\odot}$]{}, based on the broad [H$\alpha$]{} line (Section 5.1).
By contrast, and in line with other investigations of low-mass AGNs (Greene et al. 2008; Jiang et al. 2011a), the BH in UM 625 deviates strongly from the BH mass–bulge luminosity relation of inactive galaxies. Assuming a spectrum of an Sb galaxy (Kinney et al. 1996), the F606W magnitude of the bulge corresponds to $M_{V}=-19.06$ mag, from which the [$M_\mathrm{BH}$]{}–$L_{V,\rm bulge}$ relation of G[ü]{}ltekin et al. (2009; their Equation 8) predicts $2.2\times
10^{7}$ [$M_{\odot}$]{}, nearly 15 times larger than the virial estimate.
Now, we know that the stellar population of the pseudobulge in UM 625 is quite young, most likely significantly younger than the majority of the more massive galaxies used to define the [$M_\mathrm{BH}$]{}–$L_{\rm bulge}$ relation. A proper comparison, therefore, requires that we apply a correction for age or mass-to-light ratio to the bulge luminosity of UM 625. Assuming, as before, an Sb galaxy spectrum, for which $B-V\approx 0.7$ mag (Fukugita et al. 1995), which, in fact, agrees closely with $B-V=0.67$ mag as measured by Salzer et al. (1989), a simple stellar population of solar metallicity predicts an age of $\sim2$ Gyr and a mass-to-light ratio $M/L_{V}\approx 1.6$ [$M_{\odot}$]{}/[$L_{\odot}$]{} (Bruzual & Charlot 2003). For a typical elliptical galaxy, with $B-V\approx
1.0$ mag, $M/L_{V}\approx 6.3$ [$M_{\odot}$]{}/[$L_{\odot}$]{}. Applying this age correction reduces the $V$-band luminosity by about a factor of 4 and the predicted BH mass by about a factor of 5. Although the mass discrepancy is now lower, the pseudobulge of UM 625, like other pseudobulges containing low-mass BHs (Greene et al. 2008; Jiang et al. 2011a), systematically deviate from the [$M_\mathrm{BH}$]{}–$L_{\rm bulge}$ relation of classical bulges and elliptical galaxies. Kormendy et al. (2011) suggest that that there are two different modes of accretion: BHs in classical bulges and ellipticals grow rapidly via merger-driven gas infall; in contrast, BHs hosted in pseudobulges grow mainly via secular evolution through slower, stochastic processes.
Finally, we note that the pseudobulge of UM 625, as in the hosts of other low-mass AGNs (e.g. Greene et al. 2008; Jiang et al. 2011b), deviates from the Faber-Jackson relation (Faber & Jackson 1976) of classical bulges and elliptical galaxies, in the sense that it has a lower $\sigma_{\star}$ for a given luminosity. With an $I$-band absolute magnitude of $M_{\rm I}=-20.27$ (assuming, as before, a spectrum of an Sb galaxy), the Faber-Jackson relation of inactive early-type galaxies (Table 5 of Jiang et al. 2011b) predicts $\sigma_{\star}=124~{\ensuremath{\mathrm{km~s^{-1}}}}$, nearly twice the observed value inferred from $\sigma_{\rm [S~II]}$.
Summary
=======
We conducted a comprehensive, multiwavelength study of the nuclear and host galaxy properties of UM 625, a type 1 AGN with a BH mass of $1.6\times 10^6$ [$M_{\odot}$]{} determined through the detection of broad [H$\alpha$]{} emission.
Analysis of [*Chandra*]{} and [*XMM-Newton*]{} observations reveals a heavily absorbed (${\ensuremath{N_\mathrm{H}}}= 9.7_{-5.8}^{+18}\times10^{22} \, \rm cm^{-2}$) nucleus with an intrinsic X-ray luminosity of $L_{2-10\,\rm keV} = 9.5 \times 10^{40}$ [erg s$^{-1}$]{}. The source may be intrinsically X-ray weak (${\ensuremath{\alpha_{\rm{ox}}}}=-1.72$) compared to higher luminosity AGNs. UM 625 belongs to a minority of low-mass AGNs detected in the radio, but it is not strong enough to qualify as radio-loud. In combination with nuclear photometry at UV, optical, and NIR bands extracted from high-resolution [*HST*]{} images, the broad-band SED constrains the bolometric luminosity to ${\ensuremath{L\mathrm{_{bol}}}}\approx(0.5-3)\times10^{43}$ [erg s$^{-1}$]{} and ${{\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}}\approx0.02-0.15$.
We performed a comprehensive analysis of SDSS and [*HST*]{} images to quantify the structure and morphology of the host galaxy. The galaxy is an isolated, undisturbed, nearly face-on S0 galaxy with a prominent pseudobulge ([Sérsic]{} index $n = 1.60$) that accounts for $\sim$60% of the total light in the $R$ band. The pseudobulge has relatively blue colors ($B-V \approx 0.7$ mag) and is mildly disky. Embedded within the central $\sim150-400$ pc is a UV-bright semi-ring forming stars at a rate of $\sim$0.3 [$M_{\odot}$]{} yr$^{-1}$. Consistent with other low-mass AGNs, UM 625 follows the [$M_\mathrm{BH}$]{}–$\sigma_{\star}$ relation but not the [$M_\mathrm{BH}$]{}–$L_{\rm bulge}$ relation of inactive galaxies.
We thank the referee for a very thorough and helpful review of the paper. N.J. thanks Minjin Kim, Zhaoyu Li, Tinggui Wang, Hongguang Shan, Chen Cao, and Chien Peng for discussions and for help in image analysis. We thank Tomas Dahlen and the STScI Help Desk for advice concerning the [*HST*]{} photometric system. We thank Lulu Fan for his early participation in this project when he worked with X.-B.D. The research of L.C.H. is supported by the Carnegie Institution for Science and by NASA grants awarded through STScI. L.C.H. thanks the Chinese Academy of Sciences and the National Astronomical Observatories of China for their hospitality while part of this paper was written. This work is supported by the China Scholarship Council, Chinese NSF grants NSF-11033007, NSF-11133006, NSF-11073019, a National 973 Project of China (2009CB824800), and the Fundamental Research Funds for the Central Universities (USTC WK2030220004). H. Y. and J. X. W. acknowledge support from NSFC (11233002). Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[^1]: Available at http://staff.ustc.edu.cn/ xbdong/Data\_Release/IMBH\_DR4/, together with auxiliary code to explain and demonstrate the fitting procedure.
[^2]: IRAF (Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Observatory, which is operated by AURA, Inc., under cooperative agreement with the National Science Foundation.
[^3]: http://www.astro.yale.edu/dokkum/lacosmic/
[^4]: http://www.stsci.edu/hst/wfpc2/software/wfpc2-psf-form.html
[^5]: http://drizzlepac.stsci.edu/
[^6]: http://www.stsci.edu/hst/observatory/focus/FocusModel. We did not include the offset information for the F606W image because it is only available for data taken after 2003.
[^7]: http://sundog.stsci.edu/cgi-bin/searchfirst
[^8]: The specific flux at 5100 Å is derived directly from our SDSS spectral fit. If, instead, we estimate $f_{\nu}(5100$ Å) from the [H$\alpha$]{} flux (Greene & Ho 2005b), we obtain ${\ensuremath{\alpha_{\rm{ox}}}}=-1.76$ for the total (narrow plus broad)[H$\alpha$]{} flux and ${\ensuremath{\alpha_{\rm{ox}}}}=-1.53$ for the broad component alone.
[^9]: Dong et al. (2012b) give [[$L\mathrm{_{bol}}$]{}/[$L\mathrm{_{Edd}}$]{}]{} = 0.04 using the same technique, except that they estimate [$L\mathrm{_{bol}}$]{} only using the broad component of [H$\alpha$]{}, which is 3 times lower than the narrow component [H$\alpha$]{}. Here we use total [H$\alpha$]{} while the narrow [H$\alpha$]{} is seriously contaminated by the star-forming ring as calculated in section 5.3. Our strategy is to give a reliable range of [$L\mathrm{_{bol}}$]{} and [[$L\mathrm{_{bol}}$]{}/[$L\mathrm{_{Edd}}$]{}]{} taking all results into consideration.
|
---
address:
- |
Institut für Theoretische Physik, J. W. Goethe-Universität,\
D-60054 Frankfurt am Main, Germany
- |
Institut für Theoretische Physik, J. Liebig-Universität,\
D-35392 Giessen, Germany
author:
- 'C. Spieles, M. Bleicher, L. Gerland, H. Stöcker'
- 'C. Greiner'
title: Dynamics of Strangeness Production and Strange Matter Formation
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introduction
============
We want to draw the attention to the dynamics of a (finite) hadronizing quark matter drop. Strange and antistrange quarks do not hadronize at the same time for a baryon-rich system[@CG1]. Both the hadronic and the quark matter phases enter the strange sector $f_s\neq 0$ of the phase diagram almost immediately, which has up to now been neglected in almost all calculations of the time evolution of the system. Therefore it seems questionable, whether final particle yields reflect the actual thermodynamic properties of the system at a certain stage of the evolution. We put special interest on the possible formation of exotic states, namely strangelets (multistrange quark clusters). They may exist as (meta-)stable exotic isomers of nuclear matter [@Bod71]. It was speculated that strange matter might exist also as metastable exotic multi-strange (baryonic) objects (MEMO’s [@Sch92]). The possible creation — in heavy ion collisions — of long-lived remnants of the quark-gluon-plasma, cooled and charged up with strangeness by the emission of pions and kaons, was proposed in [@CG1; @Liu84; @CG2]. Strangelets can serve as signatures for the creation of a quark gluon plasma. Currently, both at the BNL-AGS and at the CERN-SPS experiments are carried out to search for MEMO’s and strangelets, e. g. by the E864, E878 and the NA52 collaborations[@qm95; @str95].
The model
=========
We adopt a model [@CG2] for the hadronization and space-time evolution of quark matter droplet. We assume a first order phase transition of the QGP to hadron gas. The expansion of the QGP droplet is described in a hybrid-like model, which takes into account equilibrium as well as nonequilibrium features of the process by the following two crucial, yet oversimplifying (and to some extent controversial) assumptions: (1) the plasma sphere is permantently surrounded by a thin layer of hadron gas, with which it stays in perfect equilibrium (Gibbs conditions) during the whole evolution; in particular the strangeness degree of freedom stays in chemical equilibrium because the complete hadronic particle production is driven by the plasma phase. (2) The nonequilibrium radiation is incorporated by a time dependent freeze-out of hadrons from the outer layers of the hadron phase surrounding the QGP droplet. During the expansion, the volume increase of the system thus competes with the decrease due to the freeze–out. The global properties like (decreasing) $S/A$ and (increasing) $f_s$ of the remaining two-phase system then change in time according to the following differential equations for the baryon number, the entropy, and the net strangeness number of the total system: $$\begin{aligned}
\label{eq1}
\frac{d}{dt}A^{tot} & = & -\Gamma \, A^{HG} \nonumber \\
\frac{d}{dt}S^{tot} & = & -\Gamma \, S^{HG} \\
\frac{d}{dt}(N_s - N_{\overline{s}})^{tot} & = & -\Gamma \,
(N_s - N_{\overline{s}})^{HG} \, \, \, , \nonumber\end{aligned}$$ where $\Gamma = \frac{1}{A^{HG}}
\left( \frac{\Delta A^{HG}}{\Delta t} \right) _{ev}$ is the effective (‘universal’) rate of particles (of converted hadron gas volume) evaporated from the hadron phase. The equation of state consists of the bag model for the quark gluon plasma and a mixture of relativistic Bose–Einstein and Fermi–Dirac gases of well established strange and non–strange hadrons up to 2 GeV in Hagedorn’s eigenvolume correction for the hadron matter [@CG1]. Thus, one solves simultaneously the equations of motion (\[eq1\]) and the Gibbs phase equilibrium conditions for the intrinsic variables, i.e. the chemical potentials and the temperature, as functions of time.
Strangelet distillation at low $\mu/T$
======================================
In [@Sp96] it was shown that large local net-baryon and net-strangeness fluctuations as well as a small but finite amount of stopping can occur at RHIC and LHC. This can provide suitable initial conditions for the possible creation of strange matter in colliders. A phase transition (e. g. a chiral one) can further increase the strange matter formation probability. In [@Sp96] it was further demonstrated with the present model that the high initial entropies per baryon do not hinder the distillation of strangelets, however, they require more time for the evaporation and cooling process.
Fig. \[abdampf\] (left) shows the time evolution of the baryon number for $S/A^{\rm init}=200$ and $f_s^{\rm init}=0.7$ for various bag constants. For $B^{1/4}<180$ MeV a cold strangelet emerges from the expansion and evaporation process, while the droplet completely hadronizes for bag constants $B^{1/4}\ge 180$ MeV (for $B^{1/4}=210 $ MeV hadronization proceeds without any significant cooling of the quark phase, although the specific entropy $S/A$ decreases by a factor of 2 from 200 to only 100). The strangeness separation works also in these cases, and leads to large final values of the net strangeness content, $f_s \stackrel{>}{\sim } 1.5-2$. However, then the volume of the drop becomes small, it decays and the strange quarks hadronize into $\Lambda $-particles and other strange hadrons. For even higher bag constants $B^{1/4}\approx 250$ MeV neither the baryon concentration effect nor strangeness distillery occurs (Fig. \[rates2\]).
Fig. \[abdampf\] (right) shows the evolution of the two-phase system for $S/A^{\rm
init}=200$, $f_s^{\rm init}=0$ and for a bag constant $B^{1/4}=160$ MeV in the plane of the strangeness fraction vs. the baryon density. The baryon density increases by more than one order of magnitude! Correspondingly, the chemical potential rises as drastically during the evolution, namely from $\mu^i=16$ MeV to $\mu^f>200$ MeV. The strangeness separation mechanism drives the chemical potential of the strange quarks from $\mu^i_s=0$ up to $\mu^f_s\approx 400$ MeV. Thus, the thermodynamical and chemical properties during the time evolution are quite different from the initial conditions of the system.
Fig. \[abdampf\] illustrates the increase of the baryon density in the plasma droplet as an inherent feature of the dynamics of the phase transition (cf. [@Wi84]). The origin of this result lies in the fact that the baryon number in the quark–gluon phase is carried by quarks with $m_{\rm q}\ll T_{\rm C}$, while the baryon density in the hadron phase is suppressed by a Boltzmann factor $\exp (-m_{\rm baryon}/T_{\rm C})$ with $m_{\rm baryon}\gg T_{\rm C}$. Mainly mesons (pions and kaons) are created in the hadronic phase. More relative entropy $S/A$ than baryon number is carried away in the hadronization and evaporation process[@CG2], i.e. $(S/A)^{HG} \gg (S/A)^{QGP}$. Ultimately, whether $(S/A)^{HG}$ is larger or smaller than $(S/A)^{QGP}$ at finite, nonvanishing chemical potentials might theoretically only be proven rigorously by lattice gauge calculations in the future. However, model equations of state do suggest such a behaviour, which would open such intriguing possibilities as baryon inhomogenities in ultrarelativistic heavy ion collisions as well as in the early universe.
Finite size effects
===================
The bag model equation of state for infinite quark matter is certainly a very rough approximation. Regarding finite size effects the leading correction to the quark matter equation of state is the curvature term. For massless quarks the volume term of the gandcanonical potential suffers the following modification (including the gluon contribution)[@Mad93]: $$\Omega_{\rm C}
=(\frac{1}{8\pi^2}\mu_{\rm q}^2+\frac{11}{72}T^2)C$$ with $C=8\pi R$ being the curvature of the spherical bag surface. From this one can easily derive all thermodynamic quantities and study the evolution of the two-phase system QGP/hadron gas in the above described model. It shows that even in the case of a favourable bag constant $B^{1/4}=145$ MeV a quark blob with an initial net baryon number of $A_{\rm B}^{\rm
init}=30$ will completely hadronize — in contrast to the calculation with the unmodified equation of state (Fig. \[finsize\]). Of course, the difference between the dynamics according to the two equations of state is reduced for larger systems. Still, it can be speculated that shell effects may allow for the formation of rather small strangelets which are stable. Moreover, the introduction of a more realistic hadronic equation of state (e. g. with the help of a relativistic mean field theory including adequate interactions for strange hadrons [@Sch92]) might modify this pessimistic picture again.
Particle rates from the hadronizing plasma
==========================================
Enhanced production of strange particles in relativistic nuclear collisions has received much attention recently[@qm95; @str95]. In particular thermal models have been developed and applied[@Ra95; @PBM] to explain (strange) particle yields and to extract the characteristic thermodynamic properties of the system (a few macroscopic parameters) from them. In our model the picture of a sudden hadronization which is supposed in these studies is only one possible outcome. Under more general assumptions the observed particle rates have to be put in relation to the whole time evolution of the system.
The integrated particle rates and the quark chemical potentials as functions of time have been calculated for two different scenarios: In Fig. \[rates1\] the results are plotted for a bag constant of $B^{1/4}=160$ MeV which is favorable for the strangeness distillation. In Fig. \[rates2\] a very high bag constant of $B^{1/4}=250$ MeV is used. This results in a very rapid (and complete) hadronization without significant cooling. Obviously, in the first case the particle rates reflect the massive changes of the chemical potentials during the evolution (which is the result of the strangeness distillation process). Note that e. g. the $\Lambda$’s are emitted mostly at the late stage, whereas the $\bar \Lambda$’s stem almost exclusively from the early stage. The $\bar \Lambda/\Lambda$ ratio is therefore not a meaningful quantity (if one takes it naively), since the two yields represent different sources! For the other choice of the bag constant the present model renders more or less the picture which is claimed by ’static’ thermal models: the plasma fireball decomposes very fast into hadrons (watch the different time scales of Figs. \[rates1\] and \[rates2\]) and the quark chemical potentials stay low compared to the temperature. Time dependent rates of the hypothetic $H^0$ Dibaryon are also shown in Figs. \[rates1\] and \[rates2\]. This particle is introduced to the hadronic resonance gas with its appropriate quantum numbers and two different assumed masses. It appears that the distillation mechanism gives rise to $H^0$ yields of the same order as the $\bar \Omega$’s (Fig. \[rates1\]) if the mass is $m_{H^0}\approx 2020$ MeV. For the high bag constant the $H^0$ yields are much more suppressed as compared to the strange (anti-)baryons. The absolute yields of the $H^0$ do not change much, since the system emits the particles at significantly higher temperature (due to the high bag constant).
Hyper–cluster formation in a microscopic model
==============================================
We now apply the Ultrarelativistic Quantum Molecular Dynamics 1.0$\beta$ [@uqmd; @bleicher], a semiclassical transport model, to calculate the abundances of strange baryon-clusters in relativistic heavy ion collisions. The model is based on classical propagation of hadrons and stochastic scattering ($s$ channel excitation of baryonic and mesonic resonances/strings, $t$ channel excitation, deexcitation and decay). In order to extract hyper-cluster formation probabilities the $\Lambda$ pair phase space after strong freeze-out is projected on the assumed dilambda wave function (harmonic oscillator) via the Wigner-function method as described in [@rafi]. According to the weak coupling between $\Lambda$’s in mean-field calculations [@Sch92] we assume the same coupling for $\Lambda\Lambda$-cluster as for deuterons[@rafi].
In Fig. \[yclust\] the calculated rapidity distributions of hyperons and $\Lambda\Lambda$-clusters are shown for central reactions of heavy systems at AGS and SPS energies. The multiplicities of $\Lambda$’s plus $\Sigma^0$’s in inelastic p+p reactions are $0.088\pm 0.003$ at 14.6 GeV/c and $0.234\pm
0.005$ at 200 GeV/c with the present version of the model. These numbers are given to assess the absolute yields in A+A collisions. The hyperon rapidity density stays almost constant when going from AGS to SPS energies, the ${\rm d}N/{\rm d}y$ of the hyper-clusters even drops slightly at midrapidity. This is due to the higher temperature which gives rise to higher relative momenta and therefore a reduced cluster probability. The $\Lambda/\Lambda\Lambda$ ratio is approximately 100, which can be compared to the $\Lambda/H^0$ ratios which result from the expanding quark gluon plasma (see last section).
References {#references .unnumbered}
==========
[99]{} C. Greiner, P. Koch and H. Stöcker, ; C. Greiner, D. H. Rischke, H. Stöcker and P. Koch, . A. R. Bodmer, ; S. A. Chin and A. K. Kerman, ; J. D. Bjorken and L. D. McLerran, . J. Schaffner, C. Greiner, C. B. Dover, A. Gal, H. Stöcker, Phys. Rev. Lett. [**71**]{}, 1328 (1993) H.-C. Liu and G.L. Shaw, . C. Greiner and H. Stöcker, . C. Spieles, L. Gerland, H. Stöcker, C. Greiner, C. Kuhn, J.P. Coffin, , E. Witten, ; E. Farhi and R. L. Jaffe, . J. Madsen, , and contribution to International Symposium on Strangeness and Quark Matter (Sept. 1-5, 1994) Crete (Greece) World Scientific, 1995. Proc. of the Int. Conf. on Ultrarelativistic Nucleus-Nucleus Collisions, Quark Matter ’95, Monterey, CA, USA, Nucl. Phys. [**A590**]{} (1995). Proc. of the Int. Conf. on Strangeness in Hadronic Matter, S’95, Tucson, AZ, USA, AIP Press, Woodbury, NY (1995). J. Letessier, A. Tounsi, U. Heinz, J. Sollfrank, J. Rafelski, . P. Braun-Munzinger, J. Stachel, J.P. Wessels, N. Xu ; P. Braun-Munzinger, J. Stachel, . S. A. Bass, M. Bleicher, M. Brandstetter, A. Dumitru, C. Ernst, L. Gerland, J. Konopka, S. Soff, C. Spieles, H. Weber, L. A. Winckelmann, N. Amelin H. Stöcker and W. Greiner;\
source code and technical documentation, to be published; S. A. Bass et al., contribution to this volume. M. Bleicher et al., contribution to this volume. R. Mattiello et al., contribution to this volume, and priv. comm.
|
---
abstract: |
This paper describes a computationally feasible approximation to the AIXI agent, a universal reinforcement learning agent for arbitrary environments. AIXI is scaled down in two key ways: First, the class of environment models is restricted to all prediction suffix trees of a fixed maximum depth. This allows a Bayesian mixture of environment models to be computed in time proportional to the logarithm of the size of the model class. Secondly, the finite-horizon expectimax search is approximated by an asymptotically convergent Monte Carlo Tree Search technique. This scaled down AIXI agent is empirically shown to be effective on a wide class of toy problem domains, ranging from simple fully observable games to small POMDPs. We explore the limits of this approximate agent and propose a general heuristic framework for scaling this technique to much larger problems.\
author:
- |
**Joel Veness joel.veness@nicta.com.au\
\
**Kee Siong Ng keesiong.ng@nicta.com.au\
\
**Marcus Hutter marcus.hutter@anu.edu.au\
\
**David Silver silver@cs.ualberta.ca\
********
date: 4 September 2009
title: |
****
------------------------------------------------------------------------
height5pt A Monte Carlo AIXI Approximation
------------------------------------------------------------------------
height2pt
---
\#1[*\#1*]{}
Reinforcement Learning (RL); Context Tree Weighting (CTW); Monte Carlo Tree Search (MCTS); Upper Confidence bounds applied to Trees (UCT); Partially Observable Markov Decision Process (POMDP); Prediction Suffix Trees (PST).
Introduction {#Introduction}
============
A main difficulty of doing research in artificial general intelligence has always been in defining exactly what [*artificial general intelligence*]{} means. There are many possible definitions [@legg07], but the AIXI formulation [@Hutter:04uaibook] seems to capture in concrete quantitative terms many of the qualitative attributes usually associated with intelligence.
Consider an agent that exists within some (unknown to the agent) environment. The agent interacts with the environment in cycles. At each cycle, the agent executes an action and receives in turn an observation and a reward. There is no explicit notion of state, neither with respect to the environment nor internally to the agent. The *general reinforcement learning problem* is to construct an agent that, over time, collects as much reward as possible in this setting.
The AIXI agent is a mathematical solution to the general reinforcement learning problem. The AIXI setup mirrors that of the general reinforcement problem, however the environment is assumed to be an unknown but computable function; i.e. the observations and rewards received by the agent given its actions can be computed by a Turing machine. Furthermore, the AIXI agent is assumed to exist for a finite, but arbitrarily large amount of time. The AIXI agent results from a synthesis of two ideas:
1. the use of a finite-horizon expectimax operation from sequential decision theory for action selection; and
2. an extension of Solomonoff’s universal induction scheme [@solomonoff64] for future prediction in the agent context.
More formally, let $U(q, a_1a_2\ldots a_n)$ denote the output of a universal Turing machine $U$ supplied with program $q$ and input $a_1a_2\ldots a_n$, $m \in \mathbb{N}$ a finite lookahead horizon, and $\ell(q)$ the length in bits of program $q$. The action picked by AIXI at time $t$, having executed actions $a_1a_2\ldots a_{t-1}$ and received the sequence of observation-reward pairs $o_1r_1o_2r_2\ldots o_{t-1}r_{t-1}$ from the environment, is given by: $$\label{aixi_eq}
a_t^* = \arg\max\limits_{a_t}\sum\limits_{o_t r_t} \dots \max\limits_{a_{t+m}}\sum\limits_{o_{t+m} r_{t+m}}[r_t + \dots + r_{t+m}]
\sum\limits_{q:U(q,a_1...a_{t+m})=o_1 r_1 ... o_{t+m}r_{t+m}}2^{-\ell(q)}.$$ Intuitively, the agent considers the sum of the total reward over all possible futures (up to $m$ steps ahead), weighs each of them by the complexity of programs (consistent with the agent’s past) that can generate that future, and then picks the action that maximises expected future rewards. Equation (\[aixi\_eq\]) embodies in one line the major ideas of Bayes, Ockham, Epicurus, Turing, von Neumann, Bellman, Kolmogorov, and Solomonoff. The AIXI agent is rigorously shown in [@Hutter:04uaibook] to be optimal in different senses of the word. (Technically, AIXI is Pareto optimal and ‘self-optimising’ in different classes of environment.) In particular, the AIXI agent will rapidly learn an accurate model of the environment and proceed to act optimally to achieve its goal.
The AIXI formulation also takes into account stochastic environments because Equation (\[aixi\_eq\]) can be shown to be formally equivalent to the following expression: $$\label{aixi_eq2}
a_t^* = \arg\max\limits_{a_t}\sum\limits_{o_tr_t} \dots \max\limits_{a_{t+m}} \sum\limits_{o_{t+m}r_{t+m}}
[r_t + \dots + r_{t+m}]
\sum\limits_{\rho \in {\cal M}}2^{-K(\rho)}\rho( o_1r_1 \ldots o_{t+m}r_{t+m} \cbar a_1\ldots a_{t+m}),$$ where $\rho( o_1r_1 \ldots o_{t+m}r_{t+m} \cbar a_1\ldots a_{t+m})$ is the probability of $o_1r_1 \ldots o_{t+m}r_{t+m}$ given actions $a_1\ldots a_{t+m}$. Class ${\cal M}$ consists of all enumerable chronological semimeasures [@Hutter:04uaibook], which includes all computable $\rho$, and $K(\rho)$ denotes the Kolmogorov complexity of $\rho$ [@li-vitanyi].
An accessible overview of the AIXI agent can be found in [@LeggPHD08]. A complete description of the agent is given in [@Hutter:04uaibook].
The AIXI formulation is best understood as a rigorous [*definition*]{} of optimal decision making in general [*unknown*]{} environments, and not as an algorithmic solution to the general AI problem. (AIXI after all, is only asymptotically computable.) As such, its role in general AI research should be viewed as, for example, the same way the minimax and empirical risk minimisation principles are viewed in decision theory and statistical machine learning research. These principles define what is optimal behaviour if computational complexity is not an issue, and can provide important theoretical guidance in the design of practical algorithms. It is in this light that we see AIXI. This paper is an attempt to scale AIXI down to produce a practical agent that can perform well in a wide range of different, unknown and potentially noisy environments.
As can be seen in Equation (\[aixi\_eq\]), there are two parts to AIXI. The first is the expectimax search into the future which we will call [*planning*]{}. The second is the use of a Bayesian mixture over Turing machines to predict future observations and rewards based on past experience; we will call that [*learning*]{}. Both parts need to be approximated for computational tractability. There are many different approaches one can try. In this paper, we opted to use a generalised version of the UCT algorithm [@kocsis06] for planning and a generalised version of the Context Tree Weighting algorithm [@ctw95] for learning. This harmonious combination of ideas, together with the attendant theoretical and experimental results, form the main contribution of this paper.
The paper is organised as follows. Section \[sec:agent setting\] describes the basic agent setting and discusses some design issues. Section \[sec:mcts\] then presents a Monte Carlo Tree Search procedure that we will use to approximate the expectimax operation in AIXI. This is followed by a description of the context tree weighting algorithm and how it can be generalised for use in the agent setting in Section \[sec:ctw\]. We put the two ideas together in Section \[sec:together\] to form our agent algorithm. Theoretical and experimental results are then presented in Sections \[sec:theory\] and \[sec:experiments\]. We end with a discussion of related work and other topics in Section \[sec:discussion\].
The Agent Setting and Some Design Issues {#sec:agent setting}
========================================
A string $x_1x_2 \ldots x_n$ of length $n$ is denoted by $x_{1:n}$. The prefix $x_{1:j}$ of $x_{1:n}$, $j\leq n$, is denoted by $x_{\leq j}$ or $x_{< j+1}$. The notation generalises for blocks of symbols: e.g. $ax_{1:n}$ denotes $a_1x_1a_2x_2\ldots a_nx_n$ and $ax_{<j}$ denotes $a_1x_1 a_2x_2\ldots a_{j-1}x_{j-1}$. The empty string is denoted by $\epsilon$. The concatenation of two strings $s$ and $r$ is denoted by $sr$.
The (finite) action, observation, and reward spaces are denoted by $\cA, \cO$, and $\cR$ respectively. Also, ${\cal X}$ denotes the joint perception space $\cO \times \cR$.
A history is a string $h\in (\mathcal{A} \times \mathcal{X})^n$, for some $n \geq 0$. A partial history is the prefix of some history.
The set of all history strings of maximum length $n$ will be denoted by $(\mathcal{A} \times \mathcal{X})^{\leq n}$.
The following definition states that the agent’s model of the environment takes the form of a probability distribution over possible observation-reward sequences conditioned on actions taken by the agent.
\[def:environment\_model\] An environment model $\rho$ is a sequence of functions $\{ \rho_0, \rho_1, \dots \}$, $\rho_n \colon \mathcal{A}^n \rightarrow \dens \; (\mathcal{X}^n)$, that satisfies:
1. $\forall a_{1:n} \forall x_{<n}: \,\rho_n(x_{<n} \cbar a_{<n}) = \sum_{x_n \in {\cal X}}\, \rho_n(x_{1:n} \cbar a_{1:n})$
2. $\forall a_{<n} \forall x_{<n}: \,\rho_n(x_{<n} \cbar a_{<n}) > 0$.
The first condition (called the chronological condition in [@Hutter:04uaibook]) captures the natural constraint that action $a_n$ has no effect on observations made before it. The second condition enforces the requirement that the probability of every possible observation-reward sequence is non-zero. This ensures that conditional probabilities are always defined. It is not a serious restriction in practice, as probabilities can get arbitrarily small. For convenience, we drop the index $t$ in $\rho_t$ from here onwards.
Given an environment model $\rho$, we have the following identities: $$\begin{gathered}
\rho( x_n \cbar ax_{<n}a_n) = \frac{\rho(x_{1:n} \cbar a_{1:n})}{ \rho(x_{<n} \cbar a_{<n})} \label{cond prob of or}\\
\rho (x_{1:n} \cbar a_{1:n}) = \rho(x_1 \cbar a_1)\rho(x_2 \cbar a_1x_1a_2) \cdots \rho( x_n \cbar ax_{<n}a_n) \label{conditional mu 1}\end{gathered}$$
We represent the notion of *reward* as a numeric value that represents the magnitude of instantaneous pleasure experienced by the agent at any given time step. Our agent is a hedonist; its goal is to accumulate as much reward as it can during its lifetime. More precisely, in our setting the agent is only interested in maximising its future reward up to a fixed, finite, but arbitrarily large horizon $m \in \mathbb{N}$.
In order to act rationally, our agent seeks a *policy* that will allow it to maximise its future reward. Formally, a policy is a function that maps a history to an action. If we define $R_k(aor_{\leq t}) := r_k$ for $1 \leq k \leq t$, then we have the following definition for the expected future value of an agent acting under a particular policy:
Given history $ax_{<t}$, the $m$-horizon expected future reward of an agent acting under policy $\pi \colon (\mathcal{A} \times \mathcal{X})^{\leq t+m} \rightarrow \cA$ with respect to an environment model $\rho$ is: $$\label{eq:val}
v_{\rho}^m(\pi, ax_{<t}) := \mathbb{E}_{x_{t:t+m} \sim \rho} \left[ \sum\limits_{i=t}^{t+m} R_i(ax_{\leq t+m} ) \right],$$ where for $t \leq k \leq t+m$, $a_k := \pi(ax_{<k})$. The quantity $v_{\rho}^m(\pi, ax_{<t}a_t)$ is defined similarly, except that $a_t$ is now no longer defined by $\pi$.
The optimal policy $\pi^*$ is the policy that maximises Equation (\[eq:val\]). The maximal achievable expected future reward of an agent with history $h \in ({\cal A} \times {\cal X})^{t-1}$ in environment $\rho$ looking $m$ steps ahead is $V_{\rho}^m(h) := v_{\rho}^m(\pi^*, h)$. It is easy to see that $$\label{value defn}
V_{\rho}^{m}(h) =
\max\limits_{a_t}\sum\limits_{x_t}{\rho(x_t \cbar h a_t)} \cdots
\max\limits_{a_{t+m}}\sum\limits_{x_{t+m}}{\rho(x_{t+m} \cbar h ax_{t:t+m-1} a_{t+m})}
\left[\sum\limits_{i=t}^{t+m}r_i\right].$$
All of our subsequent efforts can be viewed as attempting to define an algorithm that determines a policy as close to the optimal policy as possible given reasonable resource constraints. Our agent is *model based*: we learn a model of the environment and use it to estimate the future value of our various actions at each time step. These estimates allow the agent to make an approximate best action given limited computational resources.
We now discuss some high-level design issues before presenting our algorithm in the next section.
A major problem in general reinforcement learning is *perceptual aliasing* [@chrisman92], which refers to the situation where the instantaneous perceptual information (a single observation in our setting) does not provide enough information for the agent to act optimally. This problem is closely related to the question of what constitutes a state, an issue we discuss next.
A [*Markov*]{} state [@sutton-barto98] provides a sufficient statistic for all future observations, and therefore provides sufficient information to represent optimal behaviour. No perceptual aliasing can occur with a Markov state. In Markov Decision Processes (MDPs) and Partially Observable Markov Decision Processes (POMDPs) all underlying [*environmental states*]{} are Markov.
A [*compact*]{} state representation is often assumed to generalise well and therefore enable efficient learning and planning. A common approach in reinforcement learning (RL) [@sutton-barto98] is to approximate the environmental state by using a small number of handcrafted features. However, this approach requires both that the environmental state is known, and that sufficient domain knowledge is available to select the features.
In the general RL problem, neither the states nor the domain properties are known in advance. One approach to general RL is to find a compact representation of state that is approximately Markov [@mccallum96; @shani07; @psr04; @SuttonT04], or a compact representation of state that maximises some performance criterion [@Hutter:09phimdp; @Hutter:09phidbn]. In practice, a Markov representation is rarely achieved in complex domains, and these methods must introduce some approximation, and therefore some level of perceptual aliasing.
In contrast, we focus on learning and planning methods that use the agent’s history as its representation of state. A history representation can be generally applied without any domain knowledge. Importantly, a history representation requires no approximation and introduces no aliasing: each history is a perfect Markov state (or $k$-Markov for length $k$ histories). In return for these advantages, we give up on compactness. The number of states in a history representation is exponential in the horizon length (or $k$ for length $k$ histories), and many of these histories may be equivalent. Nevertheless, a history representation can sometimes be more compact than the environmental state, as it ignores extraneous factors that do not affect the agent’s direct observations.
In order to form non-trivial plans that span multiple time steps, our agent needs to be able to predict the effects of its interaction with the environment. If a model of the environment is known, search-based methods offer one way of generating such plans. However, a general RL agent does not start with a model of the environment; it must learn one over time. Our agent builds an approximate model of the true environment from the experience it gathers when interacting with the real world, and uses it for online planning.
If the problem is small, model-based RL methods such as Value Iteration for MDPs can easily derive an optimal policy. However this is not appropriate for the larger problems more typical of the real world. Local search is one way to address this problem. Instead of solving the problem in its entirety, an approximate solution is computed before each decision is made. This approach has met with much success on difficult decision problems within the game playing research community and on large-sized POMDPs [@Ross:pomdp].
The general RL problem is extremely difficult. On any real world problem, an agent is necessarily restricted to making approximately correct decisions. One of the distinguishing features of sophisticated heuristic decision making frameworks, such as those used in computer chess or computer go, is the ability of these frameworks to provide acceptable performance on hardware ranging from mobile phones through to supercomputers. To take advantage of the fast-paced advances in computer technology, we claim that *a good autonomous agent framework should naturally and automatically scales with increasing computational resources*. Both the learning and planning components of our approximate AIXI agent have been designed with scalability in mind.
One of the key resources in real world decision making is time. As we are interested in a practical general agent framework, it is imperative that our agent be able to make good approximate decisions *on demand*. Different application domains have different real-world time constraints. We seek an agent framework that can make good, approximate decisions given anything from $10$ milliseconds to $10$ days thinking time per action.
Monte Carlo Tree Search with Model Updates {#sec:mcts}
==========================================
In this section we describe , a Monte Carlo Tree Search (MCTS) technique for stochastic, partially observable domains that uses an incrementally updated environment model $\rho$ to predict and evaluate the possible outcomes of future action sequences.
The algorithm is a straightforward generalisation of the UCT algorithm [@kocsis06], a Monte Carlo planning algorithm that has proven effective in solving large state space discounted, or finite horizon MDPs. The generalisation requires two parts:
- The use of an environment model that is conditioned on the agent’s history, rather than a Markov state.
- The updating of the environment model during search. This is essential for the algorithm to utilise the extra information an agent will have at a hypothetical, *particular* future time point.
The generalisation involves a change in perspective which has significant practical ramifications in the context of general RL agents. Our extensions to UCT allow , in combination with a sufficiently powerful predictive environment model $\rho$, to implicitly take into account the value of information in search and be applicable to partially observable domains.
is a best-first Monte Carlo Tree Search technique that iteratively constructs a search tree in memory. The tree is composed of two interleaved types of nodes: decision nodes and chance nodes. These correspond to the alternating $\max$ and $\sum$ operations in expectimax. Each node in the tree corresponds to a (partial) history $h$. If $h$ ends with an action, it is a chance node; if $h$ ends with an observation, it is a decision node. Each node contains a statistical estimate of the future reward.
Initially, the tree starts with a single decision node containing $|\cA|$ children. Much like in existing MCTS methods [@chaslot08d], there are four conceptual phases to a single iteration of . The first is the [*selection*]{} phase, where the search tree is traversed from the root node to an existing leaf chance node $n$. The second is the [*expansion*]{} phase, where a new decision node is added as a child to $n$. The third is the [*simulation*]{} phase, where a playout policy in conjunction with the environment model $\rho$ is used to sample a possible future path from $n$ until a fixed distance from the root is reached. Finally, the [*backpropagation*]{} phase updates the value estimates for each node on the reverse trajectory leading back to the root. Whilst time remains, these four conceptual operations are repeated. Once the time limit is reached, an approximate best action can be selected by looking at the value estimates of the children of the root node.
During the selection phase, action selection at decision nodes is done using a policy that balances exploration and exploitation. This policy has two main effects:
- to move the estimates of the future reward towards the maximum attainable future reward if the agent acted optimally.
- to cause asymmetric growth of the search tree towards areas that have high predicted reward, implicitly pruning large parts of the search space.
The future reward at leaf nodes is estimated by choosing actions according to a heuristic policy until a total of $m$ actions have been made by the agent, where $m$ is the search horizon. This heuristic estimate helps the agent to focus its exploration on useful parts of the search tree, and in practice allows for a much larger horizon than a brute-force expectimax search.
builds a sparse search tree in the sense that observations are only added to chance nodes once they have been generated along some sample path. A full expectimax search tree would not be sparse; each possible stochastic outcome will be represented by a distinct node in the search tree. For expectimax, the branching factor at chance nodes is thus $|O|$, which means that searching to even moderate sized $m$ is intractable.
Figure \[staructtree\] shows an example tree. Chance nodes are denoted with stars. Decision nodes are denoted by circles. The dashed lines from a star node indicate that not all of the children have been expanded. The squiggly line at the base of the leftmost leaf denotes the execution of a playout policy. The arrows proceeding up from this node indicate the flow of information back up the tree; this is defined in more detail in Section \[backup\].
A decision node will always contain $|\mathcal{A}|$ distinct children, all of whom are chance nodes. Associated with each decision node representing a particular history $h$ will be a value function estimate, $\hat{V}(h)$. During the selection phase, a child will need to be picked for further exploration. Action selection in MCTS poses a classic exploration/exploitation dilemma. On one hand we need to allocate enough visits to all children to ensure that we have accurate estimates for them, but on the other hand we need to allocate enough visits to the maximal action to ensure convergence of the node to the value of the maximal child node.
Like UCT, recursively uses the UCB policy [@auer02] from the $n$-armed bandit setting at each decision node to determine which action needs further exploration. Although the uniform logarithmic regret bound no longer carries across from the bandit setting, the UCB policy has been shown to work well in practice in complex domains such as Computer Go [@Gelly06] and General Game Playing [@cadia2008]. This policy has the advantage of ensuring that at each decision node, every action eventually gets explored an infinite number of times, with the best action being selected exponentially more often than actions of lesser utility.
The visit count $T(h)$ of a decision node $h$ is the number of times $h$ has been sampled by the algorithm. The visit count of the chance node found by taking action $a$ at $h$ is defined similarly, and is denoted by $T(ha)$.
\[eq:ucb\] Suppose $m$ is the search horizon and each single time-step reward is bounded in the interval $[\alpha, \beta]$. Given a node representing a history $h$ in the search tree, the action picked by the UCB action selection policy is: $$a_{UCB}(h) :=
\arg\max_{a \in \mathcal{A}}
\begin{cases}
\frac{1}{m(\beta - \alpha)} \hat{V}(ha) + C \sqrt{\frac{\log(T(h))}{T(ha)}} & \mbox{if } T(ha) > 0;\\
\infty & \mbox{otherwise},
\end{cases}$$ where $C \in \mathbb{R}$ is a positive parameter that controls the ratio of exploration to exploitation. If there are multiple maximal actions, one is chosen uniformly at random.
Note that we need a linear scaling of $\hat{V}(ha)$ in Definition \[eq:ucb\] because the UCB policy is only applicable for rewards confined to the $[0,1]$ interval.
Chance nodes follow immediately after an action is selected from a decision node. Each chance node $ha$ following a decision node $h$ contains an estimate of the future utility denoted by $\hat{V}(ha)$. Also associated with the chance node $ha$ is a density $\rho(\cdot \cbar ha)$ over observation-reward pairs.
After an action $a$ is performed at node $h$, $\rho(\cdot \cbar ha)$ is sampled once to generate the next observation-reward pair $or$. If $o$ has not been seen before, the node $hao$ is added as a child of $ha$. We will use the notation $\cO_{ha}$ to denote the subset of $\cO$ representing the children of partial history $ha$ created so far.
\[subsec:playout\] If a leaf decision node is encountered at depth $k < m$ in the tree, a means of estimating the future reward for the remaining $m-k$ time steps is required. The agent applies its heuristic playout function $\Pi$ to estimate the sum of future rewards $\sum_{i=k}^m r_i$. A particularly simple, pessimistic baseline playout function is $\Pi_{random}$, which chooses an action uniformly at random at each time step.
A more sophisticated playout function that uses action probabilities estimated from previously taken real-world actions could potentially provide a better estimate. The quality of the actions suggested by such a predictor can be expected to improve over time, since it is trying to predict actions that are chosen by the agent after a search. This powerful and intuitive method of constructing a generic heuristic will be explored further in a subsequent section.
Asymptotically, the heuristic playout policy makes no contribution to the value function estimates of . When the remaining depth is zero, the playout policy always returns zero reward. As the number of simulations tends to infinity, the structure of the search tree is equivalent to the exact depth $m$ expectimax tree with high probability. This implies that the asymptotic value function estimates of are invariant to the choice of playout function. However, when search time is limited, the choice of playout policy will be a major determining factor of the overall performance of the agent.
\[backup\] After the selection phase is completed, a path of nodes $n_1 n_2 \dots n_k$, $k \leq m$, will have been traversed from the root of the search tree $n_1$ to some leaf $n_k$. For each $1 \leq j \leq k$, the statistics maintained for (partial) history $h_{n_j}$ associated with node $n_j$ will be updated as follows:
$$\label{eq:backup}
\hat{V}(h_{n_j}) \leftarrow \frac{T(h_{n_j})}{T(h_{n_j})+1}\hat{V}(h_{n_j})+\frac{1}{T(h_{n_j})+1} \sum\limits_{i=j}^{m} r_i$$
$$\label{eq:inc}
T(h_{n_j}) \leftarrow T(h_{n_j}) + 1$$
Note that the same backup equations are applied to both decision and chance nodes.
\[subsec:inc model update\] Recall from Definition \[def:environment\_model\] that an environment model $\rho$ is a sequence of functions $\{ \rho_0, \rho_1, \rho_2, \ldots \}$, where $\rho_t : \mathcal{A}^t \rightarrow \dens \; (\mathcal{X}^t)$. When invoking the [Sample]{} routine to decide on an action, many hypothetical future experiences will be generated, with $\rho_t$ being used to simulate the environment at time $t$. For the algorithm to work well in practice, we need to be able to perform the following two operations in time sublinear with respect to the length of the agent’s entire experience string.
- Update - given $\rho_{t}(x_{1:t} \cbar a_{1:t}), a_{t+1},$ and $x_{t+1}$, produce $\rho_{t+1}(x_{1:t+1} \cbar a_{1:t+1})$
- Revert - given $\rho_{t+1}(x_{1:t+1} \cbar a_{1:t+1})$, recover $\rho_t(x_{1:t} \cbar a_{1:t})$
The revert operation is needed to restore the environment model to $\rho_t$ after each simulation to time $t+m$ is performed. In Section \[sec:ctw\], we will show how these requirements can be met efficiently by a certain kind of Bayesian mixture over a rich model class.
We now give the pseudocode of the entire algorithm.
Algorithm \[alg:bayesmc\] is responsible for determining an approximate best action. Given the current history $h$, it first constructs a search tree containing estimates $\hat{V}_{\rho}^m(ha)$ for each $a\in \mathcal{A}$, and then selects a maximising action. An important property of Algorithm \[alg:bayesmc\] is that it is *anytime*; an approximate best action is always available, whose quality improves with extra computation time.
An environment model $\rho$ A history $h$ A search horizon $m \in \mathbb{N}$ $(\Psi)$ $(\Psi, h, m)$ $\rho \leftarrow$ [Revert]{}$(\rho, m)$ $(\Psi,h)$
For simplicity of exposition, [Initialise]{} can be understood to simply clear the entire search tree $\Psi$. In practice, it is possible to carry across information from one time step to another. If $\Psi_t$ is the search tree obtained at the end of time $t$, and $aor$ is the agent’s actual action and experience at time $t$, then we can keep the subtree rooted at node $\Psi_t(hao)$ in $\Psi_t$ and make that the search tree $\Psi_{t+1}$ for use at the beginning of the next time step. The remainder of the nodes in $\Psi_t$ can then be deleted.
As a Monte Carlo Tree Search routine, Algorithm \[alg:bayesmc\] is embarrassingly parallel. The main idea is to concurrently invoke the [Sample]{} routine whilst providing appropriate locking mechanisms for the nodes in the search tree. An efficient parallel implementation is beyond the scope of the paper, but it is worth noting that ideas [@chaslot08] applicable to high performance Monte Carlo Go programs are easily transferred to our setting.
Algorithm \[alg:sample\] implements a single run through some trajectory in the search tree. It uses the [SelectAction]{} routine to choose moves at interior nodes, and invokes the playout policy at unexplored leaf nodes. After a complete path of length $m$ is completed, the recursion takes care that every visited node along the path to the leaf is updated as per Section \[backup\].
An environment model $\rho$ A search tree $\Psi$ A (partial) history $h$ A remaining search horizon $m \in \mathbb{N}$ 0 Generate $(o,r)$ from $\rho(or \cbar h)$ Create node $\Psi(hor)$ if $T(hor) = 0$ reward $\leftarrow$ $r$ $+$ [Sample]{}$(\rho, \Psi, hor, m - 1)$ reward $\leftarrow$ [Playout]{}$(\rho, h, m)$ $a$ $\leftarrow$ [SelectAction]{}$(\Psi, h)$ reward $\leftarrow$ [Sample]{}$(\rho, \Psi, ha, m)$ $\hat{V}(h) \leftarrow \frac{1}{T(h)+1}[reward + T(h)\hat{V}(h)]$ $T(h) \leftarrow T(h)+1$ reward
The action chosen by [SelectAction]{} is specified by the UCB policy described in Definition \[eq:ucb\]. If the selected child has not been explored before, then a new node is added to the search tree. The constant $C$ is a parameter that is used to control the shape of the search tree; lower values of $C$ create deep, selective search trees, whilst higher values lead to shorter, bushier trees.
A search tree $\Psi$ A history $h$ An exploration/exploitation constant $C$ $\mathcal{U} = \{ a\in \cA \colon T(ha) = 0\}$ Pick $a \in \mathcal{U}$ uniformly at random Create node $\Psi(ha)$ a $\arg\max\limits_{a \in \cA} \left\{ \frac{1}{m(\beta - \alpha)} \hat{V}(ha) + C \sqrt{\frac{\log(T(h))}{T(ha)}} \right\}$
An environment model $\rho$ A history $h$ A remaining search horizon $m \in \mathbb{N}$ A playout function $\Pi$ $reward \leftarrow 0$ Generate $a$ from $\Pi(h)$ Generate $(o,r)$ from $\rho(or \cbar ha)$ $reward \leftarrow reward + r$ $h \leftarrow haor$ reward
Extensions of Context Tree Weighting {#sec:ctw}
====================================
Context Tree Weighting (CTW) [@ctw95; @ctw-tutorial] is a theoretically well-motivated online binary sequence prediction algorithm that works well in practice [@begleiter04]. It is an online Bayesian model averaging algorithm that computes a mixture of all prediction suffix trees [@ron96] of a given bounded depth, with higher prior weight given to simpler models. We examine in this section several extensions of CTW needed for its use in the context of agents. Along the way, we will describe the CTW algorithm in detail.
\[ctw\_enhanced\] We first look at how CTW can be generalised for use as environment models (Definition \[def:environment\_model\]), which are functions of the form $\rho_n : {\cal A}^n \to {\it Density}\; ({\cal X}^n)$. This means we need an extension of CTW that, incrementally, takes as input a sequence of actions and produces as output successive conditional probabilities over observations and rewards. The high-level view of the algorithm is as follows: we process observations and rewards one bit at a time using standard CTW, but bits representing actions are simply appended to the input sequence without updating the context tree. The algorithm is now described in detail. If we drop the action sequence throughout the following description, the algorithm reduces to the standard CTW algorithm.
We start with a brief review of the KT estimator [@kt-estimator] for Bernoulli distributions. Given a binary string $y_{1:t}$ with $a$ zeroes and $b$ ones, the KT estimate of the probability of the next symbol is as follows: $$\begin{gathered}
\text{Pr}_{kt}( Y_{t+1} = 1 \cbar y_{1:t} ) := \frac{b + 1/2}{a + b + 1} \label{kt update}\\
\text{Pr}_{kt}( Y_{t+1} = 0 \cbar y_{1:t} ) := 1 - \text{Pr}_{kt} ( Y_{t+1} = 1 \cbar y_{1:t}).\label{kt update 0}\end{gathered}$$ The KT estimator is obtained via a Bayesian analysis by putting a ($\tfrac{1}{2},\tfrac{1}{2}$)-Beta prior on the parameter of the Bernoulli distribution. From (\[kt update\])-(\[kt update 0\]), we obtain the following expression for the block probability of a string: $$\text{Pr}_{kt}( y_{1:t} ) = \text{Pr}_{kt}(y_1 \cbar \epsilon) \text{Pr}_{kt}(y_2 \cbar y_1) \cdots \text{Pr}_{kt} (y_t \cbar y_{1:t-1}).$$ Given a binary string $s$, one can establish that $\Pr_{kt}(s)$ depends only on the number of zeroes $a_s$ and ones $b_s$ in $s$. If we let $0^{a}1^{b}$ denote a string with $a$ zeroes and $b$ ones then: $$\label{kt block}
\text{Pr}_{kt}(s) = \text{Pr}_{kt}(0^{a_s}1^{b_s}) = \frac{1/2 (1 + 1/2) \cdots (a_s - 1/2) 1/2 (1 + 1/2) \cdots (b_s - 1/2)} {(a_s + b_s)!}.$$ We write $\Pr_{kt}(a,b)$ to denote $\Pr_{kt}(0^a1^b)$ in the following. The quantity $\Pr_{kt}(a,b)$ can be updated incrementally as follows: $$\begin{aligned}
\text{Pr}_{kt}(a+1,b) &= \frac{a + 1/2}{a + b + 1} \text{Pr}_{kt}(a,b) \\
\text{Pr}_{kt}(a,b+1) &= \frac{b + 1/2}{a + b + 1} \text{Pr}_{kt}(a,b), \label{kt block inc}\end{aligned}$$ with the base case being $\Pr_{kt}(0,0) = 1$.
We next describe prediction suffix trees, which are a form of variable-order Markov models.
\[defn:pst\] A prediction suffix tree (PST) is a pair $(M,\Theta)$ satisfying the following:
1. $M$ is a binary tree where the left and right edges are labelled 1 and 0 respectively; and
2. associated with each leaf node $l$ in $M$ is a probability distribution over $\{0,1\}$ parameterised by $\theta_l \in \Theta$ (the probability of 1).
We call $M$ the model of the PST and $\Theta$ the parameter of the PST, in accordance with the terminology of [@ctw95], .
A prediction suffix tree $(M,\Theta)$ maps each binary string $y_{1:n}$, where $n \geq$ the depth of $M$, to a probability distribution over $\{0,1\}$ in the natural way: we traverse the model $M$ by moving left or right at depth $d$ depending on whether the bit $y_{n-d}$ is one or zero until we reach a leaf node $l$ in $M$, at which time we return $\theta_l$. For example, the PST shown in Figure \[fig:pst\] maps the string 110 to $\theta_{10} = 0.3$. At the root node (depth 0), we move right because $y_3 = 0$. We then move left because $y_{3-1} = 1$. We say $\theta_{10}$ is the distribution associated with the string 110. Sometimes we need to refer to the leaf node holding the distribution associated with a string $h$; we denote that by $M(h)$, where $M$ is the model of the PST used to process the string.
@ur [ \_1 = 0.1 & \_1 \^0\
\_[10]{} = 0.3 & \_1 \^0\
& \_[00]{} = 0.5 & ]{}
To use a prediction suffix tree of depth $d$ for binary sequence prediction, we start with the distribution $\theta_l := \Pr_{kt}(1 \cbar \epsilon) = 1/2$ at each leaf node $l$ of the tree. The first $d$ bits $y_{1:d}$ of the input sequence are set aside for use as an initial context and the variable $h$ denoting the bit sequence seen so far is set to $y_{1:d}$. We then repeat the following steps as long as needed:
1. predict the next bit using the distribution $\theta_h$ associated with $h$;
2. observe the next bit $y$, update $\theta_h$ using Formula (\[kt update\]) by incrementing either $a$ or $b$ according to the value of $y$, and then set $h := hy$.
The above describes how a PST is used for binary sequence prediction. In the agent setting, we reduce the problem of predicting history sequences with general non-binary alphabets to that of predicting the bit representations of those sequences. Further, we only ever condition on actions and this is achieved by appending bit representations of actions to the input sequence without a corresponding update of the KT estimators. These ideas are now formalised.
For convenience, we will assume without loss of generality that $|{\cal A}| = 2^{l_{\cal A}}$ and $|{\cal X}| = 2^{l_{\cal X}}$ for some $l_{\cal A},l_{\cal X} > 0$. Given $a \in {\cal A}$, we denote by $\bstr{a} = a[1,l_{\cal A}] = a[1]a[2]\ldots a[l_{\cal A}] \in \{0,1\}^{l_{\cal A}}$ the bit representation of $a$. Observation and reward symbols are treated similarly. Further, the bit representation of a symbol sequence $x_{1:t}$ is denoted by $\bstr{x_{1:t}} = \bstr{x_1}\bstr{x_2}\ldots\bstr{x_t}$. The $i$th bit in $\bstr{x_{1:t}}$ is denoted by $\bstr{x_{1:t}}[i]$ and the first $l$ bits of $\bstr{x_{1:t}}$ is denoted by $\bstr{x_{1:t}}[1,l]$.
To do action-conditional prediction using a PST, we again start with $\theta_l := \Pr_{kt}(1 \cbar \epsilon) = 1/2$ at each leaf node $l$ of the tree. We also set aside a sufficiently long initial portion of the binary history sequence corresponding to the first few cycles to initialise the variable $h$ as usual. The following steps are then repeated as long as needed:
1. set $h := h\bstr{a}$, where $a$ is the current selected action;
2. for $i := 1$ to $l_{\cal X}$ do
1. predict the next bit using the distribution $\theta_h$ associated with $h$;
2. observe the next bit $x[i]$, update $\theta_h$ using Formula (\[kt update\]) according to the value of $x[i]$, and then set $h := hx[i]$.
Now, let $M$ be the model of a prediction suffix tree, $L(M)$ the leaf nodes of $M$, $a_{1:t} \in {\cal A}^t$ an action sequence, and $x_{1:t} \in {\cal X}^t$ an observation-reward sequence. We have the following expression for the probability of $x_{1:t}$ given $M$ and $a_{1:t}$: $$\begin{aligned}
\Pr( x_{1:t} \cbar M, a_{1:t} )
&= \prod_{i=1}^t \prod_{j=1}^{l_{\cal X}} \Pr( x_i[j] \cbar M, \bstr{ax_{<i}a_i} x_i[1,j-1]) \notag \\
&= \prod_{n \in L(M)} \text{Pr$_{kt}$} (\bstr{x_{1:t}}_{|n}), \label{tree prop1}\end{aligned}$$ where $\bstr{x_{1:t}}_{|n}$ is the (non-contiguous) subsequence of $\bstr{x_{1:t}}$ that ended up in leaf node $n$ in $M$. More precisely, $$\bstr{x_{1:t}}_{|n} := \bstr{x_{1:t}}[l_1] \bstr{x_{1:t}}[l_2] \cdots \bstr{x_{1:t}}[l_n],$$ where $1 \leq l_1 < l_2 < \cdots < l_n \leq t$ and, for each $i$, $i \in \{l_1,\ldots l_n\} \text{\;iff\;} M(\bstr{x_{1:t}}[1,i-1]) = n$.
The above deals with action-conditional prediction using a single PST. We now show how we can perform action-conditional prediction using a Bayesian mixture of PSTs in an efficient way. First, we need a prior distribution on models of PSTs.
Our prior, containing an Ockham-like bias favouring simple models, is derived from a natural prefix coding of the tree structure of a PST. The coding scheme works as follows: given a model of a PST of maximum depth $D$, a pre-order traversal of the tree is performed. Each time an internal node is encountered, we write down 1. Each time a leaf node is encountered, we write a 0 if the depth of the leaf node is less than $D$; otherwise we write nothing. For example, if $D = 3$, the code for the model shown in Figure \[fig:pst\] is 10100; if $D = 2$, the code for the same model is 101. The cost $\Gamma_{D}(M)$ of a model $M$ is the length of its code, which is given by the number of nodes in $M$ minus the number of leaf nodes in $M$ of depth $D$. One can show that $$\sum_{M \in C_D} 2^{-\Gamma_{D}(M)} = 1,$$ where $C_D$ is the set of all models of prediction suffix trees with depth at most $D$; i.e. the prefix code is complete. We remark that the above is another way of describing the coding scheme in [@ctw95]. We use $2^{-\Gamma_D(\cdot)}$, which penalises large trees, to determine the prior weight of each PST model.
The following is a key ingredient of the (action-conditional) CTW algorithm.
\[defn:context tree\] A context tree of depth $D$ is a perfect binary tree of depth $D$ where the left and right edges are labelled 1 and 0 respectively and attached to each node (both internal and leaf) is a probability on $\{0,1\}^*$.
The node probabilities in a context tree are estimated from data using KT estimators as follows. We update a context tree with the history sequence similarly to the way we use a PST, except that
1. the probabilities at each node in the path from the root to a leaf traversed by an observed bit is updated; and
2. we maintain block probabilities using Equations (\[kt block\])-(\[kt block inc\]) instead of conditional probabilities (Equation (\[kt update\])) like in a PST. (This is done for computational reasons to ease the calculation of the posterior probabilities of models in the algorithm.)
The process can be best understood with an example. Figure \[fig:ct\] (left) shows a context tree of depth two. For expositional reasons, we show binary sequences at the nodes; the node probabilities are computed from these. Initially, the binary sequence at each node is empty. Suppose $1001$ is the history sequence. Setting aside the first two bits 10 as an initial context, the tree in the middle of Figure \[fig:ct\] shows what we have after processing the third bit 0. The tree on the right is the tree we have after processing the fourth bit 1. In practice, we of course only have to store the counts of zeros and ones instead of complete subsequences at each node because, as we saw earlier in (\[kt block\]), $\Pr_{kt}(s) = \Pr_{kt}(a_s,b_s)$. Since the node probabilities are completely determined by the input sequence, we shall henceforth speak unambiguously about [*the*]{} context tree after seeing a sequence.
@ur [ & \_1 \^0 & \_1 \^0\
& & \_1 \^0\
& & ]{} @ur [ & \_1 \^0 & 0 \_1 \^0\
& 0 & 0 \_1 \^0\
& & ]{} @ur [ & \_1 \^0 & 01 \_1 \^0\
& 0 & 01 \_1 \^0\
& & 1 ]{}
The context tree of depth $D$ after seeing a sequence $h$ has the following important properties:
1. the model of every PST of depth at most $D$ can be obtained from the context tree by pruning off appropriate subtrees and treating them as leaf nodes;
2. the block probability of $h$ as computed by each PST of depth at most $D$ can be obtained from the node probabilities of the context tree via Equation (\[tree prop1\]).
These properties, together with an application of the distributive law, form the basis of the highly efficient (action-conditional) CTW algorithm. We now formalise these insights.
We first need to define the weighted probabilities at each node of the context tree. Suppose $a_{1:t}$ is the action sequence and $x_{1:t}$ is the observation-reward sequence. Let $\bstr{x_{1:t}}_{|n}$ be the (non-contiguous) subsequence of $\bstr{x_{1:t}}$ that ended up in node $n$ of the context tree. The weighted probability $P^n_w$ of each node $n$ in the context tree is defined inductively as follows: $$\begin{aligned}
P^n_w( \bstr{x_{1:t}}_{|n} & \cbar \bstr{a_{1:t}}) \\
:=& \begin{cases}
\Pr_{kt} ( \bstr{x_{1:t}}_{|n}) & \text{if $n$ is a leaf node} \\
\frac{1}{2}\Pr_{kt}(\bstr{x_{1:t}}_{|n}) +
\frac{1}{2}P^{n_l}_w(\bstr{x_{1:t}}_{|n_l} \cbar \bstr{a_{1:t}}) P^{n_r}_w(\bstr{x_{1:t}}_{|n_r} \cbar \bstr{a_{1:t}}) & \text{otherwise,}
\end{cases}\end{aligned}$$ where $n_l$ and $n_r$ are the left and right children of $n$ respectively. Note that the set of sequences $\{\, \bstr{x_{1:t}}_{|n} :\, $n$ \text{ is a node in the context tree} \,\}$ has a dependence on the action sequence $\bstr{a_{1:t}}$.
If $n$ is a node at depth $d$ in a tree, we denote by $p(n) \in \{0,1\}^d$ the path description to node $n$ in the tree.
Let $D$ be the depth of the context tree. For each node $n$ in the context tree at depth $d$, we have for all $a_{1:t} \in {\cal A}^t$, for all $x_{1:t} \in {\cal X}^t$, $$P^n_w( \bstr{x_{1:t}}_{|n} \cbar \bstr{a_{1:t}})
= \sum_{M \in C_{D-d}} 2^{-\Gamma_{D-d}(M)} \prod_{l \in L(M)} \text{{\em Pr}}_{kt}( \bstr{x_{1:t}}_{|p(n)p(l)}),$$ where $\bstr{x_{1:t}}_{|p(n)p(l)}$ is the (non-contiguous) subsequence of $\bstr{x_{1:t}}$ that ended up in the node with path description $p(n)p(l)$ in the context tree. \[model averaging\]
\#1\#2[\_[l L(\#1)]{} \_[kt]{}( \_[|p(\#2)p(l)]{})]{}
[The proof proceeds by induction on $d$. The statement is clearly true for the leaf nodes at depth $D$. Assume now the statement is true for all nodes at depth $d+1$, where $0 \leq d < D$. Consider a node $n$ at depth $d$. Letting $\overline{d} = D - d$, we have $$\begin{aligned}
P^n_w( &\bstr{x_{1:t}}_{|n} \cbar \bstr{a_{1:t}}) \\
&= \half\rootterm + \half P^{n_l}_w( \bstr{x_{1:t}}_{|n_l} \cbar \bstr{a_{1:t}})P^{n_r}_w(\bstr{x_{1:t}}_{|n_r} \cbar \bstr{a_{1:t}}) \notag\\
&= \half\rootterm + \half \left[ \sum_{M \in C_{\overline{d+1}}} \prM \cterm{M}{n_l} \right]\notag\\
& \hspace*{12em} \left[ \sum_{M \in C_{\overline{d+1}}} \prM \cterm{M}{n_r} \right] \notag\\
&= \half\rootterm \;+ \sum_{M_1 \in C_{\overline{d+1}}}\sum_{ M_2 \in C_{\overline{d+1}}} 2^{-(\Gamma_{\overline{d+1}}(M_1) + \Gamma_{\overline{d+1}}(M_2) +1)}\cdot \notag\\
&\hspace{13em} \left[ \cterm{M_1}{n_l} \right] \left[ \cterm{M_2}{n_r} \right] \\ &= \half\rootterm + \sum_{ \widehat{M_1M_2} \in C_{\overline{d}}} 2^{ -\Gamma_{\overline{d}}(\widehat{M_1M_2}) } \cterm{\widehat{M_1M_2}}{n} \\ &= \sum_{ M \in C_{D-d}} 2^{-\Gamma_{D-d}(M)} \cterm{M}{n}, \notag\end{aligned}$$ where $\widehat{M_1M_2}$ denotes the tree in $C_{\overline{d}}$ whose left and right subtrees are $M_1$ and $M_2$ respectively. ]{}
A corollary of Lemma \[model averaging\] is that at the root node $\lambda$ of the context tree we have $$\begin{aligned}
P^\lambda_w( \bstr{x_{1:t}} \cbar \bstr{a_{1:t}})
&= \sum_{ M \in C_D} 2^{-\Gamma_{D}(M)} \prod_{l \in L(M)} \text{Pr}_{kt}( \bstr{x_{1:t}}_{|p(l)}) \label{corr 1}\\
&= \sum_{ M \in C_D} 2^{-\Gamma_{D}(M)} \prod_{l \in L(M)} \text{Pr}_{kt}( \bstr{x_{1:t}}_{|l}) \label{corr 2} \\
&= \sum_{ M \in C_D} 2^{-\Gamma_{D}(M)} \Pr( x_{1:t} \cbar M, a_{1:t}), \label{corr 3}\end{aligned}$$ where the last step follows from Equation (\[tree prop1\]). Note carefully that $\bstr{x_{1:t}}_{|p(l)}$ in line (\[corr 1\]) denotes the subsequence of $\bstr{x_{1:t}}$ that ended in the node pointed to by $p(l)$ in the context tree but $\bstr{x_{1:t}}_{|l}$ in line (\[corr 2\]) denotes the subsequence of $\bstr{x_{1:t}}$ that ended in the leaf node $l$ in $M$ if $M$ is used as the only model to process $\bstr{x_{1:t}}$. Equation (\[corr 3\]) shows that the quantity computed by the (action-conditional) CTW algorithm is exactly a Bayesian mixture of (action-conditional) PSTs.
The weighted probability $P^\lambda_w$ is a block probability. To recover the conditional probability of $x_{t}$ given $ax_{<t}a_t$, we simply evaluate $$P_w^\lambda( \bstr{x_t} \cbar \bstr{ax_{<t}a_t}) = \frac{P_w^\lambda( \bstr{x_{1:t}} \cbar \bstr{a_{1:t}}) }{ P_w^\lambda( \bstr{x_{<t}} \cbar \bstr{a_{<t}}) },$$ which follows directly from Equation (\[cond prob of or\]). To sample from this conditional probability, we simply sample the individual bits of $x_t$ one by one. For brevity, we will sometimes use the following notation for $P^\lambda_w$: $$\begin{gathered}
\Upsilon(x_{1:t} \cbar a_{1:t}) := P^\lambda_w( \bstr{x_{1:t}} \cbar \bstr{a_{1:t}}) \\
\Upsilon( x_t \cbar ax_{<t}a_t) := P_w^\lambda( \bstr{x_t} \cbar \bstr{ax_{<t}a_t}).\end{gathered}$$
In summary, to do action-conditional prediction using a context tree, we set aside a sufficiently long initial portion of the binary history sequence corresponding to the first few cycles to initialise the variable $h$ and then repeat the following steps as long as needed:
1. set $h := h\bstr{a}$, where $a$ is the current selected action;
2. for $i := 1$ to $l_{\cal X}$ do
1. predict the next bit using the weighted probability $P^\lambda_w$;
2. observe the next bit $x[i]$, update the context tree using $h$ and $x[i]$, calculate the new weighted probability $P^\lambda_w$, and then set $h := hx[i]$.
Note that in practice, the context tree need only be constructed incrementally as needed. The depth of the context tree can thus take on non-trivial values. This memory requirement of maintaining a context tree is discussed further in Section \[subsec:comp considerations\].
As explained in Section \[subsec:inc model update\], the [Revert]{} operation is performed many times during search and it needs to be efficient. Saving and restoring a copy of the context tree is unsatisfactory. Luckily, the block probability estimated by CTW using a context depth of $D$ at time $t$ can be recovered from the block probability estimated at time $t+m$ in $O(mD)$ operations in a rather straightforward way. Alternatively, a copy on write implementation can be used to modify the context tree during the simulation phase.
\[subsec:predicate ctw\] As foreshadowed in [@buntine92thesis; @helmbold97], the CTW algorithm can be generalised to work with rich logical tree models [@blockeel98topdown; @kramer-widmer01; @LogicforLearning; @ng05thesis; @lloyd-ng-learnModal] in place of prediction suffix trees. A full description of this extension, especially the part on predicate definition/enumeration and search, is beyond the scope of the paper and will be reported elsewhere. Here we outline the main ideas and point out how the extension can be used to incorporate useful background knowledge into our agent.
Let ${\cal P} = \{ p_0, p_1, \ldots, p_m \}$ be a set of predicates (boolean functions) on histories $h \in ( {\cal A} \times {\cal X} )^n, n \geq 0$. A $\cP$-model is a binary tree where each internal node is labelled with a predicate in $\cP$ and the left and right outgoing edges at the node are labelled True and False respectively. A $\cP$-tree is a pair $(M_\cP,\Theta)$ where $M_\cP$ is a $\cP$-model and associated with each leaf node $l$ in $M_\cP$ is a probability distribution over $\{0,1\}$ parameterised by $\theta_l \in \Theta$.
A ${\cal P}$-tree $(M_\cP,\Theta)$ represents a function $g$ from histories to probability distributions on $\{0,1\}$ in the usual way. For each history $h$, $g(h) = \theta_{l_h}$, where $l_h$ is the leaf node reached by pushing $h$ down the model $M_\cP$ according to whether it satisfies the predicates at the internal nodes and $\theta_{l_h} \in \Theta$ is the distribution at $l_h$.
The use of general predicates on histories in $\cP$-trees is a powerful way of extending the notion of a “context” in applications. To begin with, it is easy to see that, with a suitable choice of predicate class $\cP$, both prediction suffix trees (Definition \[defn:pst\]) and looping suffix trees [@holmes06] can be represented as $\cP$-trees. Much more background contextual information can be provided in this way to the agent to aid learning and action selection.
The following is a generalisation of Definition \[defn:context tree\].
Let $\cP = \{ p_0, p_1, \ldots, p_m \}$ be a set of predicates on histories. A $\cP$-context tree is a perfect binary tree of depth $m+1$ where
1. each internal node at depth $i$ is labelled by $p_i \in \cP$ and the left and right outgoing edges at the node are labelled True and False respectively; and
2. attached to each node (both internal and leaf) is a probability on $\{0,1\}^*$.
We remark here that the (action-conditional) CTW algorithm can be generalised to work with $\cP$-context trees in a natural way, and that a result analogous to Lemma \[model averaging\] but with respect to a much richer class of models can be established. A proof of a similar result is in [@helmbold97]. Section \[sec:experiments\] describes some experiments showing how predicate CTW can help in more difficult problems.
Putting it All Together {#sec:together}
=======================
We now describe how the entire agent is constructed. At a high level, the combination is simple. The agent uses the action-conditional (predicate) CTW predictor presented in Section \[sec:ctw\] as a model $\Upsilon$ of the (unknown) environment. At each time step, the agent first invokes the routine to estimate the value of each action. The agent then picks an action according to some standard exploration/exploitation strategy, such as $\epsilon$-Greedy or Softmax [@sutton-barto98]. It then receives an observation-reward pair from the environment which is then used to update $\Upsilon$. Communication between the agent and the environment is done via binary codings of action, observation, and reward symbols. Figure \[agent\_overview\] gives an overview of the agent/environment interaction loop.
It is worth noting that, in principle, the AIXI agent does not need to explore according to any heuristic policy. This is since the value of information, in terms of expected future reward, is implicitly captured in the expectimax operation defined in Equations (\[aixi\_eq\]) and (\[aixi\_eq2\]). Theoretically, ignoring all computational concerns, it is sufficient just to choose a large horizon and pick the action with the highest expected value at each timestep.
Unfortunately, this result does not carry over to our approximate AIXI agent. In practice, the true environment will not be contained in our restricted model class, nor will we perform enough simulations to converge to the optimal expectimax action, nor will the search horizon be as large as the agent’s maximal lifespan. Thus, the exploration/exploitation dilemma is a non-trivial problem for our agent. We found that the standard heuristic solutions to this problem, such as $\epsilon$-Greedy and Softmax exploration, were sufficient for obtaining good empirical results. We will revisit this issue in Section \[sec:experiments\].
Theoretical Results {#sec:theory}
===================
Some theoretical properties of our algorithm are now explored.
\[subsec:model class apx\] We first study the relationship between $\Upsilon$ and the universal predictor in AIXI. [Using $\Upsilon$ in place of $\rho$ in Equation (\[value defn\]), the optimal action for an agent at time $t$, having experienced $ax_{1:t-1}$, is given by $$\begin{aligned}
a_t^*
=& \arg \max\limits_{a_t}\sum\limits_{x_t} \frac{ \Upsilon( x_{1:t} \cbar a_{1:t}) }{ \Upsilon( x_{<t} \cbar a_{<t})} \cdots
\max\limits_{a_{t+m}}\sum\limits_{x_{t+m}} \frac{ \Upsilon( x_{1:t+m} \cbar a_{1:t+m} ) } { \Upsilon( x_{<t+m} \cbar a_{<t+m}) }
\left[\sum\limits_{i=t}^{t+m}r_i\right] \notag\\
= & \arg \max\limits_{a_t}\sum\limits_{x_t} \cdots
\max\limits_{a_{t+m}}\sum\limits_{x_{t+m}} \left[ \sum\limits_{i=t}^{t+m}r_i \right] \prod_{i=t}^{t+m} \frac{ \Upsilon ( x_{1:i} \cbar a_{1:i}) } {
\Upsilon ( x_{<i} \cbar a_{<i}) } \notag\\
= & \arg \max\limits_{a_t}\sum\limits_{x_t}\cdots \max\limits_{a_{t+m}}\sum\limits_{x_{t+m}} \left[ \sum\limits_{i=t}^{t+m}r_i \right] \frac{ \Upsilon( x_{1:t+m} \cbar a_{1:t+m}) }{\Upsilon(x_{<t} \cbar a_{<t})}\notag\\
= & \arg \max\limits_{a_t}\sum\limits_{x_t}\cdots \max\limits_{a_{t+m}}\sum\limits_{x_{t+m}} \left[ \sum\limits_{i=t}^{t+m}r_i \right] \Upsilon( x_{1:t+m} \cbar a_{1:t+m}) \notag\\
= & \arg \max\limits_{a_t}\sum\limits_{x_t}\cdots \max\limits_{a_{t+m}}\sum\limits_{x_{t+m}} \left[ \sum\limits_{i=t}^{t+m}r_i \right] \sum_{ M \in {\mathcal
C}_{D}} 2^{-\Gamma_{D}(M)} \Pr( x_{1:t+m} \cbar M, a_{1:t+m}). \label{aipsi}\end{aligned}$$ ]{} Contrast (\[aipsi\]) now with Equation (\[aixi\_eq2\]) which we reproduce here: $$a_t = \arg\max\limits_{a_t}\sum\limits_{x_t} \dots \max\limits_{a_{t+m}} \sum\limits_{x_{t+m}}
\left[\sum\limits_{i=t}^{t+m}r_i\right]
\sum\limits_{\rho \in {\cal M}}2^{-K(\rho)}\rho(x_{1:t+m} \cbar a_{1:t+m}),$$ where ${\cal M}$ is the class of all enumerable chronological semimeasures, and $K(\rho)$ denotes the Kolmogorov complexity of $\rho$ [@Hutter:04uaibook]. The two expressions share a prior that enforces a bias towards simpler models. The main difference is in the subexpression describing the mixture over the model class. AIXI uses a mixture over all enumerable chronological semimeasures. This is scaled down to a mixture of prediction suffix trees in our setting. Although the model class used in AIXI is completely general, it is also incomputable. Our approximation has restricted the model class to gain the desirable computational properties of CTW. As indicated in Section \[subsec:predicate ctw\], the model class $C_D$ can be significantly enlarged by using predicates without sacrificing the efficient computability of mixtures.
\[subsec:Upsilon to mu\] We show in this section that if there is a good model of the (unknown) environment in the class $C_D$, then CTW will ‘find’ it. We need the following entropy inequality.
\[entropy ineq\] Let $\{y_i\}$ and $\{ z_i \}$ be two probability distributions, i.e. $y_i \geq 0, z_i \geq 0,$ and $\sum_i y_i = \sum_i z_i = 1$. Then we have $$\sum_{i} (y_i - z_i)^2 \leq \sum_i y_i \ln \frac{y_i}{z_i}.$$
\[conv to mu\] Let $\mu$ be the true environment model. The $\mu$-expected squared difference of $\mu$ and $\Upsilon$ is bounded as follows. For all $n \in \mathbb{N}$, for all $a_{1:n}$, $$\begin{gathered}
\hspace{-8em} \sum_{k=1}^n \sum_{x_{1:k}} \mu(x_{<k} \cbar a_{<k}) \biggl( \mu(x_k \cbar ax_{<k}a_k) - \Upsilon( x_k \cbar ax_{<k}a_k ) \biggr)^2 \\
\hspace{10em} \leq \min_{M \in C_D} \,\biggl\{\, \Gamma_D(M) \ln 2 + D_{KL}( \mu( \cdot \cbar a_{1:n}) \,\|\, \Pr( \cdot \cbar M, a_{1:n})) \,\biggr\},\end{gathered}$$ where $D_{KL}(\cdot \,\|\, \cdot)$ is the KL divergence of two distributions.
We adapt a proof from [@Hutter:04uaibook §5.1.3]. $$\begin{aligned}
&\;\;\;\; \sum_{k=1}^n \sum_{x_{1:k}} \mu(x_{<k} \cbar a_{<k}) \biggl( \mu(x_k \cbar ax_{<k}a_k) - \Upsilon( x_k \cbar ax_{<k}a_k ) \biggr)^2 \\
&= \sum_{k=1}^n \sum_{x_{<k}} \mu(x_{<k} \cbar a_{<k}) \sum_{x_k} \biggl( \mu(x_k \cbar ax_{<k}a_k) - \Upsilon( x_k \cbar ax_{<k}a_k ) \biggr)^2 \\
&\leq \sum_{k=1}^n \sum_{x_{<k}} \mu(x _{<k} \cbar a_{<k}) \sum_{x_k} \mu(x_k \cbar ax_{<k}a_k) \ln \frac{ \mu (x_k \cbar ax_{<k}a_k) }{ \Upsilon( x_k \cbar
ax_{<k}a_k ) } \hspace{2em} \text{[by Lemma \ref{entropy ineq}]} \\
&= \sum_{k=1}^n \sum_{x_{1:k}} \mu( x_{1:k} \cbar a_{1:k}) \ln \frac{ \mu( x_k \cbar ax_{<k}a_k) }{ \Upsilon( x_k \cbar ax_{<k}a_k ) } \hspace{8.9em} \text{[by
Eq. (\ref{cond prob of or})]}\\
&= \sum_{k=1}^n \sum_{x_{1:k}} \biggl( \sum_{x_{k+1:n}} \mu( x_{1:n} \cbar a_{1:n}) \biggr) \ln \frac{ \mu( x_k \cbar ax_{<k}a_k) }{
\Upsilon( x_k \cbar ax_{<k}a_k) } \hspace{6.5em} \text{[by Defn. \ref{def:environment_model}]} \\
&= \sum_{k=1}^n \sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \ln \frac{ \mu( x_k \cbar ax_{<k}a_k) }{ \Upsilon( x_k \cbar ax_{<k}a_k) } \\
&= \sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \sum_{k=1}^n \ln \frac{ \mu( x_k \cbar ax_{<k}a_k) }{ \Upsilon( x_k \cbar ax_{<k}a_k) } \\
&= \sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \ln \frac{ \mu( x_{1:n} \cbar a_{1:n}) }{ \Upsilon( x_{1:n} \cbar a_{1:n}) } \hspace{11em} \text{[by Eq.
(\ref{conditional mu 1})]}\\
&= \sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \ln \left[ \frac{\mu(x_{1:n}\cbar a_{1:n})}{\Pr( x_{1:n} \cbar M, a_{1:n})} \frac{\Pr( x_{1:n} \cbar
M, a_{1:n})} {\Upsilon( x_{1:n} \cbar a_{1:n})} \right] \hspace{2em} \text{[arbitrary $M \in C_D$]}\\
&= \sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \ln \frac{\mu(x_{1:n}\cbar a_{1:n})}{\Pr( x_{1:n} \cbar M, a_{1:n})} +
\sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \ln \frac{\Pr( x_{1:n} \cbar
M, a_{1:n})} {\Upsilon( x_{1:n} \cbar a_{1:n})}\\
&\leq D_{KL}( \mu( \cdot \cbar a_{1:n}) \,\|\, \Pr( \cdot \cbar M, a_{1:n})) + \sum_{x_{1:n}} \mu( x_{1:n} \cbar a_{1:n}) \ln \frac{ \Pr( x_{1:n} \cbar
M, a_{1:n}) }{ 2^{-\Gamma_D(M)} \Pr( x_{1:n} \cbar M, a_{1:n}) } \hspace{1.5em} \text{[by Eq. (\ref{corr 3})]} \\
&= D_{KL}( \mu( \cdot \cbar a_{1:n}) \,\|\, \Pr( \cdot \cbar M, a_{1:n})) + \Gamma_D(M) \ln 2.\end{aligned}$$ Since the inequality holds for arbitrary $M \in C_D$, it holds for the minimising $M$.
If the KL divergence between $\mu$ and the best model in $C_D$ is finite, then Theorem \[conv to mu\] implies $\Upsilon( x_k \cbar ax_{<k}a_k)$ will converge rapidly to $\mu( x_k \cbar ax_{<k}a_k)$ for $k \rightarrow \infty$ with $\mu$-probability 1.
The contrapositive of the statement tells us that if $\Upsilon$ fails to predict the environment well, then there is no good model in $C_D$. This result provides the motivation for looking at ways of enriching the model class in Section \[sec:discussion\].
Let $\mu$ be the true underlying environment. We now establish the link between the expectimax value $V_{\mu}^{m}(h)$ and its estimate $\hat{V}_\Upsilon^m(h)$ computed by the algorithm using $\Upsilon$ as the environment model.
In [@kocsis06], the authors show that the UCT algorithm is consistent in finite horizon MDPs and derive finite sample bounds on the estimation error due to sampling. By interpreting histories as Markov states, our general agent problem reduces to a finite horizon MDP and the results of [@kocsis06] are now directly applicable. Restating the main consistency result in our notation, we have $$\label{uctc}
\forall \epsilon \forall h \lim_{T(h) \to \infty} \Pr \left( | V_{\Upsilon}^m(h) - \hat{V}_\Upsilon^m(h) | \leq \epsilon \right) = 1.$$ Further, the probability that a suboptimal action (with respect to $V^m_\Upsilon(\cdot)$) is picked by goes to zero in the limit. Details of this analysis can be found in [@kocsis06].
Theorem \[conv to mu\] above in conjunction with [@Hutter:04uaibook Thm.5.36] implies $V_{\Upsilon}^m(h) \to V_\mu^m(h)$, as long as there exists a model in the model class that approximates the unknown environment $\mu$ well. This, and the consistency (\[uctc\]) of the algorithm, imply that $\hat{V}_\Upsilon^m(h)$ will converge to $V^m_{\mu}(h)$.
Experimental Results {#sec:experiments}
====================
In this section we evaluate our algorithm on a number of pre-existing domains. We have chosen domains that, from the agent’s perspective, have noisy perceptions, partial information, and inherent stochastic elements. In particular, we will focus on learning and approximately solving some benchmark POMDPs. The planning problem (i.e. computation of the optimal policy given the full POMDP model) associated with these POMDPs were considered challenging in the mid-nineties but can now be solved easily. We stress here that our requirement of having to learn the environment model, as well as solve the planning problem, [*significantly*]{} increases the difficulty of these problems.
As we shall see, our agent achieves state-of-the-art performance in both generality (eight separate problems with different characteristics are attempted) and optimality (the agent converges to the optimal policy in seven cases, and exhibits good scaling properties in the remaining case).
Our test domains are now described in detail. Their characteristics are summarised in Table \[table:domain\_characteristics\].
Domain Aliasing Noisy $\cO$ Noisy $\cA$ Uninformative $\cO$
----------------------------- ---------- ------------- ------------- --------------------- --
1d-maze yes no no yes
Cheese Maze yes no no no
Tiger yes yes no no
Extended Tiger yes yes no no
4 $\times$ 4 Grid yes no no yes
TicTacToe no no no no
Biased Rock-Paper-Scissor no yes yes no
Partially Observable Pacman yes no no no
: Domain characteristics[]{data-label="table:domain_characteristics"}
The 1d-maze is a simple problem from [@Cassandra94actingoptimally]. The agent begins at a random, non-goal location within a $1 \times 4$ maze. There is a choice of two actions: left or right. Each action transfers the agent to the adjacent cell if it exists, otherwise it has no effect. If the agent reaches the third cell from the left, it receives a reward of $1$. Otherwise it receives a reward of $0$. The distinguishing feature of this problem is that the observations are *uninformative*; every observation is the same regardless of the agent’s actual location.
This well known problem is due to [@mccallum96]. The agent is a mouse inside a two dimensional maze seeking a piece of cheese. The agent has to choose one of four actions: move up, down, left or right. If the agent bumps into a wall, it receives a penalty of $-10$. If the agent finds the cheese, it receives a reward of $10$. Each movement into a free cell gives a penalty of $-1$. The problem is depicted graphically in Figure \[fig:cheese\_maze\]. The number in each cell represents the decimal equivalent of the four bit binary observation the mouse receives in each cell. The problem exhibits perceptual aliasing in that a single observation is potentially ambiguous.
![The cheese maze[]{data-label="fig:cheese_maze"}](cheese-maze.eps){width="20em"}
This is another familiar domain from [@Kaelbling95planningand]. The environment dynamics are as follows: a tiger and a pot of gold are hidden behind one of two doors. Initially the agent starts facing both doors. The agent has a choice of one of three actions: listen, open the left door, or open the right door. If the agent opens the door hiding the tiger, it suffers a -100 penalty. If it opens the door with the pot of gold, it receives a reward of 10. If the agent performs the listen action, it receives a penalty of $-1$ and an observation that correctly describes where the tiger is with $0.85$ probability.
The problem setting is similar to Tiger, except that now the agent begins sitting down on a chair. The actions available to the agent are: stand, listen, open the left door, and open the right door. Before an agent can successfully open one of the two doors, it must stand up. However, the listen action only provides information about the tiger’s whereabouts when the agent is sitting down. Thus it is necessary for the agent to plan a more intricate series of actions before it sees the optimal solution. The reward structure is slightly modified from the simple Tiger problem, as now the agent gets a reward of 30 when finding the pot of gold.
The agent is restricted to a 4 $\times$ 4 grid world. It can move either up, down, right or left. If the agent moves into the bottom right corner, it receives a reward of $1$, and it is randomly teleported to one of the remaining $15$ cells. If it moves into any cell other than the bottom right corner cell, it receives a reward of $0$. If the agent attempts to move into a non-existent cell, it remains in the same location. Like the 1d-maze, this problem is also uninformative but on a much larger scale. Although this domain is simple, it does require some subtlety on the part of the agent. The correct action depends on what the agent has tried before at previous time steps. For example, if the agent has repeatedly moved right and not received a positive reward, then the chances of it receiving a positive reward by moving down are increased.
In this domain, the agent plays repeated games of TicTacToe against an opponent who moves randomly. If the agent wins the game, it receives a reward of $2$. If there is a draw, the agent receives a reward of $1$. A loss penalises the agent by $-2$. If the agent makes an illegal move, by moving on top of an already filled square, then it receives a reward of $-3$. A legal move that does not end the game earns no reward.
This domain is taken from [@farias09]. The agent repeatedly plays Rock-Paper-Scissor against an opponent that has a slight, predictable bias in its strategy. If the opponent has won a round by playing rock on the previous cycle, it will always play rock at the next timestep; otherwise it will pick an action uniformly at random. The agent’s observation is the most recently chosen action of the opponent. It receives a reward of $1$ for a win, $0$ for a draw and $-1$ for a loss.
This domain is a partially observable version of the classic PacMan game. The agent must navigate a $17 \times 17$ maze and eat the food pellets that are distributed across the maze. Four ghosts roam the maze. They move initially at random, until there is a Manhattan distance of 5 between them and PacMan, whereupon they will aggressively pursue PacMan for a short duration. The maze structure and game are the same as the original arcade game, however the PacMan agent is hampered by partial observability. PacMan is unaware of the maze structure and only receives a 4-bit observation describing the wall configuration at its current location. It also does not know the exact location of the ghosts, receiving only 4-bit observations indicating whether a ghost is visible (via direct line of sight) in each of the four cardinal directions. In addition, the location of the food pellets is unknown except for a 3-bit observation that indicates whether food can be smelt within a Manhattan distance of 2, 3 or 4 from PacMan’s location, and another 4-bit observation indicating whether there is food in its direct line of sight. A final single bit indicates whether PacMan is under the effects of a power pill. At the start of each episode, a food pellet is placed down with probability $0.5$ at every empty location on the grid. The agent receives a penalty of 1 for each movement action, a penalty of 10 for running into a wall, a reward of $10$ for each food pellet eaten, a penalty of $50$ if it is caught by a ghost, and a reward of $100$ for collecting all the food. If multiple such events occur, then the total reward is cumulative, i.e. running into a wall and being caught would give a penalty of $60$. The episode resets if the agent is caught or if it collects all the food.
Figure \[fig:pocman\] shows a graphical representation of the partially observable PacMan domain. This problem is the largest domain we consider, with an unknown optimal policy. The main purpose of this domain is to show the scaling properties of our agent with respect to a challenging problem.
![A screenshot (converted to b&w) of the partially observable PacMan domain[]{data-label="fig:pocman"}](pocman-greyscale.eps)
Table \[table:domain\_description\] outlines the parameters used in each experiment. The sizes of the action and observation spaces are given. The $\cA$ bits, $\cO$ bits and $\cR$ bits parameters specify the number of bits used to encode the action, observation and reward spaces. The context depth parameter $D$ specifies the maximum number of most recent bits used by the action-conditional CTW prediction scheme. The search horizon is specified by the parameter $m$.
The experimental results are presented in terms of average reward per time step. The key factors of interest are the performance of the agent as it accumulates more real world experience, and the performance of the agent as it is given more thinking time per decision.
All experiments were performed on a dual quad-core Intel 2.53Ghz Xeon. If computational concerns could be ignored, it would be natural to make $D$ as large as possible since CTW is robust against overfitting due to its strong bias towards simple PSTs. There are similar issues with the choice of horizon; ideally the horizon would be as large as possible if we could ignore computational concerns. In practice however, these parameters must be made much smaller for our agent to be tractable on our modest hardware. Section \[subsec:comp considerations\] discusses the asymptotic properties of our algorithms. Although the asymptotic behaviour is excellent (essentially linear in $D$ and $m$ in terms of both time and space), our prototype implementation is still pushing the boundaries of what can be done on a present day workstation. There are obvious problems if these parameters are set too small. For example, if the problem is $n$-Markov but we only use a $D<n$, or if the optimal policy requires planning ahead more than $m$ steps, then we cannot expect the agent to perform optimally.
Domain $|\cA|$ $|\cO|$ $\cA$ bits $\cO$ bits $\cR$ bits $D$ $m$
--------------------------- --------- ---------- ------------ ------------ ------------ ----- -----
1d-maze 2 1 1 1 1 32 10
Cheese Maze 4 16 2 4 5 96 8
Tiger 3 3 2 2 7 96 5
Extended Tiger 4 3 2 3 8 96 4
4 $\times$ 4 Grid 4 1 2 1 1 96 12
TicTacToe 9 19683 4 18 3 64 9
Biased Rock-Paper-Scissor 3 3 2 2 2 32 4
Partial Observable Pacman 4 $2^{16}$ 2 16 8 64 8
: Parameter Configuration[]{data-label="table:domain_description"}
Our agent has both limited thinking time and a limited amount of time to gather experience in the real world. Potentially, both of these dimensions will affect the agent’s performance. This section explores what the agent’s performance on different problem domains as we vary the two parameters.
Figure \[fig:reward\_vs\_age\] shows the performance of the agent as it accumulates more experience. Two seconds of search time per decision was used for each experiment. The label Age for the horizontal axis refers to the number of cycles that has transpired.
![Average reward vs age (measured in number of cycles). Two seconds of search were used for each action. []{data-label="fig:reward_vs_age"}](1dmaze_reward_vs_age.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs age (measured in number of cycles). Two seconds of search were used for each action. []{data-label="fig:reward_vs_age"}](cheese_maze_reward_vs_age.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs age (measured in number of cycles). Two seconds of search were used for each action. []{data-label="fig:reward_vs_age"}](tiger_reward_vs_age.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs age (measured in number of cycles). Two seconds of search were used for each action. []{data-label="fig:reward_vs_age"}](xtiger_reward_vs_age.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs age (measured in number of cycles). Two seconds of search were used for each action. []{data-label="fig:reward_vs_age"}](4x4grid_reward_vs_age.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs age (measured in number of cycles). Two seconds of search were used for each action. []{data-label="fig:reward_vs_age"}](tictactoe_reward_vs_age.eps "fig:"){width="0.49\columnwidth"}
Figure \[fig:reward\_vs\_time\] shows the performance of the agent on each problem domain by running it with varying amounts of search. The environment model used for each experiment was learned by the agent from randomly interacting with the environment for $50'000$ timesteps, with the exception of TicTacToe which used a model built from $500'000$ timesteps. Random action selection was used for computational reasons; it allowed large amounts of experience to be gathered quickly. For each data point, the agent is run for 2000 timesteps, using the best action chosen greedily by . The average reward is then calculated from the performance across these 2000 timesteps.
![Average reward vs search effort (measured in terms of the number of simulations used for picking each action).[]{data-label="fig:reward_vs_time"}](1dmaze_reward_vs_time.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs search effort (measured in terms of the number of simulations used for picking each action).[]{data-label="fig:reward_vs_time"}](cheese_reward_vs_time.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs search effort (measured in terms of the number of simulations used for picking each action).[]{data-label="fig:reward_vs_time"}](tiger_reward_vs_time.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs search effort (measured in terms of the number of simulations used for picking each action).[]{data-label="fig:reward_vs_time"}](xtiger_reward_vs_time.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs search effort (measured in terms of the number of simulations used for picking each action).[]{data-label="fig:reward_vs_time"}](4x4grid_reward_vs_time.eps "fig:"){width="0.49\columnwidth"} ![Average reward vs search effort (measured in terms of the number of simulations used for picking each action).[]{data-label="fig:reward_vs_time"}](tictactoe_reward_vs_time.eps "fig:"){width="0.49\columnwidth"}
In all cases, given sufficient thinking time and experience, the performance of our agent approaches optimality. Generally speaking, the agent’s performance gets better as it acquires more experience and is given more search time per decision. The agent’s performance on the tiger domains warrants some discussion.
The behaviour of the agent in the Tiger domain varies as the amount of interaction with the environment increases. Initially, the agent avoids selecting a door, as it is too uncertain about the environment dynamics. However, as it gathers more experience, more sophisticated behaviour emerges; the agent correctly acquires multiple pieces of information before picking a door. If some of the information is contradictory, the agent gathers more information before making its decision.
The performance of the agent in the Extended Tiger domain is sensitive to the number of simulations used by . As can be seen in Figure \[fig:reward\_vs\_age\], two seconds of thinking time were insufficient to act optimally. As indicated by figure \[fig:reward\_vs\_time\], optimal behaviour is only achieved when using a minimum of approximately $10'000$ simulations per decision. Only then does agent to understand that it is worth listening initially, then standing up, and then finally choosing the correct door according to the information it gathered whilst sitting down. With less simulations, the agent avoids picking a door. Interestingly, the performance of the agent drops after it has interacted with the world $5000$ times, yet then sharply increases. At $5000$ steps, the agent has overcome its aversion towards picking a door, without fully understanding the environment dynamics. This causes the agent to sometimes pick the wrong door. Further interaction refines the environment model and subsequently allows the agent to improve its performance.
Above we introduced the partially observable Pacman domain. In contrast to our other domains, this is an enormous problem. Even if the underlying state space were known, the learning and planning problems would still be hard because there are more than $2^{50}$ states.
Figure \[fig:pacman-scaling\] shows the scaling properties of our agent. Again, random exploration was used to build the model for computational reasons. The average reward at each data point was gathered by running the agent for 4000 timesteps, with each action being determined by .
Visually, the performance of the agent was non-optimal. However, after 2.5 million cycles of interaction, the agent had managed to learn a number of important concepts. It knows not to run into walls. It knows how to seek out food from the limited information provided by its sensors. It knows how to run away and avoid chasing ghosts. The main subtlety that it hasn’t learnt (after 2.5 million timesteps) is to aggressively chase down ghosts when it has eaten a red power pill. Also, its behaviour can sometimes become temporarily erratic when stuck in a long corridor with no nearby food or visible ghosts. Still, the ability to perform reasonably in a large domain, and exhibit consistent increases in performance with additional resources (experience or search time) makes us optimistic about the long-term potential of our work.
An important parameter in is the choice of the playout function. In MCTS-based methods for playing Computer Go, it is well known that adding knowledge to the playout function can dramatically improve performance [@gelly2006tr]. One of the benefits of MCTS methods is that if the domain is known, the playout function presents a natural way to incorporate domain knowledge. In the general agent setting, it would be desirable to automatically gain some of the benefits of expert design through online learning.
If the domain is unknown, a natural baseline playout policy is one that selects between each action uniformly at random. Although this playout policy is obviously quite poor, it does make some heuristic sense: the playouts end up guiding the search toward areas that give off larger rewards without requiring a carefully planned action sequence. In Section \[subsec:playout\], we described an intuitive method to incrementally learn a playout policy by attempting to model the real-world actions chosen by . The aim of this section is to show that our heuristic approach, using a CTW-based action predictor as a playout function, can give significant improvements to over the naive, uniformly random policy.
Figure \[fig:bootstrapped\_playouts\_cheese\] shows the impact of using the learned playout function on the cheese maze. (The other domains we tested exhibit similar behaviour.) Two versions of the same agent were run for $120'000$ cycles. Actions were selected using an $\epsilon$-greedy policy: i.e. with probability $\epsilon$ the agent moved randomly, otherwise the best action according to was chosen. The initial $\epsilon$ of $0.9$ was decayed by multiplying by $0.999$ at each time-step. A small ($100$ or $500$) simulations were used to decide on each action, to maximise the impact of the playout policy on the overall agent performance. The agent that used the self-improving playout policy learned faster and obtained a higher maximum average reward than the agent using uniform random playouts. Although the difference in average reward is small numerically, there is a qualitative difference in the performance of the agent. For example, the uniform playout policy when using 100 simulations averages approximately -1 per timestep. This is equivalent to a policy that simply runs around the maze, never finding the cheese, without ever bumping into a wall. When using 100 learned playouts however, the average reward ends up greater than zero. To achieve this, the agent must be finding the cheese, on average, in less than 11 steps every instance.
Our results demonstrate that it is both reasonable and practical for a MCTS-based general reinforcement learning agent to attempt to learn a playout function online. Our results are by no means exhaustive. The ideal action predictor may not resemble the observation/reward predictor, or it may be designed with different speed/accuracy trade-offs in mind. Online learning of playout functions for MCTS-based agents is a promising direction for future research. Building on this idea, one could also look at ways to modify the UCB policy used in to automatically take advantage of learnt playout knowledge, similar to the heuristic techniques used in Computer Go [@gelly07].
\[subsec:comp considerations\] If an agent has interacted with the world for $t$ cycles, using a context tree with depth $D$, there is at most $O(tD\log(|\cO||\cR|))$ nodes in the context tree. In practice, unless the environment is very noisy, only a subset of the $2^D$ possible contexts will be created. In our experiments, no more than a gigabyte of memory was required to store the entire environment model. The time complexity of CTW is also impressive: $O(D)$ to generate a single bit, and $O(Dm\log(|\cO||\cR|))$ to generate the $m$ observation/reward pairs needed to perform a single simulation.
This section gives an example of how Predicate CTW can be used to incorporate domain knowledge that drastically simplifies the agent’s learning task. We saw earlier in Figure \[fig:reward\_vs\_age\] that the dynamics of TicTacToe required a large amount of training examples for CTW to correctly predict the environment dynamics. Essentially, the main difficulty for the first hundred thousand steps was avoiding making illegal moves. In this experiment, the set of predicates that define CTW was augmented with a predicate that indicated whether the last move by the agent was legal. As one would expect, the agent using this augmented predicate set quickly learnt to play according to the game rules. Figure \[fig:pctw-tictactoe\] shows how a small but carefully chosen piece of domain knowledge can have a significant impact on the agent’s performance.
Discussion {#sec:discussion}
==========
We discuss some related and future work in this section. The headings reflect the general area of the literature in which those work can be found.
There have been several attempts at studying the computational properties of AIXI. In [@hutter02fastest], an asymptotically optimal algorithm is proposed that, in parallel, picks and runs the fastest program from an enumeration of provably correct programs for any given well-defined problem. A similar construction that runs all programs of length less than $l$ and time less than $t$ per cycle and picks the best output (in the sense of maximizing a provable lower bound for the true value) results in the optimal time bounded AIXI$tl$ agent [@Hutter:04uaibook Chp.7]. Like Levin search [@levin73], such algorithms are not practical in general but can in some cases be applied successfully; see e.g. [@schmidhuber97a; @schmidhuber97b; @schmidhuber03; @schmidhuber04].
In tiny domains, universal learning is computationally feasible with brute-force search. In [@poland06] the behaviour of AIXI is compared with a universal predicting-with-expert-advice algorithm [@poland05] in repeated $2\times 2$ matrix games and shows they exhibit different behaviour.
A Monte Carlo algorithm is proposed in [@pankov08] that samples programs according to their algorithmic probability as a way of approximating Solomonoff’s universal prior. A closely related algorithm is that of speed prior sampling [@schmidhuber02]. It remains an open question whether algorithms that sample from the space of general Turing machines can be made to work in practical problems.
We move on next to a discussion of related work in the general RL literature. An early and influential work is the Utile Suffix Memory (USM) algorithm by McCallum [@mccallum96]. USM uses a suffix tree to partition the agent’s history space into distinct states, one for each leaf in the suffix tree. Associated with each state/leaf is a Q-value, which is updated incrementally from experience like in Q-learning [@watkins92]. The history-partitioning suffix tree is grown in an incremental fashion, starting from a single leaf node in the beginning. A leaf in the suffix tree is split when the history sequences that fall into the leaf are shown to exhibit statistically different Q-values. The USM algorithm works well for a number of tasks but could not deal effectively with noisy environments. Several extensions of USM to deal with noisy environments are investigated in [@shani04; @shani07]. USM and their extensions are usually well-motivated but lack formal performance guarantees.
The work closest to ours in the general RL literature is the BLHT algorithm described in [@suematsu97; @suematsu99]. As in the present work, Suematsu et al. use prediction suffix trees as the model class but their suffix trees are defined at the symbol level (like in USM) as opposed to the bit level at which we operate. Another difference is that BLHT uses the maximum a posteriori (MAP) model to predict the future at any one time whereas we use a mixture of models. Having said that, the actual data structure and algorithm used in [@suematsu97; @suematsu99] to efficiently compute the MAP model bears close resemblance to CTW, and their algorithm may indeed be a general form of the context tree maximising algorithm [@volf95]. In their experiments, Suematsu et al. chose to use a uniform prior over the tree models even though their algorithm would work with an Ockham prior like that given in Equation (\[aipsi\]). It is also worth noting that our use of a Bayesian mixture admits a much stronger convergence result compared to what can be proved for BLHT. For control, BLHT uses an (unspecified) dynamic programming based algorithm.
The active LZ algorithm [@farias09] is also similar in spirit to our work. It combines a Lempel-Ziv [@lempel-ziv77] based prediction scheme with dynamic programming for control to produce an agent that is provably optimal if the environment is $n$-Markov, for some arbitrary $n$. They introduced and evaluated the performance of their agent on the ($n$-Markov) biased Rock-Paper-Scissor domain. We ran our agent on the same domain, using action-conditional CTW, $10000$ simulations and a uniform playout policy. Figure \[fig:aixi\_vs\_activelz\] shows our results overlayed with their reported results. Though it is difficult to compare implementations, it is clear that our agent has reached optimal performance using vastly less (at least two orders of magnitude) experience.
Predictive state representations (PSRs) [@psr02; @psr04] maintain predictions of future experience. Formally, a PSR is a probability distribution over the agent’s future experience, given its past experience. A subset of these predictions, the core tests, provide a sufficient statistic for all future experience. PSRs provide a Markov state representation, can represent and track the agent’s state in partially observable environments, and provide a complete model of the world’s dynamics. Unfortunately, exact representations of state are impractical in large domains, and some form of approximation is typically required. There is considerable interest in PSRs but there are at present still no satisfactory learning and discovery algorithms for PSRs.
Temporal-difference networks [@SuttonT04] are a form of predictive state representation in which the agent’s state is approximated by abstract predictions. These can be predictions about future observations, but also predictions about future predictions. This set of interconnected predictions is known as the [*question network*]{}. Temporal-difference networks learn an approximate model of the world’s dynamics: given the current predictions, the agent’s action, and an observation vector, they provide new predictions for the next time-step. The parameters of the model, known as the [*answer network*]{}, are updated after each time-step by temporal-difference learning. Some promising recent results applying TD-Networks for prediction (but not control) to small POMDPs have been reported in [@makino09].
Bayesian model averaging is a well-studied technique in statistics and machine learning [@hoeting99bma; @buntine92thesis; @oliver-hand95; @chipman98]. There is a nice connection between CTW, Buntine’s tree-smoothing algorithm [@buntine92thesis], Winnow-style online learning [@littlestone88; @littlestone94], and boosting [@freund97decision-theoretic]. The key idea behind Lemma \[model averaging\] appears in [@buntine92thesis Lemma 6.5.1]. The same technique is used in [@helmbold97] to implement an efficient version of the $P(\beta)$ online learning algorithm [@cesa-bianchi93] as a way of avoiding the problematic post-pruning step in decision-tree induction [@breiman84classification]. [@pereira99] then builds on that work to implement an efficient version of the Hedge algorithm [@freund97decision-theoretic] for constructing mixtures of the larger class of edge-based (as opposed to node-based) prunings of a tree. The algorithm in [@pereira99] can be used in conjunction with the predicate CTW idea to enlarge our agent’s model class.
There are several noteworthy ways the basic CTW algorithm can be extended. The finite depth limit on the context tree can be removed [@Willems94ctwext] without increasing the asymptotic space overhead of the algorithm. We chose to avoid this extension however due to the asymptotic time complexity increase of generating a symbol from linear in the context depth to linear in the number of observed symbols. CTW has also been extended to general non-binary alphabets, and the state-of-the-art seems to be the DE-CTW algorithm [@begleiter04; @begleiter06]. We opted not to use DE-CTW for several reasons. Firstly, DE-CTW is not a strictly online algorithm: a preprocessing phase is required to compute a way of decomposing the alphabets. Secondly, what is computed by DE-CTW isn’t really a Bayesian mixture and this is an unnecessary deviation from the theory of AIXI. Lastly, most of the effects of decomposing alphabets can in fact be realised using the predicate CTW extension.
Our experimental results have been restricted to problems of modest size. Future work will attempt to apply the algorithms presented here to more challenging domains.
The biggest limitation of our current agent is the restricted model class. Prediction suffix trees are simplistic models, inadequate to compactly represent something as simple as the rules of TicTacToe. Furthermore, the strong emphasis placed by CTW on temporally recent symbols is appropriate for only a subset of interesting real-world problems. The aim of the Predicate CTW extension is to relax this restriction somewhat, yet keep the desirable computational properties of CTW. As these predicates are arbitrary boolean functions on the agent’s history, they have the power to represent more complicated pieces of information that are useful to an agent in terms of making sensible predictions. Domain knowledge can be encoded in the form of user-supplied predicates, which seems essential for our agent to have any realistic chance of scaling to problems with real-world visual or audio data. Given a large model class $\cP$, the main learning problem in predicate CTW is in the identification of a small subset $\cP'$ of $\cP$ that is relevant to the current environment. This is a major unsolved problem in our setup and we think a suitable application of the Minimum Message Length principle [@wallace05] along the lines of [@Hutter:09phimdp] would shed much light on the key issues.
Furthermore, the performance of our agent is dependent on the amount of thinking time allowed at each time step. A crucial property of is that it is naturally parallel. A prototype parallel implementation of has been completed, with promising scaling results using between 4 and 8 processing cores. We are confident that further improvements to our prototype implementation will allow us to solve problems where the amount of search, rather than the agent’s predictive power, is the main performance bottleneck. Continuing advances in computer hardware will no doubt help address this issue as well.
Conclusion
==========
The main contribution of the paper is the extension and synthesis of two key results from online MDP planning (UCT) and information theory/machine learning (CTW) in the design of an agent that directly and scalably approximates the AIXI ideal. This is an important result. Although well established theoretically, it has previously been unclear whether AIXI could motivate the design of practical, yet theoretically well-founded algorithms. Our work answers this question strongly in the affirmative: empirically, our AIXI approximation achieves state-of-the-art performance and theoretically, we can provide some characterisation of the type of environments we expect our agent to handle.
To develop this approximation, we introduced two key algorithms:
- - a histories-as-states expectimax approximation algorithm;
- action-conditional CTW - an agent-specific generalisation of the CTW algorithm.
Furthermore, we demonstrated that our approach opens a number of future research areas:
- incorporating background knowledge through the predicate CTW extension;
- the possibility of constructing self-improving heuristic playout policies.
Although we are a long way away from being able to construct a truly powerful general agent, the future looks promising. We hope this work generates further interest from the broader artificial intelligence community in both AIXI and general reinforcement learning agents.
The authors thank Alan Blair, Bernhard Hengst, John Lloyd, Hassan Mahmud, Malcolm Ryan, Scott Sanner, and William Uther for many helpful discussions. NICTA is funded by the Australian Government’s Department of Communications, Information Technology, and the Arts and the Australian Research Council through Backing Australia’s Ability and the ICT Research Centre of Excellence programs.
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|
---
abstract: |
We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomial-time algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems.
graphs, connectivity problems, NP-hardness problems, approximation, upper and lower bounds
author:
- Erin Wolf Chambers
- 'Sándor P. Fekete'
- 'Hella-Franziska Hoffmann'
- |
\
Dimitri Marinakis
- 'Joseph S.B. Mitchell'
- Venkatesh Srinivasan
- |
\
Ulrike Stege
- Sue Whitesides
bibliography:
- 'lit.bib'
title: |
Connecting a Set of Circles\
with Minimum Sum of Radii
---
Introduction
============
We consider a natural geometric connectivity problem, arising from assigning ranges to a set of center points. In a general graph setting, we are given a weighted graph $G=(V,E)$. Each vertex $v\in V$ in the graph is assigned a radius $r_v$, and two vertices $v$ and $w$ are connected by an edge $f_{vw}$ in the connectivity graph $H=(V,F)$, if the shortest-path distance $d(v,w)$ in $G$ does not exceed the sum $r_v+r_w$ of their assigned radii. In a geometric setting, $V$ is given as a set of points $P = \{p_1,\ldots,p_n\}$ in the plane, and the respective radii $r_i$ correspond to circular ranges: two points $p_i$, $p_j$ have an edge $f_{ij}$ in the connectivity graph, if their circles intersect. The [Connected Range Assignment Problem]{} (CRA) requires an assignment of radii to $P$, such that the objective function $R = \sum_i r_i^{\alpha}, \alpha = 1$ is minimized, subject to the constraint that $H$ is connected.
Problems of this type have been considered before and have natural motivations from fields including networks, robotics, and data analysis, where ranges have to be assigned to a set of devices, and the total cost is given by an objective function that considers the sum of the radii of circles to some exponent $\alpha$. The cases $\alpha=2$ or $3$ correspond to minimizing the overall power; an example for the case $\alpha=1$ arises from scanning the corresponding ranges with a minimum required angular resolution, so that the scan time for each circle corresponds to its perimeter, and thus radius.
In the context of clustering, Doddi et al. [@Doddi00_apxMinSumDiam], Charikar and Panigraphy [@Charikar04_minSumDiam], and Gibson et al. [@Gibson08_minSumRadii] consider the following problems. Given a set $P$ of $n$ points in a metric space, metric $d(i,j)$ and an integer $k$, partition $P$ into a set of at most $k$ clusters with minimum sum of a) cluster diameters, b) cluster radii. Thus, the most significant difference to our problem is the lack of a connectivity constraint. Doddi et al. [@Doddi00_apxMinSumDiam] provide approximation results for a). They present a polynomial-time algorithm, which returns $O(k)$ clusters that are $O(\log(\frac{n}{k}))$-approximate. For a fixed $k$, transforming an instance into a min-cost set-cover problem instance yields a polynomial-time $2$-approximation. They also show that the existence of a $(2-\epsilon)$-approximation would imply $P=NP$. In addition, they prove that the problem in weighted graphs without triangle inequality cannot be efficiently approximated within any factor, unless $P=NP$. Note that every solution to b) is a $2$-approximation for a). Thus, the approximation results can be applied to case a) as well. A greedy logarithmic approximation and a primal-dual based constant factor approximation for minimum sum of cluster radii is provided by Charikar and Panigraphy [@Charikar04_minSumDiam]. In a more geometric setting, Bil[ò]{} et al. [@Bilo05_geomClustering] provide approximation schemes for clustering problems.
Alt *et al.* [@aab+-mccps-06] consider the closely related problem of selecting circle centers and radii such that a given set of points in the plane are covered by the circles. Like our work, they focus on minimizing an objective function based on $\sum_i r_i^{\alpha}$ and produce results specific to various values of $\alpha$. The minimum sum of radii circle coverage problem (with $\alpha = 1$) is also considered by Lev-Tov and Peleg [@Lev-Tov05] in the context of radio networks. Again, connectivity is not a requirement. The work of Clementi *et al.* [@Clementi04_powerAssignments] focuses on connectivity. It considers minimal assignments of transmission power to devices in a wireless network such that the network stays connected. In that context, the objective function typically considers an $\alpha > 1$ based on models of radio wave propagation. Furthermore, in the type of problem considered by Clementi *et al.* the connectivity graph is directed; *i.e.* the power assigned to a specific device affects its transmission range, but not its reception range. This is in contrast to our work in which we consider an undirected connectivity graph. See [@Fuchs08_hardnessRAP] for a collection of hardness results of different (directed) communication graphs.
Carmi et al. [@Carmi05_MAST] prove that an Euclidean minimum spanning tree is a constant-factor approximation for a variety of problems including the [*Minimum-Area Connected Disk Graph*]{} problem, which equals our problem with the different objective of minimizing the [*area*]{} of the [*union*]{} of disks, while we consider minimizing the [*sum*]{} of the [*radii*]{} (or perimeters) of all circles. In this paper we present a variety of algorithmic aspects of the problem. In Section 2 we show that for a given connectivity tree, an optimal solution can be computed efficiently. Section 3 sketches a proof of NP-hardness for the problem when there is an upper bound on the radii. Section 4 provides a number of approximation results in case there is no upper bound on the radii. In Section 5 we present a PTAS for the general case, complemented by experiments in Section 6. A concluding discussion with open problems is provided in Section 7.
CRA for a Given Connectivity Tree {#sec:cract}
=================================
For a given connectivity tree, our problem is polynomially solvable, based on the following observation.
\[l1\] Given a connectivity tree $T$ with at least three nodes. There exists an optimal range assignment for $T$ with $r_i = 0$ for all leaves $p_i$ of $T$.
Assume an optimal range assignment for $T$ has a leaf $p_i \in P$ with radius $r_i > 0$. The circle $C_i$ around $p_i$ with radius $r_i$ intersects circle $C_j$ around $p_i$’s parent $p_j$ with radius $r_j$. Extending $C_j$ to $r_j := {{\mbox{dist}}}(p_i,p_j)$ while setting $r_i := 0$ does not increase the solution value $R = \sum_{p_i\in P} r_i$.
Direct consequences of Lemma \[l1\] are the following.
\[c1\] There is an optimal range assignment satisfying Lemma \[l1\] and $r_j > 0$ for all $p_j \in P$ of height $1$ in $T$ (i.e., each $p_j$ is a parent of leaves only).
\[c2\] Consider an optimal range assignment for $T$ satisfying Lemma \[l1\]. Further let $p_j \in P$ be of height 1 in $T$. Then $r_j \geq \max_{p_i {\mbox{ \scriptsize is child of } p_j}} \{{{\mbox{dist}}}(p_i,p_j)\}$.
These observations allow a solution by dynamic programming. The idea is to compute the values for subtrees, starting from the leaves. Details are omitted.
\[t1\] For a given connectivity tree, CRA is solvable in $O(n)$.
Range Assignment for Bounded Radii
==================================
Without a connectivity tree, and assuming an upper bound of $\rho$ on the radii, the problem becomes NP-hard. In this extended abstract, we sketch a proof of NP-hardness for the graph version of the problem; for the geometric version, a suitable embedding (based on [Planar 3SAT]{}) can be used.
With radii bounded by some constant $\rho$, the problem CRA is NP-hard in weighted graphs.
![ Two variable gadgets connected to the same clause gadget. “True” and “False” vertices marked in bold white or black; auxiliary vertices are indicated by small dots; the clause vertex is indicated by a triangle. Connectivity edges are not shown. \[nphard\] ](nphard.pdf){width="70.00000%"}
See Figure \[nphard\] for the basic construction. The proof uses a reduction from [3Sat]{}. Variables are represented by closed “loops” at distance $\rho$ that have two feasible connected solutions: auxiliary points ensure that either the odd or the even points in a loop get radius $\rho$. (In the figure, those are shown as bold black or white dots. The additional small dots form equilateral triangles with a pair of black and white dots, ensuring that one point of each thick pair needs to be chosen, so a minimum-cardinality choice consists of all black or all white within a variable.) Additional “connectivity” edges ensure that all variable gadgets are connected. Each clause is represented by a star-shaped set of four points that is covered by one circle of radius $\rho$ from the center point. This circle is connected to the rest of the circles, if and only if one of the variable loop circles intersects it, which is the case if and only if there is a satisfying variable.
Solutions with a Bounded Number of Circles
==========================================
A natural class of solutions arises when only a limited number of $k$ circles may have positive radius. In this section we show that these [*k-circle solutions*]{} already yield good approximations; we start by giving a class of lower bounds.
A best $k$-circle solution may be off by a factor of $(1 + \frac{1}{2^{k+1}-1})$.
![A class of CRA instances that need $k+1$ circles in an optimal solution.\[k-disk-ex\]](k_disks_needed){width=".8\textwidth"}
Consider the example in Fig. \[k-disk-ex\]. The provided solution $r$ is optimal, as $R := \sum{r_i} = \frac{{{\mbox{dist}}}(p_0,p_n)}{2}$. Further, for any integer $k\geq2$ we have $d_1 = 2 \cdot \sum_{i=0}^{k-2}{2^i} + 2^{k-1} < 2 \cdot 2^{k} + 2^{k-1} = d_2$. So the radius $r_{k+1}$ cannot be changed in an optimal solution. Inductively, we conclude that exactly $k+1$ circles are needed. Because we only consider integer distances, a best $k$-circle solution has cost $R_k \geq R+1$, i.e., $\frac{R_k}{R} \geq 1 + \frac{1}{2^{k+1}-1}$.
In the following we give some good approximation guarantees for CRA using one or two circles.
\[l2\] Let ${\cal P}$ a longest (simple) path in an optimal connectivity graph, and let $e_m$ be an edge in ${\cal P}$ containing the midpoint of ${\cal P}$. Then $\sum r_i \geq \max\{\frac{1}{2}|{\cal P}|, |e_m|\}$.
This follows directly from the definition of the connectivity graph which for any edge $e = p_up_v$ in ${\cal P}$ requires $r_u + r_v \geq |e|$.
A best $1$-circle solution for CRA is a $\frac{3}{2}$-approximation, even in the graph version of the problem.
Consider a longest path ${\cal P} = (p_0,\ldots ,p_k)$ of length $|{\cal P}| = d_{\cal P}(p_0,\ldots ,p_k) := \sum_{i=0}^{k-1}{|p_ip_{i+1}|}$ in the connectivity graph of an optimal solution. Let $R^*~:=~\sum{r_i^*}$ be the cost of the optimal solution, and $e_m~=~p_ip_{i+1}$ as in Lemma \[l2\]. Let $\bar d_i := d_{\cal P}(p_i,\ldots ,p_k)$ and $\bar d_{i+1} := d_{\cal P}(p_0,\ldots ,p_{i+1})$. Then $\min\{\bar d_i,\bar d_{i+1}\} \leq \frac{\bar d_i+\bar d_{i+1}}{2}=\frac{d_{\cal P}(p_0,\ldots ,p_i) + 2|e_m| +
d_{\cal P}(p_{i+1},\ldots ,p_k)}{2} = \frac{|{\cal P}|}{2} + \frac{|e_m|}{2} \leq R^* +
\frac{R^*}{2} = \frac{3}{2}R^*$. So one circle with radius $\frac{3}{2}R^*$ around the point in $P$ that is nearest to the middle of path ${\cal P}$ covers $P$, as otherwise there would be a longer path.
![A lower bound of $\frac{3}{2}$ for 1-circle solutions.\[1\_disk\_strict\]](1_disk_strict){width="40.00000%"}
Fig. \[1\_disk\_strict\] shows that this bound is tight. Using two circles yields an even better approximation factor.
\[th:43\] A best $2$-circle solution for CRA is a $\frac{4}{3}$-approximation, even in the graph version of the problem.
Let ${\cal P} = (p_0,\ldots,p_k)$ be a longest path of length $|{\cal P}| = d_{\cal P}(p_0,\ldots ,p_k) := \sum_{i=0}^{k-1}{|p_ip_{i+1}|}$ in the connectivity graph of an optimal solution with radii $r_i^*$. Then $R^* := \sum r_i^* \geq \frac{1}{2}|{\cal P}|$. We distinguish two cases; see Fig. \[fig:43\].
[**Case 1.**]{} There is a point $x$ on ${\cal P}$ at a distance of at least $\frac{1}{3}|{\cal P}|$ from both endpoints. Then there is a $1$-circle solution that is a $\frac{4}{3}$-approximation.
[**Case 2.**]{} There is no such point $x$. Then two circles are needed. One of them is placed at a point in the first third of ${\cal P}$, and the other circle is placed at a point in the last third of ${\cal P}$. Let $e_m=p_ip_{i+1}$ be defined as in Lemma \[l2\]. Further, let $d_i := d_{\cal P}(p_0,\ldots ,p_i)$, and let $d_{i+1} := d_{\cal P}(p_{i+1},\ldots ,p_k)$. Then $|e_m| = |{\cal P}|-d_i-d_{i+1}$ and $d_i, d_{i+1} < \frac{1}{3}|{\cal P}|$.
![The two $\frac{4}{3}$-approximate $2$-circle solutions constructed in the proof of Theorem \[th:43\]: (Top) case 2a; (bottom) case 2b.[]{data-label="fig:43"}](2_disk_approx.pdf){width="0.6\linewidth"}
[**Case 2a.**]{} If $|e_m| < \frac{1}{2}|{\cal P}|$ then $d_i + d_{i+1} = |{\cal P}| - |e_m| > \frac{1}{2}|{\cal P}| > |e_m|$. Set $r_i := d_i$ and $r_{i+1} := d_{i+1}$, then the path is covered. Since $d_i, d_{i+1} < \frac{1}{3}|{\cal P}|$ we have $r_i + r_{i+1} = d_i + d_{i+1} < \frac{2}{3}|{\cal P}| \leq \frac{4}{3}R^*{{}}$ and the claim holds.
[**Case 2b.**]{} Otherwise, if $|e_m| \geq \frac{1}{2}|{\cal P}|$ then $d_i + d_{i+1} {{}} \leq \frac{1}{2}|{\cal P}| \leq |e_m|$. Assume $d_i \geq d_{i+1}$. Choose $r_i := d_i$ and $r_{i+1} := |e_m|-d_i$. As $d_{i+1} \leq |e_m| - d_i$ the path ${\cal P}$ is covered and $r_i + r_{i+1} = d_i + (|e_m| - d_i) = |e_m|$, which is the lower bound and thus the range assignment is optimal.
If all points of $P$ lie on a straight line, the approximation ratio for two circles can be improved.
\[overl\_lem\] Let $P$ be a subset of a straight line. Then there is a non-overlapping optimal solution, i.e., one in which all circles have disjoint interior.
An arbitrary optimal solution is modified as follows. For every two overlapping circles $C_i$ and $C_{i+1}$ with centers $p_i$ and $p_{i+1}$, we decrease $r_{i+1}$, such that $r_i + r_{i+1} = {{\mbox{dist}}}(p_i,p_{i+1})$, and increase the radius of $C_{i+2}$ by the same amount. This can be iterated, until there is at most one overlap at the outermost circle $C_j$ (with $C_{j-1}$). Then there must be a point $p_{j+1}$ on the boundary of $C_j$: otherwise we could shrink $C_j$ contradicting optimality. Decreasing $C_j$’s radius $r_{j}$ by the overlap $l$ and adding a new circle with radius $l$ around $p_{j+1}$ creates an optimal solution without overlap.
\[th:54\] Let $P$ a subset of a straight line $g$. Then a best $2$-circle solution for CRA is a $\frac{5}{4}$-approximation.
![A non-overlapping optimal solution.[]{data-label="fig:line_notation"}](line_proof_1.pdf){width="0.7\linewidth"}
According to Lemma \[overl\_lem\] we are, w.l.o.g., given an optimal solution with non-overlapping circles. Let $p_0$ and $p_n$ be the outermost intersection points of the optimal solution circles and $g$. W.l.o.g., we may further assume $p_0,p_n \in P$ and $R^* := \sum{r_i} =
\frac{{{\mbox{dist}}}(p_0,p_n)}{2}$ (otherwise, we can add the outermost intersection point of the outermost circle and $g$ to $P$, which may only improve the approximation ratio). Let $p_i$ denote the rightmost point in $P$ left to the middle of $\overline{p_0 p_n}$ and let $p_{i+1}$ its neighbor on the other half. Further, let $d_i := {{\mbox{dist}}}(p_0,p_i)$, $d_{i+1} := {{\mbox{dist}}}(p_{i+1},p_n)$ (See Fig. \[fig:line\_notation\]). Assume, $d_i \geq
d_{i+1}$. We now give $\frac{5}{4}$-approximate solutions using one or two circles that cover $\overline{p_0 p_n}$.
[**Case 1.**]{} If $\frac{3}{4} R^* \leq d_i$ then $\frac{5}{4} R^* \geq 2R^* - d_i = {{\mbox{dist}}}(p_i,p_n)$. Thus, the solution consisting of exactly one circle with radius $2R^* - d_i$ centered at $p_i$ is sufficient.
[**Case 2.**]{} If $\frac{3}{4} R^* > d_i \geq d_{i+1}$ we need two circles to cover $\overline{p_0 p_n}$ with $\frac{5}{4}R^*$.
![A $\frac{5}{4}$-approximate $2$-circle solution with $d_i < \frac{3}{4}R^*$. The cross marks the position of the optimal counterpart $p_i^*$ to $p_i$ and the grey area sketches $A_i$.[]{data-label="fig:line_solution"}](line_proof_2.pdf){width="0.7\linewidth"}
[**Case 2a.**]{}The point $p_i$ could be a center point of an optimal two-circle solution if there was a point $p_i^*$ with ${{\mbox{dist}}}(C_i,p_i^*) = {{\mbox{dist}}}(p_i^*,p_n) = R^* - d_i$. So in case there is a $p_i' \in P$ that lies in a $\frac{1}{4}R^*$-neighborhood of such an optimal $p_i^*$ we get ${{\mbox{dist}}}(C_i,p_i'), {{\mbox{dist}}}(p_i',p_n) \leq R^*-d_i+\frac{1}{4}R^*$ (see Fig. \[fig:line\_solution\]). Thus, $r(p_i) := d_i, r(p_i'):= R^*-d_i+\frac{1}{4}R^*$ provides a $\frac{5}{4}$-approximate solution.
[**Case 2b.**]{} Analogously to Case 2a, there is a point $p_{i+1}' \in P$ within a $\frac{1}{4}R^*$-range of an optimal counterpart to $p_{i+1}$. Then we can take $r(p_{i+1}):=d_{i+1}$, $r(p_{i+1}') := R^*-d_{i+1}+\frac{1}{4}R^*$ as a $\frac{5}{4}$-approximate solution.
[**Case 2c.**]{} Assume that there is neither such a $p_i'$ nor such a $p_{i+1}'$. Because $d_i, d_{i+1}$ are in $(\frac{1}{4}R^*,\frac{3}{4}R^*)$, we have $\frac{1}{4}R^* < R^* - d_j < \frac{3}{4}R^*$ for $j = i,i+1$, which implies that there are two disjoint areas $A_i$, $A_{i+1}$, each with diameter equal to $\frac{1}{2}R^*$ and excluding all points of $P$. Because $p_i$, the rightmost point on the left half of $\overline{p_0 p_n}$, has a greater distance to $A_i$ than to $p_0$, any circle around a point on the left could only cover parts of both $A_i$ and $A_{i+1}$ if it has a greater radius than its distance to $p_0$. This contradicts the assumption that $p_0$ is a leftmost point of a circle in an optimal solution. The same applies to the right-hand side. Thus, $A_i \cup A_{i+1}$ must contain at least one point of $P$, and therefore one of the previous cases leads to a $\frac{5}{4}$-approximation.
![A lower bound of $\frac{5}{4}$ for 2-circle solutions.\[2\_disk\_strict\]](2_disk_strict){width="0.5\linewidth"}
Fig. \[2\_disk\_strict\] shows that the bound is tight. We believe that this is also the worst case when points are [*not*]{} on a line. Indeed, the solutions constructed in the proof of Theorem \[th:54\] cover a longest path ${\cal P}$ in an optimal solution for a general $P$. If this longest path consists of at most three edges, $p_i (=:p_{i+1}')$ and $p_{i+1} (=:p_i')$ can be chosen as circle centers, covering all of $P$. However, if ${\cal P}$ consists of at least four edges, a solution for the diameter may produce two internal non-adjacent center points that do not necessarily cover all of $P$.
Polynomial-Time Approximation Schemes
=====================================
We now consider the problem in which each of the $n$ points of $P=\{p_1,\ldots,p_n\}$ has an associated upper bound, $\bar r_i$, on the radius $r_i$ that can be assigned to $p_i$.
Unbounded Radii
---------------
We begin with the case in which $\bar r_i=\infty$, for each $i$. Consider an optimal solution, with radius $r_i^*$ associated with input point $p_i$. We first prove a structure theorem that allows us to apply the $m$-guillotine method to obtain a PTAS. The following simple lemma shows that we can round up the radii of an optimal solution, at a small cost to the objective function:
\[lem:round\] Let $R^*=\sum_i r_i^*$ be the sum of radii in an optimal solution, ${\cal D}^*$. Then, for any fixed $\epsilon>0$, there exists a set, ${\cal D}_m$, of $n$ circles of radii $r_i$ centered on points $p_i$, such that (a). $r_i\in {\cal R}=\{D/mn, 2D/mn,\ldots, D\}$, where $D$ is the diameter of the input point set $P$ and $m=\lceil 2/\epsilon\rceil$; and (b). $\sum_i r_i \leq (1+\epsilon)R^*$.
Each of the $n$ radii $r_i^*$ can be increased by at most $D/mn\leq
\epsilon D/2n$ at a total cost of at most $\epsilon D/2$. Since increasing the radii of the circles keeps the set of circles connected and since $R^*\geq D/2$, we obtain the result.
Disks centered at the points $P$ of radii in the set ${\cal R}=\{D/mn, 2D/mn, \ldots, D\}$ will be referred to as [*${\cal
R}_{\epsilon,P}$-circles*]{}, or [*${\cal R}$-circles*]{}, for short, with the understanding that $\epsilon$ and $P$ will be fixed throughout our discussion. Consider the arrangement of all ${\cal R}$-circles. We let ${\cal I}_x$ (resp., ${\cal I}_y$) denote the $x$-coordinates of the left/right (resp., $y$-coordinates of the top/bottom) extreme points of these circles. (Specifically, ${\cal I}_x=\{x_{p_i} \pm j(D/mn):
1\leq i\leq n, 0\leq j\leq mn{\bar r_i}/D\}$ and ${\cal I}_y=\{y_{p_i}
\pm j(D/mn): 1\leq i\leq n, 0\leq j\leq mn{\bar r_i}/D\}$.)
We say that a set ${\cal D}$ of $n$ ${\cal R}$-circles is [*$m$-guillotine*]{} if the bounding box, $BB({\cal D})$, of ${\cal D}$ can be recursively partitioned into a rectangular subdivision by axis-parallel “$m$-perfect cuts” that are defined by coordinates ${\cal I}_x$ and ${\cal I}_y$, with the finest subdivision consisting of a partition into rectangular faces each of which has no circle of ${\cal D}$ strictly interior to it. An axis-parallel cut line $\ell$ is [*$m$-perfect*]{} with respect to ${\cal D}$ and a rectangle $\rho$ if $\ell$ intersects at most $2m$ circles of ${\cal D}$ that have a nonempty intersection with $\rho$.
Key to our method is a structure theorem, which shows that we can transform an arbitrary set ${\cal D}$ of circles centered on points $P$, having a connected union and a sum of radii $R$, into an $m$-guillotine set of ${\cal R}$-circles, ${\cal D}_m$, having sum of radii at most $(1+\epsilon)R^*$. More specifically, we show (proof deferred to the full paper):
\[thm:structure\] Let ${\cal D}$ be a set of circles of radii $r_i$ centered at points $p_i\in P$, such that the union of the circles is connected. Then, for any fixed $\epsilon>0$, there exists an $m$-guillotine set ${\cal
D}_m$ of $n$ ${\cal R}$-circles such that the union of the circles ${\cal
D}_m$ is connected and the sum of the radii of circles of ${\cal D}_m$ is at most $(1+(C/m))\sum_i r_i$. Here, $m=\lceil 1/\epsilon \rceil$ and $C$ is a constant.
First, by Lemma \[lem:round\], we can afford to round up the radii of the circles ${\cal D}$ to make them ${\cal R}$-circles.
Now, let $\rho$ denote the axis-aligned bounding box of the resulting circles. If there are no circles strictly interior to $\rho$ (i.e., if all circles intersecting $\rho$ intersect (touch) the boundary, $\partial\rho$, of $\rho$), then we are done: the set of circles is trivially $m$-guillotine already. Thus, we assume that there is at least one circle of ${\cal D}$ strictly interior to $\rho$.
If there exists an $m$-perfect cut of $\rho$, by a horizontal or vertical line intersecting at most $2m$ circles within $\rho$ and intersecting at least one circle that lies interior to $\rho$, then we partition $\rho$ with it and recurse on the two boxes on each side of the cut. We observe that if such an $m$-perfect cut exists, then one exists with the property that the defining coordinate of the cut is from the set ${\cal I}_x$ (for vertical cuts) or ${\cal I}_y$ (for horizontal cuts), since we can translate the cut between two consecutive such coordinates without changing the set of circles it intersects. (More precisely, we can translate a cut to be infinitesimally close to one of the discrete coordinates without changing its combinatorial type. To address this technicality, we can either augment the sets ${\cal I}_x$ and ${\cal I}_y$ with the midpoints of the intervals between consecutive coordinates, or we can replace each coordinate $z$ with three coordinates – $z$ and $z^+$ (infinitesimally greater than $z$) and $z^-$ (infinitesimally smaller than $z$).)
Thus, we now assume that no $m$-perfect cut exists for partitioning $\rho$, and that $\rho$ contains in its interior at least one circle. For a vertical line, $\ell_x$, through coordinate $x$, let $f(x)$ denote the length of the [*$m$-span*]{} of $\ell_x$ with respect to ${\cal D}$ and $\rho$: $f(x)=0$ if, within $\rho$, $\ell_x$ intersects at most $2m$ circles of ${\cal D}$; otherwise, if $\ell_x$ intersects $K>2m$ circles of ${\cal D}$ within $\rho$, then $f(x)$ is the distance (along $\ell_x$) from the first point, $a_m$, where $\ell_x$ enters into the $m$th circle, going from the top boundary of $\rho$ downwards along $\ell_x$, to the first point, $b_m$, where $\ell_x$ enters into the $m$th circle, going from the bottom boundary of $\rho$ upwards along $\ell_x$. Because $K>2m$, we know that $a_m$ is above $b_m$ and that there must be at least one circle of ${\cal D}$ whose intersection with $\ell_x$ is a proper subset of the [*bridge segment*]{} $a_mb_m$. We similarly define $g(y)$ to be the length of the (horizontal) bridge segment along a horizontal cut $\ell_y$ through coordinate $y$.
We think of $f(x)$ and $g(y)$ as the cost of augmentation for the network ${\cal N}_\rho$ consisting of the union of circles, truncated within $\rho$, that are the boundaries of the circles ${\cal D}$; by adding segments (bridges) of length $f(x)$ (resp., $g(y)$), a vertical cut $\ell_x$ (resp., horizontal cut $\ell_y$) can be made $m$-perfect. We claim that we can charge off the lengths of the bridges that would suffice to augment the network ${\cal N}_\rho$ to make it $m$-guillotine, in the usual sense of an $m$-guillotine network (subdivision), as in [@m-gsaps-99]. Specifically, we argue that we can select cuts for which the bridge lengths ($m$-spans) can be charged off to the total length of the network (sum of the circle circumferences, which is $O(\sum_i r_i)$), showing that the sum of the bridge lengths is at most $(C/m)\sum_i r_i$.
We partition each circle (bounding the circles ${\cal D}$) into four 90-degree arcs: two “vertical arcs” (with angular ranges (-45,45) and (135,225)) and two “horizontal arcs” (with angular ranges (45,135) and (225,315)). We define the “chargeable length” of a vertical cut $\ell_x$ to be the “$m$-dark” length of $\ell_x\cap
\rho$. Specifically, a subsegment $ab$ of $\ell_x\cap \rho$ is said to be $m$-dark with respect to ${\cal N}$ if for any $p\in ab$, the rightwards and leftwards rays from $p$ each cross at least $m$ vertical arcs of ${\cal N}_\rho$ before exiting $\rho$. If we cut $\rho$ along $\ell_x$, then the $m$-dark portion of the cut can be charged off to the left/right sides of the vertical arcs of ${\cal
N}_\rho$ lying to the right/left of $\ell_x$, distributing the charge to be ($1/m$)th to each of the $m$ arcs first hit.
Our charging scheme is based on the observation, following the method of [@m-gsaps-99], that there must exist a “favorable” vertical cut $\ell_x$ or horizontal cut $\ell_y$ for $\rho$ such that the chargeable length of the cut is at least as long as the cost ($f(x)$ or $g(y)$) of the cut. The existence of a favorable cut follows from the observation that $\int_{x\in \rho} f(x) dx = \int_{y\in \rho} h(y)
dy$, where $h(y)$ is the chargeable length associated with the horizontal cut $\ell_y$; thus, assuming, without loss of generality, that $\int_{x\in \rho} f(x) dx \geq \int_{y\in
\rho} g(y) dy$, we see that there must exist a value $y^*$ where $g(y^*)\leq h(y^*)$, which defines a favorable horizontal cut $\ell_{y^*}$ for which the chargeable length exceeds the length of the $m$-span. (In case $\int_{x\in \rho} f(x) dx <
\int_{y\in \rho} g(y) dy$, there exists a favorable vertical cut.) Further, we claim that there must exist a favorable cut corresponding to the coordinate sets ${\cal I}_x$, ${\cal I}_y$: the chargeable length of cut $\ell_{y^*}$ does not change as we perturb $y^*$ to the nearest coordinate in the set ${\cal I}_y$, while, by convexity of the circular arcs, the length of the $m$-span associated with a cut is locally minimized at endpoints of the intervals defined by the points of ${\cal I}_y$.
Once a favorable cut is found with respect to rectangle $\rho$, the cut partitions the problem into two subrectangles, and the argument is recursively applied to each. Since each circular arc of length $\lambda$ is charged for length at most $\lambda/2m$ on each of its two sides, we get that the overall length of all $m$-spans that are associated with favorable cuts constructed recursively in converting the network ${\cal N}_{BB({\cal D})}$ to an $m$-guillotine network is at most $(C/m)\sum_i r_i$, for a constant $C$.
Finally, we claim that the circles ${\cal D}$ can be converted to an $m$-guillotine set ${\cal D}_m$, having sum of radii at most $(C/m)\sum_i r_i$: Associated with each $m$-span $a_mb_m$ that is added to the network of circular arcs in order to make the network $m$-guillotine, we enlarge the radius of the circle defining one of the $m$-span endpoints (say, $a_m$), by at most $|a_mb_m|$, so that this circle now covers the entire $m$-span segment. Since we have enlarged one circle in a set of circles whose union is connected, the union remains connected. Thus, with a total increase in radii of at most $(C/m)\sum_i r_i$, we end up with an $m$-guillotine set ${\cal D}_m$ of circles, proving our structure theorem.
We now give an algorithm to compute a minimum-cost (sum of radii) $m$-guillotine set of ${\cal R}$-circles whose union is connected. The algorithm is based on dynamic programming. A subproblem is specified by a rectangle, $\rho$, with $x$- and $y$-coordinates among the sets ${\cal I}_x$ and ${\cal I}_y$, respectively, of discrete coordinates. The subproblem includes specification of [*boundary information*]{}, for each of the four sides of $\rho$. Specifically, the boundary information includes: (i) $O(m)$ “portal circles”, which are ${\cal
R}$-circles intersecting the boundary, $\partial \rho$, of $\rho$, with at most $2m$ circles specified per side of $\rho$; and, (ii) a connection pattern, specifying which subsets of the portal circles are required to be connected within $\rho$. There are a polynomial number of subproblems, for any fixed $m$. For a given subproblem, the dynamic program optimizes over all (polynomial number of) possible cuts (horizontal at ${\cal I}_y$-coordinates or vertical at ${\cal
I}_x$-coordinates), and choices of up to $2m$ ${\cal R}$-circles intersecting the cut bridge, along with all possible compatible connection patterns for each side of the cut. The result is an optimal $m$-guillotine set of ${\cal R}$-circles such that their union is connected and the sum of the radii is minimum possible for $m$-guillotine sets of ${\cal R}$-circles. Since we know, from the structure theorem, that an optimal set of circles centered at points $P$ can be converted into an $m$-guillotine set of ${\cal R}$-circles centered at points of $P$, whose union is connected, and we have computed an optimal such structure, we know that the circles obtained by our dynamic programming algorithm yield an approximation to an optimal set of circles. In summary, we have shown the following result:
\[thm:PTAS\] There is a PTAS for the min-sum radius connected circle problem with unbounded circle radii.
Bounded Radii
-------------
We now address the case of bounded radii, in which circle $i$ has a maximum allowable radius, $\bar r_i<\infty$. The PTAS given above relied on circle radii being arbitrarily large, so that we could increase the radius of a single circle to cover the entire $m$-span segment. A different argument is needed for the case of bounded radii.
We obtain a PTAS for the bounded radius case, if we make an additional assumption: that for any segment $pq$ there exists a connected set of circles, centered at points of $p_i\in P$ and having radii $r_i\leq \bar r_i$, such that $p$ and $q$ each lie within the union of the circles and the sum of the radii of the circles is $O(|pq|)$.
Here, we only give a sketch of the method, indicating how it differs from the unbounded radius case. The PTAS method proceeds as above in the unbounded radius case, except that we now modify the proof of the structure theorem by replacing each $m$-span bridge $a_mb_m$ by a shortest connected path of ${\cal R}$-circles. We know, from our additional assumption, that the sum of the radii along such a shortest path is $O(|a_mb_m|)$, allowing the charging scheme to proceed as before. The dynamic programming algorithm changes some as well, since now the subproblem specification must include the “bridging circle-path”, which is specified only by its first and last circle (those associated with the bridge endpoints $a_m$ and $b_m$); the path itself, which may have complexity $\Omega(n)$, is implicitly specified, since it is the shortest path (which we can assume to be unique, since we can specify a lexicographic rule to break ties).
In summary, we have
\[thm:PTAS-bounded\] There is a PTAS for the min-sum radius connected circle problem with bounded circle radii, assuming that for any segment $pq$, with $p$ and $q$ within feasible circles, there exists a (connected) path of feasible circles whose radii are $O(|pq|)$.
Experimental Results
====================
![ Ratios of the average over all enumerated trees and of the best 1-circle tree to the optimal $\sum r_i$. Results were averaged over 100 trials for each number. \[exp\_ratios\] ](ratio.pdf){width="60.00000%"}
It is curious that even in the worst case, a one-circle solution is close to being optimal. This is supported by experimental evidence. In order to generate random problem instances, we considered different numbers of points uniformly distributed in a 2D circular region. For each trial considering a single distribution of points, we enumerated all possible spanning trees using the method described in [@Avis96_reverse], and recorded the optimal value with the algorithm mentioned in Section \[sec:cract\]. This we compared with the best one-circle solution; as shown in Fig. \[exp\_ratios\], the latter seems to be an excellent heuristic choice. These results were obtained in several hours using an i7 PC.
Conclusion
==========
A number of open problems remain. One of the most puzzling is the issue of complexity in the absence of upper bounds on the radii. The strong performance of the one-circle solution (and even better of solutions with higher, but limited numbers of circles), and the difficulty of constructing solutions for which the one-circle solution is not optimal strongly hint at the possibility of the problem being polynomially solvable. Another indication is that our positive results for one or two circles only needed triangle inequality, i.e., they did not explicitly make use of geometry.
One possible way may be to use methods from linear programming: modeling the objective function and the variables by linear methods is straightforward; describing the connectivity of a spanning tree by linear cut constraints is also well known. However, even though separating over the exponentially many cut constraints is polynomially solvable (and hence optimizing over the resulting polytope), the overall polytope is not necessarily integral. On the other hand, we have been unable to prove NP-hardness without upper bounds on the radii, even in the more controlled context of graph-induced distances. Note that some results were obtained by means of linear programming: the tight lower bound for 2-circle solutions (shown in Fig. 7) was found by solving appropriate LPs.
Other open problems are concerned with the worst-case performance of heuristics using a bounded number of circles. We showed that two circles suffice for a $\frac{4}{3}$-approximation in general, and a $\frac{5}{4}$-approximation on a line; we conjecture that the general performance guarantee can be improved to $\frac{5}{4}$, matching the existing lower bound. Obviously, the same can be studied for $k$ circles, for any fixed $k$; at this point, the best lower bounds we have are $\frac{7}{6}$ for $k=3$ and $1+\frac{1}{2^{k+1}}$ for general $k$. We also conjecture that the worst-case ratio $f(k)$ of a best $k$-circle solution approximates the optimal value arbitrarily well for large $k$, i.e., $\lim_{k\to\infty}f(k) = 1$.
### Acknowledgments. {#acknowledgments. .unnumbered}
A short version of this extended abstract appears in the informal, non-competitive European Workshop on Computational Geometry. This work was started during the 2009 Bellairs Workshop on Computational Geometry. We thank all other participants for contributing to the great atmosphere.
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abstract: 'We present images of the HCN J = 1-0 emission from five nearby spiral galaxies made with the Berkeley-Illinois-Maryland Association interferometer. The HCN observations comprise the first high-resolution ($\theta$ $\sim$ 5– 10) survey of dense molecular gas from a sample of normal galaxies, rather than galaxies with prolific starburst or nuclear activities. The images show compact structure, demonstrating that the dense gas emission is largely confined to the central kiloparsec of the sources. To within the uncertainties, 70 - 100% of the single-dish flux is recovered for each source; this implies that there is not a significant contribution to the HCN flux from low-level emission in the disks of the galaxies. In one of the galaxies, NGC 6946, the ratio of HCN to CO integrated intensities ranges from 0.05–0.2 within the extent of the HCN emission ($r = 150$ pc), with an average value of 0.11 $\pm$ 0.01 over the whole region; the range and average values of the ratios in NGC 6946 are very similar to what is observed in the central $r = 250$ pc of the Milky Way. A comparison with single-dish observations allows us to place an upper limit of 0.01 on the ratio of integrated intensities in the region $150 < r < 800$ pc in NGC 6946. In NGC 6946, NGC 1068 and the Milky Way, the ratio I$_{\rm HCN}$/I$_{\rm CO}$ is 5 to 10 times higher in the bulge regions than in their disks; this suggests that the physical conditions in their bulges and disks are very different. Furthermore, the presence of dense gas on size scales of $\sim$ 500 pc in the centers of these nearby galaxies and the Milky Way suggests that the internal pressure is at least 10$^7$ cm$^{-3}$ K in their centers; this is some three orders of magnitude greater than the pressure in the local interstellar medium in the Milky Way, and it is two orders of magnitude greater than the pressure from the self-gravity of a solar neighborhood giant molecular cloud. In NGC 4826 and M51, as in the Milky Way and NGC 1068, there is a linear offset of $\sim$ 100 pc between the dense gas distribution and the peak of the radio continuum emission. We did not detect HCN towards three additional spiral galaxies.'
author:
- 'Tamara T. Helfer and Leo Blitz'
title: Synthesis Imaging of Dense Gas in Nearby Galaxies
---
Introduction
==============
Single dish millimeter observations of HCN and CS emission from nearby galaxies show that most spiral galaxies, not just starburst galaxies, have an appreciable amount of dense ($\sim$ 10$^5$ cm$^{-3}$) gas in their bulges (Helfer & Blitz 1993 and references therein). These observations are in good agreement with what is seen in the inner $r \sim 250$ parsecs of the Milky Way, where strong, diffuse emission from CS (Bally et al. 1987) and HCN (Jackson et al. 1996) is observed despite the moderate star formation rates there. (HCN and CS trace gas densities of $\ga$ 10$^5$ cm$^{-3}$ in galaxies; this is two orders of magnitude higher than the density required to excite CO.) The ubiquity of large-scale emission from HCN and CS over hundreds of parsecs in the centers of galaxies is surprising when compared with the known properties of giant molecular clouds (GMCs) in the solar neighborhood, which contain few and relatively small ($\la$ pc-scale) clumps of dense gas — and these only where stars are actively forming or where stars have formed very recently (i.e. where there is a local source of pressure, not common to the GMC as a whole).
In galaxies, single-dish millimeter wavelength beams typically cover tens to hundreds of GMC-sized diameters. As part of our program to measure the degree to which different galactic environments affect the properties of molecular clouds, we began a program with the Berkeley-Illinois-Maryland Association (BIMA) interferometer to survey eight nearby galaxies in HCN emission in order to measure the distribution and amount of dense gas on size scales of individual GMCs or small associations of GMCs. We consider these galaxies to be “normal” since they do not have prolific circumnuclear starbursts or active galactic nuclei; however, like most spiral galaxies, these galaxies have low-level nuclear line emission (Ho, Filippenko & Sargent 1993). In this paper, we present the results of the BIMA survey. We also present a comparison of the HCN emission from one of the sources, NGC 6946, with its CO J = 1-0 emission (Regan & Vogel 1995). A comparison with the CO emission from the other sources will be the topic of a future paper.
Observations
==============
BIMA Observations
-------------------
The sources and their coordinates are listed in Table 1. The sources were selected from the single-dish survey of Helfer & Blitz (1993) as galaxies with strong CO emission that were also detected in HCN with a single pointing using the NRAO 12 m telescope. The data were collected with the BIMA interferometer (Welch et al. 1996), which then comprised 6 antennas, between 1994 February 15 and 1995 May 02; the observations included data from up to three array configurations. For each source, the receivers were tuned to the redshifted frequency of the HCN J = 1-0 transition ($\nu$$_{\rm o}$ = 88.61 GHz). The data were processed using the MIRIAD package (Sault, Teuben, & Wright 1995). The time variations of the amplitude and phase gains were calibrated using short observations of nearby quasars every $\sim$ 30 minutes; a planet or strong quasar was observed to set the absolute flux scale as well as to calibrate the spectral dependence of the gains across the IF passband. The digital correlator was configured to achieve a maximum spectral resolution of 1.56 MHz (5.3 km s$^{-1}$). For some of the observations, there was an intermittent phase lock on one of the antenna receivers that was not discovered until after the observations; in these cases all baselines involving that antenna were eliminated from further data reduction.
For each source, the calibrated data were smoothed to 21.1 km s$^{-1}$ resolution, then gridded and Fourier transformed using natural weighting. The visibilities were also weighted by the inverse of the noise variance, which was determined from the system temperatures and from the gains of the individual antennas. We cleaned the maps using the standard Högbom algorithm. Maps of integrated intensity were made by summing those channels with emission after clipping the channel maps at a 1 $\sigma$ level. The beam sizes and noise levels of the final maps are listed in Table 2. The noise levels are somewhat underestimated relative to the formal uncertainties, since they were determined from the clipped moment maps. The absolute flux calibration in all maps is probably accurate to $\pm$ 30%.
Single-dish Observations of CO in NGC 6946
-------------------------------------------
We have also used BIMA to image CO in the centers of three of the galaxies (NGC 3628, NGC 4826, and M83) with detected HCN emission. A fourth galaxy, NGC 6946, has been imaged in CO using BIMA by Regan & Vogel (1995). The CO emission, unlike the HCN emission (see below), is significantly “resolved out” in the interferometric images of these galaxies; it is therefore necessary to fill in the zero-spacing flux with a single-dish telescope. We are currently still making these measurements for the first three sources; we therefore defer further discussion of the CO results for these galaxies to a future paper.
In the case of NGC 6946, we measured the CO short-spacing flux with the NRAO 12 m telescope[^1] and combined these data with Regan & Vogel’s BIMA map. We observed on 95 June 20–21 in the newly implemented “on-the-fly” (OTF) scheme at the NRAO 12 m (Emerson et al., in preparation) and covered a region roughly 5 on a side. We observed orthogonal polarizations using two 256 channel filterbanks, each with a spectral resolution of 2 MHz per channel. The data were gridded and a linear baseline removed from the resulting data cubes in AIPS; the data were then transferred to the MIRIAD package for further processing. Details of the single dish and interferometric data combination were very similar to those described in Helfer & Blitz (1995) for the case of NGC 1068.
Results
=========
We detected HCN in five of the eight galaxies: NGC 3628, NGC 4826, NGC 5194 (M51), NGC 5236 (M83), and NGC 6946; their images are presented in Figure \[hcnmaps\] along with optical images from the Digitized Sky Survey.[^2] The spectra from the positions of peak HCN emission in each of the detected sources are shown in Figure \[spectra\].
In each case, the HCN images show compact structure, with the detected emission confined to the central $\sim$ 500 pc diameter; the 5–10 FWHM synthesized beam sizes correspond to linear resolutions of 125–200 pc at the distances of the galaxies. While the HCN emission is compact, it appears resolved by the interferometer in each map. In order to investigate the effect of the lack of $uv$ sampling at small spatial visibilities, we compared the integrated intensities measured with the interferometer to the single-dish fluxes of Helfer & Blitz (1993). Table 3 lists the integrated intensities measured at BIMA along with those measured at the NRAO 12 m and the fraction of the single-dish flux recovered by the interferometer. With the possible exception of M51 (see $\S$ 3.1), the interferometer appears to recover all the single-dish flux measured by Helfer & Blitz. This means that the maps shown in Figure \[hcnmaps\] are a reliable representation of the distribution of the HCN emission, constrained of course by the usual limitation of the signal to noise ratios. Allowing for low-level, more diffuse emission detected at $< 2~\sigma$, we conclude that the HCN emission is confined to the central kiloparsec of the galaxies (it is perhaps slightly more extended in M51). The spatial extent of the HCN emission is in good agreement with what is seen in the Milky Way (Jackson et al. 1996) and in NGC 1068 (Helfer & Blitz 1995).
Individual Sources
--------------------
The galaxies in this survey were originally selected from a list of the brightest extragalactic CO emitters (see Helfer & Blitz 1993). The following are brief descriptions of the individual sources.
NGC 3628 – This galaxy is a member of the Leo triplet (along with NGC 3623 and NGC 3627). It is a nearly edge-on Sbc galaxy with a prominent and irregular dust lane. Its nuclear region contains a modest starburst (e.g. Condon et al. 1982, Braine & Combes 1992). The CO from NGC 3628 (Young, Tacconi, & Scoville 1983; Boissé, Casoli, & Combes 1987; and Israel, Baas, & Maloney 1990) is strongly peaked in the inner kiloparsec and has a similar extent and mass as that in the center of the Milky Way (Boissé et al. 1987). The HCN in NGC 3628 is resolved and appears elongated in the east-west direction. There is emission at the 2 $\sigma$ level to the northwest of the central source that may be associated with gas further along the major axis. The HCN, CO and radio continuum centers of NGC 3628 all peak some 21 to the southeast of the optical nucleus; however, the optical nucleus is heavily obscured and its position is highly uncertain (Boissé et al. 1987).
NGC 4826 – This Sab galaxy has been called variously the “Black Eye,” the “Evil Eye,” or somewhat more optimistically (Rubin 1994) the “Sleeping Beauty” galaxy for its conspicuous dust lane (see Sandage 1961; see also the cover of [*The Astronomical Journal*]{}, 1994, 107, 1). NGC 4826 has gained notoriety recently for the discovery that the inner kiloparsec-scale disk is counterrotating with respect to the larger scale rotation of the galaxy (Braun, Walterbos, & Kennicutt 1992; Rubin 1994; Braun et al. 1994). The CO emission (Casoli & Gerin 1993) is confined to the inner $r \sim
1\arcmin$. The HCN emission appears symmetric and centrally peaked in NGC 4826.
NGC 5194 (M51) – M51 is the prototypical grand design spiral and is one of the best studied galaxies in CO emission. Interferometric CO images of the nuclear region of M51 (e.g. Lo et al. 1987; Rand & Kulkarni 1990; and Adler et al. 1992; the last includes zero-spacing flux) show a notable lack of a single central concentration of CO, despite the strong molecular emission associated with the spiral arms in the nuclear region. M51 is the only galaxy of the five detected in HCN at BIMA that does not have a strong central concentration of CO. In contrast to the CO emission, the HCN emission does appear to be centrally concentrated, though there is a significant contribution to the total flux from low-level ($< 2~\sigma$) emission. We note that Kohno et al. (1996) mapped the central concentration as well as more extended structure in their HCN map of M51 observed with the Nobeyama Millimeter Array. The linear extent of the low-level ($< 2~\sigma$) HCN emission in the BIMA map is somewhat larger than those of the other four galaxies studied here, and the interferometer may have “resolved out” a nonnegligible contribution to the total flux in this source (Table 3). For M51, the flux of structures larger than $\sim$ 18 is attenuated by $\ga$ 50%.
NGC 5236 (M83) – This well-studied source is a grand design, Sc/SBb spiral galaxy with strong circumnuclear star formation within the central few hundred pc (e.g. Gallais et al. 1991). The HCN emission is strongest at the position of the peak of the radio continuum emission (Condon et al. 1982; Turner & Ho 1994); the condensations to the east and to the south of the strongest HCN emission are also apparent in the radio continuum. The synthesized beam of the interferometer is quite elongated because of the foreshortening of the north-south baselines toward this low declination source. The north-south elongation of the HCN emission is therefore almost certainly an artifact of the observations.
NGC 6946 – This late-type, grand design spiral galaxy contains a moderate starburst (Turner & Ho 1983). Within the inner 1.5 kpc diameter, the CO in this galaxy has a non-axisymmetric, north-south distribution that has been interpreted as a bar (Regan & Vogel 1995 and references therein). However, Regan & Vogel (1995) combined new CO and K-band observations and showed that the CO traces gas on the trailing side of spiral arms; their observations are consistent with what is expected for the gas and stellar response to a spiral density wave rather than a bar. The CO peaks up strongly in the inner 300 pc diameter of NGC 6946. The HCN emission is detected in this region and is resolved and slightly extended in an east-west direction, with an additional elongation towards the northwest (there is a similar northwest extension in the CO map of Regan & Vogel 1995). We discuss this source more fully in the following section.
$\bf\rm I_{HCN}/I_{CO}$ in NGC 6946
------------------------------------
The ratio of the 3 mm integrated intensities, $\rm I_{HCN}/I_{CO}$, may be used as a qualitative measure of the molecular gas density (e.g. Helfer & Blitz 1996). In order to determine any line ratio from interferometric measurements, one must first take into account the possibility that the flux measured with the interferometer is missing a significant contribution from large-scale structures in the maps. In NGC 6946, the interferometer recovers all the single-dish HCN flux to within the errors of the measurement (Table 3). For the CO, the Regan & Vogel (1995) BIMA map recovered about half the single-dish flux; we therefore modeled the short spatial frequency visibilities from the NRAO 12 m data ($\S$ 2.2) and combined these with the BIMA CO map. The CO distribution and flux in the resulting map did not change appreciably interior to r = 15; at larger radii, the most dramatic flux increases were distributed over radii from 20–50, though the shape of the structures remained about the same. With the fully-sampled CO map, we can now make a legitimate comparison of the HCN and CO intensities for this source.
To determine the ratio $\rm I_{HCN}/I_{CO}$ in NGC 6946, we convolved the CO map to match the resolution of the HCN image, and we converted both intensities to a main beam brightness scale ($\rm \int
T_{MB}$ $\Delta v$); the ratio was computed only for regions with detected HCN emission (I$_{\rm HCN} > 2.5~\sigma_{\rm mom}$, or r $\la$ 15). The resulting ratio map of I$_{\rm HCN}$/I$_{\rm CO}$ is presented in Figure \[n6946rat\]. Although the emissions from CO and HCN both peak at the center of NGC 6946, the distribution of the HCN/CO ratio is saddle-shaped, with the highest values to the east and west of the nucleus by about 7 (175 pc) and the lowest values to the northwest and southeast of the nucleus by about 5 (125 pc). At first, it seems surprising that the ratio does not rise monotonically to the central position. However, at the small size scales resolved by the interferometer ($r \approx 70$ pc), the characteristics of individual molecular clouds start to dominate the distribution, rather than the integrated effects of dozens of GMCs. It may be that the small-scale ratio is dominated by local effects from individual clouds; this effect is seen within the central few hundred pc of the Milky Way, where the I$_{\rm CS}$/I$_{\rm CO}$ ratio (Helfer & Blitz 1993) and the I$_{\rm HCN}$/I$_{\rm CO}$ ratio (Jackson et al. 1996) look very clumpy and irregularly distributed. In the Milky Way, features like the Sgr A and B GMCs are characterized by relatively high values in the ratio maps.
The average ratio over the extent of the HCN emission ($\sim 12\arcsec$, or 300 pc diameter) in NGC 6946 is $0.11 \pm 0.01$, with a peak value of 0.19 and a minimum of 0.049. The range of I$_{\rm HCN}$/I$_{\rm CO}$ in NGC 6946 is very similar to what is seen in the Milky Way by Jackson et al. (1996); on small scales within the inner few degrees of the Milky Way, I$_{\rm HCN}$/I$_{\rm CO}$ ranges from 0.04 to 0.12, and the average over the extent of the HCN emission, or r $\approx$ 300 pc, is 0.08 (see below).
We can compare the ratio we measure in the central 300 pc of NGC 6946 with those measured with larger apertures: the single dish ratios of I$_{\rm HCN}$/I$_{\rm CO}$ in NGC 6946 are 0.063 $\pm$ 0.007 at a resolution of 24 (600 pc) (Nguyen-Q-Rieu et al. 1989; Weliachew, Casoli, & Combes 1988) and 0.025 $\pm$ 0.003 at a resolution of $\sim 1\arcmin$ (1500 pc) (Helfer & Blitz 1993). Figure \[n6946.ratio\_vs\_r\]$a$ shows this radial distribution of the average integrated I$_{\rm HCN}$/I$_{\rm CO}$ ratio. It is important to note that the points shown in Figure \[n6946.ratio\_vs\_r\]$a$ represent the [*average*]{} ratios over the area enclosed at the radius $r$, i.e. that the plot represents the [*integrated*]{} ratios I$_{\rm HCN}$/I$_{\rm CO}$ as a function of $r$. The monotonic falloff in the ratio with radius is simply a result of the confinement of the HCN emission to the inner $r$ = 150 pc, while the CO is distributed over a much larger radius (there is detectable emission at least to $r$ = 3.5 kpc, Tacconi & Young 1989). What is perhaps a more interesting quantity physically is the [*annular*]{} ratio measured as a function of radius; that is, if I$_{\rm HCN}$/I$_{\rm CO}$ is $0.10 \pm 0.01$ measured as an average from $0 < r < 150$ pc, what is the value of I$_{\rm HCN}$/I$_{\rm CO}$ from $150 < r < 800$ pc (where 800 pc is the radius of the NRAO beam in Helfer & Blitz 1993)? We can set an upper limit to I$_{\rm HCN}$/I$_{\rm CO}$ in this annulus by comparing the BIMA data with the NRAO 12 m HCN flux. Since the BIMA measurement recovered $0.81 \pm 0.17$ of the single-dish HCN flux measured at the NRAO 12 m (Table 3), let us assume that 20% of the single-dish HCN flux is distributed at radii larger than the interferometer was able to measure, yet within the half power beam area of the the NRAO 12 m — that is, radii within the annulus $150 < r < 800$ pc. (This is a conservative estimate, since any “missing” large-scale flux could also contribute to the flux at the central position.) If we then measure the flux in the Regan & Vogel CO map from $150 < r < 800$ pc, we find that the ratio I$_{\rm HCN}$/I$_{\rm CO}$ in this annulus is at most 0.01. This ratio is an order of magnitude lower than that measured over the central $r = 150$ pc, as shown in Figure \[n6946.ratio\_vs\_r\]$b$.
Nondetections
---------------
We did not detect HCN emission from NGC 4321 (M100), NGC 4527, or NGC 4569. While it is possible that the dense structure in these galaxies is so extended that the interferometer resolves out the single-dish HCN emission (for these sources, the flux of structures larger than $\sim$ 20 is attenuated by $\ga$ 50%), it is also likely that the observations simply were not sensitive enough to detect the HCN from these sources. For those observations which suffered from an intermittent phase lock (see $\S$ 2.1), the antenna that was flagged was one of the two that made up the shortest baseline pair; thus the calibration solution may not have been reliable and also the zero spacing problem may have been exacerbated for these observations.
Discussion
============
The Radial Dependence of Dense Gas Ratios
-------------------------------------------
Spectroscopic studies of CS and HCN emission in normal external galaxies suggest that most spiral galaxies contain an appreciable amount of gas at densities of $\rm \ga 10^5 ~cm^{-3}$ in their centers (Mauersberger et al. 1989; Sage, Solomon, & Shore 1990; Nguyen-Q-Rieu et al. 1992; Israel 1992; Helfer & Blitz 1993). The maps in Figure 1 show the distribution of that dense gas, namely, that it is confined to the central kiloparsec of the sources imaged. This situation appears to be very similar to that seen in the Milky Way, where widespread emission from the dense gas tracers CS (Bally et al. 1987) and HCN (Jackson et al. 1996) is found only within the central $\sim$ 500 pc diameter.
What does the distribution of dense gas tell us about the physical conditions in the molecular gas as a function of its location in a galaxy? To investigate the physical conditions rather than the total gas content, we normalize the HCN to that of the CO and consider the ratio $\rm I_{HCN}/I_{CO}$. If the kinetic temperature of the gas responsible for the cospatial HCN and CO emissions is about the same, then $\rm I_{HCN}/I_{CO}$ may be considered as a qualitative indicator of the density or thermal pressure in the gas. (Indeed, since the J = 1 state lies only 5.5 K above the ground state for CO and 4.3 K above ground for HCN, even rather cold gas has the energy to populate the J = 1 state for both molecules. It is the molecular density that is more important in determining the excitation. See Helfer & Blitz 1996.) In NGC 6946 ($\S$ 3.2), $\rm I_{HCN}/I_{CO}$ is 0.11 $\pm$ 0.01 averaged over the central $r < 150$ pc, whereas we deduce an upper limit of $\rm I_{HCN}/I_{CO} \le 0.01$ averaged in the annulus $150 < r < 800$ pc, a region that includes the inner part of the disk. A similar radial dependence of $\rm I_{HCN}/I_{CO}$ is seen in the unusual Seyfert/starburst hybrid galaxy NGC 1068, where the ratio approaches 0.6 in the central $r$ = 175 pc (Helfer & Blitz 1995), and the ratio falls off monotonically to about 0.1 at the large reservoir of molecular gas at about 1 kiloparsec from the nucleus. In the Milky Way, the ratio $\rm I_{HCN}/I_{CO}$ is about 0.081[^3] $\pm$ 0.004 averaged over the central $r = 315$ pc (Jackson et al. 1996); between $3.5 < r < 7$ kpc in the plane of the Milky Way, the average ratio is $\sim$ 0.026 $\pm$ 0.008, and in solar neighborhood GMCs, we measure $\rm I_{HCN}/I_{CO}$ ratios of 0.014 $\pm$ 0.020 when averaged over $\sim$ 50 pc GMCs (Helfer 1995 – and these are upper limits to the ratio averaged over several hundred parsecs). These numbers are summarized in Table 4.
What seems apparent from these comparisons is that $\rm I_{HCN}/I_{CO}$ is a strong function of galactocentric radius – or more precisely, that there is at least a bimodal distribution in the ratio in normal galaxies: the ratio at the center is substantially higher than elsewhere in the galaxy. Even though local effects can dominate on scales of individual GMCs ($\S$ 3.2), the average ratio of dense gas emission is highest at a galaxy’s center and drops at larger distances from the center. Furthermore, the general agreement between the ratios in the central $\sim$ 500 pc of NGC 6946 and the Milky Way suggest that the physical conditions in the centers of the two galaxies are similar.
GMCs in the High Pressure Environments of Galactic Bulges
----------------------------------------------------------
The measurement that the ratio $\rm I_{HCN}/I_{CO}$ is 5 to 10 times higher in the bulge regions of the Milky Way and NGC 6946 than in their disks suggests that the physical conditions of the molecular gas in the bulge and disk regions are very different. Furthermore, the presence of dense gas on size scales of $\sim$ 500 pc suggests that the internal pressure is very high in the molecular gas. Let us assume that the intrinsic line widths of the clouds are dominated by nonthermal, bulk motions as in local clouds, and that their intrinsic linewidths are $\ge$ 1 km s$^{-1}$. Although we cannot model the density accurately with the observation of a single transition of HCN or CO, a simple LVG analysis of the line ratios suggests that a line ratio of $\rm I_{HCN}/I_{CO}$ = 0.11 implies densities of 10$^{4.2-5.2}$ cm$^{-3}$ for gas at a kinetic temperature in the range T$_{\rm K}$ = 15 – 70 K. These densities are consistent with what is measured in the Milky Way bulge molecular clouds, where n(H$_2$) $\ga 10^4 \rm ~cm^{-3}$ (Güsten 1989). If we take the gas densities in NGC 6946 to be $\rm 10^{4.6} ~cm^{-3}$ , then the typical kinetic pressure throughout the molecular gas is $\rho v^2/k \sim 1 \times 10^7$ cm$^{-3}$ K.
Spergel & Blitz (1992) considered the effects of the extended, hot coronal gas in the bulge of the Milky Way on the thin molecular layer that is embedded within it, and they argued that the pressure in the center of the Galaxy is two to three orders of magnitude greater than that in the solar neighborhood. How does such an extraordinary difference in the environmental pressure affect the properties of molecular clouds? In disk GMCs, the external pressure of the ambient ISM ($\rm \sim
10^4 ~cm^{-3}$ K) is small compared with the pressure from the self-gravity of a cloud ($\rm \sim 10^5 ~cm^{-3}$ K), and a source of “local” pressure (i.e., ongoing or recent star formation) is required to support any localized high-density clumps within the cloud. In the high-pressure ($\rm \sim 5 \times 10^6 ~cm^{-3}$ K, estimated for the Milky Way from the X-ray measurements of Yamauchi et al. 1990) environments of bulge molecular gas, on the other hand, the GMCs need not be self-gravitating – in fact, only the most massive clouds [*could*]{} have the Jeans masses required to be self-gravitating. In bulge clouds, then, it is the external pressure of the environment that dominates, and these high pressures can support the dense gas throughout the molecular component (regardless of whether there is any active star formation in the GMCs).
These arguments are easily extended to observations of external galaxies, which typically also have X-ray emission associated with their bulges (Fabbiano, Kim, & Trinchieri 1992). From our observations of HCN in the bulges of external galaxies, it appears that the physical conditions in molecular gas in the centers of galaxies are much more similar to each other than they are to local GMCs in the Milky Way.
Positional Offsets Between the Dense Gas and Radio Continuum Distributions
----------------------------------------------------------------------------
In the Milky Way, there is a pronounced offset of $\sim$ 80 pc in the position of the peak radio continuum emission (Sgr A\*, $l = 0\arcdeg, b = 0\arcdeg$) and the centroid of the dense gas (traced by CS and HCN) distribution ($l =
0.6\arcdeg, b = 0\arcdeg$). There is also evidence for an offset of $\sim$ 100 pc in NGC 1068 between the peaks of the HCN emission and the radio continuum emission (Helfer & Blitz 1995), and a kinematic analysis of the molecular gas suggests the existence of an $m = 1$ mode (i.e. a dipole asymmetry in the mass distribution) in NGC 1068. Whether it is the radio continuum emission or the molecular gas that traces the center of the mass distribution, the dynamical timescale of the offsets is brief ($<$ 10$^6$ years). If the offsets are indeed ephemeral, and not a steady-state condition of the galaxies, then it is appropriate to look for a source of the instability that causes them.
How common are such offsets between the peaks of the dense molecular gas distribution and the radio continuum emission in galaxies? We compared the positions of peak HCN emission with those of the peak radio continuum in our sample (Figure 1, Table 5). While three of the galaxies show a reasonable coincidence between the two positions, two of the five sources, NGC 4826 and M51, show significant offsets: in NGC 4826, the offset is $3\arcsec.7$ or 74 pc; in M51, it is $2\arcsec.6$ or 130 pc. (In M83, there is a $1\arcsec.6$ or 40 pc offset, but because of the elongated beams and extended distribution in both HCN and the radio continuum, the peak positions are less certain.)
It appears that these offsets are a common feature in spiral galaxies. In this sample, two out of five detected galaxies have offsets; we have already mentioned the offsets in the Milky Way and in NGC 1068. In a recent study of the K-band morphology in 18 face-on spiral galaxies, Rix & Zaritsky (1995) found that about one third of the disks have significant $m = 1$ modes at 2.5 disk exponential scale lengths.
It may be significant that the two galaxies in our sample that show offsets, NGC 4826 and M51, also share the characteristic that they have low-level nonstellar nuclear activity (both are LINERs; these are low-level active galactic nuclei, or AGN); the other three galaxies detected here (NGC 3628, M83, and NGC 6946) have low-level starburst activity instead (L. Ho, private communication). A recent study by Ho et al. (1993) suggests that up to 80% of the 500 brightest galaxies in the northern sky harbor some kind of activity in their nuclei; of these, about half are classified as LINERs, and half are starbursts. It is surprising both that [*most*]{} spiral galaxies appear to show some kind of nuclear activity and also that the activity seems to be roughly evenly divided between stellar and nonstellar mechanisms. The offsets of the dense gas from the peak of the radio continuum in NGC 4826 and M51, as well as that in NGC 1068 (a galaxy with a more energetic AGN, but similar conceptually to the LINERs) may help to distinguish empirically the kind of activity that dominates in a given galaxy’s nucleus.
Conclusions
=============
We have presented the results of a survey of HCN emission from eight nearby spiral galaxies made with the BIMA interferometer. These observations comprise the first high-resolution survey of dense ($\sim 10^5$ cm$^{-3}$) gas from a sample of relatively normal galaxies, rather than galaxies with prolific starburst or nuclear activities.
We imaged five of the eight galaxies in HCN: NGC 3628, NGC 4826, NGC 5194 (M51), NGC 5236 (M83), and NGC 6946. To within the uncertainties, the interferometer recovers all of the single-dish flux measured for each source in a single pointing at the NRAO 12 m telescope (Helfer & Blitz 1993); this implies that there is not a significant contribution to the HCN fluxes from extended emission in the disks of the galaxies. In all the galaxies observed, the HCN emission is confined to the central kiloparsec of the sources.
We added zero-spacing data from the NRAO 12 m telescope to the BIMA CO map of NGC 6946 by Regan & Vogel (1995) in order to compare the ratio of HCN to CO intensities in this galaxy. The ratio $\rm I_{HCN}/I_{CO}$ ranges from 0.05–0.2 within the central $r = 150$ pc; the average ratio over this region is $\rm I_{HCN}/I_{CO} = 0.11 \pm 0.01$. A comparison with single-dish observations allows us to place an upper limit of $\rm I_{HCN}/I_{CO}
\le 0.01$ in the annulus $150 < r < 800$ pc in NGC 6946.
The extent of HCN emission in NGC 6946, $r \sim 150$ pc, and the ratio $\rm
I_{HCN}/I_{CO} = 0.11$ in this region are similar to what is observed in the Milky Way ($r \sim 250$ pc, $\rm I_{HCN}/I_{CO} = 0.08$, Jackson et al. 1996); this suggests that the physical conditions in the centers of these two galaxies are similar. Furthermore, in NGC 6946, NGC 1068, and the Milky Way, the ratios at the centers are 5 to 10 times higher than those in the disks of these galaxies. This result is consistent with an enhancement of two to three orders of magnitude in the pressure of the bulges compared with the disks (Spergel & Blitz 1992). It appears that the physical regions in the centers of galaxies are much more like each other than the conditions in the center of a galaxy relative to its disk.
In NGC 4826 and M51, as in the Milky Way and in NGC 1068, there is a linear offset of $\sim$ 100 pc between the dense gas distribution and the peak of the radio continuum emission. These offsets appear to be a common feature in galaxies and may indicate that their disks are non-axisymmetric. It may be significant that of the five galaxies imaged in HCN, NGC 4826 and M51 are LINERs, whereas the other three are starbursts.
We did not detect HCN in three galaxies with positive single-dish HCN emission: NGC 4321 (M100), NGC 4527, and NGC 4569. It could be that the emission is extended enough in these sources that the interferometer resolved out any detectable emission; however, we cannot rule out the possibility that there was some intrinsic problem with the observations of these sources.
We thank the referee, Paul Ho, for his careful reading and suggestions; these helped us to improve the manuscript. We thank Mike Regan for providing us with the BIMA NGC 6946 CO data, and we thank Darrel Emerson, Phil Jewell, Tom Folkers and the staff of the NRAO 12 m telescope for assistance with the OTF observations and data reduction. Luis Ho helped with the early stages of the BIMA observations. We thank Kotaro Kohno for kindly providing us with the NRO map of HCN in M51 prior to publication. TTH thanks Jack Welch for hospitality while visiting UC-Berkeley. This research was partially supported by a grant from the National Science Foundation, with additional support from the State of Maryland.
Adler, D.S., Lo, K.Y., Wright, M.C.H., Rydbeck, G., Plante, R.L., & Allen, R.J. 1992, , 392, 497 Bally, J., Stark, A.A., Wilson, R.W., & Henkel, C. 1987, , 65, 13 ———–. 1988, , 324, 223 Boissé, P., Casoli, F., & Combes, F. 1987, , 173, 229 Braine, J. & Combes, F. 1992, , 264, 433 Braun, R., Walterbos, R.A.M., & Kennicutt, R.C. 1992, Nature, 360, 442 Braun, R., Walterbos, R.A.M., Kennicutt, R.C., & Tacconi, L.J. 1994, , 420, 558 Casoli, F. & Gerin, M. 1993, , 279, L41 Condon, J.J., Condon, M.A., Gisler, G., & Puschell, J.J. 1982, , 252, 102 Fabbiano, G., Kim, D.-W., & Trinchieri, G. 1992, , 80, 531 Gallais, P., Rouan, D., Lacombe, F., Tiphene, D., & Vauglin, I. 1991, , 243, 309 Helfer, T.T. 1995, Ph.D. Thesis, U. Maryland Helfer, T.T. & Blitz, L. 1993, , 419, 86 —————. 1995, , 450, 90 Ho, L.C., Filippenko, A.V., & Sargent, W.L.W. 1993, in IAU Symp. 159, Multi-Wavelength Continuum Emission of AGN, eds. A. Blecha & T.J.-L. Courvoisier (Dordrecht:Reidel), 275 Israel, F. P. 1992, , 265, 487 Israel, F.P., Baas, F., Maloney, P.R. 1990, , 237, 17 Jackson, J.M., Heyer, M.H., Paglione, T.A.D., & Bolatto, A.D. 1996, , 456, 91 Kohno, K., Kawabe, R., Tosaki, T., & Okumura, S.K. 1996, , in press Lo, K.Y., Ball, R., Masson, C.R., Phillips, T.G., Scott, S., and Woody, D.P. 1987, , 317, L63 Mauersberger, R., Henkel, C., Wilson, T. L., & Harju, J. 1989, , 226, L5 Nguyen-Q-Rieu, Jackson, J. M., Henkel, C., Truong-Bach, & Mauersberger, R. 1992, , 399, 521 Rand, R.J. & Kulkarni, S.R. 1990, , 349, L43 Regan, M., & Vogel, S.N. 1995, , 452, 21 Rix, H.-W. & Zaritsky, D. 1995, , 447, 82 Rubin, V.C. 1994, , 107, 173 Sage, L. J., Shore, S. N., & Solomon, P. M. 1990, , 351, 422 Sandage, A. 1961, The Hubble Atlas of Galaxies (Washington: Carnegie Institute of Washington) Sault, R.J., Teuben, P.J., & Wright, M.C.H. 1995, in Astronomical Data Analysis Software and Systems IV, eds. R.A. Shaw, H.E. Payne, & J.J.E. Hayes, A.S.P. Conference Series 77, 433 Spergel, D.N. & Blitz, L. 1992, Nature, 357, 665 Tacconi, L.J. & Young, J.S., 1989, , 71, 455 Turner, J.L. & Ho, P.T.P. 1983, , 268, L79 Turner, J.L. & Ho, P.T.P. 1994, , 421, 122 Welch, W.J., et al. 1996, , 108, 93 Weliachew, L., Casoli, F., & Combes, F. 1988, , 199, 29 Yamauchi, S., Kawada, M., Koyama, K., Kunieda, H., & Corbet, R.H.D. 1990, , 365, 532 Young, J.S., Tacconi, L.J., & Scoville, N.Z. 1983, , 269, 136
[lrrrr]{}
NGC 3628 & 11$^{\rm h}$20$^{\rm m}$16.$^{\rm s}$27 & 13$^{\arcdeg}$35$^{\arcmin}$39.$^{\arcsec}$0 & 9 & 847 NGC 4321 (M100)& 12 22 54.80 & 15 49 23.0 & 17 & 1550 NGC 4527 & 12 34 08.80 & 02 39 10.3 & 20 & 1734 NGC 4569 & 12 36 50.02 & 13 09 53.1 & 17 & -235 NGC 4826 & 12 56 44.25 & 21 40 52.3 & 5 & 408 NGC 5194 (M51) & 13 29 53.30 & 47 11 50.0 & 10 & 463 NGC 5236 (M83) & 13 37 00.23 & -29 52 04.5 & 5 & 516 NGC 6946 & 20 34 51.91 & 60 09 11.9 & 5 & 52
[lrrr]{}
NGC 3628 & 4.7 $\times$ 4.1 & 8.13 & 2.0 NGC 4321 (M100)& 8.2 $\times$ 6.5 & 2.91 & 2.1 NGC 4527 & 13.5 $\times$ 10.7 & 1.07 & 3.6 NGC 4569 & 5.5 $\times$ 4.2 & 6.85 & 1.0 NGC 4826 & 10.5 $\times$ 8.0 & 1.87 & 3.2 NGC 5194 (M51) & 7.9 $\times$ 6.5 & 3.03 & 1.5 NGC 5236 (M83) & 12.5 $\times$ 4.1 & 3.03 & 3.6 NGC 6946 & 5.9 $\times$ 5.0 & 5.31 & 1.7
[lrrr]{}
NGC 3628 & 49.0 $\pm$ 17.5 & 1.5 $\pm$ 0.2 & 1.01 $\pm$ 0.36 NGC 4826 & 39.0 $\pm$ 16.7 & 1.3 $\pm$ 0.2 & 0.93 $\pm$ 0.41 NGC 5194 (M51) & 26.4 $\pm$ 10.5 & 1.2 $\pm$ 0.2 & 0.68 $\pm$ 0.27 NGC 5236 (M83) & 103 $\pm$ 23 & 2.7 $\pm$ 0.2 & 1.18 $\pm$ 0.27 NGC 6946 & 39.2 $\pm$ 5.5 & 1.5 $\pm$ 0.2 & 0.81 $\pm$ 0.17
[lccc]{} Bulge& 0.081 $\pm$ 0.004 & 0.11 $\pm$ 0.01 & 0.6 Disk & 0.026 $\pm$ 0.008 & $\le$ 0.01 & 0.1 Local GMCs & 0.014 $\pm$ 0.020 & — & —
[lr]{} NGC 3628 & -0.5, -0.1 NGC 4826 & +2.2, -3.0 NGC 5194 (M51) & -2.2, +1.3 NGC 5236 (M83) & -1.3, -0.9 NGC 6946 & -0.1, -0.5
[^1]: The National Radio Astronomy Observatory is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation.
[^2]: Based on photographic data of the National Geographic Society – Palomar Observatory Sky Survey (NGS-POSS) obtained using the Oschin Telescope on Palomar Mountain. The NGS-POSS was funded by a grant from the National Geographic Society to the California Institute of Technology. The plates were processed into the present compressed digital form with their permission. The Digitized Sky Survey was produced at the Space Telescope Science Institute under US Government grant NAG W-2166.
[^3]: See note $c$ to Table 4.
|
---
abstract: 'We study the gravitational Dirichlet problem in AdS spacetimes with a view to understanding the boundary CFT interpretation. We define the problem as bulk Einstein’s equations with Dirichlet boundary conditions on fixed timelike cut-off hypersurface. Using the fluid/gravity correspondence, we argue that one can determine non-linear solutions to this problem in the long wavelength regime. On the boundary we find a conformal fluid with Dirichlet constitutive relations, viz., the fluid propagates on a ‘dynamical’ background metric which depends on the local fluid velocities and temperature. This boundary fluid can be re-expressed as an emergent hypersurface fluid which is non-conformal but has the same value of the shear viscosity as the boundary fluid. The hypersurface dynamics arises as a collective effect, wherein effects of the background are transmuted into the fluid degrees of freedom. Furthermore, we demonstrate that this collective fluid is forced to be non-relativistic below a critical cut-off radius in AdS to avoid acausal sound propagation with respect to the hypersurface metric. We further go on to show how one can use this set-up to embed the recent constructions of flat spacetime duals to non-relativistic fluid dynamics into the AdS/CFT correspondence, arguing that a version of the membrane paradigm arises naturally when the boundary fluid lives on a background Galilean manifold.'
author:
- |
Daniel Brattan$^a$[^1], Joan Camps$^a$[^2], R. Loganayagam$^b$[^3], Mukund Rangamani$^a$[^4]\
\
,\
\
,\
title: |
[**CFT dual of the AdS Dirichlet problem:\
Fluid/Gravity on cut-off surfaces**]{}
---
(0,0)(0,0) (380, 330)[DCPT-11/25]{}
Introduction {#s:intro}
============
The AdS/CFT correspondence [@Maldacena:1997re] which postulates a remarkable duality between large $N$ quantum field theories and gravitational dynamics, provides a useful theoretical laboratory to address questions underlying the dynamics of these systems. Not only has it proven useful to obtain quantitative information about the dynamics of strongly coupled field theories, but it also provides a unique perspective into the geometrization of field theoretic concepts.
Since the early days of the AdS/CFT correspondence it has been known that the radial direction of the bulk spacetime encodes in some sense the energy scale of the dual field theory [@Susskind:1998dq]. While the nature of this map is not terribly precise outside of the simple example of pure geometry (dual to the vacuum state of the field theory), it nevertheless provides valuable intuition about certain basic aspects of effective field theory dynamics [@Banks:1998dd; @Peet:1998wn], and has led to the idea of the holographic renormalisation group [@deBoer:1999xf], which relates the radial ‘evolution’ in AdS to RG flows in field theories. More recently this idea has been exploited to geometrize Wilson’s concept of integrating out momentum shells to generate field theory effective actions, in terms of integrating out regions of the bulk geometry which in turn lead to effective multi-trace boundary conditions on the cut-off surface, a fixed radial slice (in some preferred foliation) in AdS [@Heemskerk:2010hk; @Faulkner:2010jy]. One of the key features of this holographic Wilsonian approach was the emergence of multi-trace deformations of the field theory even in the planar limit, consistent with field theory expectation.
A natural question in this context is what does this RG flow mean for the gravitational equations of motion? More precisely, consider the problem of integrating out radial geometric shells in Einstein gravity with negative cosmological constant (which is a consistent truncation of string theory/supergravity). One anticipates based on the standard dictionary which relates the bulk metric to the boundary energy momentum tensor to obtain a scale dependent effective action for the energy momentum tensor, containing arbitrarily high multi-traces of the stress tensor. The reason for the generation of these multi-traces is clear, once one factors in the intrinsic non-linearity of gravity. The basic equations in this context are of course easy to write down; as explained in [@Heemskerk:2010hk; @Faulkner:2010jy] the flow is driven by the radial ADM Hamiltonian and one can in principle solve the resulting Hamilton-Jacobi like equation for the effective action on the cut-off hypersurface. Despite the conceptual simplicity of the formulation of Wilsonian RG in terms of geometric effective actions, the point still remains that gravity’s intrinsic non-linearity makes explicit solutions hard to come by.
One can ask whether there is a tractable sector of the gravitational flow equations which leads to new insight. A natural avenue for exploration is suggested by the long-wavelength regime where we restrict attention to fluctuations of low frequency in the field theory directions. As evidenced by the fluid/gravity correspondence [@Bhattacharyya:2008jc] there is an essential simplification in this regime; bulk Einstein’s equations can be explicitly solved order by order in a long-wavelength expansion along the boundary.[^5] As such one should be able to use this framework in conjunction with the fluid dynamical expansion to derive an effective action for the low frequency degrees of freedom which live on a cut-off surface in the interior of the AdS spacetime.[^6] Rather than tackle this problem directly we will take a slightly different tack in this paper, one which we believe clarifies some aspects of evolution in the radial direction and its possible connection to RG flows. One of our main conclusions will be that imposing rigid cut-offs in AdS is more naturally viewed in terms of perturbing the CFT by some non-local deformation or equivalently by introducing explicit state-dependent sources in the boundary theory.
A second motivation is the recent work [@Bredberg:2010ky; @Bredberg:2011jq] which derives an explicit map between solutions of vacuum Einstein equations (with no cosmological constant) and those of incompressible Navier-Stokes equations, thereby making direct contact with some of the ideas of the black hole membrane paradigm in asymptotically flat spacetime [@Damour:1978cg; @Thorne:1986iy]. This problem, which has been further generalized in [@Compere:2011dx], is the zero cosmological constant analog of the problem we consider (see also [@Eling:2009pb; @Eling:2009sj] for another approach and [@Cai:2011xv] for related work). The idea is to consider a fixed timelike hypersurface with Dirichlet data enforcing a flat metric on the slice. Given these boundary conditions one wants to solve vacuum Einstein equations so as to obtain a solution which has a regular future horizon.[^7] By explicit construction which involves long wavelength fluctuations around flat space in a Rindler patch the authors of [@Bredberg:2011jq; @Compere:2011dx] construct solutions to vacuum Einstein’s equations order by order in a perturbation expansion in gradients along the hypersurface directions. The resulting geometry has a regular Rindler horizon, and one obtains a regular solution to Einstein’s equations contingent on the fact that dynamics of the induced stress tensor on the hypersurface satisfies the incompressible Navier-Stokes equations.
While this development is fascinating, one is hampered from a first principles understanding of the physics from a holographic viewpoint, owing to the rather poorly understood concepts of flat space holography. Moreover, given the connection between fluid dynamics (albeit relativistic and conformal) and Einstein’s equations with negative cosmological constant as described by the aforementioned fluid/gravity correspondence [@Bhattacharyya:2008jc](and its non-relativistic extension in [@Bhattacharyya:2008kq] ), it is interesting to ask whether the construction in [@Bredberg:2011jq] can be obtained as a limit of the fluid/gravity map. If this is possible, one can then look for the field theoretic interpretation of the flat space problem.
Motivated by these issues, we consider a region of the AdS spacetime bounded by a timelike hypersurface $\Sigma_D$ at some radial position, say $r = r_D$ in the supergravity limit of AdS/CFT. We are interested in solving for the bulk dynamics where will give ourselves the freedom to specify boundary conditions on $\Sigma_D$. The second boundary condition (which is necessary to zero-in onto a unique solution) will be specified by demanding regularity in the interior of the spacetime. We have schematically depicted the set-up in . In the large $N$ limit, the specification of the problem thus is tantamount to solving classical partial differential equations (PDEs) in AdS with a Dirichlet boundary condition imposed on various fields at the hypersurface $\Sigma_D$. The question we would like to know the answer to is simply: “What is the problem that we are solving in the CFT language?”
![Schematic representation of the Dirichlet problem we consider in this paper. The Dirichlet surface is taken to be at some value $r = r_D$ where we impose boundary conditions on the fields. The solutions will further be constrained by requiring that they be regular on any putative horizon ${\mathcal H}^+$ (shown in the figure) or the origin. The question we are after is what is the boundary image of this Dirichlet data?[]{data-label="f:setup"}](Dir-schema)
(0,0) (-4.8,1.36)[$\Sigma_D$]{} (-8.4,2.3)[${\mathcal H}^+$]{} (-5.24,-0,36)[data = $\hat{\mathfrak X}$]{} (-1.7,-0.36)[data = ${\mathfrak X}$]{}
As we have reviewed above, various results exist in literature that suggest that solving such a Dirichlet problem is analogous to some kind of RG from the CFT point of view. Despite the strongly suggestive nature of this holographic RG point of view, it is also not very clear what kind of an RG is one speaking of within a CFT. A-priori, for one, it does not seem like the RG flow that arises from cutting off a CFT a la Wilson is the correct way to dualize the Dirichlet problem. Hence, our question - what is the CFT dual of a bulk Dirichlet problem?[^8]
As a warm-up we first consider the bulk Dirichlet problem for linear PDEs, using the simple setting of a Klein-Gordon field propagating in a cut-off AdS spacetime. In this case it is not hard to see that one is deforming the field theory by a non-local double-trace operator, whose precise form, we argue, can be extracted by suitable convolution of appropriate bulk-to-boundary propagators.
We then turn to the issue of setting up the problem in a gravitational setting, outlining it in general before moving on to the tractable setting of the fluid/gravity regime. In the long wavelength regime we will argue that the bulk Dirichlet problem reduces to a particular forcing of the fluid on the boundary of the asymptotically AdS spacetime. The fluid/gravity correspondence was generalized to fluids propagating on curved backgrounds with slowly varying curvatures in [@Bhattacharyya:2008ji] and the most general solutions which will prove to be of interest to us were presented in [@Bhattacharyya:2008mz]. Using these results it transpires that we can immediately write down the solution to the bulk Dirichlet problem in the long wavelength regime.
The logic is the following: we wish to prescribe on the hypersurface $r =r_D$ a Lorentzian metric which we denote as ${\hat g}_{\mu\nu}$. This is arbitrary subject to the requirement that its curvatures be slowly varying so that we can treat it with the fluid/gravity perturbation scheme. We then solve Einstein’s equations demanding regularity in the interior of the spacetime. Using standard intuition from the AdS/CFT correspondence it can be argued that the seed geometry which we need to set up the gradient expansion should simply be a black hole geometry which has a regular future event horizon, which furthermore satisfies the prescribed Dirichlet boundary condition.[^9] It is not hard to see that such a seed solution is obtained by simply performing a coordinate transformation of the well known planar Schwarzschild-AdS black hole.
But this is precisely the set-up of [@Bhattacharyya:2008ji; @Bhattacharyya:2008mz], the only difference being the fact that in these works the Dirichlet data is imposed on the boundary at $r =\infty$. Let’s call this boundary metric $g_{\mu\nu}$, which is also by definition slowly varying etc.. The solution to the asymptotic Dirichlet problem is characterized by the boundary metric $g_{\mu\nu}$, a distinguished velocity field $u^\mu$ (which is unit normalized) and a scalar function $b$ (determining the temperature or equivalently the local energy density). Let us denote these variables collectively as ${\mathfrak X}$. The boundary Brown-York stress tensor (up to counter-terms) takes the fluid dynamical form and is built out of the data contained in ${\mathfrak X}$.
Now given the space of solutions to the asymptotic Dirichlet problem, we can reparametrize that space of solutions appropriately to obtain the solutions of the new bulk Dirichlet problem. The only condition we have to satisfy is that the induced metric on $\Sigma_D$ in the solutions obtained this way[^10] be equal to ${\hat g}_{\mu\nu}$. Furthermore, we can extract the stress tensor on $\Sigma_D$[^11]. We will argue that there is a corresponding velocity field ${\hat u}^\mu$ (normalized with respect to the hypersurface metric) and a scalar function (the hypersurface temperature), which parameterize the stress tensor of the hypersurface, which not surprisingly takes the fluid dynamical form. The main novelty is that the stress tensor does not however correspond to that of a conformal fluid. The introduction of an explicit scale by way of the Dirichlet surface’s location engenders a non-vanishing trace, which curiously evolves in a highly suggestive manner under change of cut-off surface position, see .
Calling the totality of the data on the hypersurface ${\hat {\mathfrak X}}$ we further show that within the gradient expansion there is a one-to-one correspondence between the hypersurface data and the boundary data; $\varphi_D: {\mathfrak X} \to {\hat {\mathfrak X}}$ is bijective. This then has the advantage that we can immediately understand the boundary dual of the bulk Dirichlet problem as a conformal fluid which is placed on a[^12] ‘dynamical background’ whose metric depends on the same set of variables that characterizes the fluid itself (in addition to the prescribed hypersurface metric). So from the boundary viewpoint there is a complete mixing between intrinsic and extrinsic data, which is the long-wavelength non-linear analog of the double trace deformation seen for the scalar toy model. Moreover, this solution allows us to see that the dynamics of the fluid on the Dirichlet surface, as given by the conservation equation on $\Sigma_D$, ‘emerges’ as collective dynamics of the boundary CFT. In particular, the boundary fluid lives on a ‘dynamical background’, and the effects of the background can be suitably subsumed into the fluid description. This suggests that the correct way to think about the hypersurface physics is in terms of a ‘dressed fluid’ living on an inert geometry. Thus, the effective description of a fluid on this dynamical background is geometrically encapsulated in terms of the Dirichlet hypersurface dynamics.
Examining the resulting dynamics on $\Sigma_D$ we find that the hypersurface or effective fluid suffers from a possible pathology for $r_D$ smaller than some critical value $r_{D,snd}$. At $r_{D,snd}$ the sound mode of the effective fluid starts to propagate outside the inert background $\Sigma_D$’s light-cone. We suggest that in the CFT, this effect is due to the extreme forcing of the fluid on the boundary by the ‘dynamical’ metric, and moreover propose that one can obtain sensible dynamics by projecting out the sound mode. This involves looking at the fluid at a scaling limit and this can be formalized as taking the incompressible non-relativistic limit of the fluid on the hypersurface in a manner entirely analogous to the scaling limit described in for generic relativistic fluids in [@Bhattacharyya:2008kq; @Fouxon:2008tb].[^13]
Having understood the Dirichlet problem for generic $\Sigma_D$ away from the horizon ${\mathcal H}^+$, we then proceed to push this surface deeper into the spacetime and ask what happens as we approach the horizon. In this regime $\Sigma_D$ dynamics continues to be described by incompressible Navier-Stokes equations in the limit, though with some slight differences from the BMW limit mentioned above. Zooming in onto the region between $\Sigma_D$ and the horizon, we provide an embedding of the construction of [@Bredberg:2011jq; @Compere:2011dx] into the fluid/gravity correspondence [@Bhattacharyya:2008jc]. Further, we demonstrate that in this limit both the bulk metric in the region between $\Sigma_D$ and the boundary, and the boundary metric degenerate from metrics on a Lorentzian manifold to Newton-Cartan like structures. This raises interesting questions about the natural emergence of the Galilean structures in the AdS/CFT correspondence which we postpone for future work.
The plan of this paper is as follows: In we first address the Dirichlet problem for a scalar field propagating in using this linear problem to build intuition. In we pose the bulk Dirichlet problem for gravity in spacetime and solve it in the long wavelength approximation borrowing heavily on the results from the fluid/gravity correspondence. The remainder of the paper is then devoted to understanding the physics of our construction in various regimes: demonstrates how the Dirichlet surface dynamics, as governed by the conservation equation, arises from the boundary physics. Aided by this we argue that the Dirichlet dynamics is probably pathological past a critical radius and propose a non-relativistic scaling of the resulting fluid a la BMW in to cure this possible pathology. Finally, in we study the near-horizon Dirichlet problem and make contact with the recent work on the flat space Dirichlet problem (and its connection with Navier-Stokes equations). We end with a discussion in . Various appendices contain useful technical results. In particular, to aid the reader we provide a comprehensive glossary of our conventions and key formulae in . This is followed by a complete ‘Dirichlet dictionary’ relating hypersurface variables to boundary variables in for ready reference.
[*Note added:*]{} While this work being completed we received [@Kuperstein:2011fn] which has partial overlap with the results presented in . These authors also attempt to solve for bulk geometries with prescribed boundary conditions on $\Sigma_D$ in the long wavelength regime and interpret their results in terms of a RG flow of fluid dynamics.
[*Note added in v2:*]{} In the first version of the paper the non-relativistic metrics quoted in and in were incorrect; the metrics as presented do not solve the bulk Einstein’s equations to the desired order. These are now corrected in the current version. However, the full set of terms that we need to include in order to see the Naiver-Stokes equation on the boundary is quite large. Hence in the main text we only report the results for the case where the non-relativistic fluid moves on a Ricci flat spatial manifold in and present the general results in a new appendix . We note that the results of also correct the expressions originally derived in [@Bhattacharyya:2008kq].
Dirichlet problem for probe fields {#s:dscalar}
==================================
To set the stage for the discussion let us consider setting up the bulk Dirichlet problem for linear PDEs in an asymptotically spacetime. As a canonical example we will consider the dynamics of a probe scalar field $\Phi(r,x^\mu)$ of mass $m$. Generalizations to other linear wave equations such as the free Maxwell equation are straightforward.
We will let this scalar field propagate on a background asymptotically geometry with spatio-temporal translational symmetries so that the background metric can be brought to the form $$ds^2 =r^2\, g_{\mu\nu}(r) \, dx^\mu \, dx^\nu + \frac{g_{rr}(r)}{r^2} \, dr^2
\label{bggen}$$ The boundary of the spacetime is at $r \to \infty$ and we will assume that the boundary metric is the Minkowski metric on ${\mathbb R}^{d-1,1}$ for simplicity, so that asymptotically $g_{\mu\nu} \to \eta_{\mu\nu}$ and $g_{rr} \to 1$ (as $r\to \infty$).
The dynamical equation of motion for the scalar is the free Klein-Gordon equation which can be written as an ODE in the radial direction for the Fourier modes $\Phi_{k}(r)$ of $\Phi(r,x^\mu)$ $$\Phi(r,x^\mu)=\int \frac{d^dk}{(2\pi)^d} \,e^{i\, k\cdot x} \, \Phi_k(r)\ ,$$ and takes the form $$\frac{1}{\sqrt{-g\, g_{rr}}}\, \partial_r \, \left(\sqrt{-g\, g^{rr}} \, \partial_r \Phi_{k}(r) \right) - \left(g_{\mu\nu} \,k^\mu\,k^\nu +m^2 \right) \Phi_{k} = 0$$ As a second order equation we need to specify two boundary conditions. We are going to restrict attention to the finite part of the geometry as illustrated in and impose Dirichlet boundary conditions for the field at some hypersurface $\Sigma_D$ at $r = r_D$. The second boundary condition in general can take the form of a regularity boundary condition in the interior of the spacetime. If we were working in a spacetime with a horizon this would demand that the mode functions of interest are purely ingoing at the future horizon.
The question we wish to pose is the following: usually in an asymptotically spacetime we know that the solution to the scalar wave equation above has two linearly independent modes with power-law fall-off characterized by the source $J_\phi$ and vev $\phi$ of the dual boundary operator ${\cal O}_\Phi$ (which we recall is a conformal primary). We wish to ask what is the characterization of the boundary data as a functional of the Dirichlet hypersurface data. In this simple linear problem it is easy to see that there is a one-one map between the two sets of data. Essentially we are asking how to tune $J_\phi$ and $\phi$ so that the value of the scalar field on the Dirichlet hypersurface at $r=r_D$ takes on its given value.
While it is possible to derive a formal answer to the above question, it is useful to first visit the simple setting of pure spacetime where we have the luxury of being able to solve the scalar wave-equation explicitly to see an explicit answer to the question.
Probe scalar in {#s:adss}
----------------
We specialize our consideration to the pure geometry where $g_{\mu\nu} = \eta_{\mu\nu} $ and $g_{rr} = 1$ and one has enhanced Lorentz symmetry on the constant $r$ slices. The wave equation simplifies to $$\frac{1}{r^{d-1} }\, \frac{d}{dr} \, \left(r^{d+1} \, \frac{d}{dr} \Phi_{k}(r) \right) - \left(k^2 +m^2 \right) \Phi_{ k} = 0
\label{mkgads}$$ This is well known to have solutions in terms of Bessel functions, but we will proceed to examine the behavior in a gradient expansion to set the stage for the real problem of interest later.
### The $k=0$ case
The translationally invariant solution of the massive Klein-Gordon equation in the bulk is[^14] $$\Phi(r) = \frac{\phi}{(2\nu)\,r^\Delta} + r^{\Delta-d} J_\phi$$ where the dual primary has a scaling dimension $\Delta$ obeying $\Delta(\Delta-d)=m^2$ along with a source $J_\phi$ and a normalized vev[^15] $\phi$ defined via $$J_\phi \equiv \left[r^{d-\Delta}\, \Phi(r)\right]_{r\to\infty}$$ $$\phi \equiv \left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi \right)\right]_{r\to\infty} = \left[-r^\Delta\left( r\partial_r \Phi -(\Delta-d)\Phi\right)\right]_{r\to\infty}$$ with $$\nu \equiv \Delta - \frac{d}{2} = \sqrt{\frac{d^2}{4} + m^2}$$ for convenience. For simplicity, we will assume $\Delta > \frac{d}{2} $ and choose $m$ such that $\nu \notin {\mathbb Z}$ to avoid complications with logarithms. Extension to $\Delta \in [\frac{d}{2}-1, \frac{d}{2}]$ with the lower end of the interval saturating the unitary bound is possible with the added complication of taking proper account of the necessary boundary terms.
We will begin by rewriting this solution in terms of the quantities on the Dirichlet surface which we denote with a hat to distinguish them from the boundary data: $$\hat{J}_\phi \equiv \left[r^{d-\Delta}\,\Phi(r)\right]_{r\to r_D}=J_\phi+ \frac{\phi}{(2\nu)\,r_D^{2\nu}} \ ,
\label{j0}$$ $$\hat{\phi} \equiv \left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi \right)\right]_{r\to r_D} =\phi \ .
\label{ph0}$$
Since the transformation between the data on the boundary $\{J_\phi,\phi\}$ and that on the hypersurface $\{\hat{J}_\phi, \hat{\phi}\}$ is linear it is a simple matter to write the bulk solution in terms of the hypersurface variables. One simply has $$\Phi(r) =\frac{\hat{\phi}}{(2\nu)\, r^\Delta} + r^{\Delta-d} \left(\hat{J}_\phi- \frac{\hat{\phi}}{(2\nu)r_D^{2\nu}}\right) .
\label{pDsol}$$ This is the answer we seek and all that remains is to interpret this result.
It is now easy to notice that the imposition of the Dirichlet condition on a hypersurface inside the bulk is equivalent to making the boundary source a specific function of the vev. From we can read off the specific deformation of the boundary CFT action to be given by $$\delta \mathcal{L}_{CFT} = -\frac{1}{16\pi \,G_{d+1}\, } \, \left(\hat{J}_\phi\,\hat{\phi}- \frac{1}{2(2\nu)\, r_D^{2\nu}}\, \hat{\phi}^2 \right) \propto \hat{J}_\phi \, {\cal O}_\Phi - \frac{(16\pi \,G_{d+1})}{2(2\nu)\, r_D^{2\nu}}\, {\cal O}_\Phi^2$$ which happens to be an irrelevant double-trace deformation [@Witten:2001ua; @Berkooz:2002ug] of the boundary CFT. Hence, at least in this simple setup the dual of the Dirichlet problem is to make the source of a primary ${\cal O}_\Phi$ a particular joint function of the vev of the primary in the given state and another fixed (state-independent) auxiliary source.
### The $k\neq0$ case : Derivative expansion up to $k^2$
Having seen the result for the translationally invariant case $k=0$, we now proceed with $k \neq 0$. It is well known that general solution to the wave equation is given in terms of Bessel functions which we parameterize as[^16] $$\Phi_k(r) = \frac{\phi_k}{r^\Delta}\times\frac{\Gamma(\nu)}{2(k/2r)^{\nu}}I_{\nu}(k/r) + r^{\Delta-d} (J_\phi)_k \times \frac{2(k/2r)^{\nu}}{\Gamma(\nu)}K_{\nu}(k/r)$$ Note that our previous result for $k=0$ follows from just keeping the leading $x^0$ terms in the expansions $$\begin{split}
\frac{2x^{\nu}}{\Gamma(\nu)}\;K_{\nu}(2x) &= \sum_{j=0}^{\infty} \frac{\Gamma(\nu-j)}{\Gamma(\nu)} \frac{(-x^2)^j}{j!} +x^{2\nu}\sum_{j=0}^{\infty} \frac{\Gamma(-\nu-j)}{\Gamma(\nu)} \frac{(-x^2)^j}{j!}\\
\frac{\Gamma(\nu)}{2x^{\nu}}I_{\nu}(2x) &=\sum_{j=0}^{\infty} \frac{\Gamma(\nu)}{(2\nu+2j)\Gamma(\nu+j)} \frac{x^{2j}}{j!}
\end{split}
\label{eq:expn}$$ For a general $k$, we can repeat the analysis of the previous section. While this can be done generally at all orders in $k$ with some work, for simplicity we will resort to derivative expansion keeping terms upto order $k^2$. Not only will this allow us to see some of the structures emerging explicitly, but it also sets the stage for our gravitational computation in later sections.
Using the expansion above, we have $$\begin{split}
\Phi_k(r) &= \frac{\phi_k}{(2\nu)\, r^\Delta}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2r^2}+\ldots \right) + r^{\Delta-d} (J_\phi)_k \left( 1-\frac{1}{(2\nu-2)} \frac{k^2}{2r^2}+\ldots\right.\\
&\qquad\qquad \quad \left. +
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2r}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2r^2}+\ldots\right\}\right)
\end{split}$$ with the ellipses representing order $k^4$ terms and higher.
The source at the intermediate surface is as before easily determined $$\begin{split}
(\hat{J}_\phi)_k &\equiv \left[r^{d-\Delta}\Phi_k(r)\right]_{r\to r_D}\\
&= (J_\phi)_k \left(1-\frac{1}{(2\nu-2)} \frac{k^2}{2\,r_D^2}+\ldots + \frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2r_D}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2\,r_D^2}+\ldots\right\}\right)\\
&\qquad \qquad +\frac{\phi_k}{(2\nu)\, r_D^{2\nu}}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2\,r_D^2}+\ldots \right)
\end{split}
\label{j2}$$ while the normalized vev of the primary to this order in derivative expansion can be determined after subtracting an appropriate counter-term as[^17] $$\begin{split}
\hat{\phi}_k &=\left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi_k \right)+\frac{r^\Delta}{2\nu-2}\frac{k^2}{r^2}\Phi_k +\ldots \right]_{r\to r_D}\\
&=\phi_k\left(1 +\frac{2\nu-2}{(2\nu)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) +\frac{4(J_\phi)_k}{(2\nu-2)} \times\frac{\Gamma(-\nu)}{\Gamma(\nu+2)}\left(\frac{k}{2}\right)^{2\nu}+\ldots \\
\end{split}
\label{ph2}$$ To solve the Dirichlet problem,we need to solve for $\phi_k,(J_\phi)_k $ in terms of the hatted variables from and which can be inverted to get $$\begin{split}
(J_\phi)_k &= \frac{(\hat{J}_\phi)_k}{\mathfrak{D}}\left(1 +\frac{2\nu-2}{(2\nu)^2} \frac{k^2}{2r_D^2}+\ldots \right) -\frac{\hat{\phi}_k}{\mathfrak{D}\, (2\nu) \, r_D^{2\nu}}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) \\
\phi_k &=\frac{\hat{\phi}_k}{\mathfrak{D}} \left( 1-\frac{1}{(2\nu-2)} \frac{k^2}{2\,r_D^2}+\ldots +
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2\,r_D}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2\,r_D^2}+\ldots\right\}\right)\\
&\qquad \qquad -\frac{4(\hat{J}_\phi)_k}{\mathfrak{D}(2\nu-2)} \times\frac{\Gamma(-\nu)}{\Gamma(\nu+2)}\left(\frac{k}{2}\right)^{2\nu}+\ldots \\
\end{split}$$ where the momentum dependent coefficient ${\mathfrak D}$ is $$\begin{split}
\mathfrak{D}&\equiv 1-\frac{4(2\nu-1)}{(2\nu)^2(2\nu-2)} \frac{k^2}{2\,r_D^2}+
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2\,r_D}\right)^{2\nu}\left\{1-\frac{1}{\nu^2(\nu-1)} \right.\\
&\left.\qquad \qquad \qquad+\; \frac{(2 \nu +1)^4-4 (2 \nu +2)^2+7}{(2 \nu)^2
(2 \nu +2)^3}\;\frac{k^2}{r_D^2}+\ldots\right\}\\
\end{split}$$ As we saw in the $k=0$ case, we have yet again determined a state dependent source on the boundary for the primary operator ${\cal O}_\Phi$. The key feature to note from the above analysis, is that the expression for the boundary source $J_\phi$ is non-analytic in $k$ and hence non-local when Fourier-transformed back to position space. Hence, we see that in general we have a map between the non-local double trace deformation on the boundary and the Dirichlet data on $\Sigma_D$ (similar non-local double-trace deformations were explored earlier in [@Marolf:2007in]).
A general proposal for linear systems
-------------------------------------
From the analysis of the free scalar wave equation in the picture is rather clear. In the CFT, in general one can make the source a non-local functional of the vev of the primary operator. Usually such a function can be fed into the holographic dictionary via a ‘state-dependent’ boundary condition, which whilst somewhat unnatural from a field theory is a perfectly sensible boundary condition to consider. For some special classes of functionals, this state-dependent boundary condition has a very simple bulk interpretation as a Dirichlet boundary condition imposed on an intermediate surface, implying that we can trade the non-locality of the boundary sources into local behavior at some lower radius.
We just have one further question to answer before we declare victory: how do we in practice determine this special set of sources in various holographic setups? For the general backgrounds we can formally write the solution to the wave equations in terms of integrals over the Dirichlet data convolved with suitable ‘Dirichlet bulk to boundary propagators’, ${\cal K}_\text{source}$ and ${\cal K}_\text{vev}$. The former propagates the information contained in $\hat{J}_\phi$ to the boundary source, while the latter allows determination of the contribution from the vev $\hat{\phi}$ on $\Sigma_D$, i.e., formally $$J_\phi(x) = \int d^dx' \, \left\{{\cal K}_\text{source}(r_D, x; x') \, \hat{J}_\phi(x') + {\cal K}_\text{vev}(r_D, x; x') \, \hat{\phi}(x')\right\}$$ Note that implicit in our definition of these Dirichlet bulk to boundary propagators is the information of the boundary condition in the interior of the geometry and the necessary counter-terms. While it is possible to work this out in more specific geometries, such as a Schwarzschild-AdS$_{d+1}$ spacetime to see the interplay of these IR boundary conditions, we will leave this toy problem for now, and proceed to analyze the more interesting case of gravitational dynamics in wherein we do have to face-up with non-linearities of the equations of motion.[^18]
Before proceeding to the gravitational setting, however, let us make a few pertinent observations relevant to the motivation mentioned at the beginning of . The result we have obtained is quite intuitive; demanding that our fields take on the desired value at $\Sigma_D$ entails a linear relation between the two pieces of data at infinity, thereby leading to the observation about the source depending on the vev. We also see that despite some superficial resemblance to the Wilsonian RG flow where too one encounters multi-trace operators there is a crucial distinction in the physics. In the formulation of [@Heemskerk:2010hk; @Faulkner:2010jy] one finds that for fixed asymptotic data, upon integrating out the region of the geometry between the boundary and a cut-off surface (which we can for simplicity take to be $\Sigma_D$ for the sake of discussion) one obtains an effective action for a cut-off field theory living on $\Sigma_D$ with scale dependent sources. These are irrelevant double traces (which are the only terms generated in a Gaussian theory which the linear models under discussion are), and one obtains the $\beta$-functions for the double trace couplings along the flow. In the present context however what we have is a situation wherein we are forced to engineer a specific double trace deformation on the boundary so as to ensure that we satisfy the Dirichlet boundary conditions on $\Sigma_D$. This is conceptually different from usual notions of RG, where one does not conventionally consider state dependent boundary conditions in the UV. However, there is a sense in which renormalisation of sources takes place which will become quite clear when we look at the gravitational problem.
The Dirichlet problem for gravity {#s:dgrav}
=================================
Having understood the boundary meaning of the Dirichlet problem for probe fields in a fixed background, we now turn to the situation where we consider dynamical gravity in the bulk. While we could consider other matter degrees of freedom in the bulk whose backreaction we now have to take into account, we choose for simplicity to restrict attention to the dynamics in the pure gravity sector which, as is well known, is a consistent truncation of the supergravity equations of motion. From a field theory perspective, we are going to work in the planar limit and focus on the dynamics of a single operator, the stress tensor and its source, the CFT metric $g_{\mu\nu}$.
Setting up the general Dirichlet problem {#s:setdg}
----------------------------------------
First of all one should ask what does it mean to consider the Dirichlet problem at a fixed hypersurface in the bulk when gravity is dynamical. We will take the view that the location of the hypersurface $\Sigma_D$ is specified by a scalar function on the bulk manifold ${\cal M}_{d+1}$. We want to determine the metric on this spacetime by solving the dynamical equations of motion subject to the boundary condition on the prescribed hypersurface. To wit, we demand that ${\cal M}_{d+1}$ be endowed with a Lorentzian metric ${\cal G}_{MN}$ which solves Einstein’s equations with a negative cosmological constant. The equations of motion are (in units where $R_{AdS} =1$)[^19] $$E_{MN} = {\cal R}_{MN} + d \, {\cal G}_{MN} = 0
\label{eins}$$ We will adapt coordinates $X^A = \{r ,x^\mu\}$ to the hypersurface $\Sigma_D$ and take this distinguished surface to be at $r = r_D$ with intrinsic coordinates $x^\mu$.[^20] We impose the boundary condition $${\cal G}_{MN} \, dX^M \, dX^N \big|_{r\to r_D} = \;r_D^2\, \hat{g}_{\mu\nu}(x) \, dx^\mu\, dx^\nu
\label{indgh}$$ where ${\hat g}_{\mu\nu}(x)$ is the Dirichlet data we wish to specify. To complete the specification of the problem we should impose some boundary condition in the interior of the spacetime, which we will canonically take to be a regularity condition. The scaling by $r_D^2$ above whilst unconventional from the bulk perspective, is more natural in the AdS/CFT context for it makes it easy to compare with the case where we push the hypersurface to the boundary.
The general problem as stated above is quite hard. For one it is not clear that for generic choices of Dirichlet data one obtains a solution compatible with regularity in the interior of the spacetime. While a local solution in an open neighbourhood of $\Sigma_D$ can presumably be obtained by adapting a Fefferman-Graham like expansion, one is unlikely to be able to gain much insight using this procedure. Moreover, given that the map between $\Sigma_D$ and the boundary is expected to be non-local (borrowing intuition from the linear problem) one wonders whether there are causality issues as well. In particular, for generic state dependent boundary conditions causality is murky – does the source adjust itself acausally to obtain the appropriate response? Likewise on $\Sigma_D$ there is a concern that signals can propagate outside the light-cone of the metric $\hat{g}_{\mu\nu}$ (which they could for instance do through the bulk); does this imply a corresponding pathology for the boundary physics as well?[^21] In short, the well-posedness of the Dirichlet problem is a-priori unclear. Nevertheless, we will ignore all these subtleties for now and forge ahead.
After solving Einstein’s equations with the boundary conditions we have set-up, we can extract the Brown-York stress tensor on $\Sigma_D$, denoted $\hat{T}_{\mu\nu}$, using the standard set of boundary counter-terms [@Henningson:1998gx; @Balasubramanian:1999re] $$\hat{T}_{\mu\nu}=- \frac{r_D^d}{8\pi G_{d+1}} \left(\hat{K}_{\mu\nu}-\hat{K} \,\hat{g}_{\mu\nu}+(d-1)\, \hat{g}_{\mu\nu} +\ldots\right) .
\label{hypst}$$ $r_D^2\,\hat{K}_{\mu\nu}$ is the extrinsic curvature of the hypersurface $\Sigma_D$ and $\hat{K}$ is its trace, defined as usual in terms of the normal to the surface.
Note that we have written the answer in terms of the intrinsic metric on the hypersurface which is related to the induced metric from the bulk up to a rescaling by $r_D^2$. The homogeneous scaling of the stress tensor under allows us to fix an overall $r_D$ dependent pre-factor. The hypersurface stress tensor is of course covariantly conserved (the Gauss-Codacci constraint), i.e., $$\hat{\nabla}_\mu{\hat{T}^\mu}{}_{\nu}=0\,.
\label{hcons}$$ Given $\{\hat{g}_{\mu\nu}, \hat{T}_{\mu\nu} \}$ we can ask what are the corresponding boundary conditions on the boundary that lead to the same geometry. Based on our scalar problem we can conclude that the boundary source $g_{\mu\nu}$ and stress tensor $T_{\mu\nu}$ are in general non-local functionals of the Dirichlet data. We would like to characterize the map between these two sets of data[^22] $$\varphi_D: \{ \hat{g}_{\mu\nu}, \hat{T}_{\mu\nu} \} \to \{g_{\mu\nu}, T_{\mu\nu} \} \ .$$
While this problem is in general difficult, there is one context in which we can not only solve for the boundary data in terms of the hypersurface variables, but we can also investigate the issues raised above in precise terms. This is the long-wavelength hydrodynamical regime along the hypersurface as in the fluid/gravity correspondence [@Bhattacharyya:2008jc; @Bhattacharyya:2008ji; @Bhattacharyya:2008mz], wherein gravitational duals to arbitrary fluid flows on the boundary were constructed order by order in a gradient expansion. The reason this is possible, as explained in these works, is that the bulk spacetime in this long-wavelength regime is well approximated ‘tubewise’ by the boosted planar Schwarzschild- solution, see . As a result one finds that Einstein’s become ultra-local in the $x^\mu$ directions leading one to effective ODEs to determine the radial profiles. The tubes in question are centered around radially ingoing null geodesics, which can be used to translate information from any bulk hypersurface of interest to the boundary (see [@Bhattacharyya:2008xc; @Bhattacharyya:2008mz] for a discussion of the causal structure). Given this, it is actually easy to solve the problem of finding the map $\varphi_D$ and we now describe the construction in the rest of this section.
![Schematic representation of the gravitational Dirichlet problem in the fluid/gravity regime. The causal structure of the fluid/gravity spacetimes is illustrated emphasizing the tubewise approximation; in each tube the geometry resembles that of a uniformly boosted Schwarzschild-AdS$_{d+1}$ black hole. Suitable choices of the Dirichlet surface allow us to find the map between the boundary data ${\mathfrak X}$ and the Dirichlet hypersurface data $\hat{\mathfrak X}$ within each tube, rendering the problem tractable. []{data-label="f:tubes"}](Dir-tubes)
(0,0) (-4.8,2)[$\Sigma_D$]{} (-6.5,-0.5)[ $\hat{\mathfrak X}=\{\hat{g}_{\mu\nu}, \hat{u}^\mu, \hat{T}\} \quad \stackrel{\varphi_D}{\longmapsto}\quad{\mathfrak X}=\{g_{\mu\nu}, u^\mu, T\}$]{}
Dirichlet problem and the Fluid/Gravity correspondence {#s:fg1}
------------------------------------------------------
To set the stage for our discussion, let us recall that the fluid/gravity map constructs in a gradient expansion, regular solutions to the bulk Einstein’s equations which are dual to arbitrary fluid flows on the boundary of the asymptotically spacetime. For a boundary metric $g_{\mu\nu}(x)$ which is slowing varying, the boundary stress tensor in this context is a not an arbitrary symmetric traceless two tensor, but constrained to take the hydrodynamical form. It is parameterized by $d$ independent parameters - a velocity field $u_\mu(x)$ (unit normalized so that $g_{\mu\nu}\, u^\mu\,u^\nu =-1$) and a scalar function $b(x)$ which parameterizes the temperature. The bulk metric ${\cal G}_{MN}$ is determined in terms of the data ${\mathfrak X} = \{g_{\mu\nu}(x), u_\mu(x), b(x)\}$.
We wish to implement the same procedure, but starting with analogous data $\hat{{\mathfrak X}} = \{\hat{g}_{\mu\nu}(x), \hat{u}_\mu(x), b(x)\}$ on the Dirichlet hypersurface $\Sigma_D$. But given the ultra-locality inherent in the long-wavelength regime and the fact that [@Bhattacharyya:2008mz] have solved the problem for arbitrary boundary metrics (corresponding to fluids on arbitrary slowly varying curved backgrounds), we don’t need to solve any equations. The solution space of the bulk Dirichlet problem coincides with the solution space found in [@Bhattacharyya:2008mz] and the problem at hand is readily solved by slicing this solution space appropriately. With this aim, we now review the solutions constructed in [@Bhattacharyya:2008mz].
### Review of fluid/gravity {#s:fgrev}
The general solutions of the bulk equations of motion in the fluid/gravity regime take the form $$ds^2 ={\cal G}_{MN} \,dX^M \,dX^N = - 2 \, {\mathfrak u}_\mu(x) \, dx^\mu \,\left( dr + r\,{\mathfrak V}_\nu(r,x)\,\,dx^\nu\right)+ r^2\,{\mathfrak G}_{\mu \nu}(r,x) \, dx^\mu\, dx^\nu \ ,
\label{formmetw}$$ where the fields ${\mathfrak V}_\mu$ and ${\mathfrak G}_{\mu\nu}$ are functions of $r$ and $x^\mu$ which admit an expansion in the boundary derivatives and are known to second order in the gradients. For our purposes it will suffice to consider the first order metric where[^23] $$\begin{aligned}
\mathfrak{u}_\mu &=& u_\mu\ ,\qquad \mathfrak{V}_\mu = \mathcal{A}_\nu+\frac{r}{2}\, f(br)\, u_\nu \\
\mathfrak{G}_{\mu\nu} &=& P_{\mu\nu} +2b \,F(br)\ \sigma_{\mu\nu} \end{aligned}$$ with the functions $$f(x) \equiv 1 - \frac{1}{x^d} \ , \qquad F(x)\equiv \int_{x}^{\infty}\frac{y^{d-1}-1}{y(y^{d}-1)}dy\,.
\label{fFdef}$$ ${\cal A}_\mu$ is the Weyl covariant connection introduced in [@Loganayagam:2008is] which is expressed in terms of the acceleration and the expansion of the velocity $u_\mu$ $$\mathcal{A}_\mu\equiv u^\lambda\nabla_\lambda u_\mu-\frac{\nabla_\lambda u^\lambda}{d-1}u_\mu = a_\mu - \frac{\theta}{d-1} \, u_\mu ,$$ while $\sigma_{\mu\nu}$ is shear strain rate tensor of $u_\mu$: $$\sigma_{\mu\nu}\equiv{P_\mu}^\alpha {P^\beta}_\nu\,
\left[\nabla_{(\alpha} u_{\beta)}-g_{\alpha\beta}\frac{\nabla_\lambda u^\lambda}{d-1}\right],\quad\textrm{with}\quad P_{\mu\nu}\equiv g_{\mu\nu}+u_\mu u_\nu .$$ So the bulk metric to first order in derivatives explicitly takes the form: $$ds^2=-2 \,u_\mu \,dx^\mu \left( dr + r\ \mathcal{A}_\nu dx^\nu \right) + r^2 \left[ g_{\mu\nu} +\frac{u_\mu u_\nu}{(br)^d}+2b \,F(br)\ \sigma_{\mu\nu}\right] dx^\mu dx^\nu + \ldots
\label{metricsimp:eq}$$ and we have refrained from explicitly denoting the $x^\mu$ dependence of ${\mathfrak X}$ and the ellipses denote second order and higher gradient terms. The corresponding co-metric (the metric on the cotangent bundle/the inverse metric) is given by $$\begin{split}
\mathcal{G}^{AB}&\partial_A\otimes\partial_B \\
&= \left[r^2 \, f(br)-\frac{2\,r\,\theta}{d-1}\right]\partial_r\otimes\partial_r+2\left[u^\mu -r^{-1} a^{\mu}\right]\partial_\mu\otimes_s\partial_r\\
&\qquad +r^{-2}\left[P^{\mu\nu} -2b \,F(br)\ \sigma^{\mu\nu}\right]\partial_\mu\otimes\partial_\nu\\
\end{split}$$ The stress tensor on the boundary is that of a viscous relativistic fluid: $$T_{\mu\nu} = p\, g_{\mu\nu} + (\varepsilon + p) \, u_\mu\,u_\nu - 2 \, \eta\, \sigma_{\mu\nu} + \ldots
\label{Tbdy}$$ with thermodynamic state variables $$p = \frac{1}{d-1}\, \varepsilon = \frac{1}{16\pi\, G_{d+1} } \, \frac{1}{b^d} \,
\label{epbdy}$$ and shear viscosity $$\eta=\frac{1}{16\pi \,G_{d+1}} \;\frac{1}{b^{d-1}}\,.
\label{etab}$$ The expression in should be thought of as a way to generate solutions of Einstein equations when provided with hydrodynamic configurations that solve the ideal fluid equations derived form the $T_{\mu\nu}$ above, i.e., when provided with $u^\mu,b$ that satisfy $\nabla_\mu T^{\mu\nu} = 0$ to first order in gradients. Our aim is to reformulate this set of solutions as solutions to the bulk Dirichlet problem.
### Dirichlet data from fluid/gravity solutions {#s:dfgsol}
Given the solution to the bulk equations of motion, we will begin by simply slicing it at a given radial position $r=r_D$ and extract the intrinsic metric on the Dirichlet surface[^24].
The advantage of working with the Weyl covariant form of the bulk metric is that one can simultaneously deal with Dirichlet surfaces specified by slowly varying functions $r=\rho(x)$. These can always be brought to the form $r=r_D$ by working in a suitable boundary Weyl-frame (local rescaling by a conformal factor $\log(1-\rho(x)/r_D)$ will do the trick). The Weyl connection $\mathcal{A}$ eats $\rho(x)$ so that $$\mathcal{A}_{\mu}=\rho^{-1} \left(\nabla_\mu+ a_\mu - \frac{\theta}{d-1}\, u_\mu \right) \rho$$ with this understanding all our formulae hold for arbitrary $\rho(x)$.
At fixed $r=r_D$ the hypersurface metric reads (recalling ) to first order $$\hat{g}_{\mu\nu}= g_{\mu\nu} + \frac{u_\mu u_\nu}{(b\,r_D)^d}+2b \,F(br_D)\ \sigma_{\mu\nu} -\frac{2}{r_D}\, u_{(\mu} \mathcal{A}_{\nu)} + \ldots$$ While this relation was obtained by slicing a known solution with prescribed asymptotic boundary conditions, it has the nice feature of having solved the equations of motion (to the desired order) and moreover satisfies the regularity condition in the interior. We will now turn the logic around and imagine $\hat{g}_{\mu\nu}$ to be specified at $\Sigma_D$ and view the equation above as specifying the boundary intrinsic metric in terms of the hypersurface metric. Hence, the above equation giving $g_{\mu\nu}$ in terms of $\hat{g}_{\mu\nu}$ and $u_\mu$ is the Dirichlet constitutive relation we seek – when such a relation is imposed on the boundary data, the intrinsic hypersurface metric is automatically fixed to $\hat{g}_{\mu\nu}$.
To complete the specification of the map $\varphi_D$ we need to further eliminate the boundary velocity field in favour of a vector field on the hypersurface. To do this we need to examine the hypersurface stress tensor and parameterize it appropriately. The stress tensor at $r_D$ is easily obtained from to be $$\hat{T}_{\mu\nu} = \hat{p}\, \hat{g}_{\mu\nu} + \frac{1}{\hat{\alpha}^2}\, (\hat{\varepsilon}+\hat{p})\, u_\mu u_\nu - 2\, \hat{\alpha}\, \eta \, \sigma_{\mu\nu} +\frac{2}{r_D} \, (\hat{\varepsilon}+\hat{p})\, u_{(\mu} \mathcal{A}_{\nu)} +\ldots
\label{eq:rDquantities}$$ with[^25] $$\hat{\varepsilon}\equiv\frac{(d-1)}{8\pi\, G_{d+1}}\; \frac{\hat{\alpha}}{\hat{\alpha}+1}\; \frac{1}{b^d} \,,\quad\quad
\hat{\varepsilon} + \hat{p} \equiv\frac{d }{16\pi \,G_{d+1}}\, \frac{\hat{\alpha}}{b^d}\,,
\label{eqofstate:eq}$$ with $\eta$ as given before in and we have defined a hypersurface scalar $$\hat{\alpha}\equiv \frac{1}{\sqrt{f(b\, r_D)}} =\frac{1}{\sqrt{1-(b\,r_D)^{-d}}}\,.
\label{alhatdef}$$ The stress tensor is tantalizingly similar to that of a viscous fluid, but as yet, we cannot interpret $\eta$ as the shear viscosity since we have not expressed $\hat{T}_{\mu\nu}$, in terms of hypersurface variables.
This is however easy to remedy. Define $\hat{u}_\mu$ to be the unit normalized (with respect to $\hat{g}_{\mu\nu}$ of course) timelike eigenvector of $\hat{T}_{\mu\nu}$. A simple computation shows that $$\hat{u}_\mu\equiv \frac{u_\mu}{\hat{\alpha}} + \frac{\hat{\alpha}}{r_D} \mathcal{A}_\mu\,.$$ We want to express the hypersurface stress tensor in terms of $\hat{u}_\mu$ and its gradients with respect to the $\hat{g}_{\mu\nu}$ compatible connection $\hat{\nabla}_\mu$. This can be done by the standard computation of the difference of $\nabla_\mu - \hat{\nabla}_\mu$. We outline the calculation in and simply quote the relevant result here: $$\hat{\sigma}_{\mu\nu}=\hat{\alpha} \, \sigma_{\mu\nu}\,,\quad\quad \mathcal{A}_\nu =\hat{\mathcal{A}}_\nu - \frac{\frac{d}{2}(\hat{\alpha}^2-1)}{1+\frac{d}{2}(\hat{\alpha}^2-1)}\; \hat{a}_\nu\,.$$ Armed with this data we can write now the stress tensor at $r=r_D$ as $$\hat{T}_{\mu\nu} = \hat{p}\, \hat{g}_{\mu\nu} + (\hat{\varepsilon}+\hat{p})\, \hat{u}_\mu \,\hat{u}_\nu - 2\, \eta\, \hat{\sigma}_{\mu\nu}+\ldots$$ We see that the result indeed is a stress tensor of a relativistic fluid with energy density $\hat{\varepsilon}$, pressure $\hat{p}$, given in and the same value of shear viscosity as the boundary theory $\eta$, . The dynamical content of this system is still the conservation equation $$\hat{\nabla}_\mu{\hat{T}^\mu}{}_{\nu}=0\,.
\label{hypcons}$$ which follows from realizing that this is the ‘momentum constraint’ equation for the radial slicing of Einstein’s equations (or if one prefers the Gauss-Codacci constraint on the hypersurface). Its equation of state is given in and we will momentarily evaluate the trace of $\hat{T}_{\mu\nu}$ in .
Further note that the stress tensor has no contribution associated with the expansion of the fluid, i.e., the bulk viscosity vanishes identically on the hypersurface. Nevertheless, the fluid is not a conformal fluid, for the trace of the stress tensor is non-vanishing. $$\hat{T}^\mu{}_\mu = \hat{T}_{\mu\nu}\, \hat{g}^{\mu\nu}= - \hat{\varepsilon} + (d-1)\hat{p} =\frac{d(d-1)}{16\pi \, G_{d+1}}\ \frac{\hat{\alpha}-1}{\hat{\alpha}+1}\ \frac{\hat{\alpha}}{b^d} \ .
\label{eq:traceDirichlet}$$ This is not entirely surprising for we have introduced an explicit scale $r_D$ into the problem, and as required for consistency the trace vanishes in the limit $r_D \to \infty$ as $\hat{\alpha} \to 1$. More curious is the fact that rate of change of the trace with the $\Sigma_D$’s radial location is simple: $$\hat{T}^\mu{}_\mu =-r_D\frac{d\hat{\varepsilon}}{dr_D}\,
\label{trerd}$$ The evolution of the trace is highly suggestive for can be interpreted as saying the the trace is generated by the variation of the local energy density with respect to some scale. This kind of a relation probably hints at some kind of non-linear realization of scale invariance. Again this is reminiscent of the holographic RG ideas and it would be interesting to flesh this out in greater detail.
Having the notion of the hypersurface velocity field $\hat{u}_\mu$ we can now proceed to write the boundary metric in terms of hypersurface data. Inverting the relation for the velocities to obtain (see ) $$u_\mu = \hat{\alpha}\, \hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D} \,\left( \hat{\mathcal{A}}_\nu-\frac{\frac{d}{2}(\hat{\alpha}^2-1)}{1+\frac{d}{2}(\hat{\alpha}^2-1)}\; \hat{a}_\nu\right)
\label{buhu1}$$ one can show that $$g_{\mu\nu} = \hat{g}_{\mu\nu} -(\hat{\alpha}^2-1) \, \hat{u}_\mu \, \hat{u}_\nu + \frac{2\,\hat{\alpha}^2}{r_D} \, \left[ \hat{u}_{(\mu} \hat{\mathcal{A}}_{\nu)}-\frac{\frac{d}{2}(\hat{\alpha}^2-1)}{1+\frac{d}{2}(\hat{\alpha}^2-1)}\; \hat{u}_{(\mu}\hat{a}_{\nu)} \right] - \frac{2b}{\hat{\alpha}}\, F(br_D) \hat{\sigma}_{\mu\nu}
\label{bghg1}$$ The equations and together specify the map $\varphi_D$ we seek in the long-wavelength regime. Note that the hypersurface and the boundary data are determined by the same scalar function $b(x)$, which however enters non-trivially through $\hat{\alpha}$ in the determination of dynamics on $\Sigma_D$.
Note that the light-cones of $g_{\mu\nu}$ are enlarged by a factor determined by $\hat{\alpha}$ relative to that of $\hat{g}_{\mu\nu}$. This is the first signal that there is some interesting interplay between boundary causal structures and fixing boundary conditions on $\Sigma_D$; we will address this issue in some detail in . But first we finish the solution to the Dirichlet problem as stated and write down the bulk metric in the long wavelength regime.
Bulk metric in terms of Dirichlet data {#s:dbulk}
--------------------------------------
Given the map in and we are in a position to re-write the bulk metric in terms of $\Sigma_D$ data $\hat{{\mathfrak X}}$ alone. Substituting the transformations we can write the final result as in with $$\begin{split}
\mathfrak{u}_\mu &= u_\mu = \hat{\alpha}\, \hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D}\left[\frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}-\frac{\hat{\theta}}{d-1}\hat{u}_\mu\right] \\
\mathfrak{V}_\mu &= \mathcal{A}_\mu+\frac{r}{2}\, f(br)\, u_\mu \\
&= \hat{\xi}\left[\frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}-\frac{\hat{\theta}}{d-1}\hat{u}_\mu\right] +\frac{r}{2}\, f(br) \,\hat{\alpha}\, \hat{u}_\mu\\
\mathfrak{G}_{\mu\nu} &= P_{\mu\nu} +2\,b \,F(br)\ \sigma_{\mu\nu} = \hat{P}_{\mu\nu} +2\,b \,\hat{F}(br)\ \hat{\sigma}_{\mu\nu}\\
\label{dirbulk1}
\end{split}$$ and we have defined $$\hat{\xi} \equiv 1-\frac{1}{2}\,\frac{r}{r_D}\;\frac{f(br)}{f(br_D)}
= 1-\frac{\hat{\alpha}^2}{2}\, \frac{r}{r_D}\, f(br)
\label{xidef}$$ and $$\hat{F}(br) \equiv \frac{1}{\hat{\alpha}}\left(F(br)-F(br_D)\right) = \frac{1}{\hat{\alpha}} \; \int_{br}^{br_D}\; \frac{y^{d-1}-1}{y(y^{d}-1)}dy\,.$$ The factors of $\hat{u}_\mu$ are distributed between ${\mathfrak G}_{\mu\nu}$ and ${\mathfrak V}_\nu$ by the requirement that the former be transverse to ${\mathfrak u}_\mu$.
This bulk metric ${\cal G}_{MN}$ solves the gravitational Dirichlet problem in the long-wavelength regime. We have thus solved the problem posed at the beginning of this section completely in terms in this regime aided by the ultra-locality of the gradient expansion. We summarize the complete map $\varphi_D$ and the resultant dictionary between CFT variables and the hypersurface variables in the . As a consistency check note that the results agree when we send the surface $\Sigma_D$ to the boundary with those derived in [@Bhattacharyya:2008mz]; one simply sets $r_D=\infty$ and $\hat{\alpha} =1$. In the remainder of the paper we will explore the relation between the dynamics on $\Sigma_D$ and that on the boundary, with an aim towards getting better intuition for various issues raised in and .
Emergence of Dirichlet dynamics in the CFT$_d$ {#s:emergence}
==============================================
Having obtained a solution to the gravitational Dirichlet problem in the long-wavelength regime, we now turn to analyze the underlying physics of the system. From the viewpoint of the bulk, the dynamics of the system under study has two equivalent descriptions – one in terms of hypersurface (hatted) variables $\hat{\nabla}_\mu\hat{T}^{\mu\nu}=0$, and another in terms of the un-hatted boundary variables $\nabla_\mu T^{\mu\nu}=0$. While the former is more natural in the bulk (since it is $\hat{g}_{\mu\nu}$ which is fixed in the bulk for our Dirichlet boundary conditions on $\Sigma_D$), only the latter has a straightforward interpretation in the CFT$_d$. Hence, it is interesting to ask how the hypersurface description should be interpreted within the CFT.
From the CFT point of view, the fluid is living in a metric background with a Dirichlet constitutive relation (rewriting in terms of $u_\mu$) $$\begin{split}
g_{\mu\nu}&=\hat{g}_{\mu\nu} - \left(1-\frac{1}{\hat{\alpha}^2}+\frac{2\,\theta}{(d-1)\,r_D}\right)u_\mu u_\nu+\frac{2}{r_D}\, u_{(\mu} a_{\nu)} -2\,b F(br_D)\ \sigma_{\mu\nu} +\ldots\\
g^{\mu\nu}&= \hat{g}^{\mu\nu}-\left(1-\hat{\alpha}^2-\frac{2\,\hat{\alpha}^4\,\theta}{(d-1)\,r_D}\right)u^\mu u^\nu \ +2\,b F(br_D)\ \sigma^{\mu\nu} -\frac{2\,\hat{\alpha}^2}{r_D}\, u^{(\mu} a^{\nu)}+ \ldots \\
\end{split}
\label{bdyggu}$$ where $\hat{\alpha}^2$ is defined in .[^26] The function $\hat{\alpha}$ increases with increase in the local temperature (decreases with increase in the local $b$). These expressions basically tell us how the ambient spacetime background the CFT$_d$ lives on responds to the motion of the CFT fluid.
Let us first understand the physical content of the piece in the constitutive relation with zero-derivatives. This can mostly be done heuristically which we will do first and then confirm it with an explicit calculation. The zero-derivative piece of the boundary metric $g_{\mu\nu}$ is made of an inert piece $\hat{g}_{\mu\nu}$ which does not respond to the fluid motion along with an additional piece proportional to $u_\mu u_\nu$. The presence of a term proportional to $u_\mu u_\nu$ means the boundary metric effectively has a correction in its $dt^2$ piece in the local fluid rest-frame (which is responsible for opening up of the light-cone in the boundary). This kind of correction as is well known in general relativity just represents a gravitational potential well. Note that this potential well travels along with the fluid and hence it is tempting to think that there is a way to describe the collective packet of fluid and the local graviton cloud that it carries along in terms of a ‘dressed’ fluid.
To guide our intuition, let us draw analogies with another familiar physical situation where the background responds to the system locally via this kind of a relation. One analogous situation is that of a charge carrier moving in a polarizable medium. The polarizability of the medium defines the constitutive relation of the medium in exact analogy with the constitutive relations for the metric above. We know that in the case of the charge carrier moving in a polarizable medium often the polarizability can be taken in to account by [*shifting*]{} the dispersion of the charge carrier and pretending that this ‘dressed’ charge carrier is essentially moving through an inert medium.
What we want to argue out in this section is the fact that a similar ‘dressing’ phenomenon happens in the case of the CFT$_d$ fluid - we want to rewrite the problem of a fluid with $T^{\mu\nu}$ moving in the ‘polarizable’ $g_{\mu\nu}$ into the problem where a dressed fluid with $T^{\mu\nu}_{\text{dressed}}$ moving in the an inert spacetime $g_{\mu\nu,\text{inert}}$. It is clear that the inert background is just the non-dynamical part of the metric, i.e., $g_{\mu\nu,\text{inert}} =\hat{g}_{\mu\nu}$. We would like to claim that $T^{\mu\nu}_{\text{dressed}} = \hat{T}^{\mu\nu} $. This then would be a complete physical picture of how the dynamics on a Dirichlet hypersurface in the bulk emerges directly from the boundary description.
Conservation equations at the boundary and on the Dirichlet surface {#s:conseq}
-------------------------------------------------------------------
Let us now implement the dressing picture heuristically described above at the level of the equations to derive the conservation equations on the Dirichlet surface from those on the boundary. Given an arbitrary energy-momentum tensor $$T^{\mu\nu} = \varepsilon \,u^\mu u^\nu + p \,P^{\mu\nu} +\pi^{\mu\nu}
\label{piTdef}$$ with $\pi^{\mu\nu}$ capturing the dissipative terms involving at least one gradient of the velocity field or thermodynamic state variables, we have $$\begin{split}
\nabla_{\nu}T^{\mu\nu} &= u^\mu\left[ u^\nu \nabla_\nu \varepsilon + (\varepsilon+ p) \nabla_\nu u^\nu\right] + (\varepsilon+p)\, a^\mu+ P^{\mu\nu}\nabla_{\nu}\, p +\nabla_{\nu}\, \pi^{\mu\nu}\\
&= u^\mu\left[ \frac{s}{c_{snd}^2} u^\nu\nabla_\nu T + T\, s \, \theta-u_\alpha\nabla_{\beta}\pi^{\alpha\beta}\right] + T\,s \,a^\mu+ s\, P^{\mu\nu}\nabla_{\nu} T +P^{\mu\nu}\nabla_{\lambda}\pi_\nu^{\lambda}\\
\end{split}
\label{conseom}$$ where we have introduced $$c_{snd}^2\equiv \frac{dp}{d\varepsilon}= s\frac{dT}{d\varepsilon} \ ,$$ and used the Euler relation $\varepsilon + p = s\, T$, with $s$ being the entropy density of the fluid. Since the part proportional to $u^\mu$ and the part transverse to $u^\mu$ should separately vanish, we get $$\begin{split}\label{eq:dT}
s\left[\partial_\mu + a_\mu-c_{snd}^2 \,u_\mu \,\theta \right] T+P_\mu{}^{\nu}\nabla_{\lambda}\pi_\nu{}^{\lambda}-c_{snd}^2 \,u_\mu \, \pi^{\alpha\beta}\nabla_\alpha u_\beta &=0
\end{split}$$ Similarly, $\hat{\nabla}_{\nu}\hat{T}^{\mu\nu}=0$ is equivalent to the equation $$\begin{split}\label{eq:HatdT}
\hat{s}\left[\partial_\mu + \hat{a}_\mu-\hat{c}_{snd}^2 \, \hat{u}_\mu \,\hat{\theta} \,\right] \hat{T}+\hat{P}_\mu{}^{\nu}\hat{\nabla}_{\lambda}\hat{\pi}_\nu{}^{\lambda}-\hat{c}_{snd}^2\, \hat{u}_\mu \,\hat{\pi}^{\alpha\beta}\hat{\nabla}_\alpha \hat{u}_\beta &=0
\end{split}$$ Using the relations $$\hat{s}=s \ , \qquad \text{and} \;\; \hat{T}=T\, \hat{\alpha} ,
\label{hbst}$$ which are derived in we can write $$\begin{split}
\hat{s}\left[\partial_\mu + \hat{a}_\mu-\hat{c}_{snd}^2\, \hat{u}_\mu\, \hat{\theta} \right] \hat{T}
&= s \left[\partial_\mu + \hat{a}_\mu-\hat{c}_{snd}^2 \,\hat{u}_\mu \,\hat{\theta} \right] T\hat{\alpha}\\
&= \hat{\alpha} \,s \left[\left(1+\frac{d\,\ln \hat{\alpha}}{d\, \ln T}\right)\partial_\mu + \hat{a}_\mu-\hat{c}_{snd}^2 \,\hat{u}_\mu \,\hat{\theta} \right] T \\
&= \hat{\alpha} \left(1+\frac{d\ln \hat{\alpha}}{d\, \ln T}\right) \,s \left[\partial_\mu + \frac{\hat{a}_\mu-\hat{c}_{snd}^2\, \hat{u}_\mu \, \hat{\theta}}{\left(1+\frac{d\, \ln \hat{\alpha}}{d\, \ln T}\right)} \right] T \\
\end{split}$$ so that becomes $$\begin{split}\label{eq:HatdT2}
s \left[\partial_\mu + \frac{\hat{a}_\mu-\hat{c}_{snd}^2\, \hat{u}_\mu \,\hat{\theta}}{\left(1+\frac{d\,\ln \hat{\alpha}}{d\,\ln\ T}\right)} \right] T+\frac{\hat{P}_\mu{}^{\nu}\hat{\nabla}_{\lambda}\hat{\pi}_\nu{}^{\lambda}-\hat{c}_{snd}^2\, \hat{u}_\mu\, \hat{\pi}^{\alpha\beta}\hat{\nabla}_\alpha \hat{u}_\beta}{\hat{\alpha} \left(1+\frac{d\,\ln \hat{\alpha}}{d\,\ln\ T}\right)} &=0
\end{split}$$ For the equation to describe the same dynamical system as the equation , it is necessary and sufficient that $$\begin{split}\label{eq:matchEqn}
&\hat{a}_\mu-\hat{c}_{snd}^2 \,\hat{u}_\mu \,\hat{\theta} +\frac{\hat{P}_\mu{}^{\nu}\hat{\nabla}_{\lambda}\hat{\pi}_\nu{}^{\lambda}-\hat{c}_{snd}^2 \,\hat{u}_\mu\, \hat{\pi}^{\alpha\beta}\hat{\nabla}_\alpha \hat{u}_\beta}{\hat{T}\,\hat{s} } \\
&\quad\stackrel{?}{=}\left(1+\frac{d\,\ln \hat{\alpha}}{d\,\ln\ T}\right)\left[a_\mu-c_{snd}^2 \,u_\mu \,\theta +\frac{P_\mu{}^{\nu}\nabla_{\lambda}\pi_\nu{}^{\lambda}-c_{snd}^2 \,u_\mu\, \pi^{\alpha\beta}\nabla_\alpha u_\beta}{T\,s}\right]
\end{split}$$ We can show that this is indeed true at the first derivative level by using the conversion formulae (rewriting ) $$\begin{split}
{u}_\mu &= \left(1+\frac{\hat{\alpha}\,\hat{\theta}}{r_D(d-1)}\right)\hat{\alpha}\,\hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D} \frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\\
\frac{d\ln \hat{\alpha}}{d\ln\ T}&= \frac{d}{2}(\hat{\alpha}^2-1), \quad {\theta}=\frac{1}{\hat{\alpha}}\hat{\theta} \\
\hat{a}_\nu &= \left(1+\frac{d}{2}(\hat{\alpha}^2-1)\right) a_\nu
,\quad {\theta}{u}_\mu=\hat{\theta}\hat{u}_\mu \\
\hat{c}^2_{snd} &= \left(1+\frac{d}{2}(\hat{\alpha}^2-1)\right) c^2_{snd} .
\end{split}$$ Hence, till this order, we have proved that $\hat{T}^{\mu\nu}$ is indeed the dressed energy-momentum tensor that we were looking for.
It should be instructive to extend this analysis to higher orders in derivatives. In particular, it would be interesting to pin down the specific property of the Dirichlet constitutive relation which leads to the fact that the dressed viscosity $\hat{\eta}$ is same as the bare value $\eta$ and furthermore understand why the hypersurface fluid has no bulk viscosity. This would complement the analysis of [@Iqbal:2008by] who demonstrated the absence of corrections to the shear viscosity by considering a flow equation in the linearized regime between the boundary and the horizon.
Causality and relativistic fluids on the Dirichlet hypersurface {#s:csq}
---------------------------------------------------------------
Having established a clear connection between the dynamics of the dressed fluid on the Dirichlet surface and that of fluid on a ‘dynamical’ boundary metric, we now turn to examining the properties of the fluid motion. It seems a priori that all is well in the long wavelength regime with regards to the issues raised at the beginning of section viz. the issue of locality and causality of the Dirichlet problem in . However, this is probably a bit too quick; while it is true that we have local dynamical equations given by the conservation of the hypersurface stress tensor , we have not established firmly that these equations arise from a sensible thermodynamic system. We now proceed to address this issue.
The energy momentum tensor of the dressed fluid on the hypersurface is characterized by an energy density $\hat{\varepsilon}$ and a pressure $\hat{p}$ which are given in . In particular, the pressure of the fluid is $$\hat{p} \equiv \frac{\left[1+\frac{d}{2}(\hat{\alpha}-1)\right]}{8\pi G_{d+1}b^d}\frac{\hat{\alpha}}{\hat{\alpha}+1} = \frac{2\hat{\alpha}}{\hat{\alpha}+1}\left[1+\frac{d}{2}(\hat{\alpha}-1)\right] p
\label{prhyper}$$ We also note that $\hat{\varepsilon} = \frac{2\,\hat{\alpha}}{\hat{\alpha}+1} \, \varepsilon$ which is useful in what follows.
Using the thermodynamic relations $$\frac{d\hat{s}}{\hat{s}}= \frac{d\hat{\varepsilon}}{\hat{\varepsilon}+\hat{p}} ,\quad \frac{d\hat{T}}{\hat{T}}= \frac{d\hat{p}}{\hat{\varepsilon}+\hat{p}} \quad\text{and}\quad \hat{\varepsilon}+\hat{p} = \hat{T} \hat{s}$$ we get the entropy density and the temperature of this fluid as $$\hat{s}= \frac{1}{4 G_{d+1}}\, \frac{1}{b^{d-1}}=s ,\quad\text{and} \quad \hat{T}=\frac{d}{4\pi b}\hat{\alpha}=\hat{\alpha}\, T
\label{hst2}$$ as quoted above in . The first of these relations follows from the fact that the entropy of the fluids on the asymptotic boundary as well as on the Dirichlet surface are given in terms of the area of the horizon which is unchanged by the solution. To determine the temperature on the hypersurface one has to account for the fact that the surface is in the interior of the spacetime. In the planar Schwarzschild-AdS$_{d+1}$ solution we get deviation from the Hawking temperature (which is temperature in the CFT) via a red-shift factor $\hat{\alpha}$. Conversely, given the above relations , the expressions for $\hat{\varepsilon}$ and $\hat{p}$ can be deduced using $d\hat{\varepsilon}=\hat{T}d\hat{s}$ and $d\hat{p}=\hat{s}d\hat{T}$.
The speed of sound mode in this system is given by $$\label{spsnd:eq}
\begin{split}
\hat{c}_{snd}^2 &\equiv \frac{\partial\hat{p}}{\partial\hat{\varepsilon}}= \frac{1}{d-1}\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right] = {c}_{snd}^2 \left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]
\end{split}$$ This exceeds the speed of light as measured by $\hat{g}_{\mu\nu}$, i.e., we get superluminal sound propagation, for $\hat{\alpha}>\hat{\alpha}_{snd}$ where $\hat{\alpha}_{snd}\equiv \sqrt{3-\frac{4}{d}}$. This corresponds to $$\label{r_snd:eq}
\begin{split}
b\ r_{D,snd} &\equiv \left(\frac{\hat{\alpha}_{snd}^2}{\hat{\alpha}_{snd}^2-1}\right)^{1/d}\\
& =\left(\frac{3-\frac{4}{d}}{2-\frac{4}{d}}\right)^{1/d}\\
& \approx 1+ \frac{1}{d}\ln (3/2) + O(d^{-2}) \\
\end{split}$$ One can intuitively understand this result from the viewpoint of the boundary fluid. As we noted earlier the boundary fluid is subject to a gravitational potential well. Should one locally increase the strength of this well then the fluid would get sufficiently accelerated, perhaps leading to a pathology. This is manifest in the picture of the dressed fluid moving on an inert background achieved by translating over to the Dirichlet surface. In particular, this gets reflected in the fact that the effective pressure $\hat{p}$ felt by the dressed fluid increases relative to its energy density $\hat{\varepsilon}$ thus driving the dressed fluid into a regime where the dominant energy condition is violated. Such violations of the dominant energy condition are known to be susceptible to superluminal sound modes[^27] as we observed above.
How much should one be worried by this apparent acausal behavior where the dressed sound mode travels superluminally with respect to the inert part of the boundary metric? After all, as is easily verified the mode with dispersion $\omega \sim \hat{c}_{snd}\, k$ propagates within the local light-cone of the ‘dynamical’ boundary metric $g_{\mu\nu}$. This is achieved by the phenomenon we had already alluded to towards the end of the section : as we move our Dirichlet surface into the AdS, the Dirichlet constitutive relation ensures that the boundary light-cone opens up (see ) thus ensuring that the dressed sound mode is not superluminal when measured with respect to $g_{\mu\nu}$. Of course, pending a detailed analysis of the initial value problem posed by this dynamical system (and other possible global issues), one cannot assert that the boundary physics is sensible from above observations alone.
We would however like to suggest that viewing the hypersurface fluid as an autonomous dynamical system, a superluminal sound mode probably indicates a pathology. As described in [@Adams:2006sv] one should anticipate that the corresponding initial value problem[^28] for the hypersurface fluid might be ill-posed. Is this the way the long wavelength problem is telling us that the dual of the generic bulk Dirichlet problem in the CFT is ill-posed? Is it possible that the bulk Dirichlet problem in gravity is pathological the moment $r_D$ is finite and the fact that the effective dynamics of the fluid of the CFT remains sensible up to a critical radius $r_{D,snd}$ is just a long wavelength artifact? Clearly this issue deserves further investigation.
We will now take an alternate approach which sidesteps these deep questions – given that our issue is with the superluminal sound mode on $\Sigma_D$ (for $r_D<r_{D,snd}$), is it possible to project this offending mode out of our dynamics hence avoiding the entire issue? We will now argue that fortunately the answer is yes – there is indeed a way to project out the sound mode, retaining sensible physics at least within the long wavelength regime.
The way to do this is to move to the incompressible non-relativistic regime of fluid/gravity correspondence first studied in [@Bhattacharyya:2008kq]. We will now implement their construction in our setting allowing us to obtain sensible dynamics for the Dirichlet problem. We will postpone some of the more general questions raised above to the discussion section .
The Dirichlet problem for gravity with non-relativistic fluids {#s:dgravnr}
==============================================================
The discussion so far has concentrated on mapping relativistic fluid dynamics on a curved background $\Sigma_D$ to a corresponding problem on the boundary where we have a relativistic fluid on a ‘dynamical metric’. Moreover in the previous section, we cited the possible problems with the sound mode as a motivation to project it out consistently so as to get a clearly sensible dynamical system with no possible issues with the initial value problem etc.. Our goal now is to describe how this can be done consistently within our gradient expansion inspired by the non-relativistic incompressible scaling limit of [@Bhattacharyya:2008kq; @Fouxon:2008tb]. As we will see this limit has the added advantage that it naturally allows us to make contact with the metric derived in [@Cai:2011xv].
The idea as explained beautifully in [@Bhattacharyya:2008kq] is the following: every relativistic fluid has a scaling limit where we freeze out the propagating sound mode, which drives the fluid into a non-relativistic regime, while simultaneously making it incompressible. This BMW scaling can essentially be derived by the requirement that one retains the non-linearities of the conservation equation (at least at first order). The resulting conservation equation is the classic incompressible non-relativistic Navier-Stokes equations. Using the fluid/gravity map [@Bhattacharyya:2008kq] constructed a gravitational dual of this system.
The BMW scaling involves two ingredients. Firstly, the velocities and the temperatures of the fluid are taken to be slowly varying functions of a specific kind, with spatial and temporal gradients having different scaling dimensions (heuristically $\partial_t \sim \partial_x^2$). Secondly, the amplitude of the spatial velocity and the temperature fluctuation (about some constant equilibrium value) are also taken to be small and are of the same order as $\partial_x$ and $\partial_t$ respectively. It is convenient to introduce a large parameter $\aleph$ (which is inverse of the small parameter $\epsilon$ in [@Bhattacharyya:2008kq]) in terms of which the above statements can be written as $\partial_t\sim \aleph^{-1}$, $\partial_x\sim \aleph^{-2}$ etc. We will denote the corresponding parameter for the hypersurface fluid by the hatted $\hat{\aleph}$.
Under the large $\aleph$ limit it is possible to show that the relativistic conservation equations map straightforwardly into the incompressible Navier-Stokes equations. We review this scaling in for the convenience of the reader and proceed in the main text to directly implement an analogous scaling on the hypersurface $\Sigma_D$ using a large parameter $\hat{\aleph}$.
Non-relativistic fluids on the Dirichlet hypersurface {#s:nrhyp1}
-----------------------------------------------------
To keep the computation sufficiently general we will take the metric on the Dirichlet surface to have non-vanishing curvature. Furthermore, it is useful as in [@Bhattacharyya:2008kq] to allow for the background metric to be decomposed to slowly varying parts of different orders so as to recover non-relativistic fluids which are forced on $\Sigma_D$. One reason for doing so is that we are going to obtain a boundary metric, via the map $\varphi_D$ described in , which naturally contains such terms. Hence it pays to be more general to see the mixing of various contributions at the boundary.
To start off let us consider on the hypersurface a metric $\hat{g}_{\mu\nu}$ of the form: $$\hat{g}_{\mu\nu} = \hat{g}^{(0)}_{\mu\nu} + \hat{h}_{\mu\nu}
\label{Dgans}$$ with $$\hat{g}^{(0)}_{\mu\nu} = -dt^2 + \hat{g}^{(0)}_{ij}(x) \, dx^i \, dx^j
\label{Dg0}$$ where $\hat{g}^{(0)}_{ij}$ are slowly varying functions of $x^i$ and with $\hat{h}_{\mu\nu}$ are the metric perturbations which we take it as $$\hat{h}_{\mu\nu} \,dx^\mu\,dx^\nu= 2\, \hat{\aleph}^{-1}\, \hat{k}^*_{i}\, dt\, dx^i + \hat{\aleph}^{-2}\, \left(\hat{h}^*_{tt}\, dt^2 + \hat{h}^*_{ij} \, dx^i \, dx^j \right)
\label{Dhans}$$ To keep things simple it turns out to be useful to work with the background spatial metric $g^{(0)}_{ij}$ being Ricci flat i.e., $R^{(0)}_{ij} =0$. This turns out to simplify the analysis considerably for a host of terms dependent on the curvature of $g^{(0)}_{ij}$ drop out – a more comprehensive analysis for general backgrounds is presented in . We indicate the various corrections that arise from the curvature terms at appropriate stages in the main text.
All the functions which have a $*$ subscript or superscript (which we freely interchange to keep formulae clear) are of a specific functional form with anisotropic scaling of their spatial and temporal gradients. $$\hat{{\cal Y}}_*(t,x^i) : {\mathbb R}^{d-1,1} \mapsto {\mathbb R}\ , \;\; \text{such that}
\;\; \{ \partial_t \hat{{\cal Y}}_*(t,x^i), \hat{\nabla}^{(0)}_i \hat{{\cal Y}}_*(t,x^i)\} \sim \{{\cal O}(\hat{\aleph}^{-2}) ,{\cal O}(\hat{\aleph}^{-1})\}$$ where $\hat{\aleph}$ is a counting parameter introduced to implement the BMW scaling (on the boundary $\aleph^{-1} = \epsilon_\text{BMW}$ as discussed in ).
Following [@Bhattacharyya:2008kq], we parameterize the velocity field as $$\hat{u}^{\mu} = \hat{u}^t \left(1, \hat{\aleph}^{-1}\, \hat{v}_*^i \right)$$ where the function $\hat{u}^t$ is determined by requiring that $\hat{g}_{\mu\nu}\hat{u}^{\mu}\hat{u}^{\nu}=-1$. This gives the full velocity field in a large $\hat{\aleph}$ expansion as [^29] $$\begin{split}
\hat{u}^t &=1 + \frac{\hat{\aleph}^{-2}}{2} \left( \hat{h}^*_{tt} + 2 \,\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*} \right)+ {\cal O}(\hat{\aleph}^{-4})\\
\hat{u}^i &= \hat{\aleph}^{-1} \, \hat{v}_*^i + \frac{\hat{\aleph}^{-3}}{2} \left( \hat{h}^*_{tt} + 2 \,\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*} \right)\, \hat{v}_*^i + {\cal O}(\hat{\aleph}^{-4})\\
{\hat{u}}_t &= -1 - \frac{1}{2}\, {\hat{\aleph}}^{-2} \, \left(- {h}^*_{tt} + {\hat{g}}^{(0)}_{jk} \, {\hat{v}}^j_* \, {\hat{v}}^k_* \right)+ {\cal O}(\hat{\aleph}^{-4})\\
{\hat{u}}_i &= {\hat{\aleph}}^{-1}\, \left( {\hat{v}}^*_i + {k}^*_i \right)+ \hat{\aleph}^{-3}\left[{h}^*_{ij}\hat{v}^{j}_{*}+ \frac{1}{2} \left( \hat{h}^*_{tt} + 2 \,\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*} \right)\, \left( {\hat{v}}^*_i + {k}^*_i \right) \right] + {\cal O}(\hat{\aleph}^{-4})
\end{split}$$ and the velocity gradients are given by $$\begin{aligned}
\hat{\theta} &=& {\cal O}(\hat{\aleph}^{-4}) \nonumber \\
\hat{\mathcal{A}}_{\mu} dx^{\mu} &=& \hat{a}_{\mu} dx^{\mu}= \hat{\aleph}^{-3} \left[ \partial_{t} \hat{v}_{i}^{*}
+ \hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*} - \hat{f}_i^* \right] dx^{i} + {\cal O}(\hat{\aleph}^{-4}) \nonumber \\
\hat{\sigma}_{\mu \nu} dx^{\mu} dx^{\nu} &=&\hat{\aleph}^{-2}\, \hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dx^{j}
- 2\hat{\aleph}^{-3} \, \hat{v}^{j}_{*} \,\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dt + {\cal O}(\hat{\aleph}^{-4})
\label{Duder}
\end{aligned}$$ where $\hat{\nabla}^{(0)}_{\mu}$ is the covariant derivative compatible with $\hat{g}^{(0)}(x^{i})$ and we have freely raised and lowered the spatial indices with $\hat{g}^{(0)}_{ij}$ for brevity. Further, $\hat{f}^i$ is a forcing function determined as a functional of $\hat{h}_{\mu\nu}$ data $$\hat{f}_{i} = \frac{1}{2} \partial_i \hat{h}^*_{tt}- \partial_t \hat{k}^*_i + \hat{q}^{*}_{\ ij} \, \hat{v}_*^j \,.
\label{hypforce}$$ and $\hat{q}^*_{ij} = \hat{\nabla}^{(0)}_i \hat{k}^*_j -\hat{\nabla}^{(0)}_j \hat{k}^*_i$ . In deriving these expressions we have used the fact that to leading order in the $\aleph \to \infty$ expansion, the velocity field $v^i_*$ is divergenceless (see below).
We take the scaling in $b$ to be of the form $$b = b_0 + \hat{\aleph}^{-2}\, \delta b_* \ .
\label{Dbexp}$$ Using $$\begin{aligned}
\hat{\alpha} &=& \hat{\alpha}_0 + \hat{\aleph}^{-2} \, \left(\frac{d}{2}\, \hat{\alpha}_0 \left( 1 - \hat{\alpha}_0^2 \right) \,\frac{\delta b_*}{b_0} \right)
+{\cal O}(\hat{\aleph}^{-4}) \ , \qquad \hat{\alpha}_0 \equiv \frac{1}{\sqrt{f \left(b_0\, r_{D}\right)}}\end{aligned}$$ we can evaluate the non-relativistic pressure per mass density and kinematic viscosity: $$\begin{aligned}
\hat{p}_{*} &=&\frac{\delta \hat{p}}{\hat{\varepsilon}_0+\hat{p}_0} = - \hat{\aleph}^{-2} \, \left(1 + \frac{d}{2} \left(\hat{\alpha}^2_{(0)} -1\right) \right) \frac{\delta b_*}{b_0} +{\cal O}(\hat{\aleph}^{-4}) \nonumber \\
\hat{ \nu}_{0} &=& \frac{\eta_0}{\hat{\varepsilon}_0+ \hat{p}_0}= \frac{b_0}{d \, \hat{\alpha}_0}+ {\cal O}(\hat{\aleph}^{-2})
\label{bmwpnu} \end{aligned}$$ with $\hat{\rho}_0\equiv \hat{\varepsilon}_0+ \hat{p}_0 = \frac{d \,\hat{\alpha}_0}{16 \pi G_{d+1}\, b_0^d}$ playing the role of the non-relativistic mass density.
Given these data we can show that the conservation equations reduce to the incompressible Navier-Stokes equations:$$\begin{aligned}
&& \hat{\nabla}^{(0)}_i \, v^i_* = 0 \nonumber \\
&& \hat{\nabla}^{(0)}_i \hat{p}_* + \partial_t \hat{v}^*_i + \hat{v}_*^j \, \hat{\nabla}^{(0)}_j \hat{v}^*_i - 2\, \hat{\nu}_0\, \hat{\nabla}^{(0)^j} \left(\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)}\right) = \hat{f}_i
\label{hypns}\end{aligned}$$
We now proceed to derive the expressions entering into the bulk metric and the map $\varphi_D$ given in .
Bulk metric in terms of Dirichlet data {#s:nrhyp2}
--------------------------------------
Armed with the results from we can proceed to use those of to construct the bulk metric corresponding to the non-relativistic fluid on the Dirichlet hypersurface $\Sigma_D$. In principle Einstein equations need to be solved in the new gradient expansion to obtain the non-relativistic solutions. As argued by the authors of [@Bhattacharyya:2008kq], this can be done via an algorithm very similar to the algorithm used to find the metric dual of the relativistic fluid. The main difference is the anisotropic scaling of space with respect to time and the fact that the bulk metric is no-more ultra-local in space but is still ultra-local in time. We can again proceed from the space of non-relativistic solutions with asymptotic boundary conditions that we present in and reparametrize it in terms of Dirichlet data.
But we will take here instead an easier route and directly derive it from the the bulk relativistic metric written in terms of Dirichlet data. We should be careful though – given the difference in derivative counting between the relativistic scaling and the non-relativistic scaling, it is in principle possible that a higher order term according to the relativistic counting contributes at a lower order according to the non-relativistic counting. In order to obtain the metric accurate to the order where the Navier-Stokes equations can be seen, ${\cal O}(\hat{\aleph}^{-3})$, we need to have the certain terms in the relativistic metric accurate to third order in gradients.[^30]
If we however restrict to the case of Ricci flat spatial metric on the hypersurface $\Sigma_D$, then we can obtain the non-relativistic metric from the second order relativistic fluid/gravity metric obtained in [@Bhattacharyya:2008mz]. There are three terms we need to account for which give rise to non-relativistic contributions proportional to $\nabla^{(0)}_j\nabla^{(0) j} \hat{v}^*_i \equiv \nabla_{(0)}^2 \hat{v}^*_i$ and $ \nabla^{(0)}_j \hat{q}_*^{j}{}_i$. These involve new radial functions which we collect below after presenting the bulk metrics highlighting the terms that were missed in the original analyses. General expressions including spatial curvatures can be found in .
Since in the scaling limit $\hat{\aleph} \gg 1$ one has from that the $\hat{a}_\mu = \hat{\mathcal{A}}_\mu$ to leading order things simplify considerably. Using the formulae in the last subsection and including the terms from the second order metric[^31] (highlighted) it is easy to show that the bulk metric becomes[^32]
$$\label{hypbmwf1}
\begin{split}
ds^2 &=
-2 \hat{\alpha}\, \hat{u}_\mu \,dx^\mu dr
+\frac{2\hat{\alpha}^2}{r_D}\left[\frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\right]dx^\mu dr -2\,r\,\hat{\alpha}\, (2\, \hat{\xi}-1)
\, \frac{\hat{u}_{(\mu}\,\hat{a}_{\nu)} }{1+\frac{d}{2}\, (\hat{\alpha}^2 -1)}
\, dx^\mu \,dx^\nu\\
&\qquad
+\; r^2\left[\hat{g}_{\mu\nu} + \left(1-\hat{\alpha}^2\, f(br) \right)
\hat{u}_\mu \,\hat{u}_\nu + 2b\,\hat{F}(br)\, \hat{\sigma}_{\mu\nu}\right]
dx^\mu \,dx^\nu \\
&\qquad
\red{- 4b^2\kappa_L\hat{\alpha}\hat{P}_{\mu}^{\lambda}\hat{\mathcal{D}}_{\alpha}
{\hat{\sigma}^{\alpha}}_{\lambda}\,dx^\mu\,dr}\\
&\qquad
\red{-2\,\frac{\hat{\alpha}^3}{r_D^2}\hat{\mathcal{S}}_{\mu\lambda}\hat{u}^\lambda\,dx^\mu\,dr+\,2
\frac{\hat{\alpha}^3}
{r_D^2(d-2)}\left[1+\frac{2}{d\hat{\alpha}(\hat{\alpha}+1)}\right]
\hat{\mathcal{R}}_{\mu\lambda}\hat{u}^\lambda\,dx^\mu\,dr}\\
&\qquad
\red{+\; 2\,(br)^2\left[\hat{M}_1(br)\, \hat{u}_{(\mu}\hat{\mathcal{S}}_{\nu)\lambda}\hat{u}^\lambda -
\hat{M}_2(br)\, \hat{u}_{(\mu}\hat{\mathcal{R}}_{\nu)\lambda}\hat{u}^\lambda +2\, \hat{L}_1(br)\,
\hat{u}_{(\mu}\hat{P}_{\nu)}^{\lambda}\hat{\mathcal{D}}_{\alpha}{\hat{\sigma}^{\alpha}}_{\lambda}
\right]dx^\mu dx^\nu} \\
&= ds_0^2 + \hat{\aleph}^{-1} ds_1^2 + \hat{\aleph}^{-2} ds_2^2 + \hat{\aleph}^{-3} ds_3^2 + {\cal O}(\hat{\aleph}^{-4}) \\
\end{split}$$
with $$\label{hypbmwf1a}
\begin{split}
ds_0^2 &=
2\,\hat{\alpha}_0\ dt\ dr + r^2\left(-\hat{\alpha}_0^2 \,f_0 \,dt^2
+ \hat{g}^{(0)}_{ij}\,dx^i dx^j\right)\\
ds_1^2 &=
-2\, \hat{\alpha}_0\left( \hat{v}^*_i + \hat{k}^*_i \right)\ dx^i\ dr
+ 2\, r^2 \left[\hat{k}^*_i -\left(1-\hat{\alpha}_0^2\, f_0\right)
\left( \hat{v}^*_i + \hat{k}^*_i \right)\right] dx^i\, dt \\
ds_2^2 &=
2\, \hat{\alpha}_0 \left[- \frac{1}{2}\hat{h}^*_{tt}
+ \frac{1}{2}\, \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_*
+\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right]dt\ dr
+ r^2\left[\hat{h}^*_{tt}\, dt^2 + \hat{h}^*_{ij} \, dx^i \, dx^j \right]\\
&\quad
+r^2 \left(1-\hat{\alpha}_0^2\, f_0\right) \left[\left(- \hat{h}^*_{tt}
+ \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_*
+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}
\right)dt^2 \right.\\
&\qquad \left. \qquad
+ \left( \hat{v}^*_i + \hat{k}^*_i \right) \left( \hat{v}^*_j
+ \hat{k}^*_j \right)dx^i dx^j \right]
+2\,r^2\,b_0\, \hat{F}_0\,\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dx^{j}\\
ds_3^2 &=
-2\,\hat{\alpha}_0\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+
\left( \frac{1}{2}\hat{h}^*_{tt} + \,\hat{k}^*_{j}\, \hat{v}^{j}_{*}
+\frac{1}{2} \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*}
+\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right] dx^i dr \\
&\quad
+\frac{2\hat{\alpha}_0^2}{r_D\left(1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right)}\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*}
-\hat{f}_i^* \right] dx^{i}dr\\
&\quad
+ 2r\,\frac{\hat{\alpha}_0(2\,\hat{\xi}_0-1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}
\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*}
-\hat{f}_i^* \right] dx^{i}dt\\
&\quad
-2\,r^2 \left(1-\hat{\alpha}_0^2 \,f_0\right)\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}
+ \left(\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*}
+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)\,
\left( \hat{v}^*_i + \hat{k}^*_i \right) \right] dx^i dt \\
&\qquad
-4\, r^2\, b_0\, \hat{F}_0\, \hat{v}_{*}^j\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dt \;
\red{-\;2\, b_0^2\, r^2\, \hat{L}_1 \nabla^2_{(0)} v_i^* \, dt \, dx^i
-2\,b_0^2\, \hat{\kappa}_L\,\hat{\alpha}_0\, \hat{\nabla}^2_{(0)}v^*_i\, dx^i\, dr} \\
&\qquad
\red{+ \;\hat{M}_0 \,\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\, dx^i \, dt+ 2\, \frac{\hat{\alpha}_0^2}{r_D^2} \, \frac{\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}}{d\,(d-2)\, (\hat{\alpha}_0 +1) } \,
dx^i\, dr }
\end{split}$$ where $$\begin{split}
\hat{\alpha}_0 &\equiv (1-(b_0 r_d)^{-d})^{-1/2}\ ,\quad f_0 \equiv 1-(b_0 r)^{-d}\ ,\\
\hat{\xi}_0 &\equiv 1- \frac{r}{2r_D}\hat{\alpha}_0^2\, f_0\\
\hat{p}_* &\equiv -\frac{\delta b_*}{b_0}\left\{1+\frac{d}{2}\, (\hat{\alpha}^2 -1)\right\} \\
\hat{F}_0 &\equiv \frac{1}{\hat{\alpha}_0} \int_{b_0 r}^{b_0 r_D}\frac{y^{d-1}-1}{y(y^{d}-1)}dy\\
\hat{f}_i^* &\equiv \frac{1}{2}\partial_i \hat{h}^*_{tt} - \partial_t \hat{k}^*_i +\left[\hat{\nabla}^{(0)}_i \hat{k}^*_j - \hat{\nabla}^{(0)}_j \hat{k}^*_i\right] {v}_*^j = \frac{1}{2}\partial_i\hat{h}^*_{tt} - \partial_t\hat{k}^*_i +\hat{q}^*_{ij} {v}_*^j \\
\hat{L}_1 &\equiv \frac{L(br)}{(br)^d}-\frac{L(br_D)}{(br_D)^d}+\hat{\kappa}_L \left[1-\hat{\alpha}^2\, f_0\right]\\
\hat{\kappa}_L &\equiv \frac{1}{d}\left[
\xi(\xi^d-1)\frac{d}{d\xi}\left[\xi^{-d}L(\xi)\right]+
\frac{1}{\xi\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}+\frac{1}{\xi^2(d-2)}\right]_{\xi=br_D} \\
\hat{M}_0 &\equiv \frac{1}{(d-2)}\left[-\hat{\alpha}_0^2\left(1-\frac{r^2}{r_D^2}\right)+\frac{2}{d}\frac{\hat{\alpha}_0}{1+\hat{\alpha}_0}\left(1-\hat{\alpha}_0^2 f_0\right)\frac{r^2}{r_D^2}\right]
\end{split}$$ The function $L(x)$ which enters into the above formulae is given as $$L(br) = \int_{br}^\infty\xi^{d-1}d\xi\int_{\xi}^\infty dy\ \frac{y-1}{y^3(y^d
-1)}
\label{}$$ while the other functions $\hat{M}_1$ and $\hat{M}_2$ can be found in .
In the final results we have highlighted the terms appearing in $ds_3^2$ that were missed in the first version of the paper and are necessary in order to solve Einstein’s equations to ${\cal O}(\hat{\aleph}^{-3})$. Having realized the necessity of these terms it a-posteriori became clear that the BMW scaling metric introduced in [@Bhattacharyya:2008kq] to describe the bulk dual of boundary non-relativistic fluids also receives corrections. The corrected form of this metric including effects of boundary spatial curvature is now presented in . The general expressions for the Dirichlet non-relativistic fluid are collected in as they involve many more terms compared to what was originally reported in v1 of this paper.
As required this metric reduces to the metric derived by the BMW scaling on the boundary [@Bhattacharyya:2008kq] that we review in in our conventions; we simply set $r_D=\infty,\hat{\alpha}_0 = 1$ in the above metric. As remarked above, it compares with the metric given in [@Bhattacharyya:2008kq] up to the new terms mentioned above.
Inspired by the gravity duals of fluids on cut-off hypersurfaces in flat space [@Bredberg:2011jq; @Compere:2011dx], in [@Cai:2011xv] the result for the bulk dual to a non-relativistic fluid on a Dirichlet hypersurface in has been recently derived. The results there are presented for a fluid on a flat background and are entirely contained within our framework. To facilitate ease of comparison with their results we now specify our computation to the case where the metric on $\Sigma_D$ is flat and furthermore switch off the forcing. Taking $\hat{g}_{\mu\nu} = \eta_{\mu\nu}$ which amounts to setting $\hat{g}^{(0)}_{ij} = \delta_{ij}$ and $\hat{k}_i^* = \hat{h}_{tt}^* = \hat{h}^*_{ij} =0$ in the above one can show that the bulk metric reduces to the form with $$\label{hypbmwf}
\begin{split}
ds_0^2 &=
2\,\hat{\alpha}_0\ dt\ dr + r^2\left(-\hat{\alpha}_0^2\, f_0\, dt^2
+ \delta_{ij}dx^i dx^j\right)\\
ds_1^2 &=
-2\, \hat{\alpha}_0\, \hat{v}^*_i \ dx^i\ dr
- 2\, r^2 \,\left(1-\hat{\alpha}_0^2 \,f_0\right)\hat{v}^*_i dx^i dt \\
ds_2^2 &=
2 \,\hat{\alpha}_0 \left[ \frac{1}{2} \, \hat{v}^2_*
+\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\,
(\hat{\alpha}_0^2 -1)} \right]dt\ dr \\
&\quad
+r^2 \left(1-\hat{\alpha}_0^2 \,f_0\right) \left[\left( \hat{v}^2_*
+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)dt^2
+ \hat{v}^*_i \hat{v}^*_j dx^i dx^j \right]\\
&\quad
+2\,r^2\,b_0\,\hat{F}_0\,\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dx^{j}\\
ds_3^2 &=
-2\,\hat{\alpha}_0\, \left( \frac{1}{2} \hat{v}^{2}_{*}
+\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\,
(\hat{\alpha}_0^2 -1)}\right) \hat{v}^*_i\, dx^i\, dr
+\frac{2\hat{\alpha}_0^2}{r_D\left(1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right)}\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \partial_j \hat{v}_{i}^{*}
\right] dx^{i}dr\\
&\quad + 2\,r\,\frac{\hat{\alpha}_0(2\hat{\xi}_0-1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}
\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \partial_{j} \hat{v}_{i}^{*} \right] dx^{i}\,dt\\
&\quad
-2\,r^2 \left(1-\hat{\alpha}_0^2\, f_0\right) \left(\hat{v}^{2}_{*}\
+\hat{p}_* \frac{d\hat{\alpha}^2 }{1+\frac{d}{2}\, (\hat{\alpha}^2 -1)} \right)
\hat{v}^*_i \,dx^i \,dt -4\,r^2\,b_0\, \hat{F}_0\,\hat{v}_{*}^j\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dt \\
&\quad
\red{-\;2\, b_0^2\, r^2\, \hat{L}_1 \nabla^2_{(0)} v_i^* \, dt \, dx^i
-2\,b_0^2\, \hat{\kappa}_L\,\hat{\alpha}_0\, \hat{\nabla}^2_{(0)}v^*_i\, dx^i\, dr} \,,
\end{split}$$ which agrees with that derived in [@Cai:2011xv] once one accounts for some differences in convention. In particular, one has to rescale the time coordinate to absorb the factor of $\hat{\alpha}_0$, in addition to redefining $dt \rightarrow \hat{\alpha}_0^{-1}\, dt$ along with $\hat{v}^*_i \rightarrow \frac{\hat{\alpha}_0}{r_D^2} \, \hat{v}^*_i$ and $\hat{v}_*^i \rightarrow \hat{\alpha}_0\, \hat{v}^*_i$. The inhomogeneity in the scaling of the spatial velocities results from the fact that we define our hypersurface metric to be the induced metric rescaled by a factor of $r_D^{-2}$, while [@Cai:2011xv] works with the induced metric. Also, our derivation here allows us to go to ${\cal O}(\hat{\aleph}^{-3})$ which is necessary in order to see the dynamical Navier-Stokes equation on the hypersurface $\Sigma_D$.
The boundary data for the non-relativistic Dirichlet fluid {#s:dbdymet}
----------------------------------------------------------
Apart from the construction of the bulk dual to the non-relativistic fluid on the Dirichlet surface, we would also like to know what the corresponding physics on the boundary is. For instance we see that the boundary velocity field can be read off from the terms with $dr$ in and is given as $$\begin{split}
u_t &= -\hat{\alpha}_0 - \hat{\aleph}^{-2} \hat{\alpha}_0 \left[- \frac{1}{2}\hat{h}^*_{tt} + \frac{1}{2} \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_* +\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right] + {\cal O}(\hat{\aleph}^{-4}) \\
u_i &= \hat{\aleph}^{-1}\, \hat{\alpha}_0\, \left(\hat{v}^*_i + \hat{k}^*_i \right)\\
&\quad +\hat{\aleph}^{-3}\,
\hat{\alpha}_0\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+ \left( \frac{1}{2}\hat{h}^*_{tt} + \,\hat{k}^*_{j}\, \hat{v}^{j}_{*} +\frac{1}{2} \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*} +\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}\right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right]\\
&\qquad -\hat{\aleph}^{-3}\frac{\hat{\alpha}_0^2}{r_D\left(1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right)}\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*}-\hat{f}_i^* \right] \\
& \qquad
+\hat{\aleph}^{-3}\left[ b_0^2\, \hat{\kappa}_L\,\hat{\alpha}_0\, \hat{\nabla}^2_{(0)} \hat{v}^*_i -\frac{\hat{\alpha}_0^2}{r_D^2} \, \frac{\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}}{d\,(d-2)\, (\hat{\alpha}_0 +1) }\right] + {\cal O}(\hat{\aleph}^{-4})
\end{split}$$ while the boundary metric can be read off from the large $r$ behavior and is given to be $$\begin{split}
g_{tt} &= -\hat{\alpha}_0^2 + \hat{\aleph}^{-2} \, \left[\hat{h}^*_{tt}\ +
\left(1-\hat{\alpha}_0^2 \right) \left(- \hat{h}^*_{tt} + \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_*+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right) \right] + {\cal O}(\hat{\aleph}^{-4}) \\
g_{ti} &= \hat{\aleph}^{-1}\left( \hat{k}^*_i + (\hat{\alpha}_0^2-1)\, (\hat{k}_i^* + \hat{v}_i^*) \right)
- \hat{\aleph}^{-3}\frac{\hat{\alpha}_0^3}{r_D\left(1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)\right)}
\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*} -\hat{f}_i^* \right] \\
&\quad -\hat{\aleph}^{-3} \left(1-\hat{\alpha}_0^2 \right)\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+ \left(\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*}+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right]\\
&\quad +\;\hat{\aleph}^{-3} \left[\frac{2 \, b_0}{\hat{\alpha}_0}\, F(b_0\,r_D)\hat{v}_{*}^j\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} + \frac{\hat{\alpha}_0^4}{2\,(d-2)\, r_D^2)}\hat{\nabla}^j_{(0)}\hat{q}^*_{ij} -
\frac{\hat{\alpha}_0^2\,(\hat{\alpha}_0^2-1)}{2\, (d-2)\, r_D^2}\left(1+\frac{2}{d\,\hat{\alpha}_0\,(\hat{\alpha}_0+1)}\right) \hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\right] \\
&\quad
+\;\hat{\aleph}^{-3} \, \left[
-\, b_0^2\, \hat{L}_1(\infty) \, \hat{\nabla}^2_{(0)}\hat{v}^*_{i} \right] + \;{\cal O}(\hat{\aleph}^{-4}) \\
g_{ij} &= \hat{g}^{(0)}_{ij} + \hat{\aleph}^{-2} \left(\hat{h}^*_{ij} -(\hat{\alpha}_0^2 -1)\, (\hat{v}^*_i + \hat{k}^*_i) \,(\hat{v}^*_j + \hat{k}^*_j) - \frac{2 \, b_0}{\hat{\alpha}_0}\, F(b_0\,r_D) \,\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \right) + \;{\cal O}(\hat{\aleph}^{-4})
\end{split}$$ with $\hat{L}_1(\infty)$ denoting the asymptotic value of $\hat{L}_1(br)$.
Note that even in the non-relativistic limit the boundary ‘dynamical’ metric $g_{\mu\nu}$’s light-cone is being opened up relative to that of the hypersurface metric. As before this is a consequence of the red-shift effect. We are normalizing here the hypersurface metric to have flat Minkowski metric to leading order and this causes a rescaling by an amount $\hat{\alpha}_0^2$ on the boundary. As long as $\hat{\alpha}_0$ is finite this is not an issue for we are just encountering an overall rescaling of the boundary time.
What we have derived here is the AdS analog of the membrane paradigm connection recently proposed in [@Bredberg:2011jq; @Compere:2011dx]. Recall that the construction described in these papers proceeds by looking at an asymptotically flat geometry with Dirichlet boundary conditions at some timelike hypersurface (the analog of our $\Sigma_D$) and one solves vacuum Einstein’s equations. It was shown that the hypersurface dynamics is constrained to obey the incompressible Navier-Stokes equations, just as what we have shown above. However, the solutions described in this section solve Einstein’s equations with a negative cosmological constant and we furthermore have argued that the Dirichlet dynamics is obtained by suitably dressing up of a CFT fluid by allowing it to propagate on a ‘dynamical’ background metric. In our context it is clear that the interpretation of the physics is less in terms of an RG flow, and more along the lines of the medium dependent ‘dressing up’ of the boundary fluid dynamics in contrast to the Wilsonian RG perspective put forth as an interpretation of the membrane paradigm originally in [@Bredberg:2010ky].
There is however one regime where our analysis should be able to make some contact with the discussion in [@Bredberg:2011jq; @Compere:2011dx]; this is the near horizon regime where one expects to encounter a local Rindler geometry, which is the starting point for analyzing the Dirichlet problem in flat space. In the next section we show how one can embed this construction using the solutions we have described, enabling one thus to explore the AdS version of the membrane paradigm.
The near horizon Dirichlet problem {#s:nh}
==================================
So far in our discussion the Dirichlet hypersurface $\Sigma_D$ has been located at some radial position $r_D$ that is finite. We now want to investigate what happens as we push this surface closer towards the horizon, i.e, $\Sigma_D \to {\cal H}^+$ via $r_D \to b^{-1}$. To understand the resulting physics we first have to realize that we are doing something strange: the horizon is a null surface and has therefore a degenerate metric. $\Sigma_D$ on the other hand is constrained to be a timelike surface with a non-degenerate metric. So it is clear that demanding a well-behaved metric is going to result in an infinite scaling by the red-shift factor $\hat{\alpha}$; the main question is whether one can implement the scaling while retaining interesting dynamics. We will now proceed to show that there exists a scaling of parameters such that the near-horizon geometry makes sense. Furthermore, we argue that this allows us to embed the flat space constructions of [@Bredberg:2011jq] into our AdS set-up.
We outline the construction in a couple of stages to guide intuition: firstly in we will examine the conservation equation on the hypersurface and from there infer the scaling of parameters. This is the only sensible thing to do for us, since the entire dynamical system of the boundary CFT fluid has been converted into that of the hypersurface fluid living on an inert background. We then examine the consequences of the scalings we derive in focussing on the region close to the horizon; this amounts to blowing up the region of spacetime between ${\mathcal H}^+$ and $\Sigma_D$. This blown up region bears close resemblance to the solution of the vacuum Einstein’s equations dual to the incompressible Navier-Stokes system discussed in [@Bredberg:2011jq]. Indeed one should anticipate this based on the usual intuition that near any non-degenerate horizon one encounters a patch of the Rindler geometry. However, we will also encounter differences owing to the fact at the end of the day we are solving with a non-vanishing cosmological constant.
There is also the further question of what the geometry between the Dirichlet surface and the asymptotic AdS region looks like and moreover what is the boundary physics of our scalings? We argue in that the near horizon scaling regime renders the bulk metric viewed as a Lorentzian metric on a spacetime manifold nonsensical. However, it turns out that there is a nice language to describe the geometry in terms Newton-Cartan structures and we show that the bulk co-metric (which in usual Lorentzian geometry is the inverse metric) is well behaved as is the boundary co-metric. The result is quite satisfying from the physical perspective: the near horizon limit demands a drastic modification of the boundary metric, which forces one into a non-relativistic or Galilean regime. Consequently, rather than describing geometry in terms of Lorentzian structure, we are forced to use the less familiar but equally effective geometrization of the idea of a Galilean spacetime, in terms of Newton-Cartan geometry (see [@Misner:1973by] for a nice account of this subject and [@Ruede:1996sy] for a more recent review).
Scaling of the Dirichlet dynamics in the near horizon region {#s:dhypdyn}
------------------------------------------------------------
To understand the behavior of the fluid on the Dirichlet surface as we push $\Sigma_D$ closer to the horizon, we first look at the conservation equation in this limit. It turns out that demanding non-trivial dynamics on the Dirichlet surface forces one into a scaling regime of the fluid, effectively making it non-relativistic. However, the scaling we encounter is not quite the BMW scaling [@Bhattacharyya:2008kq] discussed earlier in but a slightly modified version of the same.
### A new scaling regime {#s:nhnewscale}
Ignoring for the moment the fact for $r_D < r_{D,snd}$ we are supposed to be projecting out the sound mode to ensure subluminal propagation on the Dirichlet hypersurface, let us write the relativistic conservation equations and examine them as we zoom in towards the horizon. Consider then the conservation equation on the Dirichlet hypersurface ; we have analyzed this from a generic viewpoint in but for now we will focus on the truncated equations to second order by setting $\hat{\pi}^{\mu\nu} = -2\eta\, \hat{\sigma}^{\mu\nu}$ (cf., ). Projecting these conservation equations parallel and transverse to the velocity we get (using $\eta$ as given in ) $$\begin{aligned}
&& (d-1)\, \hat{u}^\mu \, \frac{ {\hat \nabla}_\mu b}{b} + {\hat \theta} + \frac{2b}{\hat{\alpha}\ d} {\hat u}_\mu \,{\hat \nabla}_\nu {\hat \sigma}^{\mu\nu} = 0 \nonumber \\
&& - \, \, \left(1+\frac{d}{2}\,(\hat{\alpha}^2-1)\right)\, {\hat P}_\mu^{\; \alpha} \frac{{\hat \nabla}_\alpha b}{b} + {\hat a}_\mu -\frac{2\, b^d}{\hat{\alpha}\ d} \, {\hat P}_{\mu\alpha} \, {\hat \nabla}_\beta \left(\frac{1}{b^{d-1}}\,{\hat \sigma}^{\alpha \beta}\right) =0
\label{releom1}\end{aligned}$$ We are going to try to analyze these equations in the limit when $\hat{\alpha} \to \infty$.
From it is clear that if we insist on leaving the hypersurface data independent of $\hat{\alpha}$ then we find that we have contributions at different orders that need to be independently cancelled. The most constraining equation is the $ {\cal O}(\hat{\alpha}^2)$ term from the transverse equation which demands that the spatial gradients of $b$ vanish. Then at $ {\cal O}(1)$ we have to kill the acceleration and have a non-trivial equation from the longitudinal part. At order $\hat{\alpha}^{-1}$ we would need to kill all terms that show up with the shear viscosity. The upshot is that we are left with vacuous dynamics on the Dirichlet hypersurface should $b$ and $\hat{u}^\mu$ be ${\cal O}(1)$ as $\hat{\alpha} \to \infty$.
While this sounds a bit strange, a moments pause reveals that this is indeed what one should expect on physical grounds. The horizon of a black hole is a null surface (it is generated by null generators) and in the process of moving the Dirichlet hypersurface to the horizon, we are effectively doing an infinite rescaling (hence $\hat{\alpha} \to \infty$) to bring the Dirichlet metric $\hat{g}_{\mu\nu}$ to be timelike and non-degenerate. Before we do such a rescaling however, we are in the ultra-relativistic regime as far as the horizon goes – in such a regime it is natural to expect that there is no dynamics, the fluid streams along the null generators and is effectively frozen into a stationary flow.
It is then clear that in order to obtain non-trivial dynamics on the Dirichlet surface one has to scale the fields $b$ and $\hat{u}^\mu$ in some fashion. The crucial question is whether there is any scaling that retains non-trivial dynamics; operationally demanding that we obtain an ‘interesting’ non-linear equation. Consider the following scaling:[^33] $$b = \hat{\varkappa} \, b_\bullet + \frac{1}{\hat{\varkappa}^{3}}\, \delta b_\star \ , \qquad \hat{u}^\mu = \left( 1 + {\cal O}(\hat{\varkappa}^{-2} ), \;\hat{\varkappa}^{-1} \, v^i_\star\right) \ , \qquad \hat{\alpha} = \hat{\varkappa}\ \hat{\alpha}_\bullet \left( 1 + {\cal O}(\hat{\varkappa}^{-2} )\right)
\label{nhscaling}$$ where the functions with subscript $\star$ have specific functional form with anisotropic scaling of their spatial and temporal gradients. Specifically, $$\hat{{\cal Y}}_\star(t,x^i) : {\mathbb R}^{d-1,1} \mapsto {\mathbb R}\ , \;\; \text{such that}
\;\; \{ \partial_t \hat{{\cal Y}}_\star(t,x^i), \partial_i \hat{{\cal Y}}_\star(t,x^i)\} \sim \{{\cal O}(\hat{\varkappa}^{-2}) ,{\cal O}(\hat{\varkappa}^{-1})\}$$ where we assign gradient weight $1$ to the spatial derivatives and $2$ to temporal derivatives. This is inspired of course by the non-relativistic BMW scaling discussed in and as discussed there the ${\cal O}(\hat{\varkappa}^{-2})$ part of the velocity field is fixed by normalization.[^34]
However, there is a crucial difference from the BMW scaling; the leading term in the expansion of $b_\bullet$ is growing with $ \hat{\varkappa}$, which seems to be an issue. Nevertheless, it is easy to check that under this scaling (which admittedly is obtained by demanding a sensible $ {\cal O}(1)$ equation from the longitudinal part) one finds that the equations can be reduced to: $$\partial_i v^i_\star =0 \ , \qquad -\frac{d\hat{\alpha}_\bullet^2}{2\, b_\bullet} \partial_i \delta b_\star + \partial_t v_i^\star + v^j_\star \partial_j v_i^\star - \frac{b_\bullet}{\hat{\alpha}_\bullet\ d} \nabla^2 v_i^\star = 0
\label{nrnseq1}$$ where we have specified to a flat Minkowski background metric for specificity ($\hat{g}_{\mu\nu} = \eta_{\mu\nu}$) and to make the equations more familiar. These are of course, the unforced (by background) Navier-Stokes equations which we have earlier described via the BMW scaling in . The incompressibility condition as in that case arises from the longitudinal conservation equation at ${\cal O}(\hat{\varkappa}^{-2}\,\hat{\alpha})$ while the Navier-Stokes equation itself appears at ${\cal O}(\hat{\varkappa}^{-3} \, \hat{\alpha})$. A similar scaling can be done for the forced Navier-Stokes system, we simply use the BMW scaling for the fluctuating part of the hypersurface metric.
### Deconstructing the near horizon scaling {#s:dcnh}
How do we reconcile this scaling derived here for the relativistic fluid with that derived by BMW [@Bhattacharyya:2008kq]? Firstly, let’s note that the rationale of our scaling up the background value of $b \sim \hat{\varkappa}\, b_\bullet$ is that shear term survives the scaling. From the second equation of we clearly see that this is required in order to retain the shear term in the limit. On the other hand the non-relativistic pressure in the BMW limit is proportional to $\hat{\alpha}_0^2\, \frac{\delta b_*}{b_0}$ and in the near horizon limit both $b_0 \sim \hat{\varkappa}\, b_\bullet$ and $\hat{\alpha}_0 \sim \hat{\varkappa}$ diverge. The extra scaling down of $\delta b_* \sim \frac{1}{\hat{\varkappa}}\, b_\star$ (over and above the scaling by $\hat{\varkappa}^{-2}$) is necessary to offset this divergence and ensure that the pressure gradient term contributes at the same order as the convective derivative $\partial_t + v^i_*\partial_i $ and the shear. The effective pressure that enters into the equation is $$\hat{\varkappa}^{-2}\,\hat{p}_\star = -\frac{d}{2} \, \hat{\varkappa}^{2}\,\hat{\alpha}_\bullet^2 \frac{\hat{\varkappa}^{-3}\, \delta b_\star}{\hat{\varkappa} \, b_\bullet}$$ using the scaling of $\hat{\alpha}$ in so that the first term of the Navier-Stokes equation in is essentially $\partial_i \hat{p}_\star$. So the final equations of motion on the near horizon region for the hypersurface dynamics are simply: $$\partial_i v^i_\star =0 \ , \qquad \partial_i \hat{p}_\star + \partial_t v_i^\star + v^j_\star \partial_j v_i^\star - \, \hat{\nu}_\bullet \nabla^2 v_i^\star = 0
\label{nrnseq2}$$ with kinematic viscosity $$\hat{\nu}_\bullet = \frac{b_\bullet}{\hat{\alpha}_\bullet\ d}$$ We have here glossed over the fact that since we scale $b \sim \hat{\varkappa}$ we potentially have a problem. The limit seems to suggest that we are taking the zero temperature limit of the black hole geometry since the Hawking temperature (which is seen on the boundary) is $T \sim b^{-1}$ . This naively sounds like we are outside the long wavelength regime and as a result should not be using the fluid/gravity map to describe the Dirichlet problem.
A different way to say this from a dynamical equation of motion perspective is that the scalings were derived by demanding that the equations remained non-trivial, which operationally means that different terms appear to have inhomogeneous weights. Hence by suitable fiddling of amplitudes and derivatives we engineered that terms which originally scaled as $\hat{\alpha}$ to various powers all contribute homogeneously. However, we have not analyzed the higher order contributions to the equations of motion. Is it possible that relativistic two derivative terms in the stress tensor, which show up as the corrections to viscous relativistic hydrodynamics equations of show up at the same order? Note that this is not a problem for the case discussed in [@Bhattacharyya:2008kq] for they engineered their scalings about a fixed background temperature, and there were no stray factors of $\hat{\aleph}$ (equivalently $\hat{\varkappa}$) in (in front of $b_0$) to augment higher order contributions.
For us to make the argument that the higher order terms are suppressed requires knowledge of the transport coefficients at second order on the Dirichlet hypersurface. This is in principle calculable by extending the Dirichlet fluid/gravity map of to one higher order in gradients. However, we will now argue that there is an essential simplification which allows us to make the statement boldly without computation. The rationale is simply that the enhanced scaling of the zero mode part of $b$ is compensated by the suppressing the corresponding fluctuation term $\delta b$ so as to ensure that $p_\star$ is finite. Since the overall value of $b_0$ scales out in the non-relativistic regime (we will see this clearly from implementing the scalings in the next subsection) it cannot affect the resulting equations. So we conjecture that accounting for higher order corrections as well, one will obtain as the leading order equations in the $\hat{\varkappa}$ expansion (all further corrections are suppressed by higher powers of $\hat{\varkappa}$).
One physical way to motivate the correction is to first note that while the boundary temperature is being scaled to zero in the limit the hypersurface temperature can indeed be maintained to be finite:[^35] $$\hat{T}_\bullet = \frac{\hat{\alpha}_\bullet\ d}{4\pi \,b_\bullet}$$ So from the Dirichlet observer’s point of view there is still scope for a non-relativistic scaling, and in fact this is how the Dirichlet observer would carry out the BMW regime. What is clear is that due to the extra red-shifting in translating to the boundary, the asymptotic observer is going to have trouble reconciling this near horizon scaling regime in his variables. We will return to this issue in .
In summary, the non-relativistic incompressible Navier-Stokes equations determine the dynamics of the hypersurface fluid as $\Sigma_D$ approaches the horizon; we will now proceed to investigate what this means for the various metrics: first we look at the bulk metric first and then examine what is the effect on the boundary.
The bulk metric between $\Sigma_D$ and ${\mathcal H}^+$ {#s:nhdirmet}
-------------------------------------------------------
From the discussion at the end of it seems natural that we should try to ask the question as to whether we can satisfy the constraints of the Dirichlet problem in the region between $\Sigma_D$ and the horizon (for the moment forgetting about the asymptotic region). Clearly we have $\Sigma_D$ approaching the horizon, so we will be forced to consider a double-scaling regime where we zoom in close to the horizon and expand out the spacetime region in-between. Let us temporarily ignore the consequences of this on the equations of motion and formally take the limit of the bulk metric given in , .
To zoom into the region between the horizon and $\Sigma_D$, realizing that $r_D \to \frac{1}{b}$ in the limit, we take: $$r = \frac{1}{b_\bullet\, \hat{\varkappa}} \left(1+ \frac{\rho}{\hat{\varkappa}^2 b_\bullet\ \hat{\alpha}_\bullet}\right) .$$ from which it follows that $$dr = \frac{1}{\hat{\varkappa}^3\, b_\bullet^2\ \hat{\alpha}_\bullet } \,d\rho$$
Similarly we parametrize the position of our Dirichlet hypersurface via $$r_D = \frac{1}{b_\bullet\, \hat{\varkappa}} \left(1+ \frac{\rho_D}{\hat{\varkappa}^2 b_\bullet\ \hat{\alpha}_\bullet}\right) .$$ Both $\rho_D$ and $\hat{\alpha}_\bullet$ are a measure of how close we are to the horizon after one has zoomed into the near horizon region. To relate them, we subtitute $r_D$ into the expression for $\hat{\alpha}$ and do a large $\hat{\varkappa}$ expansion. Identifying the leading term as $\hat{\alpha}_\bullet$, we get $$\rho_D \equiv\frac{b_\bullet}{\hat{\alpha}_\bullet\ d} = \hat{\nu}_\bullet$$ We see that $\rho_D$ is same as the kinematic viscosity - hence, the $\rho_D\to 0$ limit is identical to the inviscid limit in hydrodynamics.
Implementing this change of variables and the scaling in the bulk metric we obtain the near-horizon metric. In fact, the fastest way to derive the metric in is to utilize the fact that the near horizon limit is essentially the non-relativistic limit together with a particular form of the pressure fluctuations; essentially we substitute $\hat{\alpha}_0 = \hat{\varkappa} \, \hat{\alpha}_\bullet$, $b_0 = \hat{\varkappa}\, b_\bullet$, $\hat{p}_* = \hat{p}_\star$ into along with the identification $\hat{\aleph} = \hat{\varkappa}$ to ensure that we have the correct scalings. We first calculate the near horizon expansions of various functions appearing in the metric $$\begin{split}
\hat{\alpha} &= \hat{\alpha}_\bullet \, \hat{\varkappa}
\left[1 + \hat{\varkappa}^{-2} \left( \hat{p}_\star + \frac{d+1}{4d\hat{\alpha}_\bullet^2}\right)+ {\cal O}(\hat{\varkappa}^{-4}) \right] \\
\hat{\varkappa}^2 b_\bullet^2 r^2 &= \left[1+\hat{\varkappa}^{-2}\frac{2\rho}{\hat{\alpha}_\bullet^2\rho_D d}+{\cal O}(\hat{\varkappa}^{-4}) \right]\\
\hat{\varkappa}^2 b_\bullet^2 r^2\left[1- \hat{\alpha}^2\, f(br) \right] &=\left(1-\frac{\rho}{\rho_D}\right)
\left[1 + 2\hat{\varkappa}^{-2} \left( \hat{p}_\star -\frac{d-3}{4d\hat{\alpha}_\bullet^2}\frac{\rho}{\rho_D}\right)+ {\cal O}(\hat{\varkappa}^{-4}) \right] \\
-\hat{\varkappa}^2 b_\bullet^2 r^2 \hat{\alpha}^2\, f(br)&= -\frac{\rho}{\rho_D} + 2\hat{\varkappa}^{-2}\hat{p}_\star\left(1-\frac{\rho}{\rho_D}\right)\\
&\quad +\hat{\varkappa}^{-2}\frac{\rho}{2\hat{\alpha}_\bullet^2\,\rho_D} \left(\frac{(d-3)\, \rho- (d+1) \, \rho_D}{\rho_D d} \right) + {\cal O}(\hat{\varkappa}^{-4}) \\
\frac{2\hat{\alpha}^2}{r_D\left(1+\frac{d}{2}(\hat{\alpha}^2-1)\right)} &= 4 \rho_D \hat{\varkappa}\hat{\alpha}_\bullet\left[1
+ {\cal O}(\hat{\varkappa}^{-2}) \right]\\
2\,r\,\frac{\hat{\alpha}(2\hat{\xi}-1)}{1+\frac{d}{2}\, (\hat{\alpha}^2 -1)} &= \frac{4\rho_D}{\hat{\varkappa}^2 b_\bullet^2}\left(1-\frac{\rho}{\rho_D}\right)\left[1
+ {\cal O}(\hat{\varkappa}^{-2}) \right]\\
\end{split}$$ Using these, we get the metric as $$b_\bullet^2\, \hat{\varkappa}^2\,ds^2 = ds_0^2 + \hat{\varkappa}^{-1} \, ds_1^2 + \hat{\varkappa}^{-2} \, ds_2^2 + \hat{\varkappa}^{-3} \, ds_3^2+ {\cal O}(\hat{\varkappa}^{-4})
\label{}$$ where $$\begin{aligned}
ds_0^2 &=&
2\, dt \, d\rho
-\,\frac{\rho}{\rho_D} dt^2 + \delta_{ij}\, dx^i\, dx^j
\nonumber \\
ds_1^2 &= &
- 2\, \hat{v}^\star_i \, d\rho\, dx^i - 2 \, \left(1-\frac{\rho}{\rho_D} \right) \hat{v}^\star_i \, dt\, dx^i
\nonumber \\
ds_2^2 &= &
2\, \left( \frac{1}{2}\,\hat{v}_\star^2 + \hat{p}_\star +
\frac{d+1}{4\,d} \frac{1}{\hat{\alpha}_\bullet^2}\right) dt \, d\rho
\nonumber \\
&& \qquad
+ \; \left[ \left(1-\frac{\rho}{\rho_D} \right)
\left( \hat{v}_\star^2
+ 2 \hat{p}_\star \right)
+ \frac{\rho}{2\hat{\alpha}_\bullet^2\,\rho_D} \left(\frac{(d-3)\, \rho- (d+1) \, \rho_D}{\rho_D d} \right)\right] dt^2
\nonumber \\
&& \qquad
+\; \left[\left(1-\frac{\rho}{\rho_D} \right) \hat{v}^\star_i\,\hat{v}^\star_j
+ \, \frac{2 \, \rho}{\hat{\alpha}_\bullet^2\rho_D d}\, \delta_{ij} \right] dx^i \, dx^j
\nonumber \\
ds_3^2 &= &
2 \left(1-\frac{\rho}{\rho_D} \right)
\left[2\rho_D \left( \partial_{t}\hat{v}_{i}^\star
+\hat{v}_\star^{j} \partial_j \hat{v}_{i}^\star\right)
- \left(\hat{v}_\star^2 + 2 \hat{p}_\star
- \frac{d-3}{2d\hat{\alpha}_\bullet^2}\, \frac{\rho}{\rho_D}\right) \hat{v}_i^\star \right] dx^{i}\, dt
\nonumber \\
&& \qquad
+\; 2\left[2\rho_D \left( \partial_{t}\hat{v}_{i}^\star
+\hat{v}_\star^{j} \partial_j \hat{v}_{i}^\star\right)-\;\,
\left( \frac{1}{2}\,\hat{v}_\star^2
+ \hat{p}_\star + \frac{d+1}{4\,d} \frac{1}{\hat{\alpha}_\bullet^2}\right) \, \hat{v}_i^\star \,\right] d\rho\, dx^i \,
\nonumber \\
&& \qquad
-\,\left[\rho^2-\rho_D^2+4\, \rho_D\,(\rho_D-\rho)\right]\,\partial_j\partial^j \hat{v}^{\star}_i \, dx^i\,dt
+ \frac{1}{d}\, \partial_j\partial^j \hat{v}^{\star}_i \, d\rho\, dx^i \,.
\label{hypnrscale}
\end{aligned}$$ We have restricted attention to the simplest setting where $\hat{g}_{\mu\nu} = \eta_{\mu\nu}$ (hence $\hat{g}^{(0)}_{ij} = \delta_{ij}$ above) and $\hat{v}_\star^2 = \delta_{ij} \, \hat{v}_\star^i \, \hat{v}_\star^j$. In deriving the expressions as in we have used the incompressibility condition, which appears as before as the leading order fluid equation of motion. The Navier-Stokes equations themselves can also be used to simplify the last term in : one can replace the convective derivative of the spatial velocity $\partial_{t}\hat{v}_{i}^\star +\hat{v}_\star^{j} \partial_j \hat{v}_{i}^\star$ by $-\partial_i \hat{p}_\star + 2\,\hat{\nu}_\bullet \, \nabla^2 v_i^\star$.
This is in fact closely related (but not identical) to the metric derived in [@Bredberg:2011jq] and generalized further in [@Compere:2011dx] to higher orders (compare with for instance Eq (3.2) and (6.5) of the latter paper). We recall that these works consider solutions to vacuum Einstein’s equations without a cosmological constant. In particular, [@Bredberg:2011jq] solves the Dirichlet problem in flat space by looking for small gradient fluctuations around a Rindler geometry, while we are dealing with solutions of which has a non-vanishing negative cosmological constant.
We see that to leading order we nevertheless should expect agreement to the analysis of [@Bredberg:2011jq; @Compere:2011dx] – this is clear from the first two lines of . This is reflecting the universal Rindler nature of a non-degenerate horizon such as that of the planar Schwarzschild-AdS$_{d+1}$ solution we started with. Moreover, we can also physically understand how a solution of reduces to that of vacuum Einstein’s equations by noting that in the limit we consider here, the cosmological constant gets diluted away. The factor of $\hat{\varkappa}$ on the l.h.s, is essentially indicative of $R_\text{AdS} \to \hat{\varkappa}^2 \, R_\text{AdS}$ in our near horizon limit. This is to be expected; by zooming in onto the region between $\Sigma_D$ and ${\cal H}^+$ we are effectively blowing up a small sliver of the spacetime and are thus losing any information about the background curvature scale $R_\text{AdS}$. In some sense the near horizon limit is like the limit we take to decouple the asymptotically flat region from the AdS throat in the D3-brane geometry; what is unclear is whether some notion of decoupling exists in the present context.
But starting at ${\cal O}(\hat{\varkappa}^{-2})$ on the l.h.s. of we start to see deviations from the metric presented in [@Bredberg:2011jq; @Compere:2011dx]. Even though we are zooming close to the horizon of a Schwarzschild-AdS$_{d+1}$ black hole, starting at second order we should expect to see curvature contributions. The various terms at this and higher orders originate from how the Rindler geometry gets corrected as we step away from the strict limit. In particular, using the near-horizon scaling of variables we see that reduces to $$R_{AB}+\frac{d}{\hat{\varkappa}^2} \; \frac{1}{(d\, \hat{\alpha}_\bullet\, \rho_D)^2}\; {\mathcal G}_{AB}=0\,.
\label{}$$ It follows that all terms which involve $\frac{1}{\hat{\alpha}_\bullet^2}$ in the denominator correspond to the corrections due to the AdS curvature. Setting such terms in to zero i.e., taking $\hat{\alpha}_\bullet \to \infty$ leads us to the metrics derived in [@Bredberg:2011jq; @Compere:2011dx].
We are here restricting attention to the metric to leading orders in the $\hat{\varkappa}$ expansion; upto ${\cal O}(\hat{\varkappa}^{-3})$. In principle since the fluid/gravity metrics are known to higher orders one can carry out the construction to higher orders and we expect that we should be able to reproduce the results of [@Compere:2011dx] who have derived the expressions in the near horizon limit to ${\cal O}(\hat{\varkappa}^{-6})$ in our notation. Note that this simplification of the near horizon region construction happens because of the correlated amplitude and gradient scaling, as already emphasized in [@Bhattacharyya:2008kq].
Near horizon limit and Galilean degenerations {#s:nhgal}
---------------------------------------------
Having seen that it is possible to take the near horizon scaling limit and retain both interesting dynamics on $\Sigma_D$ and further have a sensible geometry between $\Sigma_D$ and ${\mathcal H}^+$, our next goal is to address whether we can make sense of the metric between the hypersurface and the boundary. There are already many indications that this is going to be problematic; the vanishing of the boundary temperature $ \propto b^{-1}$ being the most prominent one, which suggests break-down of the gradient expansion.
The bulk metric turns out not to make sense, but we shall see that there is an object that does – this is the inverse bulk metric which we shall refer to as the co-metric (see below). Similarly, the boundary co-metric also is well behaved and we will argue naturally provides a Newton-Cartan like structure on the boundary so that the boundary geometry degenerates from a Lorentzian manifold into a Galilean manifold.
In the following, we will call the metric on the co-tangent bundle $g^{\mu\nu}$ as the co-metric. This is a more accurate terminology in the context of non-relativistic limit (the Galilean or Newton-Cartan limit) than the usually preferred ‘inverse metric’ since in this limit the co-metric degenerates (becomes non-invertible) and the usual metric ceases to exist. Hence, we will prefer as much as possible to work with the co-metric instead of the metric.
### Emergence of Galilean structure on the boundary
Let us first examine what happens to the boundary metric data when we try to push the Dirichlet hypersurface towards the horizon. This corresponds to $\hat{\alpha}$ tending to infinity. We will work with the relativistic expressions of to maintain covariance; it is easy to then pass over to the near horizon scaling regime discussed in and obtain explicit parameterization of the results.
From the formulae for the boundary metric in terms of $u^\mu$ we see that there are various terms which blow up. Moreover, in some cases the higher derivative terms overwhelm the lower derivatives suggesting a breakdown of derivative expansion, as already suspected from the vanishing of the boundary temperature. It is clear that this limit if it exists cannot be straightforward.
As the Dirichlet hypersurface tries to approach the horizon, it first hits the hypersurface $r_D=r_{D,snd}$. From the boundary viewpoint, the interaction between the fluid with its gravitational potential packet drives the system to have superluminal sound propagation, $\hat{c}_{snd}$ exceeds the dressed speed of light as determined by $\hat{g}^{\mu\nu}$. By this point the interaction between the boundary metric and the fluid has become so important that the bare velocities $u^\mu$ have no more physical significance. We should rather think in terms of the dressed fluid and see how we can make sense out of the approach to the horizon.
So, we will first rewrite the above formulae in terms of the dressed velocity $\hat{u}^\mu$. The dictionary for the (normalized) metric and the co-metric are $$\begin{split}
g_{\mu\nu}
&= \hat{g}_{\mu\nu} + \left[1-\hat{\alpha}^2-\frac{2\,\hat{\alpha}^3\,\hat{\theta}}{r_D(\,d-1)}\right]\hat{u}_\mu \hat{u}_\nu+\frac{2\,\hat{\alpha}^3 \, \hat{u}_{(\mu} \hat{a}_{\nu)}}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} - \frac{2\,b}{\hat{\alpha}} \,F(br_D)\,\hat{\sigma}_{\mu\nu} \\
g^{\mu\nu} &= \hat{g}^{\mu\nu}+ \left[1-\frac{1}{\hat{\alpha}^2}+\frac{2\, \hat{\theta}}{\hat{\alpha}\, r_D\, (d-1)}\right]\hat{u}^\mu \hat{u}^\nu+\frac{2\,b}{\hat{\alpha}} \,F(br_D)\, \hat{\sigma}^{\mu\nu}-\frac{2\,\hat{\alpha}\, \hat{u}^{(\mu} \hat{a}^{\nu)}}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\\
\end{split}$$
We have already remarked on the necessity to project out the dressed sound mode and take an incompressible limit of the dressed fluid a la BMW as the Dirichlet hypersurface crosses $r_D=r_{D,snd}$ in . The formulae above reinforce that intuition – we see for example that unless $\hat{\theta}$ is sent to zero, the first derivative terms in the boundary metric overwhelm the zeroth order answer thus leading to a breakdown of the derivative expansion. So, we will drop the $\hat{\theta}$ terms with the understanding that we are projecting into the incompressible sector of the dressed fluid. Even this does not seem to help the case for the metric, since it still diverges – keeping only leading order terms (and assuming none of the subsequent terms grow with $\hat{\alpha}$ – we will postpone a more careful analysis for the future) we get $$\begin{split}
g_{\mu\nu}
&= -\hat{\alpha}^2\hat{u}_\mu \hat{u}_\nu+\ldots\\
g^{\mu\nu}
&= \hat{g}^{\mu\nu}+ \hat{u}^\mu \hat{u}^\nu+\ldots\\
\end{split}$$
We are thus led to a remarkable conclusion: the near horizon Dirichlet constitutive relation is that the boundary co-metric degenerates along $t_{\mu}\equiv\hat{u}_\mu$ i.e., $g^{\mu\nu}t_\nu=0$ and the boundary metric has one divergent time-like eigenvalue along the same direction $g_{\mu\nu}= -\hat{\alpha}^2 t_\mu t_\nu$. This is the signature that the boundary metric is becoming Newtonian/non-relativistic - crudely speaking this is analogous to the $c\to\infty$ behavior of Minkowski metric/co-metric $$\eta^{\mu\nu}=\text{diag}(-\frac{1}{c^2},1,1,\ldots) \quad\text{and}\quad
\eta_{\mu\nu}=\text{diag}(-{c^2},1,1,\ldots)$$ The way to make this mathematically precise is to resort to what is called as Galilean manifold.
A Galilean manifold is a manifold with a Newtonian time co-vector $t_\mu$, and a degenerate co-metric $g^{\mu\nu}$ satisfying $g^{\mu\nu}t_\nu=0$. As in standard differential geometry, we can demand a connection that is compatible with this structure. This requires that there exists a torsionless connection defining a covariant derivative $\nabla^{\text{NC}}_\mu$ which is compatible with both the co-metric and the time co-vector, i.e., $\nabla^{\text{NC}}_\lambda g^{\mu\nu}=0$ and $\nabla^{\text{NC}}_\mu t_\nu=0$. If in addition we have such a a torsionless covariant derivative $\nabla_\mu$ compatible with the Galilean structure then we call $\nabla^{\text{NC}}$ as the Newton-Cartan connection and the corresponding Galilean manifold is termed a Newton-Cartan manifold. See [@Misner:1973by; @Ruede:1996sy] for further discussions and [@Bagchi:2009my] for another AdS/CFT perspective on the Galilean structures.
Thus we have just argued that that there is a natural emergence of Galilean structure in the boundary theory as we take our Dirichlet surface nearer and nearer to the horizon. In this case, the metric that we have to put the field theory in gets enormously simplified and we get just a CFT on a Galilean manifold with a Galilean co-metric having $\hat{u}_\mu$ as the degenerate direction. This is a very precise way of stating what the membrane paradigm is from the boundary theory viewpoint – the membrane paradigm is a particular Galilean limit of the field theory.[^36]
Having shown that we have a Galilean structure on the boundary, we further ask: “Is there a Newton-Cartan structure"? Consider the Galilean limit of the Christoffel connection $\nabla_\mu$ which is by construction torsionless and compatible with the Galilean co-metric $g^{\mu\nu}$. This could be a sensible Newton-Cartan structure if it annihilates the time co-vector; to see what $\nabla_\mu t_\nu$ is we use $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho\hat{u}_\rho &= -(1-\frac{1}{\hat{\alpha}^2})\left[ \hat{\sigma}_{\mu\nu}+\frac{\hat{\theta}}{d-1}\ \left(\hat{P}_{\mu\nu}+\frac{d}{2}\hat{\alpha}^2\hat{u}_\mu \hat{u}_\nu\right)-\frac{d\hat{\alpha}^2}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{a}_{(\mu}\hat{u}_{\nu)}\right]\\
\end{split}$$ so that $$\begin{split}
\nabla_\mu t_\nu &= \hat{\nabla}_\mu\hat{u}_\nu+\tilde{\Gamma}_{\mu\nu}{}^\rho\hat{u}_\rho\\
&= -\frac{d}{2}(\hat{\alpha}^2-1)\frac{\hat{\theta}}{d-1}\hat{u}_\mu \hat{u}_\nu+\hat{\omega}_{\mu\nu}+\hat{a}_\mu \hat{u}_\nu
+\frac{1}{\hat{\alpha}^2}\left[\hat{\sigma}_{\mu\nu}+\frac{\hat{\theta}}{d-1}\hat{P}_{\mu\nu}\right] -\frac{2\hat{a}_{(\mu}\hat{u}_{\nu)}}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}
\end{split}
\label{nablat}$$ This in fact determines a sensible answer in the infinite $\hat{\alpha}$ limit , once we factor in our earlier observation that $\hat{\theta}$ is suppressed in the near horizon limit (it is ${\cal O}(\hat{\alpha}^{-4})$) as are any other gradients of the velocity field. These scalings can be read off essentially from as the near horizon limit is the BMW limit as far as the velocities are concerned.
The leading contribution to comes at ${\cal O}(\hat{\alpha}^{-2})$ from the vorticity and expansion. It is tempting to speculate that the degeneration of the co-metric along with define a Newton-Cartan like limit and we will postpone a structural analysis of this limit for future work. In the next subsection, we will try to examine the bulk co-metric to see what this Galilean limit entails in the bulk.
The bulk co-metric in the Near-Horizon limit
--------------------------------------------
In the preceding subsection, we argued for the emergence of Galilean structures in the Near-Horizon limit thus forcing us to formulate the boundary geometry in terms of a co-metric than a metric. Since the boundary metric is ill-behaved in this limit, it is clear that the bulk metric should be ill-behaved too. This poses a conundrum since it seems as if in taking the near-horizon limit we have destroyed the -asymptopia and one might wonder whether there is any sense in talking about the bulk geometry far away from the horizon. We will in this subsection take the description of the boundary geometry via a co-metric as a clue and argue that the bulk geometry is also well-described by a bulk co-metric. Thus while the metric description might break down the co-metric description with its associated Galilean structures continue to describe the bulk and the boundary geometry.
Now we present the co-metric as a function of the Dirichlet data. $$\begin{aligned}
\mathcal{G}^{AB}\partial_A\otimes\partial_B &=&
\left[r^2\, f(br)-\frac{2\, r\, \theta}{d-1}\right]\partial_r\otimes\partial_r+2\left[u^\mu -r^{-1} a^{\mu}\right]\partial_\mu\otimes_s\partial_r \nonumber \\
&&\qquad \qquad +\; r^{-2}\left[P^{\mu\nu} -2b F(br)\ \sigma^{\mu\nu}\right]\partial_\mu\otimes\partial_\nu \nonumber \\
&=& \left[r^2\, f(br)-\frac{2\,r\,\hat{\theta}}{\hat{\alpha}\,(d-1)}\right]\partial_r\otimes\partial_r \nonumber \\
&&\quad +\; 2\left[\frac{\hat{u}^\mu}{\hat{\alpha}}\left(1-\frac{\hat{\alpha}\,\hat{\theta}}{r_D\,(d-1)}\right) - \frac{\hat{a}^\mu}{r\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\right]\partial_\mu\otimes_s\partial_r \nonumber \\
&&\quad +\;r^{-2}\left[\hat{P}^{\mu\nu} -\frac{\hat{\alpha}\left[\hat{u}^\mu \hat{a}^\nu + \hat{a}^{\mu}\hat{u}^\nu\right]}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} -2b\, \hat{F}(br)\ \hat{\sigma}^{\mu\nu}\right]\partial_\mu\otimes\partial_\nu\end{aligned}$$ where $$\begin{split}
\hat{F}(br) &\equiv \frac{1}{\hat{\alpha}}\left(F(br)-F(br_D)\right) = \frac{1}{\hat{\alpha}} \; \int_{br}^{br_D}\; \frac{y^{d-1}-1}{y(y^{d}-1)}dy\,.
\end{split}$$ As we anticipated, there are no problems evident in the infinite $\hat{\alpha}$ limit (provided one takes $\hat{\theta}$ to zero appropriately to preserve the validity of derivative expansion). Contrast this with the bulk metric (see ) whose large $\hat{\alpha}$ limit seems dubious. This supports our contention that there is a completely sensible description of the bulk and the boundary geometry in terms of co-metrics everywhere when we take the near-horizon Dirichlet problem along with an incompressible limit on the dressed fluid.
Our exercise of rewriting the near-horizon bulk metric in terms of the non-relativistic variables can be repeated in the case of co-metric and we can easily convince ourselves that this goes through without any new subtleties. In fact, most terms in the above expression are sub-dominant in the $\hat{\varkappa}$ expansion introduced in the previous subsections. In fact, at the zeroth order the spatial part of the co-metric is just that of vacuum AdS; with the temporal part contributing only at a higher order. The horizon structure encoded in $(br)^{-d}$ is completely invisible until quite high orders in $\hat{\varkappa}$ expansion, which is to be expected since the boundary temperature is being scaled to zero. To be specific using the scaling we find $$\mathcal{G}^{AB}\partial_A\otimes\partial_B = r^2 \, \partial_r \otimes \partial_r + \frac{1}{r^2} \,
\partial_i \otimes \partial_i + \hat{\varkappa}^{-1} \left(2 \, \partial_t \otimes_s \partial_r \right) + {\cal O}(\hat{\varkappa}^{-2}) \,.$$ Basically we seem to find that the co-metric description is mostly oblivious to the near-horizon geometry (which as we have seen is well-described in the metric description). Hence, we are naturally led to an effective description where there are two regions of the geometry: one well-described by a metric, and another by a co-metric.
The interesting question is to see whether there is an overlap region where the two descriptions are valid and the metric is just the inverse of the co-metric. Having detained the reader for so long, we will leave the detailed answer of this question to future work. However, we would like to draw the attention of the reader to the following fact – the near-horizon limit of the Dirichlet problem naturally seems to lead to novel geometric structures closely associated with the Galilean limit. These Galilean structures evidently call for more detailed studies especially since they might hold valuable lessons for non-relativistic holography (as previously pointed out in [@Bagchi:2009my]).
Discussion {#s:discuss}
==========
The bulk Dirichlet problem in , which we defined as the gravitational dynamics in a spacetime with negative cosmological constant, subject to Dirichlet boundary conditions on a preferred timelike hypersurface $\Sigma_D$, is interesting in the AdS/CFT context for several reasons. For one, it allows us to investigate questions in a cut-off AdS spacetime which might be dual to a field theory with a rigid UV cut-off via the usual AdS/CFT dictionary. Furthermore, it allows us to touch upon the ideas involving holographic renormalisation and the implementation of the Wilsonian RG ideas in the gravitational description. The main motivation behind our work was to ask, what is it that one is doing on the dual field theory living on the boundary of the AdS spacetime that ensures these Dirichlet boundary conditions on $\Sigma_D$, and whether such a problem is always well-posed.
For linear systems in AdS it is easy to see that there is a simple map between data on the hypersurface and that on the boundary. In particular, Dirichlet boundary conditions on $\Sigma_D$ translate into non-local multi-trace deformations of the dual field theory. One can think of the boundary theory being deformed by some ‘state dependent’ sources. Such a boundary condition while seemingly bizarre from standard field theoretic perspective, has an effective description in terms of a much simpler and local source, which is the Dirichlet data on $\Sigma_D$.
In this paper we have further argued that such bulk Dirichlet problems for full non-linear gravitational dynamics in are also amenable to solution in a certain long wavelength regime. Using results from the fluid/gravity correspondence [@Bhattacharyya:2008jc; @Bhattacharyya:2008mz] we have constructed the bulk spacetime resulting from the specification of a hypersurface metric on $\Sigma_D$. The solution is contingent upon the hypersurface dynamics, as determined by the conservation of the stress tensor induced on $\Sigma_D$, be conserved. In the long wavelength regime this stress tensor can be written as that of a non-conformal fluid propagating on the fixed geometry of $\Sigma_D$. Armed with such a solution we can examine the dual CFT to determine how one is achieving the Dirichlet dynamics.
In the long wavelength regime the boundary theory is described by fluid dynamics with what we term as Dirichlet constitutive relations. In particular, while the stress tensor of the boundary fluid is of the conventional conformal fluid form, it lives on a background geometry whose metric is ‘dynamical’ in the following sense: the boundary metric depends on the dynamical fluid degrees of freedom. One should think of this in terms of a fluid being subject to a gravitational potential well which is furthermore carried along with the fluid. A useful analogy is to think of a charge carrier in a polarizable medium. The medium or more precisely in our case the ‘dynamical’ background exerts a force on the fluid. However, one can subsume the ‘dynamical’ background’s effect on the fluid, and rewrite the dynamics as that of a ‘dressed fluid’ on a fixed background. This ‘dressed fluid’ is a collective effect and moreover from the geometry we know what its description should be – it is simply the non-conformal fluid on the Dirichlet surface. This can be independently verified by starting with the boundary fluid and the ‘dynamical’ boundary metric and showing that the boundary conservation equation can be rewritten as the hypersurface conservation equation, thus deriving the non-conformal fluid stress tensor on $\Sigma_D$.
We also note that in the long wavelength regime because one is working order by order in boundary gradients, the boundary fluid is deformed locally – at any spacetime point the source $g_{\mu\nu}$ depends only on the fluid degrees of freedom (velocity and temperature) at that point. This is what allows us to think of the fluid as carrying around with it a local gravitational cloud. From a formal point of view, the gravitational Dirichlet problem involves turning on local multi-trace deformations for the field theory.
From the above discussion one is then tempted to say that the bulk geometry provides a way to repackage non-local deformations into a local perturbation at a lower radius or scale. This is highly suggestive of some sort of renormalisation of sources as one propagates boundary conditions into the bulk. Let us compare this picture with the recent discussion of the holographic Wilsonian RG flow idea of [@Heemskerk:2010hk; @Faulkner:2010jy]. In these works starting with a field theory with given sources for say just single trace operators in the planar theory, one derives an effective action containing not just renormalized single trace sources, but also multi-traces on some chosen cut-off surface. The flow equation governing the radial evolution of such sources was argued to arise by effectively integrating out a part of the bulk geometry between the boundary and the cut-off surface. It was also important for that discussion given the particularization to the Wilsonian perspective that one does not prejudice oneself with the boundary conditions in the interior of the spacetime below the cut-off (the infra-red of the field theory).
In the current context we see that by transferring the boundary conditions onto the Dirichlet surface we have shifted the burden of multi-traces onto the boundary – as described above there is a suitable set of non-local multi-trace deformations on the boundary which ‘renormalizes’ into a single trace source on the cut-off. Whilst this is not totally surprising for linear systems such as the scalar problem discussed in the text, it is satisfying that a similar statement can be made for non-linear gravitational dynamics by invoking the long wavelength gradient expansion. We should however note that our discussion explicitly assumes knowledge of the interior (infra-red) boundary conditions; in the fluid/gravity solutions constructed herein we have demanded that there be a regular future event horizon to single out a sensible solution.
Having obtained the effective dynamics on the hypersurface, we learnt that the conservation equations result in a sound mode which travels outside the light-cone of the fixed hypersurface metric once one pushes the surface too far into the interior. Past this sonic threshold, it is probably not sensible to maintain a relativistic fluid description on $\Sigma_D$. We argued that we should pass over into a non-relativistic regime, by a suitable scaling of fluid variables (a la BMW [@Bhattacharyya:2008kq]). It is interesting to ask whether the acausality manifesting itself in the sound mode of the hypersurface dynamics can be discerned (without invoking the ‘dressed’ picture) from the boundary. More importantly, one wonders whether the gravitational Dirichlet problem suffers from such pathologies in general.[^37] Does one encounter any acausal behavior the moment the hypersurface is inside the bulk outside the long wavelength regime? What about their validity where say stringy effects are taken into account? Can we engineer such a boundary condition consistently using objects like orientifolds in string theory? What happens once we go beyond large N? Does the field theory with these non-local deformations make sense beyond large N for at least some subclass of these deformations? What is the field theory interpretation of other kinds of boundary conditions other than Dirichlet boundary conditions (say if we fix the mean extrinsic curvature as proposed in [@Lysov:2011xx]) ? These are fascinating questions which deserve to be explored further.
Finally, by examining the Dirichlet problem in the vicinity of the event horizon of the fluid/gravity solutions we made contact with the recent constructions of flat space gravitational duals to incompressible Navier-Stokes flows on a cut-off hypersurface [@Bredberg:2011jq; @Compere:2011dx], deriving in effect the membrane paradigm from the boundary field theory. Focussing on a tiny sliver of the geometry between the horizon and $\Sigma_D$ leads to the long wavelength solution around the Rindler horizon found in these works. This is reminiscent of the Penrose limit; while in the latter we focus on the geometry close to a null geodesic and blow it up, here we zoom close to a null surface and blow up the spacetime in its vicinity (perhaps a better analogy is the near horizon limit of black D3-branes without any statement about decoupling). The process of blowing up the near horizon region dilutes away the cosmological constant; hence rather than obtaining a solution to Einstein’s equations with negative cosmological constant we end up with a geometry that solves vacuum Einstein’s equations at leading and next-to-leading order. However, starting at higher orders we start to see the effect of the cosmological constant, revealing the throat region between the near-horizon Rindler geometry and the asymptotic AdS spacetime.[^38]
From our embedding of the construction of [@Bredberg:2011jq; @Compere:2011dx] into the fluid/gravity correspondence [@Bhattacharyya:2008jc; @Bhattacharyya:2008mz] we learn that the boundary fluid lives on a manifold with Galilean/Newton-Cartan like structure in the near horizon limit. This degeneration of the Lorentzian structure into an effective Galilean one is what enforces the incompressible Navier-Stokes scaling from the viewpoint of the CFT. One can simply say that the membrane paradigm is a particular Galilean limit of the field theory. This perspective also clarifies the universality of the membrane paradigm – since we are zooming down to the Rindler region in the vicinity of a non-extremal black hole horizon, we should expect the same geometry for any non-degenerate horizon. Hence the dual description should always be in terms of an incompressible Navier-Stokes fluid as long as we deal with systems carrying only conserved charges [@Bhattacharyya:2008kq]. In the presence of other light degrees of freedom, as happens in systems with spontaneously broken symmetries, say the holographic superfluids discussed in [@Sonner:2010yx; @Bhattacharya:2011ee; @Herzog:2011ec; @Bhattacharya:2011tr] we would have to project out all linearly dispersing modes to achieve the same.
In the near horizon limit we find that both the bulk metric and the boundary metric degenerate into a Newton-Cartan time-metric, but the co-metric is spacelike and well behaved. We speculate that the bulk spacetime in this limit should be described in two patches: the region between the horizon and the Dirichlet hypersurface enjoys a description in terms of a conventional metric while the region between $\Sigma_D$ and the boundary requires use of the co-metrics and Newton-Cartan structures. We conclude that the language of Galilean geometries is what is necessary to implement the membrane paradigm in the AdS/CFT correspondence.
The idea of implementing a Galilean limit of AdS/CFT correspondence was proposed recently in [@Bagchi:2009my] who were also motivated by considerations involving the incompressible Navier-Stokes equations, which turn out to enjoy an enhanced symmetry algebra [@Gusyatnikova:1989nx]. It was argued there based on this algebra that appropriate Newton-Cartan manifold should correspond to an slice of the bulk spacetime. Here in contrast we are retaining the entire manifold, not just the two dimensional slice. However, we achieve this at the expense of discarding the Lorentzian metric structure, and work with a co-metric and a time co-vector.
Clearly the role of Galilean/Newton-Cartan structures in the context of AdS/CFT requires further investigation. The bulk geometries we construct here seem to provide an interesting interpolation between Lorentzian and Galilean structures somewhat reminiscent of the Galilean limit procedure described in [@Kunzle:1976fk; @Gonzalez:2003fk]. It is also tempting to contemplate the possibility that apart from the context discussed herein, these structures could also provide useful clues on how to work with the gravity duals of non-relativistic field theories. In the context of Schrödinger spacetimes which were proposed as duals to non-relativistic conformal field theories [@Son:2008ye; @Balasubramanian:2008dm] the Galilean limit is usually implemented as a DLCQ limit which can naturally be studied in the Newton-Cartan language.
There are many directions in which the constructions we have described here can be generalized. For one it would be very useful to understand the gravitational Dirichlet problem outside of the long wavelength regime and to investigate its well-posedness. Further it would be useful to flesh out in precise detail the connections (if any) between the ideas around holographic RG flows and the Dirichlet problem and ask whether one can use the ideas developed here to implement the Wilsonian holographic flow of [@Heemskerk:2010hk; @Faulkner:2010jy] in the non-linear gravity context.
Even within the long-wavelength regime there are tantalizing similarities with the blackfold approach [@Emparan:2009at; @Emparan:2011br], where one considers hypersurfaces which have intrinsic as well as extrinsic dynamics. Freezing out the extrinsic dynamics should lead one to the gravitational Dirichlet problem (for instance in the analysis of the black string and membrane instabilities [@Camps:2010br] the extrinsic dynamics was frozen and the blackfold equations are precisely those of fluid dynamics). While the blackfold analysis is generically well suited for co-dimension three of higher hypersurfaces, it appears that one could recover some of the results discussed herein by extrapolating the equations to a co-dimension one hypersurface.[^39]
Of more immediate interest is to complete the derivation of the hypersurface dynamics from that on the boundary to higher orders in the gradient expansion. In particular, while we have shown that the ideal fluid equations on $\Sigma_D$ arise from those on the boundary, moving to second order in gradients in the conservation equation will allow us to show at the non-linear level that the shear viscosity of the hypersurface fluid is the same as that on the boundary. This statement would establish the non-linear version of the non-renormalisation of $\eta$ (or equivalently $\eta/s$ since we know that the entropy density being associated with the horizon remains unchanged) complementing the earlier analysis of [@Iqbal:2008by] who showed this in the regime of linearized hydrodynamics using a flow equation. One would hope to also understand by this study why the bulk viscosity term on the hypersurface vanishes despite the fluid being non-conformal and having a non-vanishing trace for the stress tensor. At the same time it would be interesting to understand how the higher order transport coefficients evolve from the boundary to $\Sigma_D$ which should be possible to examine within our framework with a little bit of work.
Another interesting avenue for exploration is to ask about the gravitational Dirichlet problem in the presence of degenerate horizons. All of the discussion in the present paper and indeed in the fluid/gravity correspondence has focussed on situations of thermal equilibrium at non-zero temperature and hence one naturally studies spacetimes with non-degenerate horizons. Degenerate horizons pose an intriguing challenge and it would be useful to understand how the boundary theory (which should be more than just a fluid dynamical system) reorganizes itself as we pass over to the hypersurface description. It would also be interesting to extend our considerations to stationary black holes where it would be useful to understand the Dirichlet dynamics when the hypersurface is inside the ergorsurface (as defined by the asymptotic observer).
Finally, another interesting avenue for contemplation is whether the ideas discussed herein can be ported to the context of brane-worlds, where we have gravity induced on a cut-off surface in AdS. Recently [@Figueras:2011gd] attempts to derive the brane gravitational equations by working on a hypersurface near the boundary of AdS and invoking the Fefferman-Graham expansion. It is clear from our discussion that no matter what boundary condition we choose on the hypersurface, it is most likely to involve non-local deformations on the dual boundary field theory. What this implies for induced gravity scenarios is an issue that deserves further study.
Acknowledgements {#s:acks .unnumbered}
----------------
We would especially like to thank Sayantani Bhattacharyya for collaborating with us at the various steps of this paper. It is a pleasure to thank Jyotirmoy Bhattacharya, Amar V Chandra, Atish Dabholkar, Roberto Emparan, Sean Hartnoll, Veronika Hubeny, Nabil Iqbal, Cynthia Keeler, Vijay Kumar, Hong Liu, Donald Marolf, Shiraz Minwalla, Ricardo Monteiro, Niels Obers, David Poland, Suvrat Raju, Andrew Strominger and Piotr Surowka for extremely useful discussions on ideas presented in this paper. We further would like to thank Roberto Emparan, Rajesh Gopakumar, Cynthia Keeler, Hong Liu, Donald Marolf and Andrew Strominger for comments on a draft version of the manuscript. RL and MR would like to thank ICTS, TIFR for wonderful hospitality during the Applied AdS/CFT workshop. MR in addition would also like to thank HRI and University of Amsterdam for their kind hospitality. DB is supported by a STFC studentship, while JC, MR are supported by a STFC Rolling grant. RL is supported by the Harvard Society of Fellows through a junior fellowship. Finally, RL would like to thank various colleagues at the society for interesting discussions.
Notation {#app:notation}
========
We work in the $(-++\ldots)$ signature. The dimensions of the spacetime in which the conformal fluid lives is denoted by $d$ . We usually assume $d>2$ unless otherwise specified. In the context of AdS/CFT, the dual AdS$_{d+1}$ space has $d+1$ spacetime dimensions.The lower-case Greek indices $\mu,\nu= 0,1,\ldots,d-1$ are used as boundary space-time indices, whereas the upper-case Latin indices $A,B=0,1,\ldots, d$ are used as the bulk indices. The lower-case Latin indices $i=1,\ldots,d-1$ index the different spatial directons at the boundary.
Throughout this paper, we take the extra holographic co-ordinate to be $r$ with the boundary of the bulk spacetime at $r\rightarrow\infty$.
Among the objects carrying Greek indices the hatted variables belong to a hypersurface $r=r_D$ whereas the unhatted objects naturally belong to the boundary $r=\infty$. Further we use $*$ to mark non-relativistic objects.
We use round brackets to denote symmetrisation and square brackets to denote antisymmetrisation. For example, $B_{(\mu\nu)}\equiv \frac{1}{2}\, \left(B_{\mu\nu}+B_{\nu\mu}\right)$ and $B_{[\mu\nu]}\equiv \frac{1}{2}\left( B_{\mu\nu}-B_{\nu\mu}\right)$. For tensor products we denote symmetrisation with an explicit subscript as $X \otimes_s Y = \frac{1}{2} (X \otimes Y + Y \otimes X)$.
Quick reference table {#s:tabsum}
---------------------
We have included a table with other useful parameters used in the text. In the table \[notation:tab\], the relevant equations are denoted by their respective equation numbers appearing inside parentheses.
\[notation:tab\]
[||r|l||r|l||]{}\
Symbol & Definition & Symbol & Definition\
\
$\mathcal{G}_{AB}$ & Bulk metric with & $\mathcal{G}^{AB}$ & Bulk co-metric with\
& components $\mathfrak{u}_\mu,\mathfrak{V}_\mu,\mathfrak{G}_{\mu\nu}$ & & components $\mathfrak{u}^\mu,\mathfrak{P}^{\mu\nu},\mathfrak{P}^{\mu\nu}\mathfrak{V}_\nu$\
$G_{d+1}$ & Bulk Newton constant & $\Lambda_{d+1}$ & $-\frac{1}{2}d(d-1)$ Bulk C.C.\
$b$ & Inverse Horizon radius & $f(br)$ & $1-(br)^{-d}$\
$F(br)$ & $\int_{br}^{\infty}\; \frac{y^{d-1}-1}{y(y^{d}-1)}dy$ & $\hat{F}(br)$ & $\frac{\left(F(br)-F(br_D)\right)}{\sqrt{f(br_D)}} $\
$\hat{\xi}$ &$1-\frac{rf(br)}{2r_Df(br_D)}$ & &\
\
$g_{\mu\nu}$ & Boundary metric $[r^{-2}ds_{d+1}^2]_{r=\infty}$ & $g^{\mu\nu}$ & Boundary co-metric\
$u^\mu$ & Fluid velocity & $P_{\mu\nu}$ & $g_{\mu\nu}+u_\mu u_\nu$\
$p$ & Fluid pressure & $\varepsilon$ & Fluid energy density\
$b$ & $(4 G_{d+1} s)^{-\frac{1}{d-1}}$ & $s$ & Fluid entropy density\
$\zeta$& Bulk viscosity & $\eta$ & $\frac{s}{4\pi}$ Shear Viscosity\
$\nabla$ & Christoffel covariant derivative & ${\Gamma}_{\mu\nu}{}^\lambda$ & Christoffel symbols\
$\tilde{\Gamma}_{\mu\nu}{}^\lambda$ & $\equiv\hat{\Gamma}_{\mu\nu}{}^\lambda-{\Gamma}_{\mu\nu}{}^\lambda$ & $\mathcal{A}_\mu$ & Weyl-Connection\
$\sigma_{\mu\nu}$ & Shear strain rate & $\omega_{\mu\nu}$ & Fluid vorticity\
$a_{\mu}$ & Acceleration field & $\theta$ & expansion rate\
$c^2_{snd}$ &\
\
$\hat{g}_{\mu\nu}$ & Dirichlet metric $[r^{-2}ds_{d+1}^2]_{r=r_D}$ & $\hat{g}^{\mu\nu}$ & Dirichlet co-metric\
$\hat{u}^\mu$ & Fluid velocity & $\hat{P}_{\mu\nu}$ & $\hat{g}_{\mu\nu}+\hat{u}_\mu \hat{u}_\nu$\
$\hat{p}$ & Fluid pressure & $\hat{\varepsilon}$ & Fluid energy density\
$\hat{\alpha}$ & $(1-(br_D)^{-d})^{-1/2}=f^{-1/2}(br_D)$ & $\hat{s}$ & Fluid entropy density\
$\hat{\zeta}$& Bulk viscosity & $\hat{\eta}$ & $\frac{\hat{s}}{4\pi}$ Shear Viscosity\
$\hat{\nabla}$ & Christoffel covariant derivative & $\hat{\Gamma}_{\mu\nu}{}^\lambda$ & Christoffel symbols\
$\tilde{\Gamma}_{\mu\nu}{}^\lambda$ & $\equiv\hat{\Gamma}_{\mu\nu}{}^\lambda-{\Gamma}_{\mu\nu}{}^\lambda$ & $\hat{\mathcal{A}}_\mu$ & Weyl-Connection\
$\hat{\sigma}_{\mu\nu}$ & Shear strain rate & $\hat{\omega}_{\mu\nu}$ & Fluid vorticity\
$\hat{a}_{\mu}$ & Acceleration field & $\hat{\theta}$ & expansion rate\
$r_{D,snd}$ &\
$\hat{c}^2_{snd}$ &\
\[notation2:tab\]
[||r|l||r|l||]{}\
Symbol & Definition & Symbol & Definition\
\
$\aleph^{-1}$ &\
$g^{(0)}_{ij}$ & Spatial metric backgnd. & $g^{ij}_{(0)}$ & Spatial co-metric backgnd.\
$h_{tt}^*,h_{ij}^*$ & $\delta g_{tt},\delta g_{ij}\sim \aleph^{-2}$ & $k_i^*$ & $\delta g_{ti}\sim \aleph^{-1}$\
$\nabla^{(0)}_i$ & Covariant derivative using $g^{(0)}_{ij}$ & $q^*_{ij}$ & $\nabla^{(0)}_i k_j^*-\nabla^{(0)}_j k_i^*$\
$v_*^i$ & Fluid velocity $\frac{u^i}{u^t}\sim \aleph^{-1}$ & $\rho_0$ & Mass density $\varepsilon_0+p_0$\
$b_0,\hat{\alpha}_0,\ldots$ & Backgnd. values & $\delta b_*,\hat{\alpha}_*,\ldots$ & Variation-typically $\sim \aleph^{-2} $\
$p_0$ & Backgnd. pressure & $\varepsilon_0$ & Backgnd. energy density\
$p_*$ & Pressure/mass density & $\nu_0$ & Kinematic viscosity $\frac{\eta_0}{\rho_0}$\
&$\frac{p-p_0}{\rho_0}\sim c_{snd}^2\frac{T_*}{T_0}\sim \aleph^{-2} $ & $\hat{\xi}_0$ &$1-\frac{rf(b_0r)}{2r_Df(b_0r_D)}$\
\
$\hat{\varkappa}^{-1}$ &\
&\
$\hat{g}^{(\bullet)}_{ij}$ & Spatial metric backgnd. & $\hat{g}^{ij}_{(\bullet)}$ & Spatial co-metric backgnd.\
$\hat{\nabla}^{(\bullet)}_i$ & Covariant derivative using $\hat{g}^{(\bullet)}_{ij}$ & $\rho_\bullet$ & Mass density\
$\hat{v}_\star^i$ & Fluid velocity $\frac{\hat{u}^i}{\hat{u}^t}\sim \hat{\varkappa}^{-1}$ & & $\varepsilon_\bullet+p_\bullet$ $\sim \hat{\varkappa}^{-(d-1)} $\
$b_\bullet,\hat{\alpha}$ & Backgnd. values $\sim \hat{\varkappa}$ & $\delta b_\star$ & Variation- $\sim \hat{\varkappa}^{-3} $\
$\hat{p}_\bullet$ & Backgnd. pressure $\sim \hat{\varkappa}^{-(d-1)} $ & $\hat{\varepsilon}_\bullet$ & Backgnd. energy density $\sim \hat{\varkappa}^{-d} $\
$\hat{p}_\star$ & Pressure/mass density & $\hat{\nu}_\bullet$ & Kinematic viscosity $\frac{\hat{\eta}_\bullet}{\hat{\rho}_\bullet}$\
& $\frac{\hat{p}-\hat{p}_\bullet}{\hat{\rho}_\bullet}\sim \hat{c}_{snd}^2\frac{\hat{T}_*}{\hat{T}_0}\sim \hat{\varkappa}^{-2} $ & $\rho_D$ & $\rho$ co-ordinate of Dirichlet surface\
$\rho$ &\
Notation in the Bulk
--------------------
We denote the bulk metric by $\mathcal{G}_{AB}$. The inverse metric in the bulk (we will call this the co-metric - since it is the metric on the cotangent bundle) is denoted by $\mathcal{G}^{AB}$.
We take the bulk AdS radius to be unity which is equivalent to setting the bulk cosmological constant to be $\Lambda_{d+1}=-\frac{d(d-1)}{2}$. We denote the bulk Newton constant by $G_{d+1}$. For the ease of reference, we now give the value of $G_{d+1}$ for some of the well-known CFTs with gravity duals (see [@Maldacena:1997re; @Aharony:1999ti; @Aharony:2008ug] for further details):
1. The d=4, $\mathcal{N}$=4 Super Yang-Mills theory on $N_c$ D3-Branes with a gauge group SU(N$_c$) and a ‘t Hooft coupling $\lambda\equiv g_{YM}^2 N_c$ is believed to be dual to IIB string theory on AdS$_5\times$S$^5_{R=1}$ with $G_{5}=\pi /(2 N_c^2)$ and $\alpha'=(4\pi\lambda)^{-1/2} $.
2. A d=3, $\mathcal{N}$=6 Superconformal [^40] Chern-Simons theory on $N_c$ M2-Branes with a gauge group U(N$_c$)$_k\times$ U(N$_c$)$_{-k}$ (where the subscripts denote the Chern-Simons couplings) and a ‘t Hooft coupling $\lambda\equiv N_c/k$ is conjectured to be dual to M-theory on AdS$_4\times$S$^7_{R=2}$/Z$_k$ with $G_{4}=N_c^{-2}\sqrt{9\lambda/8}=3k^{-1/2}(2N_c)^{-3/2}$.
3. A d=6, $\mathcal{N}$=(2,0) superconformal theory on $N_c$ M5-Branes is conjectured to be dual to M-theory on AdS$_7\times$S$^4_{R=1/2}$ with $G_{7}=3\pi^2/(16 N_c^{3})$.
The general bulk metric which is Weyl-covariant is given by $$ds^2 =\mathcal{G}_{AB}dx^Adx^B= -2\,{\mathfrak u}_\mu dx^\mu\left(dr+r \,{\mathfrak V} _\nu dx^\nu\right) + r^2\, {\mathfrak G}_{\mu\nu} dx^\mu dx^\nu$$ which is invariant under the boundary Weyl transformations $$\left\{r,{\mathfrak u}_\mu, {\mathfrak V}_\nu,{\mathfrak G}_{\mu\nu} \right\} \mapsto \left\{e^{-\phi}r,e^{\phi}{\mathfrak u}_\mu, {\mathfrak V}_\nu+\partial_\nu\phi,e^{2\phi}{\mathfrak G}_{\mu\nu} \right\}$$ where $\phi=\phi(x)$ is an arbitrary function at the boundary. Without loss of generality assume that ${\mathfrak G}_{\mu\nu}$ is transverse to ${\mathfrak u}_\mu$, i.e. , ${\mathfrak G}_{\mu\nu}\mathfrak{u}^\nu=0$. Further we have $${\mathfrak u}_\mu = \mathfrak{u}_\mu(x), \quad {\mathfrak V}_\nu={\mathfrak V}_\nu(r,x),\quad {\mathfrak G}_{\mu\nu}={\mathfrak G}_{\mu\nu}(r,x)$$ We will raise/lower/contract the unhatted greek indices using the boundary metric $$g_{\mu\nu} dx^\mu dx^\nu \equiv [r^{-2}ds^2]_{r\to \infty} = \left[{\mathfrak G}_{\mu\nu}-\frac{2}{r}{\mathfrak u}_{(\mu} {\mathfrak V}_{\nu)} \right]_{r\to \infty} dx^\mu dx^\nu$$ and the velocity field $\mathfrak{u}_\mu$ is a unit time-like vector of this metric $$\mathfrak{u}^\mu \mathfrak{u}_\mu \equiv g^{\mu\nu}\mathfrak{u}_\mu \mathfrak{u}_\nu = -1$$
A Weyl-covariant basis for the bulk cotangent bundle is given by $\{dr+r\,{\mathfrak V}_\nu dx^\nu\ ,dx^\mu\}$ with a corresponding dual basis $\left\{\partial_r\ , \partial_\mu - r\,{\mathfrak V}_\mu\partial_r\right\}$. In this dual basis, the co-metric (or the inverse metric) is given by $$\begin{split}
\mathcal{G}^{AB}&\partial_A\otimes\partial_B \\
&=2\, {\mathfrak u}^\mu\left[\partial_\mu - r\,{\mathfrak V}_\mu\partial_r\right]\otimes_s\partial_r + r^{-2}\,\mathfrak{P}^{\mu\nu}\left[\partial_\mu - r\,{\mathfrak V}_\mu\partial_r\right]\otimes\left[\partial_\nu - r\,{\mathfrak V}_\nu\partial_r\right] \\
\end{split}$$ where $\mathfrak{P}^{\mu\nu}$ is the unique transverse tensor that satisfies $$\mathfrak{P}^{\mu\nu} {\mathfrak u}_\nu=0 \quad\text{and}\quad \mathfrak{P}^{\mu\lambda}{\mathfrak G}_{\lambda\nu}= \delta^\mu_\nu + {\mathfrak u}^\mu \, {\mathfrak u}_\nu$$ The unit normal vector of a hypersurface $r=r_D$ is given by[^41] $$n_A dx^A = \frac{dr}{\sqrt{\mathcal{G}^{rr}}} = \frac{dr}{\sqrt{\mathfrak{P}^{\alpha\beta}{\mathfrak V}_\alpha{\mathfrak V}_\beta-2r u^\alpha{\mathfrak V}_\alpha}}$$ $$\begin{split}
n^A \partial_A &= \frac{\mathcal{G}^{rr}\partial_r+\mathcal{G}^{r\mu}\partial_\mu}{\sqrt{\mathcal{G}^{rr}}}\\
&= \frac{\left(\mathfrak{P}^{\alpha\beta}\,{\mathfrak V}_\alpha\,{\mathfrak V}_\beta-2\,r \,{\mathfrak u}^\alpha\,{\mathfrak V}_\alpha\right)\partial_r+\left({\mathfrak u}^\mu-r^{-1}\,\mathfrak{P}^{\mu\alpha}\, {\mathfrak V}_\alpha\right)\partial_\mu}{\sqrt{\mathfrak{P}^{\alpha\beta}\,{\mathfrak V}_\alpha{\mathfrak V}_\beta-2\,r\, {\mathfrak u}^\alpha\,{\mathfrak V}_\alpha}}\\
&= \frac{-r\,{\mathfrak u}^\alpha\,{\mathfrak V}_\alpha\partial_r+\left({\mathfrak u}^\mu-r^{-1}\,\mathfrak{P}^{\mu\alpha}\,{\mathfrak V}_\alpha\right)\left[\partial_\mu - r\,{\mathfrak V}_\mu\partial_r\right]}{\sqrt{\mathfrak{P}^{\alpha\beta}\,{\mathfrak V}_\alpha\,{\mathfrak V}_\beta-2\,r\, {\mathfrak u}^\alpha{\mathfrak V}_\alpha}}\\
\end{split}$$ From these expressions, it follows that the normalized induced metric and co-metric on a hypersurface $r=r_D$ is given by $$\begin{split}
\hat{g}_{\mu\nu} &\equiv \left\{ r^{-2}\mathcal{G}_{\mu\nu}
\right\}_{r\to r_D}\\
&= \left\{ \mathfrak{G}_{\mu\nu} -
\frac{2}{r}\mathfrak{u}_{(\mu}\mathfrak{V}_{\nu)} \right\}_{r\to r_D} \\
\hat{g}^{\mu\nu} &\equiv \left\{
r^2\left(\mathcal{G}^{\mu\nu} - \frac{\mathcal{G}^{\mu
r}\mathcal{G}^{r\nu}}{\mathcal{G}^{rr}}\right)\right\}_{r\to r_D}\\
&= \left\{ \mathfrak{P}^{\mu\nu} -
\frac{\left[r \, {\mathfrak u}^\mu-\mathfrak{P}^{\mu\alpha}\mathfrak{V}_\alpha\right]\left[r\,
{\mathfrak u}^\nu-\mathfrak{P}^{\nu\beta}\mathfrak{V}_\beta\right]}{\mathfrak{P}^{\alpha\beta}\mathfrak{V}_\alpha\mathfrak{V}_\beta-2\, r\,
{\mathfrak u}^\alpha\mathfrak{V}_\alpha}\right\}_{r\to r_D}
\end{split}$$ The hatted greek indices are raised/lowered/contracted using this hatted metric/co-metric.
Notation at the Boundary
------------------------
The metric on the $d$ dimensional boundary is denoted by $g_{\mu\nu}$ which is a representative the class of metrics on the conformal boundary of the bulk spacetime. $$g_{\mu\nu}\equiv \left\{r^{-2}ds^2\right\}_{r\to \infty}$$ The inverse of this metric (we will call this the co-metric – since it is the metric on the cotangent bundle) is denoted by $g^{\mu\nu}$. We denote with $\nabla$ the corresponding Christoffel connection/covariant derivative. Our conventions for Christoffel symbols and the curvature tensors are fixed by the relations $$\begin{split}
\nabla_{\mu}V^{\nu}&=\partial_{\mu}V^{\nu}+\Gamma_{\mu\lambda}{}^{\nu}V^{\lambda} \qquad \text{and}\qquad [\nabla_\mu,\nabla_\nu]V^\lambda=-R_{\mu\nu\sigma}{}^{\lambda}V^\sigma .
\end{split}$$
On this spacetime lives a conformal fluid with velocity field $u^\mu$ (with $u^\mu u_\mu =-1$) , pressure $p$, energy density $\varepsilon=(d-1)p$, and shear viscosity $\eta$. We introduce the projector $P_{\mu\nu}\equiv g_{\mu\nu}+u_\mu u_\nu$ which projects onto the space transverse to $u^\mu$.
The gradients of the velocity field are decomposed as: $$\begin{split}
\nabla_\mu u_\nu &= \sigma_{\mu\nu}+\omega_{\mu\nu}-u_\mu a_\nu + \frac{\theta}{d-1} P_{\mu\nu}\\
&= \sigma_{\mu\nu}+\omega_{\mu\nu}-u_\mu \mathcal{A}_\nu + \frac{\theta}{d-1} g_{\mu\nu}\\
\end{split}$$ where we have introduced $$\begin{aligned}
&&
\bullet \; \text{the shear strain rate:} \qquad \sigma_{\mu\nu}\equiv \left[P_{(\mu}^\alpha P_{\nu)}^\beta- \frac{P_{\mu\nu}}{d-1}P^{\alpha\beta} \right] \nabla_\alpha u_\beta \label{eqn:sigdef} \\
&&
\bullet \; \text{the vorticity:} \qquad \qquad \quad \ \;
\omega_{\mu\nu}\equiv P_{[\mu}^\alpha P_{\nu]}^\beta \nabla_\alpha u_\beta
\label{eqn:omdef} \\
&&
\bullet \; \text{the acceleration field:} \qquad \ \,a_\mu \equiv u_\alpha\nabla^\alpha u_\mu
\label{eqn:adef} \\
&&
\bullet \; \text{the expansion rate:} \qquad \qquad \theta \equiv \nabla_\alpha u^\alpha
\label{eqn:thdef}
\end{aligned}$$ The hydrodynamic Weyl-connection (see [@Loganayagam:2008is] where it was introduced for more details) is defined to be $$\mathcal{A}_\mu \equiv u_\alpha\nabla^\alpha u_\mu - \frac{\nabla_\alpha u^\alpha}{d-1} u_\mu
\label{eqn:Adef}$$ The bulk metric-dual for hydrodynamics is given by (see [@Bhattacharyya:2008mz]) $$\begin{split}
\mathfrak{u}_\mu &= u_\mu\ ,\quad\ \mathfrak{G}_{\mu\nu} = P_{\mu\nu} +2b\, F(br)\, \sigma_{\mu\nu}+\ldots \\
\mathfrak{V}_\mu &= \mathcal{A}_\mu+\frac{r}{2}\, (1-(br)^{-d})\, u_\nu +\ldots \\
\mathfrak{u}^\mu &= u^\mu\ ,\quad\mathfrak{P}^{\mu\nu} = P^{\mu\nu} -2b \,F(br)\, \sigma^{\mu\nu}+\ldots \\
\end{split}$$ where $$\hat{F}(br) \equiv \frac{1}{\hat{\alpha}}\left(F(br)-F(br_D)\right) = \frac{1}{\hat{\alpha}} \; \int_{br}^{br_D}\; \frac{y^{d-1}-1}{y(y^{d}-1)}dy\,.$$
The energy-momentum tensor of a general relativistic fluid (till first order in the gradient expansion) is given as $$\label{enmom:eq}
T^{\mu\nu} \equiv (\varepsilon+p)\, u^\mu\, u^\nu + p\, g^{\mu\nu}- 2\,\eta\, \sigma^{\mu\nu}-\zeta\, \theta\, P^{\mu\nu}+\ldots$$ which can be computed from the bulk data (of the metric dual to hydrodynamics) via $$\label{eq:BYT}
T_{\mu\nu} \equiv \left\{\frac{r^d}{16\pi \,G_{d+1}}\left[-2\,{K}_{\mu\nu}+2\,K\,{g}_{\mu\nu}-2\,(d-1)\,{g}_{\mu\nu}+\ldots \right]\right\}_{r\to\infty}$$ where $r^2 \,K_{\mu\nu}$ is the extrinsic curvature of the constant $r$ hypersurface and $K\equiv g^{\mu\nu}\, K_{\mu\nu}$ is its trace. This computation gives $$p=\frac{\varepsilon}{d-1} = \frac{1}{16\pi G_{d+1}} \; \frac{1}{b^d} ,\quad\ \eta=\frac{1}{16\pi G_{d+1}}\; \frac{1}{b^{d-1}} \quad\text{and}\quad \zeta =0$$ and the Bekenstein-Hawking argument in the bulk gives the entropy density and the temperature of this fluid as $$s= \frac{1}{4\, G_{d+1}}\; \frac{1}{b^{d-1}} ,\quad\text{and}\quad T=\frac{d}{4\pi\, b}$$ where we have found it convenient to introduce a variable $b\equiv (4 G_{d+1} s)^{-\frac{1}{d-1}}$.
Notation on the Dirichlet Hypersurface
--------------------------------------
We choose a hypersurface $r=r_D$ in the bulk to impose Dirichlet boundary condition. We will work in a boundary Weyl-frame where $r_D$ is independent of $x$. This Weyl-frame change is consistent with the gradient expansion in the boundary hydrodynamics provided the initial surface $r=\rho(x)$ had a slowly varying $\rho(x)$. We will introduce a parameter $$\hat{\alpha} \equiv \frac{1}{\sqrt{1-(b r_D)^{-d}}}$$ which parameterizes how far the Dirichlet surface is to the boundary/ how close it is to the horizon. We have $\hat{\alpha}(r_D=\infty) =1$ and $\hat{\alpha}(r_D=1/b) =\infty$.
After this we proceed as we did in the $r=\infty$ case in the previous subsection. All the same definitions can be repeated - we will just distinguish the objects in the Dirichlet hypersurface by a hat - so we have $\hat{u}^\mu ,\ \hat{g}^{\mu\nu} ,\ \hat{p}$ and so on. Unless specified, all the hatted tensors are raised and lowered by the hatted metric/co-metric.
The energy momentum tensor is calculated using the same expression as in equation except that we evaluate it at $r=r_D$ now. We get $$\label{enmomh:eq}
\hat{T}^{\mu\nu} \equiv (\hat{\varepsilon}+\hat{p})\,\hat{u}^\mu \,\hat{u}^\nu + \hat{p}\, \hat{g}^{\mu\nu}- 2\,\hat{\eta}\, \hat{\sigma}^{\mu\nu}-\hat{\zeta} \,\hat{\theta}\, \hat{P}^{\mu\nu}+\ldots$$ where $$\label{enmomh2:eq}
\begin{split}
\hat{\varepsilon} &\equiv \frac{(d-1)}{8\pi \,G_{d+1}}\; \frac{\hat{\alpha}}{\hat{\alpha}+1} ;\; \frac{1}{b^d}= \frac{2\hat{\alpha}}{\hat{\alpha}+1}\ \varepsilon\\
\hat{p} &\equiv \frac{\left[1+\frac{d}{2}(\hat{\alpha}-1)\right]}{8\pi\, G_{d+1}}\; \frac{\hat{\alpha}}{\hat{\alpha}+1} \; \frac{1}{b^d}= \frac{2\hat{\alpha}}{\hat{\alpha}+1}\left[1+\frac{d}{2}(\hat{\alpha}-1)\right]p\\
\hat{\eta} &\equiv \frac{1}{16\pi \,G_{d+1}}\; \frac{1}{b^{d-1}}=\eta \quad\text{and}\quad \hat{\zeta} \equiv 0 =\zeta\\
\end{split}$$
Dictionary for the Dirichlet Problem {#A:dirdict}
====================================
In this subsection, we collect the formulae which translate between the boundary data and the Dirichlet data to serve as a ready reference for the reader. The (normalized) metric and the co-metric on the Dirichlet hypersurface are given by $$\begin{split}
\hat{g}_{\mu\nu}&= g_{\mu\nu} + \left(1-\frac{1}{\hat{\alpha}^2}\right)u_\mu u_\nu+2b F(br_D)\ \sigma_{\mu\nu} -\frac{1}{r_D}\left[u_\mu \mathcal{A}_\nu+\mathcal{A}_\mu u_\nu \right]+\ldots\\
&= g_{\mu\nu} + \left(1-\frac{1}{\hat{\alpha}^2}+\frac{2\theta}{(d-1)r_D}\right)u_\mu u_\nu-\frac{1}{r_D}\left[u_\mu a_\nu+a_\mu u_\nu \right] +2b F(br_D)\ \sigma_{\mu\nu} +\ldots\\
\hat{g}^{\mu\nu} &= P^{\mu\nu}-2b F(br_D)\ \sigma^{\mu\nu} -\hat{\alpha}^2\left[u^\mu - r_D^{-1}a^{\mu}\right]\left[u^\nu - r_D^{-1}a^{\nu}\right]\left[1+\frac{2\hat{\alpha}^2\theta}{(d-1)r_D}\right] \\
&= g^{\mu\nu}+\left(1-\hat{\alpha}^2-\frac{2\hat{\alpha}^4\theta}{(d-1)r_D}\right)u^\mu u^\nu \ -2b F(br_D)\ \sigma^{\mu\nu} +\frac{\hat{\alpha}^2}{r_D}\left[u^\mu a^\nu + a^{\mu}u^\nu\right] \\
\end{split}$$ and the correctly normalized velocities at the hypersurface are $$\begin{split}
\hat{u}_\mu &= \frac{u_\mu}{\hat{\alpha}} + \frac{\hat{\alpha}}{r_D} \mathcal{A}_\mu = \left(1-\frac{\hat{\alpha}^2\theta}{r_D(d-1)}\right)\frac{u_\mu}{\hat{\alpha}} + \frac{\hat{\alpha}}{r_D} a_\mu \\
\hat{u}^\mu &= \hat{\alpha}u^\mu\left(1+\frac{\hat{\alpha}^2\theta}{r_D(d-1)}\right) +\ldots \\
\end{split}$$
From these it follows that the transverse projectors are related by $$\begin{split}
\hat{P}_{\mu\nu} &= P_{\mu\nu} + 2b F(br_D)\sigma_{\mu\nu}\\
\hat{P}^\mu{}_\nu &= {P}^\mu{}_\nu+\frac{\hat{\alpha}^2}{r_D}u^\mu a_\nu \\
\hat{P}^{\mu\nu} &= {P}^{\mu\nu} -2b F(br_D) \sigma^{\mu\nu}+\frac{\hat{\alpha}^2}{r_D}\left[u^\mu a^\nu + a^{\mu}u^\nu\right]\\
\end{split}$$
Given a relation of the form $\hat{U}_\mu=V_\mu$, we can always write $$\begin{split}
\hat{\nabla}_\mu \hat{U}_\nu = \nabla_\mu V_\nu -\tilde{\Gamma}_{\mu\nu}{}^\rho V_\rho
\end{split}$$ which defines the tensor $\tilde{\Gamma}_{\mu\nu}{}^\rho=\hat{\Gamma}_{\mu\nu}{}^\rho-{\Gamma}_{\mu\nu}{}^\rho$. We evaluate this tensor in the to get $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho &= (\hat{\alpha}^2-1)\left[\sigma_{\mu\nu}+\frac{\theta}{d-1}\ \left(P_{\mu\nu}+\frac{d}{2}u_\mu u_\nu\right)-da_{(\mu}u_{\nu)}\right]u^\rho\\
&\qquad + \frac{\hat{\alpha}^2-1}{\hat{\alpha}^2}\left[-2\omega^\rho{}_{(\mu}u_{\nu)}+(\frac{d}{2}-1)u_{\mu}u_{\nu}a^\rho\right] \\
\end{split}$$ in particular $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho u_\rho
&=-(\hat{\alpha}^2-1)\left[\sigma_{\mu\nu}+\frac{\theta}{d-1}\ \left(P_{\mu\nu}+\frac{d}{2}u_\mu u_\nu\right)-da_{(\mu}u_{\nu)}\right] \\
\end{split}$$ From this it follows that $$\begin{split}
\hat{\sigma}_{\mu\nu} &= \hat{\alpha} \,\sigma_{\mu\nu}\ ,\quad \hat{\omega}_{\mu\nu} = \frac{1}{\hat{\alpha}} \,\omega_{\mu\nu} ,\quad \hat{\theta}=\hat{\alpha}\,\theta,\\
\hat{a}_\nu &=\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]a_\nu ,\quad \hat{\mathcal{A}}_\nu =\mathcal{A}_\nu+\frac{d}{2}\,(\hat{\alpha}^2-1)\,a_{\nu} \\
\end{split}$$ We refer the reader to the previous subsection for the dictionary involving the energy-momentum tensor.
Now we are ready to present the inverse relations. We will first invert the equations above to get $$\begin{split}
{\sigma}_{\mu\nu} &= \frac{1}{\hat{\alpha}}\, \hat{\sigma}_{\mu\nu}\ ,\quad {\omega}_{\mu\nu} = {\hat{\alpha}} \,\hat{\omega}_{\mu\nu} ,\quad {\theta}=\frac{1}{\hat{\alpha}}\,\hat{\theta},\\
{a}_\nu &=\frac{\hat{a}_\nu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} ,\quad {\mathcal{A}}_\nu =\hat{\mathcal{A}}_\nu-\frac{(\hat{\alpha}^2-1)\frac{d}{2}}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{a}_{\nu} =\frac{\hat{a}_\nu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}-\frac{\hat{\theta}}{d-1}\hat{u}_\nu \\
\end{split}$$ and then the velocities $$\begin{split}
{u}_\mu &= \left(1+\frac{\hat{\alpha}\hat{\theta}}{r_D(d-1)}\right)\hat{\alpha}\,\hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D} \frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} \\
{u}^\mu &= \frac{\hat{u}^\mu}{\hat{\alpha}}\left(1-\frac{\hat{\alpha}\,\hat{\theta}}{r_D\,(d-1)}\right) +\ldots \\
\end{split}$$ followed by the projectors $$\begin{split}
{P}_{\mu\nu}
&= \hat{P}_{\mu\nu} - \frac{2b}{\hat{\alpha}} \,F(br_D)\,\hat{\sigma}_{\mu\nu}\\
{P}^\mu{}_\nu
&= \hat{P}^\mu{}_\nu-\frac{\hat{\alpha}}{r_D}\frac{\hat{u}^\mu\hat{a}_\nu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} \\
{P}^{\mu\nu} &= \hat{P}^{\mu\nu} +\frac{2b}{\hat{\alpha}} \,F(br_D)\hat{\sigma}^{\mu\nu}-\frac{\hat{\alpha}\left[\hat{u}^\mu \hat{a}^\nu + \hat{a}^{\mu}\hat{u}^\nu\right]}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\\
\end{split}$$
The dictionary for the (normalized) metric and the co-metric are $$\begin{split}
g_{\mu\nu}
&= \hat{g}_{\mu\nu} + \left[1-\hat{\alpha}^2-\frac{2\hat{\alpha}^3\hat{\theta}}{r_D(d-1)}\right]\hat{u}_\mu \hat{u}_\nu+\frac{\hat{\alpha}^3\left[\hat{u}_\mu \hat{a}_\nu + \hat{a}_{\mu}\hat{u}_\nu\right]}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} - \frac{2b}{\hat{\alpha}} F(br_D)\hat{\sigma}_{\mu\nu} \\
g^{\mu\nu}
&= \hat{g}^{\mu\nu}+ \left[1-\frac{1}{\hat{\alpha}^2}+\frac{2\hat{\theta}}{\hat{\alpha}r_D(d-1)}\right]\hat{u}^\mu \hat{u}^\nu+\frac{2b}{\hat{\alpha}} F(br_D)\hat{\sigma}^{\mu\nu}-\frac{\hat{\alpha}\left[\hat{u}^\mu \hat{a}^\nu + \hat{a}^{\mu}\hat{u}^\nu\right]}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\\
\end{split}$$
Finally, we can write the tensor $\tilde{\Gamma}_{\mu\nu}{}^\rho $ in terms of hatted variables as $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho
&= (1-\frac{1}{\hat{\alpha}^2})\left[ \hat{\sigma}_{\mu\nu}+\frac{\hat{\theta}}{d-1}\ \left(\hat{P}_{\mu\nu}+\frac{d}{2}\hat{\alpha}^2\hat{u}_\mu \hat{u}_\nu\right)-\frac{d\hat{\alpha}^2}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{a}_{(\mu}\hat{u}_{\nu)}\right]\hat{u}^\rho\\
&\qquad + (\hat{\alpha}^2-1)\left[-2\hat{\omega}^\rho{}_{(\mu}\hat{u}_{\nu)}+\frac{\frac{d}{2}-1}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{u}_{\mu}\hat{u}_{\nu}\hat{a}^\rho\right] \\
\end{split}$$ In particular $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho \hat{u}_\rho
&= - (1-\frac{1}{\hat{\alpha}^2})\left[ \hat{\sigma}_{\mu\nu}+\frac{\hat{\theta}}{d-1}\ \left(\hat{P}_{\mu\nu}+\frac{d}{2}\hat{\alpha}^2\hat{u}_\mu \hat{u}_\nu\right)-\frac{d\hat{\alpha}^2}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{a}_{(\mu}\hat{u}_{\nu)}\right]
\end{split}$$
Now we present the Bulk metric/co-metric as a function of the Dirichlet data. The Bulk metric is given by $$\label{appeq:gbulk}
\begin{split}
\mathcal{G}_{AB}dx^A dx^B
&=-2 u_\mu dx^\mu \left( dr + r\ \left[\mathcal{A}_\nu+\frac{r}{2}(1-(br)^{-d})u_\nu\right] dx^\nu \right) \\
& \qquad + r^2 \left[ P_{\mu\nu} +2b F(br)\ \sigma_{\mu\nu}\right] dx^\mu dx^\nu +\ldots\\
&= -2 \left[\hat{\alpha}\hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D}{\mathcal{A}}_\mu \right]dx^\mu\left( dr + r\ \left[
\hat{\xi}\mathcal{A}_\nu +\frac{r}{2}(1-(br)^{-d})\hat{\alpha}\hat{u}_\nu \right] dx^\nu \right)\\
& \qquad + r^2 \left[ \hat{P}_{\mu\nu} +2b \hat{F}(br)\ \hat{\sigma}_{\mu\nu}\right] dx^\mu dx^\nu +\ldots\\
\end{split}$$ where $$\begin{split}
\mathcal{A}_\mu &\equiv \frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}-\frac{\hat{\theta}}{d-1}\hat{u}_\mu=\hat{\mathcal{A}}_\mu-\frac{(\hat{\alpha}^2-1)\frac{d}{2}}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{a}_{\mu}\\
\hat{\xi} &\equiv 1-\frac{1}{2}\frac{r}{r_D}\frac{(1-(br)^{-d})}{(1-(br_D)^{-d})}
= 1-\frac{\hat{\alpha}^2}{r_D}\frac{r}{2}(1-(br)^{-d})\\
\hat{F}(br) &\equiv \frac{1}{\hat{\alpha}}\left(F(br)-F(br_D)\right) = \frac{1}{\hat{\alpha}} \; \int_{br}^{br_D}\; \frac{y^{d-1}-1}{y(y^{d}-1)}dy\,.
\end{split}$$ The Bulk co-metric is given by $$\begin{split}
\mathcal{G}^{AB}&\partial_A\otimes\partial_B \\
&= \left[r^2(1-(br)^{-d})-\frac{2r\theta}{d-1}\right]\partial_r\otimes\partial_r+2\left[u^\mu -r^{-1} a^{\mu}\right]\partial_\mu\otimes_s\partial_r\\
&\qquad +r^{-2}\left[P^{\mu\nu} -2b F(br)\ \sigma^{\mu\nu}\right]\partial_\mu\otimes\partial_\nu\\
&= \left[r^2(1-(br)^{-d})-\frac{2r\hat{\theta}}{\hat{\alpha}(d-1)}\right]\partial_r\otimes\partial_r\\
&+2\left[\frac{\hat{u}^\mu}{\hat{\alpha}}\left(1-\frac{\hat{\alpha}\hat{\theta}}{r_D(d-1)}\right) - \frac{\hat{a}^\mu}{r\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\right]\partial_\mu\otimes_s\partial_r\\
&\qquad +r^{-2}\left[\hat{P}^{\mu\nu} -\frac{\hat{\alpha}\left[\hat{u}^\mu \hat{a}^\nu + \hat{a}^{\mu}\hat{u}^\nu\right]}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} -2b \hat{F}(br)\ \hat{\sigma}^{\mu\nu}\right]\partial_\mu\otimes\partial_\nu
\end{split}$$ In terms of the components in the Weyl-covariant basis, we have $$\begin{split}
\mathfrak{u}_\mu &= u_\mu = \hat{\alpha}\, \hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D}\left[\frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}-\frac{\hat{\theta}}{d-1}\hat{u}_\mu\right] \\
\mathfrak{V}_\mu &= \mathcal{A}_\mu+\frac{r}{2}\, (1-(br)^{-d})\, u_\nu \\
&= \hat{\xi}\left[\frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}-\frac{\hat{\theta}}{d-1}\, \hat{u}_\mu\right] +\frac{r}{2}\, (1-(br)^{-d})\, \hat{\alpha}\,\hat{u}_\mu\\
\mathfrak{G}_{\mu\nu} &= P_{\mu\nu} +2b\, F(br)\, \sigma_{\mu\nu} = \hat{P}_{\mu\nu} +2b \,\hat{F}(br)\, \hat{\sigma}_{\mu\nu}\\
\mathfrak{u}^\mu &= u^\mu = \frac{\hat{u}^\mu}{\hat{\alpha}}\left(1-\frac{\hat{\alpha}\,\hat{\theta}}{r_D(d-1)}\right)\\
\mathfrak{P}^{\mu\nu} &= P^{\mu\nu} -2b \,F(br)\, \sigma^{\mu\nu} = \hat{P}^{\mu\nu} -\frac{\hat{\alpha}\left[\hat{u}^\mu \hat{a}^\nu + \hat{a}^{\mu}\hat{u}^\nu\right]}{r_D\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]} -2b \,\hat{F}(br)\, \hat{\sigma}^{\mu\nu} \\
\end{split}$$
Lorentz-Covariant derivative of the induced metric {#app:CovDeriv}
==================================================
We wish to find the tensor $\tilde{\Gamma}_{\mu\nu}{}^\rho$ which describes the difference between the covariant derivatives at the boundary and the Dirichlet surface, i.e., Given a relation of the form $\hat{U}_\mu=V_\mu$, we can always write $$\begin{split}
\hat{\nabla}_\mu \hat{U}_\nu = \nabla_\mu V_\nu -\tilde{\Gamma}_{\mu\nu}{}^\rho V_\rho
\end{split}$$ which defines the tensor $\tilde{\Gamma}_{\mu\nu}{}^\rho$.
For definiteness, in this subsection we will continue to raise/lower/contract using the boundary metric $g_{\mu\nu}$ - this means in particular raising/lowering/contracting do not commute with the hatted covariant derivative $\hat{\nabla}$ so we need to be a bit careful.
As usual, zero-torsion condition implies $\tilde{\Gamma}_{\mu\nu}{}^\rho=\tilde{\Gamma}_{\nu\mu}{}^\rho$ and metric compatibility with $\hat{g}_{\mu\nu}\equiv g_{\mu\nu}+h_{\mu\nu}$ gives $$\tilde{\Gamma}_{\mu\nu}{}^\rho\hat{g}_{\rho\lambda}=\frac{1}{2} \left[\nabla_\mu h_{\lambda\nu}+\nabla_\nu h_{\lambda\mu}-\nabla_\lambda h_{\mu\nu} \right]$$ where $$h_{\mu\nu} = \left\{ \frac{u_\mu u_\nu}{(br)^d}+2b F(br)\ \sigma_{\mu\nu} -\frac{1}{r}\left[u_\mu \mathcal{A}_\nu+\mathcal{A}_\mu u_\nu \right]+\ldots \right\}_{r\to r_D}$$ Since all our expressions are exact upto second derivatives, it is enough to work with just the zero derivative piece in $h_{\mu\nu}$. $$\begin{split}
\frac{\hat{\alpha}^2}{\hat{\alpha}^2-1}\tilde{\Gamma}_{\mu\nu}{}^\rho &=\hat{\alpha}^2\left[\sigma_{\mu\nu}+\frac{\theta}{d-1}\ \left(P_{\mu\nu}+\frac{d}{2}u_\mu u_\nu\right)-da_{(\mu}u_{\nu)}\right]u^\rho-2\omega^\rho{}_{(\mu}u_{\nu)}+(\frac{d}{2}-1)u_{\mu}u_{\nu}a^\rho \\
\end{split}$$
It follows that $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho u_\rho
&=-(\hat{\alpha}^2-1)\left[\sigma_{\mu\nu}+\frac{\theta}{d-1}\ \left(P_{\mu\nu}+\frac{d}{2}u_\mu u_\nu\right)-da_{(\mu}u_{\nu)}\right] \\
\end{split}$$
We can now evaluate $$\begin{split}
\hat{\nabla}_\mu u_\nu &\equiv\nabla_\mu u_\nu-\tilde{\Gamma}_{\mu\nu}{}^\rho u_\rho\\
&= \hat{\alpha}^2 \sigma_{\mu\nu} + \omega_{\mu\nu}+ \hat{\alpha}^2 \frac{\theta}{d-1}\ P_{\mu\nu}-u_\mu a_\nu\left(1+\frac{d}{2}(\hat{\alpha}^2-1)\right)-\frac{d}{2}(\hat{\alpha}^2-1)\mathcal{A}_{\mu}u_{\nu}\\
\end{split}$$
Finally, we obtain[^42] $$\begin{split}
\hat{\nabla}_\mu\hat{u}_\nu&=\hat{\nabla}_\mu \left\{\frac{u_\nu}{\hat{\alpha}}\right\}+\ldots\\
&= \hat{\alpha} \sigma_{\mu\nu} +\frac{\omega_{\mu\nu}}{\hat{\alpha}} + \frac{\hat{\alpha}\theta}{d-1}\ \hat{P}_{\mu\nu}-\hat{u}_\mu a_\nu\left(1+\frac{d}{2}(\hat{\alpha}^2-1)\right)\\
\end{split}$$ from which it follows that $$\begin{split}
\hat{\sigma}_{\mu\nu} &\equiv \hat{\alpha} \sigma_{\mu\nu}\ ,\quad \hat{\omega}_{\mu\nu} \equiv \frac{1}{\hat{\alpha}} \omega_{\mu\nu} ,\quad \hat{\theta}\equiv\hat{\alpha}\theta\\
\hat{a}_\nu &\equiv \left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]a_\nu \\
\hat{\mathcal{A}}_\nu&\equiv \mathcal{A}_\nu-(1-\hat{\alpha}^2)\frac{d}{2}a_{\nu} \\
\end{split}$$ We invert the last relation to get $$\begin{split}
\mathcal{A}_\nu &=\frac{\hat{a}_\nu}{1-\frac{d}{2}(1-\hat{\alpha}^2)}-\frac{\hat{\theta}}{d-1}\hat{u}_\nu
\end{split}$$ This can be used to write $u_\mu$ is terms of hatted variables $$\begin{split}
u_\mu = \hat{\alpha}\hat{u}_\mu-\frac{\hat{\alpha}^2}{r_D} \mathcal{A}_\mu = \left(1+\frac{\hat{\alpha}\hat{\theta}}{r_D(d-1)}\right)\hat{\alpha}\hat{u}_\mu - \frac{\hat{\alpha}^2}{r_D} \frac{\hat{a}_\mu}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}
\end{split}$$
Now, we want to write $\tilde{\Gamma}_{\mu\nu}{}^\rho$ in hatted variables. We start with $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho &= (\hat{\alpha}^2-1)\left[\sigma_{\mu\nu}+\frac{\theta}{d-1}\ \left(P_{\mu\nu}+\frac{d}{2}u_\mu u_\nu\right)-da_{(\mu}u_{\nu)}\right]u^\rho\\
&\qquad + \frac{\hat{\alpha}^2-1}{\hat{\alpha}^2}\left[-2\omega^\rho{}_{(\mu}u_{\nu)}+(\frac{d}{2}-1)u_{\mu}u_{\nu}a^\rho\right] \\
\end{split}$$ and use the Dirichlet dictionary to get $$\begin{split}
\tilde{\Gamma}_{\mu\nu}{}^\rho
&= (1-\frac{1}{\hat{\alpha}^2})\left[ \hat{\sigma}_{\mu\nu}+\frac{\hat{\theta}}{d-1}\ \left(\hat{P}_{\mu\nu}+\frac{d}{2}\hat{\alpha}^2\hat{u}_\mu \hat{u}_\nu\right)-\frac{d\hat{\alpha}^2}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{a}_{(\mu}\hat{u}_{\nu)}\right]\hat{u}^\rho\\
&\qquad + (\hat{\alpha}^2-1)\left[-2\hat{\omega}^\rho{}_{(\mu}\hat{u}_{\nu)}+\frac{\frac{d}{2}-1}{\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\hat{u}_{\mu}\hat{u}_{\nu}\hat{a}^\rho\right] \\
\end{split}$$
Non-relativistic scaling a la BMW {#A:bmw}
=================================
In this appendix we review and correct the scaling limit of [@Bhattacharyya:2008kq].The major change in notation we make is to replace $\epsilon_\text{BMW}$ by a parameter $\aleph^{-1}$ so that the BMW limit is a large $\aleph$ asymptotics of the expressions below.
For simplicity we consider a fluid without bulk viscosity (which includes conformal fluids) with an energy-momentum tensor $T^{\mu\nu}$ given by $$T_{\mu\nu} = p\, g_{\mu\nu} + (\varepsilon + p) \, u_\mu\,u_\nu - 2 \, \eta\, \sigma_{\mu\nu} + \ldots
\label{Tbdy2}$$ where we assume that this fluid lives on a background spacetime with a metric $g_{\mu\nu}$ and for the moment have just written out the stress tensor to first order in gradients.
Spacetime split for the non-relativistic scaling {#s:bmwA}
-------------------------------------------------
We begin by decomposing this metric into an ambient part $g^{(0)}_{\mu\nu}$ and a forcing part $h_{\mu\nu}$, the split being done so as to recover explicit forcing terms in the Navier-Stokes (in addition to the pressure gradient term). One picks a suitable frame for the ambient metric, and writes the geometry as $$g_{\mu\nu} = g^{(0)}_{\mu\nu} + h_{\mu\nu} \ , \qquad g^{(0)}_{\mu\nu}\, dx^\mu\, dx^\nu = -dt^2 + g^{(0)}_{ij} \, dx^i\, dx^j \ ,$$ where ${g}^{(0)}_{ij}$ are slowly varying functions of $x^i$ with ${h}_{\mu\nu}$ being treated as a perturbation. The metric perturbations which force the fluid are taken to be $${h}_{\mu\nu} \,dx^\mu\,dx^\nu= 2\, {\aleph}^{-1}\, {k}^*_{i}\, dt\, dx^i + {\aleph}^{-2}\, \left({h}^*_{tt}\, dt^2 + {h}^*_{ij} \, dx^i \, dx^j \right)$$ where $\aleph$ is the book-keeping parameter that implements the BMW scaling (note $\aleph = \epsilon_\text{BMW}^{-1}$).
We employ the notation that all the functions which have a $*$ subscript or superscript (which we freely interchange to keep formulae clear) are of a specific functional form with anisotropic scaling of their spatial and temporal gradients. $${\cal Y}_*(t,x^i) : {\mathbb R}^{d-1,1} \mapsto {\mathbb R}\ , \;\; \text{such that}
\;\; \{ \partial_t {\cal Y}_*(t,x^i), \nabla^{(0)}_i {\cal Y}_*(t,x^i)\} \sim \{{\cal O}(\aleph^{-2}) ,{\cal O}(\aleph^{-1})\}$$ The co-metric corresponding to the metric above is given as $$\begin{split}
g^{\mu\nu} \partial_\mu \otimes \partial_\nu &= - \partial_t\otimes \partial_t + g^{ij}_{(0)}\partial_i\otimes\partial_j + 2\, {\aleph}^{-1}\ {k}_*^{i}\ \partial_t \otimes_s \partial_i\\
&\qquad - {\aleph}^{-2}\left[ \left({h}^*_{tt}-k_j^*k^j_*\right)\partial_t\otimes \partial_t + \left({h}_*^{ij}+k^i_*k^j_*\right)\partial_i\otimes \partial_j \right] \\
& \qquad + 2\, {\aleph}^{-3}\left[\left({h}^*_{tt}-k_j^*k^j_*\right) {k}_*^{i} - {h}_*^{ij} k^*_j\right]\partial_t \otimes_s \partial_i + {\cal O}(\aleph^{-4})
\end{split}$$ where we have freely raised and lowered the spatial indices with ${g}^{(0)}_{ij}$.
The velocity field of the fluid is parameterized as $${u}^{\mu} = u^t\, \left(1, {\aleph}^{-1}\, {v}_*^i \right)
\label{bmwd1}$$ where the function $u^t$ is determined via the constraint $g_{\mu\nu}\, u^\mu\, u^\nu =-1$. This gives the full velocity field in a large $\aleph$ expansion as $$\begin{split}
u^t &=1 + \frac{\aleph^{-2}}{2} \left( h^*_{tt} + 2 \,k^*_{j}\, v^{j}_{*} +{g}^{(0)}_{jk} \, v^{j}_{*}\, v^{k}_{*} \right)+ {\cal O}(\aleph^{-4})\\
u^i &= \aleph^{-1} \, v_*^i + \frac{\aleph^{-3}}{2} \left( h^*_{tt} + 2 \,k^*_{j}\, v^{j}_{*} + {g}^{(0)}_{jk} \, v^{j}_{*}\, v^{k}_{*} \right)\, v_*^i + {\cal O}(\aleph^{-4})\\
{u}_t &= -1 - \frac{1}{2}\, {\aleph}^{-2} \, \left(- {h}^*_{tt} + {g}^{(0)}_{jk} \, {v}^j_* \, {v}^k_* \right)+ {\cal O}(\aleph^{-4})\\
{u}_i &= {\aleph}^{-1}\, \left( {v}^*_i + {k}^*_i \right)+ \aleph^{-3}\left[{h}^*_{ij}v^{j}_{*}+ \frac{1}{2} \left( h^*_{tt} + 2 \,k^*_{j}\, v^{j}_{*} +{g}^{(0)}_{jk} \, v^{j}_{*}\, v^{k}_{*} \right)\, \left( {v}^*_i + {k}^*_i \right) \right] + {\cal O}(\aleph^{-4})
\end{split}$$ which can alternately be written as $$\begin{split}
u^\mu \partial_\mu &=\partial_t + \aleph^{-1} \, v_*^i \partial_i \\
&\quad + \frac{\aleph^{-2}}{2} \left( h^*_{tt} + 2 \,k^*_{j}\, v^{j}_{*} + {g}^{(0)}_{jk} \, v^{j}_{*}\, v^{k}_{*} \right)\partial_t + \frac{\aleph^{-3}}{2} \left( h^*_{tt} + 2 \,k^*_{j}\, v^{j}_{*} + {g}^{(0)}_{jk} \, v^{j}_{*}\, v^{k}_{*} \right)\, v_*^i\partial_i + {\cal O}(\aleph^{-4})\\
{u}_\mu dx^\mu &= -dt +{\aleph}^{-1}\, \left( {v}^*_i + {k}^*_i \right) dx^i - \frac{1}{2}\, {\aleph}^{-2} \, \left(- {h}^*_{tt} + {g}^{(0)}_{jk} \, {v}^j_* \, {v}^k_* \right)dt\\
&\quad + \aleph^{-3}\left[{h}^*_{ij}v^{j}_{*}+ \frac{1}{2} \left( h^*_{tt} + 2 \,k^*_{j}\, v^{j}_{*} + {g}^{(0)}_{jk} \, v^{j}_{*}\, v^{k}_{*} \right)\, \left( {v}^*_i + {k}^*_i \right) \right] dx^i + {\cal O}(\aleph^{-4})\\
\end{split}$$
Now, we are ready to calculate the velocity gradients - with some foresight, we will use the fact that the BMW limit is also an incompressibility limit where $v_*^i$ is divergenceless (see below). With this in mind, we can write the velocity gradients as $$\begin{split}
{\theta} &= {\cal O}( {\aleph}^{-4}) \\
{a}_{\mu} dx^{\mu} &= {\aleph}^{-3} \left[ \partial_{t} {v}_{i}^{*}+{v}_{*}^{j} {\nabla}^{(0)}_{j} {v}_{i}^{*} -f_i^* \right] dx^{i} + {\cal O}( {\aleph}^{-4}) \\
{a}^{\mu} \partial_{\mu} &= {\aleph}^{-3} \left[ \partial_{t} {v}^{i}_{*}+{v}_{*}^{j} {\nabla}^{(0)}_{j} {v}^{i}_{*} -f^i_* \right] \partial_{i} + {\cal O}( {\aleph}^{-4}) \\
{\sigma}_{\mu \nu} dx^{\mu} dx^{\nu} &= {\aleph}^{-2}\, {\nabla}^{(0)}_{(i} {v}^{*}_{j)} \,dx^{i} dx^{j}
-2{\aleph}^{-3} \, {v}^{j}_{*} \, {\nabla}^{(0)}_{(i} {v}^{*}_{j)} \,dx^{i} dt + {\cal O}( {\aleph}^{-4}) \\
{\sigma}^{\mu \nu} \partial_\mu\otimes\partial_\nu &= {\aleph}^{-2}{\nabla}_{(0)}^{(i} {v}_{*}^{j)}\partial_i\otimes\partial_j+ 2 {\aleph}^{-3} ({v}_{i}^{*}+k_i^*) {\nabla}_{(0)}^{(i} {v}_{*}^{j)}\partial_t\otimes_s\partial_j + {\cal O}( {\aleph}^{-4})\\
{\omega}_{\mu \nu} dx^{\mu}\wedge dx^{\nu} &= {\aleph}^{-2}\, {\nabla}^{(0)}_{[i} {v}^{*}_{j]} \,dx^{i} \wedge dx^{j}
- 2{\aleph}^{-3} {v}^{j}_{*} \, {\nabla}^{(0)}_{[i} {v}^{*}_{j]} \,dx^{i}\wedge dt + {\cal O}( {\aleph}^{-4}) \\
\end{split}$$ where $\nabla^{(0)}_{i}$ is the covariant derivative compatible with $\hat{g}^{(0)}(x^{i})$ and $f_i^*$ is the ‘gravitational force’ acting on the fluid $$f_i^* \equiv \frac{1}{2}\partial_i {h}^*_{tt} - \partial_t {k}^*_i +\left[\nabla^{(0)}_i k^*_j - \nabla^{(0)}_j k^*_i\right] {v}_*^j = \frac{1}{2}\partial_i {h}^*_{tt} - \partial_t {k}^*_i +q^*_{ij} {v}_*^j
\label{fidefn}$$ where we have introduced $q^*_{ij} \equiv \nabla^{(0)}_i k^*_j - \nabla^{(0)}_j k^*_i$.
Navier-Stokes equations on a curved geometry {#s:bmwB}
--------------------------------------------
Now, we turn to the scaling of the thermodynamic variables. We define the mass density $\rho_0$, the pressure per mass density $p_*$ and the kinematic viscosity $\nu_0$ by $$\begin{split}
\rho_0 \equiv \epsilon_0 + p_0\ ,\quad p = p_0 + {\aleph}^{-2} \rho_0\ {p}_{*}\quad\text{and}\quad \eta_0 =\rho_0\, \nu_0
\end{split}$$ where as before the subscript $0$ indicates the background value. All other thermodynamic variables have similar scalings, for example, $$\varepsilon = \varepsilon_0 + \aleph^{-2}\rho_0 \, \varepsilon_* \quad\text{and}\quad b = b_0 + \aleph^{-2} \, \delta b_*
\label{bmwd2}$$ The BMW limit is taken as to be the scaling as $\aleph \to \infty$ and in this limit the conservation equation $\nabla_\mu T^{\mu\nu} = 0$ reduces to $${\cal O}(\aleph^{-2}): \, \qquad \nabla_i^{(0)} \, v^i_* = 0
\label{incombdy}$$ and then a non-relativistic forced Navier-Stokes equation: $$\begin{split}
{\cal O}(\aleph^{-3}): \qquad \left[\partial_t +v^j_*\nabla^{(0)}_j\right] v_i^* - 2\, \nu_0\, \nabla^{(0)^j} \left(\nabla^{(0)}_{(i} v^{*}_{j)}\right)= f_i^* - \nabla^{(0)}_i \left[p_*+\nu_0^2\, \frac{d\, (d-3)}{(d-1)\,(d-2)} \; R^{(0)}\right]
\end{split}
\label{nsbdy1}$$ where the kinematic viscosity $\nu_0$ is given as $$\nu_0 \equiv \frac{\eta_0}{\rho_0} = \frac{b_0}{d}
\label{}$$ Before proceeding we should explain the origin of this equation since it differs from that presented in [@Bhattacharyya:2008kq]. On the l.h.s of we see a familiar term corresponding to the convective derivative of the non-relativistic velocity. The usual Laplacian term is modified due to the background curvature into the second derivative piece multiplying $\nu_0$. Its origin can be traced back to the term $-2 \, \eta \, \sigma_{\mu\nu}$ in the relativistic stress tensor . On the r.h.s. of we have collected all the forcing terms: there is the familiar pressure gradient term along with two other terms that arise from curvature. $f_i^*$ is the forcing term that arises from the fluctuating part of the metric as is clear from ; this term has been accounted for in [@Bhattacharyya:2008kq]. However, we also should see a forcing of the fluid from the ‘background’ curvature: the spatial part of the metric $g^{(0)}_{\mu\nu}$ is a curved spatial metric $g_{ij}^{(0)}$ and its effect on the fluid turns out to be at the order ${\cal O}(\aleph^{-3})$, just the same as the other terms in the equation. However, its origins in the relativistic stress tensor are a bit more involved; it does not arise from any of the terms written down in but rather from a second order gradient term in the relativistic stress tensor $2\,\eta\, b\, C_{\mu\alpha\nu\beta}u^\alpha u^\beta$ i.e., a coupling between the fluid and background curvature [@Bhattacharyya:2008mz].
Note that we have here specialized to relativistic conformal fluids which have holographic duals, so that the transport coefficient multiplying the tensor structure $C_{\mu\alpha\nu\beta}u^\alpha u^\beta$ is fixed to be $2\,\eta\, b$. For a general fluid we can have a new transport coefficient here $\kappa \propto \eta \, b$ and the correspondingly we would replace the coefficient of $R^{(0)}$ in with $\nu_0^2 \to \frac{\kappa_0}{\rho_0\, d}$. Also, we should note that that while other tensor structures involving curvatures couplings are allowed for non-conformal fluids, these will necessarily have non-vanishing trace and as a result will only show up at sub-leading order in the BMW scaling limit.
At the risk of being overly pedantic we reiterate the fact that if we wish to place consider the non-relativistic BMW scaling of a relativistic fluid, then we must necessarily work with higher order gradient terms in the relativistic fluid stress tensor. To obtain the correct non-relativistic equations up to the order where we encounter Navier-Stokes equations the relevant part of the relativistic stress tensor is given to be $$T_{\mu\nu} = p\, g_{\mu\nu} + (\varepsilon + p) \, u_\mu\,u_\nu -2\,\eta\, \sigma_{\mu\nu}+2\,\eta\, b\, C_{\mu\alpha\nu\beta}u^\alpha u^\beta
\label{}$$ which is a subset of the full second order stress tensor derived in [@Bhattacharyya:2008mz] $$\label{enmom:eq}
\begin{split}
T_{\mu\nu}& = p\, g_{\mu\nu} + (\varepsilon + p) \, u_\mu\,u_\nu -2\,\eta\, \sigma_{\mu\nu}\\
&-2\,\eta \,\tau_\omega \, \left[u^{\lambda}\mathcal{D}_{\lambda}\sigma_{\mu \nu}+\omega_{\mu}{}^{\lambda}\sigma_{\lambda \nu}+\omega_\nu{}^\lambda \sigma_{\mu\lambda} \right]\\
&+2\,\eta\, b\left[u^{\lambda}\mathcal{D}_{\lambda}\sigma_{\mu \nu}+\sigma_{\mu}{}^{\lambda}\sigma_{\lambda \nu} -\frac{\sigma_{\alpha \beta}\sigma^{\alpha \beta}}{d-1}P_{\mu \nu}+ C_{\mu\alpha\nu\beta}u^\alpha u^\beta \right]+\ldots\\
\end{split}$$ It is a simple exercise to verify that none of the other terms involved in the second order stress tensor contribute to the BMW scaled equations at ${\cal O}(\aleph^{-3})$.
The bulk metric dual to a non-relativistic fluid on the boundary of AdS {#s:bmwC}
-----------------------------------------------------------------------
One can also construct the gravitational solutions dual to such fluids as described in [@Bhattacharyya:2008kq]. To do so we simply need to apply the scaling of parameters described earlier as for e.g., in , to the general fluid/gravity bulk metric dual to a relativistic fluid on the boundary of AdS. Such a metric correct to second order in the relativistic gradient expansion was originally derived in [@Bhattacharyya:2008mz] generalizing the original result of [@Bhattacharyya:2008jc]. It was believed that this in general would suffice to find the gravity dual for a non-relativistic fluid that satisfies the incompressible Navier-Stokes equations derived above , . In fact the original computation presented in [@Bhattacharyya:2008kq] argued that it would actually suffice to consider the relativistic metric accurate to first order in gradients. Unfortunately, this turns out to be incorrect and one needs a subset of second order gradient terms along with one particular third order gradient term to solve Einstein’s equations to order ${\cal O}(\aleph^{-3})$. Note that this is necessary because it is at ${\cal O}(\aleph^{-3})$ that we encounter the dynamical content of the boundary fluid equations, viz., the Navier-Stokes equation .
The new ingredient in our analysis is that we need to worry about a particular third order term proportional to ${\cal D}_\mu {\cal R}$ in the fluid/gravity correspondence. The term in question turns out to be computable using the original algorithm outlined for constructing bulk metrics dual to boundary fluids in [@Bhattacharyya:2008jc] and thankfully involves a decoupled tensor structure that can be sourced independently. Including this term, we find that for the non-relativistic fluid on the Dirichlet surface it suffices to consider the following truncation of the relativistic fluid/gravity metric to third order in boundary gradient expansion: $$\label{metric3trunc}
\begin{split}
ds^2&=-2 u_\mu dx^\mu \left( dr + r\ \mathcal{A}_\nu dx^\nu \right) \\
&+ \left[ r^2 g_{\mu\nu} +2u_{(\mu}\mathcal{S}_{\nu)\lambda}u^\lambda -\frac{2}{3r}\frac{u_{(\mu}\mathcal{D}_{\nu)}\mathcal{R}}{(d-1)(d-2)} \right]dx^\mu dx^\nu\\
&+r^2\left[ \frac{u_\mu u_\nu}{(br)^d} +2b F(br) \sigma_{\mu\nu}+4b^2 \frac{L(br)}{(br)^d}u_{(\mu}P_{\nu)}^{\lambda}\mathcal{D}_{\alpha}{\sigma^{\alpha}}_{\lambda}\right]dx^\mu dx^\nu \\
&-2(br)^2\left[ H_1(br) C_{\mu\alpha\nu\beta}u^\alpha u^\beta+\frac{b N(br)}{(br)^d} \frac{(d-3)u_{(\mu}\mathcal{D}_{\nu)}\mathcal{R}}{(d-1)(d-2)}\right] dx^\mu dx^\nu \\
& +\ldots\\
\end{split}$$ where the functions appearing above and their large $r$ asymptotics are given to be: $$\label{funcDreq}
\begin{split}
f(br) &\equiv 1-\frac{1}{(br)^{d}} \\
F(br)&\equiv \int_{br}^{\infty}\frac{y^{d-1}-1}{y(y^{d}-1)}dy \\
&\approx \frac{1}{br} -\frac{1}{d(br)^d}+ \frac{1}{(d+1)(br)^{d+1}}+\frac{\#}{(br)^{2d}}+\ldots\\
L(br) &\equiv \int_{br}^\infty\xi^{d-1}d\xi\int_{\xi}^\infty dy\ \frac{y-1}{y^3(y^d
-1)} \\
&\approx \frac{1}{(d+1)(br)}-\frac{1}{2(d+2)(br)^2}+\frac{1}{(d+1)(2d+1)(br)^{d+1}}\\
&\quad-\frac{1}{(d+1)(2d+4)(br)^{d+2}} +\frac{\#}{(br)^{2d+1}}+\ldots \\
H_1(br)&\equiv \int_{br}^{\infty}\frac{y^{d-2}-1}{y(y^{d}-1)}dy \\
&\approx \frac{1}{2(br)^2}-\frac{1}{d(br)^d}+ \frac{1}{(d+2)(br)^{d+2}}+\frac{\#}{(br)^{2d}}+\ldots\\
N(br) &\equiv \int_{br}^\infty\xi^{d-1}d\xi\int_{\xi}^\infty dy\ \frac{y^2-1}{y^4(y^d
-1)} \\
&\approx\frac{1}{(d+1)(br)} -\frac{1}{3(d+3)(br)^3}+\frac{1}{(d+1)(2d+1)(br)^{d+1}}\\
&\quad-\frac{1}{(d+3)(2d+3)(br)^{d+3}} +\frac{\#}{(br)^{2d+1}} +\ldots \\
\end{split}$$ There are various new curvature tensors and derivatives introduced above. ${\cal D}$ denotes the Weyl covariant derivative introduced in [@Loganayagam:2008is] which was used to present the bulk metric dual relevant for fluid/gravity correspondence in the case of curved boundaries in [@Bhattacharyya:2008mz]. ${\cal R}_{\mu\nu}$ is likewise a Weyl covariant Ricci tensor and ${\cal S}_{\mu\nu}$ is a Weyl-covariant Schouten tensor $${\cal S}_{\mu\nu} = \frac{1}{d-2} \left( {\cal R}_{\mu\nu} - \frac{1}{2\, (d-1)}\, g_{\mu\nu}\, {\cal R}\right)
\label{}$$ For further details of these objects and the complete form for the second order fluid/gravity metric accurate to second order in gradients we refer the reader to [@Bhattacharyya:2008mz].
Once we have the relativistic metric at hand it is a simple matter to employ the scalings outlined earlier. We find that the bulk metric dual to an incompressible Navier-Stokes fluid living on a spatially curved geometry at the boundary of AdS takes the form $$\label{bdybmwf}
\begin{split}
ds^2
&= ds_0^2 + {\aleph}^{-1} ds_1^2 + {\aleph}^{-2} ds_2^2 + {\aleph}^{-3} ds_3^2 + {\cal O}( {\aleph}^{-4}) \\
&\text{with}\\
ds_0^2 &= 2 \ dt\ dr + r^2\left[- f_0 dt^2 + {g}^{(0)}_{ij}dx^i dx^j\right]\\
ds_1^2 &= -2 \left( {v}^*_i + {k}^*_i \right)\ dx^i\ dr + 2 r^2 \left[ {k}^*_i -\left(1- f_0\right)\left( {v}^*_i + {k}^*_i \right)\right] dx^i dt \\
ds_2^2 &= 2 \left[- \frac{1}{2} {h}^*_{tt} + \frac{1}{2} {g}^{(0)}_{jk} \, {v}^j_* \, {v}^k_* \right]dt\ dr + r^2\left[ {h}^*_{tt}\, dt^2 + {h}^*_{ij} \, dx^i \, dx^j \right]\\
&\quad +r^2 \left(1-f_0\right) \left[\left(- {h}^*_{tt} + {g}^{(0)}_{jk} \, {v}^j_* \, {v}^k_*+ {p}_* d\right)dt^2 \right.\\
&\qquad \left. + \left( {v}^*_i + {k}^*_i \right) \left( {v}^*_j + {k}^*_j \right)dx^i dx^j \right]+2\,r^2\, b_0 {F}_0 {\nabla}^{(0)}_{(i} {v}^{*}_{j)} \,dx^{i} dx^{j}\\
&\qquad
\red{-\frac{R^{(0)}}{(d-1)\,(d-2)} \, dt^2 - 2\, H_0 \left(\text{S}_{ij}^{(0)} - \frac{R^{(0)}\, g_{ij}^{(0)}}{2\, (d-1)\, (d-2)}\right) dx^i \,dx^ j }\\
ds_3^2 &= -2 \left[ {h}^*_{ij} {v}^{j}_{*}+ \left( \frac{1}{2} {h}^*_{tt} + \, {k}^*_{j}\, {v}^{j}_{*} +\frac{1}{2} {g}^{(0)}_{jk} \, {v}^{j}_{*}\, {v}^{k}_{*}\right)\, \left( {v}^*_i + {k}^*_i \right) \right] dx^i dr \\
&\quad + 2r
\left[ \partial_{t} {v}_{i}^{*}+ {v}_{*}^{j} {\nabla}^{(0)}_{j} {v}_{i}^{*} - {f}_i^* \right] dx^{i}dt
-4r^2b_0 {F}_0 {v}_{*}^j {\nabla}^{(0)}_{(i} {v}^{*}_{j)} \,dx^{i} dt
\\
&\quad -2\,r^2 \left(1- f_0\right)\left[ {h}^*_{ij} {v}^{j}_{*}+ \left( {k}^*_{j}\, {v}^{j}_{*} + {g}^{(0)}_{jk} \, {v}^{j}_{*}\, {v}^{k}_{*}+ {p}_* d \right)\, \left( {v}^*_i + {k}^*_i \right) \right] dx^i dt \\
&\quad
\red{- 2\,\frac{L_0}{(b_0r)^{d-2}} {\nabla}^2_{(0)}v^*_{i} \,dx^i \,dt -2 \, \text{S}_{ij}^{(0)}\, v_*^j \, dx^i \,dt- \frac{1}{d-2}\, {\nabla}^j_{(0)}q^*_{ij} \, dx^i \,dt}\\
&\quad
\red{+ \frac{R^{(0)}}{(d-1)\,(d-2)}\, (v^*_i + k^*_i) \, dx^i \,dt
+ 4\, H_0 \left(\text{S}_{ij}^{(0)} - \frac{R^{(0)}\, g_{ij}^{(0)} }{2\, (d-1)\, (d-2)} \right) v_*^j \, dx^i\, dt }\\
&\quad
\red{+ 2\, \frac{b_0\, N_0}{(b_0\,r)^{d-2}} \, \frac{d-3}{(d-1)\, (d-2) }\, \nabla^{(0)}_i R^{(0)} \, dx^i \,dt}
\end{split}$$ where we have highlighted the terms that were missed in the previous analysis for quick comparison. Note that when the background spatial metric $g^{(0)}_{ij}$ is Ricci flat, many of the terms vanish except for two terms in the third order metric (which are proportional to $\nabla^2_{(0)} v_i^*$ and $\nabla^j_{(0)} q^*_{ij}$ respectively). Finally we should note that $\text{S}_{ij}^{(0)}$ is used to denote the spatial components of the Schouten tensor of the full background metric $g^{(0)}_{\mu\nu}$; in particular, it should not be confused with the Schouten tensor of the spatial metric $g_{ij}^{(0)}$. The functions that enter into the metric above are: $$\begin{split}
f_0 &\equiv 1-(b_0 r)^{-d}\ ,\quad {p}_* \equiv -\frac{\delta b_*}{b_0} \quad\text{and}\quad {F}_0 \equiv \int_{b_0 r}^{\infty}\frac{y^{d-1}-1}{y(y^{d}-1)}dy\\
{f}_i^* &\equiv \frac{1}{2}\partial_i {h}^*_{tt} - \partial_t {k}^*_i +\left[ {\nabla}^{(0}_i {k}^*_j - {\nabla}^{(0)}_j {k}^*_i\right] {v}_*^j = \frac{1}{2}\partial_i {h}^*_{tt} - \partial_t {k}^*_i + {q}^*_{ij} {v}_*^j \\
L_0 &\equiv \int_{b_0 r}^\infty\xi^{d-1}d\xi\int_{\xi}^\infty dy\ \frac{y-1}{y^3(y^d
-1)}\\
H_0 &\equiv (b_0 r)^2\, \int_{b_0r}^{\infty}\frac{y^{d-2}-1}{y(y^{d}-1)}dy \\
N_0 &\equiv \int_{b_0r}^\infty\xi^{d-1}d\xi\int_{\xi}^\infty dy\ \frac{y^2-1}{y^4(y^d
-1)} \\
\end{split}$$
Finally, let us note that the Navier-Stokes equations themselves have an interesting scaling symmetry. Given any solution to and with $g^{(0)}_{ij} = \delta_{ij}$ we can consider replacing $$p_* \to \epsilon^2 \, p_{*\epsilon} \ , \qquad v^i_* \to \epsilon\, v^i_{*\epsilon} \ , \qquad f^i_* \to \epsilon^3\, f^i_{*\epsilon}$$ where again the functions entering the dynamics with subscript $*\epsilon$ have spatial gradients $\partial_i \sim \epsilon$ and temporal gradients $\partial_t \sim \epsilon^2$. This fact makes it possible to compound the Navier-Stokes scaling which effectively allows one to replace $\aleph \to \aleph^{w}$ for some $w \geq 1$. Essentially the incompressible Navier-Stokes system of equations is a fixed point set of this scaling symmetry, a fact that we have made use of in .
Bulk dual of the non-relativistic Dirichlet fluid {#s:bmwR}
=================================================
In this appendix we present without derivation the results for the non-relativistic scaling limit of the Dirichlet problem, generalizing the result quoted in . Physically the only new content is that we allow the metric on the Dirichlet surface $\Sigma_D$ to be endowed with an arbitrarily slowly varying spatial metric. Thus in contrast to we are relaxing the constraint on $g_{ij}^{(0)}$ introduced in being Ricci flat.
To indicate the differences note that the presence of a non-Ricci flat spatial metric $g^{(0)}_{ij}$ implies that the non-relativistic metric gets contributions from various tensor structures which appear at the second order in the gradient expansion of the fluid/gravity correspondence. In terms of the metric written down in [@Bhattacharyya:2008mz] some of these are straightforward to see – any term involving curvature tensors of boundary data (and thus via the Dirichlet constitutive relation the hypersurface $\Sigma_D$ curvatures) will contribute at this order. However, we also get contribution from spatial gradients of the hypersurface curvature, i.e., in the BMW scaling limit encounter terms of the form $\nabla^{(0)}_i R^{(0)}$. To guide the reader towards a derivation, we quote simply the relativistic tensor structures which are relevant and their scaling behavior under the Dirichlet BMW scaling. $$\begin{split}
\hat{g}_{\mu\nu}\, dx^\mu\, dx^\nu &= -dt^2 + \hat{g}^{(0)}_{ij} \, dx^i\, dx^j + 2\, {\hat{\aleph}}^{-1}\, \hat{k}^*_{i}\, dt\, dx^i + {\hat{\aleph}}^{-2}\, \left(\hat{h}^*_{tt}\, dt^2 + \hat{h}^*_{ij} \, dx^i \, dx^j \right).\\
\hat{u}_\mu dx^\mu &= -dt +{\hat{\aleph}}^{-1}\, \left(\hat{v}^*_i + \hat{k}^*_i \right) dx^i - \frac{1}{2}\, {\hat{\aleph}}^{-2} \, \left(- \hat{h}^*_{tt} + \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_* \right)dt\\
&\quad + \hat{\aleph}^{-3}\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+ \frac{1}{2} \left( \hat{h}^*_{tt} + 2 \,\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*} \right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right] dx^i + {\cal O}(\hat{\aleph}^{-4})\\
\hat{a}_{\mu} dx^{\mu} &= {\hat{\aleph}}^{-3} \left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*} -\hat{f}_i^* \right] dx^{i} + {\cal O}( {\hat{\aleph}}^{-4}) \\
\hat{\sigma}_{\mu \nu} dx^{\mu} dx^{\nu} &= {\hat{\aleph}}^{-2}\, \hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dx^{j}
-2{\hat{\aleph}}^{-3} \, \hat{v}^{j}_{*} \, \hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dt + {\cal O}( {\hat{\aleph}}^{-4}) \\
\end{split}
\label{genscalefA}$$ as before along with new tensor structures: $$\begin{split}
\hat{\mathcal{S}}_{\nu\lambda}\hat{u}^\lambda dx^\nu &={\hat{\aleph}}^{-2}\frac{\hat{R}^{(0)}}{2(d-1)(d-2)}dt+{\hat{\aleph}}^{-3}\left[\hat{\text{S}}^{(0)}_{ij}\hat{v}^j_*+ \frac{1}{2(d-2)}\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\right] dx^i + {\cal O}( {\hat{\aleph}}^{-4})\\
\hat{\mathcal{R}}_{\nu\lambda}\hat{u}^\lambda dx^\nu &={\hat{\aleph}}^{-3}\left[\hat{R}^{(0)}_{ij}\hat{v}^j_*+ \frac{1}{2}\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\right] dx^i + {\cal O}( {\hat{\aleph}}^{-4})\\
\hat{P}_{\nu}^{\lambda}\hat{\mathcal{D}}_{\alpha}{\hat{\sigma}^{\alpha}}_{\lambda}dx^\nu &=\frac{{\hat{\aleph}}^{-3}}{2}\hat{\nabla}^2_{(0)}\hat{v}^*_{i} dx^i + {\cal O}( {\hat{\aleph}}^{-4}) \\
\hat{C}_{\mu\alpha\nu\beta}\hat{u}^\alpha \hat{u}^\beta dx^\mu dx^\nu &= {\hat{\aleph}}^{-2} \left[\hat{\text{S}}^{(0)}_{ij}-\frac{\hat{R}^{(0)}}{2(d-1)(d-2)}\hat{g}^{(0)}_{ij}\right] dx^i dx^j\\
&\quad- \; 2\,{\hat{\aleph}}^{-3} \left[\hat{\text{S}}^{(0)}_{ij}-\frac{\hat{R}^{(0)}}{2(d-1)(d-2)}\hat{g}^{(0)}_{ij}\right] \hat{v}^j_* dx^i dt\\
\hat{\mathcal{D}}_\nu\hat{\mathcal{R}} dx^\nu &= {\hat{\aleph}}^{-3}\,\hat{\nabla}^j_{(0)}\hat{R}^{(0)}dx^j
\end{split}
\label{genscalefB}$$ where $$\begin{split}
\hat{f}_i^* &\equiv \frac{1}{2}\partial_i \hat{h}^*_{tt} - \partial_t \hat{k}^*_i +\left[\hat{\nabla}^{(0)}_i \hat{k}^*_j - \hat{\nabla}^{(0)}_j \hat{k}^*_i\right] \hat{v}_*^j = \frac{1}{2}\partial_i \hat{h}^*_{tt} - \partial_t \hat{k}^*_i +\hat{q}^*_{ij} \hat{v}_*^j \\
\hat{q}^*_{ij} &\equiv \hat{\nabla}^{(0)}_i \hat{k}^*_j - \hat{\nabla}^{(0)}_j \hat{k}^*_i
\end{split}$$ and we use $\hat{\text{S}}^{(0)}_{ij}$ to denote the spatial (i.e. $ij$-) components of the Schouten tensor for $g^{(0)}_{\mu\nu}$ keep the expressions somewhat compact.
Boundary data for non-relativistic fluids on $\Sigma_D$ {#s:}
-------------------------------------------------------
Given the scalings to obtain the non-relativistic fluid on $\Sigma_D$, it is possible to identify the relevant terms of the third order fluid/gravity metric that we need to retain to solve the Dirichlet problem. Per se the set of terms we need in the bulk metric is still given by , though as in the main text we want to use this information to solve for the Dirichlet constitutive relations. We first quote the results for the hypersurface stress tensor which defines the hypersurface velocity $\hat{u}^\mu$ before indicating the answers for the boundary velocity field and metric in terms of the hypersurface data.
We can parameterize the hypersurface stress tensor as in the main text; to obtain the correct non-relativistic equations on $\Sigma_D$ we need to retain some second order gradient terms involving hypersurface curvature tensors (analogously to the situation at the boundary as described in ). The relevant piece of the relativistic hypersurface stress tensor turns out to be: $$\begin{split}
\hat{T}_{\mu\nu}&=(\hat{\varepsilon}+\hat{p})\,\hat{u}_\mu\hat{u}_\nu + \hat{p}\,\hat{g}_{\mu\nu}-2\,\hat{\eta} \,\hat{\sigma}_{\mu\nu}
+\hat{\kappa}_C \, \hat{C}_{\mu\alpha\nu\beta}\, \hat{u}^\alpha \hat{u}^\beta +\ldots\\
\end{split}$$ where $$\begin{split}
\hat{\varepsilon} &\equiv \frac{d-1}{8\pi G_{d+1}b^d}\frac{\hat{\alpha}}{\hat{\alpha}+1}\left[1- \frac{\hat{\alpha}\hat{R}^{(0)}}{2\,r_D^2\,(d-1)\,(d-2)} \right]\\
\hat{\varepsilon}+\hat{p} &\equiv \frac{d\hat{\alpha}}{16\pi G_{d+1}b^d}\left[1- \frac{\hat{\alpha}^2\,\hat{R}^{(0)} }{2\,r_D^2\,(d-1)\,(d-2)} \right]-\frac{1}{8\pi G_{d+1}b^d}\frac{\hat{\alpha}^2}{\hat{\alpha}+1}\frac{\hat{R}^{(0)}}{r_D^2(d-1)(d-2)}\\
\hat{\eta} &= \frac{1}{8\pi G_{d+1}b^{d-1}}\\
\hat{\kappa}_C &= \frac{1}{8\pi G_{d+1}b^{d-2}}\left[1-\frac{\hat{\alpha}^2}{\hat{\alpha}+1}\frac{(br_D)^{d-2} -1}{(br_D)^d}\right]\\
\end{split}$$ Conservation of this stress tensor together with the scaling forms introduced in , leads to the incompressible Navier-Stokes equations on $\Sigma_D$.
The boundary velocity field and metric can be expressed in terms of the Dirichlet data as before. Before we write out the exressions in their gory detail, let us introduce some new parameters $\hat{\kappa}_L$, $\hat{\kappa}_N$ which depend on the location of $\Sigma_D$ as $$\begin{split}
\hat{\kappa}_L &\equiv \frac{1}{d}\left[
\xi(\xi^d-1)\frac{d}{d\xi}\left[\xi^{-d}L(\xi)\right]+
\frac{1}{\xi\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}+\frac{1}{\xi^2(d-2)}\right]_{\xi=br_D} \\
\hat{\kappa}_N &\equiv \frac{1}{d}\left\{
(d-3)\left[\xi(\xi^d-1)\frac{d}{d\xi}\left[\xi^{-d}N(\xi)\right]+
\frac{1}{\xi\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\right] \right.\\
&\left. -\frac{d-2}{2\, \xi^3\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\right\}_{\xi=br_D}
\end{split}$$ These are in turn limiting values of certain functions which appear in various guises in the result for the Dirichlet constitutive relations and the bulk metric and are collected once and for all below. $$\begin{split}
\hat{F}(br) &\equiv \frac{1}{\hat{\alpha}}\left(F(br)-F_D\right) \\
\hat{H}_1(br) &\equiv \left(H_1(br)-H_{1D}\right)\\
\hat{\xi}_1(br) &\equiv \frac{\hat{\alpha}}{b}\left(\frac{1}{r}-\frac{1}{r_D}\right)+\frac{\hat{\alpha}}{br_D}\left[1-\hat{\alpha}^2f(br)\right]=\frac{\hat{\alpha}}{br}\left[1-\frac{r}{r_D}\hat{\alpha}^2f(br)\right]\\
\hat{M}_1(br) &\equiv \frac{\hat{\alpha}^2}{b^2}\left(\frac{1}{r^2}-\frac{1}{r_D^2}\right)+\frac{\hat{\alpha}^2}{b^2r_D^2}\left[1-\hat{\alpha}^2f(br)\right]=\frac{\hat{\alpha}^2}{(br)^2}\left[1-\frac{r^2}{r_D^2}\hat{\alpha}^2f(br)\right]\\
\hat{M}_2(br) &\equiv\frac{\hat{\alpha}^2}{b^2r_D^2(d-2)}\left[1+\frac{2}{d\hat{\alpha}(\hat{\alpha}+1)}\right]
\left[1-\hat{\alpha}^2f(br)\right] \\
\hat{L}_1(br) &\equiv \frac{L(br)}{(br)^d}-\frac{L_D}{(br_D)^d}+\kappa_L \left[1-\hat{\alpha}^2f(br)\right]\\
&\quad-\frac{\hat{\alpha}^2-1}{2b^2(d-2)}\left(\frac{1}{r^2}-\frac{1}{r_D^2}\right)-\frac{(\hat{\alpha}^2-1)}{2b\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\left(\frac{1}{r}-\frac{1}{r_D}\right)\\
\hat{N}_1(br) &\equiv \hat{\alpha}(d-3)\left(\frac{N(br)}{(br)^d}-\frac{N_D}{(br_D)^d}\right)+\kappa_N\hat{\alpha}\left[1-\hat{\alpha}^2f(br)\right]\\
&+\frac{\hat{\alpha}}{3b^3}\left(\frac{1}{r^3}-\frac{1}{r_D^3}\right)-\frac{\hat{\alpha}\left[\frac{\hat{\alpha}^2}{r_D^2}+(d-3)b^2(\hat{\alpha}^2-1)\right]}{2b^3\left[1+\frac{d}{2}(\hat{\alpha}^2-1)\right]}\left(\frac{1}{r}-\frac{1}{r_D}\right)\\
\end{split}
\label{allhatfunx}$$ where the functions entering the above expressions and their asymptotics have been previously been collected together in .
The final result of this exercise leads to: $$\begin{split}
u_t &= -\hat{\alpha}_0 - \hat{\aleph}^{-2} \hat{\alpha}_0 \left[- \frac{1}{2}\hat{h}^*_{tt} + \frac{1}{2} \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_* +\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} - \frac{\hat{\alpha}_0^2}{r_D^2} \, \frac{\hat{R}^{(0)}}{2\,(d-1)\,(d-2)}
\right] + {\cal O}(\hat{\aleph}^{-4}) \\
\end{split}$$
$$\begin{split}
u_i &= \hat{\aleph}^{-1}\, \hat{\alpha}_0\, \left(\hat{v}^*_i + \hat{k}^*_i \right)\\
&\quad +\hat{\aleph}^{-3}\,
\hat{\alpha}_0\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+ \left( \frac{1}{2}\hat{h}^*_{tt} + \,\hat{k}^*_{j}\, \hat{v}^{j}_{*} +\frac{1}{2} \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*} +\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}\right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right]\\
&\qquad -\hat{\aleph}^{-3}\, \frac{\hat{\alpha}_0^2}{r_D\left(1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right)}\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*}-\hat{f}_i^* \right] \\
&\qquad
+\hat{\aleph}^{-3}\left[ b_0^2\, \hat{\kappa}_L\,\hat{\alpha}_0\, \hat{\nabla}^2_{(0)}v^*_i - b_0^3\, \hat{\kappa}_N \, \hat{\alpha}_0^2\, \frac{\nabla^{(0)}_i\, R^{(0)}}{(d-1)(d-2)} \right] \\
&\qquad
+ \hat{\aleph}^{-3}\, \frac{\hat{\alpha}_0^3}{r_D^2} \, \left[\hat{S}^{(0)}_{ij}\hat{v}^j_* - \frac{1}{d-2}\, \left(1 + \frac{2}{d\, \hat{\alpha}_0\, (\hat{\alpha}_0 +1) } \right) \hat{R}^{(0)}_{ij}\hat{v}^j_*
- \frac{\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}}{d\,(d-2)\, \hat{\alpha}_0\, (\hat{\alpha}_0 +1) }\right]\\
&\qquad
+ {\cal O}(\hat{\aleph}^{-4})
\end{split}$$
Further the Dirichlet constitutive relation for the boundary metric is $$\begin{split}
g_{tt} &= -\hat{\alpha}_0^2 + \hat{\aleph}^{-2} \, \left[\hat{h}^*_{tt}\ +
\left(1-\hat{\alpha}_0^2 \right) \left(- \hat{h}^*_{tt} + \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_*+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right) \right] \\
& \quad
+ \hat{\aleph}^{-2} \, \left[\frac{\hat{\alpha}_0^4}{r_D^2}\,\frac{\hat{R}^{(0)}}{(d-1)\,(d-2)}\right]
+ {\cal O}(\hat{\aleph}^{-4}) \\
g_{ti} &= \hat{\aleph}^{-1}\left( \hat{k}^*_i + (\hat{\alpha}_0^2-1)\, (\hat{k}_i^* + \hat{v}_i^*) \right)
- \hat{\aleph}^{-3}\frac{\hat{\alpha}_0^3}{r_D\left(1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)\right)}
\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*} -\hat{f}_i^* \right] \\
&\quad -\hat{\aleph}^{-3} \left(1-\hat{\alpha}_0^2 \right)\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+ \left(\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*}+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right]\\
&\quad +\;\hat{\aleph}^{-3} \left[\frac{2 \, b_0}{\hat{\alpha}_0}\, F(b_0\,r_D)\hat{v}_{*}^j\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} - 2\, b_0^2{H}_{1}(b_0 r_D) \left(\hat{\text{S}}^{(0)}_{ij}-\frac{\hat{R}^{(0)}}{2(d-1)(d-2)}\hat{g}^{(0)}_{ij}\right)v_*^j \right] \\
&\quad
+\;\hat{\aleph}^{-3} \, \frac{\hat{\alpha}_0^4}{2\,r_D^2}\left[2\,\hat{\text{S}}^{(0)}_{ij}\,\hat{v}^j_*+ \frac{1}{(d-2)}\hat{\nabla}^j_{(0)}\hat{q}^*_{ij} -\frac{\hat{R}^{(0)}}{(d-1)\,(d-2)} \left(\hat{v}^*_i + \hat{k}^*_i \right) \right] \\
& \quad -\;\hat{\aleph}^{-3} \,
\frac{\hat{\alpha}_0^2\,(\hat{\alpha}_0^2-1)}{2\,r_D^2\,(d-2)}\left[1+\frac{2}{d\,\hat{\alpha}_0\,(\hat{\alpha}_0+1)}\right] \left[2\,\hat{R}^{(0)}_{ij}\hat{v}^j_*+ \hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\right] \\
&\quad
+\;\hat{\aleph}^{-3} \, \left[\frac{1}{(d-1)\, (d-2)}\, b_0^3\,\hat{N}_1(\infty) \,\hat{\nabla}^j_{(0)}\hat{R}^{(0)}
-\, b_0^2\, \hat{L}_1(\infty) \, \hat{\nabla}^2_{(0)}\hat{v}^*_{i} \right]\\
&\quad
+ \;{\cal O}(\hat{\aleph}^{-4})\\
g_{ij} &= \hat{g}^{(0)}_{ij} + \hat{\aleph}^{-2} \left(\hat{h}^*_{ij} -(\hat{\alpha}_0^2 -1)\, (\hat{v}^*_i + \hat{k}^*_i) \,(\hat{v}^*_j + \hat{k}^*_j) - \frac{2 \, b_0}{\hat{\alpha}_0}\, F(b_0\,r_D) \,\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \right) \\
& \quad + \; \hat{\aleph}^{-2} \,b_0^2\,{H}_{1}(b_0 r_D) \left(2\,\hat{\text{S}}^{(0)}_{ij}-\frac{\hat{R}^{(0)}}{(d-1)(d-2)}\hat{g}^{(0)}_{ij}\right) \\
& \quad + \;{\cal O}(\hat{\aleph}^{-4})
\end{split}$$ where $$\begin{split}
\hat{L}_1(\infty) &\equiv\frac{1}{d}\left[
\frac{b_0\, r_D}{\hat{\alpha}_0^2}\, L'(b_0\, r_D)+
\frac{1-\frac{d}{2}}{b_0\, r_D\left[1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right]}+\frac{1-\frac{d}{2}}{b_0\, r_D^2(d-2)}\right] \left(1-\hat{\alpha}_0^2\right)\\
\hat{N}_1(\infty) &\equiv \frac{1}{d}\left\{
(d-3)\left[\frac{b_0\, r_D}{\hat{\alpha}_0^2}N'(b_0\, r_D)+
\frac{1-d/2}{b_0\, r_D\left[1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right]}\right] \right.\\
&\left. -\frac{d-2}{2\, b_0\, r_D^3\left[1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right]}\right\}\hat{\alpha}_0\left[1-\hat{\alpha}_0^2\right] -\frac{\hat{\alpha}_0}{3\, b_0\, r_D^3}+\frac{\hat{\alpha}_0^3}{2\, b_0\, r_D^3\left[1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right]}\\
\end{split}$$
The bulk dual for arbitrarily spatially curved metric on $\Sigma_D$ {#s:}
-------------------------------------------------------------------
The final result for the bulk metric dual to the non-relativistic fluid living on the Dirichlet hypersurface $\Sigma_D$ is simply obtained by plugging in the scaling form , into , having eliminated the boundary data in favor of the hypersurface data using the Dirichlet One obtains: $$ds^2 = ds_0^2 + {\aleph}^{-1} ds_1^2 + {\aleph}^{-2} ds_2^2 + {\aleph}^{-3} ds_3^2 + {\cal O}( {\aleph}^{-4})
\label{}$$ with $$\label{hypbmwf1aR012}
\begin{split}
ds_0^2 &=
2\,\hat{\alpha}_0\ dt\ dr + r^2\left(-\hat{\alpha}_0^2 \,f_0 \,dt^2
+ \hat{g}^{(0)}_{ij}\,dx^i dx^j\right)\\
ds_1^2 &=
-2\, \hat{\alpha}_0\left( \hat{v}^*_i + \hat{k}^*_i \right)\ dx^i\ dr
+ 2\, r^2 \left[\hat{k}^*_i -\left(1-\hat{\alpha}_0^2\, f_0\right)
\left( \hat{v}^*_i + \hat{k}^*_i \right)\right] dx^i\, dt \\
ds_2^2 &=
2\, \hat{\alpha}_0 \left[- \frac{1}{2}\hat{h}^*_{tt}
+ \frac{1}{2}\, \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_*
+\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right]dt\ dr
+ r^2\left[\hat{h}^*_{tt}\, dt^2 + \hat{h}^*_{ij} \, dx^i \, dx^j \right]\\
&\quad
+r^2 \left(1-\hat{\alpha}_0^2\, f_0\right) \left[\left(- \hat{h}^*_{tt}
+ \hat{g}^{(0)}_{jk} \, \hat{v}^j_* \, \hat{v}^k_*
+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}
\right)dt^2 \right.\\
&\qquad \left. \qquad
+ \left( \hat{v}^*_i + \hat{k}^*_i \right) \left( \hat{v}^*_j
+ \hat{k}^*_j \right)dx^i dx^j \right]
+2\,r^2\,b_0\, \hat{F}_0\,\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dx^{j}\\
&\qquad
- \frac{\hat{\alpha}_0^3}{r_D^2} \;\frac{R^{(0)}}{(d-1)(d-2)}\; dt \,dr
-2\, b_0^2\, r^2\, \hat{H}_1(b_0r) \left[\hat{\text{S}}^{(0)}_{ij}-\frac{\hat{R}^{(0)}}{2(d-1)(d-2)}\hat{g}^{(0)}_{ij}\right] dx^i dx^j \\
& \qquad
- \, (b_0r)^2\, \hat{M}_1(b_0r) \, \frac{R^{(0)}}{(d-1)(d-2)}\, dt^2
\end{split}$$ $$\label{hypbmwf1aR3}
\begin{split}
ds_3^2 &=
-2\,\hat{\alpha}_0\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}+
\left( \frac{1}{2}\hat{h}^*_{tt} + \,\hat{k}^*_{j}\, \hat{v}^{j}_{*}
+\frac{1}{2} \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*}
+\hat{p}_* \frac{\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)\, \left( \hat{v}^*_i + \hat{k}^*_i \right) \right] dx^i dr \\
&\quad
+\frac{2\hat{\alpha}_0^2}{r_D\left(1+\frac{d}{2}(\hat{\alpha}_0^2-1)\right)}\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*}
-\hat{f}_i^* \right] dx^{i}dr\\
&\quad
+ 2r\,\frac{\hat{\alpha}_0(2\,\hat{\xi}_0-1)}{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)}
\left[ \partial_{t} \hat{v}_{i}^{*}+\hat{v}_{*}^{j} \hat{\nabla}^{(0)}_{j} \hat{v}_{i}^{*}
-\hat{f}_i^* \right] dx^{i}dt\\
&\quad
-2\,r^2 \left(1-\hat{\alpha}_0^2 \,f_0\right)\left[\hat{h}^*_{ij}\hat{v}^{j}_{*}
+ \left(\hat{k}^*_{j}\, \hat{v}^{j}_{*} + \hat{g}^{(0)}_{jk} \, \hat{v}^{j}_{*}\, \hat{v}^{k}_{*}
+\hat{p}_* \frac{d\hat{\alpha}_0^2 }{1+\frac{d}{2}\, (\hat{\alpha}_0^2 -1)} \right)\,
\left( \hat{v}^*_i + \hat{k}^*_i \right) \right] dx^i dt \\
&\qquad
-4\, r^2\, b_0\, \hat{F}_0\, \hat{v}_{*}^j\hat{\nabla}^{(0)}_{(i} \hat{v}^{*}_{j)} \,dx^{i} dt
-2\, b_0^2\, r^2\, \hat{L}_1 \nabla^2_{(0)} v_i^* \, dt \, dx^i +2\, b_0^3\, r^2\, \hat{N}_1
\, \frac{\nabla_i^{(0)} R^{(0)}}{(d-1) (d-2)} \, dx^i\, dt
\\
&\qquad
-2\left[ b_0^2\, \kappa_L\,\hat{\alpha}_0\, \hat{\nabla}^2_{(0)}v^*_i - b_0^3\, \kappa_N \, \hat{\alpha}_0^2\, \frac{\nabla^{(0)}_i\, R^{(0)}}{(d-1)(d-2)} \right] dx^i \, dr\\
&\qquad
-2\, \frac{\hat{\alpha}_0^3}{r_D^2} \, \left[\hat{\text{S}}^{(0)}_{ij}\hat{v}^j_* - \frac{1}{d-2}\, \left(1 + \frac{2}{d\, \hat{\alpha}_0\, (\hat{\alpha}_0 +1) } \right) \hat{R}^{(0)}_{ij}\hat{v}^j_*
- \frac{\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}}{d\,(d-2)\, \hat{\alpha}_0\, (\hat{\alpha}_0 +1) }\right]
dx^i\, dr
\\
&\qquad
+4 \, b_0^2\, r^2\,\hat{H}_1(b_0r) \left[\hat{\text{S}}^{(0)}_{ij}-\frac{\hat{R}^{(0)}}{2(d-1)(d-2)}\hat{g}^{(0)}_{ij}\right] \hat{v}^j_* dx^i dt \\
&\qquad
+ 2\, (b_0r)^2\, \hat{M}_1(b_0r) \, \left( (v_i^* + k_i^*)\, \frac{\hat{R}^{(0)}}{2(d-1)(d-2)} -
\hat{\text{S}}^{(0)}_{ij}\hat{v}^j_* - \frac{1}{2(d-2)}\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\right)dx^i\, dt \\
&\qquad
+ 2\, (b_0r)^2\, \hat{M}_2(b_0r) \,\left[\hat{R}^{(0)}_{ij}\hat{v}^j_*+ \frac{1}{2}\hat{\nabla}^j_{(0)}\hat{q}^*_{ij}\right] dx^i \, dt \\
&\qquad
\end{split}$$
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[^1]: d.k.brattan@durham.ac.uk
[^2]: joan.camps@durham.ac.uk
[^3]: nayagam@physics.harvard.edu
[^4]: mukund.rangamani@durham.ac.uk
[^5]: See [@Rangamani:2009xk; @Hubeny:2010wp] for reviews which describe developments in this area and [@Bhattacharyya:2008ji; @Bhattacharyya:2008mz; @Bhattacharyya:2008kq] for generalizations that will be of great use in the following.
[^6]: It is natural to expect that such a construction might provide a holographic derivation of the hydrodynamic deconstruction described in [@Nickel:2010pr].
[^7]: In the asymptotically flat spacetime one also has to specify initial data for radial evolution on the past null infinity ${\mathscr I}^-$. The boundary conditions chosen in [@Bredberg:2011jq] are such that no disturbance propagates into the bulk spacetime from ${\mathscr I}^-$.
[^8]: To avoid confusion we wish to emphasize that we will refer to the bulk Dirichlet problem as one defined in the preceding paragraph. While this reduces to the standard Dirichlet problem on the boundary of AdS when we take the surface $\Sigma_D$ to infinity, we will soon see that the bulk Dirichlet problem induces different boundary conditions at infinity when the surface $\Sigma_D$ is retained at a finite position and it therefore pays to maintain the distinction.
[^9]: The implicit idea behind the uniqueness of the seed geometry here is based on the notion that equilibrium dynamics in the field theory at finite temperature is governed by a black hole, i.e., we are always in the ‘deconfined phase’ in the field theory on the boundary.
[^10]: Note that in order to get something interesting, it is important that we have a solution space allowing for non-trivial metrics at infinity (which we do thanks to [@Bhattacharyya:2008mz]). We are not simply slicing a single spacetime but working with a family of geometries characterized by the boundary data ${\mathfrak X}$ which is being adjusted so as to agree with the Dirichlet boundary conditions.
[^11]: There is an issue of counter-terms that one can use when working at finite radial coordinate in an asymptotically AdS spacetime. We will take the conservative view that the relevant counter-terms are the same as those necessary asymptotically.
[^12]: We use the word dynamical to characterize the boundary metric in the following sense: the metric on the boundary depends on the dynamical degrees of freedom of the system, viz., $u_\mu$ and $T$, as in . This is a pre-specified constitutive relation for the boundary metric in terms of the fluid variables, not unlike the constitutive relation for the energy momentum tensor. We will call such a constitutive relation coming from bulk Dirichlet problem as the Dirichlet constitutive relation. The fluid at the boundary, therefore, sees a dynamic metric background but no new degrees of freedom are introduced and hence for example, there are no boundary Einstein’s equations that need to be solved.
[^13]: We shall refer to scaling limit as the BMW limit after the authors of [@Bhattacharyya:2008kq], despite the fact that we are focussing on sub-sonic excitations and that it has been well documented in classic textbooks, cf., [@Landau:1965pi].
[^14]: We will for the moment refrain from imposing any IR boundary condition so as to be able to see the general structure. After all in pure AdS imposing regularity at the Poincaré horizon would kill the vev which has to vanish in the vacuum.
[^15]: Note that the vev is $(16\pi\, G_{d+1})^{-1} \, \phi$ where $G_{d+1}$ is the gravitational constant in AdS$_{d+1}$.
[^16]: We are working here with space-like momenta having $k^2>0$. Results for time-like momenta follow from replacing $k$ by $i\,k$ and using the Bessel relations $$\begin{split}
\frac{2(ix)^{\nu}}{\Gamma(\nu)}\,K_{\nu}(2ix) &= -\frac{\pi}{2}\,\frac{2x^{\nu}}{\Gamma(\nu)}Y_{\nu}(2x)
+ i \sin\left[(-\nu)\pi\right]\Gamma(-\nu+1) x^{\nu}J_{\nu}(2x)\\
\frac{\Gamma(\nu)}{2(ix)^{\nu}}\, I_{\nu}(2ix) &= \frac{\Gamma(\nu)}{2x^{\nu}}J_{\nu}(2x)\\
\end{split}$$
[^17]: If we keep $k^4$ terms and higher, one needs to subtract appropriate counter-terms at that order. These counter-terms are determined by requiring that $\hat{\phi}_k$ is finite as $r_D\to\infty$. The explicit expressions for counter-terms to any required order can be determined using the expansions in (see for e.g., [@Papadimitriou:2010as]).
[^18]: In fact, even for pure it is interesting to ask what the deformation on the boundary is when we move $\Sigma_D$ close to the Poincaré horizon; in this limit it seems natural to expect that the asymptotically we obtain a a Neumann boundary condition. We thank Don Marolf for emphasizing this to us.
[^19]: We will use upper-case Latin indices for the bulk spacetime indices, reserving lower-case Greek indices for hypersurface or boundary indices. See for a list of conventions.
[^20]: It might be useful to view this scalar function as physically being specified either as the level set of the red-shift factor, or by introducing a dynamical scalar field.
[^21]: We thank Veronika Hubeny for extensive discussions on these issues.
[^22]: \[alert\] At this point it is worthwhile to get a technical point out of the way. We will have two metric structures in the story henceforth, the hypersurface metric $\hat{g}_{\mu\nu}$ and a boundary metric $g_{\mu\nu}$. To avoid confusion, we will write all equations intrinsic to the hypersurface or to the boundary consistent with the respective metric structures. In practice this simply means that one raises/lowers indices of the equations with respect to the appropriate metric. These have to be handled with care, but by judicious use of the two metrics one can relate other components if required.
[^23]: In the fluid/gravity literature one chooses to maintain ${\mathfrak u}_\mu = u_\mu$ to all orders in the gradient expansion for simplicity. We will generalize this suitably in the rest of the discussion to simplify our formulae.
[^24]: We would like to thank Sayantani Bhattacharyya for collaboration on some of the ideas in this section.
[^25]: The expressions for the hypersurface energy density and pressure $\hat{\varepsilon}$ and $\hat{p}$ can be re-written in terms of the hypersurface temperature $\hat{T}$ (see ) which is the more natural quantity on $\Sigma_D$. However, it is convenient for practical reasons to leave these definitions in terms of $b$.
[^26]: We alert the reader again to footnote \[alert\] - the expressions in should be dealt with care as the l.h.s. and r.h.s contain contributions from quantities defined with respect to different metric structures.
[^27]: For an early discussion see [@Bludman:1968zz] where similar issues for fluids models driven to high pressure regimes are discussed.
[^28]: While the initial value problem for ideal, or even viscous fluids is ill-posed as the conservation equations are parabolic, we here want to drive home the point that the pathology we want to encounter happens for the sounds modes that are usually non-problematic.
[^29]: Note that most of these expressions are readily obtained by just ‘hatting’ the formulae in .
[^30]: This point was initially missed by both our analysis and that of [@Bhattacharyya:2008kq]. It was originally thought that it would be sufficient to obtain the non-relativistic metric from just the first order relativistic metric. This unfortunately is not true and as a result the non-relativistic metrics quoted in v1 of this paper and in [@Bhattacharyya:2008kq] only solve Einstein’s equations to first order in the non-relativistic gradient expansion. The correct form of the general expressions are now collected in .
[^31]: Note that the highlighted terms involve Weyl invariant Ricci ${\hat{\mathcal R}}_{\mu\nu}$ and Schouten ${\hat{\mathcal S}}_{\mu\nu}$ curvature tensors which are defined in .
[^32]: Note that we have retained certain terms at ${\cal O}(\hat{\aleph}^{-3})$ which are actually not necessary to solve the equation of motion at this order (eg., the velocity cubed term). This is to facilitate ease of comparison of our results when we undertake the near-horizon analysis in , with those in the existing literature.
[^33]: This scaling can be compounded with the scaling symmetry of Navier-Stokes equations [@Bhattacharyya:2008kq] to change the exponents of $\hat{\varkappa}$ but we refrain from doing so for simplicity (see end of ).
[^34]: Roughly speaking $\hat{\varkappa} \sim \hat{\aleph}$ of ; this is the overall parameter that will organize for us the hierarchy necessary in the near horizon limit.
[^35]: We thank Sayantani Bhattacharyya for emphasizing this point to us.
[^36]: Once we include the new corrections highlighted for e.g., in , we find that the boundary Dirichlet constitutive relations and the boundary gradient expansion are in tension. For instance we find terms which originate at ${\cal O}(\hat{\aleph}^{-3})$ migrate down to ${\cal O}(\hat{\aleph}^{-1})$. However, to the order we have looked at all such terms are related to enforcing Landau frame choice both on the hypersurface and the boundary, and hence to a gauge choice for the bulk metric. Our statements in this section assume that such pure gauge terms are irrelevant, but this issue deserves further investigation.
[^37]: We believe the answer to this question is most likely to be in the affirmative. If we consider deforming the boundary conditions at infinity by making the boundary metric an arbitrary local function of multi-traces of the stress tensor, we can think of the problem effectively as a mixed boundary condition a la [@Compere:2008us], which has been argued to suffer from ghosts generically (see also [@Andrade:2011dg]).
[^38]: It has been pointed out in [@Bredberg:2011xw] that there is an obstruction to finding solutions with Dirichlet boundary conditions in the near horizon Rindler like region, when the fluids live on compact curved spatial manifolds, such a a sphere (as would be relevant for the near horizon of asymptotically flat Schwarzschild black hole). In our construction the near-horizon limit described in requires that we simultaneously scale the curvatures to zero, and so we are unable to see the origins of such obstructions.
[^39]: We thank Roberto Emparan for useful discussions on this issue.
[^40]: In the case of $k=1,2$, the supersymmetry should get enhanced to d=3, $\mathcal{N}$=8.
[^41]: The corresponding problem for general function $r=r_D(x)$ can be reduced to this problem by a suitable choice of Weyl frame.
[^42]: We have used $$\partial_\mu \left\{\frac{1}{\hat{\alpha}}\right\}=\frac{1}{\hat{\alpha}}\frac{d}{2}(\hat{\alpha}^2-1)\mathcal{A}_\mu$$
|
---
abstract: 'More accurate machine learning models often demand more computation and memory at test time, making them difficult to deploy on CPU- or memory-constrained devices. *Teacher-student compression (TSC)*, also known as *distillation*, alleviates this burden by training a less expensive student model to mimic the expensive teacher model while maintaining most of the original accuracy. However, when fresh data is unavailable for the compression task, the teacher’s training data is typically reused, leading to suboptimal compression. In this work, we propose to augment the compression dataset with synthetic data from a generative adversarial network (GAN) designed to approximate the training data distribution. Our *GAN-assisted TSC* ([GAN-TSC]{}) significantly improves student accuracy for expensive models such as large random forests and deep neural networks on both tabular and image datasets. Building on these results, we propose a comprehensive metric—the *TSC Score*—to evaluate the quality of synthetic datasets based on their induced TSC performance. The TSC Score captures both data diversity and class affinity, and we illustrate its benefits over the popular Inception Score in the context of image classification.'
author:
- |
Ruishan Liu\
Dept. of Electrical Engineering\
Stanford University\
Nicolo Fusi\
Microsoft Research\
Cambridge, MA, USA Lester Mackey\
Microsoft Research\
Cambridge, MA, USA
bibliography:
- 'refs.bib'
title: |
Teacher-Student Compression\
with Generative Adversarial Networks
---
Introduction {#sec:introduction}
============
Modern machine learning models have achieved remarkable levels of accuracy, but their complexity can make them slow to query, expensive to store, and difficult to deploy for real-world use. Ideally, we would like to replace such cumbersome models with simpler models that perform equally well. One way to address this problem is to perform *teacher-student compression* (*TSC*, also known as *distillation*), which consists of training a student model to mimic the outputs of a teacher model [@bucila2006model; @li2014learning; @hinton2015distilling]. For example, expensive ensemble and deep neural network (DNN) teachers have been used to train inexpensive decision tree [@craven1996extracting; @frosst2017distilling] and shallow neural network [@bucila2006model; @li2014learning; @ba2014deep; @hinton2015distilling; @urban2016deep] students. While alternative model-specific compression strategies abound (see \[sec:conclusion\]), TSC is distinguished by its broad applicability: the same framework can be used to compress any classifier, be it a random forest or a deep neural network. An important degree of freedom in the TSC problem is the *compression set* used to train the student. Ideally, fresh (unlabeled) data from the training distribution would fuel this task, but often no fresh data remains after the teacher is trained [@bucila2006model; @ba2014deep]. In this case, one branch of the literature, dating back to the pioneering work of @bucila2006model, recommends generating synthetic data for compression and proposes tailored generation schemes for tabular [@bucila2006model] and image [@urban2016deep] data. A second branch, rooted in the distillation community [@hinton2015distilling; @frosst2017distilling], simply uses the same data to train teacher and student [see also @ba2014deep]. Here, we show that the latter convention leads to suboptimal compression performance and propose a synthetic data generation strategy for both tabular and image data that improves upon standard augmentation schemes. Specifically, when fresh data is unavailable for TSC, we propose to augment the compression set with synthetic data produced by generative adversarial networks (GANs) [@goodfellow2014generative]. GANs attempt to generate new datapoints from the distribution underlying a given dataset and have achieved impressive fidelity for a variety of data types including images [@goodfellow2014generative], text [@yu2017seqgan], and electronic health records [@choi2017generating]. Here, we identify TSC as a practical downstream task for which GAN generation is consistently useful across data types and classification tasks and develop *GAN-assisted TSC* ([GAN-TSC]{}) to improve the TSC of an arbitrary classifier. Our extensive empirical evaluation demonstrates the effectiveness of [GAN-TSC]{}for tabular data (for which GANs are seldom used), image data, random forest classifiers, and DNN classifiers.
Note that there is an important distinction between training a student to mimic a teacher with synthetic data and training a student to solve the original supervised learning problem with synthetic data. The goal of the original supervised learning task is to approximate the ideal mapping $f^*$ between inputs $x$ and outputs $y$. This ideal $f^*$ is a functional of the true but unknown distribution underlying our data, and our information concerning $f^*$ is limited by the real data we have collected. The goal in TSC is to approximate the teacher prediction function $g$ which maps from inputs to predictions $z$. Because the teacher is a function of the training data alone, $g$ itself is a functional of the training data alone and is otherwise independent of the unknown distribution that generated that data. In addition, because we have access to the teacher, we have the freedom to query the function $g$ at any point, and hence our information concerning $g$ is limited only by the number of queries we can afford. In particular, when we generate a new query point $x$, we can observe the actual target value of interest, the teacher’s prediction $g(x)$; this is not true for the supervised learning task, where no new labels can be observed. The insensitivity to errors in synthetic labels and access to fine-grained teacher predictions make TSC more ideally suited to synthetic data augmentation. Indeed, we will see in \[sec:RF,sec:NN\] that the same GAN data that leads to improved TSC leads to degraded accuracy when used to augment the original supervised learning training set. This is consistent with past work that demonstrates gains from GAN-augmented supervised learning in specific data-starved situations but reports degraded accuracy when all training data is used [@bowles2018gan Tab. 4]. See [@ba2014deep] for further discussion on the distinctions between TSC and the original supervised learning task.
Since the improvement realized by [GAN-TSC]{}depends on the synthetic data quality, we further propose to use [GAN-TSC]{}to evaluate the quality of synthetic datasets and their generators. In essence, we declare a synthetic dataset to be of higher quality if a compressed model trained on that data achieves higher test accuracy. Synthetic data evaluation is a notoriously difficult problem marked by the lack of universally agreed-upon quality measures [@theis2015note]. Some standard quality measures, like *multiscale structural similarity* [@wang2003], quantify the diversity of a synthetic dataset but do not capture *class affinity*, the ability of datapoints to be correctly associated with their labels with high confidence. Others, like the popular *Inception Score* [@salimans2016improved], quantify class affinity based on the predicted label distribution of a trained neural network. However, these scores do not account for within-class diversity and are easily misled by adversarial datapoints that elicit high confidence predictions but do not resemble real data. To address these shortcomings, we develop a *TSC Score* that quantifies the true test accuracy of compressed models trained using synthetic data; this offers a robust, goal-driven metric for synthetic data quality that accounts for both diversity and class affinity. In summary, we make the following principal contributions in this paper:
1. We identify TSC as a practical downstream task for which GAN data augmentation is consistently useful across data types and classification tasks and develop [GAN-TSC]{}as a drop-in replacement for standard TSC.
2. For random forest teachers, we demonstrate 25 to 336-fold reductions in execution and storage costs with less than $1.2\%$ loss in test performance across a suite of real-world tabular datasets. In each case, [GAN-TSC]{}improves over the tabular data augmentation strategy of @bucila2006model.
3. For image classification, we show [GAN-TSC]{}consistently improves student test accuracy for a variety of deep neural network teacher-student pairings and two popular compression objectives.
4. We introduce a new TSC Score for evaluating the quality of GAN-generated datasets and illustrate its advantages over the popular Inception Score.
Teacher-Student Compression with GANs {#sec:model_compression}
=====================================
We begin by reviewing standard approaches to DNN TSC and describing our proposals for random forest TSC and improving TSC with GAN data.
**Deep Neural Network TSC ** In the standard teacher-student approach to compressing a neural network classifier, a relatively inexpensive prediction rule, like a shallow neural network, is trained to predict the unnormalized log probability values—the *logits* $z$—assigned to each class by a previously trained deep network classifier. The inexpensive model is termed the *student*, and the expensive deep network is termed the *teacher*. Given a compression set of $n$ feature vectors paired with teacher logit vectors, $\{(x^{(1)}, z^{(1)}),..., (x^{(n)}, z^{(n)}) \}$, @ba2014deep proposed framing the TSC task as a multitask regression problem with $L^2$ loss, $L (\theta) = || g(x; \theta) - z ||^2_2.
$ Here, $\theta$ represents any student model parameters to be learned (e.g., the student network weights), and $g(x; \theta)$ is the vector of logits predicted by the student model for the input feature vector $x$.
@li2014learning introduced an alternative TSC objective function, and @hinton2015distilling parameterized this objective by a temperature parameter $T > 0$. Specifically, the student is trained to mimic the annealed teacher class probabilities, $ \textstyle
q_{j} (z/T) = {\mathrm{exp} (z_{j} / T)}{/\sum_k \mathrm{exp} (z_k/T)},
$ for each class $j$ by solving a multitask regression problem with cross-entropy loss, $L_T(\theta) = -\textsum_{j}\ q_j(z/T) \log (q_j(g(x; \theta)/T)).
$ @hinton2015distilling showed that, under a zero-mean logit assumption, cross-entropy regression recovers $L^2$ logit matching as $T\to\infty$; however, the two approaches can differ for small $T$. In Sec. \[sec:NN\], we will experiment with both of these popular TSC approaches. **Random Forest TSC ** Random forests [@breiman2001random] construct highly accurate prediction rules by averaging the predictions of a diverse and often large collection of learned decision trees. Effectively mimicking a large random forest with a single decision tree or a small forest has the potential to reduce prediction computation and storage costs by multiple orders of magnitude [@bucila2006model; @joly2012l1; @pmlr-v70-begon17a; @DBLP:conf/icdm/PainskyR16; @painsky2018lossless]. Focusing on the common setting of binary classification, we propose to train a student regression random forest to predict a teacher forest’s outputted probability $p$ of a datapoint $x$ having the label $1$.
**GAN-assisted TSC ([GAN-TSC]{}) ** In a typical TSC setting, as much data as possible has been dedicated to training the highly accurate teacher model, leaving little fresh data for training the student model. While one branch of the TSC literature recommends generating synthetic data with customized augmentation algorithms for tabular [@bucila2006model] and image [@urban2016deep] data, the more common solution in the distillation literature is to simply reuse the teacher training set as the compression set [@hinton2015distilling; @frosst2017distilling]. However, we will see in Secs. \[sec:RF\] and \[sec:NN\] that compressing with training data alone leads to suboptimal student performance. This suboptimality occurs both due to teacher overfitting (there is a mismatch between a teacher’s test predictions and its overconfident training predictions) and student overfitting (the student can benefit from observing the teacher’s outputs at points other than the original training points). To boost student performance and compression efficiency, we propose a simple solution applicable to tabular and image data alike: augment the compression set with synthetic feature vectors generated by a high-quality GAN. These synthetic feature vectors are then labeled with the teacher’s outputted class probabilities or logits. We call this approach *GAN-assisted TSC* and release our Python implementation at
**AC-GAN ** To generate high-quality GAN feature vectors which capture the salient features of each class, we use the auxiliary classifier GAN (AC-GAN) of @odena2016conditional. The AC-GAN generator $G$ produces a synthetic feature vector $X_{fake} = G(W, C)$ from a random noise vector $W$ and an independent target class label $C$ drawn from the real data class distribution. For any given feature vector $x$, the AC-GAN discriminator $D$ predicts both the probability of each class label $P(C \mid x)$ and the probability of the data source being real or fake, $P(S \mid x)$ for $S \in \{ real, fake \}$. For a given training set ${\mathcal{D}}_{real}$ of labeled feature vectors, two components contribute to the AC-GAN training objective, $$\begin{aligned}
L_{source} = &\textstyle \frac{1}{|{\mathcal{D}}_{real}|}\textsum_{(x,c)\in{\mathcal{D}}_{real}} \log P(S = real \mid x)
\textstyle + \mathbb{E}_{W,C\sim p_c} [\log P(S = fake \mid G(W, C))] \text{ and } \\
\label{Eq:Lclass}
L_{class} = &\textstyle \frac{1}{|{\mathcal{D}}_{real}|}\textsum_{(x,c)\in{\mathcal{D}}_{real}} \log P(C = c \mid x)
\textstyle + \mathbb{E}_{W,C\sim p_c} [\log P(C \mid G(W, C))],\end{aligned}$$ representing the expected conditional log-likelihood of the correct source and the correct class of a feature vector, respectively. Training proceeds as an adversarial game with the generator $G$ trained to maximize $L_{class} - L_{source}$ and the discriminator $D$ trained to maximize $L_{class} + L_{source}$.
A Teacher-Student Compression Score for Evaluating GANs {#sec:compression_score}
=======================================================
The evaluation of synthetic datasets is an important but challenging task. Two criteria commonly considered essential for a high-quality synthetic dataset are datapoint diversity and [class affinity]{}. The most widely used GAN quality measure, the Inception Score (IS)of @salimans2016improved, measures across-class diversity but does not account for within class diversity. In addition, the IS measures a form of class affinity based on the predictions of a pre-trained neural network but is easily misled by datapoints that elicit high confidence predictions without resembling real data. For example, if the classification loss $L_{class}$ is heavily upweighted relative to the source loss $L_{source}$ while training an AC-GAN, the generator will be more likely to produce feature vectors classified with high confidence by neural networks. As we will see in Sec. \[sec:eval\], such feature vectors need not resemble real data but will nevertheless receive high ISs (which should be reserved for high-quality datasets). To account for both class affinity and diversity in a more robust and holistic manner, we propose to use the performance of a student trained on GAN data as a measure of GAN dataset quality. Python code to compute the TSC Score is available at
**The TSC Score ** To evaluate the quality of a generated dataset ${\mathcal{D}}$ relative to a real dataset ${\mathcal{D}}_{real}$, we define a *Teacher-Student Compression Score (TSCS)* based on the test accuracy ${\operatorname{acc}}({\mathcal{D}})$ of a student trained with compression set ${\mathcal{D}}$ to mimic a pre-trained teacher: $$\label{Eq:CS}
\textbf{TSCScore}({\mathcal{D}}; {\mathcal{D}}_{real}) = \textstyle\frac{{\operatorname{acc}}({\mathcal{D}}) - {\operatorname{acc}}_{\mathrm{mode}}}{{\operatorname{acc}}({\mathcal{D}}_{real}) - {\operatorname{acc}}_{\mathrm{mode}}},$$ where ${\operatorname{acc}}_{\mathrm{mode}}$ is the accuracy obtained by always predicting the most common class in the test set. In our experiments, we choose ${\mathcal{D}}_{real}$ to be the teacher’s training data, but any choice is equally valid, as the ranking induced by the TSCS is not affected by the choice of ${\mathcal{D}}_{real}$.
The TSCS declares a synthetic dataset to be of higher quality if a compressed model trained only on that data achieves higher accuracy on real test data. The score takes values in $[0,\infty)$ and tends to $0$ as the synthetic data distribution diverges from the real data distribution. Increased within-class diversity, increased across-class diversity, and increased class affinity all tend to increase the TSCS, as they enable the student to more accurately mimic the teacher’s output across all classes. This makes the TSCS a more holistic measure of synthetic data quality than the IS or multiscale structural similarity. However, crucially, the TSCS is only impacted by aspects of class affinity and diversity that matter for performance on real test data. Hence, unlike the IS which is completely determined by the idiosyncratic output of an imperfect network, the TSCS is robust to the idiosyncratic preferences of an imperfect teacher or student. In particular, we would not expect a student trained on unrealistic or adversarial synthetic data to perform well on real test data even if it very accurately mimics the teacher’s predictions on such data. A potential inconvenience of the TSCS is the need to train an inexpensive student model. To ensure that the TSCS can be computed efficiently, we train each student for only one epoch; our experiments suggest that this is sufficient to effectively capture GAN data quality and can be less expensive then evaluating the IS.
#### Evaluating GANs: An Illustration with CIFAR-10 {#sec:eval}
To illustrate the potential benefit of the TSCS over the commonly-used IS, we reinstate the CIFAR-10 experimental setup of Fig. \[fig:$L^2$\]. We evaluate the TSCS on 50K CIFAR-10 images (the teacher’s training data), 50K well-trained GAN images (i.e., data from the AC-GAN described in Sec. \[sec:NN\]), and 50K inferior images which have high confidence classifications under the teacher network but do not resemble real data. The inferior data is generated by training the well-trained AC-GAN for 10 additional epochs using only the classification objective $L_{class}$ (\[Eq:Lclass\]). That is, both the generator $G$ and discriminator $D$ are trained to maximize $L_{class}$, while ignoring the traditional GAN objective component $L_{source}$. We report means and standard errors across 3 independent runs. In Table \[Table:CS\], the GAN data quality degrades noticeably after the additional training with only $L_{class}$, and the TSCS decreases in accordance with our expectations. However, the IS increases for the inferior GAN images despite the evident unrealistic artifacts.
[Real Data]{} Well-trained GAN Inferior GAN
----------------------------------------------------------------- ---------------------------------------------------------------- -----------------------------------------------------------------
{width="\scoreimagesize\textwidth"} {width="\scoreimagesize\textwidth"} {width="\scoreimagesize\textwidth"}
Inception: $11.2 \pm 0.1$ Inception: $5.80 \pm 0.06$ Inception: $5.93 \pm 0.06$
Compression: $0.994 \pm 0.003$ Compression: $0.778 \pm 0.002$ Compression: $0.702 \pm 0.002$
To highlight the practicality of the TSCS, we also report a timing comparison of the IS and TSCS evaluations. To compute the IS, we perform one Inception network forward pass on 50K GAN images. To compute the TSCS, we first perform one forward pass on the same 50K images to get the NIN teacher’s logits. We then train the LeNet student for one epoch with one forward and one backward pass. We finally perform one forward pass on 10K real test images to compute student test accuracy. Using the IS code of [@salimans2016improved] and an NVIDIA Tesla V100 GPU, the IS required 1436.6s and the TSCS 350.1s.
Related and Future Work {#sec:conclusion}
=======================
To reduce the deployment costs of expensive machine learning classifiers, we introduced GAN-assisted TSC as a straightforward way to improve teacher-student compression. We demonstrated the benefits of [GAN-TSC]{}for both tabular and image data classifiers and developed a new TSC Score for evaluating the quality of synthetic datasets. While we have focused on improving the popular teacher-student paradigm of compression, we would be remiss to not mention alternative, model-specific approaches to reducing deployment costs, including parameter sharing [@chen2015compressing], network pruning [@han2015learning], and network parameter prediction [@denil2013predicting] for DNNs and indicator function selection [@joly2012l1], pre-pruning [@pmlr-v70-begon17a], and probabilistic modeling and clustering [@DBLP:conf/icdm/PainskyR16; @painsky2018lossless] for random forests. A number of exciting opportunities for future work remain. For example, [GAN-TSC]{}is readily integrated into more complex TSC approaches that currently reuse the teacher’s training data for compression. Prime examples are the recent approaches of [@WangXuXuTa2018; @xu2018training; @NIPS2018_7358]. These differ from standard TSC by employing non-standard GAN-type compression losses, in which the student acts as the discriminator [@WangXuXuTa2018] or generator [@xu2018training; @NIPS2018_7358]; @NIPS2018_7358 also train the teacher and student together. In addition, GAN development for tabular data has received much less attention than GAN development for image data, and we anticipate that significant improvements over the AC-GANs used in our experiments will result in significant performance benefits for [GAN-TSC]{}.
Supplement
==========
Random Forest [GAN-TSC]{}vs. GAN-assisted supervised learning {#sec:rf-gansl}
-------------------------------------------------------------
Consistent with our discussion in Sec. \[sec:introduction\] and our findings in Fig. \[fig:SL\], Fig. \[fig:magic-SL\] shows that the same GAN data that substantially improves student compression performance in Fig. \[fig:magic\] harms or scarcely improves test AUC when the random forest student is trained without compression for the original MAGIC supervised learning task.
![ Without compression, a random forest student is trained for the original supervised learning task on MAGIC using only training data (‘Student Only’), only GAN data, or a mixture of training and GAN data. []{data-label="fig:magic-SL"}](magic-SL.png){width="1\linewidth"}
Random Forest [GAN-TSC]{}vs. MUNGE {#sec:rf-munge-supp}
----------------------------------
On the task of compressing a 500-tree random forest into a single tree (see Sec. \[sec:RF\]), [GAN-TSC]{}outperforms TSC with MUNGE data on every dataset ((Higgs 100k: MUNGE 64.9%, [GAN-TSC]{}69.6%), (Higgs 1k: MUNGE 58.3%, [GAN-TSC]{}59.0%) (MAGIC: MUNGE 0.909, [GAN-TSC]{}0.918) (Evergreen: MUNGE 0.878, [GAN-TSC]{}0.882) ).
|
---
title: 'Exoplanet Clouds Chapter v2.1.docx'
---
****
Mark S. Marley\
*NASA Ames Research Center*
Andrew S. Ackerman
*NASA Goddard Institute for Space Studies*
Jeffrey N. Cuzzi
*NASA Ames Research Center*
Daniel Kitzmann
*Zentrum für Astronomie und Astrophysik,\
Technische Universität Berlin*
Abstract {#abstract .unnumbered}
========
Clouds and hazes are commonplace in the atmospheres of solar system planets and are likely ubiquitous in the atmospheres of extrasolar planets as well. Clouds affect every aspect of a planetary atmosphere, from the transport of radiation, to atmospheric chemistry, to dynamics and they influence - if not control - aspects such as surface temperature and habitability. In this review we aim to provide an introduction to the role and properties of clouds in exoplanetary atmospheres. We consider the role clouds play in influencing the spectra of planets as well as their habitability and detectability. We briefly summarize how clouds are treated in terrestrial climate models and consider the far simpler approaches that have been taken so far to model exoplanet clouds, the evidence for which we also review. Since clouds play a major role in the atmospheres of certain classes of brown dwarfs we briefly discuss brown dwarf cloud modeling as well. We also review how the scattering and extinction efficiencies of cloud particles may be approximated in certain limiting cases of small and large particles in order to facilitate physical understanding. Since clouds play such important roles in planetary atmospheres, cloud modeling may well prove to be the limiting factor in our ability to interpret future observations of extrasolar planets.
Introduction
============
Clouds and hazes are found in every substantial solar system atmosphere and are likely ubiquitous in extrasolar planetary atmospheres as well. They provide sinks for volatile compounds and influence both the deposition of incident flux and the propagation of emitted thermal radiation. Consequently they affect the atmospheric thermal profile, the global climate, the spectra of scattered and emitted radiation, and the detectability by direct imaging of a planet. As other chapters in this book attest, clouds and hazes are a complex and deep subject. In this chapter we will broadly discuss the roles of clouds and hazes as they relate to the study of exoplanet atmospheres. In particular we will focus on the challenge of exoplanet cloud modelling and discuss the impact of condensates on planetary climates.
The terms “clouds” and “hazes” are sometimes used interchangeably. Here we use the term “cloud” to refer to condensates that grow from an atmospheric constituent when the partial pressure of the vapor exceeds its saturation vapor pressure. Such supersaturation is typically produced by atmospheric cooling, and cloud particles will generally evaporate or sublimate in unsaturated conditions. A general framework for such clouds in planetary atmospheres is provided by *Sánchez-Lavega et al.* (2004). By “haze” we refer to condensates of vapor produced by photochemistry or other non-equilibrium chemical processes. This usage is quite different from that of the terrestrial water cloud microphysics literature where the distinction depends on water droplet size and atmospheric conditions.
Because exoplanetary atmospheres can plausibly span such a wide range of compositions as well as temperature and pressure conditions, a large number of species may form clouds. Depending on conditions, clouds in a solar composition atmosphere can include exotic refractory species such as Al~2~O~3~, CaTiO~3~, Mg~2~SiO~4~ and Fe at high temperature and Na~2~S, MnS, and of course H~2~O at lower temperatures. Many other species condense as well including CO~2~ in cold, Mars-like atmospheres and NH~3~ in the atmospheres of cool giants, like Jupiter and Saturn. In Earth-like atmospheres water clouds are likely important although Venus-like conditions and sulfuric-acid or other clouds are possibilities as well. Depending on atmospheric temperature, pressure, and composition the range of possibilities is very large. Furthermore not all clouds condense directly from the gas to a solid or liquid phase as the same species. For example in the atmosphere of a gas giant exoplanet solid MnS cloud particles are expected to form around 1400 K from the net reaction H~2~S + Mn$\rightarrow $MnS(s) + H~2~ (*Visscher et al.* 2006).
Clouds strongly interact with incident and emitted radiative fluxes. The clouds of Earth and Venus increase the planetary Bond albedo (the fraction of all incident flux that is scattered back to space) and consequently decrease the equilibrium temperature. Clouds can also “trap” infrared radiation and heat the atmosphere. Hazes, in contrast, because of their usually smaller particle sizes can scatter incident light away from a planet but not strongly affect emergent thermal radiation, and thus predominantly result in a net cooling. The hazes of Titan play such an “anti-greenhouse effect” role in the energy balance of the atmosphere. For these reasons global atmospheric models of exoplanets, including those aiming to define the habitable zone given various assumptions, must consider the effects of clouds. However clouds are just one ingredient in such planetary atmosphere models. Bulk atmospheric composition, incident flux, gravity, chemistry, molecular and atomic opacities, and more must all be integrated along with the effect of clouds in order to construct realistic models. Introductory reviews by *Burrows & Orton* (2010) and *Seager & Deming* (2010) cover the important fundamentals atmospheric modeling and place cloud models in their broader context.
In the remainder of this chapter we discuss the importance of clouds to exoplanet atmospheres, particularly considering their impact on habitability, discuss cloud modeling in general and the types of models developed for exoplanet studies, and finally briefly review observations of exoplanet clouds. Because clouds have played such a large role in efforts to understand the atmospheres of brown dwarfs, we also briefly review the findings of this field. We conclude with an overview of how the Mie opacity of particles behaves in various limits and consider the case of fluffy particles.
Importance of Clouds and Hazes in Exoplanet Atmospheres
=======================================================
Albedo, Detectability, and Characterization
-------------------------------------------
Before discussing the role on clouds on the spectra of extrasolar planets it is worthwhile to review the various albedos which enter the discussion. The Bond or bolometric albedo is the fraction of all incident light, integrated over the entire stellar spectrum, which is scattered back to space by a planet. This albedo is a single number and enters into the computation of a planet’s equilibrium temperature (the temperature an airless planet would have if its thermal emission were equal to the incident radiation which it absorbs). It is also useful to know the monochromatic ratio of all scattered to incident light as a function of wavelength, which is the spherical albedo. For historical reasons it is more common to discuss the geometric albedo, which is specifically the wavelength-dependent ratio of the light scattered by the entire planet in the direction directly back towards its star compared to that which would be so scattered by a perfectly reflecting Lambert disk of the same radius as the planet. Care must be taken to distinguish all these albedos as they can differ markedly from one another even for the same planet and the literature is rife with confusion between them.
Perhaps the single most important effect of clouds is to brighten the reflected light spectra of exoplanets, particularly at optical wavelengths ( $0.38<\lambda <1$ mm). For planets with appreciable atmospheres, Rayleigh scattering is most efficient in the blue. At longer wavelengths however absorption by either the planetary solid surface or oceans (for terrestrial planets) or by atmospheric gaseous absorbers becomes important in the red. This is because for common molecules at planetary temperatures, such as H~2~O, CO~2~, or CH~4~, vibrational-rotational transitions become important at wavenumbers below about 15,000 cm^-1^ or wavelength $\lambda >0.6$ mm. Except for diatomic species (notably O~2~) planetary atmospheres are generally not warm enough to exhibit strong electronic absorption features in the optical. Thus the reflected light or geometric albedo spectrum of a generic cloudless planet with an atmosphere would be bright at blue wavelengths from Rayleigh scattering and dark in the red and at longer wavelengths from gaseous molecular or surface absorption.
Clouds, however, tend to be bright with a fairly gray opacity through the optical. Thus a thick, scattering cloud can brighten a planet in the far red by scattering more light back to space than a cloudless planet. As a result two similar planets, one with and one without cloud cover, will have very different geometric albedos in the red, and consequently differing brightness contrasts with their parent stars. The mean Earth water clouds increase the contrast between the reflection spectrum of an Earth-like planet and its host star by one order of magnitude (*Kitzmann et al.* 2011b). This effect, first noted for giant planets by *Marley et al.* (1999) and *Sudarsky et al.* (2000) and further explored in *Cahoy et al.* (2010) is illustrated in Figure 1 which plots geometric albedo as a function of $\lambda $ for giant planets with and without clouds. A warm, cloudless atmosphere is dark in reflected light while a cooler atmosphere, sporting water clouds, is much brighter. In this case clouds affect the model spectra far more than a factor of three difference in the atmospheric abundance of heavy elements, which is also shown.
In the case of searches for planets by direct coronagraphic imaging in reflected light, clouds may even control whether or not a planet is discovered. Depending on the spectral bandpass used for planet discovery at a given distance from its primary star a cloudy planet may be brighter and more detectable than a cloudless one. Discussions of the influence of clouds and albedo on detectability include those by *Tinetti et al.* (2006a) and *Kitzmann et al.* (2011a).
Once a planet is detected, spectra are needed to characterize atmospheric abundances of important molecules. For terrestrial extrasolar planets this is best done by the analysis of the thermal emission spectrum (*Selsis,* 2004; *Tinetti et al.*, 2012). Clouds, however, may conceal the thermal emission from the surface and dampen spectral features of molecules (e.g., the bio-indicators N~2~O or O~3~). Indeed, clouds on Earth have a larger impact on the emitted infrared flux than the differences between night and day (*Hearty et al.*, 2009; *Tinetti et al.*, 2006a). Thermal emission spectra are therefore very sensitive to the types and fractional coverages of clouds present in the atmosphere. At high spectral resolution the most important terrestrial molecular spectral features in the mid-infrared, such as O~3~, remain detectable even for cloud covered conditions in many cases. Figure 2 shows thermal emission spectra affected by low-level water droplet and high-level water ice clouds of an Earth-like planet orbiting different main-sequence dwarf stars (adopted from *Vasquez et al.*, 2013). With increasing cloud cover of either cloud type the important 9.6 mm absorption band of ozone is strongly dampened, along with an overall decrease in the thermal radiation flux. For some cases presented in Fig. 2 (e.g. for the F-type star and 100% high-level cloud cover) the ozone band seems to be completely absent or even appears in emission rather than absorption (F-type star, 100% low-level clouds). At lower spectral resolution, such as could be obtained for terrestrial exoplanets in the near future, clouds render the molecular features even less detectable; thus clouds will strongly affect the determination of their atmospheric composition. For example, as shown by *Kitzmann et al.* (2011a) a substantial amount of water clouds in an Earth-like atmosphere can completely hide the spectral signature of the bio-indicator ozone in low-resolution thermal emission spectra.
Clouds can also obscure spectral features originating from the surface of a planet. In principle, signals of surface vegetation (“vegetation red edge”) are present in the reflected light spectra of Earth-like planets. However, as often pointed out (e.g., *Arnold et al.* (2002) and *Hamdani et al.* (2006)) this spectral feature can easily be concealed by clouds. For the detectability of possible vegetation signatures of terrestrial extrasolar planets *Montañés-Rodríguez et al.* (2006) and *Tinetti et al.* (2006b) concluded that clouds play a crucial role for these signatures in the reflection spectra. Thus, apart from the scattering characteristics of different planetary surface types, the presence of clouds has been found to be one of the most important factors determining reflection spectra.
Transmission spectra of transiting planets can be used to obtain many atmospheric properties, such as atmospheric composition and temperature profiles. Transmission spectroscopy of extrasolar giant planets has already proven to be a successful method for the characterization of giant exoplanets. As shown by *Pallé et al.* (2009) the major atmospheric constituents of a terrestrial planet remain detectable in transmission spectra even at very low signal-to-noise ratios. Thus, transmission spectra can provide more information about the atmospheres of exoplanets than reflection spectra. Theoretical transmission spectra of Earth-sized transiting planets have been studied by *Ehrenreich et al.* (2006) including the effects of optically thick cloud layers. Their results show that the transmission spectra only contain information about the atmosphere above the cloud layer and that clouds can effectively increase the apparent radius of the planet. The impact of clouds on the transmission spectra therefore depends strongly on their atmospheric height. If cloud layers are only located in the lower atmosphere, which is already opaque due to absorption and scattering by gas species, their overall effect on the spectrum will be small (*Kaltenegger & Traub*, 2009).
Habitability of Extrasolar Terrestrial Planets
----------------------------------------------
From the terrestrial bodies with an atmosphere in the solar system we know that clouds are a common phenomenon and should also be expected to occur in atmospheres of terrestrial extrasolar planets. Apart from the usual well-known greenhouse gases, clouds have the most important climatic impact in the atmospheres of terrestrial planets by affecting the energy budget in several ways. Firstly, clouds can scatter incident stellar radiation back to space resulting in atmospheric cooling (albedo effect). On the other hand, clouds can trap thermal radiation within the lower atmosphere by either absorption and re-emission at their local temperature (classical greenhouse effect) or by scattering thermal radiation back towards the surface (scattering greenhouse effect), which heats the lower atmosphere and planetary surface. All these effects are determined by the wavelength-dependent optical properties of the cloud particles (absorption and scattering cross-sections, single scattering albedo, asymmetry parameter, scattering phase function). These properties can differ considerably for different cloud forming species (owing to their refractive indices) and atmospheric conditions (composition and temperature structure). Note that the single scattering albedo, yet a fourth type of albedo (see Section 2.1 for the others), measures the fraction of all incident light scattered by a single cloud particle.
Life as we know it requires the presence of liquid water to form and survive. In the context of terrestrial exoplanets we are primarily concerned with habitable conditions on the planetary surface. Lifeforms may of course also exist in other environments, such as deep under the planetary surface or within a subsurface ocean (*Lammer et al.*, 2009). Since there is no possibility of detecting the presence of such habitats by remote observations our current definition of a habitable terrestrial planet assumes a reservoir of liquid water somewhere on its surface. Liquid surface water implies that these planets would also have abundant water vapor in their atmospheres. We can therefore safely assume that water (and water ice) clouds will naturally occur in the atmospheres of these exoplanets throughout the classical habitable zone.
As such, H~2~O and CO~2~ clouds are the prime focus for habitable exoplanets. Other possible condensing species also found in our solar system include C~2~H~6~ and CH~4~ (e.g., in the atmosphere of Titan), N~2~ (Triton), or H~2~SO~4~ (Venus, Earth). Any chemical species in liquid form on a planetary surface can in principle be considered as a potential cloud-forming species in the atmosphere if its atmospheric partial pressure is high enough and the temperature low enough.
Given these considerations it is clear that clouds are important for the determination of the boundaries of the classical habitable zone around different kinds of stars. In the next two subsections we discuss some of the ways that clouds influence the habitable zone boundaries.
### Inner Edge of the Habitable Zone
The closer a terrestrial planet orbits its host star the greater stellar insolation and consequently surface temperature. This leads to enhanced evaporation of surface water and increases the amount of water vapor in the atmosphere. The strong greenhouse effect by this additional water vapor further increases the surface temperature and therefore the evaporation of surface water, resulting in a positive feedback cycle (see *Covey et al.*, this volume).
The inner boundary of the classical habitable zone is usually defined either as the distance from a host star at which an Earth-like planet completely loses its liquid surface water by evaporation (runaway greenhouse limit), or as the distance from the host star when water vapor can reach the upper atmosphere (water loss limit). The latter is the definition favored by *Kasting et al.* (1993); the former definition is used by *Hart* (1979). In the present Earth atmosphere water escape is limited by the cold trap at the tropopause which loses its efficiency in a sufficiently warm, moist atmosphere (*Kasting et al.*, 1993).
Using a one-dimensional atmospheric model, *Kasting et al.* (1993) estimated that the inner boundary of the habitable zone for the runaway greenhouse scenario for an Earth around the Sun would be located at 0.84 AU. This distance, however, was calculated with the clouds treated as a surface albedo effect and neglecting any feedbacks. For example once formed clouds increase the planetary albedo which in turn partially reduces the temperature increase due to the higher stellar insolation and the cloud greenhouse effect. *Kasting* (1988) investigated some feedback effects and concluded that with a single layer of thick water clouds Earth could be moved as close as 0.5 AU (for a cloud coverage of 100%) or 0.67 AU (50% cloud coverage) from the Sun before all liquid surface water would be lost. *Goldblatt & Zahnle* (2011) explored the various feedback issues in detail and concluded that more sophisticated modeling approaches are necessary to explore habitability.
Water clouds also contribute to the greenhouse effect. Depending on the temperature at which the cloud emits thermal radiation, its greenhouse effect can match or exceed the albedo effect. Cold clouds, such as cirrus, are net greenhouse warmers. Low clouds with temperatures near the effective radiating temperature affect climate almost entirely by their albedo, which depends on such factors as patchiness, cloud thickness, and the size distributions and composition of cloud particles. The exact climatic impact of clouds therefore crucially depends on the balance between their greenhouse and albedo effects. Whether the greenhouse or albedo effect dominates for clouds forming under runaway greenhouse conditions cannot be easily determined a priori. A better understanding of the cloud microphysics and convection processes in moist atmospheres during a runaway greenhouse process is needed to determine the cloud properties and their fractional coverage near the inner habitable zone (HZ) boundary. The effect of clouds under runaway greenhouse conditions represents one of the most important unresolved issues in planetary climate.
Note that this discussion is relevant to wet planets with moist atmospheres fed by extensive seas. Dry, land or desert, planets will have unsaturated atmospheres in the tropics and thus radiate at a higher temperature and cool more efficiently than planets with a water saturated atmosphere. As a result the habitable zone for such planets may be larger (*Abe et al.,* 2011).
### Outer Edge of the Habitable Zone
The outer edge of the habitable zone is set by the point at which there is no longer liquid water available at the surface as it is locked in ice. With falling insolation planets found progressively farther away from their central star have lower atmospheric and surface temperatures. With lower surface temperatures, the removal of CO~2~ from the atmosphere due to the carbonate-silicate cycle, which controls the amount of CO~2~ in Earth’s atmosphere on time scales of order a million years, becomes less efficient (see *Covey et al.*, this volume). Thus, if the terrestrial planet is still geologically active, CO~2~ can accumulate in the atmosphere by volcanic outgassing. With decreasing atmospheric temperatures, CO~2~ will condense at some point to form clouds composed of CO~2~ ice crystals. Because condensation nuclei can be expected to be available in atmospheres of terrestrial planets, the most dominant nucleation process for the formation of CO~2~ clouds is usually assumed to be heterogeneous nucleation (*Glandorf et al 2002*), in which cloud particles form on existing seed particles.
Just like water clouds, the presence of CO~2~ clouds will result in an increase of the planetary albedo by scattering incident stellar radiation back to space. However, in contrast to water, CO~2~ ice is almost transparent in the infrared (*Hansen*, 1997, 2005) except within some strong absorption bands. Thus CO~2~ clouds are unlikely to trigger a substantial classical greenhouse effect by absorption and re-emission of thermal radiation.
On the other hand, CO~2~ ice particles can efficiently scatter thermal radiation. This allows for a scattering greenhouse effect in which a fraction of the outgoing thermal radiation is scattered back towards the planetary surface (*Forget & Pierrehumbert*, 1997). Depending on the cloud properties this scattering greenhouse effect can outweigh the albedo effect and can in principle increase the surface temperature above the freezing point of water. However, the scattering greenhouse effect is a complex function of the cloud optical depth and particle size (*Colaprete and Toon,* 2003). Furthermore, because the greenhouse effect of CO~2~ clouds depends on the scattering properties of the ice particles, the particle shape (which cannot be expected as spherical) or particle surface roughness may play an important role. However, these effects cannot be easily quantified because neither the particle shapes nor their surface properties are known. These and other uncertainties in the cloud microphysics of CO~2~ ice in cool CO~2~ dominated atmospheres makes the calculation of the position of the outer HZ boundary complicated. More details on the formation of CO~2~ clouds in the present and early Martian atmosphere can be found in *Glandorf et al.* (2002), *Colaprete & Toon* (2003), and *Määttänen et al.* (2005), as well as in *Esposito et al.* (this volume). Inferences about CO~2~ cloud particle sizes in the current martian atmosphere as constrained by a variety of datasets are presented by *Hu et al.* (2012).
For a fully cloud-covered early Mars with thick a CO~2~ dominated atmosphere and CO~2~ clouds composed of spherical ice particles *Forget & Pierrehumbert* (1997) estimated that the outer boundary of the HZ is located at 2.4 AU, in contrast to the cloud-free boundary of 1.67 AU by *Kasting et al.* (1993). This greater value has been further used by *Selsis et al*. (2007) to extrapolate the effects of CO~2~ clouds on the outer HZ boundary towards other main sequence central stars (see also *Kaltenegger & Sasselov*, 2011).
For terrestrial super-Earths it has been suggested by *Pierrehumbert & Gaidos* (2011) that these planets could have retained much of their primordial H~2~ atmosphere owing to their greater mass. According to their model study, the classical habitable zone might be far larger than expected for a CO~2~ dominated atmosphere although this study did not explore the impact of clouds.
Clouds and Radiation
====================
Clouds are important for planetary atmospheres because they interact with both incident short wavelength radiation from the parent star and emergent thermal radiation emitted by the planetary surface, if present, and the atmosphere itself. Perhaps the simplest example of such interaction occurs for spherical cloud particles composed of a single constituent. In this case the wavelength dependent optical properties can be computed from Mie calculations that depend only on particle size and the wavelength-dependent complex refractive index of the bulk condensate. In this section we briefly review the basics of the interaction of cloud particles with radiation and summarize the use of Mie theory for modeling this interaction as well as point out some useful simplifications that can be applied in certain limiting cases.
Basic Radiative Properties of Cloud Particles
---------------------------------------------
Cloud opacity ultimately depends upon the radiative properties of the constituent particles. A particle with radius *r* has a cross section to scatter losslessly or to absorb incident radiation at some wavelength *l*, given by *C~s~* or *C~a~* respectively. These cross sections are defined as $C_s=Q_s\pi r^2$ and $C_a=Q_a\pi r^2$, and their sum is the extinction cross section *C~e~*. The scattering and absorption efficiencies *Q~s~* and *Q~a~*, and, from them, the extinction efficiency *Q~e~* = *Q~s~* ** + *Q~a~* is thus defined. The particle single scattering albedo is $\omega =Q_s/Q_e$, and the scattering phase function is $P\left(\theta \right)$. These quantities can be computed by a Mie code, which computes the electromagnetic wave propagation through spherical particles, given a tabulation of the optical properties of the cloud material. Good physically-based introductions to radiative properties of particles can be found in *van de Hulst* (1957), *Hansen & Travis* (1974), and *Liou* (2002).
Modeling cloud particle radiative properties can seem forbidding; workers often think a Mie scattering code, with all its exotic predictions, is needed. However, in many applications, much simpler approaches not only can provide very good quantitative fidelity - and greater physical insight - but also can be applied directly to more complex kinds of particles than the spheres for which Mie theory is derived.
The direct beam of radiant flux vertically traversing a cloud *layer* of particles composed of some material *q* is reduced by a factor $\exp \left(-\tau _q\right)$, where the layer optical depth is $\tau _q=nQ_e\pi r^2H_q=nC_eH_q$, and *n* is the particle number density, *r* the particle radius, and *H~q~* the vertical thickness of the layer (assuming a uniform vertical distribution for simplicity). *Q~e~* ** and thus $\tau _q$ are functions of the wavelength *l*, through the *l*–dependent real and imaginary refractive indices of the material in question (*n~r~* *, n~i~;* see below, and *Draine & Lee,* 1984 or *Pollack et al.,* 1994 or [](http://www.astro.uni-jena.de/Laboratory/Database/databases.html)<http://www.astro.uni-jena.de/Laboratory/Database/databases.html> for typical values). The particle opacity *k* (in units of length-squared per mass) is the effective particle cross section per unit mass of either solids or gas.
If the phase function is strongly forward scattering, as can be the case for particles with $r/\lambda >10$ or so (*Hansen & Travis* 1974), much of the “scattering” can be approximated as unattenuated radiation by reducing the layer optical depth and renormalizing the particle albedo and phase function. This is done by applying simple corrections to both *w* and $\tau_q$ known as similarity relations. *Irvine, (*1975), *van de Hulst* (1980), and *Liou* (2002) review these, and present a number of ways to solve the radiative transfer problem in general.
Refractive Indices and Opacity
------------------------------
All calculations of particle radiative properties (*Q~e~*, *Q~s~*, *Q~a~*, $\omega $ and *P(j)*) begin with the refractive indices of the material and the particle size. While Mie codes are available for download (e.g*.*, ftp://climate1.gsfc.nasa.gov/wiscombe/Single\_Scatt/), analytical approximations that are valid in all the relevant limits can often be of great use. Such approximations rely on the fact that realistic clouds have size and shape distributions that average away the exotic fluctuations shown by Mie theory for particles of specific sizes (see *Hansen & Travis* 1974 for examples). With current computational capabilities, Mie calculations are not onerous, but if many wavelengths and/or grids of numerous models are of interest, the burden is compounded, so one needs to understand whether such detailed predictions are actually needed.
One simple approximation to scattering by equidimensional, but still irregular, particles was developed by *Pollack & Cuzzi* (1980). This approach uses Mie theory for particles with small-to-moderate $r/\lambda$, where shape irregularities are indiscernible to the waves involved and Mie calculations converge rapidly. For larger $r/\lambda $ a ** combination of diffraction, external reflection, and internal transmission is used as adjusted for shape and parametrized by laboratory experiments. The approach is easy to apply in cases where the angular distribution of scattered radiation is important, as long as a particle size distribution smears away the significant oscillations in scattering properties which Mie theory predicts for monodisperse spheres (*Hansen & Travis* 1974). This method saves on computational effort for particles with large $r/\lambda $ but does involve a Mie code for the smaller particles.
Even simpler approaches are feasible. In many cases of interest, only globally averaged emergent fluxes or reflectivities, or perhaps heating calculations, are needed; here, the details of the phase function are of less interest than the optical depth and particle albedo, which are based only on the efficiencies. For these cases one can do fairly well using asymptotic expressions for efficiencies in the limiting regimes $r/\lambda {\ll}1$ and $r/\lambda {\gg}1$, connecting them with a suitable bridging function. These expressions depend on the refractive indices of the material in question.
The simplest limit is when $r/\lambda {\gg}1$ (geometrical optics); in fact, many of the cloud models discussed in this chapter fall into this regime at 1-10 mm wavelengths. In this range it is convenient to neglect diffraction, which is strongly concentrated in the forward direction and can be lumped with the direct beam as noted above. Then, assuming the particle has density r and is itself opaque, $Q_e{\leq}1$ and *C~e~* ** reduces to the physical cross section. Thus the solids-based opacity is simply $\kappa =3\pi r^2/4\rho \pi r^3=3/4r\rho $. In this regime, particle growth drastically reduces the opacity because the surface to mass ratio decreases linearly with radius (*Miyake et al.,* 1993; *Pollack et al.,* 1994).
At the opposite extreme, when $r/\lambda {\ll}1$(the Rayleigh limit), simple analytical expressions also apply (*Draine & Lee* 1984; *Bohren & Huffman* 1983). For simplicity below we give the expressions for $n_i{\ll}n_r{\sim}1$ (appropriate for silicates, oxides, water, but not iron), but the general expressions are not much more complicated (*Draine & Lee* 1984). Thus $Q_a=24xn_rn_i/\left(n_r^2+2\right)^2$ and $Q_s=8x^4\left(n_r^2-1\right)^2/3\left(n_r^2+2\right)^2$, where $x=2\pi r/\lambda $(see also *van de Hulst,* 1957, p. 70). Because *Q~s~* decreases much faster than *Q~a~* with decreasing $r/\lambda $ in this regime*,* small ** particles become not only less effective at interacting with radiation, but increasingly absorbers/emitters rather than scatterers. In this limit, their cross section $C_a=Q_a\pi r^2$ per unit particle mass becomes constant since *Q~a~* ** is proportional to *r*.
Shape and Porosity
------------------
In the case of terrestrial cloud droplets, in which condensation dominates growth, and raindrops, in which coagulation dominates growth, spherical particles of constant density are assumed, provided they fall slowly (though drag-induced deformation is taken into account when computing terminal fall-speeds for large raindrops). However, if particles condense from their vapor phase as solids, tiny initial monomers may instead coagulate by sticking into porous aggregates, perhaps having some fractal properties where the density may depend on the size. Non-spherical particle shape adds complexity to the computation of optical properties, which is commonly the case for solid particles. For instance, the single-scattering properties of ice particles in the terrestrial atmosphere depend on not only the geometric shape of the crystals (which depend on the temperature), but also the microscopic surface roughness. As noted by *Fu et al.* (1998), the extinction and absorption cross sections depend mainly on projected areas and particle volumes (note that random orientation is typically assumed), while the scattering phase function are largely determined by the aspect ratio of the crystal components and their small-scale roughness (*Fu*, 2007).
In the general exoplanet case for tiny particles in gas of typical densities, plausible inter-particle collision velocities are very small and lead to sticking but only minor compaction (see e.g., *Cuzzi and Hogan* 2003, *Dominik and Tielens* 1997) until particles become large enough (10-100 mm) where sedimentation velocity can exceed several meters per second and compaction or bouncing arise (section 6.3; see *Dominik et al. 2007*, *Zsom et al.* 2010). The thermodynamic properties of condensates and the temperature-pressure (*T-P)* structure of the atmosphere in question determine whether the condensate appears as a liquid or a solid. Figure 3 **** shows condensation curves of a number of important cloud-forming compounds, along with the *T-P* profiles of a range of exoplanetary and substellar objects.
Models of growing aggregates have been fairly well developed in the literature of protoplanetary disks (e.g.*, Weidenschilling,* 1988; *Dominik & Tielens,* 1997; *Beckwith et al.,* 2000; *Dominik et al.,* 2007; *Blum,* 2010). The properties of clouds made of this rather different kind of particle will differ in their opacity and vertical distribution from predictions of the simplest models; these properties in turn will impact, and can in principle be constrained by, remote observations of reflected and emitted radiation.
Porous Particles and Effective Medium Theory
---------------------------------------------
The most straightforward way of modeling porous aggregates is to model their effective refractive indices based on their constituent materials and porosity. If the monomers from which the particles are made are smaller than the wavelength in question, they act as independent dipoles immersed in an enveloping medium (the medium can be another material; here we assume it is vacuum). The aggregate as a whole can then be modeled as having *effective* refractive indices which depend only on the porosity of the aggregate and the refractive indices (but not the size) of the monomers. This is the so-called *effective medium theory (EMT);* several variants are discussed by *Bohren & Huffman* (1983), *Ossenkopf* (1991), *Stognienko et al*. (1995), and *Voschinnikov et al.* (2006).
For most combinations of materials, EMTs can be even further approximated by simple volume averages such that the refractive indices of the particle as a whole can be written (for a simple two-component particle with component 2 being vacuum) as $n_i=fn_{\mathit{i1}}+\left(1-f\right)n_{\mathit{i2}}$ and $\left(n_r-1\right)=f\left(n_{\mathit{r1}}-1\right)+\left(1-f\right)\left(n_{\mathit{r2}}-1\right)$, where *n~r~* and *n~i~* are the effective real and imaginary indices of the aggregate as a whole, and component 1 has volume fraction *f*. If a material which has large refractive indices, such as iron, is involved, the full expressions are needed (see *Wright*, 1987; *Bohren & Huffman*, 1983; or other basic references). We will assume the porous aggregates are roughly equidimensional, not needle-like structures, but even that added complication can often be tractable in semi-analytical ways for most materials (*Wright* 1987; *Bohren & Huffman* 1983).
In the $r/\lambda {\ll}1$ regime, porosity plays little role, both because the opacity is simply proportional to the total mass regardless of how it is distributed (see above), and because ever-smaller particles are unlikely to be aggregates of ever-smaller monomer constituents, but are more likely to be monomers themselves. *Ferguson et al.* (2007) present numerical calculations which show that, for the tiny particles they modeled (0.35 nm - 0.17 mm) indeed this expectation is fulfilled to first order, also showing that mixing of materials within these tiny aggregates has little effect (see also *Allard et al.* 2001, *Helling et al.* 2008, *Witte et al.* 2009 and references therein). Particles this small are likely to be well-mixed at all levels of typical exoplanet atmospheres (see below). Some targets of interest may have high haze layers where this regime is of interest.
However, in the $r/\lambda {\gg}1$ regime, which covers typical planetary clouds observed in the optical or near-infrared, porosity does affect opacity. Specifically, for a particle of mass *M* and porosity $\varphi$, the opacity $\kappa =C_e/M=3/4\rho r=3/4\rho _s\left(1-\varphi \right)r$ where ** $r=\left(3M/4\rho _s\left(1-\varphi \right)\right)^{1/3}$ ** is the actual radius. Comparing the opacity of this particle to that of a “solid” particle with the same mass having radius *r~s~* *,* gives $\kappa /\kappa _s=\left(1-\varphi \right)^{-2/3}$. The porous particle thus has a larger opacity for the same mass, and indeed reaches the geometrical optics (*l* independent) limit at lower mass than a nonporous particle. In this regime, effective medium theory should be a convenient and valid way of modeling porous and/or aggregate grain properties (section 3.2; see also *Helling et al.* 2008 and references therein).
Variable composition introduces further complexity to computing single-scattering properties, requiring a mixing rule to compute the refractive index for the composite particle. Different mixing rules can introduce uncertainties of order 1 and 10% respectively in the real and imaginary refractive indices for black carbon inclusions in liquid water droplets (*Lesins et al*., 2002).
Cloud Models
============
The cloud properties required to compute radiative fluxes are rather extensive. Thermal emission requires the emissivity and temperature of the cloud particles. Vertical fluxes can be computed from knowledge of the vertical distribution of particle cross-section (for each species of particle) together with the wavelength dependent scattering phase function and the single-scattering albedo. For a basic two-stream approach these quantities can be boiled down to a vertical profile of extinction coefficient, asymmetry parameter, and single-scattering albedo.
Complexities can arise if particle temperature differs from the local atmospheric temperature by virtue of latent heat exchange and radiative heating of the particles, but this should be rare in exoplanet applications (*Woitke & Helling* 2003) and typically the particles are assumed to be at ambient atmospheric air temperature. Clouds that are not horizontally uniform are another possible complication that can be considered.
The required scattering and absorption total cross sections and overall asymmetry parameter needed for input to a radiative transfer model are found by integration of the single-scattering properties and emissivities over the particle size distributions. The size distributions in turn can be computed by a so-called bin model which tracks the number of particles in multiple different bin sizes as the particles interact with the atmosphere. Such an approach is computationally expensive, and a more efficient treatment is to assume a particular functional form with a small number of free parameters.
For terrestrial applications log-normal distributions, with three parameters (total number, geometric mean radius, and geometric standard deviation) and gamma distributions, also with three parameters (total number, slope parameter, and shape parameter) are commonly used. Separate size distributions are used for particles of different phases and bulk densities (e.g., raindrops and snowflakes) and for each mode of a multimodal size distribution. For example, the parameterized size distribution of cloud droplets, which grow principally through condensation in the terrestrial atmosphere, are treated distinctly from raindrops, which grow principally through collisions. Although a poor approximation for cloud droplets, it is often assumed that the shape parameter is zero for other cloud species, which allows the gamma distribution to collapse to an exponential distribution, thereby reducing the free parameters from three to two.
For the computation of heating rates from the divergence of radiative fluxes, the vertical distribution of any absorbing and emitting species is obviously critical. Also, any vertical redistribution by scatterers in bands with emission or absorption requires that their vertical distribution is accurate. If horizontal photon transport is unimportant, it is feasible to represent the global atmosphere--or the atmosphere within any model column of finite horizontal extent, for that matter--with two columns, one clear and one cloudy (for example see the discussion in the context of brown dwarfs in *Marley et al.* (2010)). Assumptions about the vertical overlap of clouds are critical and can induce large errors in top-of-atmosphere fluxes as well as heating rates (e.g., *Barker et al.,* 1999).
In the context of extrasolar planet atmosphere modeling we must connect a simple, usually 1D model of the atmosphere with what is potentially a complex brew of cloud properties which we would ideally need to know. However it is clear that the number of variables can quickly grow to unmanageable extent. In this chapter we discuss various approaches that have been taken to address this problem.
Conceptual Framework for Cloud Modeling
---------------------------------------
Given a profile of atmospheric temperature as a function of elevation or pressure we can ask where clouds form and what are their radiative properties. Here we give a simplified discussion of the problem following *Sanchez-Lavega et al.* (2004).
The abundance of a given atmospheric species *a* in the vapor can be given by its mass mixing ratio $m_a=\rho _a/\rho =\epsilon P_a/P$ where $\rho $ and *P* are the density and pressure of the “background” atmosphere (that is, not including species *a*) and $\rho _a$ and *P~a~* are the density and partial pressure of vapor species *a*. Here $\epsilon =\mu _a/\mu $ or the ratio of the molecular weight of species *a,* $\mu _a$, to the background gas mean molecular weight $\mu $. A not unreasonable assumption is that any vapor in excess of the saturation vapor pressure of *a,* $P_{v,a}\left(T\right)$, condenses out. We can define the saturation ratio of *a* as $f_a=P_a/P_{v,a}\left(T\right)$.
Deep in the atmosphere, below cloud base, $P_a{\ll}P_{v,a}$ so the species is found in the gas phase. In a Lagrangian framework one can imagine a rising parcel of gas that cools adiabatically as it expands, and at some point it may reach $P_a=P_{v,a}$ and a saturation ratio $f_a=1$. In an Eulerian framework this condition requires that the thermal profile intersects with the vapor pressure curve for a given constituent, for instance as the 550 K model intersects the H~2~O vapor pressure curve in Figure 3. However the thermal profile for the 1300 K model does not intersect the H~2~O vapor pressure curve. Thus H~2~O would be expected to condense in the cooler, but not the warmer, atmosphere. To this point the problem is relatively straightforward for those species that condense directly from the gas phase, such as water. For other species, however, condensation instead proceeds by a net chemical reaction when the condensed phase has a lower Gibb’s free energy than the vapor phase. One such example is H~2~S + 2Na $\rightleftharpoons $ Na~2~S(s) + H~2~.
Above the condensation level a number of issues arise. First, the assumption that all vapor in excess of saturation condenses may be faulty. Condensation may require a supersaturation $\left(f_a>1\right)$ before it proceeds, owing to the thermodynamic energy barrier of forming new particles. If so, what degree of supersaturation is required? Above cloud base, which is the lowest level at which condensation occurs, abundance of the condensed phase will depend on a balance between the convective mixing of particles from below and the downward sedimentation of the condensate particles. If the sedimentation rate of some portion of the size distribution of condensate is greater than the vertical mixing velocity scale, the scale height for the condensed phase, *H~a~*, will be smaller than the atmospheric scale height *H* evaluated at cloud base. In the solar system typical values of *H~a~/H* range from 0.05 to 0.20 (*Sanchez-Lavega et al.* 2004).
Nucleation is the starting process for the formation of a condensed phase (either liquid or solid) from a gaseous state, or the formation of solid from a liquid state. It creates an initial distribution of nuclei (embryos) which, if large enough, are stable with respect to the higher-entropy phase and tend to grow into larger particles by condensation or freezing. Such a phase transition can only occur spontaneously under thermodynamically favorable conditions (for the following discussion we first focus on the process of condensate nucleation from the vapor phase). Such conditions require the saturation ratio $f_a$ to exceed unity ($f_a=1$ characterizes phase equilibrium).
The simplest nucleation mechanism is homogeneous nucleation, where the initial nuclei are formed by random spontaneous collisions of monomers in the vapor phase (e.g. single H~2~O ~~ molecules) into a molecular cluster. This process is, however, connected to an energy barrier that can prevent the formation of a stable new phase. A supersaturated gas phase possesses a high Gibb’s free energy, whereas at the same time molecules in the condensed phase would be at a lower potential energy. Thus, removing molecules from the vapour phase and adding them into a condensed phase would in principle lower the total free energy of the entire thermodynamical system. The corresponding net change in free energy depends mainly on the volume of an embryo (as a function of the particle radius $r$) and the supersaturation of the gas.
However, the creation and growth of nuclei also implies that a new surface is generated which additionally contributes to the free energy owing to surface tension. It is only thermodynamically favourable to form a condensed phase when the net reduction of free energy in the system $\left({\sim}r^3\right)$ is greater than (or equal to) the free energy required from the surface tension $\left({\sim}r^2\right)$. Therefore, there exists a critical radius for which these two contributions balance each other. Embryos smaller than the critical size are unstable and will tend to evaporate again quickly whereas those with sizes larger than the critical radius will tend to grow freely. The critical radius is, in particular, a function of the saturation ratio. Low supersaturations yield very large critical radii while, on the other hand, for increasing values of $f_a$ the size of the critical embryos decreases.
Homogeneous nucleation usually requires a very high supersaturation and is therefore unlikely to occur in atmospheres of terrestrial planets on a large scale (observed supersaturations for water vapor in the atmosphere of Earth are normally in the range of a few percent, for example), because heterogeneous nucleation occurs at much lower supersaturations and thus quenches the supersaturation well before homogeneous nucleation occurs.
The predominant nucleation process from the vapor for terrestrial planetary atmospheres is thought to be heterogeneous nucleation. Here, the initial distribution of nuclei is formed on pre-existing surfaces. The presence of such surfaces substantially lowers the supersaturation required to form the critical clusters. Depending on the properties of these surfaces (e.g. whether they are soluble with respect to the condensed phase) nucleation is already possible for saturation ratios close to unity. Such surfaces can be provided by dust, sea salt, pollen, or even bacteria. In case of terrestrial planets these particles (condensation nuclei) are usually largely available because of mechanical process associated with the planetary surface--such as wave breaking, bubble bursting, and dust saltation--and from the formation of haze particles (such as sulfuric acid or sulfate droplets) that result from photochemical processes. Therefore, heterogeneous nucleation can reasonably be expected as the dominant nucleation mechanism for terrestrial extrasolar planets. This assumption, however, complicates the treatment of cloud formation because details on such condensation nuclei (composition, number density, size distribution) are not available in case of exoplanets.
Additionally, the formation of a solid phase (e.g. ice crystals) can in principle occur by different pathways. It can either form directly from the gas phase by homogeneous or heterogeneous nucleation. On the other hand, it is also possible to form the liquid phase as an intermediate step, followed by freezing of the supercooled liquid into the solid phase afterwards. Whether this indirect or the direct pathway occurs depends on the properties of the condensing species and on the local atmospheric conditions. Water ice cloud crystals in the Earth’s atmosphere form by homogeneous and heterogeneous freezing of liquid in mixed-phase clouds (such as cumulonimbus and Arctic stratiform clouds) and heterogeneous nucleation in cirrus clouds. In the Martian atmosphere CO~2~ clouds form directly by heterogeneous nucleation from the gas phase.
Given a composition of the cloud the task for any cloud model then becomes one of computing cloud particle sizes and their number distribution through the atmosphere above cloud base. A very simple solution would be to simply assume a mean particle size and a cloud scale height and this is effectively the approach many investigations have take to explore the effect of exoplanet clouds. For the remainder of this section we consider efforts to more rigorously derive expected particle sizes and the vertical distribution of cloud particles. We begin by considering the most thoroughly modeled clouds, terrestrial clouds of liquid water and water ice. We then move on to cloud models that have been constructed for terrestrial and giant extrasolar planets and conclude by considering the lessons learned from efforts to model clouds expected in brown dwarf atmospheres.
Perspective from Earth Science
------------------------------
Cloud modeling of terrestrial clouds comes in a wide assortment of classes. For detailed cloud studies of limited spatial extent, dynamical frameworks range from 0-D parcel models, to 1-D column models, to 2-D eddy-resolving models, to 3-D large-eddy simulations and cloud-resolving models. The difference between such models is the number of spatial dimensions represented in the governing equations. Another varying aspect of cloud models is the degree of detail in describing cloud microphysics. The simplest models simply assume that all vapor in excess of saturation condenses and assume a fixed size for the cloud particles. Others assume a functional shape for the cloud particle size distributions and parameterize the microphysical processes, prognosing one, two, or three moments of the size distribution for each condensate species (such as cloud water, rain, cloud ice, snow, graupel, and hail). The most detailed approach is to resolve the cloud particle size distributions without making any assumptions about the functional shape of the size distributions.
Global climate model frameworks similarly range from one-dimensional radiative convective models (e.g., *Manabe & Wetherald*, 1967), to two-dimensional, zonally averaged models (e.g., *Schneider*, 1972), to modern three-dimensional general circulation models (GCMs). (See *Schneider & Dickinson*, 1974 for an early, comprehensive review of approaches to global climate modeling.) With respect to clouds in global models, one-dimensional radiative convective models suffer from a major deficiency, namely predicting plausible estimates of horizontal cloud coverage (see *Ramanathan & Coakley*, 1978). A two-dimensional framework is an intermediate step, though modern climate studies rely principally on three-dimensional GCMs. The rest of this section will focus on clouds in modern GCMs.
The representation of clouds in GCMs remains a major challenge, as cloud feedbacks constitute a leading source of uncertainty in current model-derived estimates of overall climate sensitivity, which are typically cast in terms of the sensitivity of globally averaged surface temperature to changes in radiative forcings (see *Hansen et al.,* 1984). The response of tropical cirrus clouds to increasing sea surface temperatures has been a topic of great debate in the last two decades. At one extreme is the thermostat hypothesis of *Ramanathan & Collins* (1991), which suggests that increased water vapor leads to more extensive, thicker anvils that will on net cool the planet through increased albedo, while at the other extreme is the argument of *Lindzen et al.* (2001) that greater condensate loading in a warmer climate precipitate more efficiently and lead to a dryer upper atmosphere that traps less infrared energy, driving a negative water vapor feedback. Both hypotheses produce a negative climate feedback for tropical clouds, one employing increased solar reflection, the other relying on increased infrared emission. While these are provocative ideas that have spawned countless evaluations of the hypothesis (with neither withstanding scrutiny), the predominant concept currently is that the tropical climate feedback for modern GCMs is determined primarily by the response of clouds in the marine boundary layer (*Bony & Dufresne,* 2005), of which the transition from overcast stratocumulus to broken cloud fields of trade cumulus is a leading primary candidate responsible for that response. With respect to cloud feedbacks generally, attention on the climate feedback of cirrus clouds formed from the detrainment of deep tropical convection has been supplanted to a large degree by more recent focus on the climate feedback of shallow clouds.
A fundamental problem in representing clouds in the GCMs is that the native GCM grid cells are very coarse, of order 100 km horizontally and 1 km vertically. Although model resolution steadily improves as computing power advances, the problem of convection and clouds being unresolved persists in models designed to span the globe and simulate climate over time scales of decades to millennia. The basic assumption in a conventional treatment of convection and clouds in a GCM is that cloud properties and precipitation rates in a model grid cell, which is much larger horizontally than any cloud, can be computed based on the mean properties of the grid cell. The purpose of a cloud parameterization is to compute cloud properties and precipitation rates from those mean properties.
A somewhat recent version of the NASA GISS (Goddard Institute for Space Studies) GCM (*Schmidt et al.,* 2006) includes a conventional treatment of clouds and convection. The atmospheric model grid spacing is roughly 2 2.5 with 20 or 23 vertical layers, and thus all convection and cloud physics are necessarily highly parameterized. Deep convection is parameterized based on the convective instability of model columns using idealized updrafts and downdrafts, which detrain air into stratiform cloud layers. The stratiform cloud cover is computed as a diagnostic function of grid-scale relative humidity, and the relative humidity thresholds used to compute cloud cover in that diagnostic function serve as the principle tuning knobs for the GCM, with the dual tuning targets of top-of-atmosphere radiative balance and an overall albedo reasonably close to satellite-based estimates (note that such tuning can easily result in exaggerated cloudiness in some regions that make up for insufficient cloudiness in others). In the GISS GCM the only prognostic cloud variable is the mass mixing ratio of cloud condensate, which is a fundamental component of the cloud parameterization. Any precipitation is assumed to evaporate or fall out in one time step (30 minutes) and the phase of the condensate is probabilistically determined from temperature to allow for a modest amount of supercooling (liquid colder than the melting point) on average. The standard version of the GCM assumes different cloud droplet concentrations over ocean and land and also assumes a fixed number concentration for ice particles, and a fixed effective variance of the condensate size distributions is assumed for computing cloud optical thickness.
A more complex approach for stratiform cloud microphysics is used in the most recent version of the NCAR (National Center for Atmospheric Research) GCM. That scheme uses two moments (mass and number) for two prognostic (cloud water and cloud ice) and two diagnostic (rain and snow) hydrometeor species (*Morrison & Gettelman,* 2009). The rapidly sedimenting species are treated diagnostically with a tridiagonal solver to allow for long time steps (20 minutes with 2 microphysics substeps) in a manner that avoids numerical instability associated with falling through more than one layer during a time step. A novel aspect of this microphysics scheme is that by assuming a particular subgrid-scale distribution of cloud water, microphysical processes that involve cloud water take into account the problem of grid-averaging over nonlinear process rates (*Pincus & Klein*, 2000). Taking into account joint subgrid-scale distributions of just two moments for two species complicates the math considerably (*Larson & Griffin,* 2012).
An alternative to parameterizing convection within grid scales O(100 km) is the multiscale modeling framework (*Randall et al*., 2003), in which two-dimensional cloud-resolving models are embedded within each GCM column. While some convection-related aspects of the global circulation are treated well by such an approach, like the more traditional approach to GCM cloud parameterization, the pervasive and climatologically important regime of shallow marine convection is poorly represented in such models (e.g., *Marchand & Ackerman*, 2010). (Explicit resolution of such clouds requires horizontal resolution O(100 m) and vertical resolution O(10 m).) Avoiding issues related to embedding 2-D slices within GCM columns (problems including how to orient the slices and shortcomings including the use of periodic lateral boundary conditions for each embedded 2-D sub-model), the most expensive, yet perhaps straightforward approach to climate modeling is the Earth Simulator, a global cloud-resolving model with simulations run on a 3.5-km horizontal grid (*Satoh et al*., 2008). The computational demands of such an approach are vast, with order 10^19^ grid cells in such a model. Even such a brute-force approach falls far short of the grid resolution required to explicitly simulate shallow marine convection, however. It is safe to say that even on the most powerful computing platforms that global simulations of Earth will be saddled with cloud and convective parameterizations for the foreseeable future.
Exoplanet Clouds
----------------
In extreme contrast to the situation for Earth outlined above, there are currently no observational constraints for atmospheres of terrestrial exoplanets that would provide information about what kind of clouds may have to be considered for a particular exoplanet. Available observables are confined to basic planetary parameters, like radius and orbital inclination (if a transit event can be observed), planetary mass (from the radial velocity method), orbital eccentricity and distance, and additionally the type of the central star. Consequently, we are faced with the difficult problem of modeling clouds in an environment without actually knowing any further details.
Cloud formation can only be treated theoretically in compliance with the chemical composition of the atmosphere because it determines the condensing species forming a cloud. Considering how diverse atmospheres of terrestrial exoplanets can be expected to be, the self-consistent modelling of cloud formation in such atmospheres without observational constraints or theoretical predictions is somewhat ambiguous. The composition of a terrestrial planet cannot be easily deduced from simple theoretical arguments. In the case of a planet which has lost its primordial atmosphere the atmospheric composition is determined by the combination of the outgassed chemical species from the planet’s interior and the volatiles delivered by impacts of asteroids and comets and will therefore depend on the planet’s mantle composition, physical processes in the planetary interior, and the composition and sizes of impactors. Another factor with a huge impact on the atmospheric composition is also the possible existence of a biosphere that interacts chemically with the atmosphere.
The long-term evolution of the atmospheric composition, such as that resulting from the carbonate-silicate cycle as on Earth, depends also on the occurrence of plate tectonics. To date, the only known planet with plate tectonics is Earth. It is currently not quite well understood under which conditions a planetary crust will start plate tectonics and how this process is maintained over an extended period of time. This is especially true for more massive terrestrial planets like super-Earths, where there is much controversy regarding plate tectonics (see e.g. *Valencia et al.* 2007, *O’Neill et al.* 2007).
Other important processes determining the atmospheric composition are the escape mechanisms of atmospheric gas to space. While thermal escape is a function of the atmospheric temperature, planetary mass, and the molecular weight, the non-thermal escape processes (erosion of the atmosphere by a stellar wind, for example) are much more complicated (e.g. *Lammer et al.* 2008 or *Tian*, this volume). They not only depend on the activity of the central star--which itself is a function of the stellar type and its particular stellar evolution--but also on the possible existence of a planetary magnetic field (linked to the rotation rate of the planet), which can protect the planetary atmosphere against loss processes.
In contrast to a low-mass planet like Earth, a more massive terrestrial super-Earth might retain a part of its primordial hydrogen dominated atmosphere, partly enriched by volcanic outgassing or additional external delivery. Such atmospheres may be vastly different from those known within our solar system. A discussion of possible atmospheric compositions can be found in *Seager & Deming* (2010).
The secondary atmosphere of an Earth-like planet (Earth-like with respect to the chemical composition of the planetary mantle) would most likely be rich in H~2~O and CO~2~ (*Schaefer et al.* 2012). Therefore, clouds composed of these species are of prime interest for habitable Earth-like planets. Cooler atmospheres can also contain significant amounts of CH~4~ and CH~3~, or SO~2~ in case of high temperatures (*Schaefer et al.* 2012).
In this environment effective cloud models must match the problem at hand. For example highly sophisticated terrestrial cloud microphysics and dynamics models are not required in order to ascertain the range of plausible albedos for a hypothetical Earth-like terrestrial planet. However more sophisticated approaches than simple ad-hoc models may be needed to interpret the colors of a directly imaged planet.
Cloud Models for Terrestrial Exoplanets
---------------------------------------
In principle the basic mathematical description of cloud microphysics in atmospheres of terrestrial exoplanets do not deviate from their solar system counterparts. The temporal and spatial evolution of the cloud particle size distribution can be described by means of a master equation (“general dynamic equation”) incorporating all relevant gain and loss processes. This includes nucleation, evaporation, sedimentation, coagulation/coalescence, diffusion, or hydrodynamical transport (see *Pruppacher & Klett*, 1996 or *Lamb & Verlinde* 2011). While the numerical solution of the master equation is still quite challenging it can be efficiently performed by the methods summarised in *Williams & Loyalka* (1991) or by applying more advanced methods (e.g. continuous and discontinuous Galerkin methods by *Sandu & Borden*, 2003).
Self-consistent modeling of the formation and temporal and spatial evolution of clouds is an unsolved problem. An ideal treatment would require a thorough knowledge of the state of the atmosphere, including its composition, the spatial distribution of chemical species, atmospheric temperature and dynamics, and the size distribution and density of potential cloud condensation nuclei and heterogeneous freezing nuclei. However, even in the terrestrial atmosphere the formation of ice at temperatures warmer than the homogeneous freezing temperature for water (about 233 K for typical drop sizes) is not well understood (e.g., *Fridlind et al*., 2007, 2012); far less, if anything, is known about heterogeneous freezing and condensation nuclei in other atmospheres and would hardly be detectable by remote observations of terrestrial extrasolar planets. Lack of laboratory data to derive the necessary microphysical rates under atmospheric conditions more “exotic” than found in the solar system further complicates the problem of describing cloud formation in atmospheres of exoplanets.
Additionally, many atmospheric models for exoplanets are restricted to one spatial dimension and are often considered to be stationary, which makes a detailed description of cloud microphysics very difficult. While the climatic effects of clouds can be approximately treated in a one-dimensional model atmosphere, a consistent modelling of cloud formation would, in principle, require a three-dimensional dynamical atmospheric model as described in Section 4.2. In comparison to one-dimensional models, however, three-dimensional general circulation models contain many more free parameters and are very computationally intensive. Properties such as the surface orography, distribution of surface types (fractions of oceans and land mass), and the local distribution of chemical species (affected by volcanism for example) play a major role for the dynamics and chemistry of the atmosphere (*Joshi*, 2003), and affect directly the formation and evolution of clouds. Since none of these detailed properties are known for a terrestrial exoplanet many additional assumptions about the planet and its atmosphere have to enter the calculations. On the other hand, three-dimensional models are the only opportunity to obtain information about the possible temporal variation and fractional coverage of clouds and their distribution throughout the atmosphere. Such results might be required to analyze transmission spectra of terrestrial exoplanetary atmospheres containing patchy clouds.
Given the aforementioned challenges due to the lack of observational constraints, clouds in atmospheres of terrestrial exoplanets are usually treated in a simplified way. The simplest method to account for the effects of clouds in an atmospheric model is a modified surface albedo. This approach has been widely used in the past (e.g., *Kasting et al.* (1993), *Segura et al.* (2003, 2005), or *Grenfell et al.* (2007)). The surface albedo of these kind of models is modified to yield a specified surface temperature for a given reference scenario. For example, a common reference scenario is an Earth-like planet around the Sun at a known orbital distance of 1 AU. The surface albedo is then adjusted to obtain the mean surface temperature of Earth (288 K), thereby mimicking the climatic effects of clouds. This adjusted surface albedo value is then used in all subsequent model calculations, assuming that the net effect of the clouds is invariant from changes of the atmospheric conditions, or type of central star. This approach makes no assumptions about the physical nature of the clouds, their composition, size, or optical properties but instead estimates their effects based on the original tuning to surface temperature. While a modified surface albedo can crudely describe the climatic effect of clouds, it cannot simulate their impact on the planetary spectra.
A more detailed approach is to consider model scenarios where the properties of clouds can be assumed to be approximately known. For a completely Earth-like planet, one could expect that the properties of clouds in such an atmosphere would closely resemble those found on Earth. This approach was used by *Kitzmann et al.* (2010) to study the impact of mean Earth water clouds in the atmospheres of Earth-like extrasolar planets. In contrast to a modified surface albedo, the wavelength-dependent optical properties of the cloud particles are explicitly taken into account to study their effects for different incident stellar spectra and other parameters. Thus, in addition to the influence on the atmospheric and surface temperatures, their impact on the planetary spectra can also be studied in detail using this modelling approach (e.g., *Kaltenegger et al.,* 2007). If the properties of the clouds are not known (e.g., for a CO~2~ cloud in a thick CO~2~ dominated atmosphere of a super-Earth) parameter studies can be performed, varying the cloud properties over a wide range to estimate the possible effect of clouds. This approach has been used by *Forget & Pierrehumbert* (1997) for CO~2~ ice clouds in an early Martian atmosphere, for example.
More detailed treatments of cloud formation in exoplanetary atmospheres include simplified air parcel and vertically resolved one-dimensional cloud models based on those originally developed for the Earth atmosphere. Such models can be used to determine mean cloud properties under different atmospheric conditions (see *Neubauer et al.* (2011, 2012) for several cloud species (e.g. H~2~O and H~2~SO~4~) or *Zsom et al.* (2012) for water (droplet and ice) clouds, for example). However, these more detailed descriptions of cloud formation already need to include many additional assumptions, such as the distribution of cloud condensation nuclei, which strongly influence the resulting cloud properties.
While cloud models for terrestrial exoplanets so far lack many of the sophisticated and detailed cloud microphysics needed to reproduce the complicated cloud structures known from Earth observations they nonetheless offer an important first-order estimate of cloud effects in exoplanetary atmospheres. One of the largest uncertainties for one-dimensional models is the treatment of fractional clouds. Unless the atmosphere is globally supersaturated, thus resulting in a completely cloud-covered planet, the fraction of the atmosphere where clouds are present has to be introduced as a free parameter. Thus, one-dimensional models are incapable of organically describing planets with fractional cloud-cover.
Although one-dimensional models are commonly employed for many terrestrial exoplanet applications, three-dimensional models have also been used. For the terrestrial super-Earth Gliese 581d, *Wordsworth et al.* (2011) adapted a Mars global circulation model that included a simplified treatment of CO~2~ ice cloud formation. This microphysical model assumes a certain size and number density of cloud condensation nuclei and equally distributes the condensable material among them, accounting also for the sedimentation and hydrodynamical transport of the cloud particles within the atmosphere. The corresponding formation of CO~2~ clouds lead to an increase of the surface temperature of Gliese 581d in their model calculations. The same approach was also used by *Wordsworth et al.* (2010) in a one-dimensional atmospheric model for the same planet.
Giant Planet Cloud Models
-------------------------
Unlike the case for Earthlike planets where water clouds are the greatest concern, a wide variety of species may condense in the hydrogen-helium dominated atmospheres of giant planets. *Sánchez-Lavega et al.* (2004) review the standard framework for cloud formation in giant planets. Homogeneous condensation occurs when the partial pressure of a species in the gas phase exceeds its saturation vapor pressure at a given temperature in the atmosphere. *Sánchez-Lavega et al.* tabulate the vapor pressures for many relevant species. Other expressions for additional species can be found in *Ackerman & Marley* (2001) and *Morley et al.* (2012). Curves tracing the set of pressure and temperature conditions at which a given species condenses assuming equilibrium chemistry (“condensation equilibrium curves”) are shown for many species in Figure 3 and a schematic illustration of the resulting cloud decks is shown in Figure 4.
Once a cloud layer forms the condensate is removed from the overlying atmosphere and thus is no longer available to react at lower temperatures higher in the atmosphere. Thus the calculation of the chemical equilibrium state for the atmosphere must account for the presence of the cloud. Such a “condensation chemistry” is distinct from equilibrium chemistry calculations in which the condensate remains in communication with the gas phase and is available for reaction at lower temperatures. Condensation chemistry is discussed in detail by *Fegley & Lodders* (1996), *Lodders & Fegley* (2002), and *Visscher et al.* (2006) as well as by *Burrows & Sharp* (1999) in the context of brown dwarf models and *Sudarsky et al.* (2003) for extrasolar giant planets. The schematic Figure 4 accounts for condensation chemistry. If iron were not sequestered into a deep cloud layer in Jupiter’s atmosphere the Fe would react with gaseous H~2~S to form FeS, thus removing H~2~S from the observable atmosphere, in contradiction to observations (*Fegley & Lodders,* 1996).
Once the cloud base pressure is found the challenge is to describe the variation in cloud particle sizes and number densities above this level. Early attempts to develop cloud models for use in giant solar system atmospheres included the work of *Lewis* (1969), *Rossow* (1978), and *Carlson et al.* (1988). These and other early works are reviewed by *Ackerman & Marley* (2001). In this subsection we focus on more recent modeling approaches that are in use today, in particular the cloud models of *Ackerman & Marley* and of *Helling* and collaborators.
### Ackerman & Marley
Iron and silicates condense in the atmospheres of warm brown dwarfs (Figure 3) and these clouds must be accounted for in models of brown dwarf emergent spectra. Early modeling attempts for such clouds simply computed the mass of dust that would be found in chemical equilibrium for a given assumed initial abundance of gas (as if the gas were isolated at a given pressure and temperature from the rest of the atmosphere. The lower the atmospheric temperature the more dust that would be present. Atmospheric models following such a prescription (such as the “DUSTY” models of *Allard et al.* 2001) adequately reproduced the near-infrared colors of the warmest brown dwarfs but predicted far too great of a dust load in cooler objects. Thus it was apparent that an accounting for sedimentation of grain particles was required. One approach used in the literature was to set a variable “critical” temperature for a given cloud such that cloud particles would only be found between cloud base and the specified temperature (*Tsuji*, 2002). Another approach was to limit the cloud to be confined within an arbitrary distance, usually one scale-height, of cloud base. Both such approaches required the choice of an arbitrary particle size for the grains. The advantage of such approaches is that they are computationally very tractable for modeling and thus allow the exploration of a large parameter space. One disadvantage is that it is difficult to consider particle size effects and other complexities.
In order to allow for vertically-varying particle number densities and sizes a second approach was suggested by *Ackerman & Marley* (2001). In their formulation downward transport of particles by sedimentation is balanced by upwards mixing of vapor and condensate (either solid grains or liquid drops),
$$-K_{\mathrm{zz}}\frac{{\partial}q_t}{{\partial}z} - f_{\mathrm{sed}}w^* q_c = 0
\label{eq:cloud_formation_AM}$$
where *K~zz~* is the vertical eddy diffusion coefficient, *q~t~* is the mixing ratio of condensate and vapor, *q~c~* is the mixing ratio of condensate, *w*\* is the convective velocity scale, and *f~sed~* is a dimensionless parameter that describes the efficiency of sedimentation. The solution of this equation allows computation of a self-consistent variation in condensate number density and particle size with altitude above an arbitrary cloud base.
In their model the cloud base is found by determining at which point in the atmosphere the local gas abundance exceeds the local condensate saturation vapor pressure $P_{v,a}$ at which point the atmosphere becomes saturated. In cases where the formation of condensates does not proceed by homogeneous condensation an equivalent vapor pressure curve is computed, as described by *Morley et al.* (2012).
The *Ackerman & Marley* cloud model has the advantage of not requiring knowledge of microphysical processes to compute particle sizes. For a given sedimentation efficiency clouds are simply assumed to have grown large enough to provide the required downward mass flux that balances Equation . Since the solution is numerically rapid a large number of models can be computed and compared with data within a tractable amount of time. Sample model temperature-pressure profiles along with equilibrium condensation curves are illustrated in Figure 3.
Considering the simplicity of this approach, the model has fared fairly well in comparisons with data. *Stephens et al.* (2009) for example compared model spectra for L and T dwarfs to a large database of near- to mid-infrared spectra. They found that cloudy L dwarfs can generally be well fit by clouds computed with *f*~sed~ = 1 to 2 while early T dwarfs, which exhibit thinner clouds, are better fit with *f*~sed~ = 3 to 4. The model thus provides a framework for describing mean global clouds in a 1-dimensional sense, but the model lacks the ability to explain why the sedimentation efficiency might change at effective temperatures around 1200 K, where the near-infrared spectra of L dwarfs evolves over a small temperature range.
*Marley et al*. (2010) considered the effect of partial cloudiness on L dwarf spectra computed with the *Ackerman & Marley* (2001) cloud model. Their method assumed that clear and cloudy columns of atmosphere had the same temperature profile and together emitted the flux corresponding to a specified effective temperature. They found that partially cloudy L dwarfs would have emergent spectra comparable to standard models with homogeneous cloud cover but with larger values of *f*~sed~. Thus a dwarf with 50% clear skies and 50% cloudy skies with *f*~sed~ = 2 ends up with a model spectrum comparable to that of a homogeneous cloud cover with *f*~sed ~= 4.
### Helling and Collaborators
The most extensive body of work on cloud formation in giant exoplanet and brown dwarf atmospheres has been undertaken by Helling and her collaborators (*Helling & Woitke,* 2006; *Helling et al.,* 2008; *Witte et al.,* 2009, 2011; *de Kok et al.,* 2011) who follow the trajectory of seed particles from the top of their model atmospheres as they sink downwards. The seeds grow and accrete condensate material as they fall, resulting in “dirty” or compositionally layered grains. This work extends the dust moment method from *Gail et al.* (1984) and *Gail & Sedlmayr* (1988). It accounts for the microphysics of grain growth given these conditions and available relevant laboratory data. Particle nucleation is explicitly computed, taking into account barriers to grain formation.
Because condensation is envisioned in this framework to proceed downwards from the top of the atmosphere rather than upwards from the deep atmosphere and because grains are allowed to interact with the surrounding gas, the cloud composition predicted by the Helling approach differs substantially from that employed in the *Ackerman & Marley* framework. An example is shown in Figure 5 for a model brown dwarf (log *g* = 5 in CGS units, *T*~eff~ = 1600 K). Here *Helling et al.* (2008) predict that in addition to the usual Fe, MgSiO~3~ and Mg~2~SiO~4~ cloud layers, additional condensates including SiO~2~ and MgO will form. These latter species are not predicted by equilibrium condensation for a cooling gas mixed upwards from higher temperature and pressure conditions The TiO~2~ cloud seeds arising at the model at the top of the atmosphere in this approach is also evident.
In fact the presence of these initial TiO~2~ seed particles at the top of the model atmosphere in the *Helling* framework deserves some discussion. In the equilibrium chemistry condensation framework, Ti-bearing condensates (e.g., CaTiO~3~) would form much deeper in the atmosphere as a gas parcel rises vertically and cools. Precipitation of such particles would remove the condensate from the gas phase and notable amounts of refractory TiO~2~ seed particles would not be expected to arrive at the top of the atmosphere. Conversely in the *Helling* conceptual framework the formation of CaTiO~3~ at the expected equilibrium cloud base is thought to be kinetically inhibited (since multiple collisions of molecules would be required) resulting in the refractory TiO~2~ clusters being mixed further upwards to ultimately seed condensation in the cooler upper reaches of the atmosphere. Thus the *Helling* approach fundamentally assumes that vertical mixing timescales, at least in localized columns, are faster than condensation timescales. As the seeds eventually fall from the top of the atmosphere they then encounter other condensable molecules which are likewise assumed not to have not been cold trapped below and the seeds then accrete these species and grow. The model iterates to find a self-consistent solution. This top-down approach thus differs from most of the other cloud modeling approaches discussed in the literature which generally conceive of a condensation sequence operating from the depths of the atmosphere upwards with species sequentially condensing, as conceptually shown in Figure 4. Atmospheric mixing by breaking gravity waves (Freytag et al. 2010) might provide a mechanism to stir the atmosphere sufficiently to deliver the seed particles.
Because of the complexity of the computational approach required to compute cloud properties in this framework there have been fewer comparisons between model spectra computed with the *Helling* clouds and data than has been the case with the *Ackerman & Marley* cloud model. Some direct comparisons between different cloud models are shown by *Helling et al.* (2008). Ultimately only a thorough comparison of the predictions of all cloud modeling schools and data will be required to establish which conceptual framework is a better approximation over which ranges of conditions. An application of the Helling framework to the clouds of Jupiter would be enlightening.
### Other Approaches
More simplified approaches have also been taken for cloud models, such as specifying particle sizes and cloud heights. *Sudarsky et al.* (2000; 2003) computed model exoplanet albedo spectra given various particle size and cloud height assumptions. In particular they utilized the *Deirmendjian* (1964) size distribution and explored the effects of changing mean cloud particle sizes and widths of the size distribution. *Sudarsky et al.* (2000) also considered the effects of various plausible photochemical hazes on giant planet albedo spectra.
*Cooper et al.* (2003) employed the timescale comparison framework pioneered by *Rossow* (1978) to compute cloud models for brown dwarfs and extrasolar giant planets. In this approach various timescales for particle nucleation, growth, and sedimentation are compared to derive expected condensate particle sizes. As discussed by *Ackerman & Marley* (2001), a difficulty with the Rossow method is that some of the critical timescales depend upon unknown factors, particularly the assumed supersaturation. Nevertheless the *Cooper et al.* model provides a useful survey of likely particle sizes for species of interest expected under various combinations of gravity and effective temperature. For example, in agreement with most of the other cloud models *Cooper et al.* predict typical silicate grain sizes in the range of 10 to 200 mm.
Tsuji and collaborators (*Tsuji* 2002, 2005; *Tsuji et al.* 2004) have computed brown dwarf models by specifying cloud-top and cloud-base temperatures. For the directly imaged planets *Currie et al.* (2011), *Madhusudhan et al.* (2011), and *Bowler et al.* (2010) employ a variety of approaches to specify cloud properties and explore parameter space. Approaches such as these offer the advantage of quickly exploring the phase space of possible models and establishing the effect of plausible cloud models on spectra. The lack of physical complexity in such models is offset by their useful ability to offer qualitative understanding of the effect of various condensate properties.
Lessons Learned from Cloudy Brown Dwarfs
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Brown dwarfs--hydrogen-helium rich objects with masses between about 12 and 80 times that of Jupiter (M~J~)--have been a proving ground for understanding the role of clouds in exotic atmospheres. This is because the class of brown dwarfs known as the L dwarfs have atmospheric temperatures in the regime in which iron and silicate grains condense from the gas phase. It is apparent from the available data that these refractory condensates do not form a pall of particles mixed through the entire atmosphere, but rather form discrete cloud layers. As such L dwarfs were the first objects outside the solar system for which a detailed description of clouds was required in order to interpret their emitted spectra.
There have been several comparisons of cloudy brown dwarf atmosphere models to observations. The most extensive to date are presented by *Cushing et al.* (2008) and *Stephens et al.* (2009). These authors compared atmosphere models of *Saumon & Marley* (2008) computed using the *Ackerman & Marley* (2001) cloud model to a variety of L- and T-type brown dwarf spectra from 0.8 to 15 mm. For the L dwarfs and early T-dwarfs the cloudy models clearly did a better job reproducing the data than cloudless models. The tunable *f*~sed~ parameter, with typical values between 1 and 3, allowed sufficient dynamic range to generally reproduce most of the observed spectra. *Witte et al.* (2011) meanwhile compare model spectra computed with the *Helling et al.* (2008) cloud formulation to spectra of L-type brown dwarfs. Likewise their cloud model shows much better agreement with data than either cloudless or very cloudy models with no dust sedimentation. In all of these studies the matches between models and data are very good in many cases, but nevertheless there remain notable spectral mismatches and it is clear that a more sophisticated cloud model would be required to fit all objects.
The most important lesson learned from the campaign to model brown dwarfs may be that large grids of atmosphere models--including a variety of cloud descriptions--are required. Models should not be so complex that the creation of such grids are a challenge. In the next section we review the available exoplanet data relevant to clouds. As we will see the exoplanet data do not yet require large systematic model grids, but such models will undoubtedly be required as more data become available. Ideally more models will be available than has been the case for brown dwarfs, thereby permitting more systematic comparisons of various cloud modeling frameworks to large exoplanet datasets.
Observations of exoplanet clouds
================================
Transiting Planets
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### Transmission Spectra
Perhaps the most convincing evidence of high altitude clouds or hazes on any extrasolar planet to date is found in the case of the transiting planet HD 189733b. This 1.1-Jupiter mass planet orbits a bright nearby K star and is thus an excellent target for detailed studies of its atmosphere. This planet is notable because its transit radius--the apparent size of the planet as a function of wavelength--follows a smooth power law as first measured by *Pont et al.* (2008). Signatures of molecular or atomic absorption expected from a clear, solar composition atmosphere are almost absent, although Na is detected at high spectral resolution (*Huitson et al*., 2012). Figure 6 illustrates the smooth variation in atmospheric transmission as measured from 0.3 through 1 mm (*Pont et al.,* 2008; *Sing et al*., 2009; *Gibson et al*., 2011). The red curve presents a pure Rayleigh scattering model while a gas opacity only model is also shown. With the possible exception of a spectral feature at 1.5 mm the smooth variation of planet radius with wavelength extends to at least 2.5 mm (*Gibson et al.,* 2011).
The most natural explanation for the HD 189733b transmission spectrum is that a population of small, high albedo particles is present in the atmosphere at low pressures (*Lecavelier des Etangs et al.,* 2008). In the terminology of Mie scattering this requires that the scattering efficiency is large compared to the absorption efficiency or $Q_{\mathit{abs}}{\ll}Q_{\mathit{scat}}$. *Lecavelier des Etangs et al.* demonstrate that this in turn requires a material with an imaginary index of refraction that is small compared to the real index and suggest MgSiO~3~ as a possible candidate. In the condensation chemistry framework (Section 3.4) however, silicates are expected to condense much deeper in the atmosphere and the mechanism by which small grains could be transported to the upper atmosphere has yet to be fully explored, although vigorous mixing is a likely explanation.
Another transiting planet for which particulate opacity may be important is GJ 1214b. This 6.5-Earth mass planet orbits an M star. Planetary structure models that fit the observed mass and radius of the planet can be found with either a thick hydrogen atmosphere comprising less than 3% of the mass of the planet or a water rich planet surrounded by a steam atmosphere. Transmission spectra of a large scale height, hydrogen-dominated atmosphere (Figure 7) are predicted to exhibit well defined absorption bands while a water dominated atmosphere would have a much smaller scale height and consequently a very smooth transmission spectrum (*Miller-Ricci & Fortney,* 2010).
The observed flat transmission spectrum from the optical to perhaps 5 mm (*Bean et al*., 2010, 2011; *Berta et al*., 2012; *Desert et al.,* 2011) is consistent with a small scale-height atmosphere, thus apparently favoring the water-rich alternative. However high altitude clouds or hazes, as with HD 189733 b, could also be concealing atmospheric absorption bands if they lie at altitudes above 200 mbar (*Bean et al*., 2010 and Figure 7). Since the nominal model pressure-temperature profile does not cross the condensation curve of any major species at solar abundance (although ZnS and KCl do condense around 1 bar), *Bean et al.* proposed photochemical products as the likely source of the haze rather than clouds. However as shown in Figure 7, *Morley et al.* (2013) find that very extended (*f*~sed~=0.1) equilibrium sulphide clouds, including Na~2~S, in an atmosphere enriched in heavy elements can also attenuate the flux sufficiently to reproduce the data. Such small values of *f*~sed~ are seen in some terrestrial regimes (*Ackerman & Marley* 2001) but whether or not this would be plausible in the atmosphere of GJ 1214b remains to be investigated.
*Miller-Ricci Kempton et al*. (2012) explored possible photochemical pathways in the atmosphere of this planet to explore possible mechanisms for haze production and found that second order hydrocarbons, including C~2~H~2~, C~2~H~4~, and C~2~H~6~ are efficiently produced by UV photolysis of methane in the atmosphere. While their model did not explore the chemistry to hydrocarbons of order higher than C~2~H~6~, they argue that polymerization of the initial photochemical products are likely and that complex hydrocarbons including tholins and soots are likely to form. *Morley et al.* (2013) demonstrate that such a hydrocarbon haze could well explain the flat transmission spectrum observed for this planet if the atmosphere is indeed hydrogen rich with efficient methane photolysis (with results very similar to the cloudy case shown in Figure 7). In an unpublished manuscript *Zahnle et al*. (2009) argue that for solar composition atmospheres in general there is a range of atmospheric temperatures near 1000 K that would be expected to favor methane photolysis and the formation of higher order hydrocarbon soots or hazes. A definitive exploration of this point would require new generations of computer codes that can follow the fate of hydrocarbon species produced by photochemistry.
While it is still too soon to definitely characterize the atmospheres of either HD 189733b or GJ 1214b, it is apparent that cloud or haze opacity is likely important in at least some transiting planet atmospheres. The James Webb Space Telescope will obtain transit spectra of dozens of planets and characterize atmospheric absorbers as a function of planet mass and composition and the degree of stellar insolation. Such comprehensive studies will help to map out the conditions under which clouds and hazes are found.
### Transiting Planets in Reflected Light
In addition to the light transmitted through the atmosphere of transiting planets, measurements have also been made of the the apparent brightness of the day side of transiting planets. By comparing the brightness of a transiting system immediately before and after a planet is occulted by its primary star the combination of light reflected and emitted by the planet can be measured. In the limit in which the planet does not emit but rather shines only by reflected light within the passband such a measurement provides the mean geometric albedo of the planet. However since such a measurement is most tractable for large planets orbiting close to their stars, in other words the hot Jupiter class of planets, thermal emission cannot generally be disregarded.
Upper limits on the geometric albedo have been placed on many planets (see the summary in *Demory et al.* (2011)), generally finding the passband averaged geometric albedo to be less than about 30% and in some cases much less (*Kipping & Spiegel,* 2011). Such a finding is not surprising for hot giant planet atmospheres (*Marley et al.,* 1999; *Sudarsky et al.,* 2000, 2003) since in the absence of a cloud layer most of the incident flux is absorbed rather than scattered. Despite these predictions as constraints became available from studies of transiting planets the preponderance of low albedos was often treated as surprising when compared to the bright albedo of Jupiter.
Persuasive evidence has been found, however, indicating that at least one hot Jupiter has a large geometric albedo that is likely attributable to a haze or cloud layer. Kepler-7b (*Latham et al.*, 2010) has a geometric albedo averaged over the Kepler bandpass (423 to 897 nm, *Koch et al.,* 2010) of $0.32\pm 0.03$ (*Demory et al.,* 2011). Such a high albedo is more typical of the cloudy solar system giants (e.g., *Karkoschka,* 1994) than a deep scattering atmosphere. *Demory et al.* thus argue that the most likely explanation is that this particular planet has a bright photochemical haze or cloud layer, perhaps similar to that seen in the transit spectra of HD 189733b. More detailed modeling to constrain the properties of the scattering layer has not yet been done and to date transit spectra have not been obtained for Kepler 7b.
As albedos are measured for more planets with appropriate corrections for thermal emission, it may become apparent which particular combinations of planet mass, composition, and stellar incident flux (particularly including UV flux) conspire to produce high albedo planets.
Directly Imaged Planets
-----------------------
### Young Jupiters
Giant planets start their lives in a warm, extended state with luminosities much greater than at later times after they have cooled. For this reason giant planets are easier to detect at young ages of a few hundred million years or less and several have already been detected, including 2M 1207b and the planets orbiting the A star HR 8799 (*Marois et al.*, 2008; 2010). The near-infrared thermal emission of all of these objects (which have effective temperatures near 1000 K) points to the presence of substantial refractory cloud decks, most likely comprised of iron and silicate grains (e.g., *Marois et al.*, 2008; *Barman et al.*, 2011). Much of the literature on these objects has focused on constraining the properties of these clouds because in field brown dwarfs clouds have largely dissipated by 1000 K. *Currie et al.* (2011), *Madhusudhan et al.* (2011), and *Bowler et al.* (2010) also all explored models for the directly imaged planets and agreed that clouds played a critical role in shaping their thermal emission at a lower effective temperature than is typical for more massive field brown dwarfs.
*Marley et al.* (2012) also constructed model atmospheres for these planets and concurred that clouds are present in the atmospheres of HR 8799 b, c, and d. They applied the cloud model of *Ackerman & Marley* (2001) and found that they could reproduce much of the available data by using cloud parameters typically seen in warmer L dwarfs. They argue from mass balance considerations that mean cloud particle size likely varies inversely with $\sqrt{g}$ where *g* is the gravitational acceleration and that all else being equal the column optical depth of a cloud varies proportionately to $\sqrt{g}$. The net result of these scaling relationships is that clouds in lower gravity objects tend to be similar to clouds found at warmer effective temperatures in higher gravity, more massive brown dwarfs. They find that as objects cool their spectra can be modeled by increasing the sedimentation efficiency, *f*~sed~ in Equation (1). The effective temperature at which *f*~sed~ begins to increase varies with gravity such that lower gravity objects begin to lose their refractory clouds at warmer effective temperatures than low mass objects.
At cooler effective temperatures near 600 K another set of condensates become important in field dwarfs. *Morley et al.* (2012) have demonstrated that Na~2~S and other clouds (Figure 3) moderately redden the near-IR colors of late T type brown dwarfs. Once young giant planets in this effective temperature range are discovered it will be possible to test if the scaling relationships seen at higher effective temperature persist.
Ultimately understanding the nature of clouds in the warm, young giant planets hinges on understanding why the cloud clearing effective temperature varies as it does. In the next few years many more young giant planets are expected to be discovered by the upcoming GPI and SPHERE exoplanet surveys.
### Giants in Reflected Light
No giant planet has yet been imaged in reflected light although such an observation is less technically difficult than imaging a terrestrial planet in reflected starlight and a number of space telescope missions have been proposed. As with terrestrial planets, clouds play a large role in affecting the reflection spectra of a giant planet (*Marley et al.,* 1999; *Sudarsksy et al.,* 2002; *Cahoy et al.,* 2010).
One way of visualizing the effect of clouds in a giant planet atmosphere is to imagine a Jupiter twin lying at progressively closer distances to the sun (Figure 4). The optical geometric albedo spectrum of Jupiter generally consists of a bright continuum punctuated by methane absorption bands. The bright continuum is formed by scattering from gas and stratospheric hazes in the blue and the bright ammonia clouds at longer wavelengths. The methane bands are formed from the column of gas overlying the clouds. Below the ammonia cloud-tops lie cloud decks of NH~4~SH and H~2~O.
A Jupiter twin (with the same internal heat flux) lying closer to its star at 2AU would have a warmer atmosphere in which the ammonia and ammonium hydrosulphide clouds would not condense. Instead the planet would be covered by a bright global layer of water clouds which would give a very high albedo (Figure 1). At even closer distances to its star the atmosphere would be too warm for water clouds to form. Gas absorption thus overwhelms gaseous Rayleigh scattering and the planet becomes much darker (Figure 1) than the cloudless case.
Additional Topics
=================
Clouds of Low Porosity Aggregates
---------------------------------
Most of the discussion in this chapter has focused on cloud or haze particles as fully dense spheres. However fluffy or porous aggregate particles may very well form and such particles behave differently both as they interact with the atmosphere and with radiation. In this section we simplify the *Ackerman & Marley* (2001) approach to llustrate how low-porosity aggregates would behave. In this simplified version, we neglect the fraction of the condensible species present as vapor above the cloud base, so $q_t$= $q_c$. We stress that this treatment is not a replacement for the complete model.
When the particles are too small to settle (see below) the condensate mixing ratio $q=q_c=\rho _c/\rho _g$ is constant and so has an infinite scale height $H_q$ (the condensate has the same scale height as the atmosphere). However, when the cloud is significantly settled such that the condensate mixing ratio scale height $H_q{\ll}H,$ and if it is assumed that the vapor mixing ratio is negligible compared to the condensate ratio then Equation governing the vertical cloud distribution can be approximated as $K_{\mathit{zz}}\mathit{dq}/\mathit{dz}-v_fq=0$, where *K~zz~* = *Lw\** is the vertical eddy diffusion coefficient, a property of macroscopic turbulence with typical lengthscale *L* and large (energy-containing) eddy velocity *w\*; L* is usually taken to be the atmospheric scale height *H,* although this might not be valid if the associated large eddy timescale *L/w\** is much longer than the rotation period of the object (*Schubert & Zhang*, 2000) or the temperature profile is stable (*Ackerman & Marley* use an ad hoc stability correction). Then, for a settled cloud with uniform mass mixing ratio *q* and effective thickness *H~q~*, the above equation is approximated by $K_{\mathit{zz}}q/H_q-v_fq=0$, thus the cloud thickness is roughly $H_q=Hw^* / v_f$. The scale height in the simplified exponential solution for condensate mass mixing ratio of *Ackerman & Marley* (2001) is essentially the same. That is, when the convective eddy velocity is much larger than the settling velocity, the cloud particles are not able to settle ( $f_{\mathit{sed}}=0$).
Particle settling velocities depend upon the dynamical regime. The settling velocity *v~f~* is the product of the local gravity *g* and the gas drag stopping time of the particle, *t~s~.* *Ackerman & Marley* (2001), equation B2, contains a bridging expression *b* that covers both the so-called Stokes and Epstein drag regimes, in which *r* is respectively greater than, or less than, the gas molecule mean free path $l_m=m_{H_2}/\sigma _{H_2}\rho _g$, where $m_{H_2}$ and $\sigma _{H_2}$ are the mass and cross section of a hydrogen molecule and $\rho _g$ is the gas density. In equations B1 and B3 of *Ackerman & Marley* (2001) the density contrast is essentially the particle density because ** $\rho {\gg}\rho _g$ for all applications of interest. So, *v~f~* is proportional to *r*, and for particles with porosity $\varphi $, $\rho =\rho _s\left(1-\varphi \right)=\rho _sf$. In the Epstein regime (*r/l~m~* *[<]{} 1)* the expression for *t~s~* is very simple: $t_s=r\rho /c\rho _g$ where *c* is the sound speed. *Cuzzi & Weidenschilling* (2006) show how the stopping time in the Stokes regime (*r/l~m~* *[>]{} 1)* is essentially larger by a factor of *r/l~m~*.
We can thus show that for porous particles under local gravity *g*, having the same mass as a particle with no porosity, $v_f=\mathit{gr}\rho /c\rho _g=gr_s\rho _s\left(1-\varphi \right)^{2/3}/c\rho _g$, and thus $H_q=Hw^*/v_f=H_{q\left(\mathit{solid}\right)}\left(1-\varphi \right)^{-2/3}$. Porous particles are lofted to greater heights than non-porous particles of the same mass, because of their slower settling velocity. In this sense they behave like smaller particles; however, their radiative behavior is not that of smaller particles, as noted above; porous particles in the regime $r/\lambda {\gg}1$ * *indeed have large and wavelength-independent opacity.
Polarization
------------
Polarization may provide an additional avenue for characterizing cloudy planets. Unlike the stellar radiation emitted by the central star, the light scattered by clouds will in general be polarized. Thus, investigating the polarized radiation scattered by clouds in contrast to the non-polarized stellar radiation may be an opportunity for characterizing cloudy exoplanetary atmospheres at short wavelengths in the future (see *Stam* 2008).
Giant planets can also be polarized in the thermal infrared. In this case the required asymmetry in radiation emitted across the apparent planetary disk must be provided either by rotational flattening or irregularities in the global cloud cover. *Marley & Sengupta* (2011) investigated the former mechanism and *de Kok et al.* (2011) the latter. Both sets of authors found that in the most favorable circumstances polarization fractions of a few percent were plausible and that in such cases polarization confirms the presence of a scattering condensate layer.
While polarization undoubtedly provides additional constraints on cloud particle sizes (e.g., *Bréon & Colzy*, 2000), condensate phase (e.g., *van Diedenhoven et al*., 2012a), and even asymmetry parameter (*van Diedenhoven et al.,* 2012b) and atmospheric structure, it may in practice be of limited value for studies of exoplanets. Within the solar system the most well known discovery attributable to a polarization measurement is the particle size of the clouds of Venus (*Hansen & Hovenier*, 1974). An imaging photopolarimeter carried on the Pioneer 10 and 11 spacecraft also constrained the vertical structure of Jupiter’s clouds (*Gehrels et al.,* 1974). However in general other techniques have proven more valuable. Especially given the low signal to noise and difficulty inherent in any measurement of an extrasolar planet, the value of further dividing the light into polarization channels must be weighed against other potential measurements (e.g., obtaining higher resolution spectroscopy).
Conclusions
===========
As with the planetary atmospheres of solar system planets, clouds are expected to play major roles in the vertical structure, chemistry, and reflected and emitted spectra of extrasolar planets. That said, the most extensive experience to date with clouds outside of the solar system has been with the brown dwarfs. The presence of refractory clouds in L dwarfs and sulphide and salt clouds in late T dwarfs has been well established and a number of methods have been developed to model these clouds.
Compared to cloud modeling approaches within the solar system or particularly on Earth, exoplanet cloud models are still in their infancy. In many cases arbitrary clouds are employed that specify a range of plausible cloud properties that are usually sufficient to explore parameter space. More sophisticated efforts attempt to derive particle sizes, composition, and number density as a function of height through the atmosphere from a given set of assumptions will be needed once higher resolution, broad wavelength spectral data become available.
As of the time of this chapter’s writing the best evidence for clouds in extrasolar planet atmospheres lies in the spectra of the planets orbiting the nearby A star HR 8799. Planets b, c, and d each have red near-infrared colors that are best explained by global refractory cloud decks that have persisted to lower effective temperatures than in higher mass field brown dwarfs. This persistence of clouds to lower effective temperatures for lower gravity objects continues a trend that has already been recognized among brown dwarfs.
Among the transiting planets there is convincing evidence for high altitude clouds--or perhaps a photochemical haze--in the atmosphere of HD 189733b. The transit spectra of this planet lacks the deep absorption bands expected for a clear atmosphere of absorbing gas but rather exhibits a smoothly varying absorption profile likely caused by small grain scattering. A second planet, GJ 1214b also has a transit spectrum lacking absorption features. In this case the spectrum may be attributable either to a small atmospheric scale height resulting from a high mean molecular weight composition or from clouds.
These early detections are like distant clouds seen at sunset presaging a coming storm. The GPI, SPHERE, and many other direct imaging planet searches are expected to discover dozens of young, self-luminous extrasolar giant planets over the coming decade (*Oppenheimer & Hinkley*, 2009; *Traub & Oppenheimer,* 2010). Many of these planets will exist in the effective temperature range in which clouds shape their emergent spectra. Meanwhile continued studies of transiting planets, particularly by the upcoming James Webb Space Telescope will probe the atmospheres of the transiting planets and test the hypothesis that hazes of photochemical origin are important in some regimes. Finally future space based coronagraphic telescopes may eventually image extrasolar giant and terrestrial planets in reflected light. All such efforts to discover and characterize extrasolar planets will hinge upon an understanding of the role clouds play in shaping the climate and atmospheres of planets.
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|
---
abstract: 'The Wald test statistic has been shown to diverge (Dufour et al, 2013, 2017) under some conditions. This note links the divergence to eigenvalues of a polynomial matrix and establishes the divergence rate.'
author:
- 'Jean-Marie Dufour[^1]'
- 'Eric Renault [^2]'
- 'Victoria Zinde-Walsh[^3]'
title: 'A technical note on divergence of the Wald statistic[^4]'
---
The set-up and an example of divergence
=======================================
Suppose that a $p\times 1$ parameter of interest $\bar{\theta}$ satisfies
$$H_{0}:g\left( \theta \right) =0,$$
where $g\left( \theta \right) $ is a $q\times 1$ vector of differentiable functions; $g\left( \theta \right) =\left( g_{1}\left( \theta \right)
,...,g_{q}\left( \theta \right) \right) ^{\prime };$ $q\leq p.$
Let $V$ be a symmetric positive definite matrix.
**Assumption 1.** *In some open set* $\Theta \subset R^{p}$*there is a random sequence* $\hat{\theta}_{T}\in \Theta $* and* $p\times p$* random matrix sequence,* $\hat{V}_{T},$* such that as* $T\rightarrow \infty $* *$$\begin{aligned}
\sqrt{T}V^{-\frac{1}{2}}\left( \hat{\theta}_{T}-\bar{\theta}\right)
&\rightarrow &_{d}Z; \\
Z &\sim &N\left( 0,I_{p}\right) ; \\
\text{\textit{and} }\hat{V}_{T} &\rightarrow &_{p}V.\end{aligned}$$Define the usual Wald test statistic:$$W_{T}=Tg^{\prime }(\hat{\theta}_{T})\left[ \frac{\partial g}{\partial \theta
^{\prime }}(\hat{\theta}_{T})\hat{V}_{T}\frac{\partial g^{\prime }}{\partial
\theta }(\hat{\theta}_{T})\right] ^{-1}g(\hat{\theta}_{T}).
\label{wald stat}$$
For linear $g$ the statistic converges to a $\chi _{q}^{2}$ distribution, for other, e.g. polynomial restrictions the limit distribution may be not $\chi ^{2}.$ The limit results for the statistic for testing general polynomial restrictions can be found in (Dufour et al., 2013, 2017); it is also established there that under some conditions the statistic may diverge when $q>1$. Below is an example of divergence.
**Example 1. Restrictions for which the Wald statistic diverges.**
*Consider for* $\theta =\left( \mathbf{x,y,z,w}\right) ^{\prime }$* the set of restrictions,* $H_{0}:$$$\left\{
\begin{array}{ccc}
\mathbf{xy} & = & 0; \\
\mathbf{xw} & = & 0; \\
\mathbf{yz} & = & 0.\end{array}\right.$$
*Then the Wald statistic for* $\hat{\theta}_{T}=\left( x,y,z,w\right)
,$* assuming that the covariance matrix is identity* $\hat{V}=V=I_{4},$* is* $\ $$$W=T\left( w^{2}+y^{2}\right) \frac{x^{2}+z^{2}}{w^{2}+x^{2}+y^{2}+z^{2}}.$$
*Suppose that the true parameter value is* $\bar{\theta}=(0,0,1,1);$* *$H_{0}$* then holds. Suppose that the estimated parameter* $\hat{\theta}_{T}=\left( x,y,z,w\right) $* as* $T\rightarrow \infty
$* is consistent and satisfies* $$z\rightarrow _{p}1;w\rightarrow _{p}1;T^{\frac{1}{2}}x\rightarrow
_{p}Z_{1};T^{\frac{1}{2}}y\rightarrow _{p}Z_{2}$$*where* $Z_{1},Z_{2}$* are independent standard normals*$.$* Then the Wald statistic can be expressed as*$$W=\left( T+Z_{2}^{2}+o_{p}\left( 1\right) \right) \frac{Z_{1}^{2}+T+o_{p}\left( 1\right) }{2T+Z_{1}^{2}+Z_{2}^{2}+o_{p}\left( 1\right) }.$$
*This is*
$$\mathit{W=T+O}_{p}\left( 1\right) \mathit{.}$$
*As* $T\rightarrow \infty $ *the statistic diverges under* $H_{0}.$
We shall assume that each $g_{l}\left( \theta \right) $ is a polynomial of order $m_{l}$ in the components of $\theta .$ Then for any $\bar{\theta}$ each polynomial component $g_{l}\left( \theta \right) ,$ can be written around $\bar{\theta}$ as$$g_{l}(\theta )=\sum_{\gamma =0}^{m_{l}}\sum_{j_{1}+...+j_{p}=\gamma
}c_{l}(j_{1},...,j_{p},\bar{\theta})\dprod\limits_{k=1}^{p}\left( \theta
_{k}-\bar{\theta}_{k}\right) ^{j_{k}} \label{power}$$with some coefficients $c(j_{1},...,j_{p},\bar{\theta}).$
If the value $\bar{\theta}$ satisfies the null hypothesis, then$$c_{l}(0,...,0,\bar{\theta})=0$$for each $l.$
A polynomial function is eiher identically zero or non-zero a.e. with respect to the Lebesgue measure. Consider a square matrix $G(y)$ of polynomials of variable $y\in \mathbb{R}^{p}.$ We say that the polynomial matrix $G(y)$ is non-singular if its determinant is a non-zero polynomial.
The rank of the $q\times p$ matrix $G(y)$ is the largest dimension of a square non-singular submatrix.
Unlike matrices of constants for polynomial matrices the rows may be linearly independent vectors of polynomial functions, while the matrix may have defficient rank. For example, in the matrix$$\left(
\begin{array}{cc}
y & 0 \\
y^{2} & 0\end{array}\right)$$the two rows are given by independent vectors of polynomials, but the rank of this matrix of polynomials is one.
**Assumption 2.** *The* $q\times 1$* function* $g\left(
\theta \right) $* is a vector of polynomial functions; the matrix of polynomials* $G\left( \theta \right) =\frac{\partial g}{\partial \theta
^{\prime }}\left( \theta \right) $* is of rank* $q.$
This does not exclude the possibility of reduced rank at some particular point or on a low dimensional space.
Under the stated assumptions for $g\left( \bar{\theta}\right) =0$ the standard asymptotic $\chi _{q}^{2}$ distribution holds for $W_{T}$ as long as $G\left( \bar{\theta}\right) =\frac{\partial g}{\partial \theta ^{\prime }}(\bar{\theta})$ is a (numerical) matrix of rank $q.$
Each restriction $g_{l}\left( \theta \right) $ can be represented as a sum $$g_{l}\left( \theta \right) =\bar{g}_{l}\left( \theta -\bar{\theta}\right)
+r_{l}\left( \theta -\bar{\theta}\right) , \label{decomp g}$$where $\bar{g}_{l}\left( .\right) $ denotes the lowest degree non-zero homogeneous polynomial and has degree $\bar{\gamma}_{l}+1,$ the degrees of all non-zero monomials in $r_{l}\left( .\right) $ are $\bar{\gamma}_{l}+1.$ We ascribe the degree of homogeneity $\infty $ to a function that is identically zero.
Correspondingly to $\left( \ref{decomp g}\right) $, in the matrix $$G\left( \theta \right) =\frac{\partial g}{\partial \theta ^{\prime }}(\theta
)$$for each row write $$G_{l}\left( \theta \right) =\bar{G}_{l}\left( \theta -\bar{\theta}\right)
+R_{l}\left( \theta -\bar{\theta}\right) ,$$the degree of any non-zero homogenious polynomial in the row vector, $\bar{G}_{l}\left( \theta -\bar{\theta}\right) ,$ is $\bar{\gamma}_{l}$; any non-zero monomial in $R_{l}\left( \theta -\bar{\theta}\right) $ has degree higher than $\gamma _{l}.$ Then collecting the lowest degree homogeneous polynomials in each row we have $$G\left( \theta \right) =\bar{G}\left( \theta -\bar{\theta}\right) +R\left(
\theta -\bar{\theta}\right) . \label{decomp G}$$
**The property of full rank reached at lowest degrees (FRALD) and FRALD-T**
===========================================================================
**Definition (FRALD).** *If the matrix* $\bar{G}\left(
\theta -\bar{\theta}\right) $* of lowest degree polynomials for* $g\left( \theta \right) $* is of full rank* $q$* we say that the Full Rank at Lower Degrees (FRALD) property is satisfied for* $g\left(
.\right) $* and* $\bar{\theta}.$
Examples in Dufour et al (2017) illustrate the possibilities that the FRALD property may hold at some points $\bar{\theta},$ but not others, and that even if FRALD property does not hold for $g\left( .\right) $ at $\bar{\theta},$ it may hold for $Sg\left( .\right) ,$ where $S$ is a non-degenerate numerical matrix.
Recall that the distribution of the Wald statistic is invariant with respect to non-degenerate linear transformation of the restrictions.
**Definition (FRALD-T).** *There exists some numerical non-degenerate matrix* $S$* such that FRALD holds for* $Sg\left(
\theta \right) $* at* $\bar{\theta}.$
If FRALD-T holds for $g,$ then for some $S$ FRALD holds for $Sg,$ meaning that $\overline{SG}\left( .\right) $ is a full rank matrix of polynomials.
It is shown in Dufour et al (2017) that for polynomial $g\left( x\right) $ with a full rank matrix $G\left( x\right) $ there always exists a non-degenerate numerical matrix $S,$ such that $SG\left( x\right) $ has the property that $\overline{SG}\left( x\right) $ has all the rows represented by linearly independent vectors of polynomials (each row contains non-zero homogeneous polynomials); these rows could be stacked by a permutation in an “eschelon form”, with the degrees of the non-zero homogeneous polynomials in non-decreasing order.
The eschelon form is given by$$\overline{SG}\left( x\right) =\left[
\begin{array}{c}
\left[ \overline{SG}\left( x\right) \right] _{1} \\
\vdots \\
\left[ \overline{SG}\left( x\right) \right] _{i} \\
\vdots \\
\left[ \overline{SG}\left( x\right) \right] _{v}\end{array}\right] , \label{eschelon}$$where $\left[ \overline{SG}\left( x\right) \right] _{i}$ has dimension $n_{i}\times q,$ all non-zero polynomials in $\left[ \overline{SG}\left(
x\right) \right] _{i}$ have degree $\bar{s}_{i}$ for $i=1,\ldots ,\,\nu $, with $0\leq \bar{s}_{1}<\cdots <\bar{s}_{i}<\cdots <\bar{s}_{\nu },$ and all the rows of $\overline{SG}\left( x\right) $ are linearly independent functions. Once any $S$ that provides such a structure is found, the rank of $\overline{SG}\left( \theta \right) $ is either $q,$ and FRALD-T holds, or is less than $q,$ in which case this property is violated. An algorithm to find $S$ is provided in Dufour et al (2017).
**Example 2 (Example 1 continued). FRALD-T does not hold.**
*Take for* $\theta ^{\prime }=\left( x;y;w;z\right) $* the function* $g\left( \theta \right) =\left(
\begin{array}{c}
xy \\
xw \\
yz\end{array}\right) .$* With* $\bar{\theta}^{\prime }=\left( 0,0,1,1\right) $* denote* $\theta ^{\prime }=\left( x;y;1+\tilde{w};1+\tilde{z}\right) $* with* $\sqrt{T}\left( x;y;\tilde{w};\tilde{z}\right)
^{\prime }$* converging to* $N\left( 0,I_{4}\right) .$* Then* $$G\left( \theta \right) =\left(
\begin{array}{cccc}
y & x & 0 & 0 \\
1+\tilde{w} & 0 & x & 0 \\
0 & 1+\tilde{z} & 0 & y\end{array}\right) ;$$*by applying a transformation (here permutation),* $P,$* to the rows of this matrix we get* $$\begin{aligned}
PG\left( \theta \right) &=&\left(
\begin{array}{cccc}
1+\tilde{w} & 0 & x & 0 \\
0 & 1+\tilde{z} & 0 & y \\
y & x & 0 & 0\end{array}\right) ; \\
&&\text{\textit{with the eschelon form}} \\
\overline{PG}\left( \theta \right) &=&\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
y & x & 0 & 0\end{array}\right) .\end{aligned}$$*The matrix* $\overline{PG}\left( \theta \right) $* has independent polynomial row vectors, and the rows are stacked so that the degrees of “leading” polynomials do not decline from row to row (eschelon form). The rank of the matrix* $\overline{PG}\left( \theta \right) $* is not full in an eschelon form, no linear transformation applied to* $G$ *can remedy this rank defficiency. So FRALD-T does not hold for this example.*
In Dufour et al (2014, 2017) the limit distribution for the Wald statistic was established for $\bar{\theta}$ when the FRALD-T property holds.
The example 1 here of the case where the statistic was shown to diverge does not satisfy FRALD-T. The next section demonstrates the mechanism whereby the violation of the FRALD-T property leads to divergence of the statistic.
Divergence of the Wald statistic where FRALD-T does not hold
============================================================
Assume that for $\bar{\theta}$ for which the null is satisfied, $g\left(
\bar{\theta}\right) =0,$ the FRALD-T property does not hold. Without loss of generality we may assume that $G\left( x\right) =\frac{\partial g}{\partial
x^{\prime }}$ is such that the eschelon form $\left( \ref{eschelon}\right) $ applies to $\bar{G}\left( x\right) $ (so that $S$ in FRALD-T and in $\left( \ref{eschelon}\right) $ is identity).
Denote by $Y$ the Gaussian limit $\sqrt{T}(\hat{\theta}-\bar{\theta})\rightarrow _{d}Y\sim N\left( 0,V\right) .$ Define $\Delta _{T}:=\mathrm{diag}[T^{s_{1}/2}I_{n_{1}},\ldots ,\,T^{s_{q}/2}I_{n_{v}}].$ With the scaling $\Delta _{T}$ we get
$$T^{1/2}\Delta _{T}\,g(\hat{\theta}_{T})\underset{T\rightarrow \infty }{\overset{d}{\rightarrow }}\bar{g}\left( Y\right) \,, \label{Delta}$$
$$\Delta _{T}\,G(\hat{\theta}_{T})\hat{V}_{T}G(\hat{\theta}_{T})^{\prime
}\Delta _{T}\underset{T\rightarrow \infty }{\overset{d}{\rightarrow }}\bar{G}\left( Y\right) V\bar{G}\left( Y\right) ^{\prime }\,, \label{Delta2}$$
where $\bar{G}\left( Y\right) V\bar{G}\left( Y\right) ^{\prime }$ has rank $r<q$ when FRALD-T does not hold. Consequently, inverting the consistent estimator $\left[ \Delta _{T}\,G(\hat{\theta}_{T})\hat{V}_{T}G(\hat{\theta}_{T})^{\prime }\Delta _{T}\right] $ for $$W=T^{1/2}\Delta _{T}\,g(\hat{\theta}_{T})^{\prime }\left[ \Delta _{T}\,G(\hat{\theta}_{T})\hat{V}_{T}G(\hat{\theta}_{T})^{\prime }\Delta _{T}\right]
^{-1}T^{1/2}\Delta _{T}\,g(\hat{\theta}_{T})$$will lead to an explosion as $T\rightarrow \infty .$
We next examine the matrix $\bar{\Sigma}_{T}\left( \hat{\theta}_{T},V_{T}\right) =\left[ \Delta _{T}\,G(\hat{\theta}_{T})\hat{V}_{T}G(\hat{\theta}_{T})^{\prime }\Delta _{T}\right] $ and its limit eigenvalues which provide the key ingredient to prove the divergence of the Wald statistic.
Denote by $\bar{\lambda}_{1T}\left( \hat{\theta}_{T}\right) ,\bar{\lambda}_{2T}\left( \hat{\theta}_{T}\right) ,...,\bar{\lambda}_{qT}\left( \hat{\theta}_{T}\right) $ the eigenvalues of the matrix $\bar{\Sigma}_{T}\left( \hat{\theta}_{T},V_{T}\right) ,$ arranged in decreasing order: $\bar{\lambda}_{1T}\left( \hat{\theta}_{T}\right) \geq \bar{\lambda}_{2T}\left( \hat{\theta}_{T}\right) \geq ...\geq \bar{\lambda}_{qT}\left( \hat{\theta}_{T}\right) $ and denote by $$\bar{\Lambda}_{T}=diag\left[ \bar{\lambda}_{1T}\left( \hat{\theta}_{T}\right) ,\bar{\lambda}_{2T}\left( \hat{\theta}_{T}\right) ,...,\bar{\lambda}_{qT}\left( \hat{\theta}_{T}\right) \right] \label{diag}$$ the $q\times q$ diagonal matrix of these eigenvalues.
We prove several auxilliary results about eigenvalues of non-random polynomial matrices (proofs are in the next section).
Start with $B\left( x,U\right) =G(x)UG(x)^{\prime },$ where $G\left(
x\right) $ is a non-zero matrix of polynomial functions and define the characteristic polynomial, $p\left( \lambda ;B\left( x,U\right) \right)
=\det \left[ \lambda I_{p}-B\left( x,U\right) \right] $.
The next proposition describes a polynomial representation for the coefficients of $p\left( \lambda ;B\left( x,U\right) \right) $ as a polynomial in $\lambda $. Denote by $F_{p}$ the set of all real symmetric positive-definite matrices.
**Proposition 1.** *Let* $x\in R^{p}$* and* $G\left(
x\right) $* be a* $q\times p$* non-zero matrix of polynomial functions in* $x$* such that rank*$\left[ G\left( x\right) \right]
=q $* a.e.,* $U\in F_{p}$*,* $B\left( x,U\right)
=G(x)UG(x)^{\prime }$* and* $p_{B}\left( \lambda ;x,U\right) =\det \left[ \lambda I_{p}-B\left( x,U\right) \right] $* is the characteristic polynomial. Then* $p_{B}\left( \lambda ;x,U\right) $*can be written as*$$p_{B}\left( \lambda ;x,U\right) =\lambda ^{q}+\Sigma _{k=0}^{q}a_{k}\left(
x,U\right) \lambda ^{q-k}, \label{charact polyn}$$*where the coefficients* $a_{k}\left( x,U\right) $* have the following polynomial expansions*$$a_{k}\left( x,U\right) =D_{m_{k}\left( U\right) }\left( x,U\right) +\tilde{R}_{k}\left( x,U\right) , \label{Dm(k)}$$*where* $D_{m_{k}\left( U\right) }\left( x,U\right) $* is a homogeneous in* $x$* polynomial of degree* $m_{k}\left( U\right) ,$* and* $\tilde{R}_{k}\left( x,U\right) $* is a sum of polynomials with any non-zero mononomials of degree strictly greater than* $m_{k}\left( U\right) .$* Further, if* $m_{k}=\underset{Q\in F_{p}}{\min }m_{k}\left( Q\right) ,$* then* $m_{k}\left( U\right) =m_{k}$* for almost all* $U\in F_{p}.$
**Example 3 Restrictions of Example 1 but with a covariance matrix** $U$** for which** $m_{k}\left( U\right) >m_{k}.$
*In the example 1 we had divergence at the rate* $T$* when the matrix* $U$* was identity. The characteristic polynomial for the same restrictions,* $\mathit{g}\left( .\right) $* of example 1 with a covarince matrix* $U,$* possibly different from* $I,$ *has as the* $q-th$* (here for* $q=3)\,$* coefficient the determinant of* $G(x)UG(x)^{\prime }.$* With* $U=I$* the determinant of* $G(x)IG(x)^{\prime }$* provides* $$a_{3}\left( x,I\right)
=w^{2}x^{2}y^{2}+2wx^{2}y^{2}+x^{4}y^{2}+x^{2}y^{4}+x^{2}y^{2}z^{2}+\allowbreak 2x^{2}y^{2}z+2x^{2}y^{2},$$*and we note that the lowest degree monomial is* $x^{2}y^{2}.$* It can be verified that for these restrictions and* $\bar{\theta}$* we get* $m_{3}=4$* (so that there can be no* $U$* for which the degree could be smaller) and by Proposition 1* $m_{3}\left(
U\right) =m_{3}=4$* for almost every* $U\in F_{q}.$ *However, below we provide* $U$* for which* $a_{3}\left( x,U\right) $ is such that $m_{3}\left( U\right) >4.$* Consider*$$U=\left(
\begin{array}{cccc}
1 & \sqrt{.98} & 0 & 0 \\
\sqrt{.98} & 1 & .1 & .1 \\
0 & .1 & 1 & 0 \\
0 & .1 & 0 & 1\end{array}\right) .$$*For this* $U$* we get* $$\begin{aligned}
a_{3}\left( x,U\right) &=&0.01w^{2}x^{2}y^{2}-0.197\,99\allowbreak
wx^{3}y^{2}-0.2wx^{2}y^{3}-0.02wx^{2}y^{2}z+0.98x^{4}y^{2} \\
&&+1.\,\allowbreak 979\,9\allowbreak x^{3}y^{3}+0.197\,99x^{3}\allowbreak
y^{2}z+x^{2}y^{4}+0.2x^{2}y^{3}z+0.01x^{2}y^{2}z^{2}\end{aligned}$$*with the lowest degree of monomial* $m_{3}\left( U\right)
=6>m_{3}=4. $
Since the coefficients of a polynomial represent symmetric polynomials in the roots (e.g., Horn and Johnson, 1985, Section 1.2), elementary symmetric polynomials in the eigenvalues can be expressed as polynomial functions in $x.$ Denote by $I_{q}\left( k\right) $ the set of all combinations of $k$ integers out of $\left\{ 1,...,q\right\} .$ Denote by $P_{k}\left( \lambda
_{1},...\lambda _{q}\right) $ the $k-th$ elementary symmetric polynomial in $\lambda _{1},...\lambda _{q}:$$$P_{k}\left( \lambda _{1},...\lambda _{q}\right) =\Sigma _{\left\{
i_{1},...,i_{k}\right\} \in I_{q}\left( k\right) }\Pi _{j=1}^{k}\lambda
_{i_{j}}.$$
**Corollary to Proposition 1.** *For the eigenvalues* $\lambda
_{i}\left( x,U\right) ,$* *$i=1,...,q,$* that are the solutions of the characteristic polynomial, we have that* $$P_{k}\left[ \lambda _{1}\left( x,U\right) ,...\lambda _{q}\left( x,U\right) \right] =\left( -1\right) ^{k}a_{k}\left( x,U\right) \label{sym polyn}$$*and thus the representation* $\left( \ref{Dm(k)}\right) $*applies.*
In the next proposition we apply scaling to the argument $x$ by considering $x=T^{-1/2}y$ and exploit the polynomial terms from $\left( \ref{Dm(k)}\right) $ with lowest degree of homogeneity in $\left( \ref{sym polyn}\right) $ to establish the rates for the eigenvalues of a scaled polynomial matrix. Recall that from the convergence result $\left( \ref{Delta2}\right) $ the matrix scaling $\Delta _{T}$ is associated with $G\left( .\right) .$ We show that when rank of $\bar{G}\left( .\right) $ is less than $q$ (in violation of the FRALD-T condition) some eigenvalues will be converging to zero and additional scaling can be applied to have the eigenvalues converge to continuous limit functions. This additional scaling will provide the divergence rate.
**Proposition 2.** *Under the conditions of Proposition 1 consider the scaled matrix for* $y\in R^{p}:$* *$$M_{T}\left( y,U\right) =\Delta _{T}G\left( T^{-1/2}y\right) UG\left(
T^{-1/2}y\right) ^{\prime }\Delta _{T}$$*and its eigenvalues* $\bar{\lambda}_{l}^{\left( T\right) }\left(
y,U\right) =\bar{\lambda}_{l}^{\left( T\right) }\left( T^{-1/2}y,U\right) ,$* *$l=1,...,q$* in descending order. Then for some non-negative integers* $\beta _{l}=\beta _{l}\left( U\right) $* that satisfy*$$\begin{aligned}
\beta _{l} &=&0\text{ for }1\leq l\leq r; \\
\beta _{l} &\geq &1\text{ for }l>r,\end{aligned}$$*we have that*$$T^{\beta _{l}}\bar{\lambda}_{l}^{\left( T\right) }\left( y,U\right)
\rightarrow \lambda _{l}\left( y,U\right) \text{ for almost all }y\in
R^{p},l=1,...,q,$$*where* $\lambda _{l}\left( y,U\right) $* are continuous a.e. non-zero functions.*
Thus we see that for eigenvalues beyond $r$ additional non-trivial scaling provides convergence to a continuous a.e. non-zero function.
The next proposition shows that convergence with these rates to a continuous (but now in some exceptional cases possibly zero) function is preserved when $U$ is replaced with a sequence $U_{T},$ of matrices from $F_{p}$ such that $U_{T}\rightarrow U.$
**Proposition 3.** *Under the conditions of Proposition 2 consider a sequence* $U_{T}\in F_{p},$* such that* $U_{T}\rightarrow
U$*. Then*
*(a) if* $m_{l}\left( U\right) =m_{l}$* for* $l=1.,,,q$* we have that*$$\begin{aligned}
T^{\beta _{l}}\bar{\lambda}_{l}^{\left( T\right) }\left( y,U_{T}\right)
&\rightarrow &\lambda _{l}\left( y,U\right) \text{ for almost all }y\in
R^{p},l=1,...,q; \\
\lambda _{l}\left( y,U\right) &>&0\text{ a.e.;}\end{aligned}$$
*(b) if for some* $k\in \left\{ 1,...,q\right\} $* it holds that* $m_{l}\left( U\right) =m_{l}$* for* $l<k$* and* $m_{k}<m_{k}\left( U\right) ,$* then* $$T^{\beta _{l}}\bar{\lambda}_{l}^{\left( T\right) }\left( y,U_{T}\right)
\rightarrow 0\text{ with }\beta _{l}=\frac{1}{2}\left( m_{k}-m_{k-1}\right)
\geq 1\text{ for }l\geq k.$$Recall that case (a) will hold for almost all $U$ by Proposition 1. Example 3 illustrates part (b) of Proposition 2: there $m_{3}=4,$ so $\beta _{3}=2,$ but $m_{3}\left( U\right) =6>4$ and thus $\lambda _{3}^{\left( T\right)
}\left( T^{-1/2}y,U_{T}\right) $ scaled up by $T^{2}$ goes to zero. It is possible that for $U_{T}$ itself $m_{3}\left( U_{T}\right) =6;$ in that case $T^{3}\lambda _{3}^{\left( T\right) }\left(
T^{-1/2}y,U_{T}\right) $ converges to a non-zero limit. Alternatively, if $U_{T}$ has $m_{3}\left( U_{T}\right) =m_{3}$ for almost all $U_{T}$ but $U_{T}$ converges to $U$ sufficiently fast the rate could still be as high as $T^{3}.$ However, in case (b) to get a precise rate we also need to consider the convergence rate for $U_{T}.$
The next proposition applies the deterministic properties to provide limits for eigenvalues of the random matrix $\bar{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) ,$ in the diagonal eigenvalue matrix $\bar{\Lambda}_{T}\left( \hat{\theta}_{T}\right) $ of $\left( \ref{diag}\right) $. Without loss of generality consider $\bar{\theta}=0.$
**Proposition 4.** *Suppose that assumptions 1,2 hold at* $\bar{\theta}=0.$* Then there is a sequence of integers* $\beta
_{l},l=1,...,q,$* which depends on* $G$* and* $V,$*such that*$$\begin{aligned}
\beta _{l} &=&0\text{ for }l=1,..,r, \\
\bar{\beta} &=&\underset{r<l\leq q}{\max }\beta _{l}\geq 1; \\
\tilde{\Delta}_{T} &=&diag\left[ T^{\beta _{1}/2},...,T^{\beta _{q}/2}\right]
; \\
&&\tilde{\Delta}_{T}\bar{\Lambda}_{T}\left( \hat{\theta}_{T}\right) \tilde{\Delta}_{T}\underset{d}{\rightarrow }diag\left[ \lambda _{1}\left( Y\right)
,...,\lambda _{T}\left( Y\right) \right]\end{aligned}$$*with all* $\lambda _{l}\left( y\right) $* continuous non-negative functions a.e..*
We see that if FRALD-T were not violated, no additional scaling would be required, but once it is violated the extra scaling is captured by $\bar{\beta}_{l}\geq 1$ for $r<l\leq q$ that determines the rate of explosion of the Wald statistic. The Theorem below shows this.
**Theorem.** *Under the conditions of Proposition 4 if FRALD-T property does not hold, i.e.* $r<q,$* then we have for* $\bar{\beta}\geq 1$* that* $$W_{T}=W_{T}\left( \hat{\theta}_{T};g,V_{T}\right) >T^{\bar{\beta}}\mu
_{T}\left( \hat{\theta}_{T},V_{T}\right) ,$$*where* $\mu _{T}\left( \hat{\theta}_{T},V_{T}\right) \underset{d}{\rightarrow }\mu \left( Y\right) ,$* a continuous positive a.e. function.*
We thus see that if FRALD-T is violated the rate of the exlosion is at least $T$ (as in example 1 here), but could be stronger even with the same restrictions (as could be in example 3).
Proofs
======
**Proof of Proposition 1.**
First, consider the polynomial expansion for $p_{B}\left( \lambda
;x,U\right) $, given e.g. in Harville (2008, Corollary 13,7,4). Denote by $I_{q}\left( k\right) $ the set of all combinations of $k$ integers out of $\left\{ 1,...,q\right\} ;$ denote for $\left\{ i_{1},...,i_{r}\right\} \in
I_{q}\left( r\right) $ by $B^{\left\{ i_{1},...,i_{r}\right\} }$ a minor of the matrix $B$ obtained by striking out all the rows and columns numbered $i_{1},...,i_{r}.$ Then $$\begin{aligned}
P_{B}\left( \lambda ;x,U\right) &=&\Sigma _{r=0}^{q}\left( -1\right)
^{q-r}\lambda ^{r}\Sigma _{\left\{ i_{1},...,i_{r}\right\} \in I_{q}\left(
r\right) }\det \left[ B\left( x,U\right) ^{\left\{ i_{1},...,i_{r}\right\} }\right] ; \\
a_{k}\left( x,U\right) &=&\left( -1\right) ^{q-k}\Sigma _{\left\{
i_{1},...,i_{k}\right\} \in I_{q}\left( k\right) }\det \left[ B\left(
x,U\right) ^{\left\{ i_{1},...,i_{k}\right\} }\right] .\end{aligned}$$Since all the components of the $B\left( x,U\right) $ matrix are polynomials in $x$ it follows that the determinants of the minors are also polynomials in $x.$ Then, given $U,$ denote by $D_{m_{k}\left( U\right) }\left(
x,U\right) $ the homogeneous polynomial in $a_{k}\left( x,U\right) $ of the lowest degree, denoted $m_{k}\left( U\right) ,$ to obtain the polynomial expansion of the Proposition.
Next, note that $a_{k}\left( x,U\right) $ is also a polynomial function in the components of the matrix $U.$ By varying $U$ over $F_{q}$ we can find the minimum possible $m_{k}\left( U\right) ,$ denoted $m_{k}.$ Thus there is some matrix, $Q\in F_{q}$ such that for $a_{k}\left( x,Q\right) $ we have that $m_{k}\left( Q\right) =m_{k}.$ This implies that in $a_{k}\left(
x,Q\right) $ there is a homogeneous polynomial of degree $m_{k}$ in $x$ that is non-zero, thus has at least one non-zero coefficient on a monomial term of degree $m_{k}.$ Since this coefficient is a polynomial function of the components of $Q,$ considering this polynomial over the corresponding components of all $U\in F_{q}$ we note that it is non-zero a.e.. This implies that $m_{k}\left( U\right) =m_{k}$ for almost all $U\in F_{q}.$
$\blacksquare $
**Proof of Proposition 2.**
For every $T$ consider $T^{-1/2}y$ in place of $x$ and $M_{T}\left(
y,U\right) $ in place of $B\left( x,U\right) $ in Proposition 1. Then for the corresponding characteristic polynomial, $p_{M_{T}}\left( \lambda
;y,U\right) ,$ the expansion similar to $\left( \ref{charact polyn}\right) $ will provide coeffficients$$\tilde{a}_{k}\left( y,U\right) =\left( -1\right) ^{q-k}\Sigma _{\left\{
i_{1},...,i_{k}\right\} \in I_{q}\left( k\right) }\det \left[ M_{T}\left(
y,U\right) ^{\left\{ i_{1},...,i_{k}\right\} }\right] .$$Note that we have that $\det \left[ M_{T}\left( y,U\right) ^{\left\{
i_{1},...,i_{k}\right\} }\right] \rightarrow \det \left[ \bar{G}\left(
y\right) V\bar{G}\left( y\right) ^{\prime \left\{ i_{1},...,i_{k}\right\} }\right] ,$ that is a non-zero constant for $k\leq r$ and zero for $k>r.$ Therefore the coefficients can be represented as $$\tilde{a}_{k}\left( y,U;T\right) =D_{\tilde{m}_{k}\left( U\right) }\left(
y,U;T\right) +\tilde{R}_{k}\left( y,U;T\right) ,$$where $\tilde{m}_{k}\left( U\right) $ is zero for $k=1,...,r.$ But for $k=r+1,...,q$ there is some $\gamma _{k}\geq 1$ such that $D_{\tilde{m}_{k}\left( U\right) }\left( y,U;T\right) =T^{-\gamma _{k}}\bar{R}_{k}\left(
y\right) ,$ where $\gamma _{k}=\frac{1}{2}\tilde{m}_{k}\left( U\right) $ and $\bar{R}_{k}\left( y\right) $ is a (positive a.e.) homogeneous polynomial in $y$ of degree $\gamma _{k}.$ So altogether we can write
$$\tilde{a}_{k}\left( y,U;T\right) =T^{-\gamma _{k}}\bar{R}_{k}\left( y\right)
+\tilde{R}_{k}\left( y,U;T\right) ,$$
where $\tilde{R}_{k}\left( .\right) $ is a polynomial that can contain non-zero monomials only of degree strictly higher than $\gamma _{k}.$ Then apply the representation $\left( \ref{sym polyn}\right) $ to the corresponding coefficients to write for every $k=1,...,q$ $$\begin{aligned}
P_{k}\left[ \lambda ^{\left( T\right) }(T^{-1/2}y)\right] &=&T^{-\gamma _{k}}\bar{R}_{k}\left( y\right) +\tilde{R}_{k}\left( T^{-1/2}y\right) ; \\
\gamma _{k} &=&0,\text{ if }k=1,...,r; \\
\gamma _{k} &\geq &1\text{ for }k=r+1,...,q.\end{aligned}$$
The proof is by induction on $k.$
For $k=1$ consider the largest eigenvalue $\lambda _{1}^{\left( T\right)
}(T^{-1/2}y).$ Note that $r\geq 1,$ so that $\gamma _{1}$ is always zero. Since $P_{1}\left[ .\right] $ is the sum of all eigenvalues we have by replacing all the $q$ eigenvalues by the largest, $\lambda _{1}^{\left(
T\right) }(T^{-1/2}y),$ that$$q\lambda _{1}^{\left( T\right) }(T^{-1/2}y)\geq \bar{R}_{1}\left( y\right)
+O\left( T^{-1/2}\right) ,$$then since $\bar{R}_{1}\left( y\right) >0$ a.e. the limit of $T^{\beta
_{1}}\lambda _{1}^{\left( T\right) }(T^{-1/2}y)$ (with $\beta _{1}=0)$ is positive a.e..
Suppose that for $k^{\prime }\geq 1$ all $T^{\beta _{l}}\lambda _{l}^{\left(
T\right) }(T^{-1/2}y)$ for $l\leq k^{\prime }$ converge to continuous positive a.e. functions.
Then by replacing in the symmetric polynomial $P_{k^{\prime }+1}\left[
\lambda ^{\left( T\right) }(T^{-1/2}y)\right] $ all the terms by the largest, $\lambda _{k^{\prime }+1}^{\left( T\right) }(T^{-1/2}y),$ and multipying by the rate, $T^{\gamma _{k^{\prime }+1}},$ we can write that$$\begin{aligned}
&&T^{\gamma _{k^{\prime }+1}}\frac{q!}{\left( k^{\prime }+1\right) !\left(
q-k^{\prime }-1\right) !}\left[ \Pi _{l\leq k^{\prime }}\lambda _{l}^{\left(
T\right) }(T^{-1/2}y)\right] \left[ \lambda _{k^{\prime }+1}^{\left(
T\right) }(T^{-1/2}y)\right] \\
&=&\frac{q!}{\left( k^{\prime }+1\right) !\left( q-k^{\prime }-1\right) !}\left[ \Pi _{l\leq k^{\prime }}T^{\beta _{l}}\lambda _{l}^{\left( T\right)
}(T^{-1/2}y)\right] \left[ T^{\left( \gamma _{k^{\prime }+1}-\Sigma _{l\leq
k^{\prime }}\beta _{l}\right) }\lambda _{k^{\prime }+1}^{\left( T\right)
}(T^{-1/2}y)\right] \\
&\geq &T^{\gamma _{k^{\prime }+1}}P_{k^{\prime }+1}\left[ \lambda ^{\left(
T\right) }(T^{-1/2}y)\right] \\
&=&\bar{R}_{k^{\prime }+1}\left( y\right) +O\left( T^{-1/2}\right) .\end{aligned}$$Since the expression in the last line has a limit that is non-zero a.e., so does the expression in the second line; by the induction hypothesis $\left[
\Pi _{l\leq k^{\prime }}T^{\beta _{l}}\lambda _{l}^{\left( T\right)
}(T^{-1/2}y)\right] $ converges to a continuous positive a.e. function. Thus for $\beta _{k^{\prime }+1}=\left( \gamma _{k^{\prime }+1}-\Sigma _{l\leq
k^{\prime }}\beta _{l}\right) $ the function $T^{\beta _{k^{\prime
}+1}}\lambda _{k^{\prime }+1}^{\left( T\right) }(T^{-1/2}y)$ converges to a continuous positive a.e. function.
From the derivation it follows that $\beta _{l}=\frac{1}{2}\tilde{m}_{l}\left( U\right) =0$ for $l=1,...,r;$ $\beta _{r+1}=\gamma _{r+1}=\frac{1}{2}\tilde{m}_{r+1}\left( U\right) \geq 1;$ and generally $\beta _{k}=\gamma
_{k}-\gamma _{k-1}=\frac{1}{2}\left( \tilde{m}_{k}\left( U\right) -\tilde{m}_{k-1}\left( U\right) \right) \geq 1$ for $k>r.\blacksquare $
**Proof of Proposition 3.**
Under the condition in (a) the degrees of homogeneity $m_{k}\left( U\right) $ for the coefficients of the characteristic polynomial of $M_{T}\left(
y,G,U\right) $ and $m_{k}\left( U_{T}\right) $ for the corresponding coefficient in $M_{T}\left( y,G,U_{T}\right) $ has to be the same for large enough $T$ and thus by the proof of Proposition 2 we conclude that $T^{\beta
_{l}}\bar{\lambda}_{l}^{\left( T\right) }\left( y,U_{T}\right) $ have the same positive a.e. limit as $T^{\beta _{l}}\bar{\lambda}_{l}^{\left(
T\right) }\left( y,U\right) .$
Under the condition in (b) we can write$$T^{m_{k+1}}P_{k+1}\left[ \lambda ^{\left( T\right) }\left(
T^{-1/2}y,U_{T}\right) \right] \geq \left[ \Pi _{i\leq k}T^{\beta
_{l}}\lambda _{l}^{\left( T\right) }\left( T^{-1/2}y,U_{T}\right) \right] \left[ T^{(m_{k+1}-m_{k})}\lambda _{k+1}^{\left( T\right) }\left(
T^{-1/2}y,U_{T}\right) \right] ,$$where $\left[ \Pi _{i\leq k}T^{\beta _{l}}\lambda _{l}^{\left( T\right)
}\left( T^{-1/2}y,U_{T}\right) \right] $ converges to a function that is positve a.e., but by the condition for $k$ the left-hand side converges to zero. Thus $\left[ T^{(m_{k+1}-m_{k})}\lambda _{k+1}^{\left( T\right)
}\left( T^{-1/2}y,U_{T}\right) \right] $ converges to zero and so does $\left[ T^{(m_{k+1}-m_{k})}\lambda _{l}^{\left( T\right) }\left(
T^{-1/2}y,U_{T}\right) \right] $ for any $l>k.\blacksquare $
**Proof of Proposition 4.**
Consider the scaling matrix $\tilde{\Delta}$ and the scaled matrix $\hat{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) =\tilde{\Delta}\Delta
_{T}G\left( \hat{\theta}_{T}\right) \hat{V}_{T}G\left( \hat{\theta}_{T}\right) ^{\prime }\Delta _{T}\tilde{\Delta}.$ By Assumption 1 since $T^{\frac{1}{2}}\left( \hat{\theta}_{T}\right) \underset{d}{\rightarrow }Y$ that is absolutely continuous by Proposition 2 the eigenvalues of $\tilde{\Delta}\Delta _{T}G\left( \hat{\theta}_{T}\right) VG\left( \hat{\theta}_{T}\right)
^{\prime }\Delta _{T}\tilde{\Delta}$ converge in distribution to continuous functions in $Y,$ some of which are non-zero a.e.. Additionally, for any sequence of $\hat{V}_{T}$ we can select a subsequence $\hat{V}_{T^{\prime }}$ that converges a.s. to $V$ and by Proposition 3 we have the same result for the limits of eigenvalues of $\tilde{\Delta}\Delta _{T}G\left( \hat{\theta}_{T}\right) \hat{V}_{T}G\left( \hat{\theta}_{T}\right) ^{\prime }\Delta _{T}\tilde{\Delta}.\blacksquare $
**Proof of the Theorem.**
Consider now the matrix $\hat{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) =\tilde{\Delta}\Delta _{T}G\left( \hat{\theta}_{T}\right) \hat{V}_{T}G\left( \hat{\theta}_{T}\right) ^{\prime }\Delta _{T}\tilde{\Delta}$ as defined in Proposition 4. The eigenvalues of the scaled matrix, $\hat{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) ,$ denoted $\tilde{\lambda}_{iT},$ $i=1,...,q$ by Proposition 4 converge in distribution$$\tilde{\lambda}_{i,T}\underset{d}{\rightarrow }\lambda _{i}\left( Y\right) ,\text{ }i=1,...,q.$$Then rewrite the Wald statistic as$$W_{T}=\left[ \tilde{\Delta}_{T}T^{1/2}\Delta _{T}g\left( \hat{\theta}_{T}\right) \right] ^{\prime }\left[ \hat{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) \right] ^{-1}\left[ \tilde{\Delta}_{T}T^{1/2}\Delta
_{T}g\left( \hat{\theta}_{T}\right) \right] .$$Since $\tilde{\lambda}_{1T}^{-1},...,\tilde{\lambda}_{qT}^{-1}$ are the eigenvalues of the non-negative definite matrix $\left[ \hat{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) \right] ^{-1},$ for any vector $\xi $ we have$$\xi ^{\prime }\left[ \hat{\Sigma}_{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) \right] ^{-1}\xi \geq \xi ^{\prime }\xi \underset{1\leq i\leq q}{\min }\left\{ \tilde{\lambda}_{iT}^{-1}\right\} ,$$thus$$W_{T}\geq \frac{\left\Vert \tilde{\Delta}_{T}T^{1/2}\Delta _{T}g\left( \hat{\theta}_{T}\right) \right\Vert }{\underset{1\leq i\leq q}{\max }\left\{
\tilde{\lambda}_{iT}\right\} }.$$We have that $$T^{1/2}\Delta _{T}g\left( \hat{\theta}_{T}\right) \underset{d}{\rightarrow }\bar{g}\left( Y\right)$$with all components of the vector function $\bar{g}\left( Y\right) $ non-zero a.e. for absolutely continuous $Y.$ Then $$\begin{aligned}
\left\Vert \tilde{\Delta}_{T}T^{1/2}\Delta _{T}g\left( \hat{\theta}_{T}\right) \right\Vert ^{2} &=&\Sigma _{i=1}^{r}\left[ T^{\left(
s_{k_{i}}+1\right) /2}g_{i}\left( \hat{\theta}_{T}\right) \right]
^{2}+\Sigma _{i=r+1}^{q}T^{\beta _{i}}\left[ T^{\left( s_{k_{i}}+1\right)
/2}g_{i}\left( \hat{\theta}_{T}\right) \right] ^{2} \\
&\geq &T^{\bar{\beta}}\underset{1\leq i\leq q}{\min }\left\{ \left[
T^{\left( s_{k_{i}}+1\right) /2}g_{i}\left( \hat{\theta}_{T}\right) \right]
^{2}\right\} .\end{aligned}$$
Define $$\mu _{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) =\frac{T^{\bar{\beta}}\underset{1\leq i\leq q}{\min }\left\{ \left[ T^{\left( s_{k_{i}}+1\right)
/2}g_{i}\left( \hat{\theta}_{T}\right) \right] ^{2}\right\} }{\underset{1\leq i\leq q}{\max }\left\{ \tilde{\lambda}_{iT}\right\} },$$then $$W_{T}\geq T^{\bar{\beta}}\mu _{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) .$$
By Proposition 4, continuity of the eigenvalue function, and of the maximum of continuous functions $$\underset{1\leq i\leq q}{\max }\left\{ \tilde{\lambda}_{iT}\right\} \underset{d}{\rightarrow }\lambda _{\max }\left( Y\right) ,$$with the limit functions non-zero a.e.. Also, $$\underset{1\leq i\leq q}{\min }\left\{ \left[ T^{\left( s_{k_{i}}+1\right)
/2}g_{i}\left( \hat{\theta}_{T}\right) \right] ^{2}\right\} \underset{d}{\rightarrow }\bar{g}_{i\min }\left( Y\right) ,$$which is a piece-wise polynomial continuous function. The ratio $$\mu \left( Y\right) =\frac{\bar{g}_{i\min }\left( Y\right) }{\lambda _{\max
}\left( Y\right) }$$exists and is non-zero a.e. and $$\mu _{T}\left( \hat{\theta}_{T},\hat{V}_{T}\right) \underset{d}{\rightarrow }\mu \left( Y\right) .$$When the FRALD-T condition is violated $\bar{\beta}\geq 1$ and the Wald statistic diverges to $+\infty .$
$\blacksquare $
[9]{} Dufour, J.-M., Renault, E. and V. Zinde-Walsh, 2013, Wald tests when restrictions are locally singular, working paper, ArXiV
Dufour, J.-M., Renault, E. and V. Zinde-Walsh, 2017, Wald tests when restrictions are locally singular, working paper, https://monde.cirano.qc.ca/dufourj/Web\_Site/Dufour\_Renault\_ZindeWalsh\_2012\_WaldTestsLocallySingularRestrictions\_W.pdf
Harville, D.A., 2008, Matrix Algebra from a Statistician’s Perspective, Springer-Verlag, New York
Horn, R. G. and Johnson, C. A. (1985), Matrix Analysis, Cambridge University Press, Cambridge, U.K.
[^1]: William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO) and Centre interuniversitaire de recherche en économie quatative (CIREQ).
[^2]: Brown University
[^3]: McGill University and CIREQ
[^4]: This work was supported by the Willam Dow Chair in Political Economy (McGill University), the Bank of Canada Research Fellowship, The Toulouse School of Economics Pierre-de-Fermat Chair of Excellence, A Guggenheim Fellowship, Conrad-Adenauer Fellowship from Alexander-von-Humboldt Foundation, the Canadian Network of Centres of Excellence program on Mathematics of Information Technology and Complex Systems, the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada and the Fonds de recherche sur la société et la culture (Québec). The authors also thank the research centres CIREQ and CIRANO for providing support and meeting space for the joint work. We thank Purevdorj Tuvaandorj for very useful comments.
|
epsf
[**Single Charged and Neutral Supersymmetric Higgs Bosons Production with Jet at pp Colliders**]{}\
\
\
[Alikhanian Brothers St.2, Yerevan 375036, Armenia\
alanak @ lx2.yerphi.am]{}
[ grabsky @ atlas.yerphi.am]{}
[**[Abstract]{}**]{}
In the framework of Minimal Supersymmetric Standard Model Higgs bosons production via gluino/squark loop in the processes $gu \rightarrow
H^{+}d$, $gd \rightarrow
H^{-}u$,$gq \rightarrow
H^{0}_iq$ are studied.
[**[Yerevan Physics Institute]{}**]{}
[**[Yerevan 1997]{}**]{}
If Higgs sector of the gauge theory contains additonal Higgs fields besides standard doublet, after spontaneous symmetry violation more than one physical Higgs bosons arise.
In particular, in the Minimal Supersymmetric Standard Model(MSSM)(see [@GH], [@HK] and references therein) the Higgs sector contains two doublets of Higgs bosons with opposite hypercharge (Y=$\pm$ 1) and after spontaneous symmetry violation the following physical states appear: charged Higgs bosons $H^{ \pm}$, and three neutral ones, $H^0_1, H^0_2, H^0_3 $.
If the mass of the $H^{\pm}$-boson is less than $m_t-m_b$, it may be produced in t-quarks decays [@GJ]- [@DR]: $$\label{AA}
t\rightarrow H^{+}b$$ However, if $m_t-m_b<m_H$ this decay is kinematically forbidden.
That is why it is necessary to study new mechanisms for the production of heavy (i.e. with masses $m_t-m_b<m_H$) charged Higgs bosons.
Here we study the following mechanisms of SUSY Higgs bosons production in $pp$-collisions:
$$\label{AB}
gu \rightarrow H^{+}d,$$
$$\label{AC}
gd\rightarrow H^{-}u ,$$
$$\label{AD}
gq \rightarrow H^{0}_i q$$
which proceed via squark/gluino loop (Fig.1) .
It must be noted that besides the squark/gluino loop contribution there is also the tree contribution [@BHS]. It is of interest to compare both contributions.The cross section of the tree contribution to the subprocess (2) is of order $O(\alpha_s \alpha \frac{m_q^2}{m_W^2} \tan^2 \beta)$, whereas the loop contribution is of order $O(\alpha_s^3 \alpha
\sin^22 \beta)$.Besides the tree contribution is supressed by the smallness of the heavy quarks (s,c,b,...) inside protons.Thus we expect that the loop contribution will dominate over the tree at not very large $\tan \beta$ . It must be noted that besides the squark/gluino loop contribution there are also contributions from squarks and t-quarks loops in the process (4).The heavy quark and squark contribution in the process (4) and the process : $$\label{AE}
gg\rightarrow H^{0}_i g$$ has been considered in [@CLLR]- [@DK].
Using Higgs bosons interactions with scalar quarks (see formula (4.19) in ref. [@GH]) : $$\begin{aligned}
\label{AF}
&& L=- \frac{g}{\sqrt{2}}m_W \sin 2 \beta(H^+\tilde {u_L^*}
\tilde{d_L}+H.c.)-\frac{gm_Z}{\cos \theta_W} \sum((T_{3i}-e_i
\sin^2\theta_W)^2 \tilde{q_{iL}^*}
\tilde{q_{iL}}+\nonumber\\
&& e_i \sin^2 \theta_W)
\tilde{q_{iR}^*}
\tilde{q_{iR}}) (H^{0}_1 \cos(\alpha+ \beta)-H^{0}_2
\sin(\alpha+ \beta))\end{aligned}$$ for the gauge invariant amplitude of the process (2) we obtain: $$\label{AW}
M= \sqrt{2} \frac{\alpha_s^{\frac{3}{2}}\alpha^{\frac{1}{2}}}{\sin\theta_W}m_W
\sin 2 \beta \bar{u}(k_1)T^a \gamma_{\mu}
P_Lu(k_2)(G_{\mu \nu}^a(F_1k_1^{\nu}+F_2k_2^{\nu})+
\epsilon_{\mu \nu \lambda \rho}G_{\lambda \rho}^a(F_3k_1^{\nu}+F_4k_2^{\nu})).$$
Here
$$\label{AG}
G_{\mu \nu}^a=k_{\mu}A^a_{\nu}-k_{\nu}A^a_{\mu}$$
$$\label{AH}
F_1=Nf_3+\frac{1}{N}f_1,$$
$$\label{AI}
F_2=Nf_4+\frac{1}{N}f_2 ,$$
$$\label{AJ}
F_3=\frac{N}{2}\int\limits_{0}^{1}dx\int\limits_{0}^{1-x}dy A_3 ,$$
$$\label{AK}
F_4=\frac{N}{2}\int\limits_{0}^{1}dx\int\limits_{0}^{1-x}dy xD ,$$
(where $N=3$ number of colours),
$$\label{AL}
f_1=\int\limits_{0}^{1}dx\int\limits_{0}^{1-x}dy(-(x+y)A_1-yA_2),$$
$$\label{AM}
f_2=\int\limits_{0}^{1}dx\int\limits_{0}^{1-x}dy((x+y)A_2-yA_1),$$
$$\label{AN}
f_3=\int\limits_{0}^{1}dx\int\limits_{0}^{1-x}dy(x+y)(-A_3),$$
$$\label{AO}
f_4=\int\limits_{0}^{1}dx\int\limits_{0}^{1-x}dyx(1-x-y)D,$$
$$\label{AP}
A_i= \frac{1}{a_i^2}
(\log(\frac{a_i(1-x-y)+b_i}{b_i})+\frac{a_i(1-x-y)}{a_i(1-x-y)+b_i}),$$
$$\label{AQ}
D=\frac{1-x-y}{b_3(a_3(1-x-y)+b_3)},$$
$$\label{AR}
a_1=(m_H^2-t)y+sx,$$
$$\label{AS}
b_1=-m^2_{\tilde{g}}x-m^2_{\tilde{u}}(1-x)+ty(1-x-y)+i\epsilon,$$
$$\label{AT}
a_2=(m_H^2-t)y+ux,$$
$$\label{A1}
b_2=b_1,$$
$$\label{A2}
a_3=(m_H^2-u)x+sy+m^2_{\tilde{g}}-m^2_{\tilde{u}},$$
$$\label{A3}
b_3=-m^2_{\tilde{g}}(1-x)-m^2_{\tilde{u}}x+ux(1-x-y)+i\epsilon.$$
In our calculations we use $m_{\tilde{u}}=m_{\tilde{d}}$ approximation because in MSSM for left scalar quarks (if we neglect masses of the light quarks) we have:
$$\label{A4}
m^2_{ \tilde{u}}-m^2_{ \tilde{d}}=( 1- \sin^2 \theta_W)m^2_Z\cos{2\beta}.$$
The processes of the neutral Higgs bosons production via gluino/squark loop may be obtained from formulas (7)-(21), taking into account (6), by the following replacements:
$$\label{A5}
m_W\sin(2\beta)\rightarrow m_Z((T_{3i}-e_i
\sin(\theta_W))^2)\tilde{q_{iL}}
\tilde{q_{iL}}+e_i
\sin(\theta_W))^2)\tilde{q_{iR}} \tilde{q_{iR}} \cos(\alpha+ \beta)$$
$$\label{A6}
m_W\sin(2\beta)\rightarrow m_Z((T_{3i}-e_i
\sin(\theta_W))^2)\tilde{q_{iL}}
\tilde{q_{iL}}+e_i
\sin(\theta_W))^2)\tilde{q_{iR}} \tilde{q_{iR}} \sin(\alpha+ \beta)$$
for $H_{1,2}^0$ -bosons respectively.In the case of the charged Higgs bosons production only left scalar quarks in loop contribute to the processes (2),(3).It must be noted also that in case of the neutral scalar Higgs bosons production in the subprocess (4) the total amplitude is the sum of both pure both left and right scalar quarks loop contributions, the pure$t$-quark loop contribution and the gluino/squark loop contribution.The tree and loop contribution do not interfere with each other.
It must be noted also ,in case of the neutral Higgs bosons production the total amplitude is the sume of the gluino/squark amplitude and pure t-quark and squark amplitude.
For the differential cross section of the subprocess (2) we obtain the following result:
$$\label{A7}
\frac{d\sigma }{dt}=-\frac{\alpha \alpha^3_s m_W^2}{128
\pi \sin^2\theta_W} (\sin 2\beta)^2 \frac{t}{s}(s^2 (\mid (F_1
\mid ^2+\mid F_3 \mid ^2)+
u^2 (\mid F_2 \mid^2+\mid F_4\mid ^2))$$
Here we use the following notations: $s=(k_1+k)^2$, $t=(k_1-k_2)^2$, $u=(k_1-k_3)^2$, $m_H$ is the mass of $H^{+}$ or $H^{0}_i$-bosons,
$$\label{A8}
\sigma(pp \rightarrow H^+ +jet+X)=
\int\limits_{\frac{m_H^2}{s_0} }^{1}dx_1
\int\limits_{\frac{m_H^2}{x_1s_0}}^{1}dx_2
\sigma(x_1x_2s_0)(u(x_1)g(x_2)+u(x_2)g(x_1))$$
with replacements
$$\label{A9}
u(x) \rightarrow d(x),u(x)+d(x)$$
for $H^-$ and $H^0_i$ Higgs bosons respectively. For structure functions we use parametrizations of ref. [@E].
The numerical result will be presented in the nearest future. The work in this direction is in progress.
The authors express their sincere gratitude to H.Hakopian for helpful discussions.
[99]{}
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Phys.Rev.Lett.V.70(1993)1372; R.P.Kauffman,Phys.Rev.D44(1991)1415;D45(1992)1512 C.P.Yuan,Phys.Lett.B283(1992)395 S.Dawson,R.P.Kauffman,Phys.Rev.Lett.V.68(1992)2273 E.Eichten et.al.,Rev. of Mod.Phys.v.56(1984)579
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abstract: |
=0.6 cm
Quintessence field is a widely-studied candidate of dark energy. There is “tracker solution” in quintessence models, in which evolution of the field $\phi$ at present times is not sensitive to its initial conditions. When the energy density of dark energy is neglectable ($\Omega_\phi\ll1$), evolution of the tracker solution can be well analysed from “tracker equation”. In this paper, we try to study evolution of the quintessence field from “full tracker equation”, which is valid for all spans of $\Omega_\phi$. We get stable fixed points of $w_\phi$ and $\Omega_\phi$ (noted as $\widehat w_\phi$ and $\widehat\Omega_\phi$) from the “full tracker equation”, i.e., $w_\phi$ and $\Omega_\phi$ will always approach $\widehat w_\phi$ and $\widehat\Omega_\phi$ respectively. Since $\widehat w_\phi$ and $\widehat\Omega_\phi$ are analytic functions of $\phi$, analytic relation of $\widehat w_\phi\sim\widehat\Omega_\phi$ can be obtained, which is a good approximation for the $w_\phi\sim\Omega_\phi$ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities $\widehat w_\phi<w_\phi$ and $\widehat\Omega_\phi<\Omega_\phi$ are statisfied if the $w_\phi$ (or $\widehat w_\phi$) is decreasing with time. In this way, the potential $U(\phi)$ can be constrained directly from observations, by no need of solving the equations of motion numerically.
author:
- 'Mingxing Luo$^{1}$'
- 'Qi-Ping Su$^{1,2}$'
title: '[**Approximate $w_\phi\sim\Omega_\phi$ Relations in Quintessence Models**]{}'
---
=0.8 cm
Introduction
============
Present astronomical observations require the existence of dark energy, a significant component of the universe with a negative pressure [@Riess:1998cb; @Riess:2004nr; @Riess:2006fw; @Spergel:2003cb; @Komatsu:2008hk]. Though it has been more than ten years since its discovery, one is yet to tell what the dark energy is. We are still analyzing properties of dark energy from observational data and seeking suitable candidates. Most properties of dark energy depend on two parameters: the equation of state $w_{de}$ and the fractional energy density $\Omega_{de}$. Once the $w_{de}\sim\Omega_{de}$ relation is obtained, we know almost all we need. At present, it is still not possible to constrain the evolution of dark energy from observations [@Daly:2004gf; @Mignone:2007tj; @Sullivan:2007pd]. There are only definite constraints of present values of $w_{de}$ and $\Omega_{de}$ from observations: $w_{de}^{(0)}$ is rather close to $-1$ and $\Omega_{de}^{(0)}$ is dominating (about 70%) [@Lazkoz:2007zk; @Copeland:2006wr; @Mantz:2007qh]. More constraints on $w_{de}$ and $\Omega_{de}$ will be forthcoming from future observations, to get the evolution of the $w_{de}\sim\Omega_{de}$ relation from the observations, more theoretical efforts should be made.
At present, the most economical candidate of dark energy is still the cosmological constant $\Lambda$, whose equation of state $w_\Lambda=-1$. There is only a free parameter $\Omega_\Lambda$ in the flat $\Lambda$CDM model. But it suffers from several problems, such as the coincidence problem and the fine tuning problem. Another well studied candidate is the quintessence $\phi$, a slowly rolling scalar field, analogous to the inflaton. Its equation of state is $w_\phi=(\dot{\phi}^2/2-U)/(\dot{\phi}^2/2+U)$ so one has $-1\leq w_\phi\leq1$. In quintessence models, the coincidence problem and the fine tuning problem can be alleviated [@Copeland:2006wr]. For example, there are tracker solutions for certain type of quintessence models, in which the evolution of $\phi$ today is not sensitive to its initial conditions at early times [@Steinhardt:1999nw]. The coincidence problem thus becomes less severe. But it is difficult to find quintessence models with analytic solutions of equation-of-motion, due to the existence of background matters (dark matter, baryon and radiations). To study evolutions of quintessence models and to be compared with observations, one usually has to solve the equations numerically. There are efforts to find analytic approximations for solutions of equations of motions, such as [@Watson:2003kk] which gives a first order approximation solution for inverse power law potentials.
In this paper, we will try to approximate the $w_\phi\sim\Omega_\phi$ relation at the recent $\Omega_\phi$ dominating period in a semi-analytic way. To make sure that the evolution of $\phi$ at present only depends on $U(\phi)$, we assume there was tracking solution at early times. In [@Steinhardt:1999nw], conditions for the existence of tracker solution was given by the “tracker equation”, which is a differential equation for $w_\phi$. But this “tracker equation” are only valid as $\Omega_\phi\ll1$. For our purpose, we need a full tracker equation that is valid for all $\Omega_\phi$ without conditions attached. Such an equation has been obtained [@Scherrer:2005je; @Chiba:2005tj; @Lee:2006gx] and will be used here to study evolutions of quintessence models.
The paper is organized as follows. In section II, we introduce two new functions $\widehat w_\phi$ and $\widehat \Omega_\phi$ which are fixed points of the full tracker equation. Assuming that $\Gamma\equiv U''U/U'^2$ and $\epsilon\equiv (U'/U)^2/2$ are nearly constant, we find that the fixed points are stable for $w_\phi$ and $\Omega_\phi$ if $\Gamma\geq1$. If $\Gamma$ and $\epsilon$ do not evolve extremely fast, the relation of $w_\phi\sim\Omega_\phi$ will always approach to that of $\widehat w_\phi\sim\widehat \Omega_\phi$. In section III we show comparisons between {$\widehat w_\phi(\phi)$, $\widehat \Omega_\phi(\phi)$} and {$w_\phi(\phi)$, $\Omega_\phi(\phi)$} numerically for several typical quintessence models. The relation of $\widehat w_\phi\sim\widehat \Omega_\phi$ is shown to be a good approximation for the $w_\phi\sim\Omega_\phi$ relation. In section IV we show how to constrain $U(\phi)$ directly from observational conditions on $w_\phi$ and $\Omega_\phi$ through $\widehat w_\phi$ and $\widehat \Omega_\phi$. Observational conditions are converted to simple inequalities for $U(\phi)$. We conclude in section V with discussions.
Get the approximation of $w_\phi\sim\Omega_\phi$ relations
==========================================================
The equations of motion for quintessence field are &+3H+U’=0&\
&H\^2()\^2=(\_m+\_r+\^2+U)& from which one gets the equations for $w_\phi$ and $\Omega_\phi$: &=&(1+)\^2\[e2\]\
-1&=& - -\
&&-\_\[e3\] where $$x\equiv\frac{1+\omega_\phi}{1-\omega_\phi}=\frac{1}{2}{\dot{\phi}^2\over U}, ~~~
\dot{x}\equiv {d\ln{x} \over d\ln{a}}, ~~~ \ddot{x}\equiv
{d^2\ln{x} \over d\ln{a}^2}$$ and $a$ is the expansion factor. We have assumed a flat universe ($\Omega_{b}+\Omega_{\phi}=1$) and set $M_{pl}\equiv1/\sqrt{8\pi G}=1$. The subscript $b$ represents the dominating background matter. As $\Omega_\phi\ll1$ at early times, Eq.(\[e3\]) reduces to the “tracker equation” in [@Steinhardt:1999nw]. At the recent acceleration era, $\Omega_\phi$ is dominating and can not be neglected. One must use the full tracker equation Eq.(\[e3\]). Note also $w_b=0$ in this case. In this paper, we assume that there was a long enough tracking period at early times, so that the evolution of the field at present depends only on $U(\phi)$.
Eliminating $\Omega_\phi$ in Eq.(\[e3\]) by using Eq.(\[e2\]), one gets: -1&=&- -\
&&- +(6+x) \[e4\] For constant $\epsilon$ and $\Gamma$, the fixed point (also called critical point) of Eq.(\[e4\]) (obtained by setting $\dot x=0$ and $\ddot x=0$): \_= (-3-2+4-) \[e6\] is stable only if \[e5\] where the $\widehat\Omega_\phi$ value of the fix point is obtained from Eq.(\[e6\]) and (\[e2\]) (also setting $\dot x=0$): \_= (3-2+4-) \[e7\] When Eq.(\[e5\]) is satisfied, $\widehat\Omega_\phi$ is also stable. In this case, ${\omega}_\phi$ and $\Omega_\phi$ will always approach $\widehat{\omega}_\phi$ and $\widehat\Omega_\phi$ respectively. In this paper, we will only study the case of $\Gamma\geq1~~(i.e.,~w_\phi\leq w_b)$, so Eq.(\[e5\]) is guaranteed for all spans of $\widehat\Omega_\phi$.
$\Gamma$ and $\epsilon$ generally are not constants, as they are functions of $U(\phi)$. The above results are still valid if the evolution of $\widehat{w}_\phi$ is not extremely fast, which can be satisfied in the most quintessence models. In this case, $w_\phi$ and ${\Omega}_\phi$ will keep on chasing the dynamic $\widehat{w}_\phi$ and $\widehat{\Omega}_\phi$. Giving the form of $U(\phi)$ of a quintessence model, one gets parametric functions $\widehat{w}_\phi(\phi)$ and $\widehat{\Omega}_\phi(\phi)$ from Eq.(\[e6\]) and (\[e7\]), and thus the analytic relation of $\widehat{w}_\phi\sim\widehat{\Omega}_\phi$. For certain models, there are simple and explicit relations of $\widehat{w}_\phi\sim\widehat{\Omega}_\phi$. For example, for power law potentials $U=U_0/\phi^n~~(n>0)$ one has: \_=- The $\widehat{w}_\phi\sim\widehat{\Omega}_\phi$ relation is a good approximation for that of $w_\phi\sim{\Omega}_\phi$, as the evolution of $w_\phi\sim{\Omega}_\phi$ will approach that of $\widehat{w}_\phi\sim\widehat{\Omega}_\phi$. We will show this in the next section. In this way, evolutions of quintessence models can be studied directly from $U(\phi)$.
Compared with numerical results
===============================
In this section we will show that ${\widehat w}_\phi$ and ${\widehat\Omega}_\phi$ are good approximations for $w_\phi$ and $\Omega_\phi$, and so is the $\widehat{w}_\phi\sim\widehat{\Omega}_\phi$ relation for that of $w_\phi\sim{\Omega}_\phi$. We have checked it for a variety type of quintessence potentials, and typical examples are shown in Fig. \[f1\] and Fig. \[f\]. The accuracy of this approximation is precise enough to study the evolution properties of quintessence models, especially the models that are favored by present observations. As $w_\phi$ must decrease from its tracking value (close to $w_b$) to present value (close to $-1$), we will only study models in which $w_\phi$ decreases monotonously ($\dot{x}<0$).
{width="14cm" height="8cm"}
At first we estimate differences between $\widehat{w}_\phi$ and $w_\phi$ and between $\widehat{\Omega}_\phi$ and ${\Omega}_\phi$. The Eq.(\[e2\]) can be rewritten as: =(1+)\^2&\
()()\^[-1]{}=(1+)\^2 \[t1\] For a variety of quintessence models, we have seen numerically that $(1+\widehat w_\phi)/(1+w_\phi)$ and $\widehat\Omega_\phi/\Omega_\phi$ have the similar evolving forms as that of $(1+\dot{x}/6)^2$ and $1>\widehat\Omega_\phi/\Omega_\phi\gtrsim(1+\dot{x}/6)^2>(1+\widehat w_\phi)/(1+w_\phi)$. Typical examples are shown in Fig. \[f1\]. If the evolution of $\widehat w_\phi$ (and $w_\phi$) is slower, the value of $\dot{x}$ will be closer to $0$, and the differences between $w_\phi,~\Omega_\phi$ and their fixed points will be smaller.
{width="14cm" height="8cm"}
There is a lower bound $\dot{x}>6w_\phi/(1-2w_\phi)$ given in [@Scherrer:2005je; @Chiba:2005tj]. As $\widehat\Omega_\phi/\Omega_\phi$ is much closer to $1$ compared with $(1+\widehat w_\phi)/(1+w_\phi)$, one gets a upper bound for the deviation $\Delta$ of ${\widehat w}_\phi$ from $w_\phi$ by setting $\widehat\Omega_\phi/\Omega_\phi\simeq1$ in Eq.(\[t1\]): \_m= which is rather small when $w_\phi$ is close to $-1$, as shown in Fig. \[f1\]. Present observations indicate that $w_\phi$ is rather close to $-1$ at low redshift. For most models $\Delta$ is much smaller than this bound as $w_\phi$ is not so close to $-1$, as shown in Fig. \[f1\]. The deviation of $\widehat\Omega_\phi$ from $\Omega_\phi$ is also small.
The $\widehat{w}_\phi\sim\widehat{\Omega}_\phi$ relation thus is a good approximation for the $w_\phi\sim{\Omega}_\phi$ relation. Several examples are shown in Fig. \[f\]. At the early tracking era, $\widehat\Omega_\phi<<1$ and the relation of $w_\phi\sim\Omega_\phi$ is almost the same as that of $\widehat w_\phi\sim\widehat\Omega_\phi$. When $\widehat\Omega_\phi$ becomes unnegligible, the curve of $\widehat w_\phi\sim\widehat\Omega_\phi$ will begin to get away from that of $w_\phi\sim\Omega_\phi$ in the $w-\Omega$ space. The curve of $w_\phi\sim\Omega_\phi$ will chase after that of $\widehat w_\phi\sim\widehat\Omega_\phi$. Normally $\widehat w_\phi$ will tend to $-1$ and $\widehat\Omega_\phi$ will tend to $1$ at last, and the two curves will be close to each other once again. Empirically, we have also found a better approximation for the relation of $w_\phi\sim\Omega_\phi$ on the basis of $\widehat w_\phi$ and $\widehat\Omega_\phi$: w\_=w\_+(1+w\_), \_=\_\[e17\] The curve of $\widetilde{w}_\phi\sim\widetilde \Omega_\phi$ is much closer to that of $w_\phi\sim\Omega_\phi$, as shown in Fig. \[f\].
Constrain quintessence potentials
=================================
In the above, we have obtained approximations $\widehat w_\phi$ and $\widehat\Omega_\phi$ for $w_\phi$ and $\Omega_\phi$ which are analytic functions of $U(\phi)$. We will show how to constrain $U(\phi)$ directly from observational results on $w_{de}$ and $\Omega_{de}$ through $\widehat w_\phi$ and $\widehat\Omega_\phi$. Present data seems to indicate that $w_{de}^{(0)}<-0.8$ and $0.7 \lesssim \Omega_{de}^{(0)}<0.8$ [@Lazkoz:2007zk; @Mantz:2007qh]. As more conditions on dark energy to be obtained in future observations, more quintessence models can be checked with directly by using our method.
At the early tracking era $w_\phi$ was close to $w_r=1/3$ [@Bludman:2004az; @Zlatev:1998tr], and present $w_{de}^{(0)}$ is very close to $-1$. Taking this for guidance, here we consider only quintessence models in which $w_\phi~$(and $\widehat w_\phi$) keeps on decreasing monotonously ($\dot x<0$). This is guaranteed if $U(\phi)$ satisfies the equation: <(1-) In this case, one finds the following inequalities w\_7/5 \[e15\] which is a necessary condition for inequalities (\[e14\]).
\
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**[$U(\phi)~(n>0,\phi>0)$]{} & $\epsilon\equiv\frac{1}{2}(\frac{U'}{U})^2$ & $\Gamma\equiv \frac{U''U}{U'^2}$ & $\Gamma(\epsilon=\frac{3}{8})>\frac{7}{5}$ & $\Gamma(\epsilon=\frac{3}{28})<\frac{77}{20}$ \
$\frac{U_0}{\phi^n}$ & $\frac{n^2}{2\phi^2}$ & $1+\frac{1}{n}$ & $n<\frac{5}{2}$ & $n>\frac{20}{57}$\
$U_0e^{n/\phi}$&$\frac{n^2}{2\phi^4}$&$1+\frac{2\phi}{n}$ & $n<\frac{50}{\sqrt{3}}$ & $n>1$\
$\frac{U_0}{\phi^n}e^{\phi^2/2}$ & $\frac{(n-\phi^2)^2}{2\phi^2}$ & $1+\frac{(n+\phi^2)}{(n-\phi^2)^2}$ & $n>0$ & $\emptyset$\
**
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
If $w_\phi$ is too close to $-1$, it will be difficult to distinguish quintessence models from the cosmological constant [@Caldwell:2005tm]. Take $w_{\phi}>-0.95$ for illustration. It is then easy to see that $\widehat w_\phi(\widehat\Omega_\phi=0.7)>-0.95$ is a sufficient condition for ($w_{\phi}^{(0)}>-0.95$, $\Omega_{de}^{(0)}>0.7$). Equivalently, (=3/28)<77/20 \[e16\] Listed in Table I are the constraints on parameters of typical potentials by Eq.(\[e15\]) and (\[e16\]).
We note that for certain potentials $\widehat\Omega_\phi$ will tend to a maximum $\widehat\Omega_{max}$ smaller than 1 at last, such as $U(\phi)=U_0e^{\phi^2/2}/\phi^n~(n>0,\phi>0)$ [@Brax:1999yv]. These potentials always have a positive minimum $U_{min}$ at a finite $\phi$. According to Eq.(\[e7\]), as the potential rolls to $U_{min}$, $\eta=\epsilon\Gamma$ will tend to a nonzero minimum $\eta_{min}$ with $\Gamma\rightarrow\infty$ and $\epsilon\rightarrow0$. In this case, Eq.(\[e15\]) and (\[e16\]) are still valid though $\widehat\Omega_{\phi max}$ may be smaller than $0.7$.
Discussions
===========
We have gotten stable fixed points $\widehat w_\phi$ and $\widehat\Omega_\phi$ from the full tracker equation, and shown that they are good approximations for $w_\phi$ and $\Omega_\phi$ even in the $\Omega_\phi$ dominating period. $\widehat w_\phi$ and $\widehat\Omega_\phi$ are analytic functions of $U(\phi)$. The relation of $\widehat w_\phi\sim\widehat\Omega_\phi$ thus is gotten from the parametric functions $\widehat w_\phi(\phi)$ and $\widehat\Omega_\phi(\phi)$, which is also a good approximation to the relation of $w_\phi\sim\Omega_\phi$.
Formally, functions of $\widehat{w}_\phi$ and $\widehat{\Omega}_\phi$ with respect to expansion factor $a$ can also be obtained. Substituting Eq.(\[e6\]),(\[e7\]) into the equation =-3w\_\_(1-\_) one gets the function of the field $\phi$ with respect to $a$ upon integration. For example, for $U=U_0/\phi^2~~(\phi>0)$ one has: (a)=(120a\^3+49a\^6)\^[1/4]{} where we have set present $\widehat \Omega_\phi^{(0)}=0.7$ and $a_0=1$. Substituting $\phi(a)$ into Eq.(\[e6\]) and Eq.(\[e7\]) one gets: \_&=&(-7a\^3)\
w\_&=&-- For most potentials, it is not easy to get explicit functions of $\phi(a)$, $\widehat w_\phi(a)$ and $\widehat\Omega_\phi(a)$.
The critical points $\widehat w_\phi$ and $\widehat\Omega_\phi$ can also be used to constrain the potential of quintessence directly from observational conditions on ($w_{de},\Omega_{de}$). We have adopted two conditions on present ($w_{de}^{(0)},\Omega_{de}^{(0)}$) for illustration. Further astronomical observations will yield more properties of dark energy. It may give conditions on ($w_{de},\Omega_{de}$) at other redshifts, or even the exact shape of the $w_{de}\sim\Omega_{de}$ relation. In that case, our method can be still usable to constrain the potential and study the properties of the quintessence models that are fit with observations directly.
![The curve of the quintessence model with $U(\phi)=U_0(e^{-\phi/2}+e^{-20\phi})$ in the $w'-w$ phase space. This curve crosses the the boundary for thawing and freezing fields [@Caldwell:2005tm] and the lower bound $w'=-(1-w)(1+w)$ in [@Scherrer:2005je; @Chiba:2005tj].[]{data-label="f2"}](cross.eps){width="8cm" height="5cm"}
In this paper, we have only studied the case that $w_\phi$ (and $\widehat w_\phi)$ keeps on decreasing monotonously, from which the inequality (\[e12\]) is obtained. In fact, there are quintessence models in which $w_\phi$ is increasing at present. One example is the case with $U(\phi)=U_0(e^{-\phi/2}+e^{-20\phi})$ [@Barreiro:1999zs]. In this type of models, $w_\phi$ will decrease to a minimum close to $-1$ and then begin to increase. So the boundary for thawing and freezing fields in [@Caldwell:2005tm] will be crossed, as shown in Fig. \[f2\]. It can be shown that when $$\frac{d\ln{(\Gamma-1)}}{d\ln{U}}>\frac{3}{2\epsilon}$$ $\widehat w_\phi$ will be increasing, so will be $w_\phi$. It requires a rapid decrease of $\Gamma$. As $\Gamma$ at early times must be close to 1 to get enough tracking, usually there is a rapid increase of $\Gamma$ at recent times. In this case the lower bound $w'>-(1-w)(1+w)$ for quintessence models [@Scherrer:2005je; @Chiba:2005tj] may be crossed too. It is because $w=(w_b-2\Gamma+2)/(2\Gamma-1)$ will no longer be larger than $w_\phi$ if the increase of $\Gamma$ is too fast. It can be seen in Fig. \[f2\] that the line of $w'\sim w$ with the double exponential potential is very close to the strict lower bound $w'>3w(1+w)$ given in [@Caldwell:2005tm]. For this type of potential, as $w_\phi$ and $\widehat w_\phi$ are increasing, there is an inequality similar to (\[e12\]): w\_>w\_, \_>\_This inequality can be used to constrain $U(\phi)$ from conditions on ($w_{de},\Omega_{de}$). The methods used in this paper can also be extended to Phantom and K-essence models.
This work is supported in part by the National Science Foundation of China (10425525).
[99]{}
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abstract: 'In this essay a generalized notion of flavor–oscillation clocks is introduced. The generalization contains the element that various superimposed mass eigenstates may have different relative orientation of the component of their spin with respect to the rotational axis of the the gravitational source. It is found that these quantum mechanical clocks do not always redshift identically when moved from the gravitational environment of a non–rotating source to the field of a rotating source. The non–geometric contributions to the redshifts may be interpreted as quantum mechanically induced fluctuations over a geometric structure of space–time.'
address: |
P–25 Subatomic Physics Group, Mail Stop H–846\
Los Alamos National Laboratory, Los Alamos, NM 87545 USA\
and\
Global Power Division, ANSER Inc.,\
1215 Jefferson Davis Highway\
Arlington, VA 22202, USA
author:
- 'D. V. Ahluwalia'
title: ' On a New Non–Geometric Element in Gravity'
---
[^1]
Empirically observed equality of the inertial and gravitational masses leads to the theory of general relativity [@SW]. This theory of gravity lends itself to a geometric interpretation. In the framework of this theory, all clocks, independent of their workings, redshift in exactly the same manner for a given source. It is further assumed that when these clocks move from the gravitational environment of one source to another, identically running clocks run identically.
The primary purpose of this essay is to introduce the notion of the [*generalized flavor–oscillation clocks*]{}, and study their evolution in a weak gravitational environment of a [*rotating*]{} source. In the process we will uncover an inherently non-geometric aspect in gravity. Specifically, we will construct two intrinsically quantum mechanical clocks that do not redshift identically when introduced in the gravitational environment of a [*rotating*]{} source.
It is a direct consequence of (a) the 1939 paper of Wigner [@EPW1939], and (b) the fact that [*locally*]{} space–time carries Poincaré symmetries; that a general quantum test particle, for evolution over “sufficiently small” space–time distances,[^2] can be described by $$\begin{aligned}
\vert \mbox Q \rangle\, \equiv \,
\sum_{k,j} A_{kj}\,\vert m_k, s_j; \vec p_k \,\rangle,
\quad &&s_j= -s\hbar, (-s+1)\hbar, \ldots, (s-1)\hbar, +s\hbar\quad,
\nonumber \\
&& k= 1, 2, \ldots n\quad,
\label{a}\end{aligned}$$ with $A_{kj}$ as some appropriately chosen superposition coefficients. That is, a general quantum test particle is described by a linear superposition of the Casimir invariants associated with the Poincaré group — the $m_k$ are the masses, $s_j$ are the $(2 s + 1)$ spin projections (along the $\vec J$, for convenience) of spin $\bf s$, and $\vec p_k$ is the momentum of the [*kth*]{} mass eigenstate. To avoid certain complicated conceptual questions, I have refrained from summing over $s$ (which takes integral and half–integral values).
The quantum test particle as introduced here is a slight generalization of the particles introduced by Wigner. Wigner’s 1939 paper suggests that quantum particles are to be specified by the Casimir invariants associated with the Poincaré group. These are the $\vert m_k, s_j; \vec p_k \,\rangle$ that appear in Eq. (\[a\]). The notion of a quantum test particle as presented in Eq. (\[a\]) is a generalization based on the existence of empirically observed particles that are a linear superposition of mass eigenstates. The neutral $K^0$–$\overline{K}^0$ mesons, and the weak–flavor eigenstates of neutrinos as suggested by various data [@vd], fall in a subclass with, $\sum_{k,j} \rightarrow \sum_{k}$, in Eq. (\[a\]). With the existing laser technology, atomic systems can be easily constructed in a state similar to that given by Eq. (\[a\]), with $m_k$ replaced by $E_k$ ($E_k$ being an atomic state). For the sake of simplicity, we will set ${\vec p_k}= {\vec 0}$.
Definition of a general quantum test particle via Eq. (\[a\]) allows us to introduce two class of “flavors;” the first class for which the sum in Eq. (\[a\]) involves a [*single*]{} spin–projection independent of $m_k$. For the second class, the sum in Eq. (\[a\]) must contain at least [*two distinct*]{} spin–projections. Eqs. (\[Q1\]) and (\[Q2\]), below, provide an example of the first and second class, respectively.
It will be seen that by making appropriate “flavor measurements” one can use the flavor states to make flavor–oscillation clocks. We will show that flavor states of both classes redshift identically in the gravitational field of a spherically symmetric mass — i.e., as expected on the basis of the geometric interpretation of general relativity. However, when these identically redshifting clocks are introduced in the gravitational field of a rotating source, a splitting in the redshift of the flavor–oscillation clocks of the second class takes place with respect to the flavor–oscillation clocks of the first class.
We will consider the flavor states to be at rest at a fixed position in the gravitational environment.[^3] Therefore, for the situation under consideration one need not worry about whether the particles are described by Klein–Gordon equation, or Dirac equation, or an equation for some higher spin [@hs].[^4] It will suffice to know that each mass eigenstate has $(2s+1)$ spinorial degrees of freedom, and that each of these degrees of freedom evolves in time as $$\vert m_k,s_j\rangle \rightarrow
\exp\left[-\frac{i H t}{\hbar}\right] \vert m_k,s_j\rangle\quad.$$ The redshift of the flavor–oscillation clocks is determined by the gravitationally induced relative phases between various mass eigenstates. Each of the mass eigenstates, $\vert m_k,s_j\rangle$, picks up a gravitationally induced phase. This phase, in general, depends on $m_k$, and the relative orientation of $s_j$ with respect to $\vec J$ and $\vec r$ ($\vec r\,\, =$ position of the test paricle). Various mass eigenstates develop relative phases as the quantum test particle evolves in a given gravitational environment. [*These relative phases depend not only on the gravitational source but also on the specific quantum mechanical characteristics of the quantum test particle as contained in $A_{kj}$. This introduces the essential [*non-geometric*]{} element when gravitational and quantum phenomena are considered simultaneously.*]{} Something quite close to this was already realized by Sakurai [@JJS p. 129] when, in the context of the celebrated Colella, Overhauser, and Werner experiment on neutron interferometry [@COW] he wrote “This experiment also shows that gravity is not purely geometric at the quantum mechanical level because the effect depends on $(m/\hbar)^2$,” but noting immediately, “However, this does not imply that the equivalence principle is unimportant in understanding an effect of this sort. If the gravitational mass ($m_{grav}$) and inertial mass ($m_{inert}$) were unequal, $(m/\hbar)^2$ would have to be replaced by $ m_{grav}\,m_{inert}/\hbar^2$.”
We take the following as working definitions. [*Geometrical elements*]{} are those that are completely specified by the gravitational source. [*Non-geometrical*]{} elements are those that crucially depend on the specific details of quantum test particles and do not follow from general relativity alone.
In a weak gravitational field of mass,[^5] $M$, with spin angular momentum $\vec J(=J\hat z,\,\,\mbox{for convenience})$, the evolution of the mass eigenstates is governed by the Hamiltonian [@LT] $$H= {\bf m} c^2 - \frac{\hbar^2}{2 m}\,
\nabla^2 -\,\frac{G {M} {\bf m}} {r} -
\left(\frac{{\vec{\bf s}}}{2}\right)\cdot \vec b\quad,\label{h}$$ and the non-relativistic Schrödinger equation (recall that we shall set $\vec p = \vec 0$ for all mass eigenstates). In Eq. (\[h\]), the gravitomagnetic field $\vec b$ is given by [@CW Eq. 6.1.25] $$\vec b \equiv\frac{2 G}{c^2} \left[
\frac{ {\vec J} - 3({\vec J}\cdot\hat { r})\hat { r} }{r^3}\right]
\quad,$$ and $\vec {\bf s}= {\bf s_z}\,\hat z$. The quantum mechanical operators that appear in Eq. (\[h\]) are defined as follows: ${\bf m} \vert m_k,s_j\rangle = m_k \vert m_k,s_j\rangle$ and ${{\bf s}}_z \vert m_k,s_j\rangle = s_j \vert m_k,s_j\rangle$.[^6] $M$ and $\vec J$ will be treated as classical gravitational sources. This framework is a natural extension of arguments that were first put forward by Overhauser, and Colella [@OC], and Sakurai [@JJS pp. 126-129].
Consider the simplest case with two distinct mass eigenstates of spin one half. That is, set $n=2$ and $s=1/2$. Further, introduce two sets of “flavor” states, one set where both the mass eigenstates have $s_j$ in the same relative orientation, and the other set where $s_j$ are oriented in opposite directions. First set: $$\vert Q_a \rangle \equiv \cos \theta \vert m_1, \uparrow \rangle +
\sin \theta \vert m_2, \uparrow \rangle\,, \,\,
\vert Q_b \rangle \equiv -\sin \theta \vert m_1, \uparrow \rangle +
\cos \theta \vert m_2, \uparrow \rangle \,\,.\label{Q1}$$ Second set: $$\vert Q_A \rangle \equiv \cos \theta \vert m_1, \uparrow \rangle +
\sin \theta \vert m_2, \downarrow \rangle\,,\,\,
\vert Q_B \rangle \equiv -\sin \theta \vert m_1, \uparrow \rangle +
\cos \theta \vert m_2, \downarrow \rangle \,\,.\label{Q2}$$
It should be emphasized that $
m_1 \ne m_2.
$ In addition, without a loss of generality, we take $m_2 > m_1$. For convenience and simplicity of the arguments, we have set $\vec p_k = \vec 0$; $\uparrow$ indicates $s_j = + \hbar/2$, and $\downarrow$ represents $s_j = - \hbar/2$.
To arrive at the stated result we now proceed in three steps.
[**I.**]{} In the absence of gravity, let us, at time $t=0$, prepare a system in state $\vert Q_a\rangle$. The flavor–oscillation probability at a later time $t$ that the system is found in state $\vert Q_b\rangle$ is: $$\begin{aligned}
&&P_{a\rightarrow b}(t) \nonumber\\ &&\quad= \left\vert
\langle Q_b \vert \left\{
\exp\left[-\frac{i m_1 c^2 t}{\hbar}\right]
\cos \theta \vert m_1, \uparrow \rangle +
\exp\left[-\frac{i m_2 c^2 t}{\hbar}\right]
\sin \theta \vert m_2, \uparrow \rangle
\right\}\right\vert^{\,2} \nonumber\\
&& \quad =
\sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0\right]\quad,\label{fo}\end{aligned}$$ where the kinematically induced phase, $\varphi^0$, is $$\varphi^0\equiv \frac{(m_2-m_1) c^2 t}{2\hbar}\quad.\label{d}$$
The similarly defined probability of flavor oscillation for $\vert Q_A\rangle\,\rightarrow\,\vert Q_B\rangle$ is the same as above: $$P_{A\rightarrow B}(t) = P_{a\rightarrow b}(t)\quad.\label{pab}$$
The characteristic time of flavor–oscillations, $ a \rightleftharpoons b$ and $A \rightleftharpoons B$, is $$T^0=\frac{2\hbar}{(m_2-m_1)c^2}\quad.$$
Thus, the phenomenon of the flavor–oscillation provides a quantum mechanical clock. In the absence of gravity, the flavor–oscillation clocks, $\{ a \rightleftharpoons b,\,\,A
\rightleftharpoons B\}$, are characterized by the same characteristic time of flavor–oscillations.[^7]
[**II.**]{} Next, we study the test particle evolution in the vicinity of a non–rotating source of gravity.[^8]
The above defined probabilities are now modified by the gravitationally induced relative phases (each of the $m_k$ picks up a [*different*]{} phase from the gravitational field): $$\begin{aligned}
&&P_{a\rightarrow b}(t) =
P_{A\rightarrow B}(t)
\nonumber\\ &&\quad
= \left\vert
\langle Q_b \vert \left\{
\exp\left[-i\varphi_1 \right]
\cos \theta \vert m_1, \uparrow \rangle +
\exp\left[-i\varphi_2\right]
\sin \theta \vert m_2, \uparrow \rangle
\right\}\right\vert^{\,2} \quad,\end{aligned}$$ where $$\begin{aligned}
\varphi_1
= \left(
m_1 c^2 - \frac{GM m_1}{r} - \frac{\hbar \hat z}{4}\cdot\vec b
\right)\frac{t}{\hbar},\,\,
\varphi_2
= \left(
m_2 c^2 - \frac{GM m_2}{r} - \frac{\hbar \hat z}{4}\cdot\vec b
\right)\frac{t}{\hbar}\,\,.\nonumber\\\end{aligned}$$ The gravitationally induced phases in $\varphi_1$ and $\varphi_2$ that arise from the $\vec{\bf s}\cdot\vec b$ part of $H$, Eq. (\[h\]), are identical. Therefore, they do not contribute to the redshift–giving relative phase. The only contribution to the redshift–giving relative phase comes from the part of the phases that are proportional to $[GM m_k/r]t/\hbar$; $k=1,2$. A straightforward calculation yields: $$P_{a\rightarrow b}(t) =
P_{A\rightarrow B}(t) = \sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0-\varphi^M\right]\quad.$$ The gravitationally induced phase, $\varphi^M$, reads: $$\varphi^M \equiv \frac{GM}{c^2 r}\,\varphi^0\quad. \label{gm}$$ In $\varphi^M$, $
{GM}/{c^2 r}
$ is the dimensionless gravitational potential, $- \Phi$, due to a gravitational source of mass $M$. The flavor–oscillation clocks $\{a \rightleftharpoons b,\,
A \rightleftharpoons B\}$ redshift, and redshift identically, as expected in the geometric interpretation of general relativity. The phase $\varphi^M$ is similar to the one first considered by Good for the $K^0$–$\overline{K}^0$ mesons [@Good] and studied in greater detail by Aronson, Bock, Cheng, and Fischbach [@ABCF], and Goldman, Nieto, and Sandberg [@GNS].[^9] The gravitationally induced neutrino–oscillation phase given in Eq. (12) of Ref. [@GRF96] is a generalization of $\varphi^M$ to the relativistic case. However the gravitationally induced fractional change in the kinematic phase $\varphi^0$, $- \,\varphi^M/\varphi^0$, is still found to be the same, i.e., equal to $- \,{GM}/{c^2 r}$, for both the relativistic and non-relativistic cases. This equality assures that in the environment of a spherically symmetric non–rotating gravitational source, flavor–oscillation clocks, $\{a \rightleftharpoons b,\,
A \rightleftharpoons B\}$, redshift identically.
[**III.**]{} Finally, let us consider the test particle evolution in the vicinity of a rotating source of gravity.
The above defined probabilities are now modified by the gravitationally induced relative phases, and these phases depend not only on $m_k$ but also on the $s_j$ structure of the test particle (as contained in $A_{kj}$). In the first case, i.e., $a \rightleftharpoons b$ oscillation, the phase due to the $\vec {\bf s}\cdot \vec b$ interaction is same for both the $m_k$ and hence does not contribute to the flavor–oscillation probability. In the second case, i.e., $A \rightleftharpoons B$ oscillation, the phase due to the $\vec {\bf s}\cdot \vec b$ interaction is opposite for the two $m_k$’s (and hence becomes relative) and contributes to the flavor–oscillation probability. The gravitationally modified flavor–oscillation probabilities are obtained to be: $$\begin{aligned}
&&P_{a\rightarrow b}(t) = \sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0-\varphi^M\right]\quad,\label{clockab1}\\
&&P_{A\rightarrow B}(t) = \sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0-\varphi^M - \varphi^J\right]\quad.\label{clockab2}\end{aligned}$$ The new gravitationally induced phase that appears in $A \rightleftharpoons B$ flavor oscillations via Eq. (\[clockab2\]) is $$\varphi^J =
\left( 2\,\frac{\vec {\bf s}_z}{2}\cdot \vec b\right)
\frac{t}{2\hbar} =
\frac{G}{c^2}\left(
\frac{J-3({\vec J}\cdot\hat { r})(\hat { r}\cdot{\hat z})}{r^3}
\right)
\frac{t}{2}\quad.\label{gj}$$ This new contribution, $\varphi^J$, to the gravitationally induced phases is a natural, but conceptually important, extension of Good’s original considerations.[^10] Comparison of Eqs. (\[fo\]) and (\[pab\]) with Eqs. (\[clockab1\]) and (\[clockab2\]) yields the result that the flavor–oscillation clock $a \rightleftharpoons b$ redshifts as if $\vec J$ were absent, while the redshift of the flavor–oscillation clock $A \rightleftharpoons B$ depends on both $M$ and $\vec J$. If the clocks $a \rightleftharpoons b$ and $A \rightleftharpoons B$ are reintroduced in the environment of a non–rotating source they will run in synch, i.e., identically. This is the central result of this essay. Conceptually, this situation may be considered as a rough gravitational analog of the Zeeman effect of atomic physics.[^11]
The quantum–mechanically induced gravitational phase $\varphi^J$ does not depend on $\hbar$. This $\hbar$ independence is a generic feature of all interaction Hamiltonians that depend linearly on the Planck constant. However, what is remarkable here is that the relevant interaction Hamiltonian that gives rise to the non–geometric element obtained here, turns out, as a consequence of the equality of the inertial and gravitational masses, to be precisely of the form that removes $\hbar$ dependence in the redshift–splitting phase $\varphi^J$.
We now discuss in a little greater detail the origin of the gravitationally induced phases. First, consider a non-rotating gravitational source. For a single mass eigenstate the classical effects of gravitation may be considered to depend on a force, $\vec{F}$, while the quantum–mechanical effects are determined by the gravitational interaction energy, $H^M_{int}$. In the weak field limit, the interaction energy and the force are given, respectively, by $$H^M_{int} = m\, \phi\,,\quad
\vec{F} = - \vec{\nabla}\,H^M_{int}\quad,$$ where the gravitational potential $\phi = - GM/r$. Along an eqi–$\phi$ surface the $\vec{F}$ vanishes and there are no classical effects in this direction. The constant potential along a segment of an eqi–$\phi$ surface can be removed by going to an appropriately accelerated frame. Quantum mechanically, the mass eigenstate picks up a global phase factor $\exp\left[- i\, m \,\phi\, t/\hbar \right]$. Again, there are no physical consequences.
If we now consider a physical state that is in a linear superposition of different mass eigenstates, then relative phases are induced between various mass eigenstates. This happens because the phase, $\exp\left[- i\, m \,\phi\, t/\hbar \right]$, depends on mass, $m$, of the mass eigenstate, and by construction each mass eigenstate carries a different mass. These relative phases are observable as flavor–oscillation phases.[^12] Specifically, along an eqi–$\phi$ surface the gravitational force $\vec{F}$ vanishes, while the relative quantum mechanical phases induced in the evolution of a linear superposition of mass eigenstates do not. For the flavor–oscillation clocks, $a \rightleftharpoons b$ and $A \rightleftharpoons B$, the general expression of the gravitationally induced flavor–oscillation phase, $\varphi^M$, is given by Eq. (\[gm\]).
For a rotating gravitational source, all of the above observations still remain valid. However, in addition, one must now consider the gravitomagnetic interaction energy and the torque, given, respectively, by $$H_{int}^J = -\,\left(\frac{ {\vec{\bf s}}}{2}\right) \cdot \vec b\,,\quad
\vec \tau =
\left(\frac{ {\vec{\bf s}}} {2}\right) \times \vec b\quad.$$
For the set of flavor states $\{\vert Q_A \rangle,\vert Q_B \rangle\}$, where ${\vec{\bf s}}
= (\sigma_z \hbar/2)\hat z$ and $\vec J= J\hat z$, , to empasize the differences between quantum and classical considerations, we study the evolution at $\vec r = r \hat z$, it is immediately clear that there is no classical effect (as far as their precession is concerned[^13]) on the individual spins because of the vanishing $\vec \tau$, whereas quantum mechanically the flavor–oscillation evolution is determined by $H^J_{int}$–dependent phases, and these phases are non-zero (and opposite for the two spin configurations superimposed in the set $\{\vert Q_A \rangle,\vert Q_B \rangle\}$). The general expression for the gravitationally induced flavor–oscillation phase $\varphi^J$ is given by Eq. (\[gj\]).[^14]
In this essay we presented a generalized notion of flavor–oscillation clocks. The generalization contains the element that various superimposed mass eigenstates may have different relative orientation of the component of their spin with respect to the rotational axis of the the gravitational source. It is found that these quantum mechanical clocks do not always redshift identically when moved from the gravitational environment of a non–rotating source to the field of a rotating source. The gravitationally induced non–geometric phase $\varphi^J$ is independent of $\hbar$ despite its origins in quantum mechanical phases. Finally, by interchanging the spin–projections associated with $m_1$ and $m_2$ in the the states $\vert Q_A\rangle$ and $\vert Q_B\rangle$ we introduce a [*third*]{} set of flavors: $$\begin{aligned}
\vert Q_{A^\prime} \rangle \equiv \cos \theta \vert m_1, \downarrow
\rangle +
\sin \theta \vert m_2, \uparrow \rangle\,,\,\,
\vert Q_{B^\prime} \rangle \equiv -\sin \theta \vert m_1, \downarrow \rangle +
\cos \theta \vert m_2, \uparrow \rangle \,\,.\label{Q3}\nonumber\\\end{aligned}$$ This results in replacing Eqs. (\[clockab1\]) and (\[clockab2\]) by: $$\begin{aligned}
&&P_{a\rightarrow b}(t) = \sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0-\varphi^M\right]\quad,\label{clockab1p}\\
&&P_{A\rightarrow B}(t) = \sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0-\varphi^M - \varphi^J\right]\quad,\label{clockab2p}\\
&&P_{A^\prime\rightarrow B^\prime}(t)
= \sin^2\left(2\theta\right)\,\sin^2\left[
\varphi^0-\varphi^M + \varphi^J\right]\quad.\label{clockab3p}\end{aligned}$$ The flavor–oscillation clocks $\{ a \rightleftharpoons b, \,A \rightleftharpoons B,\,
A^\prime \rightleftharpoons B^\prime\}$ form the maximal set of flavors for spin one half.[^15] The sign of the gravitational phase induced by the $\vec{\bf s}\cdot\vec J$ term in the Hamiltonian carries [*opposite*]{} signs for the $\,A \rightleftharpoons B$ and $A^\prime \rightleftharpoons B^\prime$ flavor–oscillations. If one agrees to measure redshift average with respect to an equally weighted ensemble of the three flavor types, then the non–geometric element averages out to zero. Extension to higher spins being obvious, we propose that the non–geometric element in redshifts may be interpreted as a quantum mechanically induced fluctuation over a geometric structure of space–time. In the weak field limit, the amplitude of these fluctuations is directly proportional to the product of the rotation, as measured by $\vec J$, and the spin of the test particle.
[**Acknowledgements**]{} The author is grateful to Drs. Mikkel Johnson (LANL) and Nu Xu (LBNL) for their comments, and ensuing discussions, on several drafts of this manuscript.
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Overhauser, A. W., and Colella, R. (1974). [*Phys. Rev. Lett.* ]{} [**33**]{}, 1237.
Good, M. L. (1961). [*Phys. Rev.*]{} [**121**]{}, 311.
Aronson, S. H., Bock, G. J., Cheng, H.-Y., and Fischbach, E. (1983). [*Phys. Rev. D*]{} [**28**]{}, 495; and references therein.
Goldman, T., Nieto, M. M., and Sandberg, V. D. (1992). [*Mod. Phys. Lett. A*]{} [**7**]{}, 3455.
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Ahluwalia, D. V., and Burgard, C. (1996). [*Gen. Rel. and Grav.*]{} [**28**]{}, 1161. Errata: In Eqs. (7), (8), (11), and (12) the $2\hbar$ should be replaced by $4 \hbar$.
Ahluwalia, D. V., and Burgard, C. (1996). [*Phys. Rev. D*]{}, submitted. Also see Ahluwalia, D. V. (1994). [*Phys. Lett. B*]{} [**339**]{}, 301; Kempf, A., and Mangano, G. (1996). hep-th/9612084; and references therein. The following related papers came to the author’s attention after the present manuscript was completed: Viola, L., and Onofrio, R. (1997). [*Phys. Rev. D*]{} [**55**]{}, 455; Wajima, S., Kasai, M., and Futamase, T. (1997). [*Phys. Rev. D*]{} [**55**]{}, 1964.
[^1]: E-mail: av@p25hp.lanl.gov
[^2]: For a global evolution one may need to consider test particles that are characterized by Casimir invariants associated with global space–time symmetries of the relevant gravitational source [*a la*]{} Feza Gürsey [@FG].
[^3]: See, for example, Ref. [@OR Sec. 3.6] for the usual operational procedure for such a general relativistic setup. In particular, note [@OR pp. 166,167]:
> “Since the behavior of freely falling clocks is completely predictable from the principle of equivalence, we will use freely falling clocks for all our measurements in the gravitational field, even measurements at a fixed position, for instance, a measurement at a fixed position on the surface of the Earth. For this purpose, we use a freely falling clock, instantaneously at rest at the fixed position. As soon as the clock has fallen too far from our fixed position and acquired too much speed, we must replace it by a new clock, instantaneously at rest. Whenever we speak of the time as measured by “a clock located at a fixed position” in a gravitational field, this phrase must be understood as shorthand for a complicated measurement procedure, involving many freely falling, disposable clocks, used in succession.”
[^4]: The essential elements of the structure that are needed are not the vector, nor the spinor fields, but their spin–independent time evolution as $\exp[\pm i\,p_\mu x^\mu]$.
[^5]: That is, we keep terms that are of first order in the dimensionless parameter $-GM/c^2\,r$.
[^6]: We follow the notation that boldface letters represent quantum mechanical operators.
[^7]: Without the requirement $m_1\ne m_2$, $\varphi^0$ would identically vanish and no flavor–oscillation clock shall exist. The flavor–oscillation clocks have no classical counterpart. If $m_1=m_2$, then the nearest classical counterpart is a gyroscope. For instance with $m_1=m_2=m$ and $\theta=\pi/4$, $\vert Q_A\rangle$ becomes $\vert m,\rightarrow\rangle$ and $\vert Q_A\rangle$ becomes $\vert m,\leftarrow\rangle$. These are the spin polarized states along the positive and negative x-direction.
[^8]: We shall assume that various parameters are so chosen that the time scale, $T\left(\vert m_k,\uparrow\rangle\rightleftharpoons
\vert m_k,\downarrow\rangle\right) $, associated with the gravitationally induced transitions, is much larger compared with the characteristic time of flavor–oscillations, i.e., $
T\left(\vert m_k,\uparrow\rangle \rightleftharpoons
\vert m_k,\downarrow\rangle\right) \gg T^0
$, and that clocks can be discarded and replaced with the new ones following a simple extension of the operational procedure [@OR] outlined in footnote 3. I am thankful to Dr. A. Mondragón (IFUNAM) for a remark on this matter.
[^9]: However, based on the general considerations of Grossman and Lipkin (where they consider neutrino oscillations), note should be made that the sign of the phase is determined by whether one is considering oscillations in time or distance [@GL].
[^10]: For a detailed discussion of the gravitational phase $\varphi^M$ and its relationship with the pioneering work of Colella, Overhauser, and Werner on neutron interferometry [@COW], and its extension to neutrino oscillations in astrophysical contexts, the reader is referred to Refs. [@GRF96; @AB].
[^11]: I thank Dr. Nu Xu (LBNL) for bringing this analogy to my attention.
[^12]: The observability of these phases is not for a local observer, but for an observer making measurements stationed at a different eqi-$\phi$ surface.
[^13]: Note, the force exerted on the spin $$\vec F =\left(\frac{1}{2} \vec{\bf s}\cdot \nabla\right)\vec b\quad,$$ is non-zero.
[^14]: For a detailed discussion of gravitational redshift in the presence of rotation, though confined to classical test particles, the reader is referred to Ciufolini and Wheeler [@CW].
[^15]: The reversing of spin–projections in states $\vert Q_a\rangle$ and $\vert Q_b\rangle$ may be used to include a fourth flavor. Such an addition to the maximal set does not alter our conclusions.
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abstract: 'In this paper we derive the Markowitz-optimal, deterministic-execution trajectory for a trader who wishes to buy or sell a large position of a share which evolves as a geometric Brownian motion in contrast to the arithmetic model which prevails in the existing literature. Our calculations include a general temporary impact, rather than a specific function. Additionally, we point out—under our setting—what are the necessary ingredients to tackle the problem with adaptive execution trajectories. We provide a couple of examples which illustrate the results. We would like to stress the fact that in this paper we use understandable user-friendly techniques.'
address:
- |
Statistics Department, Columbia University\
1255 Amsterdam Ave. Room 1005, New York, N.Y., 10027.
- |
Mathematics Department, CINVESTAV-IPN\
Av. Instituto PolitŽcnico Nacional \#2508, Col. San Pedro Zacatenco, México, D.F., C.P. 07360
author:
- 'Gerardo Hernandez-del-Valle'
- 'Carlos G. Pacheco-Gonzalez'
date: 'October 14, 2009'
title: Optimal execution of equity with geometric price process
---
Introduction
============
The problem of optimal execution is a very general problem in which a trader who wishes to buy or sell a [*large*]{} position $K$ of a given asset $S$—for instance wheat, shares, derivatives, etc.—is confronted with the dilemma of executing slowly or as quick as possible. In the first case he/she would be exposed to volatility, and in the second to the laws of offer and demand. Thus the trader most hedge between the [*market impact*]{} (due to his trade) and the [*volatility*]{} (due to the market).
The key ingredients to study this optimization problem are: (1) The modeling of the asset—which is typically modeled as a geometric Brownian process. (2) The modeling of the so-called market impact which heuristically suggests the existence of an instantaneous impact—so-called [*temporary*]{}—and a cumulative component referred to as [*permanent*]{}. And finally (3) one should establish a criteria of optimality.
The main [*aim*]{} of this paper is to study and characterize the so-called Markowitz-optimal open-loop execution trajectory, in terms of nonlinear second order, ordinary differential equations (Theorems \[thm1\] and \[thm2\]). The above is done for a trader who wishes to buy or sell a large position $K$ of shares $S$ which evolve as a geometric Brownian motion (although in the existing literature it is considered an arithmetic Brownian motion for this problem). In this paper we only deal with deterministic strategies, also called [*open loop*]{} controls; however, we point out—in Section \[cinco\]—the key ingredients to address the problem with adaptive strategies, also termed [*Markovian*]{} controls (work in progress).
The main [*motivation*]{} of this work is on one hand economic, but on the other the effect of [*market impact*]{} in the valuation of contingent claims, and its connection with optimal execution of [*derivatives*]{}. Intuitively, for this kind of problem, one would expect to consider adaptive strategies to tackle the questions, although it seems natural to first understand the deterministic case. An important element in our analysis, will be the use of a linear stochastic differential equation, first used—to our knowledge—by Brennan and Schwartz (1980) in their study of interest rates.
The problem of minimizing expected overall liquidity costs has been analyzed using different market models by Bertsimas and Lo (1998), Obizhaeva and Wang, and Alfonsi et al. (2007a,2007b), just to mention a few. However, these approaches miss the volatility risk associated with time delay. Instead, Almgren and Chriss (1999,2000), suggested studying and solving a mean-variance optimization for sales revenues in the class of deterministic strategies. Further, on Almgren and Lorenz (2007) allowed for intertemporal updating and proved that this can [*strictly*]{} improve the mean-variance performance. Nevertheless, in Schied and Schöneborn (2007), the authors study the original problem of expected utility maximization with CARA utility functions. Their main result states that for CARA investors there is surprisingly no added utility from allowing for intertemporal updating of strategies. Finally, we mention that the Hamilton-Jacobi-Bellman approach has also recently been studied in Forsyth (2009). Our paper is organized as follows: In Section 2 we introduce our model, assumptions and auxiliary results. Namely through a couple of subsequent Propositions we characterize and compute—by use of a Brennan-Schwartz type process (\[brennan\])—the moments of certain random variable that is relevant in our optimization problem. Section 3 is devoted to deriving and proving the characterization of our optimal trading strategies and the optimal value function as well. After that, in Section 4, we present a couple of examples in order to exemplify the procedure derived in Section 3. We first compare Almgren and Chriss trajectory with ours, and in Example 2 we use a temporary market impact $h$ to the power $3/5$ as suggested in the empirical study of Almgren [*et. al.*]{}’s (2005). We conclude the paper, in Section 5, by pointing out the key ingredients in the study of adaptive execution strategies.
Auxiliary Results
=================
In this section, we describe the dynamics of the asset $S$, the so-called market impact, and we introduce the Brennan-Schwartz process. This process will allow us not only to compute the moments of the optimization argument, but also to represent it in terms of an SDE. For the remainder of this section, let $c(t)$ be a fixed and differentiable function for $0\leq t\leq T<\infty$.
The model.
----------
Let the price of the share $S$ of a given company evolve as a geometric process, where the random component $B$ is standard Brownian motion, i.e. $$\begin{aligned}
dS_t&=&S_t\left[\left(g(c(t))+\frac{dh(c(t))}{dt}\right)dt+\sigma dB_t\right],\qquad S_0=s,\end{aligned}$$ and where $g$ and $h$ represent respectively the permanent—which accumulates over time—and instantaneous temporary impact. Thus, the [*future*]{} effective price per share due to our trade can be modelled as $$\begin{aligned}
\label{price}
S_t=s\exp\left\{\int_0^ug(c(v))dv+h(c(u))-\frac{1}{2}\sigma^2u+\sigma B_u\right\},\end{aligned}$$ where $\sigma>0$ is an estimable parameter.\
\(a) Note that that process $\ln(S)$, where $S$ is as in (\[price\]), coincides precisely with the “standard” notion of both permanent and temporary impact. That is, if we model the price process as arithmetic Brownian motion: $$\begin{aligned}
dS_t=\sigma dB_t+g(c(t))dt+\frac{dh}{dt}(c(t)),\qquad S_0=s\end{aligned}$$ then $$\begin{aligned}
S_t-s=\int_0^tg(c(u))du+h(c(t))+\sigma B_t.\end{aligned}$$ Hence, the first term in the right-hand side of the equality is [*accumulating*]{} over time, on the other hand, the second term is [*not*]{}.\
(b) Next, observe that the process $c$ can be thought of as a control, which in turn may be:
1. an admissible process $c$ which is adapted to the natural filtration $\mathcal{F}^S$ of the associated process (\[price\]) is called a [*feedback control*]{},
2. an admissible process $c$ which can be written in the form $c_t=u(t,S_t)$ for some measurable map $u$ is called [*Markovian control*]{}, notice that any Markovian control is a feedback control,
3. a deterministic processes of the family of admissible controls are called [*open loop controls*]{}.
\(c) In this paper we will only deal with open loop controls, yet to study feedback controls it will become quite useful to introduce the so-called Brennan-Schwartz process, introduced in the next subsection. The reason being, that to derive the Hamilton-Jacobi-Bellman equation—see Section 5—it is more convenient to have a diffusion instead of an average of a diffusion.
Averaged geometric and Brennan-Schwartz processes
-------------------------------------------------
In order to study our problem we will introduce the following averaged geometric Brownian process: $$\begin{aligned}
\label{chi}
\xi_t&:=&\int_0^tc(u)S_udu,\qquad t\geq 0.\end{aligned}$$ By (\[price\]) we can express $\xi_t$ as $$\begin{aligned}
\nonumber \xi_t&=&\int_0^t c(u)se^{\int_0^ug(c(v))dv+h(c(u))-\frac{1}{2}\sigma^2u+\sigma B_u}du.\end{aligned}$$ Thus, if $c(u)$ represents the number of shares bought or sold at time $u$ at price $S_u$, then $\xi_t$ represents the total amount spent or earned by the trader up to time $t$.
To compute the moments of $\xi$ we will use the following linear non-homogeneous stochastic differential equation $$\begin{aligned}
\label{brennan}
dX_t&=&\left[c(t)-\left(g(c(t))+\frac{dh(c(t))}{dt}-\sigma^2\right)X_t\right]dt-\sigma X_tdB_t\\
\nonumber X_0&=&0,\end{aligned}$$ which has been used for instance by Brennan and Schwartz (1980) in the modeling of interest rates, by Kawaguchi and Morimoto (2007) in environmental economics, and which may also be used to study the density of averaged geometric Brownian motion \[see for instance, Linetsky (2004)\].
In general, its usefulness is due to the fact that one may construct a Brennan-Schwartz process $X$ which satisfies $$X\stackrel{\mathcal{D}}{=}\xi,$$ where $\stackrel{\mathcal{D}}{=}$ stands for equality in distribution. In this paper, it is used, together with Itô’s lemma to show that $\xi=S\cdot X$, which alternatively will allow us to compute the second moment of $\xi$ in terms of an iterated integral, indeed:
\[prop1\] Let processes $S$, $\xi$, and $X$ be as in (\[price\]), (\[chi\]), and (\[brennan\]), respectively, then $$\begin{aligned}
\xi_t=S_t\cdot X_t,\qquad t\geq 0\end{aligned}$$ and $$\begin{aligned}
d\xi^2_t=2\xi_tS_t c(t)dt.\end{aligned}$$
By Itô’s lemma $$\begin{aligned}
d(S_t\cdot X_t)&=&S_tdX_t+X_tdS_t+dX_tdS_t\\
&=&S_tc(t)dt-S_tX_t\left(g(c(t))+\frac{dh(c(t))}{dt}\right)dt+S_tX_t\sigma^2dt\\
&&-\sigma S_tX_tdB_t+X_tS_t\left(g(c(t))+\frac{dh(c(t))}{dt}\right)dt\\
&&+\sigma X_tS_tdB_t-\sigma^2S_tX_tdt\\
&=&c(t)S_tdt\\
&=&\xi_t.\end{aligned}$$ Furthermore, for the second moment of $\xi$ it follows that $$\begin{aligned}
d\xi^2_t&=&2X_tS^2_tdX_t+2X^2_tS_tdS_t+S^2_t(dX_t)^2+X^2_t(dS_t)^2\\
&&+4X_tS_t(dX_t\cdot dS_t)\\
&=&2X_tS^2_tsc(t)dt-2X_tS^2_t\left(g(c(t))+\frac{dh(c(t))}{dt}\right)X_tdt\\
&&+2X_tS^2_t\sigma^2X_tdt-2X_tS^2_t\sigma X_tdB_t\\
&&+2X^2_tS_t\left(g(c(t))+\frac{dh(c(t))}{dt}\right)S_tdt+2X^2_tS_t\sigma S_tdB_t\\
&&+S^2_t\sigma^2X^2_tdt+X^2_t\sigma^2S^2_tdt-4X_tS_t\sigma^2X_tS_tdt\\
&=&2X_tS^2_tc(t)dt\\
&=&2\xi_t S_t c(t)dt,\end{aligned}$$ as claimed.
The previous proposition may be derived directly from the integration by parts formula. Yet, this characterization will be useful in the study, for instance, of the optimal trading schedule of derivatives or in the determination of Markovian controls.
Moments of $\xi$
----------------
Now, by Proposition \[prop1\], it is straightforward to compute the first two moments of $\xi_t$ which will be used to solve our optimal execution problem.
Let $\xi$ be as in (\[chi\]). Then $$\begin{aligned}
\label{uno}\mathbb{E}[\xi_t]&=&\int_0^t c(u)s\exp\left\{\int_0^ug(c(v))dv+h(c(u))\right\}du \\
\label{dos}\mathbb{E}[\xi^2_t]&=&2\int_0^tc(u)se^{\int_0^ug(c(n))dn+h(c(u))}\\
\nonumber&&\quad \times\left(\int_0^uc(v)se^{\int_0^vg(c(w))dw+h(c(v))+\sigma^2v}dv\right)du.\end{aligned}$$
By (\[chi\]): $$\begin{aligned}
\mathbb{E}[\xi_t]&=&\int_0^tc(u)s\mathbb{E}[S_u]du\\
&=&\int_0^t c(u)s\exp\left\{\int_0^ug(c(v))dv+h(c(u))\right\}du.\end{aligned}$$ From Proposition \[prop1\]: $$\begin{aligned}
\mathbb{E}[\xi^2_t]&=&2\mathbb{E}\left[\int_0^tc(u)sS_u\xi_udu\right]\\
&=&2\mathbb{E}\left[\int_0^tc(u)s^2S_u\left(\int_0^uc(v)sS_vdv\right)du\right]\\
\nonumber&=&2\int_0^tc(u)s^2\left(\int_0^uc(v)s\mathbb{E}[S_uS_v]dv\right)du.\end{aligned}$$ Therefore, since $$\begin{aligned}
\mathbb{E}\left[e^{\sigma B_u+\sigma B_v}\right]&=&\mathbb{E}\left[e^{\sigma(B_u-B_v)+2\sigma B_v}\right]\\
&=&\mathbb{E}\left[e^{\sigma(B_u-B_v)}\right]\mathbb{E}\left[e^{2\sigma B_v}\right]\\
&=&e^{\frac{1}{2}\sigma^2(u-v)}e^{2\sigma^2 v},\end{aligned}$$ it follows that $$\begin{aligned}
\mathbb{E}[\xi^2_t]&=&2\int_0^tc(u)se^{\int_0^ug(c(n))dn+h(c(u))}\\
&&\quad \times\left(\int_0^uc(v)se^{\int_0^vg(c(w))dw+h(c(v))+\sigma^2v}dv\right)du.\end{aligned}$$
Markowitz Optimal open-loop Trading trajectory
==============================================
In this section we derive a Markowitz-optimal open-loop trading strategy, Theorem \[thm1\] and \[thm2\], employing the auxiliary results derived in the previous section.
Execution shortfall
-------------------
If the size of the trade $K$ is “relatively” small we would expect the market impact to be negligible, that is, the trader should execute $K$ immediately. Thus, it seems natural to compare the actual total gains (losses) $\xi_T$ with the impact-free quantity $Ks$ by introducing the so-called execution shortfall $Y$ defined as $$\begin{aligned}
\label{short}
Y:=\xi_T-Ks.\end{aligned}$$
If we use Markowitz optimization criterion, then our problem is equivalent to finding the trading trajectory $\{c(t)|0\leq t\leq T\}$ which minimizes simultaneously the expected shortfall given a fixed risk-aversion level $\lambda$ characterized by the volatility of $Y$: $$\begin{aligned}
\label{optimize}
\nonumber J[c(\cdot)]&:=&\mathbb{E}[Y]+\lambda \mathbb{V}[Y]\\
&=&\lambda\mathbb{E}[\xi^2_T]+\mathbb{E}[\xi_T]-\lambda (\mathbb{E}[\xi_T])^2-Ks.\end{aligned}$$ In fact, if $\lambda>0$ then (\[optimize\]) has a unique solution, which may be represented in the following integral form:
\[prop3\] Suppose that the permanent impact $g$ is linear, i.e. $$g(x)=\alpha x,$$ for some $\alpha>0$ as suggested by Almgren [*et. al.*]{} (2005) empirical study. Let $$\begin{aligned}
\label{f}
f(x):=\int_0^x c(u)du,\qquad f'(x):=c(x)\end{aligned}$$ and $$\begin{aligned}
\gamma_1(u,f,f')&:=&\int_0^usf'(v)e^{\alpha f(v)+ h(f'(v))+\sigma^2 v}dv,\\
\gamma(u,f,f')&:=&\int_0^usf'(v)e^{\alpha f(v)+ h(f'(v))}dv.
%A&:=&(1+2\lambda Ks-2Ks).\end{aligned}$$ Then $J[c(\cdot)]$ in (\[optimize\]) can be expressed as: $$\begin{aligned}
\int_0^T\left\{\left[2\lambda \left(\gamma_1(u)-\gamma(u)\right)+1\right]f'(u)se^{\alpha f(u)+ h(f'(u))}-\frac{\lambda Ks}{T}\right\}du.\end{aligned}$$
Setting $$\begin{aligned}
f(x):=\int_0^x c(u)du,\end{aligned}$$ we have $f'(x):=c(x)$. Hence, using the integration by parts formula, $$\begin{aligned}
&&\int_0^tsf'(x)e^{\alpha f(x)+h(f'(x))}\left(\int_0^xsf'(y)e^{\alpha f(y)+h(f'(y))}dy\right)dx\\
&&=\left(\int_0^tsf'(x)e^{\alpha f(x)+h(f'(x))}dx\right)\left(\int_0^tsf'(x)e^{\alpha f(x)+h(f'(x))}dx\right)\\
&&\enskip -\int_0^tsf'(x)e^{\alpha f(x)+ h(f'(x))}\left(\int_0^xsf'(y)e^{\alpha f(y)+h(f'(y))}dy\right)dx\end{aligned}$$ implies $$\begin{aligned}
&&\nonumber\left(\int_0^tsf'(x)e^{\alpha f(x)+h(f'(x))}dx\right)^2\\
\nonumber&&\quad=2\int_0^tsf'(x)e^{\alpha f(x)+h(f'(x))}\left(\int_0^xsf'(y)e^{\alpha f(y)+h(f'(y))}dy\right)dx\\
\nonumber &&\quad=\left(\mathbb{E}[\xi_t]\right)^2.\end{aligned}$$ Thus, by (\[uno\]) and (\[dos\]), $$\begin{aligned}
\gamma_1(u)&:=&\int_0^usf'(v)e^{\alpha f(v)+h(f'(v))+\sigma^2 v}dv,\\
\gamma(u)&:=&\int_0^usf'(v)e^{\alpha f(v)+h(f'(v))}dv,
%A&:=&(1+2\lambda Ks-2Ks).\end{aligned}$$ It follows that $$\begin{aligned}
&&J[c(\cdot)]\\
&&\enskip=2\lambda\int_0^Tf'(u)se^{\alpha f(u)+ h(f'(u))}\gamma_1(u)du+\int_0^T f'(u)se^{\alpha f(u)+ h(f'(u))}du\\
&&\qquad-\int_0^T2\lambda f'(u)se^{\alpha f(u)+ h(f'(u))}\gamma(u)du- K s\\
&&\enskip=\int_0^T\left\{\left(2\lambda\gamma_1(u)+1-2\lambda\gamma(u)\right)f'(u)se^{\alpha f(u)+ h(f'(u))}-\frac{Ks}{T}\right\}du.\end{aligned}$$ as claimed.
Observe that this last expression has the following functional form in terms of $f$: $$\begin{aligned}
\label{j}
J(f)=\int_0^t\mathcal{L}(\gamma_1(u,f,f'),\gamma(u,f,f'),f(u),f'(u))du.\end{aligned}$$ Letting $$\begin{aligned}
F(f(u),f'(u)):=sf'(u)\exp\left\{\alpha f(u)+h(f'(u))\right\},\end{aligned}$$ we may re-express (\[j\]) as $$\begin{aligned}
J(f)=\int_0^T\left(2\lambda\int_0^uF(f(v),f'(v))(e^{\sigma^2 v}-1)dv+1\right)F(f(u),f'(u))du.\end{aligned}$$ In particular
\[thm1\] Suppose that $\lambda=0$. Then, the open-loop trading schedule $c^*$ is determined by a function $f_1$ which solves the following system $$\begin{aligned}
\label{ee}
\frac{\partial F}{\partial f_1}-\frac{d}{dz}\frac{\partial F}{\partial f'_1}=0,
\end{aligned}$$ with $f_1(0)=0$ and $f_1(T)=K$.
If $\lambda=0$, that is, if we only wish to minimize expected execution shortfall, then: $$\begin{aligned}
J(f)=\int_0^TF(f(u),f'(u))du,
\end{aligned}$$ and thus equation (\[ee\]) follows from the Euler-Lagrange equation \[see for instance Gelfand and Fomin (2000)\].
\[ex1\] [Let $T=\alpha=1$ and both temporary and permanent impact be linear, i.e. $g(x)=x$, $h(x)=x$, hence: $$\begin{aligned}
F(f(u),f'(u))=f'(u)\exp\left\{f(u)+f'(u)\right\}.\end{aligned}$$ Thus, if one wishes to minimize the [*expected*]{} execution shortfall one should execute according to: $$\begin{aligned}
f'(u)-f''(u)-(1+f'(u))(f'(u)+f''(u))=0,\end{aligned}$$ given that $f(0)=0$, and $f(1)=K$. ]{}
Note that as $\lambda$ increases the client is willing to be more exposed to risk. This idea is equivalent to saying that he/she is willing to increase the speed of execution. Hence we have that for $\lambda>0$, the optimal trajectory will dominate $f_1$: $$f(s)\geq f_1(s),\qquad 0\leq s\leq T$$ In other words we may decompose $f(s)=f_1(s)+f_2(s)$. This last observation together with our constraint $f(0)=0$, $f(T)=K$, lead to the following two facts: $$\begin{aligned}
f_2(t)&\geq& 0,\qquad 0\leq t\leq T\\
f_2(0)=f_2(T)&=&0.\end{aligned}$$
The previous remark and equation (\[j\]) suggest a 2 step procedure to find the optimal trajectory $f$. Namely, first find $f_1$ and given that information solve for $f=f_1+f_2$
\[thm2\] The optimal differentiable trajectory $f$ which solves (\[j\]) is given by $f_1+f_2$, where $f_1$ is given in Theorem 3.3 and $f_2$ satisfies for $0\leq v\leq T$: $$\begin{aligned}
&&f_2(u)\left(2\lambda\int_0^uF^1(f(v),f'(v))dv+1\right)\cdot\left[\frac{\partial F}{\partial f}-\frac{d}{du}\left(\frac{\partial F}{\partial f'}\right)\right]\\
&&\qquad +2\lambda\int_0^uf_2(v)\left[\frac{\partial F^1}{\partial f}-\frac{d}{dv}\left(\frac{\partial F^1}{\partial f'}\right)\right]dv\cdot F(f(u),f'(u))=0,\end{aligned}$$ where $f_2(0)=f_2(T)=0$.
The idea is to follow the derivation of the Euler-Lagrange equation \[see for instance, Gelfand and Fomin (2000)\], but in this case, the unknown function $f_2$ will play the role of the perturbation. Thus, it is essential the fact that $f_2(0)=f_2(T)=0$.
Let $$\begin{aligned}
f(v)&=&f_1(v)+f_2(v)\\
g_\epsilon(v)&=&f_1(v)+\epsilon f_2(v)\end{aligned}$$ where $f_2(0)=0=f_2(T)$ and $$\begin{aligned}
J(\epsilon)=\int_0^T\left(2\lambda\int_0^uF(g_\epsilon(v),g_\epsilon'(v))(e^{\sigma^2v}-1)dv+1\right)F(g_\epsilon(u),g_\epsilon'(u))du\end{aligned}$$ then $$\begin{aligned}
\frac{dJ}{d\epsilon}(\epsilon)&=&\int_0^T\left(2\lambda\int_0^u\frac{dF}{d\epsilon}(g_\epsilon(v),g_\epsilon'(v))(e^{\sigma^2v}-1)dv\right)\\
&&\enskip\times F(g_\epsilon(u),g_\epsilon'(u))du\\
&&+\int_0^T\left(2\lambda\int_0^uF(g_\epsilon(v),g_\epsilon'(v))(e^{\sigma^2v}-1)dv+1\right)\\
&&\enskip\times\frac{dF}{d\epsilon}(g_\epsilon(u),g_\epsilon'(u))du\\
&=&\int_0^T\left(2\lambda\int_0^u\left(f_2(v)\frac{\partial F}{\partial g_\epsilon}+f_2'(v)\frac{\partial F}{\partial g_\epsilon'}\right)(e^{\sigma^2v}-1)dv\right)\\
&&\enskip\times F(g_\epsilon(u),g_\epsilon'(u))du\\
&&+\int_0^T\left(2\lambda\int_0^uF(g_\epsilon(v),g_\epsilon'(v))(e^{\sigma^2v}-1)dv+1\right)\\
&&\enskip\times\left(f_2(u)\frac{\partial F}{\partial g_\epsilon}+f_2'(u)\frac{\partial F}{\partial g_\epsilon'}\right)du\end{aligned}$$ if we set $$\begin{aligned}
\gamma_\epsilon(u)=2\lambda\int_0^uF(g_\epsilon(v),g_\epsilon'(v))(e^{\sigma^2v}-1)dv+1\end{aligned}$$ and by the integration by parts formula we have that $$\begin{aligned}
\frac{dJ}{d\epsilon}(\epsilon)&=&\int_0^T\Bigg{(}2\lambda\int_0^uf_2(v)(e^{\sigma^2v}-1)\left[\frac{\partial F}{\partial g_\epsilon}-\frac{d}{dv}\left\{\frac{\partial F}{\partial g_\epsilon'}\right\}\right]dv\\
&&\quad-2\lambda\int_0^uf_2(v)\frac{\partial F}{\partial g_\epsilon'}\sigma^2e^{\sigma^2v}dv\\
&&\quad+2\lambda f_2(u)\frac{\partial F}{\partial g_\epsilon'}(e^{\sigma^2u}-1)\Bigg{)}F(g_\epsilon(u),g_\epsilon'(u))du\\
&&+\int_0^T\gamma_\epsilon(u)f_2(u)\left[\frac{\partial F}{\partial g_\epsilon}-\frac{d}{du}\left(\frac{\partial F}{\partial g_\epsilon'}\right)\right]du\\
&&-\int_0^Tf_2(u)\frac{\partial F}{\partial g_\epsilon'}\frac{d}{du}\left(\gamma_\epsilon(u)\right)du\end{aligned}$$ Let us compute $$\begin{aligned}
\frac{d}{du}(\gamma_\epsilon(u))=2\lambda F(g_\epsilon(u),g_\epsilon'(u))(e^{\sigma^2u}-1)\end{aligned}$$ which yields $$\begin{aligned}
J'(\epsilon)&=&\int_0^T\Bigg{(}2\lambda\int_0^uf_2(v)(e^{\sigma^2v}-1)\left[\frac{\partial F}{\partial g_\epsilon}-\frac{d}{dv}\left\{\frac{\partial F}{\partial g_\epsilon'}\right\}\right]dv\\
&&\qquad-2\lambda\int_0^uf_2(v)\frac{\partial F}{\partial g_\epsilon'}\sigma^2e^{\sigma^2v}dv\Bigg{)} F(g_\epsilon(u),g_\epsilon'(u))du\\
&&+\int_0^T\gamma_\epsilon(u)f_2(u)\left[\frac{\partial F}{\partial g_\epsilon}-\frac{d}{du}\left(\frac{\partial F}{\partial g_\epsilon'}\right)\right]du.\end{aligned}$$ But observe that when $J'(1)$ we have $$\begin{aligned}
J'(1)&=&\int_0^T\Bigg{(}2\lambda\int_0^uf_2(v)(e^{\sigma^2v}-1)\left[\frac{\partial F}{\partial f}-\frac{d}{dv}\left\{\frac{\partial F}{\partial f'}\right\}\right]dv\\
&&\qquad-2\lambda\int_0^uf_2(v)\frac{\partial F}{\partial f'}\sigma^2e^{\sigma^2v}dv\Bigg{)} F(f(u),f'(u))du\\
&&+\int_0^T\gamma_1(u)f_2(u)\left[\frac{\partial F}{\partial f}-\frac{d}{du}\left(\frac{\partial F}{\partial f'}\right)\right]du\\
&=&0.\end{aligned}$$ Now, given that $f_2(0)=f_2(T)=0$ and from the fundamental lemma of Calculus of variations we have that: $$\begin{aligned}
&&f_2(u)\gamma_1(u)\left[\frac{\partial F}{\partial f}-\frac{d}{du}\left(\frac{\partial F}{\partial f'}\right)\right]\\
&&\qquad +2\lambda\int_0^uf_2(v)\left[\frac{\partial F^1}{\partial f}-\frac{d}{dv}\left(\frac{\partial F^1}{\partial f'}\right)\right]dv\cdot F(f(u),f'(u))=0\end{aligned}$$ with the constraint that $f_2(u)>0$ for $u\in(0,T)$ and $f_2(T)=f_2(0)=0$. or $$\begin{aligned}
&&f_2(u)\left(2\lambda\int_0^uF^1(f(v),f'(v))dv+1\right)\cdot\left[\frac{\partial F}{\partial f}-\frac{d}{du}\left(\frac{\partial F}{\partial f'}\right)\right]\\
&&\qquad +2\lambda\int_0^uf_2(v)\left[\frac{\partial F^1}{\partial f}-\frac{d}{dv}\left(\frac{\partial F^1}{\partial f'}\right)\right]dv\cdot F(f(u),f'(u))=0\end{aligned}$$ as claimed.
\[ex2\] ([*cont*]{}.) [With linear and temporary impact as before, that is $g(x)=x$ and $h(x)=x$. You may find the optimal trading trajectory $f$ for arbitrary $\lambda\geq 0$ by first determining $f_1$, using Theorem \[thm1\], next you determine $f_2$ by use of Theorem \[thm2\]. That is, letting: $$\begin{aligned}
F(f(v),f'(v))&=&f'(v)\exp\left\{f(v)+f'(v)\right\}\\
F^1(f(v),f'(v))&=&F(f(v),f'(v))(e^{\sigma^2v}-1)\end{aligned}$$ Theorem \[thm2\] states that $f_2$ satisfies the following identity $$\begin{aligned}
&&-f_2(u)\left\{2f''(u)+f'(u)\left[f'(u)+f''(u)\right]\right\}\\
&&\qquad\times\left(2\lambda\int_0^uF^1(f(v),f'(v))dv+1\right)\\
&&\qquad+ 2\lambda f'(u)\int_0^uf_2(u)\left[\frac{\partial F^1}{\partial f}-\frac{d}{dv}\left(\frac{\partial F^1}{\partial f'}\right)\right]dv=0\end{aligned}$$ where $$\begin{aligned}
&&\frac{\partial F^1}{\partial f}-\frac{d}{du}\left(\frac{\partial F^1}{\partial f'}\right)\\
&&\quad=e^{f(u)+f'(u)}\\
&&\qquad\times-\left[\left\{2f''(u)+f'(u)(f'(u)+f''(u))\right\}(e^u-1)+(1+f'(u))e^u\right].\end{aligned}$$ ]{}
Examples
========
Example
-------
The first example we want to study is the case in which both the temporary and the permanent impact are linear as in Examples \[ex1\] and \[ex2\]. The motivation is to compare our results with those obtained by Almgren and Chriss (2000). In this example we have set $T=1$ and $K=3$. The solutions have been numerically calculated using Theorems \[thm1\] and \[thm2\] and then plotted in Figures 1 and 2.
It may be observed that, as one would expect, the solutions are not the same and in fact—under the present conditions—our strategy dominates Almgren and Chriss.
Example
-------
The next example we want to study is the case in which the permanent impact is some power less than 1. Namely $h(x)=x^{3/5}$, and the permanent is linear. First from Theorem \[thm1\] we have that $$\begin{aligned}
F(f_1(u),f'_1(u)):=sf'_1(u)\exp\left\{f_1(u)+(f'_1(u))^{3/5}\right\}\end{aligned}$$ and $f_1$ is the solution to: $$\begin{aligned}
f'_1-\left(f'_1+\frac{3}{3}\frac{f''_1}{(f'_1)^{2/5}}\right)-\left(\frac{3}{5}\right)^2\frac{f''_1}{(f'_1)^{2/5}}-\frac{3}{5}(f'_1)^{3/5}\left(f'_1+\frac{3}{5}\frac{f''_1}{(f'_1)^{2/5}}\right)=0,\end{aligned}$$ $f_1(0)=0$ and $f_1(T)=K$. In particular, with $T=1$ and $K=3$ as in the previous example we have plotted our result in Figure 3. Note that the sublinear impact has increased the speed of execution. Next, we computed the Markowitz-optimal open-loop trajectory, with $\lambda=1$ by first computing $f_2$ as described in Theorem \[thm2\].
The previous exercise suggests that if one chooses the temporary impact to be sub-linear, the solution—with all the other parameters fixed—will always dominate its linear counterpart. On the other hand, if the temporary impact is super-linear, the solution will be dominated by its linear counterpart.
A natural question is: What is the correct form of $h$ given our model?
Remarks on Markovian controls {#cinco}
=============================
As pointed out in the introduction, we have only dealt with [*differentiable*]{} deterministic controls—also known as open loop controls. Furthermore, our criteria of optimal is in the [*Markowitz*]{} sense. A couple of natural and reasonable question arise: how can we study [*feedback controls*]{}? How can we optimize with respect to general utility functions? Namely, given some utility function $U$ we want to find a trajectory $c$ such that $$\begin{aligned}
\sup_{c\in\mathcal{U}}\mathbb{E}_{t,x}[U(Y)]\end{aligned}$$ where $\mathcal{U}$ is the set of admissible controls and $Y=\xi-KT$ is the execution shortfall. But $\xi$ is in general a very difficult creature to characterize, unless you [**observe**]{} that you may construct a diffusion with the following dynamics: $$\begin{aligned}
dX_t=\left(c_t+\left[g(c_t)+\frac{dh}{dt}(c_t)\right]X_t\right)dt+\sigma X_tdB_t,\qquad X_0=0\end{aligned}$$ that is equal in distribution to $\xi$, i.e. $$\begin{aligned}
\xi_t\stackrel{\mathcal{D}}{=}X_t,\qquad \forall t.\end{aligned}$$ Thus $$\begin{aligned}
\sup_{c\in\mathcal{U}}\mathbb{E}_{t,x}[U(Y)]=\sup_{c\in\mathcal{U}}\mathbb{E}_{t,x}[U(X_T-KT)]\end{aligned}$$ and now you may proceed to derive the Hamilton-Jacobi-Bellman equation. It is precisely this question, which the authors are investigating presently.
[xx]{} A. Alfonsi, A. Schied and A. Schulz (2007a) [*Optimal execution strategies in limit order books with general shape functions*]{}, Preprint, QP Lab and TU Berlin. A. Alfonsi, A. Schied and A. Schulz (2007b) [*Constrained portfolio liquidation in a limit order book model*]{} Preprint, QP Lab and TU Berlin. R. Almgren and N. Chriss (1999) Value under liquidation, [*Risk*]{}. R. Almgren and N. Chriss (2000) Optimal Execution of Portfolio Transactions, [*J. Risk*]{}. [**3**]{} (2). R. Almgren, C. Thum , E. Hauptmann, and H. Li (2005) [*Direct Estimation of Equity Market Impact*]{} D. Bertsimas and D. Lo (1998) Optimal control of execution costs, [*Journal of Financial Markets*]{}. [**1**]{}. M.J. Brennan, and E. Schwartz (1979) A Continuous Time Approach to the Pricing of Bonds, [*J. of Banking and Finance*]{}. [**3**]{}. P.A. Forsyth (2009) Hamilton Jacobi Bellman Approach to Optimal Trade Schedule, Preprint. I.M. Gelfand, and S.V. Fomin (2000) [*Calculus of Variations*]{} New York: Dover Publications. K. Kawaguchi, and H. Morimoto (2007) Long-run average welfare in a pollution accumulation model, [*J. of Economic Dynamics and Control*]{}, [**31**]{}. V. Linetsky (2004) Spectral Expansions for Asian (Average Price) Options, [*Operations Research*]{}, [**52**]{}, pp.856–867. A. Obizhaeva, and J. Wang (2005) Optimal Trading Strategy and Supply/Demand Dynamics. [*J. Financial Markets*]{} A. Schied, and T. Schöneborn (2007) [*Optimal portfolio liquidation for CARA investors*]{}, Preprint, QP Lab and TU Berlin
\[fig1\] {height="12cm" width="12cm"}
\[fig2\] {height="12cm" width="12cm"}
{height="12cm" width="12cm"}
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|
---
abstract: 'We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of ${{\mathbb P}}^1$. The first family consists of pairs of genus-$2$ curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family also consists of pairs of genus-$2$ curves, but generically the curves in this family have absolutely simple Jacobians. The third family consists of pairs of genus-$3$ curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. Examples from these families show that in general it is impossible to tell from the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ whether or not the curve has rational points — or indeed whether or not it has real points. Our constructions depend on earlier joint work with Franck Leprévost and Bjorn Poonen, and on Peter Bending’s explicit description of the curves of genus $2$ whose Jacobians have real multiplication by ${{\mathbb Z}}[\sqrt{2}]$.'
address: 'Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967, USA.'
author:
- 'Everett W. Howe'
date: 12 May 2003
title: 'Infinite families of pairs of curves over ${{\mathbb Q}}$ with isomorphic Jacobians'
---
Introduction {#S-intro}
============
Torelli’s theorem shows that a curve is completely determined by its polarized Jacobian variety, but it has been known since the late 1800s that distinct curves can have isomorphic unpolarized Jacobians. In particular, the unpolarized Jacobian of a curve may not reflect all of the curve’s geometric properties. Proving that a particular property of curves cannot always be determined from the Jacobian is equivalent to showing that there exist two curves, one with the given property and one without, whose Jacobians are isomorphic to one another. Thus, for example, the pairs of curves written down in [@Howe:PAMS] show that one cannot tell whether or not a curve of genus $3$ over the complex numbers is hyperelliptic simply by looking at its Jacobian.
One would also like to find [*arithmetic*]{} properties of curves that are not determined by the Jacobian, but from an arithmetic perspective the heretofore-known explicit examples of distinct curves with isomorphic Jacobians (catalogued in the introduction to [@Howe:PAMS]) are not entirely satisfying. The primary complaint is that none of the examples involves curves that can be defined over ${{\mathbb Q}}$; in addition, for any given number field only finitely many of the examples can be defined over that field. Furthermore, all of the explicit examples in characteristic $0$ known before now involve curves with geometrically reducible Jacobians, and the arithmetic of such curves differs qualitatively from that of curves whose Jacobians are irreducible.
In this paper we address these concerns by providing three new explicit families of pairs of non-isomorphic curves with isomorphic Jacobians. Each family is parametrized by an open subset of ${{\mathbb P}}^1$, so each family gives an infinite number of examples over ${{\mathbb Q}}$. Also, the Jacobians of the curves in one of the families are typically absolutely simple. Using examples from these families, we show that the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ does not determine whether or not the curve has rational points, or indeed whether or not the curve has real points. Liu, Lorenzini, and Raynaud [@LLR] have used our results to show that the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ does not determine the number of components on the reduction of a minimal model of the curve modulo a prime.
Our first family of pairs of curves can be defined over an arbitrary field $K$ whose characteristic is not $2$. If $t$ is an element of $K$ with $t(t+1)(t^2+1)\neq 0$ then the equation $$(t + 1) y^2 = (2 x^2 - t) (4 t^2 x^4 + 4(t^2 + t + 1)x^2 + 1)$$ defines a curve of genus $2$ that we will denote $C(t)$. Clearly the quotient of $C(t)$ by the involution $(x,y)\mapsto (-x,y)$ is an elliptic curve, so the Jacobian of $C(t)$ splits over $K$.
\[T-nonsimple2\] Let $K$ be a field of characteristic not $2$ and suppose $t$ is an element of $K$ such that $t(t^2-1)(t^2+1)$ is nonzero. Then $C(t)$ and $C(-t)$ are curves of genus $2$ over $K$ whose Jacobians are isomorphic over $K$. Furthermore, $C(t)$ and $C(-t)$ are geometrically non-isomorphic unless $K$ has characteristic $11$ and $t^2 \in \{-3, -4\}$.
Our next family takes a little more effort to describe. In order to do so we must define the [*Richelot duals*]{} of a genus-$2$ curve over a field $K$ of characteristic not $2$ (see [@Cassels-Flynn Ch. 9], [@Bending §3]). Suppose $C$ is a genus-$2$ curve over $K$ defined by an equation $\delta y^2 = f$, where $\delta\in K^*$ and where $f$ is a monic separable polynomial in $K[x]$ of degree $6$. Let ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ be a separable closure of $K$, and suppose $f$ can be factored as a product $g_1 g_2 g_3$ of three monic quadratic polynomials in ${{K\llap{$\overline{\phantom{\rmK}}$}}}[x]$ that are permuted by ${\operatorname{Gal}}({{K\llap{$\overline{\phantom{\rmK}}$}}}/K)$. For each $i$ write $g_i = x^2 - t_i x + n_i$ and suppose the determinant $$d = \left|
\begin{matrix}
1 & t_1 & n_1 \\
1 & t_2 & n_2 \\
1 & t_3 & n_3
\end{matrix}
\right| ,$$ which is an element of $K$, is nonzero. Define three new polynomials by setting $$\begin{aligned}
h_1 &= g_3 \frac{dg_2}{dx} - g_2 \frac{dg_3}{dx}\\
h_2 &= g_1 \frac{dg_3}{dx} - g_3 \frac{dg_1}{dx}\\
h_3 &= g_2 \frac{dg_1}{dx} - g_1 \frac{dg_2}{dx}.\end{aligned}$$ Then the product $h_1 h_2 h_3$ is a separable polynomial in $K[x]$ of degree $5$ or $6$.
The [*Richelot dual of $C$ associated to the factorization $f = g_1 g_2 g_3$*]{} is the genus-$2$ curve $D$ defined by $d \delta y^2 = h_1 h_2 h_3.$
\[T-simple2\] Let $K$ be a field of characteristic not $2$, let $v$ be an element of $K\setminus\{0,1,4\}$ such that $$(v^2 - v + 4)
(v^2 + v + 2)
(v^2 + 3 v + 4)
(v^3 - 6 v^2 - 7 v - 4)
(v^3 - 4 v^2 + 7 v + 4)
\neq 0,$$ let $w$ be a square root of $v$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$, and define numbers $\rho_1,\ldots,\rho_6$ by setting $$\begin{aligned}
\rho_1 & = \frac{(-2+w)(1+w)}{2w^2}
& \qquad \qquad \rho_2 & = \frac{(-2-w)(1-w)}{2w^2}\\
\rho_3 & = \frac{-2(2+w)}{(-2+w)(1+w)}
& \qquad \qquad \rho_4 & = \frac{(-2-w)(1-w)}{(-w)(-1-w)} \\
\rho_5 & = \frac{(-2+w)(1+w)}{w(-1+w)}
& \qquad \qquad \rho_6 & = \frac{-2(2-w)}{(-2-w)(1-w)}.\end{aligned}$$ The $\rho_i$ are distinct from one another, so that if we let $f = \prod(x-\rho_i)$ then the curve $D$ over $K$ defined by $y^2 = f$ has genus $2$. Set $$\begin{aligned}
g_1 & = (x-\rho_1)(x-\rho_5)
& \qquad \qquad g_1' & = (x-\rho_1)(x-\rho_3)\\
g_2 & = (x-\rho_2)(x-\rho_4)
& \qquad \qquad g_2' & = (x-\rho_2)(x-\rho_6)\\
g_3 & = (x-\rho_3)(x-\rho_6)
& \qquad \qquad g_3' & = (x-\rho_4)(x-\rho_5).\end{aligned}$$ The Richelot duals $C$ and $C'$ of $D$ with respect to the factorizations $f = g_1 g_2 g_3$ and $f = g_1' g_2' g_3'$ exist, and their Jacobians become isomorphic to one another over $K(\sqrt{v(v-4)}\,)$. The curves $C$ and $C'$ are geometrically non-isomorphic unless one of the following conditions holds[:]{}
- ${\operatorname{char}}K = 3$ and $v^{10} - v^8 + v^7 - v^6 - v^5 + v + 1 = 0$[;]{}
- ${\operatorname{char}}K = 19$ and $v + 1 = 0$[;]{}
- ${\operatorname{char}}K = 89$ and $v + 36 = 0$[;]{}
- ${\operatorname{char}}K = 1033$ and $v + 508 = 0$.
Furthermore, if $K$ has characteristic $0$ and if $v$ is not an algebraic number, then the Jacobians of $C$ and $C'$ are absolutely simple.
In fact, when $K$ has characteristic $0$ it is very easy to find [*algebraic*]{} numbers $v$ in $K$ for which the Jacobians in Theorem \[T-simple2\] are absolutely simple. For example, suppose $R$ is a subring of $K$ for which there is a homomorphism $\varphi$ to an extension of ${{\mathbb F}}_{13}$. We show in the proof of Theorem \[T-simple2\] that in this case the Jacobians of $D$ and $D'$ are geometrically irreducible whenever $v$ lies in $\varphi^{-1}(2)$ or $\varphi^{-1}(6)$.
Theorem \[T-simple2\] gives a $1$-parameter family of pairs of non-isomorphic curves with isomorphic Jacobians. In fact, we shall see that there is a family of such pairs of curves parametrized by an elliptic surface; over ${{\mathbb Q}}$, this surface has positive rank.
Our third family of pairs of curves with isomorphic Jacobians is again easy to write down. Suppose $K$ is a field of characteristic not $2$ and suppose $t$ is an element of $K$ with $t(t+1)(t^2+1)(t^2+t+1)\neq 0$. Let $H(t)$ be the genus-$3$ hyperelliptic curve defined by the homogeneous equations $$\begin{aligned}
W^2 Z^2 &= - \frac{ (t^2+1) }{t(t+1)(t^2+t+1)}X^4
- \frac{4(t^2+1) }{ (t+1)(t^2+t+1)}Y^4
+ \frac{ 1}{t }Z^4 \label{E-H1} \\
0 &= - X^2 + 2 t Y^2 + (t+1) Z^2 \label{E-H2}\end{aligned}$$ and let $Q(t)$ be the plane quartic $$\begin{gathered}
\label{E-Q}
X^4 + 4t^2 Y^4 + (t+1)^2 Z^4 + (8t^2 + 4t + 8)X^2Y^2 \\
- (4t^2 + 2t + 2)X^2Z^2 + (4t^2 + 4t +8)Y^2Z^2 = 0.\end{gathered}$$
\[T-nonsimple3\] Let $K$ be a field of characteristic not $2$ and let $t$ be an element of $K$ such that $t(t+1)(t^2+1)(t^2+t+1)\neq 0$. Then the Jacobians of the two genus-$3$ curves $H(t)$ and $Q(t)$ are isomorphic to one another over $K$.
In Section \[S-curves\] we mention some simple facts about abelian surfaces with two non-isomorphic principal polarizations and we show how Richelot isogenies can in principle be used to produce such surfaces from an abelian surface that has nontrivial automorphisms. In Section \[S-nonsimple\] we prove Theorem \[T-nonsimple2\]. In Section \[S-RM\] we review a result of Bending that shows how to obtain genus-$2$ curves over a given field $K$ whose Jacobians have real multiplication by $\sqrt{2}$ over $K$, and we show how to adapt Bending’s result to obtain curves over $K$ with real multiplication by $\sqrt{2}$ over a quadratic extension of $K$. In Section \[S-Galois\] we give some Galois restrictions on our generalization of Bending’s construction that ensure that the curves we construct have two rational Richelot isogenies to curves with isomorphic Jacobians. In Section \[S-application\] we show that there is a positive-rank elliptic surface whose points give rise to pairs of genus-$2$ curves with isomorphic Jacobians, and we prove Theorem \[T-simple2\]. In Section \[S-genus3\] we prove Theorem \[T-nonsimple3\]. Finally, in Section \[S-examples\] we provide some explicit examples of curves over ${{\mathbb Q}}$ produced by our theorems, and we show that the Jacobian of a curve over ${{\mathbb Q}}$ does not determine whether or not the curve has rational points, or even whether or not it has real points.
We relied heavily on the computer algebra system Magma [@Magma] while working on this paper. Some of our Magma routines are available on the web: to find them, start at [http://alumni.caltech.edu/\~however/biblio.html]{}
and follow the links related to this paper.
Abelian surfaces with non-isomorphic polarizations {#S-curves}
==================================================
Weil [@Weil] showed that an abelian surface with an indecomposable principal polarization is a Jacobian, so one of our goals in this paper is to write down abelian surfaces with two non-isomorphic principal polarizations. In this section we will make a few observations about such surfaces.
Suppose $B$ is an abelian surface with two principal polarizations $\mu$ and $\mu'$, which we view as isogenies from $B$ to its dual variety $\hat{B}$. The polarized varieties $(B,\mu)$ and $(B,\mu')$ are isomorphic to one another if and only if there is an automorphism $\beta$ of $B$ such that $\mu' = \hat{\beta} \mu \beta$, where $\hat{\beta}$ is the dual of $\beta$. We would like to write down an abelian surface $B$ with two non-isomorphic principal polarizations $\mu$ and $\mu'$, so we would like to avoid the existence of such an automorphism $\beta$. We will accomplish this by obtaining $\mu'$ from $\mu$ through the use of an automorphism of a surface [*isogenous*]{} to $B$. Our main tool is the following well-known construction:
Suppose $(A,\lambda)$ is a principally-polarized abelian surface over a field $K$, suppose $n$ is a positive integer, and suppose $G$ is a rank-$n^2$ subgroupscheme of the $n$-torsion $A[n]$ of $A$ that is isotropic with respect to the $\lambda$-Weil pairing on $A[n]$. Let $B$ be the quotient abelian surface $A/G$ and let $\varphi:A\to B$ be the natural map. Then there is a unique principal polarization $\mu$ of $B$ that makes the following diagram commute: $$\begin{matrix}
A & {\mathop{{\relbar\joinrel\longrightarrow}}\limits^{n\lambda}} & \hat{A} \\
{\vbox{\vbox to 4pt{}\vbox{\hbox{
\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\varphi$}}$}}\vfill}}} & & {\vbox{\vbox to 4pt{}\vbox{\hbox{
\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle\hat{\varphi}$}}$}}\vfill}}} \\
B & {\mathop{{\relbar\joinrel\longrightarrow}}\limits^{\mu}} & \hat{B}
\end{matrix}$$
Now suppose that $A$ has an automorphism $\alpha$ such that $G':=\alpha(G)$ is also an isotropic subgroup of $A[n]$, and let $(B',\mu')$ be the principally polarized abelian surface obtained from $G'$ as above.
\[P-basic\] The automorphism $\alpha$ of $A$ provides an isomorphism $B\to B'$. If we identify $B'$ with $B$ via this automorphism, then $\mu' = \hat{\beta} \mu \beta$, where $\beta$ is the image of $\alpha^{-1}$ in $({\operatorname{End}}B)\otimes {{\mathbb Q}}$.
We have a commutative diagram with exact rows: $$\begin{matrix}
0 &{\longrightarrow}& G &{\longrightarrow}& A &{\mathop{{\relbar\joinrel\longrightarrow}}\limits^{\varphi}} &B &{\longrightarrow}& 0\\
& &{\vbox{\vbox to 4pt{}\vbox{\hbox{
\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\alpha$}}$}}\vfill}}} & &{\vbox{\vbox to 4pt{}\vbox{\hbox{
\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\alpha$}}$}}\vfill}}} & & & & \\
0 &{\longrightarrow}& G' &{\longrightarrow}& A &{\mathop{{\relbar\joinrel\longrightarrow}}\limits^{\varphi'}}&B' &{\longrightarrow}& 0,
\end{matrix}$$ where $\varphi$ and $\varphi'$ are the natural maps from $A$ to $B$ and $B'$, respectively. Completing the diagram, we find an isomorphism $B\to B'$. This proves the first statement of the theorem. The second statement follows by an easy diagram chase.
This proposition leaves us some hope, because the $\beta$ in the proposition will not be an element of ${\operatorname{End}}B$ if $G\neq G'$. Also, if we consider the case $n=2$ and if the principally-polarized surface $(A,\lambda)$ is given to us explicitly as either a Jacobian or a product of polarized elliptic curves, then the theory of the Richelot isogeny will allow us to write down $(B,\mu)$ and $(B,\mu')$ explicitly as Jacobians, as we explain below. Thus, we would like to explicitly write down abelian surfaces $A$ with non-trivial automorphisms. In later sections we will consider two families of such explicitly-given surfaces: products of isogenous elliptic curves, and Jacobians with real multiplication by $\sqrt{2}$.
We close this section with a comment about Richelot duals and maximal isotropic subgroups. Suppose $C$ is a genus-$2$ curve defined by an equation $\delta y^2 = f$ and suppose $D$ is the Richelot dual of $C$ corresponding to a factorization $f = g_1 g_2 g_3$. Then there is an isogeny from the Jacobian of $C$ to the Jacobian of $D$ whose kernel is the order-$4$ subgroup $G$ of ${\operatorname{Jac}}C$ containing the classes of the divisors $(a_i,0) - (b_i,0)$, where $a_i$ and $b_i$ are the roots of $g_i$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ (see [@Cassels-Flynn Ch. 9]). The subgroup $G$ is a maximal isotropic subgroup of the $2$-torsion of ${\operatorname{Jac}}C$ under the Weil pairing. Conversely, every $K$-defined maximal isotropic subgroup $G$ of $({\operatorname{Jac}}C)[2]$ arises in this way. Thus, given a $K$-defined maximal isotropic subgroup $G$, we can define the [*$G$-Richelot dual*]{} of $C$ to be the Richelot dual of $C$ with respect to the factorization $f = g_1 g_2 g_3$ that gives rise to $G$.
Proof of Theorem \[T-nonsimple2\] {#S-nonsimple}
=================================
In this section we will prove Theorem \[T-nonsimple2\] by following the outline given in Section \[S-curves\] in the case where $A$ is a split abelian surface.
Let $E$ and $E'$ be elliptic curves over a field $K$ of characteristic not $2$ and suppose there is a $2$-isogeny $\psi$ from $E$ to $E'$. Let $Q$ be the nonzero element of $E[2](K)$ in the kernel of $\psi$, and let $P$ and $R$ be the other two geometric $2$-torsion points of $E$. Let $Q' = \psi(P) = \psi(R)$, so that $Q'$ is a nonzero element of $E'[2](K)$, and let $P'$ and $R'$ be the other geometric $2$-torsion points of $E'$. Suppose the discriminants of $E$ and $E'$ are equal up to squares, so that the fields $K(P)$ and $K(P')$ are the same.
Let $A$ be the surface $E\times E'$ and let $\lambda$ be the product principal polarization on $A$. Let $\alpha$ be the automorphism of $A$ that sends a point $(U,V)$ to $(U, V+\psi(U))$. Let $G$ be the $K$-defined subgroup $$G = \{(O,O), (P, P'), (Q, Q'), (R, R')\}$$ of $A[2]$ and let $G' = \alpha(G)$, so that $G'$ is the $K$-defined subgroup $$G'= \{(O,O), (P, R'), (Q, Q'), (R, P')\}.$$ Let $(B,\mu)$ and $(B,\mu')$ be the principally-polarized surfaces obtained from $G$ and $G'$ as in Section \[S-curves\]. The polarizations $\mu$ and $\mu'$ will be indecomposable except in unusual circumstances, so there will usually be curves $C$ and $C'$ whose polarized Jacobians are isomorphic to $(B,\mu)$ and $(B,\mu')$, respectively. If $E$ and $E'$ are given to us by explicit equations, then $C$ and $C'$ can also be given by explicit equations — see [@HLP §3.2], where the unusual circumstances are also explained.
To make this outline explicit and to prove Theorem \[T-nonsimple2\] we must start with an explicit $2$-isogeny $\psi:E\to E'$, where the discriminants of $E$ and $E'$ are equal up to squares. Let $t$ be an element of $K$ such that $t (t^2 + 1) (t^2 - 1)$ is nonzero, and let $E$ and $E'$ be the elliptic curves $$\begin{aligned}
E: \quad y^2 & = x (x^2 - 4(t^2+1) x + 4(t^2+1))\\
E': \quad y^2 & = x (x^2 + 8(t^2+1) x + 16 t^2(t^2+1)).\end{aligned}$$ It is easy to check that the discriminants of $E$ and $E'$ are both equal to $t^2 + 1$, up to squares.
Let $s$ be a square root of $t^2 + 1$ in an algebraic closure of $K$, so that the $2$-torsion points of $E$ are $$\begin{aligned}
P &= (2t^2 + 2 + 2st, 0)\\
Q &= (0,0)\\
R &= (2t^2 + 2 - 2st,0)\\
\intertext{and the $2$-torsion points of $E'$ are}
P' &= (-4t^2 - 4 + 4s, 0)\\
Q' &= (0,0)\\
R' &= (-4t^2 - 4 - 4s,0).\end{aligned}$$ It is easy to check that the map $$(x,y) \mapsto \left( \frac{y^2}{x^2},
\frac{(x^2 - 4(t^2+1))y}{x^2} \right)$$ defines a $2$-isogeny $\psi:E\to E'$ that kills $Q$ and that sends $P$ to $Q'$ (see [@Silverman Example III.4.5]). Let $G$ and $G'$ be the subgroups of $A[2]$ defined above and let $(B,\mu)$ and $(B,\mu')$ be the principally-polarized surfaces obtained from $G$ and $G'$ as above. If we apply [@HLP Prop. 4] we find that $(B,\mu)$ is isomorphic over $K$ to the polarized Jacobian of the curve $y^2 = h_t$, where $$h_t = 2^{38} t^6 (t+1)^3 (t^2+1)^{12}
(2 x^2 - t)
(4 t^2 x^4 + 4(t^2 + t + 1)x^2 + 1).$$ Furthermore, $(B,\mu')$ is isomorphic to the polarized Jacobians of the curve $y^2 = h_{-t}$. Scaling $h_t$ and $h_{-t}$ by squares in $K$, we find that $y^2 = h_t$ is isomorphic to the curve $C(t)$ of Theorem \[T-nonsimple2\] and that $y^2 = h_{-t}$ is isomorphic to the curve $C(-t)$.
To complete the proof of Theorem \[T-nonsimple2\] we must show that $C(t)$ and $C(-t)$ are geometrically non-isomorphic, except for the special cases listed in the theorem. The simplest way to do this is to use Igusa invariants (see [@Igusa], [@Mestre]). Facilities for computing Igusa invariants are included in the computer algebra package Magma [@Magma].
Let us begin by working over the ring ${{\mathbb Z}}[t]$, where $t$ is an indeterminate. Let $J_2(t), J_4(t), J_6(t), J_8(t),$ and $J_{10}(t)$ be the Igusa invariants of the twist $y^2 = (2x^2 - t)(4x^4 + 4(t^2 + t + 1)x^2 + 1)$ of $C(t)$. (The invariants $J_{2i}(t)$ of this curve, scaled by $4^i$, can be computed in Magma using the function [ScaledIgusaInvariants]{}.) Let $$\begin{aligned}
R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2}
{ t (t^2+1)^3 } \\
R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3}
{ t^3 (t^2+1)^3} \\
R_{5} &= \frac{J_{10}(t) J_2(-t)^5 - J_{10}(-t) J_2(t)^5}
{t^3 (t^2+1)^7},\end{aligned}$$ all of which we view as elements of ${{\mathbb Z}}[t]$. If $C(t)$ and $C(-t)$ are isomorphic for a given value of $t$ in a given field $K$, then the polynomials $R_{2}, R_{3},$ and $R_{5}$ must all evaluate to $0$ at this value. But we compute that $$\gcd({\operatorname{resultant}}(R_{2},R_{3}), {\operatorname{resultant}}(R_{2},R_{5}))
= 2^{980} 3^{48} 11^8,$$ so if the characteristic of $K$ is neither $3$ nor $11$ then the two curves $C(t)$ and $C(-t)$ are geometrically non-isomorphic for every value of $t$ in $K$ with $t(t^2+1)(t^2-1)\neq 0$.
We repeat the above calculation in the ring ${{\mathbb F}}_3[t]$, only now we define $$\begin{aligned}
R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2}
{t (t^2-1)^2 (t^2+1)^7} \\
R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3}
{t^3 (t^2+1)^9}. \end{aligned}$$ We find that $\gcd(R_{2},R_{3}) = 1$, so the two curves $C(t)$ and $C(-t)$ are geometrically non-isomorphic for every value of $t$ in characteristic $3$, as long as $t(t^2+1)(t^2-1)\neq 0$.
Next we repeat the above calculation in the ring ${{\mathbb F}}_{11}[t]$, with $$\begin{aligned}
R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2}
{ t (t^2+1)^3 }\\
R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3}
{ t^3 (t^2+1)^3}. \end{aligned}$$ We find that $\gcd(R_{2},R_{3}) = (t^2 + 3)(t^2 + 4)$, so that the two curves are geometrically non-isomorphic in characteristic $11$ except possibly when $t^2$ is $-3$ or $-4$.
Finally we note that in characteristic $11$, when $t^2$ is $-3$ or $-4$ the curve $C(t)$ is geometrically isomorphic to the supersingular curve $y^2 = x^6 + x^4 + 4 x^2 + 7$. Thus, $C(t)$ and $C(-t)$ are geometrically isomorphic for these values of $t$.
It was not critical in our construction that the isogeny $\psi:E\to E'$ have degree $2$. Similar constructions can be made with other kinds of isogenies.
Jacobians with real multiplication by $\sqrt{2}$ {#S-RM}
================================================
In this section we review a construction of Bending [@Bending] that produces every genus-$2$ curve over a given field $K$ whose Jacobian has a $K$-rational endomorphism that is fixed by the Rosati involution and whose square is $2$. We will give a variant of Bending’s construction that produces curves over $K$ with a not-necessarily $K$-rational endomorphism that is fixed by Rosati and whose square is $2$. We do not claim that our construction will produce all such curves.
First we recall Bending’s construction. Let $K$ be a field of characteristic not $2$ and let $A$, $P$, and $Q$ be elements of $K$ with $P$ nonzero. Define $$\begin{aligned}
B &= (APQ - Q^2 + 4P^2 + 1)/P^2\\
C &= 4(AP - Q)/P\\
R &= 4P\end{aligned}$$ and let $\alpha_1$, $\alpha_2$, $\alpha_3$ be the roots of $T^3 + AT^2 + BT + C$ in a separable closure ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ of $K$. For $i = 1,2,3$ let $$G_i = X^2 - \alpha_i X + P\alpha_i^2 + Q\alpha_i + R,$$ and suppose that the product $G_1 G_2 G_3 \in K[X]$ has nonzero discriminant. Let $D$ be a nonzero element of $K$.
\[T-Bending\] The Jacobian of the genus-$2$ curve $D Y^2 = G_1 G_2 G_3$ has a $K$-rational endomorphism that is fixed by the Rosati involution and whose square is $2$. Furthermore, if $\#K>5$ then every curve over $K$ whose Jacobian has such an endomorphism is isomorphic to a curve that arises in this way from some choice of $A$, $P$, $Q$, and $D$ in $K$.
See [@Bending Theorem 4.1]. Bending assumes that the base field $K$ has characteristic $0$, but his proof works over an arbitrary field $K$ of characteristic not $2$ so long as every genus-$2$ curve over $K$ can be written in the form $y^2 = \text{(sextic)}$. This is the case for every field with more than $5$ elements.
The endomorphism of $D Y^2 = G_1 G_2 G_3$ whose existence is claimed by Theorem \[T-Bending\] is obtained by noting that the obvious Richelot dual of the curve is isomorphic over $K$ to the curve itself. Thus the degree-$4$ isogeny from the Jacobian of the curve to the Jacobian of its dual can be viewed as an endomorphism of the curve’s Jacobian, and this endomorphism has the properties claimed in the theorem.
We will want to consider curves over $K$ whose Jacobians have real multiplication by $\sqrt{2}$ that is not necessarily defined over $K$. For this reason, we will require the following variant of Bending’s construction:
Suppose $r$, $s$, and $t$ are elements of a field $K$ of characteristic not $2$, with $s\neq 0$, $s\neq 1$, and $t\neq 1$. Let $$\begin{aligned}
c_2 &= r + 4t\\
c_1 &= 4 t (r + s^3 - s^2 t - 2 s^2 + 5 s + t)\\
c_0 &= 4 t (s - 1) (r s^2 - r s t - r s - r t - 8 s t)\end{aligned}$$ and suppose that the polynomial $T^3 - c_2 T^2 + c_1 T - c_0$ has three distinct roots $\beta_1$, $\beta_2$, $\beta_3$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$. For $i = 1,2,3$ let $$g_i = x^2 -
2\beta_i x +
(1-s)\beta_i^2 - 4 s (s - 1)^2 t (s - t - 1),$$ and suppose that the discriminant of the product $f = g_1 g_2 g_3$ is nonzero. Let ${{\mathcal C}}(r,s,t)$ be the curve over $K$ defined by $y^2 = f$.
\[T-RM\] The Richelot dual of ${{\mathcal C}}(r,s,t)$ associated to the factorization $f = g_1 g_2 g_3$ is isomorphic over $K(\sqrt{st}\,)$ to ${{\mathcal C}}(r,s,t)$. The endomorphism of ${\operatorname{Jac}}{{\mathcal C}}(r,s,t)$ [(]{}over $K(\sqrt{st}\,)$[)]{} obtained by composing the Richelot isogeny with the natural isomorphism from the dual curve to ${{\mathcal C}}(r,s,t)$ is fixed by the Rosati involution, and its square is the multiplication-by-$2$ endomorphism.
This theorem can be proven by direct calculation, but here we will prove it by relating the curve ${{\mathcal C}}(r,s,t)$ back to Bending’s construction. Let $Q$ be a square root of $st$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$, let $P = (1 - s)/4$, let $A = (r + 6 s t - 2 t)/(4PQ)$, let $D=1$, and let ${{\mathcal C}}'$ be the curve over $K(Q)$ defined by using this $A$, $P$, $Q$, and $D$ in Bending’s construction. Then the $\alpha_i$ are related to the $\beta_i$ by $$Q\alpha_i = \beta_i/(s-1) + 2t,$$ and the curve $Y^2 = G_1 G_2 G_3$ is isomorphic $y^2 = g_1 g_2 g_3$ via the relation $x = 2(s-1)(QX + t)$. This shows that ${{\mathcal C}}(r,s,t)$ is isomorphic to its Richelot dual over $K(Q)$. The rest of the theorem follows from Bending’s theorem.
Bending’s family of curves has three “geometric” parameters $A$, $P$, and $Q$ and one “arithmetic” parameter $D$ (which parametrizes quadratic twists of the curve determined by $A$, $P$, and $Q$). Since the moduli space ${{\mathcal M}}$ of genus-$2$ curves with real multiplication by $\sqrt{2}$ is a two-dimensional rational variety, one might hope to replace Bending’s three-geometric-parameter family with a two-parameter family. But there is an obstruction, which stems from the fact that ${{\mathcal M}}$ is a coarse moduli space and not a fine one: A $K$-rational point on ${{\mathcal M}}$ does not necessarily give rise to a curve over $K$. Indeed, Mestre [@Mestre] has shown that to every $K$-rational point $P$ on the moduli space of genus-$2$ curves there is naturally associated a genus-$0$ curve over $K$, and $P$ corresponds to a curve over $K$ if and only if the genus-$0$ curve has a $K$-rational point.
Galois restrictions {#S-Galois}
===================
In order to prove Theorem \[T-simple2\] we will apply the construction outlined in Section \[S-curves\] to a Jacobian with real multiplication by $\sqrt{2}$ that we will obtain from Theorem \[T-RM\]; we will take the automorphism $\alpha$ to be $1+\sqrt{2}$. The construction requires that we find a Galois-stable maximal isotropic subgroup $G$ of the $2$-torsion of the Jacobian such that $G' = (1+\sqrt{2})(G)$ is a maximal isotropic subgroup different from $G$. This requirement imposes some restrictions on the values of $r$, $s$, and $t$ that we will be able to use in Theorem \[T-RM\]. In this section we will make these restrictions explicit, and in Section \[S-application\] we will find an elliptic surface that parametrizes a subset of the allowable values of $r$, $s$, and $t$.
Recall the basic outline of Theorem \[T-RM\]: Given three elements $r$, $s$, $t$ of our base field $K$, we define a polynomial $h = T^3 - c_2 T^2 + c_1 T - c_0$ in the polynomial ring $K[T]$, and we assume that $h$ is separable. We use the roots $\beta_1$, $\beta_2$, $\beta_3$ of $h$ to define three polynomials $g_1$, $g_2$, $g_3$ in the polynomial ring ${{K\llap{$\overline{\phantom{\rmK}}$}}}[x]$, we assume that the product $f = g_1 g_2 g_3$ is separable, and we define a curve $C$ by $y^2 = f$. Then we show that the Richelot dual of $C$ corresponding to the factorization $f = g_1 g_2 g_3$ is geometrically isomorphic to $C$ itself.
Let $L$ be the quotient of the polynomial ring $K[T]$ by the ideal generated by the polynomial $h$ and let $\beta$ be the image of $T$ in $L$. Since $h$ is separable, the algebra $L$ is a product of fields. Let $g\in L[x]$ be the polynomial $$g = x^2 - 2\beta x + (1-s)\beta^2 - 4 s (s-1)^2 t (s - t - 1).$$ Let $\Delta\in K^*$ be the discriminant of $h$ and let $\Delta'\in L^*$ be the discriminant of $g$.
\[T-Galois\] There are distinct Galois-stable maximal isotropic subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ with $G' = (1+\sqrt{2})(G)$ if and only if $\Delta\Delta'$ is a square in the algebra $L$.
Let the roots of $g_1$ (respectively, $g_2$, $g_3$) be $r_1$ and $r_2$ (respectively, $r_3$ and $r_4$, $r_5$ and $r_6$). For each $i$ let $W_i$ be the Weierstraß point of $C$ corresponding to the root $r_i$ of $f = g_1 g_2 g_3$. The kernel $H$ of the Richelot isogeny multiplication-by-$\sqrt{2}$ on the Jacobian $J$ of $C$ is the order-$4$ subgroup containing the divisor classes $[W_1 - W_2]$, $[W_3 - W_4]$, and $[W_5 - W_6]$.
Suppose there are distinct Galois-stable maximal isotropic subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ with $G' = (1+\sqrt{2})(G)$. Then clearly $G\neq H$, so $\#(G\cap H)$ is either $2$ or $1$. Suppose $\#(G\cap H)=2$. By renumbering the polynomials $g_i$ and by renumbering their roots, we may assume that $G$ is the order-$4$ subgroup $$G = \{ 0, [W_1 - W_2], [W_3 - W_5], [W_4 - W_6]\}.$$ Then $\sqrt{2}$ kills the first two elements of $G$ and sends the second two elements to $[W_1 - W_2]$, and it follows that $(1+\sqrt{2})(G) = G$, contradicting our assumption that $G$ and $G'$ are distinct.
So now we know that $G\cap H = \{0\}$. By renumbering the polynomials $g_i$ and by renumbering their roots, we may assume that $G$ is the order-$4$ subgroup $$G = \{ 0, [W_1 - W_5], [W_2 - W_4], [W_3 - W_6]\}.$$ It is not hard to show that the automorphism $1 + \sqrt{2}$ of $J$ sends $[W_1 - W_5]$ to $[W_2 - W_6]$, $[W_2 - W_4]$ to $[W_1 - W_3]$, and $[W_3 - W_6]$ to $[W_4 - W_5]$, so we have $$G'= \{ 0, [W_1 - W_3], [W_2 - W_6], [W_4 - W_5]\}.$$
Suppose $\sigma$ is an element of the Galois group such that $r_1^\sigma = r_2$. Since $G$ is Galois stable, it follows that $\sigma$ sends $[W_1 - W_5]$ to $[W_2 - W_4]$, and therefore $r_5^\sigma = r_4$. But since $G'$ is Galois stable, we see that $\sigma$ must send $[W_4 - W_5]$ to itself, and it follows that $r_4^\sigma = r_5$. Continuing in this manner, we find that $r_2^\sigma = r_1$ and $r_6^\sigma = r_3$ and $r_3^\sigma = r_6$. Thus, $\sigma$ acts on the roots of $f$ according to the permutation $(1 2)(3 6)(4 5)$ of the subscripts.
By considering the other choices for $r_1^\sigma$ and using the same reasoning as above, we find that the image of the absolute Galois group of $K$ in the symmetric group on the the roots of $f$ is contained in the subgroup $$S = \{ {\operatorname{Id}}, (1 2)(3 6)(4 5), (1 3)(2 4)(5 6), (1 4 6)(2 3 5),
(1 5) (2 6) (3 4), (1 6 4) (2 5 3)\};$$ here of course we identify the root $r_i$ with the integer $i$. In particular, note that the action of $\sigma$ on the $r_i$ is determined by the action of $\sigma$ on the $\beta_i$.
To show that $\Delta\Delta'$ is square in the algebra $L$, we will consider three cases, depending on the splitting of the polynomial $h$.
*Case 1.* Suppose $h$ is irreducible. Then $L$ is a field, and the condition that $\Delta\Delta'$ be a square in $L$ is equivalent to saying that $g$ defines the Galois closure $M$ of $L$ over $K$. So suppose, to obtain a contradiction, that $g$ does not define $M$ over $L$. There are two ways that this can happen: either the roots of $g$ do not lie in $M$, or $M$ is a quadratic extension of $L$ and the roots of $g$ lie in $L$.
Suppose that the roots of $g$ do not lie in $M$. Then there is an element $\sigma$ of the absolute Galois group of $K$ that fixes $M$ but that moves the roots of $g$. But this contradicts the fact that the the action of $\sigma$ on the roots of $g$ is determined by the action of $\sigma$ on the $\beta_i$.
Suppose $M$ is a quadratic extension of $L$, so that the image of the absolute Galois group in the symmetric group on the $\beta_i$ is the full symmetric group. Then the image of the absolute Galois group in the symmetric group on the $r_i$ must be the entire group $S$ given above, which acts transitively on the $r_i$. But if the roots of $g$ lie in $L$, then the $r_i$ will form two orbits under the action of the absolute Galois group, giving a contradiction.
*Case 2.* Suppose $h$ factors as a linear polynomial times an irreducible quadratic. Then one of the $\beta_i$, say $\beta_1$, lies in $K$, while $\beta_2$ and $\beta_3$ are conjugate elements in a quadratic extension $M$ of $K$. With the labelings we have chosen, this means that the image of the absolute Galois group in the symmetric group on the $r_i$ must be equal to the two-element group $$S' = \{ {\operatorname{Id}}, (1 2)(3 6)(4 5)\};$$ In particular, we see that $r_3$ and $r_6$ (and $r_4$ and $r_5$) are quadratic conjugates of one another, so they all must be elements of $M$. This means that the image of $g$ in $K[x]$ (obtained by sending $\beta$ to $\beta_1$) is an irreducible polynomial that defines $M$, while the image of $g$ in $M[x]$ (obtained by sending $\beta$ to $\beta_2$) splits into two linear factors. Thus, the discriminant $\Delta'$ of $g$ in $L = K\times M$ is equal to the discriminant of $M$ (up to squares) in the first component, and is a square in the second component. But $\Delta$ has this same property, so the product $\Delta\Delta'$ is a square in $L$.
*Case 3.* Suppose $h$ splits over $K$ into three linear factors, so that $\Delta$ is a square in $K$. Then the absolute Galois group of $K$ acts trivially on the $\beta_i$, so it must act trivially on the $r_i$ as well. This means that the discriminant $\Delta'$ of $g$ must be a square in each factor of $L = K\times K\times K$, so $\Delta\Delta'$ is a square in $L$ as well.
We see that if there are subgroups $G$ and $G'$ as in the statement of the theorem then $\Delta\Delta'$ must be a square in $L$. We leave the details of the proof of the converse statement to the reader; the point is that in each of the three cases above, the reasoning is reversible.
Application of our construction to\
curves with real multiplication {#S-application}
===================================
In this section we will follow the outline given in Section \[S-curves\] in the case where $A$ is a Jacobian with real multiplication by $\sqrt{2}$ that has appropriate Galois-stable subgroups. Theorem \[T-simple2\] will follow quickly from the result we obtain.
We will continue to use the notation from previous sections:
- $r$, $s$, and $t$ will be elements of a field $K$;
- $h$ will be a polynomial in $K[T]$ defined in terms of $r$, $s$, and $t$;
- $L$ will be the algebra $K[T]/(h)$;
- $\beta$ will be the image of $T$ in $L$;
- $g$ will be a polynomial in $L[x]$ defined in terms of $r$, $s$, $t$, and $\beta$;
- $\Delta\in K^*$ will be the discriminant of $h$; and
- $\Delta' \in L^*$ will be the discriminant of $g$.
\[T-gensimple\] Let $K$ be a field of characteristic not $2$, suppose $r$, $s$, and $t$ are elements of $K$ that satisfy the hypotheses appearing before the statement of Theorem [\[T-RM\]]{}, and let $C$ be the curve ${{\mathcal C}}(r,s,t)$ from Theorem [\[T-RM\]]{}. Suppose further that the product $\Delta\Delta'$ is a square in $L^*$, so that there are Galois-stable subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ as in the statement of Theorem [\[T-Galois\]]{}. Then the Jacobian of the $G$-Richelot dual of $C$ is isomorphic over $K(\sqrt{st}\,)$ to the Jacobian of the $G'$-Richelot dual of $C$.
Let $D$ be the $G$-Richelot dual of $C$ and let $D'$ be the $G'$-Richelot dual of $C$. Theorem \[T-RM\] and Theorem \[T-Galois\], combined with the argument in Section \[S-curves\], show that the Jacobian of $D$ becomes isomorphic to the Jacobian of $D'$ when the base field is extended to $K(\sqrt{st}\,)$.
Since $D$ and $D'$ are defined over $K$, and since their Jacobians become isomorphic over $K(\sqrt{st}\,)$, it is tempting to think that ${\operatorname{Jac}}D$ must be isomorphic over $K$ to either the Jacobian of $D'$ or the Jacobian of the standard quadratic twist of $D'$ over $K(\sqrt{st}\,)$. But in fact this is not the case. It is true that ${\operatorname{Jac}}D'$ is a $K(\sqrt{st}\,)/K$-twist of ${\operatorname{Jac}}D$, but the twist is by an automorphism of ${\operatorname{Jac}}D$ that does not come from an automorphism of $D$. Indeed, generically the automorphism group of $D$ contains $2$ elements, while the automorphism group of ${\operatorname{Jac}}D$ is isomorphic to the unit group of ${{\mathbb Z}}[2\sqrt{2}]$.
Suppose $t = s-1$ and let $u = r + 2$. Then $\Delta\Delta'$ is a square in $L$ if and only if $(u^2 + a)^2 + 8bu + 4c$ is a square in $K$, where $$\begin{aligned}
a &= -4 s (s^2 + 11s - 11)\\
b &= -8s^2 (s-1) (4s-1) \\
c &= -16s^2 (s-1) (28s^2 - 19s + 1).\end{aligned}$$
When $t = s-1$ and $r = u - 2$, we find that the coefficients of the polynomial $h$ used to define the algebra $L$ are $$\begin{aligned}
c_2 &= 4s + u - 6 \\
c_1 &= - 4 (s-1) (s^2 - 6s - u + 3) \\
c_0 &= 4 (s-1)^3 (-8s - u + 2),\end{aligned}$$ and we compute that $$\Delta = 16 s (s-1)^2 ((u^2 + a)^2 + 8bu + 4c),$$ where $a$, $b$, and $c$ are as in the statement of the proposition. Furthermore, the polynomial $g\in L[x]$ defined in Section \[S-Galois\] is $x^2 - 2\beta x + (1-s)\beta^2$, so that $\Delta' = 4 s \beta^2$. We see that $\Delta\Delta'$ is a square in $L$ if and only if the element $\delta = (u^2 + a)^2 + 8bu + 4c$ of $K$ is a square in $L$. If $L$ is a field then it is a cubic extension of $K$, and $\delta$ is a square in $L$ if and only if it is a square in $K$. If $L$ is not a field then it has $K$ as a factor, and again $\delta$ is a square in $L$ if and only if it is a square in $K$.
\[P-ellipticsurface\] Let $K={{\mathbb Q}}(s)$ be the function field in the variable $s$ over ${{\mathbb Q}}$, let $$\begin{aligned}
a &= -4 s (s^2 + 11s - 11)\\
b &= -8s^2 (s-1) (4s-1) \\
c &= -16s^2 (s-1) (28s^2 - 19s + 1),\end{aligned}$$ let $F$ be the curve over $K$ defined by $$z^2 = (u^2 + a)^2 + 8bu + 4c,$$ and let $E$ be the elliptic curve over $K$ defined by $$y^2 = x^3 - a x^2 - c x + b^2.$$ Then
- the map $u = (y-b)/x$, $z = 2x - u^2 - a$ gives an isomorphism from $E$ to $F$, whose inverse is $x = (z + u^2 + a)/2$, $y = ux + b$[;]{}
- the point $P = (0, b)$ on $E$ has infinite order[;]{}
- the point $T = (4s^2(1-s), 0)$ on $E$ has order $2$[;]{}
- the isomorphism in statement [(a)]{} takes the involution $(u,z)\mapsto (u,-z)$ on $F$ to the involution $Q \mapsto -Q-P$ on $E$[;]{}
- the isomorphism in statement [(a)]{} takes $-P$ and the origin of $E$ to the two infinite points on $F$.
An easy calculation shows that statement (a) is true; the particular values of $a$, $b$, and $c$ are irrelevant to the calculation.
To show that the point $P$ has infinite order, it suffices to show that when we specialize $s$ to a particular value the specialized $P$ has infinite order. For example, if we set $s = 2$, then $E$ becomes the curve $$y^2 = x^3 + 120 x^2 + 4800 x + 50176$$ and $P$ becomes the point $(0, -224)$. Translating $x$ by $40$, we find a new equation for $E$: $y^2 = x^3 - 13824,$ where now $P = (40,-224)$. But $7$ divides $224$ and $7$ does not divide $13824$, so by the Lutz-Nagell theorem [@Silverman Cor. VIII.7.2] the point $P$ has infinite order. This proves statement (b).
Statement (c) is clear.
Let $R$ be a point $(u,z)$ on $F$ and let $\tilde{R} = (u,-z)$ be its involute. Let $Q$ and $\tilde{Q}$ be the images of $R$ and $\tilde{R}$ on $E$. Clearly $Q$ and $\tilde{Q}$ both lie on the line $y = ux + b$, and the third intersection point of this line with $E$ is easily seen to be $P$. Thus, the involution on $E$ satisfies $\tilde{Q} + Q = -P$, and this is statement (d).
The equations for the isomorphism show that $-P$ is mapped to an infinite point on $F$, and statement (d) shows that $O_E$ gets mapped to an infinite point as well.
If we view the curve $F\cong E$ as an elliptic surface $S$ over ${{\mathbb Q}}$, then the points $P$ and $T$ of Proposition \[P-ellipticsurface\] can be viewed as rational curves on $S$. By adding multiples of $P$ and $T$ together, we get a countable family of rational curves on $S$. But $S$ contains more rational curves than just the ones in this family. For example, we have the curves $$\begin{aligned}
s &= 5/4 \\
u &= (4w^2 + 5w + 40)/(4w) \\
z &= (2w^4 + 5w^3 - 50w - 200)/(2w^2)\end{aligned}$$ and $$\begin{aligned}
s &= -1 \\
u &= (2w^2 - 10w - 4)/w \\
z &= (4w^4 - 40w^3 - 80w - 16)/w^2,\end{aligned}$$ where $w$ is a parameter; the curve $$\begin{aligned}
s &= (5-w^2)/4\\
u &= (-w^4 + w^3 + 7w^2 - 5w - 10)/(4w + 8) \\
z &= (w^8 + 9w^7 + 22w^6 - 18w^5 - 135w^4 - 135w^3)/(8w^2 + 32w + 32),\end{aligned}$$ which corresponds to a $3$-torsion point on $E$ defined over a genus-$0$ extension of the function field ${{\mathbb Q}}(s)$; five curves in which $u$ is a linear expression in $s$, for example $$\begin{aligned}
s &= 4(w^2 + 9w + 19)/w\\
u &= -s\\
z &= 16(16w^6 + 283w^5 + 1555w^4 - 29545w^2 - 102163w - 109744)/w^3;\end{aligned}$$ and three curves in which $u$ is a quadratic expression in $s$, for example $$\begin{aligned}
s &= (w^2 + 3w + 1)/w\\
u &= 4s^2 - 6s\\
z &= 4(4w^8 + 35w^7 + 105w^6 + 119w^5 - 119w^3 - 105w^2 - 35w - 4)/w^4.\end{aligned}$$
One can check that the image of the elliptic surface $S$ in the moduli space of genus-$2$ curves is $2$-dimensional. To check this, one need only write explicitly the Igusa invariants of the genus-$2$ curve obtained from a pair $(s,u)$ and verify that the rank of the Jacobian matrix of the mapping from $(s,u)$-pairs to Igusa invariants at some arbitrary point is $2$.
We now have enough machinery available to prove Theorem \[T-simple2\].
Consider the point $P = (0,b)$ from statement (b) of Proposition \[P-ellipticsurface\]. The $u$-coördinate of its image on the curve $F$ is $$u = (28s^2 - 19s + 1)/(1-4s).$$ So given any $s\in K$, we will obtain a curve satisfying the conclusion of Theorem \[T-gensimple\] if we set $$\begin{aligned}
t &= s - 1\\
r &= -2 + (28s^2 - 19s + 1)/(1-4s) \end{aligned}$$ and set $C = {{\mathcal C}}(r,s,t)$. Given a $v \in K\setminus\{0,1,4\}$, let us apply the preceding observation with $s = v/4$, so that $$\begin{aligned}
s &= v/4\\
t &= (v - 4)/4\\
r &= (7 v^2 - 11 v - 4)/(4-4v).\end{aligned}$$ The coefficients of the polynomial $h$ used in the construction of Section \[S-RM\] are $$\begin{aligned}
c_2 &= \frac{3v^2 + 9v - 20}{4(1 - v)} \\
c_1 &= \frac{(v-4)(v^3 + 3v^2 - 4v - 32)}{16(1-v)} \\
c_0 &= \frac{(v-4)^3 (v^2 + 3v + 4)}{64(1-v)},\end{aligned}$$ and over $K(w)$ the roots of $h$ are $$\begin{aligned}
\beta_1 &= \frac{(2 + w)(2 - w)}{4} \\
\beta_2 &= \frac{-(2+w)^2(2 - w + w^2)}{4(1 + w)}\\
\beta_3 &= \frac{-(2-w)^2(2 + w + w^2)}{4(1 - w)}.\end{aligned}$$ Each polynomial $g_i$ is $x^2 - 2\beta_i x + (1-s)\beta_i^2$ and has roots $\beta_i (1 \pm w/2)$, so we calculate that the roots of $g_1$ are $$\begin{aligned}
r_1 &= -(1/8) (2 - w)^2 (2 + w) \\
r_2 &= -(1/8) (2 - w) (2 + w)^2, \\
\intertext{the roots of $g_2$ are}
r_3 &= -(1/8) (2 + w)^3 (2 - w + w^2) / (1 + w) \\
r_4 &= -(1/8) (2 - w) (2 + w)^2 (2 - w + w^2) / (1 + w),\\
\intertext{and the roots of $g_3$ are}
r_5 &= -(1/8) (2 - w)^2 (2 + w) (2 + w + w^2) / (1 - w) \\
r_6 &= -(1/8) (2 - w)^3 (2 + w + w^2) / (1 - w).\end{aligned}$$ These roots are indexed in a manner consistent with the indexing of the roots in the proof of Theorem \[T-Galois\]. Note that the $r_i$ are related to the $\rho_i$ of Theorem \[T-simple2\] by the relation $$r_i = 4s(s-1)\rho_i - 2(s-1)^2,$$ so the $r_i$ are distinct exactly when the $\rho_i$ are distinct. It is easy to check that when $(v^2 + 3 v + 4)(v^2 - v + 4) (v^3 - 6 v^2 - 7 v - 4)\neq 0$ the $\rho_i$ are distinct, so in this case the curve $D$ of Theorem \[T-simple2\] has genus $2$. Furthermore, we see that $D$ is isomorphic to the curve ${{\mathcal C}}(r,s,t).$
For each $i$ let $W_i$ be the point $(\rho_i,0)$ of $D$. Let $G$ be the Galois-stable subgroup of the Jacobian of $D$ that consists of the divisor classes $$\{ [0], [W_1 - W_5], [W_2 - W_4], [W_3 - W_6]\}$$ and let $G'$ be the Galois-stable subgroup $$\{ [0], [W_1 - W_3], [W_2 - W_6], [W_4 - W_5]\}.$$ An easy computation shows that when $v^3 - 4 v^2 + 7 v + 4$ and $v^2 + v + 2$ are nonzero the $G$-Richelot dual of $D$ and the $G'$-Richelot dual of $D$ are defined (that is, the determinants mentioned in the definition of the two Richelot duals are nonzero). Then the results of Section \[S-RM\] show that the $G$-Richelot dual of $D$ and the $G'$-Richelot dual of $D$ become isomorphic over $K(\sqrt{st}\,) = K(\sqrt{v(v-4)}\,)$.
The proof that these two Richelot duals of $D$ are geometrically non-isomorphic to one another (except in the special cases listed in the theorem) is a computation along the same lines as the proof of the corresponding statement of Theorem \[T-nonsimple2\]. We leave the details to the reader.
Finally, suppose that $K$ has characteristic $0$ and suppose that there is a ring homomorphism from ${{\mathbb Z}}[v]$ to ${{\mathbb F}}_{13}$ that takes $v$ to either $2$ or $6$. Then the curve $D$ reduces modulo $13$ to one of the curves over ${{\mathbb F}}_{13}$ obtained when $v=2$ or $v=6$. One can compute that the characteristic polynomials of Frobenius for the Jacobians of these two curves are $t^4 - 4 t^3 + 22 t^2 - 52 t + 169$ and $t^4 + 4 t^3 + 22 t^2 + 52 t + 169$, respectively. Then [@Howe-Zhu Thm. 6] shows that the Jacobians are absolutely simple. Since $D$ modulo $13$ is absolutely simple, so is $D$ itself. And finally, since $C$ and $C'$ have Jacobians isogenous to that of $D$, we see that their Jacobians are absolutely simple too. The final statement of the theorem then follows from the observation that if $v$ is not algebraic, then there is a homomorphism ${{\mathbb Z}}[v]\to{{\mathbb F}}_{13}$ that sends $v$ to any given element.
We used the field ${{\mathbb F}}_{13}$ at the end of the proof simply because it is the smallest prime field that contains values of $v$ that give rise to absolutely simple Jacobians. Other prime fields have a larger proportion of good values of $v$. For example, there are $341$ values of $v$ in ${{\mathbb F}}_{769}$ that give rise to absolutely simple Jacobians. For three-digit primes $p$ the number of good $v$-values is typically greater than $0.3 p$. This implies that for a “randomly chosen” rational number $v$, it is almost certainly the case that $v$ will give rise to an absolutely simple Jacobian.
Proof of Theorem [\[T-nonsimple3\]]{} {#S-genus3}
=====================================
The proof of Theorem \[T-nonsimple3\] is very much like the proof of Theorem \[T-nonsimple2\]: We will produce three elliptic curves $E_1$, $E_2$, $E_3$, two maximal isotropic subgroups $G$, $G'$ of the $2$-torsion of $A = E_1\times E_2\times E_3$, and an automorphism $\alpha$ of $A$ that takes $G$ to $G'$. Then we will produce a hyperelliptic curve whose Jacobian is $A/G$ and a plane quartic whose Jacobian is $A/G'$. To produce these curves we will use the results of [@HLP §4]. Our notation will be chosen to match that of [@HLP], except that we will continue to call our base field $K$, instead of $k$.
Let $K$ be an arbitrary field of characteristic not $2$ and let $t$ be an element of $K$ with $t (t + 1) (t^2 + 1) (t^2 + t + 1) \neq 0$. Let $s = -(t^2 + t + 1)$ and let $r$ be a square root of $t^2 + 1$ in an algebraic closure of $K$. Let
---------------------------- --------------------------- --
$A_1 = -2(t^2+1)s$ $B_1 = (t^2+1)s^2$
$A_2 = 4(t^2+1)s$ $B_2 = 4t^2(t^2+1)s^2$
$A_3 = -2(t^2+t+1)s\qquad$ $B_3 = (t+1)^2(t^2+1)s^2$
---------------------------- --------------------------- --
and for each $i$ let $$\begin{aligned}
\Delta_1 &= A_1^2 - 4B_1 = 4t^2(t^2+1)s^2 \\
\Delta_2 &= A_2^2 - 4B_2 = 16(t^2+1)s^2 \\
\Delta_3 &= A_3^2 - 4B_3 = 4t^2s^2.\end{aligned}$$ Note that the $\Delta_i$ and the $B_i$ are nonzero, so we may define for each $i$ an elliptic curve $E_i$ by $$y^2 = x(x^2 + A_i x + B_i).$$ We define $2$-torsion points $P_i$ on the $E_i$ by setting $$\begin{aligned}
P_1 &= \bigl((t^2+1)s - rts, 0\bigr)\\
P_2 &= \bigl( -2(t^2+1)s - 2rs, 0\bigr)\\
P_3 &= \bigl((t^2+t+1)s - ts, 0\bigr)\end{aligned}$$ and for each $i$ we let $Q_i$ be the $2$-torsion point $(0,0)$ on $E_i$ and we let $R_i = P_i + Q_i$.
Let $A = E_1\times E_2\times E_3$ and let $G$ be the subgroup of $A[2]$ generated by $(P_1,P_2,P_3)$, $(Q_1,Q_2,0)$, and $(Q_1,0,Q_3)$. Associated to these choices of $A$ and $G$ there is a quantity called the [*twisting factor*]{} $T$ (see [@HLP §4]). Using the formula in [@HLP §4] we find that for our $A$ and $G$ the twisting factor is $0$, so we may apply [@HLP Prop. 14] to find a hyperelliptic genus-$3$ curve whose Jacobian is isomorphic over $K$ to $A/G$. The curve given by [@HLP Prop. 14] is defined by two equations in ${{\mathbb P}}^3$, namely $$\begin{aligned}
W^2 Z^2 &= a X^4 + b Y^4 + c Z^4 \label{E-oldH1}\\
0 &= d X^2 + e Y^2 + f Z^2 \label{E-oldH2}\end{aligned}$$ where $$\begin{aligned}
a &= 4 t (t+1) (t^2+1)^3 s^5 \\
b &= 16 t^2 (t+1) (t^2+1)^3 s^5 \\
c &= 4 t (t+1)^2 (t^2+1)^2 s^6 \\
1/d &= -2 t (t+1) (t^2+1) s^2 \\
1/e &= (t+1) (t^2+1) s^2 \\
1/f &= 2 t (t^2+1) s^2.\end{aligned}$$ If we replace $W$ by $2t(t+1)(t^2+1)s^3 W$ in Equation \[E-oldH1\] and divide out common factors, we get Equation \[E-H1\], and if we multiply Equation \[E-oldH2\] by $2 t (t+1) (t^2+1) s^2$ we get Equation \[E-H2\]. This shows that the Jacobian of $H(t)$ is isomorphic to $A/G$.
Now let $G'$ be the subgroup of $A[2]$ generated by $(P_1,P_2,R_3)$, $(Q_1,Q_2,0)$, and $(Q_1,0,Q_3)$, and let $T'$ be the twisting factor associated to $A$ and $G'$. The formula in [@HLP §4] shows that $$T' = -64(t^2+1)^2(t^2+t+1)s^3 = 64(t^2+1)^2s^4,$$ so the twisting factor is a nonzero square. Then [@HLP Prop. 15] shows that there is a plane quartic whose Jacobian is isomorphic (over $K$) to $A/G'$. The plane quartic is given by $$\label{E-oldQ}
B_1 X^4 + B_2 Y^4 + B_3 Z^4 + d' X^2Y^2 + e' X^2Z^2 + f'Y^2Z^2 = 0$$ where $$\begin{aligned}
d' &= 4 (t^2 + 1) (2t^2 + t + 2) s^2\\
e' &= -2 (t^2 + 1) (2t^2 + t + 1) s^2\\
f' &= 4 (t^2 + 1) ( t^2 + t + 2) s^2.\end{aligned}$$ Dividing Equation \[E-oldQ\] by $(t^2+1)s^2$ gives Equation \[E-Q\], so the Jacobian of $Q(t)$ is isomorphic to $A/G'$.
To complete the proof we must show that $A/G \cong A/G'$. Note that there is a $2$-isogeny $\psi$ from $E_1$ to $E_2$ that kills $Q_1$ and that takes $P_1$ and $R_1$ to $Q_2$ (see [@Silverman Example III.4.5]). Consider the automorphism $\alpha$ of $A$ that sends a point $(S_1,S_2,S_3)$ to $(S_1,S_2+\psi(S_1),S_3)$. It is easy to check that $\alpha(G) = G'$, and it follows that $A/G\cong A/G'$, as desired, so the Jacobians of $H(t)$ and $Q(t)$ are isomorphic over $K$.
Examples {#S-examples}
========
The curves $$3y^2 = (x^2 - 4) (x^4 + 7x^2 + 1)$$ and $$- y^2 = (x^2 + 4) (x^4 + 3x^2 + 1)$$ over ${{\mathbb Q}}$ are geometrically non-isomorphic, and yet their Jacobians are isomorphic to one another over ${{\mathbb Q}}$.
If we take the two curves obtained by taking $t = 2$ in Theorem \[T-nonsimple2\], replace $x$ by $x/2$ in each equation, and twist both curves by $2$, we get the two curves given above.
The curves $$5 y^2 = - 6 x^6 - 64 x^5 - 113 x^4 + 262 x^3
- 331 x^2 + 584 x + 232$$ and $$2 y^2 = - 21 x^6 - 236 x^5 + 45 x^4 - 440 x^3
- 615 x^2 - 76 x - 553$$ are geometrically non-isomorphic, but their Jacobians become isomorphic to one another over ${{\mathbb Q}}(\sqrt{-1})$. Furthermore, their Jacobians are absolutely simple.
Take $v = 2$ in Theorem \[T-simple2\]. We find that $\rho_1 = -w/4$, $\rho_2 = w/4$, $\rho_3 = 2w + 2$, $\rho_4 = w - 1$, $\rho_5 = -w - 1$, and $\rho_6 = -2w + 2$, where $w = \sqrt{2}$. The curves $C$ and $C'$ in the theorem are $y^2 = f_1$ and $y^2 = f_2$, where $$\begin{aligned}
f_1 &= -(30625/32) x^6 - (67375/16) x^5 - (305025/64) x^4 \\
& \qquad - (23765/16) x^3 + (28665/16) x^2 + (1715/2) x - (735/2)\end{aligned}$$ and $$f_2 = -(553/2) x^6 + 38 x^5 - (615/2) x^4 + 220 x^3
+ (45/2) x^2 + 118 x - (21/2).$$ Replacing $x$ with $-2/(x+1)$ in $f_1$ and multiplying the result by $(1/5)(2/7)^2 (x+1)^6$ gives rise to the first curve given in the example. Replacing $x$ with $-1/x$ in $f_2$ and multiplying the result by $2 x^6$ gives rise to the second curve. Thus the Jacobians of the two curves become isomorphic to one another over ${{\mathbb Q}}(\sqrt{v(v-4)}\,) = {{\mathbb Q}}(\sqrt{-1})$. The Jacobians are simple because we chose our $v$ to be $2$ modulo $13$.
\[EX-realpoints\] The curves $$y^2 + (x^3 + x^2 + x) y = 31 x^6 - 38 x^5 - 217 x^4 - 380 x^3
+ 304 x^2 + 501 x - 366$$ and $$11 y^2 = - 49 x^6 - 378 x^5 - 755 x^4 + 110 x^3
- 2285 x^2 + 732 x - 1368$$ are geometrically non-isomorphic, but their Jacobians are isomorphic to one another over ${{\mathbb Q}}$. Furthermore, their Jacobians are absolutely simple.
Take $v = -4/3$ in Theorem \[T-simple2\]. The curves $C$ and $C'$ we obtain are $y^2 = f_1$ and $y^2 = f_2$, where $$\begin{aligned}
f_1 &= (28125/268912) x^6 - (11250/16807) x^5 \\
& \qquad + (3154875/1882384) x^4 - (812325/470596) x^3 \\
& \qquad\qquad - (57675/470596) x^2 + (26325/16807) x - (2025/2401)\end{aligned}$$ and $$\begin{aligned}
f_2 &= -(131769/38416) x^6 + (11979/343) x^5 \\
& \qquad - (5595645/38416) x^4 + (62535/196) x^3 \\
& \qquad\qquad - (3735435/9604) x^2 + (86229/343) x - (23199/343).\end{aligned}$$ If we replace $x$ with $(2x+2)/(x+2)$ in $f_1$, multiply the result by $(343/5)^2 (x+2)^6$, and twist by $3$, we get the curve $$y^2 = 125 x^6 - 150 x^5 - 865 x^4 - 1518 x^3
+ 1217 x^2 + 2004 x - 1464;$$ replacing $y$ with $2y + (x^3 + x^2 + x)$ gives the first curve in the example. If we replace $x$ with $(x+2)/(x+1)$ in $f_2$, multiply the result by $(1/11) 196^2 (x+1)^6$, and twist by $3$, we get the second curve in the example. The Jacobians of the two curves become isomorphic to one another over ${{\mathbb Q}}(\sqrt{v(v-4)}\,) = {{\mathbb Q}}$. The Jacobians are absolutely simple because their reductions modulo $17$ are absolutely simple.
It is easy to see that the first curve in Example \[EX-realpoints\] has real-valued points, while the second curve does not. It follows that the real topology of a curve over ${{\mathbb Q}}$ is not determined by its Jacobian. Furthermore, suppose we choose a positive integer $d$ such that the quadratic twist of the second curve by $d$ has rational points. The quadratic twist of the first curve by $d$ will still not have any real points, let alone any rational points, so we see that the existence of rational points on a genus-$2$ curve over ${{\mathbb Q}}$ is not determined by its Jacobian, even if the Jacobian is absolutely simple.
There are also triples $(r,s,t)$ that satisfy the hypotheses of Theorem \[T-gensimple\] but that do not lie on the elliptic surface discussed in Section \[S-application\].
\[EX-twiceseen\] The curves $$y^2 = x^6 - 24 x^4 + 80 x^3 - 63 x^2 - 24 x - 2$$ and $$y^2 = -2 x^6 + 6 x^5 + 9 x^4 - 48 x^3 + 162 x - 171$$ are geometrically non-isomorphic, but their Jacobians become isomorphic to one another over ${{\mathbb Q}}(\sqrt{2})$. Furthermore, their Jacobians are absolutely simple.
We take $r = -7/4$ and $s = 1/2$ and $t = 1/4$ in Theorem \[T-gensimple\]. Let $\xi$ be a root of the irreducible polynomial $$x^6 + 6x^4 + 9x^2 + 16$$ and let $K$ be the number field generated by $\xi$. The polynomial $h$ of Section \[S-Galois\] is $$h = T^3 + 3/4 T^2 + 9/16 T + 3/64$$ and its roots are the elements $$\begin{aligned}
\beta_1 &= ( -\xi^4 -7\xi^2 -12)/16 \\
\beta_2 &= ( - \xi^5 +\xi^4 -5\xi^3 +7\xi^2 -10\xi )/32 \\
\beta_3 &= (\phantom{-}\xi^5 +\xi^4 +5\xi^3 +7\xi^2 +10\xi )/32\end{aligned}$$ of $K$. The polynomials $g_i$ are given by $$g_i = x^2 - 2\beta_i x + \beta_i^2 / 2 + 3/32,$$ and their roots (indexed in accordance with the proof of Theorem \[T-Galois\]) are $$\begin{aligned}
r_1 &=( - \xi^4 - \xi^3 - 7\xi^2 - 7\xi - 12)/16 \\
r_2 &=( - \xi^4 + \xi^3 - 7\xi^2 + 7\xi - 12)/16 \\
r_3 &=( - \xi^5 - \xi^4 - 3\xi^3 - 3\xi^2 - 8\xi - 8)/32 \\
r_4 &=( - \xi^5 + 3\xi^4 - 7\xi^3 + 17\xi^2 - 12\xi + 8)/32 \\
r_5 &=(\phantom{-}\xi^5 + 3\xi^4 + 7\xi^3 + 17\xi^2 + 12\xi + 8)/32 \\
r_6 &=(\phantom{-}\xi^5 - \xi^4 + 3\xi^3 - 3\xi^2 + 8\xi - 8)/32\end{aligned}$$ We note that the two subgroups $G$ and $G'$ that appear in the proof of Theorem \[T-Galois\] are indeed Galois stable. We compute that the two Richelot duals are $y^2 = f_1 $ and $y^2 = f_2$, where $$\begin{aligned}
f_1 &= - (81/512) x^6 - (1215/1024) x^5
- (21141/8192) x^4 - (8991/8192) x^3 \\
& \qquad - (19683/131072) x^2 - (2187/262144) x + (729/2097152)\\
f_2 &= - (1863/256) x^6 - (3159/512) x^5
- (26973/4096) x^4 - (11421/4096) x^3 \\
& \qquad - (76545/65536) x^2 - (28431/131072) x - (13851/1048576).\end{aligned}$$ Evaluating $f_1$ at $(2-x)/(4x)$ and multiplying the result by $(256/9)^2 x^6$ gives the first curve mentioned in the example; evaluating $f_2$ at $-x/(4x-8)$ and multiplying the result by $(128/9)^2 (x-2)^6$ gives the second curve. These curves are geometrically non-isomorphic, and their Jacobians become isomorphic over ${{\mathbb Q}}(\sqrt{st}\,) = {{\mathbb Q}}(\sqrt{2})$. Furthermore, their Jacobians are absolutely simple because their reductions modulo $7$ are absolutely simple.
We obtained Example \[EX-twiceseen\] from a triple $(r,s,t)$ that does not lie on the elliptic surface from Section \[S-application\], but the same example can be obtained from the triple $(r,s,t) = (-10,-1,-2)$, which *does* lie on the surface.
We computed all triples $(r,s,t)$ of naïve height at most $20$ for which the curve ${{\mathcal C}}(r,s,t)$ has two ${{\mathbb Q}}$-rational Richelot duals whose Jacobians are absolutely simple and isomorphic over ${{\mathbb Q}}$. Of all the examples we found, the triple $(r,s,t) = (-19/3, -6, -1/6)$ gave rise to the curves with the smallest coefficients:
\[EX-smallest\] The curves $$y^2 = -9 x^6 + 6 x^5 - 47 x^4 - 14 x^3 - 5 x^2 - 36 x - 72$$ and $$y^2 = 8 x^6 - 60 x^5 + 235 x^4 - 186 x^3 - 239 x^2 - 30 x - 1$$ are geometrically non-isomorphic, but their Jacobians are isomorphic to one another over ${{\mathbb Q}}$. Furthermore, their Jacobians are absolutely simple.
\[EX-g3\] The Jacobian of the hyperelliptic curve $$3 v^2 = - 17 u^8 + 56 u^7 - 84 u^6 + 56 u^5
- 70 u^4 - 56 u^3 - 84 u^2 - 56 u - 17$$ and the Jacobian of the plane quartic $$x^4 + 4 y^4 + 4 z^4 + 20 x^2 y^2 - 8 x^2 z^2 + 16 y^2 z^2 = 0$$ are isomorphic to one another over ${{\mathbb Q}}$.
We take $t=1$ in Theorem \[T-nonsimple3\]. The plane quartic $Q(1)$ from the theorem is the plane quartic given in the example. The hyperelliptic curve from the theorem is given by the pair of homogeneous equations $$\begin{aligned}
W^2 Z^2 &= -(1/3) X^4 - (4/3) Y^4 + Z^4 \label{E-exH1}\\
0 &= -X^2 + 2Y^2 + 2Z^2. \label{E-exH2}\end{aligned}$$ We dehomogenize the equations by setting $Z=1$, and we parametrize the conic given by Equation \[E-exH2\] by setting $$\begin{aligned}
X &= 2(u^2+1)/(u^2 + 2u - 1)\\
Y &= (u^2 - 2u - 1)/(u^2 + 2u - 1).\end{aligned}$$ Taking $W = v / (u^2 + 2u - 1)^4$ then gives us the hyperelliptic curve in our example.
We note that the discriminant of the degree-$8$ polynomial used to define the hyperelliptic curve in Example \[EX-g3\] is $2^{94}$!
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|
---
author:
- 'John Coates, Minhyong Kim [^1], Zhibin Liang [^2], Chunlai Zhao[^3]'
title: 'On the 2-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication'
---
Introduction
============
Let $E$ be an elliptic curve defined over ${\mathbb Q}$, with complex multiplication by the ring of integers of an imaginary quadratic field $K$. Thus, by the theory of complex multiplication, $K$ must be either ${\mathbb Q}(\sqrt{-1}), {\mathbb Q}(\sqrt{-2}), {\mathbb Q}(\sqrt{-3})$, or one of the fields $$\label{kay}
{\mathbb Q}(\sqrt{-q}) \, \, (q=7, 11, 19, 43, 67, 163).$$ Recently, Y. Tian [@T1], [@T2] made the remarkable discovery that one could prove deep results about the arithmetic of certain quadratic twists of $E$ with root number $-1$, by combining formulae of Gross-Zagier type for these twists, with a weak form of the 2-part of the conjecture of Birch and Swinnerton-Dyer for certain other quadratic twists of $E$, where the root number is $+1$. We recall that, when the complex $L$-series of an elliptic curve with complex multiplication does not vanish at $s=1$, the $p$-part of the conjecture of Birch and Swinnerton-Dyer has been established, by the methods of Iwasawa theory, for all primes $p$ which do not divide the order of the group of roots of unity of $K$ (see [@R]). However, at present we do not know how to extend such methods to cover the case of the prime $p=2$. Nevertheless, when $K = {\mathbb Q}(\sqrt{-1})$, one of us [@Z1], [@Z2], [@Z3], [@Z4] did establish a weaker result in this direction for the prime $p=2$, by combining the classical expression for the value of the complex $L$-series as a sum of Eisenstein series (see Corollary \[z\]), with an averaging argument over quadratic twists, and happily this weaker result has sufficed for Tian’s work in [@T1], [@T2]. The aim of the present note is to show that the rather elementary method developed in the papers [@Z1], [@Z2], [@Z3], [@Z4] works even more simply for quadratic twists of those elliptic curves $E$ having good reduction at the prime 2, and with complex multiplication by the ring of integers of the fields $K$ given by . We hope that one can use some of the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer established here (see, in particular, our Corollary \[ap\]) to extend the deep results of [@T1], [@T2], [@T3], to certain infinite families of quadratic twists of our curves $E$, having root number equal to $-1$ . It is also interesting to note that, in [@T3], Tian and his collaborators introduce a new and completely different method for establishing weak forms of the 2-part part of the conjecture of Birch and Swinnerton-Dyer for curves with $K = {\mathbb Q}(\sqrt{-1})$, by using a celebrated formula of Waldspurger, and they believe that this new method can eventually be applied to a much wider class of elliptic curves, including those without complex multiplication. Needless to say, the rather elementary methods used here seem to be special to elliptic curves with complex multiplication. Finally, we wish to thank Y. Tian for his ever helpful comments on our work.
The averaging argument
======================
Let $K$ be an imaginary quadratic field of class number 1, which we assume is embedded in ${\mathbb C}$, and let $ \mathcal{O}_K$ its ring of integers. Let $E$ be any elliptic curve defined over $K$, whose endomorphism ring is isomorphic to $\mathcal{O}_K$. Fix once and for all a global minimal generalized Weierstrass equation for $E$ over $\mathcal{O}_K$ $$\label{1}
y^2+a_1xy+a_3y =
x^3+a_2x^2+a_4x+a_6 \, \, (a_i\in\mathcal{O}_K).$$ Let $\frak L$ be the period lattice of the Neron differential $dx/(2y+a_{1}x+a_{3})$. Then $\frak L$ is a free $\mathcal{O}_{K}$-module of rank 1, and we fix $\Omega_{\infty}\in\mathbb{C}^{\times}$ such that $\frak {L}=\Omega_{\infty}\mathcal{O}_{K}$. Denote by $\psi_E$ the Grossencharacter of $E/K$ in the sense of Deuring-Weil, and write $\frak f$ for the conductor of $\psi_E$ (thus the prime divisors of $\frak f$ are precisely the primes of $K$ where $E$ has bad reduction). Now let $\frak g$ be any integral multiple of $\frak f$, and fix $g \in {{\mathcal O}_K}$ such that $\frak g = g{{\mathcal O}_K}$. Let $S$ be the set of primes ideals of $K$ dividing $\frak g$, and denote by $$L_{S}(\bar{\psi}_{E},s)=\displaystyle\sum_{(\mathfrak{a}, \frak g)=1}\frac{\bar{\psi}_{E}(\mathfrak{a})}{(N\mathfrak{a})^{s}}$$ the imprimitive Hecke $L$-function of the complex conjugate Grossencharacter of ${\psi}_{E}$. Our subsequent induction argument is based on the following expression for $L_{S}(\bar{\psi}_{E},s)$, which goes back to the 19th century. Let $z$ and $s$ be complex variables. For any lattice $L$ in the complex plane $\mathbb{C}$, define the Kronecker-Eisenstein series by $$H_{1}(z,s,L):=\sum_{w\in L}\frac{\bar{z}+\bar{w}}{|z+w|^{2s}},$$ where the sum is taken over all $w \in L$, except $-z$ if $z \in $L. This series converges to define a holomorphic function of $s$ in the half plane $Re(s)>3/2$, and it has an analytic continuation to the whole $s$-plane. Let $\frak R$ denote the ray class field of $K$ modulo $\frak g$, and let $\cal B$ be any set of integral ideals of $K$, prime to $\frak g$, whose Artin symbols give precisely the Galois group of $\frak R$ over $K$ (in other words, $\cal B$ is a set of integral ideals of $K$ representing the ray class group of $K$ modulo $\frak g$). Since the conductor of $\psi_E$ divides $\frak g$, it is well known that $\frak R$ is equal to the field $K(E_g)$, which is obtained by adjoining to $K$ the coordinates of the $g$-division points on $E$.
\[2\] We have $$L_{S}(\bar{\psi}_E,s) =
\frac{|\Omega_\infty/g|^{2s}}{\overline{(\Omega_\infty/g)}}\displaystyle\sum_{\frak b\in\cal{B}}H_1(\psi_E(\frak b)\Omega_\infty/g,s,\frak{L}).$$
As mentioned above $\cal B$ is a set of integral representatives of the ray class group of $K$ modulo $\frak g$, and so it follows that, fixing any generator of each $\frak b$ in $\cal B$, we obtain a set of representatives of $(\mathcal{O}/\frak{g})^*/\tilde{\mu}_K,$ where $\tilde{\mu}_K$ denotes the image under reduction modulo $\frak{g}$ of the group $\mu_K$ of roots of unity of $K$. Moreover, the very existence of $\psi_E$ shows that the reduction map from $\mu_K$ to $\tilde{\mu}_K$ must be an isomorphism of groups. For each $\frak b$ in $\cal B$, we choose the generator of $\frak b$ given by $\psi_E(\frak b)$. It follows that, as $\frak b$ runs over $\cal B$ and $c$ runs over $\frak g$, the principal ideals $(\psi_E(\frak b)+c)$ run over all integral ideals of $K$, prime to $\frak g$, precisely once. Thus $$L_{S}(\bar{\psi}_E,s) =
\sum_{\frak b\in\cal{B}}\sum_{c\in \frak{g}}\frac{\bar{\psi}_E((\psi_E(\frak b)+c))}{|\psi_E(\frak b)+c|^{2s}}.$$ Note that, since $c \in \frak{g}$, we have $$(\psi_E(\frak b)+c) = (\psi_E(\frak b))(1+ c/{\psi_E(\frak b)}) =
\frak{b}(1+ c/{\psi_E(\frak b)}),$$ so that $$\psi_E((\psi_E(\frak b)+c)) =
\psi_E(\frak b)(1+ c/{\psi_E(\frak b)}) = \psi_E(\frak b)+c.$$ Hence $$L_{S}(\bar{\psi}_E,s) =
\sum_{\frak b \in B}\sum_{c \in \frak{g}}\frac{\overline{\psi_E(\frak b)+c}}{|\psi_E(\frak b)+c|^{2s}},$$ which can easily be rewritten as $$\frac{|\Omega_{\infty}/g|^{2s}}{\overline{(\Omega_{\infty}/g})}\sum_{\frak {b} \in\cal B}\sum_{w \in \frak{L}}\frac{\overline{\psi_{E}\frak b)\Omega_{\infty}/g+w}}{|\psi_{E}(\frak b)\Omega_{\infty}/g+w|^{2s}} \, ,$$ completing the proof of the theorem.
We recall that, for any lattice $L$, the non-holomorphic Eisenstein series ${\cal E}_1^*(z, L)$ is defined by $${{\cal E}_1^*}(z, L) = H_1(z, 1, L).$$ Then the above proposition immediately implies that $$L_{S}(\bar{\psi}_E,1)/\Omega_\infty = g^{-1}\sum_{\frak {b} \in \cal B}{{\cal E}_1^*}(\psi_E(\frak b)\Omega_\infty/g, \frak L).$$ Also, it is well known (see, for example, [@GS]) that ${{\cal E}_1^*}(\psi_E(\frak b)\Omega_\infty/g, \frak L)$ belongs to the field $\frak R$, and satisfies $${{\cal E}_1^*}(\psi_E(\frak b)\Omega_\infty/g, \frak L) = {{\cal E}_1^*}(\Omega_\infty/g, \frak L)^{\sigma_{\frak b}},$$ where $\sigma_{\frak b}$ denotes the Artin symbol of $\frak b$ in the Galois group of $\frak R$ over $K$. Thus the above proposition has the following immediate corollary, where $Tr_{\frak R/K}$ denotes the trace map from $\frak R$ to $K$.
\[z\] We have $$L_{S}(\bar{\psi}_E,1)/\Omega_\infty = Tr_{\frak R/K}(g^{-1}{{\cal E}_1^*}(\Omega_\infty/g, \frak L)).$$
We next consider the twisting of $E$ by certain quadratic extensions of $K$. A non-zero element $M$ of $\mathcal{O}_K$ is said to be square free if it is not divisible by the square of any non-unit element of this ring.
Let $M$ be any non-zero and non-unit element of $\mathcal{O}_K$ , which satisfies (i) $M$ is square free, (ii) $M$ is prime to the discriminant of $K$, and (iii) $M \equiv 1 \, \, mod \, 4.$ Then the extension $K(\sqrt M)/K$ has conductor equal to $M\mathcal{O}_K$.
Since $M$ is square free and $M \equiv 1 \, \, mod \, 4$, the extension $K(\sqrt M)/K$ is totally and tamely ramified at all primes dividing $M$. Thus the assertion of the lemma will follow once we have shown that the primes of $K$ above 2 are not ramified in this extension. Let $v$ be any place of $K$ above 2. Let $w$ be such that $w^2 = M$, and put $z = (w-1)/2$. Then $z$ is a root of the polynomial $f(X) = X^2 - X - (M-1)/4$, so that $z$ is an algebraic integer. But $f'(z) = 2z-1$ is then clearly a unit at $v$, and so $v$ is unramified in our extension $K(\sqrt M)/K$, completing the proof.
Let $M$ be as in the above lemma, and assume in addition that $(M, \frak f)=1$. We write $\chi_M$ for the abelian character of $K$ defining the quadratic extension $K(\sqrt M)/K$, and let $E^{(M)}$ denote the twist of $E$ by $\chi_M$. Thus $E^{(M)}$ is the unique elliptic curve defined over $K$, which is isomorphic to $E$ over $K(\sqrt M)$, and which is such that $$E^{(M)}(K) = \{P \in E(K(\sqrt M))\, : \sigma(P) = \, \chi_M(\sigma)(P), \, \sigma \in Gal(K(\sqrt M)/K) \}.$$ The curve $E^{(M)}$ also has endomorphism ring isomorphic to ${{\mathcal O}_K}$, and its Grossencharacter, which we denote by $\psi_{E^{(M)}}$, is equal to the product $\psi_E \chi_M$. We write $\frak f_M$ for the conductor of $\psi_{E^{(M)}}$. In view of the above lemma, we have $\frak f_M = M\frak f$, because $(\frak f, M)=1$ and $\chi_M$ has conductor $M{{\mathcal O}_K}$. Finally, putting $$\frak p(z, \frak L) = x + (a_1^2 + 4a_2)/12, \, {\frak p}'(z, \frak L) = 2y + a_1x + a_3,$$ we obtain a classical Weierstrass equation for $E$ over ${\mathbb C}$ of the form $$Y^2 = 4X^3 - g_2(\frak L)X - g_3(\frak L),$$ with $X = \frak p(z, \frak L), \, Y = {\frak p}'(z, \frak L)$. The corresponding classical Weierstrass equation for $E^{(M)}$ over ${\mathbb C}$ is then given by $$Y^2 = 4X^3 - M^2g_2(\frak L)X - M^3g_3(\frak L).$$ Hence the period lattice for the curve $E^{(M)}$ over ${\mathbb C}$ is given by $$\label{6}
{\frak L}_M = \frac{ \Omega_{\infty}}{\sqrt M}{{\mathcal O}_K}.$$
We now suppose that we are given an infinite sequence $$\pi_1, \pi_2, \ldots, \pi_n, \ldots$$ of distinct prime elements of ${{\mathcal O}_K}$. We shall say that this sequence is [*admissible*]{} for $E/K$ if, for all $n \geq 1$, we have that $\pi_n$ is prime to the discriminant of $K$, and $$\label{8}
\pi_n \equiv 1 \, \, mod \, 4, \, \, \, \, (\pi_n, \frak f) = 1.$$ For each integer $n \geq 0$, define $$\label{3}
{{\mathcal M}_n}= \pi_1\cdots \pi_n, \, \, \, \frak g_n = {{\mathcal M}_n}\frak f.$$ We now take $\frak R_n$ to be the ray class field of $K$ modulo $\frak g_n$. Since $\pi_j \equiv 1 \, \, mod \, 4$, the above lemma shows that the extension $K(\sqrt{\pi_j})/K$ has conductor $\pi_j {{\mathcal O}_K}$, and so is contained in $\frak R_n$, for all $j$ with $1\leq j \leq n$. Hence the field $\frak J_n$ defined by $$\label{4}
\frak J_n = K(\sqrt{\pi_1},..., \sqrt{\pi_n})$$ is always a subfield of $\frak R_n$. Let $S_n$ be the set of prime ideals of $K$ dividing $\frak g_n$. Also, writing $f$ for any ${{\mathcal O}_K}$ generator of the ideal $\frak f$, we put $g_n = f{{\mathcal M}_n}$, so that $\frak g_n = g_n{{\mathcal O}_K}$. Finally, we define $\mathcal D_{n}$ to be the set of all divisors of ${{\mathcal M}_n}$ which are given by the product of any subset of $\{\pi_1,..., \pi_n\}$. The averaging theorem which follows is essentially contained in the earlier paper of one of us [@Z1], and is the basis of all of our subsequent arguments. For simplicity, we write just $\psi_M$ for the Grossencharacter of the curve $E^{(M)}$ for any $M \in \mathcal D_n$.
\[av\] Let $\{\pi_1,..., \pi_n,...\}$ be any admissible sequence for $E/K$. Then, for all integers $n \geq 1$, we have $$\label{5}
\sum_{M \in {\mathcal D}_{n}} L_{S_n}(\bar{\psi}_M, 1)/{\Omega_{\infty}} = 2^nTr_{\frak R_n/\frak J_n}({g_n}^{-1}{{\cal E}_1^*}(\Omega_\infty/g_n, \frak L)),$$ where $Tr_{\frak R_n/\frak J_n}$ denotes the trace map from $\frak R_n$ to $\frak J_n$.
Let $M$ be any element of $\mathcal D_{n}$. Applying Corollary \[z\] to the curve $E^{(M)}$ with $\frak g = \frak g_n$, and using , we conclude that $$L_{S_n}(\bar{\psi}_M, 1)\sqrt{M}/{\Omega_{\infty}} = Tr_{\frak R_n/K}({g_n}^{-1}{{\cal E}_1^*}(\frac{\Omega_\infty}{\sqrt{M}g_n}, \frak L_M)).$$ Now, for any non-zero complex number $\lambda$, we have $${{\cal E}_1^*}(z, \frak L_M) = \lambda {{\cal E}_1^*}(\lambda z, \lambda \frak L_M).$$ Hence, taking $\lambda = \sqrt{M}$, and writing $G_n$ for the Galois group of $\frak R_n/K$, we conclude that $$\label{7}
L_{S_n}(\bar{\psi}_M, 1)/{\Omega_{\infty}} = \sum_{\sigma \in G_n}(\sqrt{M})^{\sigma - 1}{g_n}^{-1}({{\cal E}_1^*}(\Omega_\infty/g_n, \frak L))^\sigma.$$ It is now clear that the assertion of the theorem is an immediate consequence of the following lemma.
\[div\] Let $H_n = Gal({\frak R_n}/{\frak J}_n)$. If $\sigma$ is any element of $G_n$, then $\sum_{M \in {\mathcal D}_n}(\sqrt{M})^{\sigma - 1}$ is equal to $2^n$ if $\sigma$ belongs to $H_n$, and is equal to $0$ otherwise.
The first assertion of the lemma is clear. To prove the second assertion, suppose that $\sigma$ maps $k \geq 1$ elements of the set $\{\sqrt{\pi_1},..., \sqrt{\pi_n\}}$ to minus themselves, and write $V(\sigma)$ for the subset consisting of all such elements. If $M$ be any element of $\mathcal D_n$, it is clear that $\sigma$ will fix $\sqrt M$ if and only if $M$ is a product of an even number of elements of $V(\sigma)$, with an arbitrary number of elements of the complement of $V(\sigma)$ in $\{\sqrt{\pi_1},..., \sqrt{\pi_n}\}$. Hence the total number of $M$ in $\mathcal D_n$ such that $\sigma$ fixes $\sqrt M$ is equal to $$2^{n-k}( (k, 0) + (k, 2) + (k, 4) + \ldots) = 2^{n-1},$$ where $(n,r)$ denotes the number of ways of choosing $r$ objects from a set of $n$ objects. Similarly, the total number of $M$ in $\mathcal D_n$ such that $\sigma$ maps $\sqrt M$ to $- \sqrt M$ is equal to $$2^{n-k}((k,1) + (k, 3) + (k, 5) + \dots) = 2^{n-1}.$$ Since these last two expressions are equal, the second assertion of the lemma is now clear.
Integrality at 2
================
We use the notation and hypotheses introduced in the last section. Our aim in this section is to prove the following result.
\[int\] Assume that $E$ has good reduction at the primes of K above 2, and that $\{\pi_1,..., \pi_n,...\}$ is any admissible sequence for $E/K$. For all $n \geq 1$, define $$\Psi_n =Tr_{\frak R_n/\frak J_n}({g_n}^{-1}{{\cal E}_1^*}(\frac{\Omega_\infty}{g_n}, \frak L)).$$ Then $2\Psi_n$ is always integral at all places of $\frak J_n$ above 2. Moreover, if the coefficient $a_1$ in is divisible by 2 in ${{\mathcal O}_K}$, then $\Psi_n$ is integral at all places of $\frak J_n$ above 2.
Before giving the proof of the theorem, we recall some classical identities involving elliptic functions (see. for example, [@D]). Let $L$ be any lattice in the complex plane, and write $\frak p(z, L)$ for the Weierstrass $\frak p$ -function attached to $L$. For each integer $m \geq 2$, we define the elliptic function $B_m(z,L)$ by $$2B_m(z, L) = \frac{\frak p''(z,L)}{\frak p'(z, L)} + \sum _{k=2}^{k=m-1} \frac{\frak p'(kz, L) -\frak p'(z,L)}{\frak p(kz, L) -\frak p(z,L)}.$$
For all integers $m \geq 2$, we have $$B_m(z, L) = {{\cal E}_1^*}(mz, L) - m{{\cal E}_1^*}(z, L).$$
Let $\zeta(z, L)$ denote the Weierstrass zeta function of $L$. The following identity is classical $${{\cal E}_1^*}(z, L) = \zeta(z, L) - z s_2(L) - \bar{z}A(L)^{-1},$$ (see, for example, Prop. 1.5 of [@GS], where the definitions of the constants $s_2(L)$ and $A(L)$ are also given). It follows immediately that $${{\cal E}_1^*}(mz, L) - m{{\cal E}_1^*}(z, L) = \zeta(mz, L) - m\zeta(z, L).$$ But now we have the addition formula $$\zeta(z_1+z_2, L) = \zeta(z_1, L) + \zeta(z_2, L) + \frac{1}{2} \frac{\frak p'(z_1, L) -\frak p'(z_2,L)}{\frak p(z_1, L) -\frak p(z_2,L)}.$$ Taking the limit as $z_1$ tends to $z_2$, we obtain the statement of the lemma for $m=2$. For any $m \geq 2$, the above addition formula also shows that $$\zeta((m+1)z, L) - (m+1)\zeta(z, L) = \zeta(mz, L) - m\zeta(z, L) + \frac{1}{2} \frac{\frak p'(mz, L) -\frak p'(z,L)}{\frak p(mz, L) -\frak p(z,L)},$$ whence the assertion of the lemma follows by induction on $m$.
The next lemma is attributed in [@D] to unpublished notes of Swinnerton-Dyer.
Let $w$ be any complex number such that $w$ is not in $L$, but $mw$ does belong to $L$ for some integer $m\geq 2$. Then ${{\cal E}_1^*}(w, L) = -B_{m-1}(w, L)/m$.
By the previous lemma, we have $$B_{m-1}(w, L) = {{\cal E}_1^*}((m-1)w, L) - (m-1){{\cal E}_1^*}(w, L).$$ But, as a function of $z$, ${{\cal E}_1^*}(z,L)$ is periodic with respect to $L$ and odd, whence it follows that ${{\cal E}_1^*}((m-1)w, L) = -{{\cal E}_1^*}(w, L)$. This completes the proof.
Now we have the addition formula $$\frak p(z_1+z_2, L) + \frak p(z_1, L) + \frak p(z_2, L) = \frac{1}{4} ((\frak p'(z_1, L) -\frak p'(z_2,L))/(\frak p(z_1, L) -\frak p(z_2,L)))^2,$$ whence we immediately obtain the following corollary.
\[11\] Let $w$ be any complex number such that $w$ is not in $L$, but $w$ does have finite order in ${\mathbb C}/L$. Let $m$ be the exact order of $w$ in ${\mathbb C}/L$. Assuming $m \geq 3$, we have $$m{{\cal E}_1^*}(w,L) = \sum _{k=1}^{k=m-2}\epsilon_k(\frak p((k+1)w, L) + \frak p(kw, L) + \frak p(w, L))^{1/2},$$ where $\epsilon_k$ denotes the sign $+1$ or $-1$.
We can now give the proof of Theorem \[int\]. Recall that the period lattice of the Neron differential of our fixed global minimal Weierstrass equation is $\frak L= \Omega_{\infty}{{\mathcal O}_K}$. Take $w= \psi (\frak b)\Omega_{\infty}/g_n$, where $\frak b$ is any fixed integral ideal of $K$ prime to $\frak g_n$. Thus ${{\cal E}_1^*}(w, \frak L)$ is any one of the conjugates of ${{\cal E}_1^*}(\Omega _\infty/g_n, \frak L)$ over $K$. Let $m$ be the smallest positive rational integer lying in the ideal $\frak g_n$, so that $m$ is also the smallest positive rational integer with the property that $mw$ lies in $\frak L$. Moreover, since $E$ has good reduction at the primes of $K$ above 2, the ideal $\frak f$ is not divisible by any prime of $K$ above 2. This means that the smallest positive rational integer in the ideal $\frak g_n$ must be odd. It follows that $m$ is odd, and it must then be $>2$. Let $P$ be the point on $E$ defined by $w$. Then we have $$\label{12}
{\frak p}(rw, \frak L) = x(rP) + (a_1^2 + 4a_2)/12, \, \, (r=1,..., m-1).$$ But, as $E$ has good reduction at all primes of $K$ above 2 and the point $rP$ has odd order, it follows that $x(rP)$ is integral at each prime of $\frak R_n$ above 2. Thus we can immediately conclude from Corollary \[11\] and that the following two assertions. Firstly, if $a_1/2$ lies in ${{\mathcal O}_K}$, then every conjugate of ${{\cal E}_1^*}(\Omega_{\infty}/g_n, \frak L)$ over $K$ is integral at all places of $\frak R_n$ above 2. In general, if we drop the assumption that $a_1/2$ lies in ${{\mathcal O}_K}$, all we can say is that every conjugate of $2{{\cal E}_1^*}(\Omega_{\infty}/g_n, \frak L)$ over $K$ is integral at every place of $\frak R_n$ above $2$. Taken together, these two assertions clearly imply Theorem \[int\]. .
The induction argument
======================
Let $E$ be an elliptic curve defined over $K$, with complex multiplication by the ring of integers of $K$, and global minimal Weierstrass equation given by . We fix once and for all any place of the algebraic closure of ${\mathbb Q}$ above 2, and write $ord_2$ for the order valuation at this place, normalized so that $ord_2(2) = 1$. Define $\alpha_E$ to be 0 or 1, according as 2 does or does not divide $a_1$ in ${{\mathcal O}_K}$, where we recall that $a_1$ is one of the coefficients in the equation . For any admissible sequence $\{\pi_1,..., \pi_n,...\}$ for $E/K$, we define $\frak M_n = \pi_1\ldots\pi_n$, and $$\label{13}
L^{(alg)}(\bar{\psi}_{\frak M_n}, 1) = L(\bar{\psi}_{\frak M_n}, 1)\sqrt{\frak M_n}/\Omega_\infty,$$ which is an element of $K$. Moreover, we define $$\label{13*}
\phi_E = \alpha_E \, \, \rm{or} \, \, \, max\{\alpha_E, \, -ord_2(\it{L^{(alg)}}(\bar{\psi}_E, 1))\},$$ according as $L(\bar{\psi}_E, 1) = 0$, or $L(\bar{\psi}_E, 1) \neq 0$. Our goal in this section is to prove the following theorem.
\[21\] Assume that $K \neq {\mathbb Q}(\sqrt{-1}), {\mathbb Q}(\sqrt{-3})$, and that $E$ has good reduction at all places of $K$ above 2. Then, for all admissible sequences $\{\pi_1,..., \pi_n,...\}$ for $E/K$, and all integers $n \geq 1$, we have $$\label{14}
ord_2(L^{(alg)}(\bar{\psi}_{\frak M_n}, 1)) \geq n - \phi_E.$$
We shall prove the theorem by induction on $n$, and we begin with an obvious remark. Let $r$ be any integer $\geq 0$, and recall that $\psi_{\frak M_r}$ denotes the Grossencharacter of the twisted curve $E^{(\frak M_r)}$. For each $n > r$, write $\frak p_n = \pi_n{{\mathcal O}_K}$. Then $\frak p_n$ is prime to the conductor of $\psi_{\frak M_r}$, and we have $$\label{15}
ord_2(1- \bar{\psi}_{\frak M_r}( \frak p_n)/N \frak p_n) \geq 1.$$ Indeed, we have $\psi_{\frak M_r}( \frak p_n) = \zeta \pi_n$, where $\zeta = 1$ or $-1$ because $K \neq {\mathbb Q}(\sqrt{-1}), {\mathbb Q}(\sqrt{-3})$. Thus $\zeta \equiv 1\, mod \, \, 2$, and then follows easily because $\pi_n \equiv 1 \, mod \, 4$ and $ N \frak p_n = \psi_{\frak M_r}( \frak p_n) \bar{\psi}_{\frak M_r}( \frak p_n).$ Note also that, on combining Theorems \[int\] and \[av\], we conclude that, for all integers $n \geq 1$, we have $$\label{16}
ord_2(\sum_{M \in {\mathcal D}_{n}} L_{S_n}(\bar{\psi}_M, 1)/\Omega_{\infty}) \geq n - \alpha_E.$$ It is clear that, on combining for $r=0$ and for $n=1$, we immediately obtain for $n=1$. Suppose now that $n > 1$, and that has been proven for all integers strictly less than $n$. Combining this inductive hypothesis with assertion , we conclude that for all proper divisors $M$ of $\frak M_n$, we have $$ord_2(L_{S_n}(\bar{\psi}_M, 1)/\Omega_\infty) \geq n - \phi_E,$$ whence again shows that holds for the integer $n$. This completes the proof of the theorem.
We next investigate which rational primes $p$ split in $K$, and have the additional property that they can be written as $p=\pi\pi^*$, with $\pi$ in ${{\mathcal O}_K}$ satisfying $\pi \equiv 1 \, mod \, 4$ (and thus automatically also satisfying $\pi^* \equiv 1 \, mod \, 4$). We call primes $p$ with this property [*special*]{} split primes for $K$. Obviously, a necessary condition for $p$ to be a special split prime for $K$ is that $p \equiv 1 \, mod \, 4$. We remark that it is clear from the Chebotarev density theorem that there are always infinitely many special split primes for $K$.
\[split\] Assume that $K \neq {\mathbb Q}(\sqrt{-1}), {\mathbb Q}(\sqrt{-2}), {\mathbb Q}(\sqrt{-3})$. Let $p$ be any rational prime which splits in $K$, and which satisfies $p \equiv 1\, mod \, 4$. If $K = {\mathbb Q}(\sqrt{-7})$, then $p$ is always a special split prime for $K$. If $K = {\mathbb Q}(\sqrt{-q})$, where $q = 11, 19, 43, 67, 163$, then such a $p$ is a special split prime for $K$ if and only if we can write $p = \pi\pi^*$ in ${{\mathcal O}_K}$ with $\pi + \pi^* \equiv 0 \, mod \, 2$.
Let $K = {\mathbb Q}(\sqrt{-q})$, and put $\tau = (1+\sqrt{-q})/2$, so that $1, \tau$ form an integral basis of ${{\mathcal O}_K}$. Assume first that $K = {\mathbb Q}(\sqrt{-7})$. Then $p = a^2 + ab + 2b^2$, with $a$ an odd integer, whose sign can be chosen so that $a\equiv 1 \, mod \, 4$, and with $b$ an even integer, which has necessarily to be divisible by 4 since $p \equiv 1 \, mod \, 4$. We then clearly have that $ \pi = a + b\tau$ satisfies $\pi \equiv 1\, mod \, 4$. Finally, assume that $K = {\mathbb Q}(\sqrt{-q})$, where $q$ is any of $11, 19, 43, 67, 163$. Then $p = a^2 + ab + mb^2$, where $a$ and $b$ are integers, and $m = (q+1)/4$ is now an odd integer. Since $p \equiv 1\, mod \, 4$, we see that $\pi = a + b\tau$ satisfies $\pi \equiv 1 \, mod \, 4$ if and only if $a \equiv 1\, mod \, 4 \,$ and $b$ is even. But $\pi + \pi^* = 2a + b$, and so $\pi + \pi^*$ will be even if and only if $b$ is even. By if $b$ is even, then $a$ is odd, and then we can always choose the sign of $a$ so that $a \equiv 1\, mod \, 4$. This completes the proof.
Now assume that our elliptic curve $E$ is in fact defined over ${\mathbb Q}$, and take to be a global minimal Weierstrass equation for $E$ over ${\mathbb Q}$. Then the conductor $N(E)$ of $E$ is given by $$N(E) = d_KN\frak f,$$ where $d_K$ denotes the absolute value of the discriminant of $K$. Moreover, the complex $L$-series $L(E, s)$ of $E$ over ${\mathbb Q}$ coincides with the Hecke L-seres $L(\bar{\psi}_E, s)$. If $R$ is a non-zero square free integer, $E^{(R)}$ will now denote the twist of $E$ by the extension ${\mathbb Q}(\sqrt{R})/{\mathbb Q}$. Write $$\label{17}
L^{(alg)}(E^{(R)}, 1) = L(E^{(R)}, 1)\sqrt{R}/\Omega_\infty.$$ Finally, $\alpha_E$ has the same definition as earlier, and $\phi_E$ is again defined by .
\[sq1\] Assume that $E$ is defined over ${\mathbb Q}$, and has complex multiplication by the ring of integers of any of the fields $K = {\mathbb Q}(\sqrt{-q})$, where $q = 7, 11, 19, 43, 67, 163$. Suppose further that $E$ has good reduction at 2. Then the conductor $N(E)$ of $E$ is a square.
Let $p$ be any prime dividing $N(E)$. Since $E$ has potential good reduction at $p$, we must have that $p^2$ exactly divides $N(E)$ whenever $p > 3$. Also $p \neq 2$, because $E$ has good reduction at 2. Thus we only have to check that an even power of 3 must divide $N(E)$. But, since $q > 3$, it is well known (see [@GR]) that $E$ is the quadratic twist of an elliptic curve of conductor $q^2$, whence it follows immediately that either 3 does not divide $N(E)$, or $3^2$ exactly divides $N(E)$, according as 3 does not, or does, divide the discriminant of the twisting quadratic extension. This completes the proof.
We now introduce a definition which for the moment is motivated by what is needed to deduce the next theorem from our earlier induction argument (but see also the connexion with Tamagawa factors discussed in the next section). Write $w_E$ for the sign in the functional equation of $L(E,s)$. We continue to assume that $E$ is defined over ${\mathbb Q}$, and satisfies the hypotheses of Lemma \[sq1\]. If $D$ is any square free integer which is prime to $N(E)$, it is well known that the root number of the twist $E^{(D)}$ of $E$ by the quadratic extension ${\mathbb Q}(\sqrt{D})/{\mathbb Q}$ is given by $\chi_D(-N(E))w_E$, where $\chi_D$ denotes the Dirichlet character of this quadratic extension. Thus, in view of Lemma \[sq1\], we are led to make the following definition.
Assume that $E$ satisfies the hypotheses of Lemma \[sq\]. A square free positive integer $M$ is said to be [*admissible*]{} for $E$ if it satisfies (i) $(M, N(E))=1$, (ii) $M \equiv 1 \, mod \, 4$ or $M \equiv 3 \, mod \, 4$, according as $w_E = +1$ or $w_E = -1$, and (iii) every prime factor of $M$ which splits in $K$ is a special split prime for $K$.
\[18\] Assume that $E$ is defined over ${\mathbb Q}$, has complex multiplication by the ring of integers of $K={\mathbb Q}(\sqrt{-q})$, where $q = 7, 11, 19, 43, 67, 163$, and has good reduction at 2. Let $M$ be a square free positive integer, which is admissible for $E$, and let $r(M)$ denote the number of primes of $K$ dividing $M$. Put $\epsilon$ equal to $+1$ or $-1$, according as $M \equiv 1 \, or \, 3 \,
mod \, 4$. Then, for $w_E = \epsilon$, we have $$\label{19}
ord_2(L^{(alg)}(E^{(\epsilon M)}, 1)) \geq r(M) - \phi_E.$$
Let $M$ be any square free integer which is admissible for $E$, and let $p$ be any prime dividing $M$. If $p$ is inert in $K$, define $\pi$ to be $p$ or $-p$, according as $p$ is congruent to 1 or 3 $mod \, 4$. If $p$ splits in $K$, then Lemma \[split\] shows that we can then write $p = \pi\pi^*$, where $\pi$ and $\pi^*$ are elements of ${{\mathcal O}_K}$, which are both congruent to 1 $mod \, 4$. Since every $p$ with $p \equiv 3 \, mod \, 4$, and $p$ dividing $M$, is inert in $K$, it is now clear that we can write $$\epsilon M = \pi_1\ldots \pi_{r(M)}, \, \ ,$$ where the $\pi_i$ are distinct prime elements of ${{\mathcal O}_K}$, which are all congruent to 1 $mod \, 4$, and which are also prime to $\frak f$ and the discriminant of $K$. Hence the above theorem is an immediate consequence of Theorem \[21\].
The following is an immediate corollary of the above theorem. Of course, the hypothesis made in the corollary that $L(E, 1) \neq 0$ implies that the root number $w_E = 1$, and so the admissible $M$ in this case are $\equiv 1 \, mod \, 4.$
\[ap\] Assume that $E$ is defined over ${\mathbb Q}$, has complex multiplication by the ring of integers of $K$, and has good reduction at 2. Suppose further that we have (i) $K \neq {\mathbb Q}(\sqrt{-3})$, (ii) $L(E, 1) \neq 0$, and (iii) $ord_2(L^{(alg)}(E, 1)) < 0$. Let $M$ be any square free positive integer which is admissible for $E$, and which is divisible only by rational primes which split in $K$. Then $$ord_2(\frac{L^{(alg)}(E^{(M)}, 1)}{L^{(alg)}(E, 1)}) \geq 2k(M),$$ where $k(M)$ denotes the number of rational prime divisors of $M$.
We now discuss some numerical examples of this theorem. For basic information about the curves discussed below, see, for example, [@GR]. As a first example, let $E$ be the elliptic curve defined by $$\label{22}
y^2 + xy = x^3 - x^2 - 2x - 1.$$ It has conductor 49, and complex multiplication by the ring of integers of $K = {\mathbb Q}(\sqrt{-7})$. In fact, this curve is isomorphic to the modular curve $X_0(49)$. By the Chowla-Selberg formula, the period lattice $\frak L$ of the Neron differential on $E$ is given by $\frak L = \Omega_\infty{{\mathcal O}_K}$, where $$\Omega_\infty = \frac{\Gamma(\frac{1}{7})\Gamma(\frac{2}{7})\Gamma(\frac{4}{7})}{2\pi i\sqrt{-7}}.$$ Moreover, $\alpha_E = 1$ because $a_1 = 1$, and $L^{(alg)}(E, 1) = 1/2$, so that $\phi_E = 1$. Note that any positive square free integer $M$ with $(M, 7) = 1$ and $M \equiv 1 \, mod \, 4$, will be admissible for $E$, provided each of its prime factors which splits in $K$ (thus a prime factor which is congruent to any of 1, 2, or 4 $mod \, 7$) is congruent to 1 $mod \, 4$. Theorem \[18\] therefore implies that, for such admissible integers $M$, we have $$\label{23}
ord_2(L^{(alg)}(E^{(M)}, 1)) \geq r(M) - 1.$$ We see from Table I at the end of this paper that this estimate is in general best possible. As a second example, take for $E$ the elliptic curve defined by $$\label{22'}
y^2 + y = x^3 - x^2 - 7x +10.$$ It has conductor 121, and complex multiplication by the ring of integers of $K = {\mathbb Q}(\sqrt{-11})$. Again by the Chowla-Selberg formula, the period lattice $\frak L$ of the Neron differential on $E$ is given by $\frak L = \Omega_\infty{{\mathcal O}_K}$, where $$\Omega_\infty = \frac{\Gamma(\frac{1}{11})\Gamma(\frac{3}{11})\Gamma(\frac{4}{11})\Gamma(\frac{5}{11})\Gamma(\frac{9}{11})}{2\pi i\sqrt{-11}}.$$ Moreover, $\alpha_E = 0$ because $a_1 = 0$, and $w_E = -1$, so that $\phi_E = 0$. The split primes for $K$ are those which are congruent to $1, 3, 4, 5, 9 \, mod \, 11$. For example, all special split primes $< 1000$ for this curve are:- $$\begin{aligned}
53,257,269,397,401,421,617,757,773,929.\end{aligned}$$ Let now $M$ be any square free positive integer which is admissible for $E$ (in particular, since we are only interested in twists $E^{(-M)}$ having root number equal to $+1$, we assume that $M \equiv 3 \, mod \, 4$ and $(M, 11) = 1$). Then Theorem \[18\] implies that $$\label{24}
ord_2(L^{(alg)}(E^{(-M)}, 1)) \geq r(M).$$ However, in this example, Table II at the end of this paper suggests that this estimate is not, in general, best possible. It seems plausible to speculate from Table II that the lower bound of $\eqref{24}$ could be improved to $r(M)+1$.
Tamagawa Factors
================
Our goal in this last section is to relate the estimate given by Theorem \[18\] to the Tamagawa factors which arise in the Birch-Swinnerton-Dyer conjecture for the twists of our given elliptic curve with complex multiplication. Suppose first that $E$ is any elliptic curve $E$ defined over ${\mathbb Q}$, and any prime $p$ of bad reduction for $E$, let $E({\mathbb Q}_p)$ denote the group of points on $E$ with coordinates in the field of $p$-adic numbers ${\mathbb Q}_p$, and $E_0({\mathbb Q}_p)$ the subgroup of points with non-singular reduction modulo $p$. We define $$\frak C_p(E) = E({\mathbb Q}_p)/E_0({\mathbb Q}_p),$$ and recall that the Tamagawa factor $c_p(E)$ is defined by $$\label{25}
c_p(E) = [E({\mathbb Q}_p):E_0({\mathbb Q}_p)].$$ If $A$ is any abelian group, $A[m]$ will denote the kernel of multiplication by a positive integer $m$ on $A$. The following lemma is very well known, but we give it for completeness.
\[use\] Let $E$ be any elliptic curve over ${\mathbb Q}$, and let $p$ be a prime number where $E$ has bad additive reduction. Then, for all positive integers $m$ with $(m, p) = 1$, we have $$\frak C_p(E)[m] = E({\mathbb Q}_p)[m].$$
Let $E_1({\mathbb Q}_p)$ denote the group of points on the formal group of $E$ at $p$. Since $E$ has additive reduction modulo $p$, the group of non-singular points on the reduction of $E$ modulo $p$ is isomorphic to the additive group of the field $\mathbb{F}_p$. As $E_1({\mathbb Q}_p)$ is pro-$p$, and we have the exact sequence $$0 \to E_1({\mathbb Q}_p) \to E_0({\mathbb Q}_p) \to \mathbb{F}_p \to 0,$$ it follows immediately that multiplication by $m$ is an isomorphism on $E^0({\mathbb Q}_p)$, whence the assertion of the lemma follows easily from a simple application of the snake lemma to the sequence $$0 \to E_0({\mathbb Q}_p) \to E({\mathbb Q}_p) \to \frak C_p(E) \to 0.$$
As earlier, let $E$ now be our elliptic curve defined over ${\mathbb Q}$ with complex multiplication by the ring of integers of the imaginary quadratic field $K$, and write $N(E)$ for the conductor of $E$. Once again, we will assume that $E$ has good reduction at 2, and so we cannot have $K = {\mathbb Q}(\sqrt{-1})$, or $K = {\mathbb Q}(\sqrt{-2})$. Let $M$ denote an odd positive square free integer with $(M, N(E)) =1$. We put $\epsilon$ equal to $+1$ or $-1$, according as $M$ is congruent to 1 or 3 $mod \, 4$. Thus 2 is always unramified in the quadratic extension ${\mathbb Q}(\sqrt{\epsilon M})/{\mathbb Q}$.
\[eq\] Let $p$ be any prime number dividing $N(E)$ or $M$. If $p$ divides $N(E)$, then $ord_2(c_p(E^{(\epsilon M)})) = ord_2(c_p(E))$. If $p$ divides $M$, then the value of $ord_2(c_p(E^{(\epsilon M)}))$ is independent of $M$.
Let $p$ be any prime factor of $N(E)$ or $M$, so that, in particular, $p$ is odd. Since the $j$-invariant of $E$, and so also the $j$-invariant of $E^{(\epsilon M)}$, are integral, it follows from the table on p. 365 of [@S] that the 2-primary subgroups of $\frak C_p(E)$ and $\frak C_p(E^{(\epsilon M)})$ are either $0, {\mathbb Z}/2{\mathbb Z}$, or ${\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2Z$. Now when $p$ divides $N(E)$, both $E$ and $E^{(\epsilon M)}$ have additive reduction at $p$, and so we conclude from Lemma \[use\] that, in this case, $$ord_2(c_p(E)) = ord_2(\#(E({\mathbb Q}_p)[2])), \, ord_2(c_p(E^{(\epsilon M)})) = ord_2(\#(E^{(\epsilon M)}({\mathbb Q}_p)[2])),$$ Also when $p$ divides $M$, we have, again from Lemma \[use\], that $$\label{cr}
ord_2(c_p(E^{(\epsilon M)})) = ord_2(\#(E^{(\epsilon M)}({\mathbb Q}_p)[2])).$$ But the Galois group of ${\mathbb Q}(\sqrt{\epsilon M})/{\mathbb Q}$ clearly acts trivially on points of order 2 on $E^{(\epsilon M)}$, and so we always have $$\label{cr'}
\#(E({\mathbb Q}_p)[2]) = \#(E^{(\epsilon M)}({\mathbb Q}_p)[2])).$$ The assertions of the lemma now follow immediately.
\[26\] Assume that $E$ is defined over ${\mathbb Q}$ and has good reduction at 2, and that $K \neq {\mathbb Q}(\sqrt{-3})$. Let $M$ be an odd positive square free integer with $(M, N(E))=1$, and having the property that every prime factor of M which is inert in $K$ is congruent to 1 $mod \, 4$. Let $p$ be any prime dividing M. Then (i) $ord_2(c_p(E^{(\epsilon M)})) = 1$ if $p$ is inert in $K$, (ii) $ord_2(c_p(E^{(\epsilon M)})) = 0$ if $p$ splits in $K$ and the trace of the Frobenius endomorphism of the reduction of $E$ modulo $p$ is odd, and (iii) $ord_2(c_p(E^{(\epsilon M)})) = 2$ if $p$ splits in $K$ and the trace of the Frobenius endomorphism of the reduction of $E$ modulo $p$ is even.
Before giving the proof of this theorem, we state an important corollary.
Assume that $E$ is defined over ${\mathbb Q}$ and has good reduction at 2, and that $K \neq {\mathbb Q}(\sqrt{-3})$. Let $M$ be a positive integer which is admissible for $E$ in the sense of Definition \[ad\], and has the property that every prime factor of $M$ is congruent to 1 $mod \, 4$. Write $r(M)$ for the number of primes divisors of $M$ in $K$. Then $$\label{27}
ord_2(\prod_{p|M} {c_p(E^{(\epsilon M)}})) = r(M).$$
Let $p$ be any prime factor of $M$. Since $(p, N(E))=1$, $E$ has good reduction at $p$ and $p$ does not ramify in $K$. Recalling and , we have to compute the order of $E({\mathbb Q}_p)[2]$. Now, since $p$ is odd, the Galois module $E[2]$ is unramified at $p$. Let $\tilde{E}$ denote the reduction of $E$ modulo $p$. Since the formal group of $E$ at $p$ is a ${\mathbb Z}_p$-module, it follows easily that reduction modulo $p$ defines an isomorphism $$E({\mathbb Q}_p)[2] = \tilde{E}(\mathbb{F}_p)[2].$$ Now the order of $\tilde{E}(\mathbb{F}_p)$ is $1+p$ or $1 - a_p + p$, according as $p$ is inert or splits in $K$, where $a_p$ is the trace of the Frobenius endomorphism of $\tilde{E}$. In particular, when $p$ splits in $K$ and $a_p$ is odd, we see immediately that $E({\mathbb Q}_p)[2] =0$. Similarly, if $p$ is inert in $K$, then, as $p \equiv 1 \, mod \, 4$, we conclude that $E({\mathbb Q}_p)[2]$ must have order 2. Suppose next that $p$ splits in $K$ and $a_p$ is even. Let $\tau_p$ be any Frobenius automorphism at $p$. Since $p$ splits in $K$, we can view $\tau_p$ as an element of the absolute Galois group of $K$, and we write $\phi_p$ for its image in the ${{\mathcal O}_K}$-automorphism group of the module $E[2]$, which is equal to $({{\mathcal O}_K}/2{{\mathcal O}_K})^*$. Then $\phi_p$ must have order dividing 2 because, since $a_p$ is even, its characteristic polynomial is equal to $X^2 - 1$. But 2 is not ramified in $K$ because $E$ has good reduction at 2. Thus the group $({{\mathcal O}_K}/2{{\mathcal O}_K})^*$ has no element of order 2, whence we must have $\phi_p =1$ and $E({\mathbb Q}_p)[2] = E[2]$. The assertions of the theorem are now clear from and .
Finally, we now compare some of our estimates with those predicted by the conjecture of Birch and Swinnerton-Dyer. The next proposition is an immediate consequence of Theorems \[18\] and \[26\].
\[bsd\] Assume that $E$ is defined over ${\mathbb Q}$ and has good reduction at 2, and that $K \neq {\mathbb Q}(\sqrt{-3})$. Assume further that $L(E, 1) \neq 0$, and that $ord_2(L^{(alg)}(E, 1)) < 0$. Then, for all positive integers $M$, which are admissible for $E$, and have the property that all of their prime factors are $\equiv 1 \, mod \, 4$, we have $$\label{28}
ord_2(\frac{L^{(alg)}(E^{(M)}, 1)}{L^{(alg)}(E, 1)}) \geq ord_2(\displaystyle\prod_{p|M} {c_p(E^{(M)})}).$$
As we shall now explain, the lower bound given by is exactly what the conjecture of Birch and Swinnerton-Dyer would predict for elliptic curves satisfying the hypotheses of this proposition. We first establish a preliminary result.
\[29\] Let $E$ be an elliptic curve defined over ${\mathbb Q}$, with complex multiplication by the ring of integers of $K$. Assume that $E$ has good reduction at 2, and that $K \neq {\mathbb Q}(\sqrt{-3})$. Let $M$ denote an odd positive square free integer with $(M, N(E)) =1$, and put $\epsilon$ equal to $+1$ or $-1$, according as $M$ is congruent to 1 or 3 $mod \, 4$. Then the 2-primary subgroups of $E({\mathbb Q})$ and $E^{(\epsilon M)}({\mathbb Q})$ have the same order, and this order is equal to 2 or 1, according as the prime 2 splits or is inert in $K$.
Let $A$ denote the elliptic curve $E$ or $E^{(\epsilon M)}({\mathbb Q})$, so that $A$ also has good reduction at $2$. In order to show that the 2-primary subgroup of $A({\mathbb Q})$ is annihilated by 2, it suffices to prove that the 2-primary subgroup of $A(K)$ is annihilated by 2. Now, as $E$ has good reduction at 2, the prime 2 does not ramify in $K$, and thus it either splits or is inert in $K$. Let $v$ denote any prime of $K$ above 2. Since $A$ has good reduction at $v$, the formal group of $A$ at $v$ is a Lubin-Tate formal group with parameter $\pi = \psi_A(v)$. Let $n$ be any integer $\geq 1$. As the group $A[{\pi^n}]$ of $\pi^n$-division points on $A$ lies on the formal group of $A$ at $v$, it follows from Lubin-Tate theory the extension $K(A[{\pi^n}])/K$ has Galois group isomorphic to $({{\mathcal O}_K}/\pi^n{{\mathcal O}_K})^*$, which is non-trivial for all $n\geq 1$ if $2$ is inert in $K$, and which is non-trivial for all $n \geq 2$ if 2 splits in $K$. In particular, the 2-primary subgroup of $A(K)$ must be trivial if 2 is inert in $K$, and it must be killed by 2 when 2 splits in $K$. But 2 splits in $K$ happens precisely when $K = {\mathbb Q}(\sqrt{-7})$, and then $A$ must be a quadratic twist of the curve given by $\eqref{22}$. Now the curve $\eqref{22}$ has a unique rational point of order 2 given by $(2, -1)$. It follows that $A({\mathbb Q})$ must also have a unique point of order 2, because $A$ is a quadratic twist of $\eqref{22}$. This completes the proof.
Now assume that $E$ satisfies the hypotheses of Proposition \[bsd\]. Since $L(E, 1)\neq 0$, we know that both $E({\mathbb Q})$ and the Tate-Shafarevich group of $E/{\mathbb Q}$ are finite, and we write $w(E)$ and $t(E)$ for their respective orders. Then the conjecture of Birch and Swinnerton-Dyer predicts that $$\label{30}
ord_2(L^{(alg)}(E, 1)) = ord_2(c_\infty(E)\displaystyle\prod_{p|N(E)} {c_p(E)}) + ord_2(t(E)) - 2ord_2(w(E)).$$ where $c_\infty(E)$ denotes the number of connected components of $E(\mathbb{R})$. If we recall Proposition \[29\], and the fact that the Cassels-Tate theorem implies that $t(E)$ is the square of an integer, we see that the combination of our hypothesis that $ord_2(L^{(alg)}(E, 1)) < 0$ and the conjectural formula imply that necessarily $$\label{31}
ord_2(t(E)) = 0.$$ Suppose now that $L(E^{(M)}, 1)\neq 0$. Again, we then know that both $E^{(M)}({\mathbb Q})$ and the Tate-Shafarevich group of $E^{(M)}/{\mathbb Q}$ are finite, and we write $w(E^{(M)})$ and $t(E^{(M)})$ for their respective orders. Then, in this case, the conjecture of Birch and Swinnerton-Dyer predicts that $$\label{32}
ord_2(L^{(alg)}(E^{(M)}, 1)) = ord_2(c_\infty(E^{(M)})\displaystyle\prod_{p|N(E)M} {c_p(E^{(M)})}) + ord_2(t(E^{(M)})) - 2ord_2(w(E^{(M)})).$$ where $c_\infty(E^{(M)})$ denotes the number of connected components of $E^{(M)}(\mathbb{R})$. Obviously, $c_\infty(E)= c_\infty(E^{(M)})$ since ${\mathbb Q}(\sqrt{M})$ is a real quadratic field. Moreover, Proposition \[29\] shows that $ord_2(w(E))=ord_2(w(E^{(M)}))$, and Lemma \[eq\] tells us that, for primes $p$ dividing $N(E)$, we have $ ord_2(c_p(E)) = ord_2(c_p(E^{(M)})$. Hence, recalling , we conclude that the conjecture of Birch and Swinnerton-Dyer predicts that $$\label{33}
ord_2(\frac{L^{(alg)}(E^{(M)}, 1)}{L^{(alg)}(E, 1)}) = ord_2(\displaystyle\prod_{p|M} {c_p(E^{(M)})}) +
ord_2(t(E^{(M)})).$$ This shows that, under the above hypotheses, the lower bound given by is precisely what the conjecture of the Birch and Swinnerton-Dyer would predict if we ignore the unknown term $ord_2(t(E^{(M)}))$ giving the order of the 2-primary subgroup of the Tate-Shafarevich group of the curve $E^{(M)}$.
Tables
======
In this section, we include some short tables of numerical examples of our results for two elliptic curves $E$ defined over ${\mathbb Q}$. We use the same notation as earlier. For the curve of conductor 49 in Table I, the root number of the curve is $+1$, and for the curve of conductor 121 in Table II the root number is $-1$. As always, $M$ will denote a square free positive integer which is admissible for the elliptic curve $E$, and $r(M)$ will denote the number of prime divisors of $M$ in the field of complex multiplication $K$.
[|l@|l@|l@|l@|l@|l@|]{} $29$&$0.7180139420$&$2$&$1$&$2$&$c_{29}=4$,\
$37$&$0.6356689731$&$2$&$1$&$2$&$c_{37}=4$,\
$109$&$0.3703553538$&$2$&$1$&$2$&$c_{109}=4$,\
$113$&$1.454965333$&$8$&$3$&$2$&$c_{113}=4$,\
$137$&$0.3303479321$&$2$&$1$&$2$&$c_{137}=4$,\
$145$&$0.6422111932$&$4$&$2$&$3$&$c_{5}=2$, $c_{29}=4$,\
$185$&$2.274238456$&$16$&$4$&$3$&$c_{5}=2$, $c_{37}=4$,\
$233$&$2.279798298$&$18$&$1$&$2$&$c_{233}=4$,\
$265$&$4.275446184$&$36$&$2$&$3$&$c_{5}=2$, $c_{53}=4$,\
$277$&$0.9292915388$&$8$&$3$&$2$&$c_{277}=4$,\
$281$&$0.2306634143$&$2$&$1$&$2$&$c_{281}=4$,\
$285$&$1.832312031$&$16$&$4$&$3$&$c_{3}=2$, $c_{5}=2$, $c_{19}=2$,\
$317$&$0.8686848279$&$8$&$3$&$2$&$c_{317}=4$,\
$337$&$0.2106283985$&$2$&$1$&$2$&$c_{337}=4$,\
$377$&$0.3982824745$&$4$&$2$&$3$&$c_{13}=2$, $c_{29}=4$,\
$389$&$1.764410302$&$18$&$1$&$2$&$c_{389}=4$,\
$401$&$1.737809629$&$18$&$1$&$2$&$c_{401}=4$,\
$421$&$0.7537907774$&$8$&$3$&$2$&$c_{421}=4$,\
$449$&$2.919635854$&$32$&$5$&$2$&$c_{449}=4$,\
$457$&$0.7234920569$&$8$&$3$&$2$&$c_{457}=4$,\
$481$&$1.410422816$&$16$&$4$&$3$&$c_{13}=2$, $c_{37}=4$,\
$545$&$0.3312558988$&$4$&$2$&$3$&$c_{5}=2$, $c_{109}=4$,\
$557$&$0.6553363680$&$8$&$3$&$2$&$c_{557}=4$,\
$565$&$0.3253401390$&$4$&$2$&$3$&$c_{5}=2$, $c_{113}=4$,\
$569$&$0.1620972858$&$2$&$1$&$2$&$c_{569}=4$,\
$613$&$0.1561714487$&$2$&$1$&$2$&$c_{613}=4$,\
$617$&$0.1556643972$&$2$&$1$&$2$&$c_{617}=4$,\
$629$&$1.233378974$&$16$&$4$&$3$&$c_{17}=2$, $c_{37}=4$,\
$641$&$0.1527224426$&$2$&$1$&$2$&$c_{641}=4$,\
$653$&$0.1513126668$&$2$&$1$&$2$&$c_{653}=4$,\
$673$&$1.341426413$&$18$&$1$&$2$&$c_{673}=4$,\
$701$&$0.1460403507$&$2$&$1$&$2$&$c_{701}=4$,\
$705$&$1.165003700$&$16$&$4$&$3$&$c_{3}=2$, $c_{5}=2$, $c_{47}=2$,\
$709$&$0.1452140903$&$2$&$1$&$2$&$c_{709}=4$,\
$757$&$0.1405348183$&$2$&$1$&$2$&$c_{757}=4$,\
$809$&$0.5437729586$&$8$&$3$&$2$&$c_{809}=4$,\
$821$&$0.5397843500$&$8$&$3$&$2$&$c_{821}=4$,\
$877$&$1.175099358$&$18$&$1$&$2$&$c_{877}=4$,\
$901$&$1.030527220$&$16$&$4$&$3$&$c_{17}=2$, $c_{53}=4$,\
$953$&$0.5010088727$&$8$&$3$&$2$&$c_{953}=4$,\
$965$&$0.2489420234$&$4$&$2$&$3$&$c_{5}=2$, $c_{193}=4$,\
$969$&$0.9937107192$&$16$&$4$&$3$&$c_{3}=2$, $c_{17}=2$, $c_{19}=2$,\
$977$&$1.113338183$&$18$&$1$&$2$&$c_{977}=4$,\
$985$&$2.217615590$&$36$&$2$&$3$&$c_{5}=2$, $c_{197}=4$,\
[|l@|l@|l@|l@|l@|l@|]{} $7$&$1.094573405$&$4$&$2$&$1$&$c_{7}=2$,\
$43$&$0.4416311353$&$4$&$2$&$1$&$c_{43}=2$,\
$79$&$0.3258219706$&$4$&$2$&$1$&$c_{79}=2$,\
$83$&$0.3178738964$&$4$&$2$&$1$&$c_{83}=2$,\
$107$&$0.2799638923$&$4$&$2$&$1$&$c_{107}=2$,\
$119$&$0.5309460896$&$8$&$3$&$2$&$c_{7}=2$, $c_{17}=2$,\
$127$&$1.027902784$&$16$&$4$&$1$&$c_{127}=2$,\
$131$&$0.2530219881$&$4$&$2$&$1$&$c_{131}=2$,\
$139$&$0.2456328864$&$4$&$2$&$1$&$c_{139}=2$,\
$151$&$0.2356706165$&$4$&$2$&$1$&$c_{151}=2$,\
$203$&$0.4065143570$&$8$&$3$&$2$&$c_{7}=2$, $c_{29}=2$,\
$211$&$0.7974669169$&$16$&$4$&$1$&$c_{211}=2$,\
$227$&$0.1922122148$&$4$&$2$&$1$&$c_{227}=2$,\
$239$&$0.1873246635$&$4$&$2$&$1$&$c_{239}=2$,\
$247$&$0.3685321923$&$8$&$3$&$2$&$c_{13}=2$, $c_{19}=2$,\
$263$&$0.1785730998$&$4$&$2$&$1$&$c_{263}=2$,\
$271$&$0.7036703591$&$16$&$4$&$1$&$c_{271}=2$,\
$287$&$0.3418872925$&$8$&$3$&$2$&$c_{7}=2$, $c_{41}=2$,\
$307$&$2.644506912$&$64$&$6$&$1$&$c_{307}=2$,\
$323$&$0.3222720533$&$8$&$3$&$2$&$c_{17}=2$, $c_{19}=2$,\
$347$&$0.6218550501$&$16$&$4$&$1$&$c_{347}=2$,\
$371$&$0.6014048805$&$16$&$4$&$3$&$c_{7}=2$, $c_{53}=4$,\
$427$&$0.2802915271$&$8$&$3$&$2$&$c_{7}=2$, $c_{61}=2$,\
$431$&$0.1394939193$&$4$&$2$&$1$&$c_{431}=2$,\
$439$&$0.5528682408$&$16$&$4$&$1$&$c_{439}=2$,\
$491$&$0.5227730093$&$16$&$4$&$1$&$c_{491}=2$,\
$503$&$0.1291248765$&$4$&$2$&$1$&$c_{503}=2$,\
$511$&$0.2562202539$&$8$&$3$&$2$&$c_{7}=2$, $c_{73}=2$,\
$547$&$1.981163103$&$64$&$6$&$1$&$c_{547}=2$,\
$551$&$0.2467448562$&$8$&$3$&$2$&$c_{19}=2$, $c_{29}=2$,\
$559$&$0.2449728774$&$8$&$3$&$2$&$c_{13}=2$, $c_{43}=2$,\
$563$&$0.4882021701$&$16$&$4$&$1$&$c_{563}=2$,\
$607$&$1.057893809$&$36$&$2$&$1$&$c_{607}=2$,\
$659$&$0.1128109364$&$4$&$2$&$1$&$c_{659}=2$,\
$707$&$0.8713129959$&$32$&$5$&$2$&$c_{7}=2$, $c_{101}=2$,\
$731$&$0.2142225669$&$8$&$3$&$2$&$c_{17}=2$, $c_{43}=2$,\
$739$&$0.1065299425$&$4$&$2$&$1$&$c_{739}=2$,\
$743$&$0.4249711969$&$16$&$4$&$1$&$c_{743}=2$,\
$763$&$0.8387289424$&$32$&$5$&$2$&$c_{7}=2$, $c_{109}=2$,\
$787$&$1.651682351$&$64$&$6$&$1$&$c_{787}=2$,\
$811$&$0.9152210367$&$36$&$2$&$1$&$c_{811}=2$,\
$887$&$0.09723712323$&$4$&$2$&$1$&$c_{887}=2$,\
$919$&$0.09552920333$&$4$&$2$&$1$&$c_{919}=2$,\
[99]{}
, , Acta Arithmetica [**19**]{} (1971), 311-317. , , Crelle [**327**]{} (1981), 184-218. , , Lecture Notes in Math. [**776**]{} (1980), Springer. , , Graduate Texts Math. 151 (1994), Springer. , , Invent. Math. [**103**]{} (1991), 25-68. , , Proc. Natl. Acad. Sci. USA [**109**]{}(2012), 21256-21258. , , to appear. , , to appear. , , Proc. Cambridge Phil. Soc. [**121**]{} (1997), 385-400. , , Proc. Cambridge Phil. Soc. [**131**]{} (2001), 385-404. , , Proc. Cambridge Phil. Soc. [**134**]{} (2003), 407-420. , , Acta Mathematica Sinica, [**21**]{} (2005), 961-976.
[^1]: Supported by EPSRC grant EP/G024979/2
[^2]: Supported by NSFC11001183 and NSFC11171231.
[^3]: Supported by NSFC01272499.
|
---
abstract: 'The integral relations formalism introduced in [@bar09; @rom11], and designed to describe 1+$N$ reactions, is extended here to collision energies above the threshold for the target breakup. These two relations are completely general, and in this work they are used together with the adiabatic expansion method for the description of 1+2 reactions. The neutron-deuteron breakup, for which benchmark calculations are available, is taken as a test of the method. The $s$-wave collision between the $^4$He atom and $^4$He$_2$ dimer above the breakup threshold and the possibility of using soft-core two-body potentials plus a short-range three-body force will be investigated. Comparisons to previous calculations for the three-body recombination and collision dissociation rates will be shown.'
author:
- 'E. Garrido'
- 'C. Romero-Redondo'
- 'A. Kievsky and M. Viviani'
title: 'Integral relations and the adiabatic expansion method for 1+2 reactions above the breakup threshold: Helium trimers with soft-core potentials'
---
Introduction
============
Calculation of continuum states corresponding to processes where a particle hits a bound $N$-body system requires in principle knowledge of the corresponding $(1+N)$-body wave function at large distances. Needless to say, the technical difficulties one has to face in order to obtain the wave function increase dramatically with $N$. In fact, already for $N=2$, calculation of the three-body wave function is far of being trivial.
However, even if knowledge of the $(1+N)$-body wave function is unavoidable, the possibility of reducing the distance at which such wave function is needed is in itself an important step forward in the description of the reaction. As shown in [@bar09; @rom11], this can indeed be done by means of two integral relations that are based on the Kohn Variational Principle. These two integral relations are a generalization to more than two particles of the integral relation given in [@har67; @hol72], and they permit to obtain the ${\cal K}$- (or ${\cal S}$)-matrix of the reaction by using only the internal part of the wave function. Therefore, all the physical information concerning a given $1+N$ reaction can be obtained without an accurate knowledge of the asymptotic part of the wave function.
The fact that the asymptotic part is not needed anymore leads to a drastic reduction of the computer effort required to extract the ${\cal K}$-matrix. The only condition necessary to obtain accurate second-order estimates of the ${\cal K}$-matrix through the integral relations is that the trial wave function must fulfill the Schrödinger equation only in the interaction region. This means that, for instance, scattering states can be described with bound-state-like trial wave functions [@kie10].
In [@bar09; @rom11] the integral relations were implemented to describe $1+2$ reactions below the breakup threshold. When used in combination with the hyperspherical adiabatic expansion method the dimension of the ${\cal K}$-matrix describing the reaction is dictated by the number of adiabatic potentials related to the possible outgoing elastic or inelastic channels, which typically is very small. Furthermore, thanks to integral relations, the convergence of the ${\cal K}$-matrix in terms of the adiabatic channels included in the expansion of the wave function is highly accelerated. Actually, the pattern of convergence is similar to the one found when the same method is used for the description of bound states [@bar09; @rom11]. This is in fact not a minor issue, since as proved in [@bar09b], when used to describe low-energy scattering states, the convergence of the adiabatic expansion slows down significantly, even to the point that an accurate calculation of the ${\cal K}$-matrix would require infinitely many adiabatic terms in the expansion, what in practice makes the procedure useless.
The success of the method for energies below the breakup threshold immediately suggests its extension to describe low-energy breakup reactions. In this case the dimension of the ${\cal K}$-matrix is not finite, since contrary to the elastic and inelastic channels or transfer reactions, the full three-body continuum states are described by infinitely many adiabatic potentials.
The first goal of this work is to generalize the method described in [@bar09; @rom11] for $1+N$ reactions to energies above the threshold for breakup of the bound target. This generalization will be shown in Section II, where a short summary of the method described in [@rom11] will be given. In Section III, the $n-d$ reaction, for which a series of benchmark calculations are available [@fri90; @fri95], will be used as a test of the method.
The second goal, which will be discussed in Section IV, concerns the use of the new method to investigate the $^4$He+$^4$He$_2$ atomic reaction above the dimer breakup threshold. The case of the $0^+$ state will be considered. In particular, we will focus on the use of soft-core $^4$He-$^4$He potentials. As shown in [@kie11], the use of an attractive gaussian potential reproducing the same two-body properties as a standard hard-core potential (like for instance the LM2M2 potential) leads to equivalent bound three-body systems and phase-shifts for the elastic $^4$He+$^4$He$_2$ reaction, but only once that the soft-core potential is used together with a three-body short-term force. The possibility of using the same kind of potentials also to describe the breakup channel is interesting, since it automatically eliminates all the technical difficulties arising from the presence of a hard-core repulsion in the potential (see for instance Ref.[@pen03]).
Finally, we close this work with a short summary and the conclusions.
Formalism
=========
In Refs.[@bar09; @rom11] $1+2$ reactions were described within the framework of the hyperspherical adiabatic expansion method for energies below the threshold for breakup of the dimer. Therefore, only elastic, inelastic, and transfer processes were possible. Since the formalism is described in great detail in Ref.[@rom11], here we just summarize its main aspects, which are given in the first part of this section. In particular, we will focus on those key points that will permit an easier understanding of the generalization of the method to energies above the two-body breakup threshold, which will be shown in the second part of the section. In the last part we will describe the integral relations, which are actually the tools that permit to extract the ${\cal K}$-matrix of the reaction from the internal part of the wave functions.
Sketch of the method for energies below the breakup threshold {#sub2a}
-------------------------------------------------------------
Given a three-body system, the corresponding wave function within the frame of the adiabatic expansion method is written as: $$\Psi({ \mbox{\boldmath $x$} },{ \mbox{\boldmath $y$} })=\frac{1}{\rho^5} \sum_{n=1}^\infty f_n(\rho) \Phi_n(\rho,\Omega),
\label{eq1}$$ where ${ \mbox{\boldmath $x$} }$ and ${ \mbox{\boldmath $y$} }$ are the usual Jacobi coordinates, and $\{\rho,\Omega\}$ are the hyperradius and the five hyperangles obtained from ${ \mbox{\boldmath $x$} }$ and ${ \mbox{\boldmath $y$} }$ [@nie01]. The wave function has a well defined total angular momentum $J$, but for simplicity in the notation we omit this index (and its projection) in the expression above.
In hyperspherical coordinates the Hamiltonian operator $\hat{{{\mathcal H}}}$ takes the form: $$\hat{{{\mathcal H}}} = -\frac{\hbar^2}{2 m} \hat{T}_\rho + \hat {\cal H}_\Omega ,$$ where $\hat{T}_\rho$ is the hyperradial kinetic energy operator, $\hat {\cal H}_\Omega$ contains all the dependence on the hyperangles, and $m$ is an arbitrary normalization mass.
The angular functions $\Phi_n(\rho,\Omega)$, which form the complete basis used for the wave function expansion (\[eq1\]), are actually the eigenfunctions of $\hat {\cal H}_\Omega$, $$\hat { \cal H}_\Omega \Phi_n(\rho,\Omega)=\frac{\hbar^2}{2 m} \frac{1}{\rho^2}
\lambda_n(\rho) \Phi_n(\rho,\Omega),
\label{eq3}$$ in such a way that the adiabatic effective potentials $$V^{(n)}_{eff}(\rho)=\frac{\hbar^2}{2m}\left( \frac{\lambda_n(\rho)+\frac{15}{4}}{\rho^2}-Q_{nn}(\rho) \right)
\label{eq4}$$ enter in the coupled set of radial equations that permit to obtain the radial functions $f_n(\rho)$, $$\begin{aligned}
&&\left[ -\frac{d^2}{d\rho^2} + \frac{2m}{\hbar^2}(V^{(n)}_{eff}(\rho)-E)
\right] f_n(\rho) \nonumber \\
&&+ \sum_{n'\neq n} \left( -2 P_{n n'} \frac{d}{d\rho} - Q_{n n'} \right)f_{n'}(\rho)
= 0 \label{eq4b}\end{aligned}$$ where $$\begin{aligned}
Q_{n n'}(\rho)&=&\Big{\langle}\Phi_n(\rho,\Omega) \Big|\frac{\partial^2}{\partial \rho^2} \Big|
\Phi_{n^\prime}(\rho,\Omega) \Big{\rangle}_\Omega,\\
P_{n n'}(\rho)&=&\Big{\langle}\Phi_n(\rho,\Omega) \Big|\frac{\partial}{\partial \rho} \Big|
\Phi_{n^\prime}(\rho,\Omega) \Big{\rangle}_\Omega,\end{aligned}$$ where the subscript $\Omega$ indicates integration over the hyperangles only (see [@nie01] for details).
A typical behavior of the adiabatic potentials is shown in Fig.\[fig1\]. They correspond to a three-body system where two of the two-body subsystems have a bound state. This is reflected in the fact that the two lowest effective adiabatic potentials go asymptotically to the binding energies $E_{2b}^{(1)}$ and $E_{2b}^{(2)}$ of each bound two-body system.
In the figure the different regions defined by the energy of the incident particles are depicted. All the three-body energies $E$ such that $E_{2b}^{(1)} < E < E_{2b}^{(2)}$ (like $E^{(1)}$ in the figure) correspond to processes where only one channel is open. Only the elastic collision between the third particle and the bound two-body state with energy $E_{2b}^{(1)}$ is possible. When the three-body energy increases up to the region $E_{2b}^{(2)} < E < 0$ ($E^{(2)}$ in the figure) a second channel is open. Two different collisions are now possible, the one where a particle hits the bound state with binding energy $E_{2b}^{(1)}$, and the one where a particle hits the state with binding energy $E_{2b}^{(2)}$. In the same way, each of these reactions has two possible outgoing channels, corresponding to the two allowed bound two-body states and the third particle in the continuum. In other words, in this energy range the inelastic (if the two bound two-body states correspond to the same subsystem) or transfer (if the two bound two-body states correspond to different subsystems) channel is open. When $E>0$ (like $E^{(3)}$ in the figure), the breakup channel is also open, and it is described by the remaining infinitely many adiabatic potentials.
The coupled system of radial equations (\[eq4b\]) decouple asymptotically, and each of the radial wave functions behave at large distances as dictated by: $$\left(-\frac{\hbar^2}{2 m} \frac{d^2}{d\rho^2} +V_{eff}^{(n)}(\rho)-E \right)
f_{n}(\rho)=0.
\label{eq5}$$
When $n$ corresponds to a closed channel the radial wave function $f_n$ vanishes asymptotically. For values of $E$ below the two-body breakup threshold, as considered in [@rom11], and for a given incoming $1+2$ channel, only outgoing $1+2$ channels are allowed (either elastic, inelastic, or transfer). In this case, as shown in [@nie01], the equation above describing the asymptotic behavior of the open-channel wave functions becomes $$\left(\frac{d^2}{dy_n^2} +(k_y^{(n)})^2-\frac{\ell_y (\ell_y+1)}{y_n^2} \right)
f_{n}(\rho)=0,
\label{eqas}$$ where $\ell_y$ is the relative angular momentum between the dimer and the third particle, $y_n$ refers to the modulus of the Jacobi coordinate between the center of mass of the outgoing bound two-body system and the third particle, and $$k_y^{(n)}=\sqrt{ \frac{2m}{\hbar^2}(E-E_{2b}^{(n)}) },
\label{eq10}$$ with $E_{2b}^{(n)}$ being the binding energy of the bound two-body system associated to the open channel $n$.
With this in hand, it is not difficult to see [@rom11] that the asymptotic form of the corresponding three-body wave function is given by: $$\Psi_i \rightarrow
\sum_{n=1}^{n_0}
\left(
A_{in}^{(K)} F_{n}^{(K)} + B_{in}^{(K)} G_{n}^{(K)}
\right),
\label{eq7}$$ where $i$ refers to the incoming channel, $n_0$ is the number of open channels (all of them $1+2$ channels), and $$F_{n}^{(K)}=\sqrt{k_y^{(n)}} j_{\ell_y}(k_y^{(n)}y_n)\frac{1}{\rho^{3/2}} \Phi_n(\rho,\Omega)
\label{eq8}$$ $$G_{n}^{(K)}=\sqrt{k_y^{(n)}} \eta_{\ell_y}(k_y^{(n)}y_n)\frac{1}{\rho^{3/2}} \Phi_n(\rho,\Omega),
\label{eq9}$$ where $j_\ell$ and $\eta_\ell$ are the usual regular and irregular Bessel functions (provided that we are dealing with short range potentials).
As shown in [@rom11], Eq.(\[eq7\]) can be written in a compact matrix form as: $$\Psi \rightarrow A^{(K)} F^{(K)} + B^{(K)} G^{(K)}=
A^{(K)} \left(F^{(K)} - {\cal K} G^{(K)}\right),
\label{eq11}$$ where $\Psi$ is a column vector with $n_0$ terms corresponding to the $n_0$ open channels, $A^{(K)}$ and $B^{(K)}$ are $n_0 \times n_0$ matrices made by the $A_{in}^{(K)}$ and $B_{in}^{(K)}$ elements in Eq.(\[eq7\]), and $F^{(K)}$ and $G^{(K)}$ are again column vectors whose $n_0$ terms are given by Eqs.(\[eq8\]) and (\[eq9\]).
From the equation above it is clear that the ${\cal K}$-matrix of the reaction is given by: $${\cal K}= -{A^{(K)}}^{-1} B^{(K)}.$$ This matrix is real, and from it one can easily obtain the ${\cal S}$-matrix as $(1+i{\cal K})(1-i{\cal K})^{-1}$.
Generalization to energies above the breakup threshold {#gener}
------------------------------------------------------
When the total three-body energy $E$ in a $1+2$ reaction is above the threshold for breakup of the dimer, the first consequence is that infinitely many adiabatic channels are then open (see Fig. \[fig1\] for $E=E^{(3)}$). Of course, still a finite number of them correspond to elastic, inelastic, or transfer processes, and the infinitely many remaining ones describe the breakup channel. Therefore, in this case the corresponding ${\cal K}$- (or ${\cal S}$-) matrix has infinite dimension.
In any case, for the finite open channels corresponding to outgoing $1+2$ structures, the expressions given in the previous subsection are still valid. This means that for these particular outgoing channels the three-body wave function behaves asymptotically as given by Eq.(\[eq7\]), and Eqs.(\[eq8\]) and (\[eq9\]) are still valid.
On the other hand, the breakup channels are characterized by the fact that the effective potentials $V_{eff}^{(n)}$ associated to them go asymptotically to zero as: $$V^{(n)}_{eff}(\rho) \stackrel{\rho \rightarrow \infty}{\rightarrow}
\frac{\hbar^2}{2m} \frac{\left( K+\frac{3}{2} \right) \left( K+\frac{5}{2} \right)}{\rho^2},
\label{eq13}$$ where $K$ is the grand-angular quantum number defined as $2\nu+\ell_x+\ell_y$, where $\ell_x$ and $\ell_y$ are the orbital angular momenta associated to the Jacobi coordinates ${ \mbox{\boldmath $x$} }$ and ${ \mbox{\boldmath $y$} }$, respectively, and $\nu=0,1,2,\cdots$. Therefore, asymptotically, each breakup adiabatic potential is associated to a fixed value of $K$. In fact, the corresponding angular eigenfunction $\Phi_n(\rho,\Omega)$ is, also asymptotically, a linear combination of hyperspherical harmonics with that particular value of $K$.
When inserting (\[eq13\]) into (\[eq5\]), we easily obtain that, asymptotically, the radial wave function for an outgoing breakup channel $n$ satisfies the equation: $$\left[\frac{d^2}{d\rho^2}+\kappa^2-\frac{(K+\frac{3}{2})(K+\frac{5}{2})}{\rho^2}
\right] f_{n}(\rho)=0,$$ which is formally identical to the Eq.(\[eqas\]), which is satisfied by the radial wave functions associated to outgoing $1+2$ channels, but replacing $\ell_y$ by $K+3/2$, $k_y^{(n)}$ by $\kappa=\sqrt{2mE/\hbar^2}$, and $y_n$ by $\rho$.
Therefore, in the more general case where the breakup channel is open, and assuming an incoming $1+2$ channel $i$, the asymptotic form of the corresponding three-body wave function is given by: $$\Psi_i \rightarrow
\sum_{n=1}^{\infty}
\left(
A_{in}^{(K)} F_{n}^{(K)} + B_{in}^{(K)} G_{n}^{(K)}
\right),
\label{eq15}$$ where $F_n^{(K)}$ and $G_n^{(K)}$ are still given by Eqs. (\[eq8\]) and (\[eq9\]), but of course, with the understanding that when $n$ corresponds to an outgoing breakup channel the replacements $\ell_y \rightarrow K+3/2$, $k_y^{(n)} \rightarrow \kappa$, and $y_n \rightarrow \rho$ have to be made (note that the relations $j_{K+\frac{3}{2}}(z) = \sqrt{\frac{\pi}{2z}} J_{K+2}(z)$ and $\eta_{K+\frac{3}{2}}(z) = \sqrt{\frac{\pi}{2z}} Y_{K+2}(z)$ permit to write Eqs.(\[eq8\]) and (\[eq9\]) in terms of the Bessel functions $J_{K+2}$ and $Y_{K+2}$, which is how the asymptotic form of the breakup channels is usually presented in the literature).
Of course, the matrix form in Eq.(\[eq11\]) of the asymptotic wave function can still be used. As before, the ${\cal K}$-matrix of the reaction is given by ${\cal K}=-{A^{(K)}}^{-1} B^{(K)}$, but now the matrices $A^{(K)}$ and $B^{(K)}$ have in principle infinite dimension and some truncation is then required.
Due to the fact that the hyperradius $\rho$ appears as the natural radial coordinate describing the asymptotic behavior of the breakup channels, we then have that, for these channels, the adiabatic expansion is able to reach the correct asymptotic behavior for a sufficiently large, but finite, value of $\rho$. For this reason, for processes without $1+2$ open channels ($3 \rightarrow 3$ processes), the ${\cal K}$-matrix could in principle be extracted with sufficient accuracy from the asymptotic behavior. However, this is not true when $1+2$ channels are open. For these channels the correct asymptotic form (described by (12) and (13)) can not be reached until the Jacobi coordinate $y$ and hyperradius $\rho$ are equal, which only happens at infinity. Therefore it is essential in this case to use a formalism in which the ${\cal K}$-matrix is not extracted from the asymptotic part of the wave function but from its internal part.
Integral relations
------------------
The derivation of the integral relations has been shown in Ref.[@rom11]. This derivation is completely general, and it is not particularized to the case of incident energies below the breakup threshold. Therefore, the same expressions derived in [@rom11] apply when the breakup channel is open. According to it, by making use of the Kohn Variational Principle, it has been proved that when using a trial three-body wave function $\Psi^t$, one can obtain the matrices $A^{(K)}$ and $B^{(K)}$ accurate up to second order in $\delta(\Psi-\Psi^t)$, and their matrix elements are given by: $$\begin{aligned}
B_{ij} & = & \frac{2m}{\hbar^2}
\langle\Psi_i^t | \hat{\cal H}-E |F_j^{(K)} \rangle \label{eq16} \\
A_{ij} & = & - \frac{2m}{\hbar^2}
\langle \Psi_i^t|\hat{\cal H}-E |G_j^{(K)} \rangle \label{eq17}, \end{aligned}$$ where $\Psi_i^t$ describes each possible incoming channel and the index $j$ refers to each possible outgoing channel (either $1+2$ or breakup).
In this work, the trial three-body wave function will be the one obtained as sketched in subsection \[sub2a\]. To be precise, it will be obtained by solving the Faddeev equations by means of the hyperspherical adiabatic expansion method (see Ref.[@nie01] for details).
Since the regular and irregular functions $F^{(K)}$ and $G^{(K)}$ are asymptotically solutions of $({\cal H}-E)F^{(K)},G^{(K)}=0$, it is then clear that the integral relations (\[eq16\]) and (\[eq17\]) depend only on the short-range structure of the scattering wave function $\Psi^t$.
It is important to note that the function $G^{(K)}$, defined in Eq.(\[eq9\]), is irregular at the origin. In order to avoid the problems arising from this fact, the $G^{(K)}$ function is regularized. This means that in Eq.(\[eq17\]) the Bessel function $\eta$ contained in $G^{(K)}$ is actually replaced by another function that goes to zero at the origin and behaves exactly as $\eta$ at large distances. In [@rom11] this was done by using $\tilde{\eta}_\ell(z)=(1-e^{-\gamma z})^{\ell+1} \eta_\ell(z)$, where $\gamma$ ($>0$) is a parameter. However, in our case, where the index $\ell$ can reach pretty high values ($\ell=K+3/2$ in the breakup channels) this procedure is not appropriate.
In this work we have regularized the irregular Bessel function by solving Eq.(\[eq5\]) for each individual adiabatic potential. The solutions of this equation behave asymptotically as $f_n(z) \rightarrow z j_\nu(z)-\tan \delta_n z \eta_\nu(z)$, where $\delta_n$ is the phase-shift, and the index $\nu$ is equal to $\ell_y$ for the $1+2$ channels, and $K+3/2$ for the breakup channels. Then, the irregular Bessel function implicitly contained in Eq.(\[eq17\]) is replaced by: $$\tilde{\eta}_\nu(z)=\frac{z j_\nu(z)-f_n(z)}{z \tan \delta_n},$$ which by construction is regular at the origin and goes asymptotically to $\eta_\nu(z)$.
A test case: neutron-deuteron scattering
========================================
The benchmark solutions for neutron-deuteron breakup amplitudes are shown in [@fri90; @fri95]. For this reason we take this case as a test for the method shown in this work.
In particular, the nucleon-nucleon interaction is chosen to be the revised Malfliet-Tjon I-III model $s$-wave potential [@fri90], which for the spin triplet and singlet cases takes the form: $$\begin{aligned}
V_t(r)&=&\frac{1}{r}\left( -626.885 e^{-1.55 r} +1438.72 e^{-3.11 r} \right) \label{eq19}\\
V_s(r)&=&\frac{1}{r}\left( -513.968 e^{-1.55 r} +1438.72 e^{-3.11 r} \right) \label{eq20},\end{aligned}$$ where $r$ is given in fm, and the potential in MeV. Also, $\hbar^2/m$=41.47 MeV fm$^2$. The potential above leads to a binding energy for the deuteron of 2.2307 MeV.
In the calculation only $s$-waves are considered. Therefore, two different total angular momenta are possible, the quartet case ($J=3/2$), for which only the triplet $s$-wave potential (\[eq19\]) enters, and the doublet case ($J=1/2$), for which both, the singlet and the triplet potentials contribute. In Fig.\[fig2\] we show the computed adiabatic effective potentials (Eq.(\[eq5\])) for the doublet case. As you can see, they follow the general trend of the potentials shown in Fig.\[fig1\], although in this case there is only one $1+2$ channel, which corresponds to the neutron-deuteron reaction that we want to investigate. The corresponding adiabatic potential is given by the thick-solid line in the figure, and it will be labeled as channel 1. As expected, the asymptotic value of this potential corresponds to the binding energy of the deuteron.
Inelasticity and phase-shifts
-----------------------------
The unitarity of the ${\cal S}$-matrix implies that given an incoming channel, for instance channel 1 ($n+d$ channel), we have that $\sum_1^\infty |{\cal S}_{1n}|^2=1$, or, in other words, $$\sum_{n=2}^\infty |{\cal S}_{1n}|^2=1-|{\cal S}_{11}|^2,$$ which means that an accurate calculation of the elastic term ${\cal S}_{11}$ amounts to an accurate calculation of the infinite summation of the $|{\cal S}_{1n}|^2$ terms ($n>1$) corresponding the breakup channels.
Also, the complex value of ${\cal S}_{11}$ can be written in terms of a complex phase-shift $\delta$ as: $${\cal S}_{11}=e^{2i\delta}=e^{-2 \mbox{\scriptsize Im}(\delta)} e^{2i \mbox{Re}(\delta)}
=|{\cal S}_{11}|e^{2i \mbox{Re}(\delta)}.
\label{eq22}$$
$K_{\mbox{\scriptsize max}}$
------------------------------ ---------------- ---------------- ---------------- ----------------
14.1 MeV 42.0 MeV 14.1 MeV 42.0 MeV
4 [**0.4662**]{} [**0.4929**]{} [**0.9794**]{} [**0.8975**]{}
[(0.4710)]{} [(0.4719)]{} [(0.9809)]{} [(0.8865)]{}
8 [**0.4637**]{} [**0.4993**]{} [**0.9784**]{} [**0.9026**]{}
[(0.4670)]{} [(0.4985)]{} [(0.9794)]{} [(0.9050)]{}
12 [**0.4640**]{} [**0.5014**]{} [**0.9783**]{} [**0.9030**]{}
[(0.4664)]{} [(0.5041)]{} [(0.9792)]{} [(0.9071)]{}
16 [**0.4643**]{} [**0.5019**]{} [**0.9782**]{} [**0.9031**]{}
[(0.4666)]{} [(0.5051)]{} [(0.9792)]{} [(0.9071)]{}
20 [**0.4644**]{} [**0.5021**]{} [**0.9782**]{} [**0.9033**]{}
[(0.4666)]{} [(0.5052)]{} [(0.9791)]{} [(0.9069)]{}
24 [**0.4645**]{} [**0.5022**]{} [**0.9782**]{} [**0.9033**]{}
[(0.4666)]{} [(0.5055)]{} [(0.9790)]{} [(0.9071)]{}
28 [**0.4645**]{} [**0.5022**]{} [**0.9782**]{} [**0.9033**]{}
[(0.4667)]{} [(0.5056)]{} [(0.9790)]{} [(0.9071)]{}
Ref.[@fri95] [**0.4649**]{} [**0.5022**]{} [**0.9782**]{} [**0.9033**]{}
: Inelasticity parameter $|{\cal S}_{11}|$ for the neutron-deuteron scattering for two different laboratory neutron beam energies (14.1 MeV and 42.0 MeV) for the doublet and quartet cases. The value of $K_{\mbox{\scriptsize max}}$ is the $K$-value associated to the last adiabatic potential included in the calculation. The numbers within parenthesis have been obtained without use of the integral relations. The last row gives the value quoted in Ref.[@fri95].[]{data-label="tab1"}
The value of $|{\cal S}_{11}|^2$ gives the probability of elastic neutron-deuteron scattering, and $|{\cal S}_{11}|$ is what usually referred to as the inelasticity parameter (denoted by $\eta$ in [@fri90; @fri95]). Obviously, the closer the inelasticity to 1 the more elastic the reaction. In fact, for energies below the breakup threshold the phase-shift is real and $|{\cal S}_{11}|=1$.
In Table \[tab1\] we give the inelasticity parameter $|{\cal S}_{11}|$ for the two laboratory neutron energies used in [@fri95], i.e., 14.1 MeV and 42.0 MeV. We have computed the inelasticity parameter for both, the doublet and the quartet states. We give the computed values of $|{\cal S}_{11}|$ for different truncations in the infinite summation (\[eq15\]). In particular, in the table we give the value $K_{\mbox{\scriptsize max}}$ of the asymptotic grand-angular quantum number associated to the last adiabatic potential included in the calculation (see Eq.(\[eq13\])). The values given in the table without parenthesis have been obtained by using the integral relations (\[eq16\]) and (\[eq17\]), while the ones within parenthesis have been calculated from the $A^{(K)}$ and $B^{(K)}$ matrices extracted directly from the asymptotic part of the three-body wave function, as indicated in Eq.(\[eq11\]). The last row in the table gives the value obtained in [@fri95].
As seen in the table, when using the integral relations the agreement with the results in [@fri95] is very good. Actually, we obtain precisely the same result for the two energies in the quartet case, and a tiny difference clearly smaller than 0.1% in the doublet case. Furthermore, the pattern of convergence is rather fast, specially in the quartet case, for which already for $K_{\mbox{\scriptsize max}}=8$ we obtain a result that can be considered very accurate. In the doublet case the convergence is a bit slower, and a value of $K_{\mbox{\scriptsize max}}$ of about 16 is needed. As we can see from the values within parenthesis, when the integral relations are not used, the value of $|{\cal S}_{11}|$ seems to converge more slowly, and even if converged, the result is less accurate.
$K_{\mbox{\scriptsize max}}$
------------------------------ ---------------- --------------- --------------- ---------------
14.1 MeV 42.0 MeV 14.1 MeV 42.0 MeV
4 [**105.82**]{} [**42.66**]{} [**69.04**]{} [**38.98**]{}
[(97.62)]{} [(28.89)]{} [(60.88)]{}
8 [**105.57**]{} [**41.65**]{} [**68.99**]{} [**37.95**]{}
[(99.91)]{} [(32.88)]{} [(63.24)]{}
12 [**105.53**]{} [**41.49**]{} [**68.98**]{} [**37.77**]{}
[(101.01)]{} [(34.80)]{} [(64.43)]{}
16 [**105.53**]{} [**41.46**]{} [**68.97**]{} [**37.73**]{}
[(101.68)]{} [(36.05)]{} [(65.11)]{}
20 [**105.53**]{} [**41.45**]{} [**68.96**]{} [**37.72**]{}
[(102.12)]{} [(36.81)]{} [(65.54)]{}
24 [**105.53**]{} [**41.44**]{} [**68.96**]{} [**37.71**]{}
[(102.41)]{} [(37.30)]{} [(65.84)]{}
28 [**105.53**]{} [**41.44**]{} [**68.96**]{} [**37.71**]{}
[(102.49)]{} [(37.90)]{} [(65.88)]{}
Ref.[@fri95] [**105.50**]{} [**41.37**]{} [**68.96**]{} [**37.71**]{}
: The same as Table \[tab1\] for Re($\delta$).[]{data-label="tab2"}
From Eq.(\[eq22\]) we have that, together with the inelasticity parameter $|{\cal S}_{11}|$, a complete specification of the matrix element ${\cal S}_{11}$ requires knowledge of the real part of the phase-shift Re($\delta$). The corresponding computed values are shown in Table \[tab2\], where the meaning of the different columns is the same as in Table \[tab1\]. The behavior of Re($\delta$) is similar to what shown in Table \[tab1\] for $|{\cal S}_{11}|$. For the quartet case the same results as in [@fri95] are obtained, while for the doublet again a very small difference smaller than 0.1% is again found. The pattern of convergence is also similar. A $K_{\mbox{\scriptsize max}}$ value of 12 is already enough to get a quite accurate value. The main difference compared to the results for $|{\cal S}_{11}|$ shown in Table \[tab1\] is that now the values of Re($\delta$) obtained without use of the integral relations are much less converged and much less accurate. The difference with the true result can reach up to 10%. This behavior was already observed in [@bar09] for the phase-shift in a reaction below the breakup threshold. This is due to the fact that the hyperradius $\rho$ and the Jacobi coordinate $y$ entering in the asymptotic forms (\[eq8\]) and (\[eq9\]) are equivalent only at infinity as commented at the end of section \[gener\] . This means that a correct extraction of the phase-shift from the asymptotic part of the wave function requires to impose the boundary condition at infinity, for which also infinite adiabatic terms would be needed.
Therefore, for energies above the breakup threshold, the conclusion from Tables \[tab1\] and \[tab2\] is similar to the one reached in [@bar09] for $1+2$ elastic processes, that is, the use of the integral relations is crucial from two different points of view. First, it accelerates drastically the convergence of the values of the ${\cal S}$-matrix elements, and second, they are needed in order to obtain the correct result.
Soft-core $\mbox{$^4$He}$-$\mbox{$^4$He}$ potential and the $\mbox{$^4$He}$-$\mbox{$^4$He}_2$ reaction
======================================================================================================
Once the integral relations have been proved to be efficient in order to describe $1+2$ reactions above the breakup threshold, in this section we shall use them to study the atomic $\mbox{$^4$He}$-$\mbox{$^4$He}_2$ process. In particular, we shall focus on three-body states with spin and parity $0^+$, and the possibility of using simple two-body soft-core potentials will be investigated.
The $\mbox{$^4$He}$-$\mbox{$^4$He}$ molecule is known to be one of the biggest diatomic molecules. Its binding energy has been estimated to be around 1 mK with a scattering length $a$ around 190 a.u [@luo96; @gri00]. Different accurate investigations of the $\mbox{$^4$He}$-$\mbox{$^4$He}$ interaction are available in the literature [@azi91; @tan95; @kor97; @jez07]. All of them present the common feature of a sharp repulsion below an interparticle distance of approximately 5 a.u.. The presence of the repulsive core is the source of a series of important technical difficulties. For instance, the wave function in the inner regions, which is very small due to the large potential repulsion, is decisive for the energy of the bound states or the asymptotic properties of the continuum states. Therefore, the wave function must be calculated with high accuracy in this region, which typically requires a very important increase of the basis size. Furthermore, in case of using the adiabatic expansion method, the angular eigenvalues $\lambda_n(\rho)$ (see Eq.(\[eq3\])) also diverge for small $\rho$, and this divergence provokes very frequent crossings between them that sometimes are not easy to handle [@nie98].
In Ref.[@kie11] the possibility of using a soft-core potential was investigated in the context of $^4$He-$\mbox{$^4$He}_2$ collisions below the threshold for breakup of the dimer. In particular the gaussian potential suggested in [@nie98] $$V_{2b}(r)=-1.227 e^{-r^2/10.03^2}
\label{soft}$$ was used (the strength of the potential is in K and the range in a.u.).
LM2M2 [@kol09] SAPT [@jez07; @sun08]
----------------------------- ---------- ---------------- ---------- ----------------------- -------------- ----------
$E_{2b}$ (mK) $-1.302$ $-1.564 $
$a$ (a.u.) 189.05 173.50
$r_0$ (a.u.) 13.84 13.79
$\langle r \rangle$ (a.u.) 98.2 90.3
($W_0$, $\rho_0$) (K, a.u.) $(0,-)$ $(18.314,6)$ $(0,-)$ $(17.760,6)$
$E_{3b}^{(g.s.)}$ (mK) $-150.0$ $-126.4$ $-126.4$ $-154.9$ $-130.9$ $-130.9$
$E_{3b}^{(exc.)}$ (mK) $-2.467$ $-2.287$ $-2.265$ $-2.805$ $-2.612$ $-2.588$
$a_0$ (a.u.) 165.9 210.6 224.3 181.7 226.0 226.8
The dimer properties obtained with this potential are given in the second column in the upper part of table \[tab3\]. In particular, the dimer binding energy ($E_{2b}$), the two-body scattering length ($a$), the effective range ($r_0$), and the interatomic distance ($\langle r \rangle$) are given. The two parameters in the soft-core gaussian potential (\[soft\]) were fitted to reproduce the scattering length and the effective range of the hard-core potential LM2M2 [@azi91]. As seen in the third column of Table \[tab3\] (upper part), when this is done the binding energy $E_{2b}$ and the interatomic distance $\langle r \rangle$ are also well reproduced.
However, even if both potentials have the same two-body properties, when moving to the three-body states important differences appear. The soft-core potential overbinds the two bound states in $\mbox{$^4$He}_3$ and the atom-dimer scattering length is clearly smaller (second and fourth columns in the lower part of Table \[tab3\]). In fact, as shown in [@kie11], the phase-shifts obtained with these two potentials for the $\mbox{$^4$He}-\mbox{$^4$He}_2$ elastic scattering (below the breakup threshold) clearly differ from each other. As an example, for an incident energy of 1 mK, the phase-shifts obtained with the gaussian and the LM2M2 potentials are $-56$ and $-63$ degrees, respectively.
As also shown in [@kie11], this anomalous behavior of the soft-core potential at the three-body level can be corrected by using a short-range effective three-body force, depending only on the hyperradius, that is added to the effective potential (\[eq4\]). In particular we choose the simple three-body force $$W(\rho) = W_0 e^{-\rho^2/\rho_0^2},
\label{3b}$$ where the strength $W_0$ is adjusted to reproduce the trimer ground-state binding energy obtained with the LM2M2 potential. When this is done the results are fairly independent of the range parameter $\rho_0$, at least within a reasonable value from $\rho_0=4$ a.u. to $\rho_0=10$ a.u.. The results given in the third column (lower part) of Table \[tab3\] have been obtained after inclusion of the three-body force. The precise values of $W_0$ and $\rho_0$ used are given in the first row of the lower part of the table. As we can see, once the binding energy of the ground state has been corrected, the binding energy of the excited state and the atom-dimer scattering length automatically agree with the corresponding values obtained with the LM2M2 potential. Furthermore, as seen in Fig.3 of [@kie11], the low energy phase-shifts are also corrected when the effective three-body force is included.
$K_{\mbox{\scriptsize max}}$
------------------------------ ---------------- ---------------- --------------- ---------------
$E=5$ mK $E=25$ mK $E=5$ mK $E=25$ mK
4 [**0.9988**]{} [**0.9351**]{} [**69.30**]{} [**35.09**]{}
[(0.9946)]{} [(0.9645)]{} [(75.52)]{} [(40.59)]{}
8 [**0.9988**]{} [**0.9111**]{} [**69.23**]{} [**34.80**]{}
[(0.9946)]{} [(0.9403)]{} [(75.45)]{} [(40.28)]{}
12 [**0.9989**]{} [**0.9104**]{} [**69.20**]{} [**34.63**]{}
[(0.9947)]{} [(0.9394)]{} [(75.41)]{} [(40.09)]{}
16 [**0.9989**]{} [**0.9110**]{} [**69.17**]{} [**34.60**]{}
[(0.9947)]{} [(0.9402)]{} [(75.39)]{} [(40.05)]{}
20 [**0.9989**]{} [**0.9110**]{} [**69.16**]{} [**34.59**]{}
[(0.9947)]{} [(0.9402)]{} [(75.38)]{} [(40.04)]{}
24 [**0.9989**]{} [**0.9109**]{} [**69.15**]{} [**34.58**]{}
[(0.9947)]{} [(0.9402)]{} [(75.38)]{} [(40.04)]{}
28 [**0.9989**]{} [**0.9109**]{} [**69.15**]{} [**34.58**]{}
[(0.9947)]{} [(0.9402)]{} [(75.37)]{} [(40.04)]{}
40 [**0.9989**]{} [**0.9109**]{} [**69.15**]{} [**34.58**]{}
[(0.9947)]{} [(0.9402)]{} [(75.37)]{} [(40.04)]{}
: Inelasticity ($|{\cal S}_{11}|$) and real part of the phase-shift (Re($\delta$)) for the $^4$He-$\mbox{$^4$He}_2$ collision at three-body energies (above threshold) $E=5$ mK and $E=25$ mK. The value of $K_{\mbox{\scriptsize max}}$ is the $K$-value associated to the last adiabatic potential included in the calculation. The numbers within parenthesis have been obtained without inclusion of the three-body force.[]{data-label="tab4"}
The importance of the inclusion of the three-body force is also seen when investigating the $\mbox{$^4$He}-\mbox{$^4$He}_2$ reaction for incident energies ($E_i=E-E_{2b}$) above the threshold for breakup of the dimer, i.e., $E_i>|E_{2b}|$ (or $E>0$). In Table \[tab4\] we give the inelasticity ($|{\cal S}_{11}|$) and the real part of the phase-shift (Re($\delta$)) for energies $E=5$ and $E=25$ mK. As in Tables \[tab1\] and \[tab2\], $K_{\mbox{\scriptsize max}}$ is the grand-angular quantum number associated to the last adiabatic term included in the expansion (\[eq1\]). The boldface numbers have been obtained when the three-body force has been included in the calculation ($W_0=18.314$ K and $\rho_0=6$ a.u.), and the results within parenthesis have been obtained without the three-body force.
As we can see, the pattern of convergence is similar to the one observed in Tables \[tab1\] and \[tab2\] for the neutron-deuteron reaction. A $K_{\mbox{\scriptsize max}}$ value of around 12 is enough to get a rather well converged inelasticity, while Re($\delta$) requires a few more adiabatic terms in order to reach convergence. Similarly to what found in [@kie11] for energies below the breakup threshold, the inclusion of the three-body force gives rise to relevant changes in the computed values. These changes are particularly noticeable for Re($\delta$), which for the two energies under consideration increases up to 6 degrees when the three-body force is not included in the calculation. Also, we observe that for $E=5$ mK the breakup probability ($1-|{\cal S}_{11}|^2$) is still rather small, clearly smaller than 1% (with three-body force), while for $E=25$ mK this probability rises up to 17% (12% without three-body force).
With the ${\cal S}$-matrix in hand, we can now compute the dissociation rate for the $\mbox{$^4$He}-\mbox{$^4$He}_2$ collision. The analytic form of this rate is given in Ref.[@sun08], and for $0^+$ states it becomes: $$D_3= \frac{\hbar \pi}{\mu_{1,23} k_{1,23}} \left( 1- |{\cal S}_{11}|^2\right) ,
\label{disso}$$ where $$\mu_{1,23}=\frac{2M_{\mbox{\scriptsize He}}}{3}\; , \hspace*{5mm}
k_{1,23}^2=\frac{2\mu_{1,23} E_i}{\hbar^2},$$ $M_{\mbox{\scriptsize He}}$ is the mass of the $^4$He atom, and $E_i=E-E_{2b}$ is the incident energy in the center of mass frame ($E_{2b}$ is the binding energy of the $\mbox{$^4$He}_2$ dimer).
In Fig.\[fig3\] we show the dissociation rate as a function of the three-body energy $E$. The thin-dashed and thin-solid lines are the results obtained with the gaussian soft-core potential (\[soft\]) without and with the additional three-body force (\[3b\]), respectively. The low energy behavior of the dissociation rate follows the $E^{K_{\mbox{\scriptsize m}}+2}$ rule derived in [@esr01], where $K_{\mbox{\scriptsize m}}$ is the smallest grand-angular quantum number associated to the continuum adiabatic channels ($K_{\mbox{\scriptsize m}}=0$ in our case of three indistinguishable bosons coupled to $J^\pi=0^+$).
From the figure we can see that the effect of the three-body force is quite important. In fact, for small energies the three-body force reduces the rate by a factor of 2. At higher energies, beyond 10 mK, the effect is the opposite, and the three-body force increases the rate by a factor close to 1.5.
In Ref.[@sun08] the same dissociation rate has been computed. The corresponding curve is shown in Fig.\[fig3\] by the dotted curve. As we can see, there is an important difference compared to our calculation. Except at high energies ($E \gtrsim 10$ mK), where our result (with three-body force) and the one in [@sun08] basically coincide, for small energies our rate is about a factor of 3 bigger.
However, we have to note that the hard-core two-body potential used in [@sun08] gives rise to somewhat different dimer properties compared to the soft-core potential (\[soft\]) and the LM2M2 potential [@azi91]. In [@sun08] they have used the potential based on the symmetry-adapted perturbation theory (SAPT) derived in [@jez07]. The two-body properties obtained with this potential are given in the last column (upper part) of Table \[tab3\]. The two-body dimer is 20% more bound than with the LM2M2 potential, and therefore the scattering length and the interatomic distance are smaller than in the LM2M2 case.
To investigate the sensitivity of the dissociation rate to the details of the two-body interaction we have then constructed a second gaussian soft-core potential reproducing the two-body properties of the SAPT potential. This potential takes the form: $$V_{2b}(r)=-1.234 e^{-r^2/10.03^2},
\label{soft2}$$ where the strength is given in K and the range in a.u.. The corresponding two-body properties are given in the fourth column (upper part) of Table \[tab3\] under the label “soft-core 2”. Again, when moving to the three-body system, this new soft-core potential presents the same deficiencies as the previous one, i.e., overbinding of the three-body bound states and a too small atom-dimer scattering length. As before, this problem is solved by inclusion of the three-body force, whose parameters are again fitted to reproduce the ground state binding energy of the helium trimer provided by the SAPT potential. The three-body properties with and without three-body force, as well as the parameters used for the three-body force are given in the last three columns (lower part) of Table \[tab3\].
Making use of this new soft-core potential and the corresponding three-body force, we can compute again the dissociation rate (\[disso\]). The results are shown in Fig.\[fig3\] by the thick-dashed curve (no three-body force) and the thick-solid curve (with three-body force). The new rates are now smaller than the ones obtained with the previous soft-core potential. Furthermore, when the three-body force is included, and therefore not only the two-body properties but also the three-body ones agree with the ones obtained with the SAPT potential, our dissociation rate and the one given in [@sun08] agree well for the whole range of energies. Only a small difference can be seen at about 1 mK. We can therefore see that the dissociation rate is very sensitive to the details of the two-body interaction. An additional 20% binding of the dimer, and the corresponding decrease in the scattering length, leads to a factor of 3 decrease in the rate.
The same conclusions are reached when investigating the recombination rate corresponding to the inverse process $\mbox{$^4$He}+\mbox{$^4$He}+\mbox{$^4$He}\rightarrow \mbox{$^4$He}_2+\mbox{$^4$He}$. This rate is given by the same ${\cal S}$-matrix elements as in the dissociation case, and its analytical form for three identical bosons with angular momentum and parity $0^+$ is given by [@sun08]: $$K_3=3! \frac{32\hbar \pi^2}{\mu k^4} \left( 1 - |{\cal S}_{11}|^2\right),
\label{recom}$$ where $$\mu^2=\frac{ M_{\mbox{\scriptsize He}}^2 }{3}
\; , \hspace*{5mm} k^2=\frac{2\mu E}{\hbar^2},$$ $M_{\mbox{\scriptsize He}}$ is the mass of the $^4$He atom, and $E$ is the total three-body energy.
The computed recombination rate is shown in Fig.\[fig4\]. The meaning of the curves is as in Fig.\[fig3\]. In Ref.[@esr01] the recombination rate $K_3$ was proved to behave at low energies as $E^{K_{\mbox{\scriptsize m}}}$ ($K_{\mbox{\scriptsize m}}$ is the smallest grand-angular quantum number associated to the continuum adiabatic channels), which means that in our case, where $K_{\mbox{\scriptsize m}}=0$, $K_3$ should be constant at very low energies, as observed in the figure. Again, inclusion of the three-body force reduces the rate by a factor of 2 to 3 at small energies (compare the dashed curves and the corresponding solid curves in the figure). Also, when the soft-core potential (\[soft2\]) together with the three-body force is used, the computed recombination rate agrees well with the rate given in [@sun08] (dotted curve in the figure). This fact reveals again the importance of the fine details of the two-body interaction.
Finally, it is important to emphasize that in this work the ${\cal S}$-matrix has been obtained by use of the integral relations (\[eq16\]) and (\[eq17\]). As explained, they have the great advantage of needing only the internal part of the wave functions. In fact, the integrals involved in the calculations shown in this section can be safely performed with a maximum value for the hyperradius $\rho$ of 4000 a.u.. Even less could be enough. However, in [@sun08], where the adiabatic expansion was also used, the radial wave functions in Eq.(\[eq1\]) had to be expanded up to $5 \times 10^5$ a.u., which is definitely very delicate from the numerical point of view.
Summary and conclusions
=======================
In this paper we have extended the use of the integral relations derived in [@bar09; @rom11] to describe 1+2 reactions above the threshold for breakup of the dimer. As in [@bar09; @rom11], the integral relations are used in combination with the adiabatic expansion method in order to construct the trial wave function. The main consequence of moving up to energies above the breakup threshold is that the ${\cal S}$-matrix describing the process has now infinite dimension. This was not the case for energies below the threshold, where the elastic, inelastic, or transfer channels were described by a finite (and small) number of adiabatic terms.
The applicability of the method has been tested with the neutron-deuteron reaction, for which benchmark calculations are available. The agreement with these calculations is good, and the pattern of convergence is similar to the one found in [@bar09; @rom11] for energies below the breakup threshold. The integral relations accelerate the convergence significantly, and typically about 10 adiabatic terms are enough to get a rather well converged result.
The method has then been used to investigate reactions involving three helium atoms. In particular we have focused on the possibility of using soft-core atom-atom potentials in order to describe the full process. These potentials permit to avoid all the technical problems arising from the use of more sophisticated potentials where a hard-core repulsion is always present. However, as already shown in [@kie11], even if the soft-core potentials reproduce properly the dimer properties, an effective three-body force is needed in order to reproduce as well the properties of the trimer bound states and the atom-dimer scattering length obtained with the hard-core potentials.
We have found that the three-body force also modifies significantly the inelasticity and the phase-shift for energies above the breakup threshold. In fact, when computing the reaction rates for dissociation of the dimer ($\mbox{$^4$He}+\mbox{$^4$He}_2 \rightarrow \mbox{$^4$He}+\mbox{$^4$He}+\mbox{$^4$He}$) and for the recombination process ($\mbox{$^4$He}+\mbox{$^4$He}+\mbox{$^4$He}\rightarrow \mbox{$^4$He}_2+\mbox{$^4$He}$) the three-body force decreases the rates even by a factor of 3 at low energies.
These two rates are also very sensitive to the details of the two-body interaction. We have found that a two-body potential, like the SAPT potential, providing a dimer state about 20% more bound than the one obtained with the LM2M2 potential, also reduces the rates by a factor of around 3 at small energies. The soft-core potential reproducing the two-body properties of the SAPT potential, together with the corresponding three-body forced designed to reproduce the trimer properties as well, gives then rise to reaction rates in very good agreement with the ones of the SAPT potential.
In summary, the integral relations are also useful in order to describe reactions above the breakup threshold. They accelerate significantly the convergence of the ${\cal S}$-matrix terms. For reactions involving hard-core two-body potentials, the use of soft-core potentials with the same two-body properties are a very good alternative, provided that they are used together with an effective three-body force designed to fit as well the bound state three-body energies. When this is done, the reaction rates obtained with the hard-core and the soft-core potentials agree pretty well. These rates are very sensitive to the details of the two-body interaction. Small variations of the dimer properties can produce sizable changes in the reaction rates at low energies.
This work was partly supported by funds provided by DGI of MINECO (Spain) under contract No. FIS2011-23565.
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abstract: 'It has been shown\[19\] that in a loop of length u to which a single stub of length v is attached (fig. 1 in ref. 19), the parity effect is completely destroyed when v/u$>2$. It was also shown that such minute topological defects (v/u$<<1$) act as singular perturbations. However ref. 19 studies the effect of a single topological defect and says that for v/u$<<1$ parity effect is not violated in the ring. In this paper we show that topological defects of the type v/u$<<1$ can also violate the parity effect depending on the exact value of v/u and the parity effect is significantly destroyed if we have many such geometric scatterers. This paper brings out the physical reasons for the destruction of parity effect. We show that the generic feature of topological defects as this is that they can produce discontinuous phase change (with change in energy) of the electron wavefunction in the ring for special value of v/u and then Legett’s cojecture breaks down. So Legett’s conjecture which generalises parity effect in presence of any arbitrary 2 body scattering and any arbitrary 1 body scattering need not be true when the one body scattering can produce discontinuous phase change of the electron wave function in the ring. This may have implications on the experimental observations.'
address: 'Institute of Physics, Bhubaneswar 751005, India'
author:
- 'P. Singha Deo[@eml]'
title: '**[How general is Legett’s conjecture for a mesoscopic ring? ]{}**'
---
A normal metal ring pierced by a magnetic field carries a persistent current and has a magnetic response. The ring shows strong parity effect in the sense that the nature of response of a ring (paramagnetic or diamagnetic) is exteremely sensitive to the number of electrons present in the ring. For spinless electrons, a clean ring with odd number of electrons has a diamagnetic response and that with an even number of electrons has paramagnetic response\[1\]. This happens because when the number of electrons in the ring changes from odd to even, there is a statistical half flux quantum which shifts the energy flux dependence by exactly $\phi_{0}$/2\[2\]. It was conjectured by Legett that this fact follows just from symmetry property of the wavefunction (the electrons being fermions, the wave function must be antisymmetric) and is independent of electron electron interaction and impurity scattering\[3\]. Refs. \[4,5,6\] are devoted to proving rigorously the so called Legett’s conjecture. At high temperature and disorder the parity effect shows up as a shift by $\pi$ of the persistent current versus flux curve with the change of number of electrons from odd to even\[1,2\]. For electrons with spin we get double parity effect\[2\]. Only in case of electrons with spin as well as interactions the parity effect may be destroyed because of fractional Aharonov Bohm(AB) effect\[7-12\]. But such fractional Aharonov Bohm effect has not yet been observed experimentally. Parity effect is seen in multichannel simulations too\[13\].
The first experiment\[14\] was done with N=$10^7$ rings and due to parity effect one expects the ensemble averaged response to scale with $\sqrt N$. Contradictory to it the actual response measured in the experiment is quiet high along with a $\phi_0/2$ periodicity. There are many possible effcts that can give the $\phi_0/2$ periodicity\[15\] but the magnitude is a puzzle\[16\] and so it is for a single ring (where alternate levels have opposite response) experiment\[17\]. In a recent single loop experiment\[18\] one observes fair agreement.
It has been shown\[19\] that in a loop of length u to which a single stub of length v is attached (fig. 1 in ref. 19), the parity effect is completely destroyed when v/u$>2$. It was also shown that minute topological defects (v/u$<<1$) act as singular perturbations. However ref. 19 studies the effect of a single topological defect and says that for v/u$<<1$ parity effect is not violated in the ring. In this paper we show that topological defects of the type v/u$<<1$ can also violate the parity effect depending on the exact value of v/u and the parity effect is significantly destroyed if we have many such geometric scatterers and that can well affect the experimental observations. This paper brings out the physical reasons for the destruction of parity effect. We show that the generic feature of topological defects as this is that they can produce discontinuous phase change of the electron wavefunction in the ring for special value of v/u and then Legett’s cojecture breaks down. So Legett’s conjecture which generalises parity effect in presence of any arbitrary 2 body scattering and any arbitrary 1 body scattering need not be true when the one body scattering can produce discontinuous phase change of the electron wave function in the ring.
The thickness of the experimental ring could not have been uniform. If there are some sharp variations in thickness then some resonant cavities may be formed at certain places (at random) in the ring. 1-D modelling of the ring helps to understand the basic physics and its result can be easily extended to the multichannel case. Resonant cavities can be taken as stubs\[20\] and width of the resonant cavitites result only in lowering the energy\[21\].
The allowed modes in the system are given by the following condition\[19\].
$$cos(\alpha)={sin(ku)cot(kv)\over 2} + cos(ku)$$
k is the allowed wave vectors and $\alpha= 2\pi\phi/\phi_{0}$, is the AB phase, $\phi$ being the flux through the ring and $\phi_{0}$ is the flux quantum. The above condition is the simplified form of the following condition\[19\].
$$cos(\alpha)=re(1/T)$$
where T is the transmission amplitude across the ring when the ring is cut open. The above equation immediately suggests something. For the clean ring the bound state condition is $e^{i(ku+\alpha)}=1$. Whereas eqn(2) is just the condition (It is worth mentioning that $cos(\alpha)=cos(2n\pi-\alpha)$)
$$e^{i[cos^{-1}(re{1\over T})+\alpha]}=1$$
B$\ddot u$ttiker et al\[22\] has shown that an electron in a ring with a random potential is effectively moving in a periodic system whose unit cell is the ring when cut open. It is also well known that $cos^{-1}(re{1\over T})$ is the Bloch phase (Ku where K is the Block momentum) aquired by the electron in traversing an unit cell of an infinite periodic system where T is the transmission amplitude across the unit cell of the periodic system\[19\]. Hence eqn(3) is just $e^{i(Ku+\alpha)}$=1 and is in perfect agreement with ref. \[22\]. It also suggests that inside the ring the electron moves with the momentum K and not with the free particle momentum $\pm k$. Hence it is not surprising that $\pm k$ states are degenerate even in the presence of magnetic field (because both +k and -k satisfy eqn (3) ). Inside the ring the electron is moving anticlockwise or clockwise with momentum K (Ku=$cos^{-1}(re{1 \over T})$). One of them is a diamagnetic (anticlockwise moving) state and the other is a paramagnetic (clockwise moving) state. Initially as the magnetic field is increased then the two states move away from each other, however the diamagnetic state and the paramagnetic state are not degenarate for any value of $\phi$ for reasons explained later. In a clean ring (ku+$\alpha$) is the phase aquired by the electron in going round the ring once whereas in a ring with scatterers (potential or geometric) $Ku+\alpha$ is the phase aquired in moving round the ring once and eqn (3) is due to the single valuedness of the wavefunction. So eqn (2) suggests that a particular mode is allowed in the ring if the phase aquired by an electron wave fn. in that mode (apart from the AB phase) in going round the ring once,i.e. $cos^{-1}(re{1 \over T})$, equals ($\alpha$). We can alternately state it as - if the Block phase of an electron in travelling a unit cell of an infinite periodic system i.e., Ku equals $\alpha$ then single valuedness of wavefunction is obtained in the ring made of the unit cell and we get a bound state. So the boundstates can be determined by graphically solving re(1/T)=cos($\alpha$). This occurs at certain k values and then $k^2$ is the energy of the electron in that particular state K. It has an analogy with a scattering problem where k is the momentum outside the potential where it vanishes, whereas the momentum K inside the potential can be quiet different. However the energy throughout is $k^2$ (we have set $\hbar=1$ and 2m=1).
So in fig(1) we show a simple plot where the solid curve is a plot of y=re(1/T) with ku for v/u=.2. Wherever this curve intersects the straight line y=cos($\alpha$), the corresponding k value is a bound state for the system. Let us start with $\alpha$=0 and then y=cos(0) curve is shown in fig. 1 by dotted lines. Two consequitive points where the curve y=re(1/T) intersects the straight line y=cos(0) are denoted by A and B in the fig. The corresponding k values are denoted as $k_1$ and $k_2$ in the fig. If $\alpha$ is increased gradually then the straight curve y=cos($\alpha$) shifts gradually downwards towards the dashed curve. As the curve y=cos($\alpha$) gradually go downwards the allowed wave vectors $k_1$ and $k_2$ slowly drift rightwards and leftwards respectively along the k axis. As $k_1$ drifts rightwards with $\alpha$ i.e., towards higher energy, $k_1$ is a diamagnetic state. Similarly $k_2$ is a paramagnetic state. That $k_1$, $k_2$, etc. gradually increase or decrease with $\alpha$ gives rise to a dispersion with $\alpha$ (E vs $\alpha$) with close by alternate states going further away from each other with $\alpha$ upto $\alpha= \pi$. y=cos($\pi$) is also shown in fig. 1 with dashed lines. If we increase $\alpha$ further then the straight curve y=cos($\alpha$) start moving upwards and comes back to its original position at $\alpha =2 \pi$. This ensures $\phi_0$ periodicity of the dispersion curves. Since cos$(\alpha$) can vary from -1 to +1 (dotted lines to dashed lines) the dispersion curve for any two consequtive states can never cross (see fig. 1). So the dispersion curve is exactly similar to that of a ring with a random potential. The cause of gaps in the dispersion curve in that case is the breakdown of rotational symmetry of the ring by the random potential and hence the removal of degenaracy of states that cross over for a clean ring. In our case the rotational symmetry is destroyed by the topological defect. Some gaps (the ones around kv=$n \pi$) are very large but most gaps are very small. In fact some special gaps may actually go to zero for reasons explained later. Hence from fig. 1 it is evidient that alternate states carry persistent currents with opposite signs and have opposite magnetic properties up to infinite energy. This is exactly the same as in case of potential scattering.
But this effect is not observed when we plot the same curves for different values of v/u=.21 (fig. 2) (in fact v/u=.2$\pm
\epsilon$ is sufficient). Consider the intersections between the graphs y=re(1/T) and y=cos(0). The first few alternate states have opposite magnetic properties but the fifth and the sixth states (two consequtive states marked A and B) are both diamagnetic disobeying the parity effect. Slowly increase $\alpha$ to see that. Parity effect is again violated for the 11th and the 12th states both of which are paramagnetic. After a regular spacing of five levels we always find two consequtive levels that violate the parity effect. Hence parity effect due to the antisymmetric property of the electron wave function is not generic to a ring with topological defects. We shall soon see why it does not happen for specific values of v/u.
Note that for kv=n$\pi$, the transmission across a stub is zero\[23\] due to the formation of a node at the junction between the ring and the stub. This mode always lie in a gap of the dispersion curve and is never an allowed mode.
Scattering by a topological defect like a stub is still a poorly understood phenomenon. Ref. \[23\] tries to explain scattering by a stub on the same footing as scattering by potentials. Here we intend to understand the problem by mapping it onto an effective delta potential. A special feature of the delta potential is that $\mid T \mid ^2$= re(T). This feature is not seen for any other potential. However this feature is also observed in case of a stub. This makes it possible to map a single stub onto an effective delta potential V(x)=k cot(kv)$\delta$(x). So the strength of the delta potentials depend on the fermi energy. That is why at certain energies V(x) becomes zero and then the gap vanishes. Now let us start with k=0 and then slowly increase k continuously. For k=0 V(x)=1/v which means it starts with a small positive value. Then it decreases and soon goes to zero. After this the strength of the potential monotonously increase on the negative side and finally becomes -$\infty$ at kv=$\pi$. After this V(x) undergo a discontinuous jump from $-\infty$ to $+\infty$. If the strength of the $\delta$ potential at kv=$\pi$ and kv=$\pi + \epsilon$ are discontinuous the scattering phase shift and hence the Block phase will also undergo discontinuous jump. re(1/T) also make a discontinuous jump from $-\infty$ to $\infty$ and hence the Block phase jumps by $\pi$ (see fig. 2) (Block phase of the infinite periodic system has to be defined to a modulo of $2
\pi$ i.e., $-1<re(1/T)<1$). The next allowed Block phase of the infinite periodic system of stubs after that at D is that at B and they differ by $\pi$. This is markedly different from the next allowed Block phase at any other gap, e.g., the Block phase at C is same as that at A. We have seen that if the Block phase of the periodic system of stubs equals the AB phase $\alpha$ (for the time being we have taken $\alpha$=0) then the single valuedness of the wave function gets satisfied in the ring and we get a bound state. This additional phase results in satisfying this condition and creating a state at B close to the value kv=$\pi $ which otherwise would not have been there had the phase change across kv=$\pi$ been continuous. If it so happens that this singularity in the Bloch phase due to a singularity in the effective potential V(x) is cancelled by another singularity then the phase difference between two consequitive Block phase would not have been $\pi$ and then this state at B would not exist because total phase aquired in this state is not enough to satisfy the single valuedness condition in the ring. All other states would have been as usual and would have been qualitatively same as that of a ring with a random potential. This is what happens in case of fig. 1. For v/u=.2 at kv=$n \pi$ cot(kv)=$\pm \infty$ but sin(ku)=0. And so there is no discontinuty in re(1/T). Hence the state at B of fig. 2 will not exist. See fig. 3 where we have superposed the two graphs of fig. 1 and fig. 2. The dotted curve is for v/u=.2 whereas the solid curve is for v/u=.21. All the states for both the cases are very close to each other except that the solid curve shows a state at A (which is the state at B in fig. 2) where the dashed curve does not show any state at all. Other states of the solid curve are very close but slightly at lower energies than those of the dashed curve because an increase in v/u means an increase in the phase space of the electrons. The state at A has no partner and locked between a diamagnetic state and a paramagnetic state it has to break the parity effect. It is easy to see from fig 2 and 3 that slope of re(1/T) is such that if it jumps from $-\infty$ to $+\infty$ then the broken parity state is diamagnetic and paramagnetic for the other case. Specific values of the parameter v/u at which these two singularities exactly cancell are negligibly few compared to the values where they do not and is hardly likely in a real situation.
We then study the spectrum of a ring with four small topological defects or stubs present in it. To find the spectrum we have to solve eqn (2) numerically using the transfer matrix mechanism to compute T\[24\]. A portion of plot is shown in figs. 4a, 4b (solid lines). Within a certain energy range (3400$>E u^2 >$150) there are much more diamagnetic levels than paramagnetic. Higher above there are however more paramagnetic levels than diamagnetic. The length of the stubs as well as the separation between the stubs has been chosen by random number genarating subroutines (.1$<v/u<$.3). We have plotted for only one configuration because we do not intend to take an average over configurations and compare with the experimentally observed numbers, because our calculations are in 1-D where localisation effects are very strong. But the usual alternate paramagnetic and diamagnetic states apart from the parity breaking states are almost unaltered by the defects because they feel a very weak V(x). We have checked that the graphs qualitatively remain the same for other configurations and for each case there is a substantial breakdown of parity effect. In a 3D multichannel ring there are many subbands and very close by levels. Each subband exhibits the parity effect\[3\]. Topological defects will destroy the parity effect of each subband with the first one being diamagnetic for each. Even one appropriate topological defect in that case can give many broken parity states.
The purpose of this paper has been to show the breakdown of Legett’s conjecture and parity effect due to topological defects (v/u$<<$1). We have also shown that a discontinuity in the scattering phase is the physical reason for it because nature of a state depend on how its boundary conditions get tuned by the phases. The discontinuty in the phase shifts the origin of the total phase by $\pi$ but does not determine wheather the total phase should increase or decrease with $\alpha$ and k. However we would like to say that any breakdown in parity effect is sure to enhance the persistent current in the single ring as well as in the many ring experiment specially when the usual states (not the parity breaking ones) are not affected much by the topological defects. Also initially upto a certain energy there is no breakdown of parity effect which means that the persistent current in a ring with very few propagating channels (only four in ref. 18) will hardly be affected by these defects. Also the abundant paramagnetic states much higher up may not be occupied and hence may be of no consequence. No theories proposed so far try to find a common explanation for the three experiments. Also all except one\[GK in 16\] completely overlook the fact that the rings can exhibit coexistence of large persistent currents and small conductances\[17\]. In our case as for every parity breaking state there is a state at which the conductance across the ring becomes zero ($\mid T \mid ^2$=0) the same reason that enhances the persistent current can decrease the conductance.
v/u values taken in this paper for qualitative analysis are much higher than realistic values. Smaller is v/u the higher up will be the diamagnetic states. Again more the width of the resonant cavities lower again will be the parity breaking states\[22\]. A 3D simulation is needed to study the real situation and it will be reported in the near future.
The author thanks Prof. A. M. Jayannavar for usefull discussions. The author also acknowledges Prof. A. M. Srivastava for a brief discussion on topology.
[99]{} prosen@iopb.ernet.in H. Cheung, Y. Gefen, E. K. Riedel, W. Shih, Phys. Rev. B 37 (1988) 6050. J. F. Weisz, R. Kishore and F. V. Kumartsev in cond-mat 9401008 and ref. therein. A. J. Leggett, in Granular Nano-electronics ed. D. K. Ferry, J. R. Barker and C. Jacoboni, NATO ASI Ser. [**B251**]{}, Plenum, New York, 1991, p.297 F. V. Kumartsev, Phys. Lett. A. [**161**]{}, 433(1992). D. Loss, Phys. Rev. Lett. [**69**]{} 343(1992). A. O. Gogolin and N. V. Prokof’ev, Phys. Rev. B. [**50**]{} 4921(1994). H. Bouchiat and G. Montambaux, J. Phys. (Paris) [**50**]{}, 2695(1989). R. A. Smith and V. Ambegaokar, Europhys. Lett. [**20**]{} 161(1992). F. V. Kumartsev. J. Phys. C. [**3**]{} 3199(1991). F. V. Kumartsev, J. Weisz, R. Kishore and M. Takahashi, Phys. Rev. B(1993) submitted. N. Yu and M. Fowler, Phys. Rev. B [**45**]{}, 11795(1992). D. Haldene, Phys. Rev. Lett. [**67**]{} 937 (1991). H. Bouchiat, G. Montambaux, D. Siegeti and R. Friesner Phys. Rev. B [**42**]{} 7647(1990). L. P. Levy, G. Dolan, J. Dunsmuir and H. Bouchiat, Phys. Rev. Lett, 64 (1990) 2074. L. Wendler, V. M. Fomin and A. A. Krokhin Phys. Rev. B [**50**]{} 4642(1994) and references therein. H. Kato and D. Yashiyoka Phys. Rev. B [**50**]{} 4943(1994); G. Kirczenow submitted to Journal of Physics C and ref. therein; G. Vignale Phys. Rev. Lett. [**72**]{}, 433(1994); P. Kopeitz Phys. Rev. Lett. [**72**]{}, 434(1994). V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. G. Gallagher and A. Kleinasser, Phys. Rev. Lett. [**67**]{}, 3578(1991). D. Mailly, C. Chapelier and A. Benoit, Phys. Rev. Lett. [**70**]{}, 2020(1993). P. Singha Deo, Phys. Rev. B [**51**]{}, 5441(1995). E. Takeman and P. F. Bagwell, Phys. Rev. B [**48**]{}, 2553(1993). F. Sols, M. Macucci, U. Ravaioli and K. Hess, Journ. of Appl. Phys. [**66**]{}, 3892(1989) and ref. there. M. B$\ddot u$ttiker, Y. Imry and R. Landauer, Phys. Lett. A 96 (1983) 365. W. Porod, Z. Shao and C. S. Lent, Phys. Rev. B. [**48**]{}, 8495(1993). P. Singha Deo and A. M. Jayannavar, Phys. Rev. B. [**50**]{}, 11629(1994).
**FIGURE CAPTIONS**
Fig. 1. To show graphical solutions for the allowed modes for v/u=.2.
Fig. 2. To show the graphical solutions for the allowed modes for v/u=.21.
Fig. 3. Superposition of fig. 1. and fig. 2.
Fig. 4a. E versus $\phi$ dispersion curves in the range $1000<Eu^2<2000$.
Fig. 4b. E versus $\phi$ dispersion curves in the range $2000<Eu^2<3400$.
|
---
abstract: 'I report the detection of circular polarisation, associated with relativistic ejections, from the ‘microquasar’ GRS 1915+105. I further compare detections and limits of circular polarisation and circular-to-linear polarisation ratios in other X-ray binaries. Since in at least two cases the dominance of linear over circular polarisation or vice versa is a function of frequency, this seems to indicate that this is a strong function of depolarisation in the source. Furthermore, I note that circular polarisation has only been detected from sources whose jets lie close to the plane of the sky, whereas we have quite stringent limits on the circular polarisation of jets which lie close to the line of sight.'
author:
- Ron Fender
title: 'Circularly polarised radio emission from GRS 1915+105 and other X-ray binaries'
---
Introduction
============
Studies of circular polarisation from jets, and indeed of relativistic jets in general, have traditionally focussed on active galactic nuclei (AGN). However, in the past decade or so it has become clear that relativistic jets from X-ray binaries, also known as ‘microquasars’ (Mirabel et al. 1992), share many of the properties (observationally, and therefore almost certainly physically) of their extragalactic cousins. Furthermore, due to the huge mass ratio $10^{5} \leq M_{\rm
AGN} \leq 10^{8}$, we may probe timescales associated with accretion and jet formation by observing X-ray binaries which would be humanly impossible for AGN (e.g. Sams, Eckart & Sunyaev 1996). For recent reviews of jets from X-ray binaries, see Mirabel & Rodriguez (1999), Fender (2003). In this paper I shall first discuss in detail observations of circularly polarised radio emission from the ‘microquasar’ GRS 1915+105 (§2), and then compare it to observations of circularly polarised radio emission from other X-ray binaries (§3).
Circular polarisation from GRS 1915+105
=======================================
In this section we present observations of circular polarisation associated with relativistic ejections from the X-ray binary jet source (‘microquasar’) GRS 1915+105, as well as some preliminary interpretations. These results have been published in Fender et al. (2002b).
Observations
------------
In Fig 1 we show radio and soft X-ray monitoring of GRS 1915+105, over a 150-day period. The radio monitoring data were obtained with the Ryle Telescope (RT), at a frequency of 15 GHz; for a more detailed description of this monitoring program see Pooley & Fender (1997). The X-ray data are from the [*Rossi*]{}XTE All-Sky Monitor (ASM) and measure the total flux in the 2-12 keV band. The [*Rossi*]{}XTE ASM is described in Levine et al. (1996) and the public data can be obtained at [**http://xte.mit.edu**]{}.
Indicated in the top panels of Fig 1 are the times of our two ATCA and multiple MERLIN observations of GRS 1915+105.
### ATCA
The Australia Telescope Compact Array (ATCA; Frater, Brooks & Whiteoak 1992) has a number of design features which enable very accurate circular polarization measurements. The low antenna cross-polarization and high polarization stability enable accurate calibration of polarization leakage terms, and the linearly-polarized feed design largely isolates Stokes V from contamination by Stokes I.
ATCA observed GRS 1915+105 twice, for six hours each, on 2001 January 17 and 2001 March 23. During the January observations, simultaneous observations at 1384 MHz and 2496 MHz were interleaved with observations at 4800 MHz and 8640 MHz; for the March observations, only 4800 MHz and 8640 MHz were observed. For both epochs the array was in a ‘6 km’ configuration, for which the lack of short baselines served to reduce confusion from other galactic sources. The observation and calibration procedures were similar to those described in Fender et al. (2000).
As discussed in Fender et al. (2000), calibration of circular polarization data requires the “strongly-polarized” calibration equations (Sault, Killeen & Kesteven, 1991), using a point-source with a few percent linear polarization. This is needed to calibrate the leakage of linear polarization into circular. For the 4800 MHz and 8640MHz observations, the VLA calibrator 1923+210 was used as a polarization calibrator for both epochs. Calibrator confusion and low linear polarization, however, precluded the use of any of the observed calibrators as polarization calibrators for the January 1384 MHz and 2496 MHz observations. As a result, we were forced to use calibration solutions derived using the “weakly-polarized” equations with the ATCA primary calibrator, 1934-638.
The use of the “weakly-polarized” equations will cause a time-varying leakage of linear polarization into circular. In tests, peak leakages of 5% of the linear polarization into circular have been observed. For the 1384 MHz observations, the low linear polarization of GRS 1915+105 implies the effect of such leakage is negligible. Even for the 2496 MHz observations, where the linear polarization rises rapidly during the observation, in the [*worst-case*]{} the leakage would be only half the Stokes V error due to thermal noise. The full polarisation ATCA data for both epochs are presented in Fig 2.
### MERLIN
The Multi Element Radio Linked Interferometer Network (MERLIN) consists of six individual antennae with a typical diameter of 25m and a maximum baseline of 217 km (Thomasson 1986). The observations presented here were undertaken in continuum mode at a frequency of 4994 MHz with a total bandwidth of 16 MHz. As MERLIN measures all four correlation products as a matter of course when in this mode, full polarimetric information can be derived from all images. Ongoing work is seeking to establish the reliability of Stokes V measurements with MERLIN; these will not be reported here.
GRS 1915+105 was observed eleven times with MERLIN following the flare observed on 2001, March 22/23. The first five epochs, corresponding to daily observations between 2001 March 24 and March 27 and again on March 29, are presented in this paper (Fig 3); further details and analysis of the full set of MERLIN observations will be published in McCormick et al. (in prep). In each case a flux calibrator, 3C286, a point source, OQ208, and a phase calibrator, 1919+086 , were included in the observing schedule. The flux calibrator and point source calibrator were observed at the beginning and end of the run whilst the rest of the observation was devoted to a cycle of 1.5 minutes on the phase calibrator and 5 minutes on GRS 1915+105.
Initial data editing and calibration were performed using the standard MERLIN d-programs and the data were then transferred to the NRAO Astronomical Image Processing System (AIPS). Within AIPS the data were processed via the MERLIN pipeline, which calibrates and images the phase reference source and then applies these solutions to the target source. This process also derives instrumental polarisation corrections and calibrates the linear polarisation position angle, using 3C286 as the calibrator and assuming a position angle of $33^{\circ}$ for its E vector. The position angles measured by MERLIN and ATCA are consistent with the same value, independently confirming the position angle calibration of each array.
Further self calibration was then carried out within AIPS and GRS 1915+105 imaged in total intensity and stokes Q and U. These maps were then combined using the AIPS task PCNTR to produce the final maps with total intensity contours and vectors denoting the direction and strength of linear polarisation.
Note that we can be confident both from previous studies (e.g. Mirabel & Rodriguez 1994; Fender et al. 1999) and these data (McCormick et al. in prep) that the component(s) to the south east (labelled in Fig 3 as ‘SE1’) is ‘approaching’, and component(s) to the north west are ‘receding’ (although in fact both sides of the jet have Doppler factors $\delta < 1$).
Variable circular polarisation
------------------------------
In both sets of ATCA observations, GRS 1915+105 is unambiguously detected as a source of circularly polarised radio emission (Stokes V).
### 2001 January 17
In 2001 January (Fig 2, left panels), significant CP is measured at all four ATCA frequencies, from 1–9 GHz. The total flux density is clearly declining, indicating the decay phase of a major flare, but there is also significant variability superposed on the relatively smooth decline, preferentially at higher frequencies. This is almost certainly indicative of repeated activity in the core, corresponding to fresh ejection events. The spectral indices support this interpretation; between 1.4–2.9 GHz the spectrum is significantly flatter than expected for optically thin synchrotron emission; between 4.8–8.6 GHz it is displaying the rapidly varying behaviour associated with ‘core’ ejection events (Fender et al. 2002a).
Inspection of the total flux and spectral index light curves indicates there were at least four separate ejection events contributing to the light curve at this epoch. Fig 1 also indicates that this outburst was more prolonged than that in 2001 March (see below).
The CP flux is clearly rising to lower frequencies, but the exact fractional spectrum is difficult to determine as the multiple components contributing to the observed emission are unresolved with ATCA. Table 1 lists the mean total, linearly polarised and circularly polarised flux densities, and Fig 4 plots these both as total and fractional spectra. We also note that there are measurements when the Stokes V flux is not significantly non-zero, and even a few points where it appears to have changed sign. However, (i) the mean Stokes V fluxes are significant, and negative (at both epochs), (ii) the apparent Stokes V sign change has a significance $<2\sigma$ and so we do not consider it convincing.
\[march01\]
------------- -------------- --------------- ------------------ -------------- --------------- ------------------
$\nu$ (MHz) I (mJy) LP (mJy) V (mJy) I (mJy) LP (mJy) V (mJy)
1384 $425 \pm 40$ $1.0\pm 0.2$ $-1.57\pm 0.21$
2496 $368 \pm 37$ $3.7 \pm 0.6$ $-0.96\pm 0.22$
4800 $262 \pm 26$ $8.5 \pm 1.1$ $-0.63 \pm 0.11$ $173 \pm 17$ $5.2 \pm 0.3$ $-0.56 \pm 0.07$
8640 $193 \pm 11$ $9.2 \pm 0.7$ $-0.48 \pm 0.10$ $113\pm 11$ $3.2\pm 0.2$ $-0.31 \pm 0.07$
------------- -------------- --------------- ------------------ -------------- --------------- ------------------
### 2001 March 23
In 2001 March, GRS 1915+105 was again observed to flare in our 15 GHz monitoring program, reaching $\sim 160$ mJy at MJD 51990.43. This time we triggered both ATCA and MERLIN – in fact the first epoch of MERLIN observations started at almost exactly the same time as the ATCA run concluded (Figs 3,5). As a result, we were able to definitively associate the outburst with relativistic ejections from the system (Fig 3).
The full polarisation ATCA data are presented in Fig 2 (right panels), and it is clear that there is less variability in the light curve than in 2001 January, with the smooth decay in radio flux at both frequencies only interrupted by the temporary increase around MJD 51991.82. Assuming the emission observed is associated with the radio event on which we triggered, our ATCA observations commence $\sim 1.4$ days after ejection (assuming a Doppler factor of $\sim
0.3$ – Fender et al. 1999 – this corresponds to only $\sim 10$ hr of evolution in the rest frame of the ejecta). The 4.8–8.6 GHz spectral index clearly demonstrates that the majority of the emission is coming from optically thin regions. The mean total intensity, linearly polarised and circularly polarised flux densities are presented in Table 1 and Fig 4.
A linear polarisation ‘rotator’ event
-------------------------------------
In the lower two panels of Fig 2 the linearly polarised flux densities and electric vector position angles are indicated. It is quite evident from the lower panel that over the $\sim 6$ hr of the observation in January 2001, both 4800 and 8640 MHz electric vectors rotate smoothly through $\sim 50$ degrees. Note that while there is evidence for changing absorption on the scales over which the ejecta from GRS 1915+105 can be tracked (Dhawan, Goss & Rodriguez 2000), variable Faraday rotation cannot be responsible for this effect since the [*separation*]{} between the vectors at the two frequencies remains constant (if Faraday rotation, which varies as $\lambda^2$, a rotation of $\sim 50$ degrees at 8.6 GHz should have a corresponding rotation of $\sim 100$ degrees at 4.8 GHz).
The smooth rotation in the electric vector position angle seems to be a little at odds with both the MERLIN observations of Fender et al. (1999) in which the electric vector was varying seemingly erratically from day-to-day, and the observations presented here (both ATCA and MERLIN) for 2001 March, in which the vector remains approximately constant. Such smooth rotation indicates either a genuinely rotating jet or, perhaps more likely, a smooth change in the (projected) position angle of the magnetic field in the emitting region – such as a global curved structure in the jet.
Similar behaviour, ‘polarisation rotator events’ – see Saikia & Salter (1988) and references therein - has also been observed in AGN. While initially interpreted as physical rotation of the magnetic field structure (which could directly link a jet to e.g. removal of angular momentum from an accretion flow) the more favoured interpretation is the formation of a shock inclined at some angle to the line of sight. The lack of the repeat of this phenomenon in 2001 March would seem to indicate it does not reflect physical rotation of the jet, which we would assume either always happens or never happens. However, more recently Gomez et al. (2001) have interpreted the steady rotation of the linear polarisation vector of a superluminal component in the jet of the AGN 3C120 as indicating an underlying twisted (helical) magnetic field structure.
Discussion
----------
The origin of a circularly polarised component in the radio emission from AGN and X-ray binaries remains uncertain. However, in both classes of object the bulk of the radio emission can be confidently assumed to arise from similar physical processes, namely synchrotron emission from relativistic electrons in a magnetised plasma flowing away from the central black hole (or neutron star) in collimated jets. Since we have every reason to believe that the Stokes I, and probably Q and U, fluxes from these objects arise via the same processes, we can hope that by studying the origin of Stokes V in X-ray binaries we may shed light on its origin in AGN, and vice versa.
### Association of CP with young ejections
What can we learn from these observations of GRS 1915+105 ? From analysis of the combined ATCA and MERLIN data sets for 2001 March, we are confident that the measured circularly polarised radio emission is associated with the relativistic ejection SE1. Our reasoning is as follows:
1. [The decreasing Stokes I flux measured at both epochs (most obviously 2001 March) arises from ejected components whose radio flux decays steadily, probably due to adiabatic expansion losses, with time. This can be inferred both from past experience and directly from the MERLIN observations which directly image the fading ejecta as they propagate away from the core. In Fig 5 we show the ATCA light curves, plus the total flux light curve from MERLIN and the fluxes of the two components (core and ejection SE1). The core stays at roughly the same flux level over all of the first five epochs of MERLIN imaging, whereas the flux of SE1 continues to fade – from this we can infer that the bright and decreasing flux observed by ATCA is dominated by emission from ejected component SE1.]{}
2. [There is a significant correlation between the Stokes I and (-)Stokes V fluxes, especially at 8640 MHz. For all epochs, and the combined data sets, there is a significant rank correlation between Stokes I and V at 8640 MHz; at 4800 MHz the correlation is marginal. Since the Stokes I, as argued above, is associated with SE1, we can therefore be confident that the Stokes V flux also arises primarily in SE1. Note that we cannot rule out a Stokes V flux of amplitude $|V| \leq 0.2$ mJy associated with the core, based on Fig 6.]{}
3. [Furthermore, the MERLIN imaging clearly shows that the linearly polarised radio emission also arises in component SE1. Therefore we are able to accurately measure the fractional linear and circular polarisation (ie. all Stokes parameters) for the synchrotron emission from a single optically thin component]{}
### Comparison of GRS 1915+105 with AGN, Sgr A\* and M81\*
Rayner et al. (2000) and Homan et al. (2001) have established that most AGN have ratios of linear to (absolute) circular polarisation $\geq 10$, whereas the low-luminosity radio cores Sgr A\* and M81\* have ratios $\leq 1$ (Bower et al. 1999; Sault & Macquart 1999; Brunthaler et al. 2001). It is interesting that for the two XRBs for which we have so far measured CP, we find both situations, depending on the frequency observed. In both SS 433 (Fender et al. 2000) and GRS 1915+105, LP $<$ CP at the lowest frequency (1.4 GHz), and LP $\geq$ CP at higher ($\geq 2.5$ GHz) frequencies. In GRS 1915+105, it is clear from Figs 2,4 that there is significant opacity (foreground or internal) at the lower frequencies, and so we consider it most likely that the reduction in LP is due to Faraday depolarisation. While in SS 433 there is less obvious indication of opacity at the lowest frequencies, it would seem likely that the same mechanism is operating there to reduce the observed LP. Furthermore, M81\* (Brunthaler et al. 2001) and a number of AGN (Rayner 2000) also seem to show flat/inverted fractional CP spectra, as we seem to have measured from GRS 1915+105 in 2001 January. However, we should also note that in Sgr A\* there is no optically thin power law component, unlike the two X-ray binaries discussed here, which will further reduce the expected LP (Bower et al. 1999).
In addition, the majority of AGN are found to maintain the same sign of Stokes V on timescales from months to decades (Komesaroff et al. 1984; Rayner et al. 2000; Homan et al. 2001). Bower et al. (2002) have more recently shown that Sgr A\* has maintained the same sign (and level) of CP over 20 years. In comparison, for X-ray binaries there are two observations of SS 433 separated by 10 days (Fender et al. 2000), and two observations of GRS 1915+105 separated by 65 days presented here; in both cases the sign remains the same. Of course the X-ray binary sample is extremely small, but should expand rapidly in the near future, with several events per year bright enough to mean CP at the $<0.1$% level (Fender & Kuulkers 2001). It is interesting to note that [*if*]{} accretion timescales scale linearly with mass from X-ray binaries to AGN (e.g. Sams, Eckart & Sunyaev 1996) then timescales of tens of days for SS 433 and GRS 1915+105 would correspond to timescales of thousands of years for Sgr A\* and millions of years for some AGN – ie. we may have already probed longer in ‘accretion time’ than in all the studies of AGN to date ! [*However*]{}, McCormick, Fender & Spencer (these proceedings) report an apparent secular sign-change in Stokes V from SS 433 over longer timescales than those reported in Fender et al. (2000), possibly indicating some long-term evolution of the magnetic field geometry (see also Ensslin, these proceedings).
### Origins of the CP component
The CP spectrum detected from GRS 1915+105 is rather similar to that observed from SS 433 (Fender et al. 2000), being observed over a broad range (1–9 GHz) and with a decreasing Stokes V (although not necessarily V/I) spectrum. The broadband nature of the CP spectrum suggests that coherent emission mechanisms are unlikely. Furthermore, we do not consider the birefringent scintillation mechanism of Macquart & Melrose (2000) very likely either, since the CP component seems to be associated with a physical event in the source, yet because of the high velocities of the ejecta it is likely to be a large distance from the source during the periods in which we measured CP.
This leads us (once again) to consider one of two mechanisms most viable, an intrinsic CP component to the synchrotron emission or LP$\rightarrow$CP conversion (‘repolarisation’). Do we have any evidence in favour of either of these ? The intrinsic synchrotron mechanism should, naively, produce a well-defined $\nu^{-0.5}$ V/I spectrum in a homogenous, optically thin, source. Our observations in 2001 March match these criteria quite closely (supported by direct imaging of a single ejection event with MERLIN), but unfortunately we only have a two-point CP spectrum at this epoch. In addition, the relatively low signal-to-noise ratio of the CP detections only allows us to constrain the (-V)/I spectral index to be $\alpha_{\rm -V/I} =
-0.3 \pm 0.3$. In 2001 January the situation is rather more complex, the mean flux and polarisation spectra certainly containing the contributions of multiple components with different optical depths. The data may suggest that the CP arises preferentially in ‘core’ components with the highest densities and optical depths, similar to the situation in AGN. This interpretation may favour instead the LP$\rightarrow$CP ‘repolarisation’ mechanism, which will operate most efficiently at higher optical depths, but currently the data are not sufficiently constraining.
However, both mechanisms require a significant population of low-energy electrons. In principle, a strong probe of this requirement could be obtained by measuring the low-frequency extent of radio emission during outbursts of GRS 1915+105 and other systems. Prompt observations at MHz frequencies could probe the electron distributions at Lorentz factors of 30 and lower (based on calculations in Fender et al. 1999) although, depending on the energy and magnetic field density in the ejecta, this emission may be self-absorbed. The consequences for the energetics of ejection events would be significant, especially in the case of a neutral baryonic plasma with one proton associated with each electron.
CP from other X-ray binaries
============================
In this section I shall present a brief compilation of measurements (in most cases limits) on radio CP from other X-ray binaries, and also compare the CP:LP ratio in X-ray binaries with that observed from blazars and low-luminosity AGN.
Measurements of other systems
-----------------------------
As well as the clear detections of CP from the three X-ray binary jet sources SS 433, GRO J1655-40 and GRS 1915+105 (see also McCormick et al., and Macquart, these proceedings, and references therein), there are also limits on CP from other X-ray binaries, some of which are stringent. In table 3 I list the measurements and limits on CP from X-ray binaries.
\[allsources\] Source Freq. (GHz) % CP Ref
----------------------- ------------- --------------- ------------------------ --
SS 433 1.384 $\sim$0.8 Fender et al. (2000)
2.497 $\sim$0.5
4.800 $\sim$0.3
8.640 $\sim$0.1
GRS 1915+105 1.384 $\sim$0.3 Fender et al. (2002b)
2.496 $\sim$0.2
4.800 $\sim$0.2–0.3
8.640 $\sim$0.2–0.3
GRO J1655-40 1.384 $\sim 0.2$ Macquart et al. (2002)
2.378 $\sim 0.2$
4.800 $<0.08$
8.640 $<0.1$
GX 339-4 8.64 $<0.7$ Corbel et al. 2000
Cir X-1 4.80 $<0.3$ Fender et al. in prep
8.60 $<0.3$
Cyg X-3 5.0 $<0.08$ de Bruyn (priv. comm.)
(Sept 2001)
V 4641 Sgr 4.80 $<0.04$ Sault (priv. comm.)
(2002) 8.64 $<0.04$
Cyg X-1 4.86 $<0.49$ Brocksopp, Fender,
8.46 $<0.33$ Bower & Clarke (2003)
14.94 $<0.63$
: Measurements and limits (3$\sigma$) on CP from radio-emitting X-ray binaries. All the systems are believed to host a black hole, except Cir X-1 which probably contains a neutron star, and Cyg X-3 for which there is little evidence either way.
It is interesting to note that the three sources from which CP has been strongly detected have jets which are close to the plane of the sky ($60^{\circ} < \theta < 90^{\circ}$). However, the jets from Cir X-1, V 4641 Sgr and Cyg X-3 are believed to be very close to the line of sight ($\theta < 15^{\circ}$). The limits on the CP from Cyg X-3 and V 4641 are particularly stringent. The limits obtained on CP in both cases are between 5–10 times below the levels of V/I detected from SS 433, GRS 1915+105 and GRO J1655-40. This indeed seems like a hint that CP, at least from X-ray binaries, is stronger when the jet is viewed approximately ‘side on’. However, it should also be noted that both sources show evidence (either on the specific occasion or others) for a strongly self-absorbed outburst, unlike e.g. SS 433 which is nearly always optically thin. In addition, the orientation effect would be rather contrary to the observations of AGN, in which strong CP is observed from ‘blazars’ with approximately face-on jets (e.g. Homan et al. 2001).
The CP:LP ratio and comparison with AGN
---------------------------------------
Both Homan et al. (2001) and Brunthaler et al. (2001) investigate the ratio of circular to linear polarisation. Specifically, Brunthaler et al. (2001) define R$_{\rm CL}$ as the ratio of fractional CP to fractional LP, and find that R$_{\rm CL} > 1$ for the two low-luminosity AGN (LLAGN) M81\* and Sgr A\* between 1–15 GHz (in fact no LP is detected at all from these sources). Homan et al., (2001) present 5 GHz measurements for many blazars, with significant measurements of both LP and CP for sixteen sources, and find for all of these that R$_{\rm CL} < 1$.
We have already seen (Fig 4) that for GRS 1915+105, at the lowest frequency R$_{\rm CL} > 1$, whereas for the other three higher frequencies R$_{\rm CL} < 1$. In Fig 7 I plot R$_{\rm CL}$ as a function of frequency for both GRS 1915+105 and SS 433, two epochs for each source. In both cases R$_{\rm CL} > 1$ for the lowest frequency (1.4 GHz) and R$_{\rm CL} < 1$ for the three higher frequencies. This seems to imply that whether R$_{\rm CL}$ is greater or less than unity depends on the degree of Faraday depolarisation in the emitting plasma. It should be noted that for the X-ray binary GRO J1655-40, which showed a high degree of linear polarisation, R$_{\rm CL} < 1$ at all frequencies (Macquart et al. 2002).
In Fig 8 I plot fractional CP against fractional LP for these two X-ray binaries, plus the two LLAGN and the most significant detections from the blazar sample of Homan et al. (2001). The two groups of AGN are separated by the line corresponding to LP=CP, whereas the X-ray binaries lie either side of the line depending on frequency. It is interesting to note that, since for both LLAGN there are only upper limits on LP, it is only for the low-frequency observations of the X-ray binaries that the exact value of R$_{\rm CL} < 1$, when it is less than unity, has been measured.
Conclusions
===========
Circularly polarised radio emission has been clearly detected from three X-ray binaries, all of which are associated with powerful jets which share many of the characteristics of AGN. In the cases of both GRS 1915+105 and GRO J1655-40, strong and variable circular polarisation was associated with clearly resolved ejection events. Comparing the fractional circular polarisation spectrum, and circular to linear polarisation ratio, clear similarities with AGN are noted. In particular, multi-frequency measurements of X-ray binaries, which reveal that the circular to linear polarisation ratio increases with wavelength, support interpretations in which the dominant factor for this ratio is the degree of Faraday depolarisation in the source.
The author would like to thank all the participants at the workshop in Amsterdam for many discussions, some useful, some useless but amusing. In addition, he would like to thank Ger de Bruyn and Bob Sault for providing information on unpublished results.
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---
abstract: |
Holographic representations of data enable distributed storage with progressive refinement when the stored packets of data are made available in any arbitrary order. In this paper, we propose and test patch-based transform coding holographic sensing of image data. Our proposal is optimized for progressive recovery under random order of retrieval of the stored data. The coding of the image patches relies on the design of distributed projections ensuring best image recovery, in terms of the $\ell_2$ norm, at each retrieval stage. The performance depends only on the number of data packets that has been retrieved thus far.
Several possible options to enhance the quality of the recovery while changing the size and number of data packets are discussed and tested. This leads us to examine several interesting bit-allocation and rate-distortion trade offs, highlighted for a set of natural images with ensemble estimated statistical properties.
address:
- 'Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel.'
- 'School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371.'
author:
- 'A. M. Bruckstein'
- 'M. F. Ezerman'
- 'A. A. Fahreza'
- 'S. Ling'
title: Holographic Image Sensing
---
holographic representation ,mean squared error estimation ,stochastic image data ,Wiener Filter.
Introduction {#sec:intro}
============
Holographic sensing and representations aim to capture and describe signals, images, videos and other information sources in a way that enables their recovery at various levels of resolution. The quality of the recovered data is dependent only on the size of the data portion that is available. Often, a mere preview of the visual information at some crude resolution level is sufficient to decide whether one needs to have the full detailed description. In such cases, holographic storage of the data is clearly beneficial.
Holographic representations of images were first proposed by Bruckstein [*et al.* ]{}in [@Bruckstein1998]. Their main ideas exploited the redundancy in images, based on either subsampling strategies or on Fourier transforms using a random phase mask, to ensure diffusion of low frequency information over all portions of the data. These ideas were then tested on individual images, but no theoretical evaluations on the expected achievable quality in the image recovery process were provided.
Several follow-up works, [*e.g.*]{}, [@Dovgard2004; @Dolev2010; @Dolev2012], came up with improved transforms for more efficient holographic encoding processes. Another approach, given in [@BHN00], proposed jittered quantization to achieve better precision with multiple instances of compressed image data exploiting differently quantized transform coefficients. A subsampling method for generating holographic data streams that ensure a uniform spatial spread of the sampled locations using low discrepancy sampling patterns can be found in [@Bruckstein2005].
In information theory, the issue of multiple representations has often been raised. However, rather few results are available in the literature for the case of many channels. Notable among them are the works of Goyal [*et al.* ]{}in [@GKAV98] and of Kutyniok [*et al.* ]{}in [@Kutyniok2009]. They deal with coding into packets of data when communication processes may drop some of the packets in unpredictable ways. An earlier work [@Vaish93] of Vaishampayan discussed multiple representations of a random variable via shifted or staggered quantization tables, a very nice idea resulting in better data fidelity as more and more quantized data are provided. Servetto [*et al.* ]{}continued on this trend of ideas in [@Servetto2000] by proposing some high-performance wavelet-based multiple description image coding algorithms.
Recently we provided an in-depth analysis of a holographic sensing paradigm in the classical setting of Wiener filtering in [@Bruckstein2019]. In our proposal, probings are performed by projection operators that eventually enable graceful successive refinement of random vector data. Our probing projection designs ensure the nice property that only the *number of available probings* determines the quality of the recovery. The *order* of their arrival is irrelevant. Another interesting recent approach to holographic coding of data combines standard data compression procedures with shifts in the image plane to obtain a set of compressed versions of images that form the distributed packets of data [@DB19]. In this case, again, random combinations of the compressed versions will yield improvement in the quality of the decompressed images, depending on the number of versions available.
In this paper, we describe in detail the design of probing projections, based on the statistics derived from a given data set of images in Sections \[sec:model\] and \[sec:design\]. The probings are guided either by the joint statistical properties of all the images in the set or by the properties of the patches of the image that is being considered for distributed holographic storage or communication. We regard images as ensembles of smaller patches and then design the probings to be optimized for the statistical properties of the patches, seen as realizations of a random process.
We ran our projection design process, along with the associated optimal bit allocation rule, on a set of natural images from various sources, described in Section \[sec:database\], and tested the image retrieval quality for various levels of noise, assumed to affect the holographic data acquisition process. The lowest noise level was actually modeling relatively noise-free projections while the higher levels were considerably deteriorating the projections that provide the distributed packets of images. The system that we implemented allows us to set the size of the image patches considered, the computation of the statistics for the patch ensemble to be encoded, and the number of holographic projections desired. Then, based on the ensemble’s autocorrelation’s spectrum of eigenvalues, we derive the projection operators and compute the predicted performance curves for recovery from any number of projections packets selected in either random or incremental orderings. The theoretical predictions are then compared to actual image recovery performance. We describe the implementation routines used and report the outcomes in Sections \[sec:comp\] and \[sec:control\].
The following notational conventions are used throughout. Let $0 \leq k < \ell $ be integers. Denote by ${\left\llbracket{\ell}\right\rrbracket}$ the set $\{1,2,\ldots,\ell\}$ and by ${\left\llbracket{k,\ell}\right\rrbracket}$ the set $\{k,k+1,\ldots,\ell\}$. Let $\bZ$, $\bN$ and $\bR$ denote, respectively, the set of integers, the set of positive integers, and the field of real numbers. Vectors are expressed as columns and denoted by bold lowercase letters. Matrices are represented by either bold uppercase letters or upper Greek symbols. An $n \times n$ diagonal matrix with diagonal entries $v_j: j \in {\left\llbracket{n}\right\rrbracket}$ is denoted by $\operatorname{diag}(v_1,v_2,\ldots,v_n)$. The identity matrix is $\BI$ or $\BI_n$ if the dimension $n$ is important. Concatenation of vectors or matrices is signified by the symbol $|$ between the components. The transpose of a matrix $\BA$ is $\BA^{\top}$.
Our Distributed Sensing Model {#sec:model}
=============================
The data of interest in our investigation is a random process, [*i.e.*]{}, a set of random column vectors $\{\bx_{\omega} : \omega \in \Omega \}$ indexed by a set $\Omega \subset \bN$. The process, of dimension $M$, is characterized by its expectation $\overline{\bx}$ and its $M \times M$ autocorrelation matrix $$\label{eq:Rxx}
\BR_{xx}:=\operatorname*{\mathbb{E}}\limits_{\{\omega \in \Omega\}} (\bx_{\omega}-\overline{\bx}) (\bx_{\omega}-\overline{\bx})^{\top}.$$ It is assumed that $\overline{\bx}$ is stored and available, and we deal with the centered case of $\overline{\bx} = \0$ and consider the diagonalization of $\BR_{xx}$, which is a positive definite symmetric matrix, with spectral decomposition given by $\BR_{xx} = \Psi \Lambda \Psi^{\top}$. Here $\Lambda= \operatorname{diag}(\lambda_1,\lambda_2,\ldots,\lambda_M)$, with $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_M > 0$, is a diagonal matrix and $\Psi = \left[\Psi_1 | \Psi_2 | \ldots | \Psi_M \right]$ is a unitary matrix whose columns, $\Psi_j$ for $j \in {\left\llbracket{M}\right\rrbracket}$, are the eigenvectors corresponding to the respective eigenvalues $\lambda_j$.
We denote by $\BP_{k}$ the projection matrix $\BU_k \BU_k^{\top}$, where $\BU_k$ is the $M \times m$ matrix that displays the $k^\text{th}$ element of a basis of the $m$-dimensional subspace of $\bR^{M}$ that $\BP_{k}$ projects onto. The assumed model to sense realizations of the process $\{ \bx_{\omega} : \omega \in \Omega\}$ is the set of $N \in \bN$ vectors $$\{\bz_k := \BU_k^{\top} \Psi^{\top} \bx_{\omega} + \bn_k : k \in {\left\llbracket{N}\right\rrbracket}\}.$$ Here $\{\Psi^{\top} \bx_{\omega}\}$ consists of column vectors with $M$ entries with autocorrelation $$\operatorname*{\mathbb{E}}_{\{\omega \in \Omega\}} [\Psi^{\top} \bx_{\omega} \bx_{\omega}^{\top} \Psi]
= \Psi^{\top} \BR_{xx} \Psi = \Lambda,$$ hence, the realizations of the random process $\{\Psi^{\top} \bx_{\omega}\}$ are vectors with uncorrelated entries having variances $\lambda_1, \lambda_2, \ldots, \lambda_M$. The noise vectors $\bn_k$ affecting the probings are realizations of a random process with zero mean and autocorrelation $\sigma_n^2 \BI_{m}$ and is assumed to be independent of $\bx_{\omega}$. This model of sensing implements, first, a *decorrelating transform coding* on the data vector $\bx_{\omega}$, then projects $\Psi^{\top} \bx_{\omega}$ via $\BU_k^{\top}$ onto an $m$-dimensional subspace of $\bR^M$ for a fixed $m <M$.
There are many possible ways to define the projections, [*e.g.*]{}, $\BU_k$ can be any $M \times m$ matrix whose $m$ column vectors, each having $M$ entries, are orthogonal. We choose $\BU_k$ to have columns selected from the standard basis $$\Big\{ \bb_1:=
\begin{bmatrix}
1 & 0 & 0 &\ldots & 0
\end{bmatrix}^{\top},
\bb_2:=
\begin{bmatrix}
0 & 1 & 0 & \ldots & 0
\end{bmatrix}^{\top},
\ldots,
\bb_M:=
\begin{bmatrix}
0 & \ldots & 0 & 1
\end{bmatrix}^{\top}
\Big\}$$ of $\bR^M$. In this case the projection $\BP_{k}$ is a diagonal $M \times M$ matrix with $m$ ones at specific locations on the diagonal as determined by the selection of the standard basis vectors in $\BU_k$. Thus, our sensing model provides $N$ vectors, each of dimension $m$, that convey the sensing information on the vectors $\Psi^{\top} \bx_{\omega}$.
When all $N$ vectors $\{\bz_k : k \in {\left\llbracket{N}\right\rrbracket} \}$ are available, we can recover $\bx_{\omega}$ by computing the best estimate under the given probing model at the noise level $\sigma_n^2$ in all of the observations. This recovery process provides the expected least-squared optimal solution for the recovery of the data $\{\bx_{\omega} : \omega \in \Omega\}$ when both the data and the noise, seen as random processes, are Gaussian.
The best estimator for $\bx_{\omega}$, given a [*partial*]{} set of $\ell$ out of $N$ packets $$\label{eq:sensed}
\Big\{ \bz_{k_j} = \BU_{k_j} \Psi^{\top} \bx_{\omega} + \bn_{k_j} : j \in {\left\llbracket{\ell}\right\rrbracket} \Big\}$$ is expressed, in the classical Wiener filter, by $$\label{eq:recovered}
\widehat{\bx}_{\omega} := \BR_{xz_{\mbox{combi}}} \left(\BR_{z_{\mbox{combi}} z_{\mbox{combi}}}\right)^{-1} \bz_{\mbox{combi}}$$ from the observation $$\bz_{\mbox{combi}}=
\underbrace{\begin{bmatrix}
\bz_{k_1}\\
\bz_{k_2}\\
\ldots\\
\bz_{k_{\ell}}
\end{bmatrix}}_{(\ell \cdot m) \times 1}
=\underbrace{\begin{bmatrix}
\BU_{k_1}^{\top}\\
\BU_{k_2}^{\top}\\
\ldots\\
\BU_{k_{\ell}}^{\top}\\
\end{bmatrix}}_{(\ell \cdot m) \times M} \Psi^{\top} \bx_{\omega} +
\underbrace{\begin{bmatrix}
\bn_{k_1}\\
\bn_{k_2}\\
\ldots\\
\bn_{k_{\ell}}\\
\end{bmatrix}}_{(\ell \cdot m) \times 1}.$$ The matrices $\BR_{xz_{\mbox{combi}}}$ and $\BR_{z_{\mbox{combi}} z_{\mbox{combi}}}$ are readily obtained, respectively, as follows $$\begin{aligned}
\BR_{xz_{\mbox{combi}}} &= \operatorname*{\mathbb{E}}_{\{\omega\}} \left[\bx_{\omega} \bz_{\mbox{combi}}^{\top}\right] =
\BR_{xx} \left[ \Psi \BU_{k_1} | \Psi \BU_{k_2} | \ldots | \Psi \BU_{k_{\ell}} \right]\\
&= \BR_{xx} \Psi \left[\BU_{k_1} | \BU_{k_2} \ldots | \BU_{k_{\ell}} \right], \mbox{ and}
\\
\BR_{z_{\mbox{combi}} z_{\mbox{combi}}} &= \operatorname*{\mathbb{E}}_{\{\omega\}} \left[\bz_{\mbox{combi}} \bz_{\mbox{combi}}^{\top}\right] =
\begin{bmatrix}
\BU_{k_1}^{\top}\\
\BU_{k_2}^{\top}\\
\ldots\\
\BU_{k_{\ell}}^{\top}\\
\end{bmatrix}
\Lambda
\left[\BU_{k_1} | \BU_{k_2} \ldots | \BU_{k_{\ell}} \right] + \sigma_n^2 \BI_{(\ell \cdot m)}.\end{aligned}$$ Since $\BR_{xx} = \Psi \Lambda \Psi^{\top}$, the best reconstructed $\bx_{\omega}$ from the given data is therefore $$\label{eq:bestrec}
\widehat{\bx}_{\omega} =
\Psi \Lambda \left[\BU_{k_1} | \BU_{k_2} \ldots | \BU_{k_{\ell}} \right]
\left(\begin{bmatrix}
\BU_{k_1}^{\top}\\
\BU_{k_2}^{\top}\\
\ldots\\
\BU_{k_{\ell}}^{\top}\\
\end{bmatrix}
\Lambda
\left[\BU_{k_1} | \BU_{k_2} \ldots | \BU_{k_{\ell}} \right] + \sigma_n^2 \BI_{(\ell \cdot m)}\right)^{-1} ~ \bz_{\mbox{combi}}
$$ The expected squared-error covariance in the reconstruction is therefore given by $$\begin{aligned}
\label{eq:Ree}
\BR_{ee} &= \operatorname*{\mathbb{E}}_{\{\omega\}} \left[\left(\bx_{\omega} - \widehat{\bx}_{\omega}\right)
\left(\bx_{\omega} - \widehat{\bx}_{\omega} \right)^{\top} \right]
=\left(\BR_{xx}^{-1}+\frac{1}{\sigma_n^2} \Psi
\left[\BU_{k_1} | \BU_{k_2} | \ldots | \BU_{k_{\ell}}\right]
\begin{bmatrix}
\BU_{k_1}^{\top}\\
\BU_{k_2}^{\top}\\
\ldots\\
\BU_{k_{\ell}}^{\top}\\
\end{bmatrix}
\Psi^{\top} \right)^{-1}
\\ \notag
& = \left(\Psi \Lambda^{-1} \Psi^{\top}+ \frac{1}{\sigma_n^2}
\Psi \sum_{j =1}^{\ell} \BP_{k_j} \Psi^{\top}\right)^{-1}=
\Psi \left(\Lambda^{-1}+\frac{1}{\sigma_n^2} \sum_{j =1}^{\ell} \BP_{k_j}\right)^{-1} \Psi^{\top},\end{aligned}$$ where $\BP_{k_j} = \BU_{k_j} \BU_{k_j}^{\top}$. Our choice of the basis for $\bR^{M}$ implies $\BU_{k_j}:=\left[ \bb_{t_1} | \bb_{t_2} | \ldots | \bb_{t_m} \right]$ is a selection of $m$ out of $M$ standard basis vectors of $\bR^{M}$. This structure yields an explicit form for the expected mean squared error ($\operatorname{MSE}$) since we have $$\sum_{j =1}^{\ell} \BP_{k_j} = \operatorname{diag}(s_1,s_2,\ldots,s_M),$$ where $s_i$ is the total number of times that the basis vector $\bb_i$ appears in the combined matrix $\left[\BU_{k_1} | \BU_{k_2} | \ldots | \BU_{k_{\ell}}\right]$. Thus, the error covariance in Equation (\[eq:Ree\]) becomes $$\begin{aligned}
\BR_{ee} &= \Psi \left(\Lambda^{-1} + \frac{1}{\sigma_n^2} \operatorname{diag}(s_1,s_2,\ldots,s_M) \right)^{-1} \Psi^{\top}\\
&= \Psi ~\operatorname{diag}\left(\frac{\lambda_1}{1+\frac{\lambda_1}{\sigma_n^2} s_1} ,
\frac{\lambda_2}{1+\frac{\lambda_2}{\sigma_n^2} s_2}, \ldots, \frac{\lambda_M}{1+\frac{\lambda_M}{\sigma_n^2} s_M}\right)~\Psi^{\top}.\end{aligned}$$ The total $\operatorname{MSE}$ in estimating $\bx_{\omega}$ is the trace of the error covariance matrix, [*i.e.*]{}, $$\operatorname{MSE}_{\mbox{total}} = \operatorname{Tr}(\BR_{ee})=\sum_{i=1}^{M} \frac{\lambda_i}{1+\frac{\lambda_i}{\sigma_n^2} s_i},$$ where $s_i$ counts the number of times the $i$^th^ entry of $\Psi^{\top} \bx_{\omega}$ has been probed in the sensing process. In summary, we have explicit expressions for the $\operatorname{MSE}$ incurred in the recovery procedure from any set of combined sensing results. We can therefore use this result to guide the design of the actual sensing process. The sensing process yields $N$ vectors of length $m$ to be distributed as packets of acquired sensing results. As mentioned earlier, the data consist of random vectors of $M$ entries with zero mean and known autocorrelation of a stochastic process. We now turn our focus to the design of the $N$ projections $\BP_{1}, \BP_{2}, \ldots, \BP_N$, [*i.e.*]{}, on selecting the matrices $\BU_{1}, \BU_{2}, \ldots, \BU_N$ that satisfy the following holographic criteria. First, the distributed restoration must have the progressive refinement property as more packets are utilized. Second, the process must be smooth in the sense that it achieves maximal uniformity and low $\operatorname{MSE}$ estimation when any $\ell$ out of $N$ packets of representation are made available for every $\ell \in {\left\llbracket{N}\right\rrbracket}$.
The Holographic Sensing Design {#sec:design}
==============================
The design is presented here in three subsections. First, we think of the design as a general resource allocation problem, where we are given some number $K$ of probings to allocate. Second, we include several practical considerations. Third, we propose a simple way to build the projection matrices for the actual sensing.
The Optimal Probing Trade-Offs {#subsec:optprobe}
------------------------------
We regard the problem of sensing design as a resource allocation, [*i.e.*]{}, rate distortion trade-off process. We have random vectors $\by_{\omega} = \Psi^{\top} \bx_{\omega} \in \bR^M$ with uncorrelated entries of variances $\lambda_1,\ldots, \lambda_M$ and we can probe batches of $m$ entries, with some small additive independent noise of zero mean and variance $\sigma_n^2$. We want to recover $\bx_{\omega}$, from a certain number of $m$-probes, via the optimal Wiener filter estimation process. It turns out that the combined total expected mean squared error in estimating the entries of $\bx_{\omega}$ depends on the number of times $s_1,s_2,\ldots,s_M$ each entry is probed since $$\label{eq:MSE}
\operatorname{MSE}\left(\Lambda, \sigma_n^2, \{s_1,s_2,\ldots,s_M\}\right) =
\sum_{i=1}^{M} \frac{\lambda_i}{1+\frac{\lambda_i}{\sigma_n^2} s_i}.$$ A natural question arises. How should we design the probes so as to obtain the smallest $\operatorname{MSE}$, given that we select $m$ locations to probe with each $\BU_k \BU_k^{\top} = \BP_k$ projection matrix?
It is clear that if we have $N$ projections and each of them selects $m$ locations, then we have $\sum_{i=1}^M s_i = N \cdot m$. For any subset of size $\ell$ out of the $N$ projections, [*i.e.*]{}, for $\BU_{k_1}, \BU_{k_2}, \ldots, \BU_{k_{\ell}}$ where $k_j \in {\left\llbracket{N}\right\rrbracket}$, we have a total of $\ell \cdot m$ probings. This leads us to consider the following questions.
1. What is the lowest $\operatorname{MSE}$ attainable with a given total number of $K$ probings?
2. What is the optimal distribution of probings, [*i.e.*]{}, the allocation of the $\ell \cdot m$ probings in total to the set $\{s_i\}$ of nonnegative integers that yields the lowest $\operatorname{MSE}$?
We are thus led to the following optimization. Let $K$ denote the total number of probings to be suitably distributed into $\ell$ probing projections. Given $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_M > 0$ and $\sigma_n^2$, minimize, over all suitable $\{s_j : j \in {\left\llbracket{M}\right\rrbracket}\}$ allocations, the $\operatorname{MSE}$ in Equation (\[eq:MSE\]), subject to the requirements that $0 \leq s_j \leq N$ must be an integer for all $j \in {\left\llbracket{M}\right\rrbracket}$ and $\sum_{j=1}^M s_j = K$.
To address this resource allocation problem, we begin with the last condition on $s_j$, [*i.e.*]{}, we relax the optimization over $\bR$, instead of $\bZ$. To make this relaxation clear, we replace $s_j \in \bZ$ by $\zeta_j \in \bR$. This reduces the problem into a straightforward Lagrangian optimization problem, requiring us to solve $$\label{eq:Lagrange}
\frac{\partial}{\partial \zeta_j}
\left(
\sum_{j=1}^{M} \frac{\lambda_j}{1+\frac{\lambda_j}{\sigma_n^2} \zeta_j} + \gamma
\left(K - \sum_{j =1}^M \zeta_j \right)\right)=0.$$ Solving it yields, for each $j \in {\left\llbracket{M}\right\rrbracket}$, $$\frac{\lambda_j^2 \cdot \sigma_n^2} {\left(\sigma_n^2 + \lambda_j \zeta_j \right)^2} = \gamma
\iff
\left(\sigma_n^2 + \lambda_j \zeta_j\right)^2 = \frac{1}{\gamma} \lambda_j^2 \cdot \sigma_n^2,$$ which implies $$\sigma_n^2 + \lambda_j \zeta_j = \frac{1}{\sqrt{\gamma}} \lambda_j \cdot \sigma_n.$$ Thus, the solution to the optimization problem is the set $$\label{eq:SSet}
\bigg\{\zeta_j = \frac{\sigma_n}{\sqrt{\gamma}} - \frac{\sigma_n^2}{\lambda_j} : j \in {\left\llbracket{M}\right\rrbracket}\bigg\}.$$ Now, adding the constraint $\sum_{k=1}^M \zeta_k =K$ means $$\begin{aligned}
\frac{\sigma_n}{\sqrt{\gamma}} \cdot M - \sigma_n^2 \sum_{k=1}^M \frac{1}{\lambda_k} = K
& \iff
\frac{1}{\sqrt{\gamma}} = \frac{1}{M \cdot \sigma_n} \left(K + \sigma_n^2 \sum_{k=1}^M \frac{1}{\lambda_k}\right) \\
& \iff
\frac{1}{\sqrt{\gamma}} = \frac{K}{M \cdot \sigma_n} + \frac{\sigma_n}{M} \sum_{k=1}^M \frac{1}{\lambda_k}.\end{aligned}$$ Thus, we obtain $$\label{eq:KSset}
\zeta_j = \frac{K}{M} + \sigma_n^2 \left( \frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_j}\right).$$
The result we have just obtained is quite nice. The $\zeta_j$s have to be distributed about the even distribution of the $K$ probings to the $M$ entities in the probed vector, [*i.e.*]{}, in the ratio $\frac{K}{M}$ according to how far their inverse variance is from the average inverse variances of all entries. The distribution, in other words, depends on the [**harmonic mean of the variances**]{}. Clearly, this result has not yet ensured that the other conditions imposed on the $s_j$s, namely $s_j$ must be a nonnegative integer, upper bounded by $N$, for each $j$. Looking at the solutions for $\zeta_j$s, however, we realize that the function $$\label{eq:rho}
\rho(j) = \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_j}$$ is a decreasing function of $j$. We therefore need to have $$\begin{aligned}
\label{eq:K1}
\frac{K}{M} + \sigma_n^2 \left( \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_M} \right) \geq 0 &\iff
\frac{K}{M} \geq \sigma_n^2 \left( \frac{1}{\lambda_M} - \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} \right) \notag\\
& \iff K \geq M \cdot \sigma_n^2 \left( \frac{1}{\lambda_M} - \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} \right)\end{aligned}$$ to guarantee that $\zeta_M$ and, hence, $\zeta_1, \ldots, \zeta_{M-1}$ will all be nonnegative. If the above conditions are satisfied, then we will also have $$\label{eq:K2}
\zeta_1 = \frac{K}{M} + \sigma_n^2 \left( \frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_1}\right) \leq K.$$ To see this, we notice that $$\frac{1}{\lambda_1} + (M-1) \frac{1}{\lambda_M} \geq \sum_{k =1}^M \frac{1}{\lambda_k}
= \frac{M-1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} +
\frac{1}{M}
\sum_{k =1}^M \frac{1}{\lambda_k}$$ implies $$(M-1) \left(\frac{1}{\lambda_M} - \frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k}\right) \geq
\frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_1}.$$ Hence, $$\frac{1}{\lambda_M} - \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} \geq \frac{1}{M-1}
\left(\frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_1} \right).$$ Substituting this last expression into the inequality in Equation (\[eq:K1\]) gives us $$K \geq \frac{M}{M-1} \cdot \sigma_n^2 \left(\frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_1} \right),$$ which becomes the inequality in Equation (\[eq:K2\]) after a simple rearrangement. Thus, we have determined a condition on $K$ to ensure the nonnegativity of all the optimal values assigned to the nonincreasing sequence of real values $\zeta_1, \zeta_2,\ldots, \zeta_m$ whose sum is $K$.
Reflecting on our journey so far, minimizing the total $\operatorname{MSE}$ in estimating $\by_{\omega}$, hence $\bx_{\omega}$, from $K$ probings of its entries is achieved by distributing the probing projections according to the set of integers $s_1,s_2,\ldots,s_M$ determined via $\displaystyle{
\min_{\{s_j\}} \sum_{j =1}^M \frac{\lambda_j}{1+ \frac{\lambda_j}{\sigma_n^2} s_j}}$ such that $\sum_{j =1}^M s_j = K$. The minimal value is attained when the integer $s_j$, for each $j \in {\left\llbracket{M}\right\rrbracket}$, is “near” the real positive value $$\label{eq:zeta}
\zeta_j = \frac{K}{M} + \sigma_n^2 \left(\frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_j}\right) \in \bR.$$ Note that by using the real values of $\zeta_j$s to compute for $\operatorname{MSE}$ in Equation (\[eq:MSE\]) we obtain $$\begin{aligned}
\label{eq:bestMSE}
\operatorname{MSE}_{\mbox{best}}&=
\sum_{j=1}^M \frac{\lambda_j}{1 + \frac{\lambda_j}{\sigma_n^2}
\left(\frac{K}{M} + \sigma_n^2 \left(\frac{1}{M} \sum_{k =1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_j}\right)\right)}
= \sum_{j=1}^M \frac{\lambda_j}{\frac{\lambda_j}{\sigma_n^2} \frac{K}{M} +
\lambda_j \left(\frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k}\right)} \notag \\
&= \sum_{j=1}^M \frac{1}{\frac{K}{M \cdot \sigma_n^2} + \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k}} = M \cdot \frac{\sigma_n^2}{\frac{K}{M} + \sigma_n^2 \left(\frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k}\right)}.\end{aligned}$$ Thus, the optimal assignment is achieved when all of the errors in estimating the components of the random vector $\by_{\omega}$ are equalized. This is a very natural and oft-encountered condition in distributed estimation. Keep in mind that this is the case when $K$ is large enough to make all of the $s_j$s nonnegative, [*i.e.*]{}, $\displaystyle{K \geq M \cdot \sigma_n^2 \left(\frac{1}{\lambda_M} - \frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k}\right)}$ from Equation (\[eq:K1\]). Such a $K$ ensures that the best $\operatorname{MSE}$ in estimating [*each component*]{} of $\by_{\omega}$ is bounded above by $\lambda_M$, [*i.e.*]{}, it stays no more than the smallest variance $\lambda_M$ in the uncorrelated random vector $\by_{\omega}$.
One can plot the function $\rho(j)=\frac{1}{M} \sum_{k=1}^M \frac{1}{\lambda_k} - \frac{1}{\lambda_j}$ (see Equation (\[eq:rho\])), given the $\lambda_j$ values. The curve displaying the allocation of $\zeta_j$ is then influenced by $\frac{K}{M}$, which serves as the constant level around which each $s_j$ lives, starting above it before decreasing as $j$ goes to $M$. The shape is determined by $\sigma_n^2 \cdot \rho(j)$ where $\rho(j)$ depends only on the diagonal entries $\lambda_1,\lambda_2, \ldots, \lambda_M$ of $\Lambda$. A plot of $\rho(j)$ based on actual $\lambda_j$ values calculated from a set of images, alongside the plots of $\zeta_j$ for several values of $\sigma_n^2$, can be found in Figure \[fig:rho\_zeta\].
Practical Calibrations
----------------------
Given the nonincreasing sequence of variances $S:=\lambda_1, \lambda_2, \ldots, \lambda_M$, the probing distribution is governed by the function $\rho(j)$ in Equation (\[eq:rho\]). Letting $\Delta := \frac{1}{M} \sum_{j =1}^M \frac{1}{\lambda_j}$, [*i.e.*]{}, $\Delta$ is the [*harmonic mean*]{} of the elements in $S$, we write $\rho(j) = \Delta - \frac{1}{\lambda_j}$. Together with the assumed noise level and the total number $K$ of probings, $\rho(j)$ provides us with the resource allocation strategy that achieves the best $\operatorname{MSE}$ performance.
In practical situations the sequence $S$ of $\lambda_j$s, the noise’s standard deviation $\sigma_n^2$, and the total number of probings $K$ are given to us as system parameters. It may happen, therefore, that $K$ is not large enough or $\sigma_n^2$ not small enough to ensure that all $\zeta_j$s are nonnegative. When this is the case, we shall allocate no probings for a subsequence of small $\lambda_j$s at the tail end of $S$. We accomplish this by repeatedly solving the probing allocation problem for leading subsequences of $S$, say for $\lambda_1, \lambda_2, \ldots \lambda_L$ for an $L < M$. We recalculate $\zeta_1,\zeta_2,\ldots,\zeta_L$ for various $L$ until we arrive at the [*largest*]{} $L$ that makes $$\label{eq:indexL}
\zeta_{i} := \frac{K}{L} + \sigma_n^2 \left(\frac{1}{L} \sum_{j=1}^L \frac{1}{\lambda_j} - \frac{1}{\lambda_i}\right) > 0 \mbox{ for all } i \in {\left\llbracket{L}\right\rrbracket},$$ implying $s_{i} \geq 0$, and fix $\zeta_{L+1} = \ldots = \zeta_M=0$. The formula in Equation (\[eq:indexL\]) is derived by solving the suitably modified optimization problem (see Equation (\[eq:Lagrange\])) $$\begin{aligned}
\label{eq:MSE_L}
\theta\left(\zeta_1,\ldots,\zeta_L,0,\ldots,0\right) &=
\sum_{j=1}^{M} \frac{\lambda_j}{1+\frac{\lambda_j}{\sigma_n^2} \zeta_j} + \gamma
\left(\sum_{j =1}^M \zeta_j - K\right) \notag \\
& =
\sum_{j=1}^{L} \frac{\lambda_j}{1+\frac{\lambda_j}{\sigma_n^2} \zeta_j} +
\sum_{j=L+1}^M \lambda_j +
\gamma \left(\sum_{j =1}^L \zeta_j - K\right).\end{aligned}$$
The allocation of the $K$ probings is then done to the leading $L$ entries of the vector $\by_{\omega}$ to achieve the goal of having holographic reconstructions for $K = N \cdot m$. In our setup, $N$ is the number of packets created by the projections for each $\by_{\omega}$ and $m$ is the fixed size of each probing packet, [*i.e.*]{}, the number of entries in each packet, which is the dimension of the image vector after a projection. These $N$ packets are to be stored or distributed in the environment or sent via some lossy communication channel that drops packets or randomizes their delivery time.
For practical examples that we encounter in the statistics of natural image patches, the elements in $S$ are highly skewed towards the leading part, [*i.e.*]{}, the first variance $\lambda_1$ is, relative to the rest of the entries, much larger. The next few variances tend to be high, then all the remaining variances decrease rapidly as the index goes to $M$. We use the following three modes of determining $s_j$ from $\zeta_j$ for $j \in {\left\llbracket{L}\right\rrbracket}$. Their respective advantages become apparent in their actual deployment.
1. As much as possible, this mode assigns $s_j$ to be the closest integer to $\zeta_j$ for $j \in {\left\llbracket{M}\right\rrbracket}$. It starts by listing the absolute distances from $\zeta_j$ to its nearest integer and sorts these distances from smallest to largest, keeping tab of the respective indices. Guided by this sorted list, the mode assigns $s_j$ to be the nearest integer to $\zeta_j$, choosing between ${\left\lfloor{\zeta_j}\right\rfloor}$ and $\lceil \zeta_j \rceil$, as $i$ goes through the indices in the list. The enforcement decreases in priority, [*i.e.*]{}, it may reverse the assignment to keep the condition $\sum_{j =1}^M s_j = K$ satisfied.
2. The mode starts by letting $s_j := {\left\lfloor{\zeta_j}\right\rfloor} + \epsilon_j$ for all $j \in {\left\llbracket{M}\right\rrbracket}$. Since $\lambda_1$ has the highest variance coefficient, we assign a new value $s_1 \gets \min\left(N, s_1 + \sum_{j =1}^M \epsilon_j\right)$. With the updated $s_1$, let $\delta := K - \sum_{j =1}^M s_j$. If $\delta > 0$, then assign $s_2 \gets \min\left(N, s_2 + \delta\right)$. One repeats this process until there is no more left-over quantity to assign.
3. This mode simply assigns $s_1=\ldots=s_m=N$ and $s_k=0$ for $k > m$.
Designing the Packet Projections
--------------------------------
After obtaining the optimal allocation that yields the best distribution of the given $K$ probings in terms of minimizing the $\operatorname{MSE}$, [*i.e.*]{}, having $$s_1 \geq s_2 \geq \ldots \geq s_L \geq 0 \mbox{ and } s_{L+1}= \ldots = s_{M}=0$$ for some $1 < L < M$, we need to probe only $L$ first entries of $\by_{\omega}$. Recall that the vector has a given autocorrelation $\BR_{yy} = \Lambda$. We now proceed to designing the $N$ packets, each of size $m$, that will optimally distribute the $K=N \cdot m$ probings with the goal of reaching the best-possible recovery with the desired progressive refinement property.
For $\ell$ out of the total $N$ packets, the number of available probings is $K_{\ell} := \ell \cdot m$. We combine Equation (\[eq:MSE\]) and the optimal $\zeta_j$s to infer that the best theoretically possible expected error for $\ell$ packets is given by $$\label{eq:best_ell}
\operatorname{MSE}_{\mbox{best}, \ell} = \sum_{j=1}^L \frac{\lambda_j}{1 + \frac{\lambda_j}{\sigma_n^2} \zeta_j} + \sum_{j=L+1}^M \lambda_j.$$ Given specific $\Lambda$, $m$, and $\sigma_n^2$, we can explicitly compute $\operatorname{MSE}_{\mbox{best}, \ell}$ for each $\ell \in {\left\llbracket{N}\right\rrbracket}$. Plots for actual images are given in Section \[sec:comp\].
We shall never attain this optimal performance for every $\ell$-set of projections, since we optimize the allocation of $s_j$s for best recovery when [*all*]{} packets are available. We should, however, strive to design the $N$ packets of projections in such a way that a random selection of $\ell$ out of $N$ packets, [*i.e.*]{}, selecting any one of the $\binom{N}{\ell}$ possible combinations of $\ell$ distinct packets, exhibits *smoothness*. In our context, it means that we recover the vector $\by_{\omega}$ as uniformly well as possible and as close as it can be to the $\operatorname{MSE}_{\mbox{best}, \ell}$ value.
Given the sequence $S$ whose elements are the diagonal entries of $\Lambda$, $\sigma_n^2$, and $K = N \cdot m$, we want to find the best way in assigning the projections into $N$ boxes satisfying the following conditions.
1. The projections sample the $\nth{1}$, the $\nth{2}$, and so on, up to the $L \textsuperscript{th}$ entry of $\by_{\omega}$.
2. The $s_j : j \in {\left\llbracket{L}\right\rrbracket}$ add up, by design, to a total of $N \cdot m$ projections, [*i.e.*]{}, $$s_1 + s_2 + \ldots + s_L = N \cdot m.$$
3. Choices of any $\ell$ out of $N$ boxes, for each $\ell \in {\left\llbracket{N}\right\rrbracket}$, yield nearly as good a recovery of $\by_{\omega}$ as possible, without choices that yield significantly bad mean squared errors.
We adress the problem as follows. Every projection $\BP_{k} = \BU_k \BU_k^{\top}$ is determined by a set of $m$ basis vectors selected from the standard basis $\{\bb_1, \bb_2, \ldots,\bb_M\}$ of $\bR^M$. The numbers $s_1,s_2,\ldots,s_L$ tell us how many times each vector $\bb_i$ will appear in the combined $N$ projections, written as $\sum_{k=1}^N \BP_k$. The key to a good projection is to have as many low-indexed probings as possible since the various projections have different influence on the expected mean squared error of recovery. However, we want every pair of projections to provide similarly good results. The same should hold for any triplet, any quadruplet, and so on, of projections.
There are several options to distribute the $N \cdot m$ balls of $L$ different colors into $N$ boxes, each with capacity $m$. There are $s_j$ balls of label $j$ for each $j \in {\left\llbracket{L}\right\rrbracket}$ $$\underbrace{1 \qquad 1 \qquad \ldots \qquad 1}_{s_1 \mbox{ balls labelled } 1} \qquad
\underbrace{2 \qquad 2 \qquad \ldots \qquad 2}_{s_2 \mbox{ balls labelled } 2} \qquad
\ldots \ldots \qquad
\underbrace{L \qquad L \qquad \ldots \qquad L}_{s_L \mbox{ balls labelled } L}.$$ Here is a simple yet effective distribution process. What we want is to have, in each of the $N$ boxes, balls with low labels. Such balls contribute the most to the [**reduction**]{} of the expected $\operatorname{MSE}$ since they correspond to probing the low-indexed components of $\by_{\omega}$, which have the highest variances.
We mark the $N$ boxes $B_1$ to $B_N$ and stack them vertically with $B_1$ on top and $B_N$ at the bottom. We then distribute the $s_1$ balls, labelled $1$, one by one to boxes $B_1$ to $B_{s_1}$. As each ball is distributed, move the just-filled box from the very top to the bottom. Once all of these $s_1$ balls are taken care of, the (still empty) top box is now $B_{s_1+1}$ and the bottom box is $B_{s_1}$. Repeat the distribution process on the remaining balls, with the boxes moved from top to bottom as before. Notice that at every instance after the top box is moved to the bottom position, the boxes are ordered in decreasing $\operatorname{MSE}$, with the box on top corresponding to the highest current $\operatorname{MSE}$, if the recovery process is to commence immediately. This way of allocating the probings to the projection boxes will eventually result in a complete allocation of $K = N \cdot m$ probings to the $N$ boxes while making all of these $N$ boxes as similar as possible in terms of their respective $\operatorname{MSE}$ measure of recovery quality.
The Image Data Set Used for Testing {#sec:database}
===================================
---------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
![Six of the images from the image data set, to be used as illustrations.[]{data-label="fig:images"}](dragon.png "fig:"){width="0.4\linewidth"} ![Six of the images from the image data set, to be used as illustrations.[]{data-label="fig:images"}](flood.jpg "fig:"){width="0.4\linewidth"}
\(a) [dragon]{} \(b) [flood]{}
![Six of the images from the image data set, to be used as illustrations.[]{data-label="fig:images"}](merlion.jpg "fig:"){width="0.4\linewidth"} ![Six of the images from the image data set, to be used as illustrations.[]{data-label="fig:images"}](oldlady.jpg "fig:"){width="0.4\linewidth"}
\(c) [merlion]{} \(d) [oldlady]{}
![Six of the images from the image data set, to be used as illustrations.[]{data-label="fig:images"}](mandrill.png "fig:"){width="0.35\linewidth"} ![Six of the images from the image data set, to be used as illustrations.[]{data-label="fig:images"}](lena.jpg "fig:"){width="0.35\linewidth"}
\(e) [mandrill]{} \(f) [lena]{}
---------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
For a software implementation in Section \[sec:comp\], we use a data set of $49$ images available at <https://github.com/adamasstokhorst/holographic/tree/master/img>. Most of them are either part of the standard images used in image processing or taken from free-to-use online repositories <https://www.pexels.com/> and <https://pixabay.com/>, with no attributions required. Figure \[fig:images\] presents $6$ of the $49$.
The image [flood]{} in Figure \[fig:images\] (b), was taken by Fardin Oyan. Originally titled [*Happiness on a Rainy Day*]{}, it won him the Young Environmental Photographer of the Year 2018 award. Two images, namely [oldlady]{} in Figure \[fig:images\] (d) and [buffalo]{} were included in the data set with the permission of their photographer Rarindra Prakarsa. The data set also contains two iconic images, used under the “fair use principle” for academic purposes. The award-winning photographer Steve McCurry shot [*Procession of Nuns*]{} in Rangoon, Burma (now Yangon, Myanmar), in 1994. We call the image [monks]{}, for brevity. The image [refugee]{} of a Syrian refugee carying his daughter while crossing the border of Macedonia and Greece in 2015 was taken by the late, decorated photographer, Yannis Behrakis. A few of the images, [*e.g.*]{}, [dragon]{} in Figure \[fig:images\] (a), belong to the corresponding author.
A Software Implementation {#sec:comp}
=========================
To demonstrate the efficacy and versatility of holographic sensing on images, we design and thoroughly test an implementation of a distributed image sensing system. The images are then shown to be progressively recovered with the recovery being insensitive to and independent of the order in which the packets became available. Our software is written in [pyton 2.7]{} with [numpy]{} [@numpy; @Oli06], Python Imaging Library [PIL]{} [@PIL], and [matplotlib]{} [@matplotlib; @Hunter07], as the minimum required libraries.
The software implementation routines fall into three types, namely preparatory routines, recovery routines, and performance analysis routines. We welcome readers who are interested to implement the tools for themselves to access the source files at <https://github.com/adamasstokhorst/holographic>. Users can tweak the input parameters to better suit their favourite implementation scenarios.
Preparatory Steps
-----------------
In the preprocessing stage, the system analyzes a database of images to obtain the necessary statistical data to use in the sensing design. Dealing with typically large size images, as is customary, we break them up into small patches of $r \times r$ pixels, where $r$ can be $4$, $8$, or $16$. Each two-dimensional image is row-stacked into vectors of $M=r^2$ entries. Taken sequentially, from all images in the database, these patches form our ensemble of $\Omega$ vectors. These vectors are then analyzed to get their average and, subsequently, the centered autocorrelation. One can easily handle both black-and-white $8$-bit per pixel images and color images, which are simply triplets of RGB $8$-bit per pixel color planes of red, blue, and green added together. The resulting centered ensemble $\{ \bx_{\omega} : \omega \in \Omega \}$ consists of vectors with $M$ entries and zero mean.
The original value range for the centered images, for $8$-bit quantization, is between $0$ to $255$, with quantization step $1$. The range is then converted in our software to the interval $-1$ to $1$, with the quantization step adjusted accordingly. We set the noise level at $\sigma_n^2 \in \{0.01, 0.25, 0.64, 1.00\}$ for implementation on actual images. Recall that in most of the process, we are looking at $\by_{\omega}$, instead of $\bx_{\omega}$, and, thus, using the diagonalized $\BR_{xx} = \Psi \Lambda \Psi^{\top}$ via $\BR_{yy} = \Lambda$. Once we have the average patch vector and the autocorrelation $\BR_{xx}$ of the ensemble, we perform, for this ensemble, the holographic sensing and recovery processes as described in Sections \[sec:model\] and \[sec:design\].
[*The $\Lambda$ and $\Psi$ Matrices*]{}
[r]{}[0.45]{}
{width="40.00000%"}
The autocorrelation matrices are of size $M \times M$, for a given $M$. We typically use $M =64$ or, less frequently, $M=256$. The standard Singular Value Decomposition function in [numpy]{} allows for a fast computation of $\Lambda$ and $\Psi$. Figure \[fig:lambda\] presents the ordered eigenvalues for the patches of each of the $49$ images in the data set separately as well as for the complete ensemble, [*i.e.*]{}, the [aggregate]{} in black. The patches are of sizes, respectively, $8 \times 8$ and $16 \times 16$. For $M=64$, Table \[table:lambda\] gives the first $8$ eigenvalues, rounded to three decimal places, of the six images from Figure \[fig:images\] and of [aggregate]{}.
![The $\Lambda$ profiles when $M \in \{64,256\}$. The vertical axis is labeled in logarithmic scale.[]{data-label="fig:lambda"}](lambda_M64.png "fig:"){width="0.78\linewidth"}\
![The $\Lambda$ profiles when $M \in \{64,256\}$. The vertical axis is labeled in logarithmic scale.[]{data-label="fig:lambda"}](lambda_M256.png "fig:"){width="0.78\linewidth"}
--------------------------------------------------------------------------------
No. Image $\lambda_1 \quad \lambda_2 \quad \ldots \quad \lambda_8$
----- --------------- ----------------------------------------------------------
[aggregate]{} $10.216 \quad 0.244 \quad 0.238 \quad 0.096 \quad
0.080 \quad 0.078 \quad 0.051 \quad 0.045$
1 [merlion]{} $15.125 \quad 0.245 \quad 0.142 \quad 0.096 \quad
0.056 \quad 0.043 \quad 0.039 \quad 0.035$
2 [dragon]{} $10.590 \quad 0.454 \quad 0.333 \quad 0.200 \quad
0.164 \quad 0.122 \quad 0.107 \quad 0.078$
3 [lena]{} $10.115 \quad 0.078 \quad 0.044 \quad 0.014 \quad
0.010 \quad 0.005 \quad 0.004 \quad 0.002$
4 [flood]{} $~~8.714 \quad 0.216 \quad 0.167 \quad 0.047 \quad
0.042 \quad 0.041 \quad 0.017 \quad 0.016$
5 [mandrill]{} $~~4.276 \quad 0.326 \quad 0.280 \quad 0.201 \quad
0.154 \quad 0.138 \quad 0.114 \quad 0.100$
6 [oldlady]{} $~~2.578 \quad 0.080 \quad 0.043 \quad 0.025 \quad
0.020 \quad 0.014 \quad 0.014 \quad 0.012$
--------------------------------------------------------------------------------
: The First $8$ Eigenvalues when $M=64$.[]{data-label="table:lambda"}
Figure \[fig:visual\] displays the orthonormal set of eigenvectors as $8 \times 8$ patches in a combined image as a typical example that can be replicated for other values of $M$. As expected, these eigenpatches look similar to the patches defined by the classical orthonormal basis used in the Discrete Cosine Transform (DCT) [@Ahmed1974]. These are the patches to invoke in reconstructing the images from the data packets that represent the sensed images.
The plot $\rho(j)$ for $j \in {\left\llbracket{64}\right\rrbracket}$ based on Equation (\[eq:rho\]) using the $\lambda_j$ values of [aggregate]{} is in Figure \[fig:rho\_zeta\] (a). If we are given just enough number $K$ of probings to allocate such that $\zeta_j \geq 0$ for all $j \in {\left\llbracket{64}\right\rrbracket}$, as was discussed in Subsection \[subsec:optprobe\] above, then the respective plots of $\zeta_j$ for $\sigma_n^2 \in \{0.01,0.25,0.64\}$ are those given in Figure \[fig:rho\_zeta\] (b)–(d). Each plot shows how the curve is influenced by $\frac{K}{M}$, which serves as the constant level around which each $\zeta_j$ lives, starting above it before decreasing as $j$ goes to $M$. The shape is determined by $\sigma_n^2 \cdot \rho(j)$ since $\zeta_j = \frac{K}{M} + \sigma_n^2 \cdot \rho(j)$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The plot of the function $\rho(j)$ and the respective distribution of $\zeta_j$ for $j \in {\left\llbracket{64}\right\rrbracket}$ based on the $\Lambda$ profile of [aggregate]{} for the indicated $\sigma_n^2$.[]{data-label="fig:rho_zeta"}](rho.png "fig:"){width="0.45\linewidth"} ![The plot of the function $\rho(j)$ and the respective distribution of $\zeta_j$ for $j \in {\left\llbracket{64}\right\rrbracket}$ based on the $\Lambda$ profile of [aggregate]{} for the indicated $\sigma_n^2$.[]{data-label="fig:rho_zeta"}](zeta_001.png "fig:"){width="0.45\linewidth"}
\(a) The plot $\rho(j)$ of [aggregate]{} \(b) $\zeta_j$ when $\sigma_n^2=0.01$
![The plot of the function $\rho(j)$ and the respective distribution of $\zeta_j$ for $j \in {\left\llbracket{64}\right\rrbracket}$ based on the $\Lambda$ profile of [aggregate]{} for the indicated $\sigma_n^2$.[]{data-label="fig:rho_zeta"}](zeta_025.png "fig:"){width="0.45\linewidth"} ![The plot of the function $\rho(j)$ and the respective distribution of $\zeta_j$ for $j \in {\left\llbracket{64}\right\rrbracket}$ based on the $\Lambda$ profile of [aggregate]{} for the indicated $\sigma_n^2$.[]{data-label="fig:rho_zeta"}](zeta_064.png "fig:"){width="0.45\linewidth"}
\(c) $\zeta_j$ when $\sigma_n^2=0.25$ \(d) $\zeta_j$ when $\sigma_n^2=0.64$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[*The Distribution of Probings*]{}
----- ------------- -------------- -------------------------------------------------------- -------------------------------------------------------
No. $(M,m,N)$ $\sigma_n^2$
Mode $1$ Mode $2$
1 $(64,4,8)$ $0.01$ $[2]^4 [1]^{24}$ $[8][7][1]^{17}$
2 $0.25$ $[7][6][5][4][3]^2[2][1]^2$ $[8]^2[5][3]^3[1]^2$
3 $0.64$ $[8]^3[4][2]^2$ $[8]^3[5][2][1]$
4 $1.00$ $[8]^4$ $[8]^4$
5 $(64,4,16)$ $0.01$ $[3]^2[2]^{23} [1]^{12}$ $[16][7][2]^{12}[1]^{17}$
6 $0.25$ $[10][9][8][7][6]^2[5][4]^2[3][1]^2$ $[16][8]^2[6]^3[4][3]^3[1]$
7 $0.64$ $[15][12]^2[8][7]^2[2][1]$ $[16][14][12][8][6]^2[2]$
8 $1.00$ $[16]^2[14][8][5]^2$ $[16]^2[15][7][5]^2$
9 $2.00$ $[16]^4$ $[16]^4$
10 $(64,8,8)$ $0.01$ $[3]^2 [2]^{23} [1]^{12}$ $[8]^3 [3] [2]^{10} [1]^{17}$
11 $0.25$ $[8]^5[6][5][4]^2[3][1]^2$ $[8]^6[6][3]^3[1]$
12 $0.64$ $[8]^8$ $[8]^8$
13 $(64,8,16)$ $0.01$ $[4]^{12}[3]^{18} [2]^{10}[1]^6$ $[16][15][3]^{22}[2]^{12} [1]^7$
14 $0.25$ $[14][13]^2[11]^3[9][8]^2[7][6][5]^2[4][2][1]$ $[16]^3[11][10]^2[8]^2[7]^2[5]^2[4][3][2]$
15 $0.64$ $[16]^6[14][7][6][5]$ $[16]^6[14][7][6][5]$
16 $1.00$ $[16]^7[12][3][1]$ $[16]^7[13][2][1]$
17 $2.00$ $[16]^8$ $[16]^8$
No. $(M,m,N)$ $\sigma_n^2$
Mode $1$ Mode $2$
1 $(64,4,8)$ $0.01$ $[5]^4 [4] [3]^2 [2]$ $[8] [5] [4]^3 [3] [2]^2$
2 $0.25$ $[8]^4$ $[8]^4$
3 $(64,4,16)$ $0.01$ $[8]^4 [7]^2 [6]^2 [3][2][1]$ $[13] [8]^2 [7]^2 [6] [5]^2 [3][2]$
4 $0.25$ $[16]^4$ $[16]^4$
5 $(64,8,8)$ $0.01$ $[8]^4 [7]^2 [6]^2 [3][2][1]$ $[8]^6 [6] [5] [3][2]$
6 $0.25$ $[8]^8$ $[8]^8$
7 $(64,8,16)$ $0.01$ $[13]^4 [12][11]^3 [8] [7][5]^2[3][2][1]$ $[16]^2 [13][12]^2 [11] [10]^2 [7]^2 [5]^2 [2][1]^2$
8 $0.25$ $[16]^8$ $[16]^8$
No. $(M,m,N)$ $\sigma_n^2$
Mode $1$ Mode $2$
1 $(64,4,8)$ $0.01$ $[1]^{32}$ $[8]^3 [7] [1]$
2 $0.25$ $[4]^3 [3]^3 [2]^3 [1]^5$ $[8] [7] [3]^2 [2]^3 [1]^5$
3 $0.64$ $[7][6][5][4][3]^2[2][1]^2$ $[8]^2[5][4][3][2][1]^2$
4 $1.00$ $[8][7][6][5][3][2][1]$ $[8]^3 [4][2]^2$
5 $2.00$ $[8]^3[7][1]$ $[8]^4$
6 $(64,4,16)$ $0.01$ $[2]^{15} [1]^{34}$ $[16] [11][1]^{37}$
7 $0.25$ $[6]^2 [5]^4 [4]^3 [3]^4 [2]^2 [1]^4$ $[15] [5]^3 [4]^3 [3]^4 [2]^3 [1]^4$
8 $0.64$ $[10][8]^2[7][6]^2[5][4][3]^2[2][1]^2$ $[16][8][7]^2[6][5][4][3]^2[2]^2[1]$
9 $1.00$ $[13][10][9][8][7][6][4][3][2][1]^2$ $[16][10][9][8][6][5][4][3][2][1]$
10 $2.00$ $[16]^2[12][9][6][4][1]$ $[16]^2[13][9][5][4][1]$
11 $4.00$ $[16]^3[15][1]$ $[16]^3[15][1]$
12 $(64,8,8)$ $0.01$ $[2]^{15} [1]^{34}$ $[8]^3 [5][1]^{35}$
13 $0.25$ $[6]^2 [5]^4 [4]^3 [3]^4 [2]^2 [1]^4$ $[8]^3 [6] [4]^3 [3]^4 [2]^3 [1]^4$
14 $0.64$ $[8]^4[7][6][5][4][3]^2[2][1]^2$ $[8]^6[5][3]^2[2]^2[1]$
15 $1.00$ $[8]^7[4][2][1]^2$ $[8]^7[5][2][1]$
16 $2.00$ $[8]^8$ $[8]^8$
17 $(64,8,16)$ $0.01$ $[3]^{19} [2]^{31}[1]^9$ $[16]^2 [5][2]^{37}[1]^{17}$
18 $0.25$ $[9][8]^4 [7]^3 [6]^4 [5]^2 [4]^5 [3]^2[2]^2[1]^2$ $[16] [12][8][7]^3 [6]^5 [5]^3 [4]^3 [3]^2[2]^3[1]^2$
19 $0.64$ $[14][12]^2[11][10]^2[9][8]^2[7][6]^2[5][4][2]^2[1]^2$ $[16]^3[11][10][9][8][7]^2[6]^2[5][4]^2[1]^3$
20 $1.00$ $[16]^2[15][13][12][11][9][8][7][6]^2[4][3][2]$ $[16]^4[11][10][9][7]^2[6][5][4][3][2]$
21 $2.00$ $[16]^7[7][5][3][1]$ $[16]^7[9][4][2][1]$
22 $4.00$ $[16]^8$ $[16]^8$
----- ------------- -------------- -------------------------------------------------------- -------------------------------------------------------
: The assignments $s_1,\ldots, s_{L}$ based on the $\Lambda$ profiles of [aggregate]{}, [lena]{}, and [mandrill]{}. For Mode $3$, the distribution is always $[N]^m$, [*i.e.*]{}, the first $m$ indices are each probed $N$ times.[]{data-label="table:distro"}
For the various levels of noise we next determine the projection packet designs to holographically sense the images in the data set, based on the learned second order statistics $\BR_{xx}$. Since we aim to achieve the lowest $\operatorname{MSE}$ when all $N$ packets, [*i.e.*]{}, $K=N \cdot m$ probings, are available, we solve for $\zeta_j$ for $j \in {\left\llbracket{L}\right\rrbracket}$. Calibrations to conform to Modes $1$ to $3$ are subsequently carried out.
Table \[table:distro\] presents the optimal sensing distributions obtained for the indicated noise levels, based on either the $\Lambda$ of [aggregate]{}, or the $\Lambda$ of [lena]{}, or that of [mandrill]{}, for illustrative comparison. The set $\{s_j : j \in {\left\llbracket{L}\right\rrbracket}\}$ is written in shorthand with $[x_1]^{y_1}[x_2]^{y_2} \ldots [x_r]^{y_r}$ denoting $s_j=x_1$ for $j \in {\left\llbracket{y_1}\right\rrbracket}$ followed by $s_j=x_2$ for $j \in {\left\llbracket{y_1+1,y_1+y_2}\right\rrbracket}$ and so on until $s_j=x_r$ for $j \in {\left\llbracket{L-y_r+1,L}\right\rrbracket}$. We remove the superscript if $y_i=1$. Notice that $L = \sum_{i=1}^r y_r$. For example, Entry 3 in Table \[table:distro\] for the [aggregate]{}’s statistics has the distribution $[8]^3[4][2]^2$ in Mode $1$. We read this as $L=6$ with $s_1=s_2=s_3=8$, $s_4=4$, and $s_5=s_6=2$, when all $N$ packets are available.
Let $(M,m,N)=(64,8,8)$. Using Mode $1$ on the $\Lambda$ profile of [aggregate]{}, we obtain the following packet sensing allocations for $\sigma_n^2 \in \{0.01, 0.25, 0.64\}$, where locations refer to the indices $i$ where $\BP_{k}$ has entry $1$ on its diagonal.
The Sensing and Recovery Steps
------------------------------
Now that all of the ingredients to sense any input image are in place, we are ready to showcase the holographic image sensing in action. Several functionalities are included in the software for analytical purposes. Aside from allowing users to vary the basic parameters $M$, $m$, $N$, and $\sigma_n^2$, there are options to perform the sensing and complete the recovery process by using either the $\Lambda$ of [aggregate]{} or that of the individual image. The modes can also be set as desired. Recall that we represent $\by_{\omega}$, instead of $\bx_{\omega}$, holographically, as $N$ packets of data $\bz_k$ given by $$\Big\{ \bz_{k} = \BU_{k} \Psi^{\top} \bx_{\omega} + \bn_{k} : j \in {\left\llbracket{N}\right\rrbracket} \Big\}.$$ The packets can be stored for later recovery, once needed, or transmitted over some channels for reconstruction at another location.
In the recovery process, we assume that an arbitrary $\ell$ out of $N$ packets have been made available. We have $\BU_{\mbox{combi},\kappa} := \left(\BU_{k_1}| \BU_{k_2}|\ldots|\BU_{k_{\ell}}\right)$, where $\kappa:=\{k_1,k_2,\ldots,k_{\ell}\}$ is the corresponding index set. The recovered estimate $\widehat{\by}_{\omega}$ of $\by_{\omega}$ is given by $$\widehat{\by}_{\omega,\kappa} = \Lambda ~\BU_{\mbox{combi},\kappa}~ \BM^{-1} ~\bz_{\mbox{combi},\kappa} \mbox{ where }
\BM := \BU_{\mbox{combi},\kappa}^{\top}~\Lambda ~\BU_{\mbox{combi},\kappa} + \sigma_n^2 \BI_{(\ell \cdot m)}.$$ The original vector $\bx_{\omega}$ is thus estimated by $\widehat{\bx}_{\omega,K} := \Psi \widehat{\by}_{\omega,K}$. We make available two procedures to output a recovered image.
1. Begin by randomly choosing any $1$ out of the $N$ packets and perform the image reconstruction. At each $\ell$, as $\ell$ goes from $2$ to $N$, choose $1$ packet at random from among the remaining $N - (\ell-1)$ packets. Combine the packet with the existing $\bz_{\mbox{combi}}$ and use the result to output an improved image. In this build up, once a packet had been chosen it remains in use for an image reconstruction as $\ell$ increases.
2. As $\ell$ goes from $1$ to $N$, randomly choose any one of the $\binom{N}{\ell}$ possible index sets $\kappa$ of size $\ell$. Use the corresponding $\bz_{\mbox{combi},\kappa}$ in the image reconstruction. Here, a packet which has been included earlier may be dropped in the next iteration.
Figure \[fig:recovered\] presents recovered [dragon]{} images using a run of the randomized procedure on Mode $1$ for the supplied input parameters. For a fixed $\ell$, we can see how the resulting images are very similar on recovery based on the two $\Lambda$ profiles, one belonging to [dragon]{} itself while the other is that of [aggregate]{}. As the noise level rises, $L$ tends to decrease, favouring the probings of the first few coordinates where the corresponding $\lambda$ values are higher. When $(M,m,N, \sigma_n^2)=(64,8,8,1.00)$, for example, all three modes based on the $\Lambda$ profile of [aggregate]{} coincide, with sensing distribution $[8]^8$. Figure \[fig:noise1\] presents incrementally recovered [dragon]{} images, on the specified input parameters, for all $\ell \in {\left\llbracket{8}\right\rrbracket}$.
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![Recovered [dragon]{} in the randomized procedure with $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$ on Mode $1$. Recovery for the images in the upper row uses the $\Lambda$ of [dragon]{}, with distribution $[8]^7[5][2][1]$, while for those in the lower row uses the $\Lambda$ of [aggregate]{}, with distribution $[8]^8$.[]{data-label="fig:recovered"}](MS_r1_a.png "fig:"){width="0.3\linewidth"} ![Recovered [dragon]{} in the randomized procedure with $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$ on Mode $1$. Recovery for the images in the upper row uses the $\Lambda$ of [dragon]{}, with distribution $[8]^7[5][2][1]$, while for those in the lower row uses the $\Lambda$ of [aggregate]{}, with distribution $[8]^8$.[]{data-label="fig:recovered"}](MS_r1_b.png "fig:"){width="0.3\linewidth"} ![Recovered [dragon]{} in the randomized procedure with $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$ on Mode $1$. Recovery for the images in the upper row uses the $\Lambda$ of [dragon]{}, with distribution $[8]^7[5][2][1]$, while for those in the lower row uses the $\Lambda$ of [aggregate]{}, with distribution $[8]^8$.[]{data-label="fig:recovered"}](MS_r1_c.png "fig:"){width="0.3\linewidth"}
\(a) $\ell=1$ \(b) $\ell=2$ \(c) $\ell=8$
![Recovered [dragon]{} in the randomized procedure with $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$ on Mode $1$. Recovery for the images in the upper row uses the $\Lambda$ of [dragon]{}, with distribution $[8]^7[5][2][1]$, while for those in the lower row uses the $\Lambda$ of [aggregate]{}, with distribution $[8]^8$.[]{data-label="fig:recovered"}](MS_r2_a.png "fig:"){width="0.3\linewidth"} ![Recovered [dragon]{} in the randomized procedure with $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$ on Mode $1$. Recovery for the images in the upper row uses the $\Lambda$ of [dragon]{}, with distribution $[8]^7[5][2][1]$, while for those in the lower row uses the $\Lambda$ of [aggregate]{}, with distribution $[8]^8$.[]{data-label="fig:recovered"}](MS_r2_b.png "fig:"){width="0.3\linewidth"} ![Recovered [dragon]{} in the randomized procedure with $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$ on Mode $1$. Recovery for the images in the upper row uses the $\Lambda$ of [dragon]{}, with distribution $[8]^7[5][2][1]$, while for those in the lower row uses the $\Lambda$ of [aggregate]{}, with distribution $[8]^8$.[]{data-label="fig:recovered"}](MS_r2_c.png "fig:"){width="0.3\linewidth"}
\(d) $\ell=1$ \(e) $\ell=2$ \(f) $\ell=8$
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![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_a.png "fig:"){width="0.4\linewidth"} ![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_b.png "fig:"){width="0.4\linewidth"}
\(a) $\ell=1$ \(b) $\ell=2$
![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_c.png "fig:"){width="0.4\linewidth"} ![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_d.png "fig:"){width="0.4\linewidth"}
\(c) $\ell=3$ \(d) $\ell=4$
![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_e.png "fig:"){width="0.4\linewidth"} ![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_f.png "fig:"){width="0.4\linewidth"}
\(e) $\ell=5$ \(f) $\ell=6$
![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_g.png "fig:"){width="0.4\linewidth"} ![Incrementally recovered [dragon]{} using the $\Lambda$ of [aggregate]{} with $(M,m,N,\sigma_n^2)=(64,8,8,1.00)$. The noise forces all three modes to use the distribution $[8]^8$, [*i.e.*]{}, $s_j=8$ for $j \in {\left\llbracket{8}\right\rrbracket}$ and $s_j=0$ for $j >8$.[]{data-label="fig:noise1"}](N1_h.png "fig:"){width="0.4\linewidth"}
\(g) $\ell=7$ \(h) $\ell=8$: all packets
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Performance Analysis
--------------------
To analyze the recovery performance, one can consider several types of $\operatorname{MSE}$ computations in each of the procedures. Let the parameters $M$, $m$, $N$, and $\sigma_n^2$ as well as the mode be given for a chosen image. Hence, we have the $\Lambda$ and $\Psi$ matrices as well as the probing distribution. First, there is the theoretical best $\operatorname{MSE}_{\mbox{best}, \ell}$ from Equation (\[eq:best\_ell\]) to be used as a benchmark. Based on this equation, for a fixed mode, one can compute for the best case $\operatorname{MSE}_{\ell}$ by replacing each $\zeta_j$ by the corresponding $s_j$. Second, for each $\ell$, one can compute the expected value $$\label{eq:expectedMSE}
\operatorname*{\mathbb{E}}(\operatorname{MSE},\ell) := \frac{1}{\binom{N}{\ell}} \sum_{i=1}^{\binom{N}{\ell}}
\left(\frac{1}{{\left\lvert\Omega\right\rvert}}\sum_{\omega=1}^{{\left\lvert\Omega\right\rvert}}
\left(\bx_{\omega}-\widehat{\bx}_{\omega,i}\right)^2\right)$$ with $$\left(\bx_{\omega}-\widehat{\bx}_{\omega,i}\right)^2 =
\left(\bx_{\omega}-\widehat{\bx}_{\omega,i}\right)^{\top} \left(\bx_{\omega}-\widehat{\bx}_{\omega,i}\right).$$ The second index $i$ of $\widehat{\bx}_{\omega,i}$ refers to the $i\textsuperscript{th}$ recovery simulation. For each $\ell$ we compute the average $\operatorname{MSE}$, taken over $\min\left(100,\binom{N}{\ell}\right)$ simulations, since averaging over $\binom{N}{\ell}$ simulations may not be practical as $N$ grows.
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![A comparison of the plots for the theoretical best (thr. best), computed based on $\zeta_j$ for $j \in {\left\llbracket{M}\right\rrbracket}$, for $\ell \in {\left\llbracket{N}\right\rrbracket}$, in Equation (\[eq:best\_ell\]), with the computed best case $\operatorname{MSE}$ in the three modes, when $\zeta_j$ is replaced by the respective $s_j$. We use $(M,m,N)=(64,8,8)$ and $\sigma_n^2 \in \{ 0.01, 0.64 \}$.[]{data-label="fig:comparemode"}](mode_001.png "fig:"){width="0.48\linewidth"} ![A comparison of the plots for the theoretical best (thr. best), computed based on $\zeta_j$ for $j \in {\left\llbracket{M}\right\rrbracket}$, for $\ell \in {\left\llbracket{N}\right\rrbracket}$, in Equation (\[eq:best\_ell\]), with the computed best case $\operatorname{MSE}$ in the three modes, when $\zeta_j$ is replaced by the respective $s_j$. We use $(M,m,N)=(64,8,8)$ and $\sigma_n^2 \in \{ 0.01, 0.64 \}$.[]{data-label="fig:comparemode"}](mode_064.png "fig:"){width="0.48\linewidth"}
\(a) $\operatorname{MSE}$ when $\sigma_n^2=0.01$ \(b) $\operatorname{MSE}$ when $\sigma_n^2=0.64$
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Figure \[fig:comparemode\] shows how the best case $\operatorname{MSE}$ curves of the three modes compare with respect to the theoretical best. Mode $1$ comes closest to the theoretical ideal at $\ell=N$, as per the design. When $\ell$ is very small, however, its best case $\operatorname{MSE}$ is considerably higher than those of the other two modes. Mode $2$ has been designed to improve the performance for such $\ell$ at the relatively small cost of decreased performance in the second half of $\ell$ values compared with Mode $1$. Mode $3$ has the best performance at $\ell=1$ but then does not improve as much as the other two modes as more packets become available.
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![The trend from a single instance of recovery for each of the indicated setting choices on [dragon]{} for $(M,m,N)=(64,8,8)$ and $\sigma_n^2 \in \{0.01, 0.64\}$.[]{data-label="fig:single"}](dragon_001.png "fig:"){width="0.48\linewidth"} ![The trend from a single instance of recovery for each of the indicated setting choices on [dragon]{} for $(M,m,N)=(64,8,8)$ and $\sigma_n^2 \in \{0.01, 0.64\}$.[]{data-label="fig:single"}](dragon_064.png "fig:"){width="0.48\linewidth"}
\(a) A [dragon]{} recovery at $\sigma_n^2=0.01$ \(b) A [dragon]{} recovery at $\sigma_n^2=0.64$
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As we switch from Mode $1$ to $2$ and then to $3$, the $\operatorname{MSE}$ variances among the simulations for small $\ell$ decreases, [*i.e.*]{}, the recovery is smoother. There is less chance of recovering the image badly in the early stages of any single simulation. On the other hand, the gain in $\operatorname{MSE}$ reduction as $\ell$ goes toward $N$ also lessens. Higher noise forces the plots of the three modes closer together, often to the point of merging into one plot. This is because the sensing mechanism assigns more probings to the portion of $\by_{\omega}$ with larger covariances to mitigate the effect of higher noise.
For an easier inspection on how a change in noise level affects the performance in an actual recovery, Figure \[fig:single\] illustrates the trend on [dragon]{} for $(M,m,N)=(64,8,8)$ and $\sigma_n^2 \in \{0.01, 0.64\}$. For each mode, we run a single recovery procedure based on the $\Lambda$ of [dragon]{} and the $\Lambda$ of [aggregate]{}. Relevant plots for other images and parameter combinations can be similarly generated.
There are insights to gain from the average recovery performance. While Mode $1$ is the closest to the original design philosophy of reaching the best $\operatorname{MSE}$ when all $N$ packets are available, its recovery performance fluctuates rather widely for $\ell < \frac{N}{2}$. Figure \[fig:perf\] presents the average $\operatorname{MSE}$ and its variance, simulated on the indicated images, for both the incremental and randomized procedures when $(M, m,N,\sigma_n^2) = (64,4,8,0.25)$.
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![The average $\operatorname{MSE}$ plots and their respective variances obtained from recovery simulations based on two options of second order statistics, namely the [aggregate]{} (aggregate) and the individual image (img). The plots in the upper row is based on the incremental recovery while those in the lower row come from the randomized one. We use Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$.[]{data-label="fig:perf"}](perf_c.png "fig:"){width="0.48\linewidth"} ![The average $\operatorname{MSE}$ plots and their respective variances obtained from recovery simulations based on two options of second order statistics, namely the [aggregate]{} (aggregate) and the individual image (img). The plots in the upper row is based on the incremental recovery while those in the lower row come from the randomized one. We use Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$.[]{data-label="fig:perf"}](perf_d.png "fig:"){width="0.48\linewidth"}
\(a) Average of Incremental $\operatorname{MSE}$ \(b) Variance in Incremental $\operatorname{MSE}$
![The average $\operatorname{MSE}$ plots and their respective variances obtained from recovery simulations based on two options of second order statistics, namely the [aggregate]{} (aggregate) and the individual image (img). The plots in the upper row is based on the incremental recovery while those in the lower row come from the randomized one. We use Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$.[]{data-label="fig:perf"}](perf_e.png "fig:"){width="0.48\linewidth"} ![The average $\operatorname{MSE}$ plots and their respective variances obtained from recovery simulations based on two options of second order statistics, namely the [aggregate]{} (aggregate) and the individual image (img). The plots in the upper row is based on the incremental recovery while those in the lower row come from the randomized one. We use Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$.[]{data-label="fig:perf"}](perf_f.png "fig:"){width="0.48\linewidth"}
\(c) Average of Randomized $\operatorname{MSE}$ \(d) Variance in Randomized $\operatorname{MSE}$
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Recovery and Performance Analysis on Control Images {#sec:control}
===================================================
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![Two images from the control data set for illustrations in recovery and analysis. Taken from <https://pixabay.com/users/magee-830963/>, [twobirds]{} depicts two rainbow lorikeets. The image [boy]{} was taken by the corresponding author.[]{data-label="fig:control"}](twobirds.jpg "fig:"){width="0.44\linewidth"} ![Two images from the control data set for illustrations in recovery and analysis. Taken from <https://pixabay.com/users/magee-830963/>, [twobirds]{} depicts two rainbow lorikeets. The image [boy]{} was taken by the corresponding author.[]{data-label="fig:control"}](boy.jpg "fig:"){width="0.44\linewidth"}
\(a) [twobirds]{} \(b) [boy]{}
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In <https://github.com/adamasstokhorst/holographic/tree/master/control> we collect some natural images for control purposes. They are sensed and recovered by using the $\Lambda$ of [aggregate]{} obtained in Section \[sec:comp\], since their second order statistics are assumed to be unknown. We use two control images, namely [twobirds]{} and [boy]{}, in Figure \[fig:control\] to illustrate and analyze their recovery. Their respective incrementally recovered images, for the indicated parameters, can be found in Figure \[fig:controlrec\].
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![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](tb_1.png "fig:"){width="0.2\linewidth"} ![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](tb_2.png "fig:"){width="0.2\linewidth"} ![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](tb_4.png "fig:"){width="0.2\linewidth"} ![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](tb_8.png "fig:"){width="0.2\linewidth"}
\(a) $\ell=1$ \(b) $\ell=2$ \(c) $\ell=4$ \(d) $\ell=8$
![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](boy_1.png "fig:"){width="0.2\linewidth"} ![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](boy_4.png "fig:"){width="0.2\linewidth"} ![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](boy_8.png "fig:"){width="0.2\linewidth"} ![Incrementally recovered [twobirds]{} and [boy]{}. On [twobirds]{}, $(M,m,N,\sigma_n^2)=(64,8,8,0.64)$. By Table \[table:distro\] Entry 12, all modes have probing distribution $[8]^8$. Mode $2$ with $(M,m,N,\sigma_n^2)=(64,4,16,0.64)$ is used to recover [boy]{}, where the probing distribution is as given in Table \[table:distro\] Entry 7.[]{data-label="fig:controlrec"}](boy_16.png "fig:"){width="0.2\linewidth"}
\(e) $\ell=1$ \(f) $\ell=4$ \(g) $\ell=8$ \(h) $\ell=16$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We see in Figure \[fig:ctrl\] how the recovered images exhibit the expected properties of progressive refinement as more and more packets are used. The plots confirm that using the $\Lambda$ of [aggregate]{} on control images results in a very similar $\operatorname{MSE}$ values as using the individual image’s profile, had it been known and used.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The $\operatorname{MSE}$ plots capture the performance of a single recovery run on Mode $1$, for each of the indicated settings, on [boy]{} and [twobirds]{}. We use both the $\Lambda$ profile of [aggregate]{} (aggr) and that of the image, as well as both the incremental (build-up) and randomized (random) procedure.[]{data-label="fig:ctrl"}](boy_ctrl.png "fig:"){width="0.44\linewidth"} ![The $\operatorname{MSE}$ plots capture the performance of a single recovery run on Mode $1$, for each of the indicated settings, on [boy]{} and [twobirds]{}. We use both the $\Lambda$ profile of [aggregate]{} (aggr) and that of the image, as well as both the incremental (build-up) and randomized (random) procedure.[]{data-label="fig:ctrl"}](twobirds_ctrl.png "fig:"){width="0.44\linewidth"}
\(a) Recovery instances on [boy]{} \(b) Recovery instances on [twobirds]{}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Concluding Remarks {#sec:conclusion}
==================
We briefly reviewed the theory of holographic sensing and then explained in detail the process of designing the sensing packets before presenting the expected performance of the distributed holographic encodings of databases of images. We have outlined the full design of the system implementing the idea of holographic representations and tested the system’s performance on a data set of $49$ natural images. This data set can be easily enlarged into a big database if desired. Similarly with the types of images. The system has been demonstrated to also perform well on randomly selected control images not present in the data set.
In conclusion we also wish to mention two distance-based correlation models that work quite well in estimating $\BR_{xx}$ for natural images. We call them the [grid]{} model, when the distance is the Manhattan length, and the [line]{} model, when the $\ell_2$-norm is used. The respective models can be formulated as follows. Let $\BR^{\mathrm{G}}$ and $\BR^{\mathrm{L}}$ be the respective $M \times M$ matrices that approximate $\BR_{xx}$ in the [grid]{} and [line]{} models. Without loss of generality, we can assume $M$ to be a square, [*e.g.*]{}, $8^2$ as in our typical implementation, and let $\nu :=\sqrt{M}$. Since correlation decays as a function of distance, the entries in the model matrices must faithfully reflect this fact. Let $i := \nu \cdot a + b$ and $j:= \nu \cdot p + q$. For $i,j \in {\left\llbracket{M}\right\rrbracket}$, let $$\label{eq:matmod}
\BR^{\mathrm{G}}_{i,j} = A \cdot \gamma^{|a-p| + |b-q|} \mbox{ and }
\BR^{\mathrm{L}}_{i,j} = A \cdot \gamma^{\sqrt{(a-p)^2 + (b-q)^2}}.$$ One can then proceed to determine the values of the constants $A$ and $\gamma$ that make the respective matrices closely resemble some experimentally obtained entries of $\BR_{xx}$ for a class of images. For $M=64$, based on our input data set of images, we obtain $A=0.18$ and $\gamma = 0.98$. The $\Lambda$ profiles as well as the corresponding $\Psi$ matrices can then be computed as above. Figures \[fig:lambda\_comp\] and \[fig:comp\_model\] provide, respectively, a useful $\Lambda$ comparison plot and a visualization of each of the $\Psi$ matrices for inspection. They show that the [line]{} model fits the values from our image data set better than the [grid]{} model.
![Comparison of the $\Lambda$ profiles of [aggregate]{} and of the two models for $M=64$.[]{data-label="fig:lambda_comp"}](lambda_comp.png){width="0.50\linewidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The visualized $\Psi$ matrices to be compared with the one for [aggregate]{} in Figure \[fig:visual\].[]{data-label="fig:comp_model"}](psi_grid.png "fig:"){width="0.3\linewidth"} ![The visualized $\Psi$ matrices to be compared with the one for [aggregate]{} in Figure \[fig:visual\].[]{data-label="fig:comp_model"}](psi_line.png "fig:"){width="0.3\linewidth"}
\(a) $\Psi$ in [grid]{} model \(b) $\Psi$ in [line]{} model
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Using a reasonably fitting model may remove the need to compute for the second order statistics from some database prior to deployment. Figure \[fig:comp\_rec\] shows that there is indeed not much of a difference in the recovery performance, either on a single recovery run or on average.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The first row displays the *average* $\operatorname{MSE}$ plots for the *randomized* recovery simulations on [boy]{}, [dragon]{}, [mandrill]{}, and [twobirds]{} on Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$. The second row displays the performance on a single recovery instance for the indicated settings on [mandrill]{} and [lena]{}. Notice that the captured randomized recovery instance on Mode $1$ based on the $\Lambda$ of [mandrill]{} is particularly bad. While such an instance is unlikely to occur, it highlights the potential drawback of using the randomized procedure in Mode $1$ on some images. Taken together, the plots allow for a quick comparison on the recovery performance across the statistical properties of a specific image, of [aggregate]{}, and in the [grid]{} and [line]{} models, for various settings.[]{data-label="fig:comp_rec"}](last_mse.png "fig:"){width="0.44\linewidth"} ![The first row displays the *average* $\operatorname{MSE}$ plots for the *randomized* recovery simulations on [boy]{}, [dragon]{}, [mandrill]{}, and [twobirds]{} on Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$. The second row displays the performance on a single recovery instance for the indicated settings on [mandrill]{} and [lena]{}. Notice that the captured randomized recovery instance on Mode $1$ based on the $\Lambda$ of [mandrill]{} is particularly bad. While such an instance is unlikely to occur, it highlights the potential drawback of using the randomized procedure in Mode $1$ on some images. Taken together, the plots allow for a quick comparison on the recovery performance across the statistical properties of a specific image, of [aggregate]{}, and in the [grid]{} and [line]{} models, for various settings.[]{data-label="fig:comp_rec"}](last_var_mse.png "fig:"){width="0.44\linewidth"}
\(a) Average of Randomized $\operatorname{MSE}$ \(b) Variance in Randomized $\operatorname{MSE}$
![The first row displays the *average* $\operatorname{MSE}$ plots for the *randomized* recovery simulations on [boy]{}, [dragon]{}, [mandrill]{}, and [twobirds]{} on Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$. The second row displays the performance on a single recovery instance for the indicated settings on [mandrill]{} and [lena]{}. Notice that the captured randomized recovery instance on Mode $1$ based on the $\Lambda$ of [mandrill]{} is particularly bad. While such an instance is unlikely to occur, it highlights the potential drawback of using the randomized procedure in Mode $1$ on some images. Taken together, the plots allow for a quick comparison on the recovery performance across the statistical properties of a specific image, of [aggregate]{}, and in the [grid]{} and [line]{} models, for various settings.[]{data-label="fig:comp_rec"}](mandrill_4L.png "fig:"){width="0.44\linewidth"} ![The first row displays the *average* $\operatorname{MSE}$ plots for the *randomized* recovery simulations on [boy]{}, [dragon]{}, [mandrill]{}, and [twobirds]{} on Mode $1$ with $(M,m,N,\sigma_n^2)=(64,4,8,0.25)$. The second row displays the performance on a single recovery instance for the indicated settings on [mandrill]{} and [lena]{}. Notice that the captured randomized recovery instance on Mode $1$ based on the $\Lambda$ of [mandrill]{} is particularly bad. While such an instance is unlikely to occur, it highlights the potential drawback of using the randomized procedure in Mode $1$ on some images. Taken together, the plots allow for a quick comparison on the recovery performance across the statistical properties of a specific image, of [aggregate]{}, and in the [grid]{} and [line]{} models, for various settings.[]{data-label="fig:comp_rec"}](lena_4L.png "fig:"){width="0.44\linewidth"}
\(c) Performance on [mandrill]{} \(d) Performance on [lena]{}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Acknowledgements {#acknowledgements .unnumbered}
================
Singapore National Research Foundation and Israel Science Foundation joint program NRF2015-NRF-ISF001-2597 and Nanyang Technological University Grant Number M4080456 support the work of the authors. The authors declare no competing interest.
References {#references .unnumbered}
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---
abstract: 'The paper presents an explicit form of the resolvent for the class of generators of $C_0$-groups with purely imaginary eigenvalues, clustering at $i\infty$, and complete minimal non-basis family of eigenvectors, constructed recently by the authors in [@Sklyar3]. The growth properties of the resolvent are described. The discrete Hardy inequality serves as the cornerstone for the proofs of the corresponding results. Moreover, it is shown that the main result on the Riesz basis property for invariant subspaces of the generator of the $C_0$-group, obtained a decade ago by G.Q. Xu, S.P. Yung and H. Zwart in [@Xu], [@Zwart], is sharp.'
address: 'G.M. Sklyar: Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70451, Szczecin, Poland; V. Marchenko$^{*}$: B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Mathematical Division, Prospekt Nauky 47, 61103, Kharkiv, Ukraine; $^{*}$ - Corresponding author'
author:
- 'Grigory M. Sklyar'
- 'Vitalii Marchenko$^{*}$'
date: '26.07.2018'
title: 'Resolvent of the generator of the $C_0$-group with non-basis family of eigenvectors and sharpness of the XYZ theorem'
---
Introduction
============
Problems of spectral theory for nonselfadjoint (NSA) operators attract more and more growing interest of experts in different fields of mathematics and natural sciences, see, e.g., [@Bagarello], [@Davies1], [@Davies2], [@Davies3], [@Henry1], [@Henry2], [@Henry3] [@Mityagin], [@Sklyar3], [@Xu], [@Zwart] and the references therein. This is primarily caused by the recent progress in theoretical physics of non-Hermitian systems [@Bagarello] on the one hand, and, on the other, by the fact that many mathematical models of dynamical processes lead to the study of linear evolution equations $$\label{Cauchy problem}
\left \{
\begin{array}{l}
\dot{x}(t)=Ax(t),\quad t \geq 0,\\
x(0)=x_0\in H,\\
\end{array}
\right .$$ in Hilbert spaces $H$ with unbounded NSA operator $A$.
In last years NSA Schrödinger operators are studied very intensively, see [@Davies3], [@Davies4], [@Henry1], [@Henry2], [@Henry3], [@Mityagin] and, especially, [@Bagarello], [@Davies1], [@Davies2]. In 2000 E.B. Davies [@Davies3] studied NSA anharmonic oscillators $$\label{L alpha}
\mathcal{L}_{\alpha} = -\frac{d^2}{d x^2}+c |x|^{\alpha},$$ defined on $L_2\left(\mathbb{R}\right)$ as the closure of the associated quadratic form defined on $C_{0}^{\infty}\left(\mathbb{R}\right)$, where $\alpha>0,$ $c\in\mathbb{C}\setminus \mathbb{R},$ $|\arg c|<C(\alpha)$. He proved that for all $\alpha>0$ the spectrum of $\mathcal{L}_{\alpha}$ consists of discrete simple eigenvalues and, if we denote them in nondecreasing modulus order by $\lambda_n,$ $|\lambda_n|\to \infty$, and consider corresponding one-dimensional spectral projections $P_n$, then the norms $\left\|P_n\right\|$ grow more rapidly than any polynomial of $n$ as $n\to\infty$, see [@Davies2], [@Davies3]. Davies called operators with such spectral behavior by spectrally wild ones. A family of eigenvectors of spectrally wild operator, although can be complete and minimal in a space, cannot constitute a Schauder basis. E.g., the eigenvectors of $\mathcal{L}_{\alpha}$, where $\Re (c)>0$, are dense in $L_2\left(\mathbb{R}\right)$ if either $\alpha\geq 1$, or $0<\alpha<1$ and $|\arg c|<\alpha\pi/2$, see [@Davies3]. We recall that a sequence $\{\phi_n\}_{n=1}^{\infty}$ of a Banach space $X$ forms a Schauder basis of $X$ provided each $x\in X$ has a unique norm-convergent expansion $$x=\sum\limits_{n=1}^{\infty} c_n \phi_n.$$
In 2004 E.B. Davies and A. B. J. Kuijlaars proved that spectral projections $P_n$ of the operator $\mathcal{L}_2$, where $c=e^{i\theta}$, $0<|\theta| < \pi,$ grow exponentially [@Davies4]: $$\lim\limits_{n\to\infty} \frac{1}{n}\ln\left\|P_n\right\|=2 \Re \left\{f\left(r(\theta)e^{\frac{i\theta}{4}} \right)\right\},$$ where $f(z)= \ln \left(z+ g(z)\right) - z g(z)$, $g(z)=(z^2-1)^{1/2}$, $r(\theta)= \left(2\cos \left(\theta/2\right) \right)^{-1/2}$.
These studies were continued by R. Henry, who determined exponential growth rates of spectral projections of the so-called complex Airy operator $\mathcal{L}_1$, where $c=e^{i\theta}$, $0<|\theta| <\frac{3\pi}{4}$, and anharmonic oscillators $\mathcal{L}_{2k},$ $k\in\mathbb{N},$ where $c=e^{i\theta}$, $0<|\theta| <\frac{(k+1)\pi}{2k},$ see [@Henry1], [@Henry2]. Moreover, in [@Henry3] Henry studied spectral projections $P_n$ of the complex cubic oscillator $ \mathcal{C}_{\beta}=-\frac{d^2}{d x^2}+ i x^3 +i\beta x,\: \beta\geq 0$ with domain $H^2\left(\mathbb{R}\right)\cap L_2\left(\mathbb{R};x^6 dx\right)\subset L_2\left(\mathbb{R}\right)$ and showed that for all $\beta\geq 0,$ $$\lim\limits_{n\to\infty} \frac{1}{n}\ln\left\|P_n\right\|=\frac{\pi}{\sqrt{3}}.$$
Recently, B. Mityagin et al. considered NSA perturbations of selfadjoint Schrödinger operators with single-well potentials and demonstrated that norms of spectral projections $P_n$ of these operators can grow at intermediate levels, from arbitrary slowly to exponentially fast [@Mityagin]. In particular, natural classes of operators with projections obeying $$\lim\limits_{n\to\infty} \frac{1}{n^{\gamma}}\ln\left\|P_n\right\|=C,$$ where $C\in (0,\infty)$ and $\gamma \in (0,1),$ were found.
On the other hand, in “good” situation, i.e. when the operator $A$ has a Riesz basis of $A$-invariant subspaces, the system (\[Cauchy problem\]) can be split into countable family of subsystems (each subsystem lives in a corresponding $A$-invariant subspace) and we can make conclusions on the behavior of (\[Cauchy problem\]) on the basis of the study of its subsystems, see, e.g., [@Miloslavskii], [@Rabah1], [@Rabah2], [@Sklyar1], [@Sklyar2], [@Xu], [@Zwart] and the references therein. That is why Riesz bases are convenient tools of infinite-dimensional linear systems theory and the following question is important.
\[q\] Which conditions are sufficient to guarantee that $$\label{question}
A\:\textit{has a Riesz basis of eigenvectors ($A$-invariant subspaces)?}$$
For equivalent definitions and stability properties of Schauder bases of subspaces (Schauder decompositions) and Riesz bases of subspaces we refer to [@Marchenko1], [@Marchenko2], [@Marchenko3] and the references therein.
A number of recent papers are devoted to Question \[q\] in the case when $A$ is a perturbation of selfadjoint, nonnegative operator with discrete spectrum, including perturbations of harmonic oscillator type operators. We refer to [@Mityagin], Section 4.3, for the brief overview of the corresponding results.
In the study of (in fact, quite old) Question \[q\] a breakthrough was made by G.Q. Xu, S.P. Yung and H. Zwart – the XYZ Theorem:
If the following three conditions hold:
1. The operator $A$ generates the $C_0$-group on a Hilbert space $H$;
2. The eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ of $A$ is a union of $K<\infty$ interpolation sequences $\Lambda_k$, $1\leq k\leq K.$ In other words, $\{\lambda_n\}_{n=1}^{\infty}=\bigcup\limits_{k=1}^K \Lambda_k$, where $$\label{1}
\min_{k}\inf\limits_{\lambda_{n},\lambda_{m}\in \Lambda_k:\: n\neq m} |\lambda_{n}-\lambda_{m}|>0;$$
3. The span of the generalized eigenvectors (eigen- and rootvectors) of $A$ is dense,
then the condition (\[question\]) holds.
More precisely [@Zwart], under the three conditions above, there exists a certain sequence of (multidimensional, if $K>1$) spectral projections $\{P_n\}_{n=1}^{\infty}$ of $A$ such that $\{P_n H\}_{n=1}^{\infty}$ forms a Riesz basis of subspaces in $H$ with $$\sup\limits_{n\in\mathbb{N}} \dim P_n H\leq K.$$
Operators satisfying conditions 1-3 of the XYZ Theorem naturally arise from applications, e.g., in the analysis of neutral type systems [@Rabah1], [@Rabah2], [@Sklyar1], [@Sklyar2]. Since it is usually not hard to verify conditions 1-3 for a concrete system (\[Cauchy problem\]), while the important Riesz basis property of eigenvectors ($A$-invariant subspaces) is difficult to prove, the XYZ Theorem provides us a powerful machinery for the analysis of various applications.
From the other hand, it was totally unclear: What if the eigenvalues of $A$ lie in a strip, parallel to imaginary axis, and do not satisfy the condition of separation (\[1\])? In particular:
\[qq\] Is it possible to construct the unbounded generator of the $C_0$-group with eigenvalues $\{\lambda_n\}_{n=1}^{\infty}\subset i\mathbb{R}$ not satisfying (\[1\]) and dense family of eigenvectors, which does not form a Riesz basis?
In [@Sklyar3] the authors obtained an affirmative answer to the Question \[qq\] and presented the class of infinitesimal operators with such eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ and complete minimal family of eigenvectors, which, however, does not form even a Schauder basis. To formulate the corresponding result we need to consider the following classes of sequences.
\[Class Sk\]([@Sklyar3]) Let $k\in\mathbb{N}$ and $\Delta$ stands for the difference operator. Then we define $$\mathcal{S}_k=\Bigl\{\left\{f(n)\right\}_{n=1}^{\infty}\subset\mathbb{R}:\: \lim\limits_{n\rightarrow \infty} f(n)=+\infty; \left\{n^j \Delta^j f(n) \right\}_{n=1}^{\infty}\in \ell_{\infty}\:\: \text{for} \:\: 1\leq j\leq k\Bigr\}.$$
Clearly $\left\{\ln n\right\}_{n=1}^{\infty}\in \mathcal{S}_k$ for all $k$.
\[construction\]([@Sklyar3]) Assume that $\{e_n\}_{n=1}^{\infty}$ is a Riesz basis of a Hilbert space $H$ and $k\in\mathbb{N}$. Then:
1. The space $H_k\left(\{e_n\}\right)=\left\{x=(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n:\: \{c_n\}_{n=1}^{\infty}\in \ell_2(\Delta^k)\right\}$, where $(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n$ denotes a formal series and $\ell_2(\Delta^k)=\left\{\{c_n\}_{n=1}^{\infty}:\:\Delta^k \{c_n\}_{n=1}^{\infty} \in \ell_2\right\},$ is a separable Hilbert space.
2. The sequence $\{e_n\}_{n=1}^{\infty}$ is dense and minimal in $H_k\left(\{e_n\}\right)$, but it is not uniformly minimal in $H_k\left(\{e_n\}\right)$. Hence $\{e_n\}_{n=1}^{\infty}$ does not form a Schauder basis of $H_k\left(\{e_n\}\right)$.
3. The operator $A_k:H_k\left(\{e_n\}\right) \supset D(A_k) \mapsto H_k\left(\{e_n\}\right),$ defined by $$A_k x=A_k (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n= (\mathfrak{f})\sum\limits_{n=1}^{\infty} i f(n) \cdot c_{n} e_n,$$ where $\left\{f(n)\right\}_{n=1}^{\infty}\in\mathcal{S}_k$, with domain $$\label{Domain_k}
D(A_k)=\left\{x= (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n \in H_k\left(\{e_n\}\right):\: \{f(n) \cdot c_{n}\}_{n=1}^{\infty}\in \ell_2(\Delta^k)\right\},$$ generates the $C_0$-group $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ on $H_k\left(\{e_n\}\right)$, which acts for every $t\in\mathbb{R}$ by the formula $$e^{A_k t} x=e^{A_k t}(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n =(\mathfrak{f})\sum\limits_{n=1}^{\infty} e^{i t f(n)} c_{n} e_n.$$
Similar results hold for the case of operators with the same spectral behaviour on certain Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$, see [@Sklyar3]. Note that if we take, for example, $f(n)=\sqrt{n},$ $n\in\mathbb{N},$ and define the operator $A_1$ on $H_1\left(\{e_n\}\right)$ as in Theorem \[construction\], then $A_1$ will not generate a $C_0$-semigroup on $H_1\left(\{e_n\}\right)$, see [@Sklyar3], Proposition 7, Proposition 8.
The main objective of the paper is to obtain an explicit form of the resolvent for the class of generators $A_k$ of $C_0$-groups from Theorem \[construction\] and to characterize the asymptotic properties of the resolvent on a complex plane. Theorem \[construction\] together with the XYZ Theorem show that Theorem 1.1 from [@Zwart] dealing with the case of simple eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ in the XYZ Theorem, satisfying $$\inf\limits_{n\neq m} |\lambda_{n}-\lambda_{m}|>0,$$ is sharp, see also Example 1.3 in [@Zwart]. In the present paper we will demonstrate that Theorem \[construction\] means that the XYZ Theorem is also sharp, see Section 2.
Operators with simple eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ not satisfying the condition 2 of the XYZ Theorem and non-basis family of eigenvectors are considered in recent applications. In [@Almog] the author study the stability of the normal state of superconductors in the presence of electric currents in the large domain limit using the time-dependent Ginzburg-Landau model. The study involves spectral analysis of the operator $\mathcal{L}: D(\mathcal{L}) \mapsto L_2 \left(\mathbb{R},\mathbb{C}\right)$, defined by $$\mathcal{L} = - \frac{d^2}{d x^2} +i x ,$$ where $D(\mathcal{L})=\left\{\psi\in L_2 \left(\mathbb{R_{+}},\mathbb{C}\right):\: x\psi \in L_2 \left(\mathbb{R_{+}},\mathbb{C}\right), \psi\in H_{0}^{2} \left(\mathbb{R_{+}},\mathbb{C}\right)\right\}.$ Let $\{\mu_n\}_{n=1}^{\infty} \subset \mathbb{R}$ denotes the non-increasing sequence of zeroes of $Ai(z)$, Airy function. Then $\{\lambda_n\}_{n=1}^{\infty}$, where $\lambda_n=e^{-\frac{2\pi}{3} i} \mu_n,\:n\in\mathbb{N},$ is a sequence of eigenvalues of $\mathcal{L}$ [@Almog]. Since $\lim\limits_{n\rightarrow \infty} \mu_n=-\infty$ and $\lim\limits_{n\rightarrow \infty} |\mu_{n+1}-\mu_n|=0$ (see [@Vallee]), the eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ of $\mathcal{L}$ obey the condition $$\lim\limits_{n\rightarrow \infty} |\lambda_{n+1}-\lambda_n|=0$$ and, hence, the set $\{\lambda_n\}_{n=1}^{\infty}$ cannot be decomposed into a finite number of sets $\Lambda_k$ satisfying (\[1\]).
The eigenfunctions of $\mathcal{L}$ are $$\tilde{\psi}_n=Ai\left(e^{\frac{\pi i}{6}} x +\mu_n\right)\in H_{0}^{2} \left(\mathbb{R_{+}},\mathbb{C}\right),\: n\in\mathbb{N}.$$ Normalized eigenfunctions $\psi_n=\frac{\tilde{\psi}_n}{\left\| \tilde{\psi}_n\right\|},$ $n\in\mathbb{N},$ are dense in $L_2\left(\mathbb{R},\mathbb{C}\right)$, as it is proved in [@Almog], but do not form a Schauder basis of $L_2 \left(\mathbb{R},\mathbb{C}\right)$, since $\mathcal{L}$ is spectrally wild [@Davies3].
The sharpness of the XYZ Theorem
================================
We will use the notation from [@Sklyar3], see also Theorem \[construction\]. By Proposition 3 of [@Sklyar3] we have that for any $k\in\mathbb{N}$ the sequence $\{e_n\}_{n=1}^{\infty}$ is dense and minimal in $H_k\left(\{e_n\}\right)$, but it is not uniformly minimal in $H_k\left(\{e_n\}\right)$. It means that for each $n\in\mathbb{N}$ $$\varrho\left(e_n, \overline{Lin}\{e_j\}_{j\neq n} \right)>0,$$ but $$\inf\limits_{n\in\mathbb{N}}\varrho\left(e_n, \overline{Lin}\{e_j\}_{j\neq n} \right)=0,$$ where $\varrho\left(x,Y\right)$ denotes a standard distance from the point $x$ to a set $Y$, defined by $$\varrho\left(x,Y\right)=\inf\limits_{y\in Y} \|x-y\|.$$
Let $\{\phi_n\}_{n=1}^{\infty}$ be dense and minimal sequence in a Hilbert space $H$, but is not uniformly minimal in $H$. Then it can happen that there exists a splitting of $\{\phi_n\}_{n=1}^{\infty}$ into infinite number of disjoint groups with at most $K<\infty$ elements in each of them, i.e. $$\{\phi_n\}_{n=1}^{\infty}=\left\{\{\phi_j\}_{j\in A_n}\right\}_{n=1}^{\infty},$$ where $$\label{disj sets}
\mathbb{N}=\bigcup\limits_{n=1}^{\infty} A_n,\quad A_n\cap A_m=\emptyset\:\:\text{if}\:\: n\neq m,\quad \left|A_n\right|\leq K\:\:\text{for all}\:\: n,$$ such that the corresponding sequence of subspaces $\{Lin \{\phi_n\}_{n\in A_n}\}_{n=1}^{\infty}$ constitute a Riesz basis of subspaces of $H$ with uniform bound of dimensions of all subspaces not exceeding $K$. See e.g. Example 1.3 in [@Zwart] for details. In order to show that the XYZ Theorem is sharp we will prove that this situation is impossible for our construction from Theorem \[construction\]. More precisely, thereby we demonstrate that if the eigenvalues of the generator of the $C_0$-group in a Hilbert space do not satisfy (\[1\]), then the conclusion of the XYZ Theorem can be false. Furthermore, we will prove a little more.
\[sharp\] Let $k\in\mathbb{N}$ and $\{e_n\}_{n=1}^{\infty}\subset H_k\left(\{e_n\}\right)$ be a sequence from Theorem \[construction\]. Suppose that $\{A_n\}_{n=1}^{\infty}$ is an arbitrary decomposition of $\mathbb{N}$ into disjoint sets, i.e. $$\mathbb{N}=\bigcup\limits_{n=1}^{\infty} A_n,\quad A_n\cap A_m=\emptyset, n\neq m.$$ Then $\left\{\overline{Lin} \{e_j\}_{j\in A_n}\right\}_{n=1}^{\infty}$ does not form a Schauder decomposition of $H_k\left(\{e_n\}\right)$.
Fix $k\in\mathbb{N}$ and assume the opposite, i.e. let there exists a decomposition of $\mathbb{N}$ into disjoint sets, $\mathbb{N}=\bigcup\limits_{n=1}^{\infty} A_n,$ $A_n\cap A_m=\emptyset$ if $n\neq m$, such that $\left\{\mathfrak{M}_n=\overline{Lin} \{e_j\}_{j\in A_n}\right\}_{n=1}^{\infty}$ constitutes a Schauder decomposition of $H_k\left(\{e_n\}\right)$. Then, by the definition of the Schauder decomposition, every $x\in H_k\left(\{e_n\}\right)$ can be uniquely represented in a series $$x=\sum\limits_{n=1}^{\infty} x_n,$$ where $x_n\in \mathfrak{M}_n$ for each $n\in\mathbb{N}$, and there exists an associated sequence of coordinate linear projections $\{P_n\}_{n=1}^{\infty}$ defined by $P_n x=P_n \sum\limits_{m=1}^{\infty} x_m=x_n,$ where $x_n\in \mathfrak{M}_n,$ $n\in\mathbb{N}$. It follows that for every $n,j\in\mathbb{N}$ $$\label{proj}
P_n e_j =\left \{
\begin{array}{l}
e_j,\quad j\in A_n,\\
0,\quad \:j\notin A_n.\\
\end{array}
\right .$$
Consider an element $x^{\ast}=(\mathfrak{f})\sum\limits_{j=1}^{\infty}e_j\in H_k\left(\{e_n\}\right).$ Then, taking into account (\[proj\]), we have that for every $n\in\mathbb{N}$ $$\label{pn_proj}
P_n x^{\ast}= P_n \left((\mathfrak{f})\sum\limits_{j=1}^{\infty}e_j\right)= \sum\limits_{j\in A_n} e_j.$$
We recall that the norm in a Hilbert space $H_k\left(\{e_n\}\right)$ is defined by $$\begin{aligned}
\|x\|_k &=\left\|(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n\right\|_k\\
&=\left\|\sum\limits_{n=1}^{\infty}\left(c_{n} -C_{k}^{1} c_{n-1}+\dots+ (-1)^{k+1} C_{k}^{k-1} c_{n-k+1} +(-1)^{k} c_{n-k}\right) e_n\right\|,\end{aligned}$$ where $x=(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n\in H_k\left(\{e_n\}\right),$ $C_k^m$ are binomial coefficients, $\|\cdot \|$ denotes the norm in an initial Hilbert space $H$ and $c_{1-j}=0$ for all $j\in\mathbb{N}$, see [@Sklyar3]. Since $\{e_n\}_{n=1}^{\infty}$ is a Riesz basis of $H$ (see Theorem \[construction\]), there exist two constants $M\geq m>0$ such that for every $y=\sum\limits_{n=1}^{\infty} \alpha_n e_n\in H$ we have $$\label{1Riesz}
m \|y\|^2 \leq \sum\limits_{n=1}^{\infty} |\alpha_n|^2 \leq M \|y\|^2.$$
By virtue of (\[pn\_proj\]) and (\[1Riesz\]) we obtain that for every $n\in\mathbb{N}$ $$\left\|P_n x^{\ast} \right\|_k^2=\left\|\sum\limits_{j\in A_n} e_j \right\|_k^2=\left\|(\mathfrak{f})\sum\limits_{j=1}^{\infty}\xi_j(n) e_j \right\|_k^2\geq \frac{1}{M},$$ where for every $n,j\in\mathbb{N}$ $$\xi_j(n)=\left \{
\begin{array}{l}
1,\quad j\in A_n,\\
0,\quad j\notin A_n.\\
\end{array}
\right .$$
Thus $\left\|P_n x^{\ast} \right\|_k \nrightarrow 0$ as $n\to\infty$, which means that $x^{\ast}$ can not be represented in a convergent series $$\sum\limits_{n=1}^{\infty} x_n^{\ast}=\sum\limits_{n=1}^{\infty} P_n x^{\ast},$$ where $x_n^{\ast}\in \mathfrak{M}_n$ for each $n\in\mathbb{N}$. So we arrived at a contradiction with the definition of the Schauder decomposition.
Theorem \[sharp\] leads to the following.
\[cor sharp\] The XYZ Theorem is sharp. None of its conditions can be weakened.
Indeed, condition $3$ of the XYZ Theorem obviously can not be weakened. If one weakens condition $2$ but conditions $1$ and $3$ are fulfilled, then, by virtue of Theorem \[sharp\], the class of counterexamples are given by Theorem \[construction\].
Let us weaken condition $1$. Suppose that conditions $2$ and $3$ are satisfied, operator $A$ does not generate the $C_0$-group on $H$ but $A$ generates the $C_0$-semigroup on $H$. Then the counterexample is given as follows.
Let $\{\phi_n\}_{n=1}^{\infty}$ be a bounded non-Riesz basis of $H$, i.e. bounded conditional basis. It means that $\{\phi_n\}_{n=1}^{\infty}$ constitutes a Schauder basis of $H$, but does not form a Riesz basis of $H$, and we have $$0<\inf\limits_{n}\|\phi_n\|,\quad \sup\limits_{n}\|\phi_n\|<\infty.$$ Since $\{\phi_n\}_{n=1}^{\infty}$ is a Schauder basis of $H$, every $x\in H$ has a unique norm-convergent expansion $$x=\sum\limits_{n=1}^{\infty} c_n \phi_n.$$ Then we define the operator $A:H\supset D(A)\mapsto H$ as follows, $$Ax=A \sum\limits_{n=1}^{\infty} c_n \phi_n = -\sum\limits_{n=1}^{\infty} n c_n \phi_n,$$ where $$D(A)=\left\{x=\sum\limits_{n=1}^{\infty} c_n \phi_n \in H:\: \sum\limits_{n=1}^{\infty} n c_n \phi_n \in H\right\}.$$
It can be easily shown that $A$ generates the $C_0$-semigroup on $H$, the spectrum of $A$ is pure point and consists of simple eigenvalues $-n$, $n\in\mathbb{N}$, with corresponding eigenvectors $\{\phi_n\}_{n=1}^{\infty}$, see, e.g., [@Haase]. Finally, it is not hard to prove that, if $\{A_n\}_{n=1}^{\infty}$ is a decomposition of $\mathbb{N}$ into disjoint sets with at most $K$ elements in each of them, such that (\[disj sets\]) holds, then $$\left\{\overline{Lin} \{\phi_j\}_{j\in A_n}\right\}_{n=1}^{\infty}$$ does not form a Riesz basis of subspaces of $H$.
For our construction of generators of $C_0$-groups with complete minimal non-basis family of eigenvectors in special classes of Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$ (see Theorem 16 in [@Sklyar3]), we have a result similar to the Theorem \[sharp\]. Here $\{e_n\}_{n=1}^{\infty}$ denotes an arbitrary symmetric basis of an initial Banach space $\ell_p$, $p\geq 1.$ Recall that Schauder basis $\{e_n\}_{n=1}^{\infty}$ is called symmetric provided any its permutation $\{e_{\theta(n)}\}_{n=1}^{\infty}$, $\theta(n): \mathbb{N}\mapsto \mathbb{N}$, also forms a Schauder basis, equivalent to $\{e_n\}_{n=1}^{\infty}$. For any $p\geq 1$ and $k\in\mathbb{N}$ the space $$\ell_{p,k}\left(\{e_n\}\right)=\left\{x=(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n:\: \{c_n\}_{n=1}^{\infty}\in \ell_p(\Delta^k)\right\},$$ where $(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n$ also denotes a formal series and $$\ell_p(\Delta^k)=\left\{\{c_n\}_{n=1}^{\infty}:\:\Delta^k \{c_n\}_{n=1}^{\infty} \in \ell_p\right\},$$ is a separable Banach space, isomorphic to $\ell_p$, see [@Sklyar3]. If $p>1$, then the sequence $\{e_n\}_{n=1}^{\infty}$ is dense and minimal in $\ell_{p,k}\left(\{e_n\}\right)$, $p>1,$ $k\in\mathbb{N}$, but it is not uniformly minimal in $\ell_{p,k}\left(\{e_n\}\right)$, so $\{e_n\}_{n=1}^{\infty}$ does not form a Schauder basis of $\ell_{p,k}\left(\{e_n\}\right)$. Using similar arguments we obtain the following result, analogous to Theorem \[sharp\].
\[sharp 2\] Let $k\in\mathbb{N}$ and $\{e_n\}_{n=1}^{\infty}\subset \ell_{p,k}\left(\{e_n\}\right)$, $p\geq 1,$ be a sequence defined above. Suppose that $\{A_n\}_{n=1}^{\infty}$ is an arbitrary decomposition of $\mathbb{N}$ into disjoint sets. Then $\left\{\overline{Lin} \{e_j\}_{j\in A_n}\right\}_{n=1}^{\infty}$ does not form a Schauder decomposition of $\ell_{p,k}\left(\{e_n\}\right)$.
An explicit form of the resolvent of the class of generators of $C_0$-groups
============================================================================
Recall that if $p>1$ and $a_n\geq0$ for $n\in\mathbb{N}$, then the discrete Hardy inequality states that $$\label{Hardy}
\sum\limits_{n=1}^{\infty} \left(\frac{1}{n} \sum\limits_{k=1}^{n} a_k\right)^p < \left(\frac{p}{p-1}\right)^p \sum\limits_{n=1}^{\infty} a_n^p$$ with the exception of the case when $a_n=0$ for all $n\in\mathbb{N}$. Moreover, the constant $\left(\frac{p}{p-1}\right)^p$ is the best possible.
The following theorem is a central result of the present paper. It provides an explicit form of the resolvent for the class of generators $A_k:H_k\left(\{e_n\}\right) \supset D(A_k) \mapsto H_k\left(\{e_n\}\right)$, $k\in\mathbb{N}$, of $C_0$-groups from Theorem \[construction\] and the description of the spectrum $\sigma(A_k)$ of generators $A_k.$
\[Expl\] Let $k\in\mathbb{N}$ and $A_k$ be the operator from Theorem \[construction\]. Then:
1. $\sigma(A_k)= \sigma_p(A_k)=\{i f(n)\}_{1}^{\infty}$.
2. The resolvent of $A_k$ is given by the following formula: $$\label{resolvent not basis fk}
\left(A_k- \lambda I \right)^{-1} x= (\mathfrak{f})\sum\limits_{n=1}^{\infty} \frac{c_{n} e_n}{i f(n)-\lambda}, \:\:\lambda \in \rho(A_k)=\mathbb{C}\setminus \{i f(n)\}_{1}^{\infty},$$ where $x=(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n\in H_k\left(\{e_n\}\right).$
First we prove the Theorem for the case $k=1$.
Let us prove that $\rho(A_1)=\mathbb{C}\setminus \{i f(n)\}_{1}^{\infty}$ is the resolvent set of the operator $A_1$ and the operator $$A(\lambda) x = (\mathfrak{f})\sum\limits_{n=1}^{\infty} \frac{1}{i f(n)-\lambda} c_n e_n,$$ where $\lambda \neq i f(n)$ for all $n\in \mathbb{N}$ and $x=(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n\in H_1\left(\{e_n\}\right),$ is the resolvent of $A_1$.
To this end denote $\lambda_n=i f(n),$ $n\in \mathbb{N}$. Recall that the norm in Hilbert space $H_1\left(\{e_n\}\right)$ is $$\|x\|_1=\left\|(\mathfrak{f})\sum\limits_{n=1}^{\infty}c_n e_n\right\|_1=\left\|\sum\limits_{n=1}^{\infty}\left(c_{n} - c_{n-1}\right) e_n\right\|,$$ where $\|\cdot \|$ denotes the norm in an initial Hilbert space $H$ and $c_0=0$, see [@Sklyar3]. Observe that $$\begin{aligned}
\left\|A(\lambda) x\right\|_1^2 &= \left\|(\mathfrak{f})\sum\limits_{n=1}^{\infty}\frac{c_n e_n}{\lambda_n-\lambda} \right\|_1^2=\left\|\frac{c_1 e_1}{\lambda_1-\lambda} + \sum\limits_{n=2}^{\infty}\left( \frac{c_n}{\lambda_n-\lambda}-\frac{c_{n-1}}{\lambda_{n-1}-\lambda} \right) e_n \right\|^2\\
&=\left\|\frac{c_1 e_1}{\lambda_1-\lambda}+\sum\limits_{n=2}^{\infty}\left( \frac{c_n}{\lambda_n-\lambda} - \frac{c_{n-1}}{\lambda_n-\lambda} + \frac{c_{n-1}}{\lambda_n-\lambda} -\frac{c_{n-1}}{\lambda_{n-1}-\lambda} \right) e_n \right\|^2\\
&\leq \left( \left\|\sum\limits_{n=1}^{\infty} \frac{c_n-c_{n-1}}{\lambda_n-\lambda} e_n \right\| + \left\|\sum\limits_{n=2}^{\infty}\left( \frac{1}{\lambda_n-\lambda}-\frac{1}{\lambda_{n-1}-\lambda} \right) c_{n-1} e_n \right\| \right)^2\\
&= \left(\Sigma_1 +\Sigma_2\right)^2 \leq 2 \Sigma_1^2 + 2\Sigma_2^2.\end{aligned}$$
Now consider $\lambda:\:\inf\limits_{n\in \mathbb{N}} |\lambda_n-\lambda|\geq a>0$. Since $\{e_n\}_{n=1}^{\infty}$ is a Riesz basis of a Hilbert space $H$ (see Theorem \[construction\]), there exist two constants $M\geq m>0$ such that for every $y=\sum\limits_{n=1}^{\infty} \alpha_n e_n\in H$ we have $$\label{Riesz}
m \|y\|^2 \leq \sum\limits_{n=1}^{\infty} |\alpha_n|^2 \leq M \|y\|^2.$$ Applying (\[Riesz\]) we obtain that $$\label{est Sigma 1}
\Sigma_1^2 \leq \frac{1}{m} \sum\limits_{n=1}^{\infty} \frac{|c_n-c_{n-1}|^2}{|\lambda_n-\lambda|^2}\leq \frac{1}{m a^2} \sum\limits_{n=1}^{\infty}|c_n-c_{n-1}|^2 \leq \frac{M}{m a^2} \|x\|_1^2.$$
Since $$\frac{1}{\lambda_n-\lambda}-\frac{1}{\lambda_{n-1}-\lambda} = \frac{\lambda_{n-1}-\lambda_n}{(\lambda_n-\lambda)(\lambda_{n-1}-\lambda)}$$ for $n\geq 2$, by virtue of (\[Riesz\]) we conclude that $$\Sigma_2^2 \leq \frac{1}{m} \sum\limits_{n=2}^{\infty} \frac{|\lambda_{n-1}-\lambda_n|^2 |c_{n-1}|^2}{|\lambda_n-\lambda|^2 |\lambda_{n-1}-\lambda|^2}\leq \frac{1}{m a^4} \sum\limits_{n=2}^{\infty} \frac{|c_{n-1}|^2}{n^2} n^2 |\Delta f(n)|^2.$$ Note that $\left\{f(n)\right\}_{n=1}^{\infty}\in\mathcal{S}_1$, hence $n|\Delta f(n)|\in \ell_{\infty}$ by the definition of the class $\mathcal{S}_1$, see Definition \[Class Sk\]. Denote $$C=\sup\limits_{n\in\mathbb{N}} n|\Delta f(n)|.$$ Then, since for $n\geq 2$ $$c_{n-1}= \sum\limits_{j=1}^{n-1} \left(c_j-c_{j-1}\right),$$ we obtain that $$\begin{aligned}
\Sigma_2^2 &\leq \frac{C^2}{m a^4} \sum\limits_{n=2}^{\infty} \frac{|c_{n-1}|^2}{n^2} = \frac{C^2}{m a^4} \sum\limits_{n=2}^{\infty} \left(\frac{1}{n} \left|\sum\limits_{j=1}^{n-1}(c_j-c_{j-1})\right|\right)^2 \\
&\leq \frac{C^2}{m a^4} \sum\limits_{n=1}^{\infty} \left(\frac{1}{n} \sum\limits_{j=1}^{n}|c_j-c_{j-1}|\right)^2.\end{aligned}$$
By virtue of the Hardy inequality (\[Hardy\]) for $p=2$ and (\[Riesz\]) we obtain $$\Sigma_2^2 \leq \frac{4C^2}{m a^4} \sum\limits_{n=1}^{\infty} |c_n-c_{n-1}|^2 \leq \frac{4 MC^2}{m a^4} \|x\|_1^2.$$ Combining this with (\[est Sigma 1\]) we finally arrive at the estimate $$\label{alambda}
\left\|A(\lambda) x\right\|_1^2 \leq \left(\frac{2}{a^2}+\frac{8C^2}{a^4} \right) \frac{M}{m} \|x\|_1^2$$ and $A(\lambda)\in[H_1\left(\{e_n\}\right)],$ i.e. $A(\lambda)$ is a linear bounded operator.
Further we choose arbitrarily $$\lambda:\:\inf\limits_{n\in \mathbb{N}} |if(n)-\lambda|\geq a>0,$$ fix $x=(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n\in H_1\left(\{e_n\}\right)$ and demonstrate that $A(\lambda) x \in D(A_1).$ For this purpose, taking into consideration (\[Domain\_k\]), it is sufficient to prove that $$\label{da}
\left\{\frac{i f(n) \cdot c_n}{i f(n)-\lambda}\right\}_{n=1}^{\infty}\in \ell_2(\Delta),$$ where $\ell_2(\Delta)=\left\{s=\{\alpha_n\}_{n=1}^{\infty}:\:\Delta s \in \ell_2\right\}$ and $\Delta$ is a difference operator given by $$\Delta=\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots\\ -1 & 1 & 0 & 0 & \dots\\ 0 & -1 & 1 & 0 & \dots\\ 0 & 0 & -1 & 1 & \dots\\ \vdots & \vdots & \vdots & \vdots & \ddots\end{array} \right).$$
To this end observe that $$\begin{aligned}
\sum\limits_{n=2}^{\infty} &\left| \frac{i f(n)\cdot c_n}{i f(n) -\lambda} - \frac{if(n-1) \cdot c_{n-1}}{if(n-1)-\lambda} \right|^2\\
&=\sum\limits_{n=2}^{\infty} \left| \left(c_n+\frac{\lambda c_n}{if(n) -\lambda}\right) -\left(c_{n-1}+\frac{\lambda c_{n-1}}{if(n-1) -\lambda}\right)\right|^2\\
& \qquad\leq 2\sum\limits_{n=1}^{\infty} |c_n-c_{n-1}|^2 + 2|\lambda|^2\sum\limits_{n=2}^{\infty} \left|\frac{ c_n}{if(n) -\lambda} -\frac{c_{n-1}}{if(n-1) -\lambda} \right|^2\\
&\qquad\qquad\leq 2M \|x\|_1^2 + 2|\lambda|^2 \Xi.\end{aligned}$$
For any $n\geq 2$ we have $$\begin{aligned}
\frac{ c_n}{if(n) -\lambda} -\frac{c_{n-1}}{if(n-1) -\lambda} &= \frac{ c_n}{if(n) -\lambda} -\frac{ c_{n-1}}{if(n) -\lambda} + \frac{ c_{n-1}}{if(n) -\lambda} - \frac{c_{n-1}}{if(n-1) -\lambda}\\
&= \frac{c_n- c_{n-1}}{if(n) -\lambda} + c_{n-1} \left( \frac{i(f(n-1)- f(n))}{(if(n) -\lambda)(if(n-1) -\lambda)}\right).\end{aligned}$$ It follows that $$\begin{aligned}
\Xi &\leq 2 \sum\limits_{n=2}^{\infty} \left|\frac{c_n- c_{n-1}}{if(n) -\lambda} \right|^2 + 2 \sum\limits_{n=2}^{\infty} \left|c_{n-1} \frac{f(n-1)- f(n)}{(if(n) -\lambda)(if(n-1) -\lambda)} \right|^2\\
&\leq \frac{2M}{a^2}\|x\|_1^2+ \frac{2}{a^4} \sum\limits_{n=2}^{\infty} n^2\left|\Delta f(n)\right|^2 \frac{\left|c_{n-1}\right|^2}{n^2}\\
&\leq \frac{2M}{a^2} \|x\|_1^2 + \frac{2C^2}{a^4} \sum\limits_{n=2}^{\infty}\frac{\left|c_{n-1}\right|^2}{n^2}\\
&\leq \frac{2M}{a^2} \|x\|_1^2 + \frac{2C^2}{a^4} \sum\limits_{n=1}^{\infty} \left(\frac{1}{n} \sum\limits_{j=1}^{n}|c_j-c_{j-1}|\right)^2.\end{aligned}$$
By virtue of the Hardy inequality (\[Hardy\]) for $p=2$ we have $$\Xi \leq \frac{2M}{a^2} \|x\|_1^2 + \frac{8 MC^2}{a^4} \|x\|_1^2.$$ Hence (\[da\]) holds. Therefore $A(\lambda) x \in D(A_1)$ and thus, $$\label{18888}
\left(A_1-\lambda I \right)A(\lambda) x = \left(A_1-\lambda I \right)(\mathfrak{f})\sum\limits_{n=1}^{\infty} \frac{1}{\lambda_n-\lambda} c_n e_n =(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n =x.$$
Now take $z\in D(A_1)$ and consider $x=\left(A_1-\lambda I \right) z.$ Then by (\[18888\]) we have that $$x=\left(A_1-\lambda I \right)A(\lambda) x = \left(A_1-\lambda I \right)A(\lambda)\left(A_1-\lambda I \right) z.$$ Consequently, $$\left(A_1-\lambda I \right)(z - A(\lambda)\left(A_1-\lambda I \right) z)=x-x=0.$$ Since $\lambda \neq if(n),$ $n\in \mathbb{N},$ then for every $z\in D(A_1)$ we have $$z = A(\lambda)\left(A_1-\lambda I \right)z,$$ and, combining this equality with (\[18888\]), we infer that $\lambda \in \rho(A_1)$ and $A(\lambda) = \left(A_1-\lambda I \right)^{-1}$ is the resolvent of $A_1$. Besides, we proved that $$\left\{ \lambda \in \mathbb{C}: \:\lambda \neq if(n), \: n\in \mathbb{N}\right\}\subset \rho(A_1).$$
Finally we observe that since $\lambda_n \in \sigma(A_1),\: n\in\mathbb{N},$ operator $A_1$ is closed as the generator of the $C_0$-group by Theorem \[construction\], the spectrum of closed operator is closed set and the set $\{i f(n)\}_{1}^{\infty}$ contains all its limit points, then $\sigma(A_1)= \sigma_p(A_1)= \{i f(n)\}_{1}^{\infty}$ and $$\rho(A_1)=\left\{ \lambda \in \mathbb{C}: \:\lambda \neq if(n), \: n\in \mathbb{N} \right\}=\mathbb{C} \setminus \sigma(A_1).$$
The proof in the case $k\geq 2$ is based on a combination of ideas of the proof for the case $k=1$ with technical combinatorial elements like in the proof of Theorem 11 from [@Sklyar3] and can be performed similarly to the above.
Define operators $$\widetilde{A_k}:\ell_{p,k}\left(\{e_n\}\right) \supset D\left(\widetilde{A_k}\right) \mapsto \ell_{p,k}\left(\{e_n\}\right)$$ on a class of Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$, see [@Sklyar3] or Section 2, as follows: $$\label{operator on Banach space}
\widetilde{A_k} x=\widetilde{A_k} (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n= (\mathfrak{f})\sum\limits_{n=1}^{\infty} i f(n) \cdot c_{n} e_n,$$ where $\left\{f(n)\right\}_{n=1}^{\infty}\in\mathcal{S}_k$, with domain $$\label{Domain_k p}
D\left(\widetilde{A_k}\right)=\left\{x= (\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n \in \ell_{p,k}\left(\{e_n\}\right):\: \{f(n) \cdot c_{n}\}_{n=1}^{\infty}\in \ell_p(\Delta^k)\right\}.$$
Then, by virtue of Theorem 16 in [@Sklyar3], $\widetilde{A_k}$ generates the $C_0$-group $\left\{e^{\widetilde{A_k} t} \right\}_{t\in \mathbb{R}}$ on $\ell_{p,k}\left(\{e_n\}\right)$, which acts on $\ell_{p,k}\left(\{e_n\}\right)$ for every $t\in\mathbb{R}$ by the formula $$\label{group}
e^{\widetilde{A_k} t} x=e^{\widetilde{A_k} t}(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_n e_n =(\mathfrak{f})\sum\limits_{n=1}^{\infty} e^{i t f(n)} c_{n} e_n.$$ An explicit form of the resolvent and the description of the spectrum $\sigma\left(\widetilde{A_k}\right)$ of generators $\widetilde{A_k}$ are provided by the following theorem, similar to the Theorem \[Expl\].
\[Expl B\] Let $k\in\mathbb{N}$, $p>1,$ and $\widetilde{A_k}$ be the operator defined above. Then:
1. $\sigma\left(\widetilde{A_k}\right)= \sigma_p\left(\widetilde{A_k}\right)=\{i f(n)\}_{1}^{\infty}$.
2. The resolvent of $\widetilde{A_k}$ is given by the following formula: $$\label{resolvent not basis fk p}
\left(\widetilde{A_k}- \lambda I \right)^{-1} x= (\mathfrak{f})\sum\limits_{n=1}^{\infty} \frac{c_{n} e_n}{i f(n)-\lambda}, \:\:\lambda \in \rho\left(\widetilde{A_k}\right)=\mathbb{C}\setminus \{i f(n)\}_{1}^{\infty},$$ where $x=(\mathfrak{f})\sum\limits_{n=1}^{\infty} c_{n} e_n\in \ell_{p,k}\left(\{e_n\}\right).$
If $\{e_n\}_{n=1}^{\infty}$ is a symmetric basis of $\ell_p$, then there exist two constants $\widetilde{M}\geq \widetilde{m}>0$ such that for every $\widetilde{y}=\sum\limits_{n=1}^{\infty} \alpha_n e_n\in \ell_p$ we have $$\widetilde{m} \|\widetilde{y}\|^p \leq \sum\limits_{n=1}^{\infty} |\alpha_n|^p \leq \widetilde{M} \|\widetilde{y}\|^p,$$ see [@Sklyar3] for details. Thus the proof repeats ideas and lines of the proof of Theorem \[Expl\].
Asymptotic behaviour of the resolvent
=====================================
For any closed linear operator $A$ on a Hilbert space $H$ the following bound for the norm of the resolvent is true: $$\label{resolvent bound}
\left\| \left(A-\lambda I\right)^{-1}\right\| \geq \frac{1}{\varrho\left(\lambda, \sigma(A)\right)},$$ provided that $\lambda\in \rho(A)$. Here $\varrho\left(\lambda, \sigma(A)\right)$ is the standard Euclidean distance between the point $\lambda$ and the spectrum $\sigma(A)$. If $A$ is normal operator on $H$, then by the spectral theorem for normal operators we immediately obtain that $$\label{resolvent b}
\left\| \left(A-\lambda I\right)^{-1}\right\| = \frac{1}{\varrho\left(\lambda, \sigma(A)\right)},$$ i.e. the inequality (\[resolvent bound\]) turns into an equality. However, equality (\[resolvent b\]) is not satisfied even for $2\times2$ nonselfadjoint matrix $$B=\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right),$$ since $\sigma(B)=\{1\}$ and $$\left(B-\lambda I\right)^{-1}=\left(\begin{array}{cc} \frac{1}{1-\lambda} & -\frac{1}{(1-\lambda)^2}\\ 0 & \frac{1}{1-\lambda} \end{array} \right),\: \lambda\neq 1.$$
This observation partially confirms the following commonly known thought: the spectrum does not contain much information about the behaviour of NSA operator $A$, see also [@Davies1], [@Davies2]. For this reason the notion of pseudospectra was introduced and came into play. The pseudospectra of $A$ is the family of sets $$\left\{\lambda\in\mathbb{C}:\: \left\| \left(A-\lambda I\right)^{-1}\right\| \geq \frac{1}{\varepsilon}\right\}_{\varepsilon>0},$$ see [@Davies1], [@Davies2], and it describes the behaviour of NSA operator $A$ much more effectively than the spectrum.
Another way to control the resolvent is to obtain for it estimates from above, see [@Davies1], [@Davies3], and works [@Malejki], [@Eisner1], [@Eisner2], [@Eisner3], where direct links between the polynomial growth of the $C_0$-semigroup $\left\{e^{At} \right\}_{t\geq 0}$ in $t$ and the behaviour of the corresponding resolvent $\left(A-\lambda I\right)^{-1}$ were established. Note that $C_0$-semigroups and $C_0$-groups with polynomial growth condition naturally appear in theory and applications of evolution equations, see, e.g., [@Goldstein], [@Sklyar2].
In 1985 A.I. Miloslavskii [@Miloslavskii] obtained sufficient conditions for $C_0$-semigroup on a Hilbert space to be polynomially bounded in terms of the behaviour of eigenvalues of the corresponding generator, under the assumption (\[question\]). B.A. Barnes [@Barnes] in 1989 obtained a number of interesting properties for generators of polynomially bounded $C_0$-groups on Banach spaces, but only in the case when generators are bounded. In 2001 M. Malejki [@Malejki] obtained necessary and sufficient conditions for a closed densely defined operator on a Banach space to be the generator of polynomially bounded $C_0$-group, in terms of the behaviour of the resolvent. In 2005 T. Eisner generalized results of Malejki to the case of polynomially bounded $C_0$-semigroups [@Eisner2]. The keystone of main results of works [@Malejki], [@Eisner2] are certain conditions on the integrability for the resolvent $$\left(A - (a+i\cdot)I \right)^{-1},$$ or the square of it, along lines parallel to the imaginary axis, where $a>0$.
Finally, in 2007 T. Eisner and H. Zwart [@Eisner1] obtained more simple characterizations of polynomial growth of a $C_0$-semigroup in terms of the first power of the resolvent of the generator. This was done in the class of operators, which have $p$-integrable resolvent for some $p>1.$ This class includes $C_0$-semigroups on Hilbert spaces and analytic $C_0$-semigroups on Banach spaces, see [@Eisner1] for details. For the overview and the prehistory of these results we refer to the Chapter III of the monograph [@Eisner3], where open questions, useful remarks and illustrative examples may also be found.
To describe the growth properties of the resolvent of operators from Theorem \[construction\] (see also Theorem \[Expl\]) and Theorem \[Expl B\] we use Proposition 12 from [@Sklyar3] on the polynomial boundedness of the constructed $C_0$-groups $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ on $H_k\left(\{e_n\}\right)$ and apply results from [@Eisner1], [@Eisner2], [@Malejki].
\[res asympt\] Let $k\in\mathbb{N}$ and $A_k$ be the operator from Theorem \[construction\].
Then the following assertions are true:
1. For every $a>0$ there exists a constant $C>0$ such that $$\label{r est}
\left\|\left(A_k- \lambda I \right)^{-1}\right\|\leq \frac{C}{\left|\Re (\lambda)\right|^{k+1}}\quad\text{for all}\quad \lambda: 0<\left|\Re (\lambda)\right|<a,$$ $$\label{r est 2}
\left\|\left(A_k- \lambda I \right)^{-1}\right\|\leq C\quad\text{for all}\quad \lambda:\left|\Re (\lambda)\right| \geq a.\qquad\qquad\:$$
2. There exists a constant $M>0$ such that for every $a>0$ and all $x,y\in H_k\left(\{e_n\}\right)$ we have $$\int\limits_{-\infty}^{\infty} \left|\bigl<\left(A_k\pm (a+is) I \right)^{-2}x,y\bigr> \right| ds\leq \frac{M}{a} \left(1+\frac{1}{a^{2k}} \right)\|x\|\|y\|.$$
3. There exists a constant $K>0$ such that for every $a>0$ and all $x,y\in H_k\left(\{e_n\}\right)$ we have $$\int\limits_{-\infty}^{\infty} \left\| \left(A_k\pm (a+is) I \right)^{-1}x\right\|^2 ds \leq \frac{K}{a} \left(1+\frac{1}{a^{2k}} \right)\|x\|^2,$$ and $$\int\limits_{-\infty}^{\infty} \left\| \left(A_k^{\ast}\pm (a+is) I \right)^{-1}y\right\|^2 ds \leq \frac{K}{a} \left(1+\frac{1}{a^{2k}} \right)\|y\|^2.$$
1\. Let $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ be the $C_0$-group corresponding to the operator $A_k$, see Theorem \[construction\]. Then, by virtue of Proposition 12 from [@Sklyar3], $C_0$-groups $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ grow in norm as $t\to\pm\infty$ but there exists a polynomial $\mathfrak{p}_k$ with positive coefficients such that $$\deg \mathfrak{p}_k =k$$ and for every $t\in\mathbb{R}$ we have $$\left\|e^{A_k t}\right\| \leq \mathfrak{p}_k (|t|).$$ So $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ belongs to the class of polynomially bounded $C_0$-groups. Hence the growth bound $\omega_{0,k}$ of the $C_0$-group $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ equals to zero, i.e. $$\omega_{0,k}=\lim\limits_{t\to\pm\infty} \frac{\ln\left\|e^{A_k t} \right\|}{t}=0,$$ see also Corollary 14 in [@Sklyar3]. Therefore the first part of the Theorem follows from the well-known representation of the resolvent of the generator in the form of the Laplace transform of the $C_0$-semigroup (group): $$\label{representation res}
\left(A_k- \lambda I \right)^{-1}=\int\limits_{0}^{\infty} e^{-\lambda t} e^{A_k t} dt,\quad \left|\Re(\lambda)\right|>0,$$ see, e.g., Theorem 11 in [@Dunford], see also Theorem 2.1 of [@Eisner1].
2\. Follows from Theorem 2.6 of [@Eisner2].
3\. Follows from Theorem 3 of [@Malejki].
Consider the case $k=1$ in Theorem \[res asympt\]. Then the conclusion of the Theorem \[res asympt\] follows from the proof of the Theorem \[Expl\]. Indeed, by (\[alambda\]) we have that for all $\lambda \in \rho\left(A_1\right)=\mathbb{C}\setminus \{i f(n)\}_{1}^{\infty}$ $$\left\|\left(A_1- \lambda I \right)^{-1}\right\| \leq \sqrt{\frac{2M}{m}}\frac{\sqrt{\left(\inf\limits_{n\in \mathbb{N}} |if(n)-\lambda|\right)^2+4 \left(\sup\limits_{n\in\mathbb{N}} n|\Delta f(n)| \right)^2}}{\left(\inf\limits_{n\in \mathbb{N}} |if(n)-\lambda|\right)^2},$$ which obviously leads to estimates (\[r est\]) and (\[r est 2\]). Thus, in general, the first part of the Theorem \[res asympt\] may be verified by direct computations and subtle estimates based on the Hardy inequality (\[Hardy\]) similar to those provided by the proof of the Theorem \[Expl\].
For the case of generators of $C_0$-groups acting on a class of Banach spaces $\ell_{p,k}\left(\{e_n\}\right),$ $p>1,$ $k\in\mathbb{N}$, we have the following result, analogous to the first part of the Theorem \[res asympt\].
\[res asympt p\] Let $k\in\mathbb{N}$, $p>1,$ and $\widetilde{A_k}:\ell_{p,k}\left(\{e_n\}\right) \supset D\left(\widetilde{A_k}\right) \mapsto \ell_{p,k}\left(\{e_n\}\right)$ be the operator defined by (\[operator on Banach space\]), (\[Domain\_k p\]), see also Theorem \[Expl B\].
Then for every $a>0$ there exists a constant $C>0$ such that $$\label{r est p}
\left\|\left(\widetilde{A_k}- \lambda I \right)^{-1}\right\|\leq \frac{C}{\left|\Re(\lambda)\right|^{k+1}}\quad\text{for all}\quad \lambda: 0<\left|\Re(\lambda)\right|<a,$$ $$\label{r est 2 p}
\left\|\left(\widetilde{A_k}- \lambda I \right)^{-1}\right\|\leq C\quad\text{for all}\quad \lambda:\left|\Re(\lambda)\right| \geq a.\qquad\qquad\:$$
Denote by $\left\{e^{\widetilde{A_k} t} \right\}_{t\in \mathbb{R}}$ the $C_0$-group corresponding to the operator $\widetilde{A_k}$, see Theorem 16 of [@Sklyar3]. Then, by virtue of Proposition 17 from [@Sklyar3], there exists a polynomial $\mathfrak{p}_k$ with positive coefficients such that $$\deg \mathfrak{p}_k =k$$ and for every $t\in\mathbb{R}$ we have $$\left\|e^{\widetilde{A_k} t}\right\| \leq \mathfrak{p}_k (|t|).$$ Therefore the required estimates follow from the formula for representation of the resolvent (\[representation res\]), see also Theorem 2.1 of [@Eisner1].
The weak spectral mapping theorem holds for our classes of $C_0$-groups, since they are polynomially bounded.
\[wsmp\] Let $k\in\mathbb{N}$, $p>1,$ $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ is the $C_0$-group corresponding to the operator $A_k$, see Theorem \[construction\], and $\left\{e^{\widetilde{A_k} t} \right\}_{t\in \mathbb{R}}$ is the $C_0$-group corresponding to the operator $\widetilde{A_k}$, see (\[group\]), Section 3.
Then for all $t\in\mathbb{R}$ $$\sigma\left(e^{A_k t}\right) = \overline{e^{t\sigma(A_k)}},$$ $$\sigma\left(e^{\widetilde{A_k} t}\right) = \overline{e^{t\sigma\left(\widetilde{A_k}\right)}}.$$
Propositions 12 and 17 from [@Sklyar3] yield that $\left\{e^{A_k t} \right\}_{t\in \mathbb{R}}$ and $\left\{e^{\widetilde{A_k} t} \right\}_{t\in \mathbb{R}}$ are polynomially bounded $C_0$-groups. The application of Theorem 7.4 from [@Nagel] (p. 91) completes the proof.
A measurable and locally bounded function $f:\mathbb{R}\mapsto \mathbb{R}$ is called by a non-quasianalytic weight provided that for all $t,s\in\mathbb{R}$ we have $$f(t)\geq 1,\quad f(t+s)\leq f(t) f(s) \quad\text{and} \quad\int\limits_{-\infty}^{\infty} \frac{\ln f(t)}{1+t^2}dt<\infty.$$
Clearly, every polynomially bounded $C_0$-group satisfy the non-quasianalytic growth condition, i.e. the condition $$\left\|T(t)\right\| \leq f(t),\quad t\in\mathbb{R},$$ where $f$ is a non-quasianalytic weight. Note that there exist $C_0$-groups $\left\{T(t) \right\}_{t\in \mathbb{R}}$, which do not satisfy the non-quasianalytic growth condition, such that the corresponding generators have empty spectrum. Thus the weak spectral mapping theorem does not hold in the class of $C_0$-groups, which do not satisfy the non-quasianalytic growth condition. This result was first proved in [@Huang]. For details see also Chapter 2, Section 2.4 of [@vanNeerven].
*Acknowledgments.* The research was partially supported by the State Fund For Fundamental Research of Ukraine (project no. $\Phi83/82-2018$).
[SK]{}
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abstract: 'The problem of estimating the eddy diffusivity from Lagrangian observations in the presence of measurement error is studied in this paper. We consider a class of incompressible velocity fields for which is can be rigorously proved that the small scale dynamics can be parameterised in terms of an eddy diffusivity tensor. We show, by means of analysis and numerical experiments, that subsampling of the data is necessary for the accurate estimation of the eddy diffusivity. The optimal sampling rate depends on the detailed properties of the velocity field. Furthermore, we show that averaging over the data only marginally reduces the bias of the estimator due to the multiscale structure of the problem, but that it does significantly reduce the effect of observation error.'
author:
- 'C.J. Cotter [^1]'
- 'G.A. PAVLIOTIS [^2]'
bibliography:
- '../bibtex\_files/mybib.bib'
title: Estimating eddy diffusivities from noisy Lagrangian observations
---
Parameter estimation, stochastic differential equations, multiscale analysis, Lagrangian observations, subsampling, oceanic transport.
[**Subject classifications.**]{} 62M05, 86A05, 86A10, 60H10, 60H30, 62F12
Introduction {#sec:intro}
============
Many phenomena in the physical sciences involve a multitude of characteristic temporal and spatial scales. In most cases it is not only impossible to study the behavior of the phenomenon at all scales, but it is also unnecessary, since usually one is interested in the evolution of a few variables which describe the dynamics at large scales. It is, therefore, important to develop systematic methods for deriving simplified coarse grained models that capture the essential features of the systems at long scales, while accurately parameterising the small scales. In recent years it has become clear that the use of data, together with coarse graining procedures, is essential for the accurate parameterisation of small scales [@GKS04; @CVE06a; @CVE06b; @IKDS07; @HS08]. The aim of this paper is to study problems of this form in the context of transport of passive tracers.
We are particularly motivated by the challenge of using Lagrangian float data to inform the design of subgrid mixing schemes for advected tracers in ocean models. The vast amount of Lagrangian float data available (for example, the ARGO project has 3000 floats in current operation [@ARGO06]) presents the opportunity to develop data-driven model reduction techniques. Lagrangian data are particularly suitable for statistical studies of the transport of passively advected substances in the ocean, with the simplest statistical description of transport phenomena provided by the average concentration of a passive tracer.
In this paper, we assume that the Lagrangian trajectories are given by the following stochastic differential equation: $$\label{e:lagrange}
\dot{x} = v(x,t) + \sqrt{2 \kappa} \dot{W}.$$ Here $x(t) \in {\mathbb{R}}^d$ represents the Lagrangian path, $v(x,t)$ is a (prescribed) incompressible velocity field, $\kappa$ is the small-scale diffusivity and $W(t)$ denotes standard Brownian motion in ${\mathbb{R}}^d$. More sophisticated models have been proposed for oceanographic applications, for example [@BeMc02; @BeMc03; @Pi02; @Wi05; @Gr+95].
We wish to extract the coarse grained (large length scale and long time scale) dynamics of solutions of equation (\[e:lagrange\]). For a wide class of velocity fields (deterministic space-time periodic, Gaussian random fields *etc.*), it is well known [@kramer; @PavlSt08; @lions] that, at sufficiently long length and time scales, the dynamics of becomes Brownian and can be described by the [**eddy diffusivity tensor**]{}. More precisely, it is possible to prove that solutions of converge, under the diffusive rescaling and assuming that the velocity field has zero mean, to an effective Brownian motion $$\label{e:weak_limit}
\lim_{{\epsilon}\rightarrow 0} {\epsilon}x(t/{\epsilon}^2) = \sqrt{2 {\mathcal{K}}} W(t),$$ weakly on $C([0,T]; {\mathbb{R}}^d)$, where $W(t)$ is a standard Brownian motion on ${\mathbb{R}}^d$ and ${\mathcal{K}}$ denotes the eddy (effective) diffusivity tensor. The tensor ${\mathcal{K}}$ represents the effective diffusivity caused by the interaction of molecular diffusion with the transport properties of $v$. Consequently, at large length scales and long time scales the dynamics of the passive tracer is governed by an equation of the form $$\label{e:coarse_grained}
\dot{X} = \sqrt{2 {\mathcal{K}}} \dot{W}.$$ It is quite often the case (in designing subgrid mixing schemes for example) that we only wish to calculate the eddy diffusivity, rather than the detailed properties of the velocity field $v(x,t)$ at all scales. It is then necessary to estimate the eddy diffusivity of a passive tracer from Lagrangian observations. In this paper we address precisely this issue: given a Lagrangian trajectory which is consistent with in the presence of observation noise, how can we estimate the eddy diffusivity ${\mathcal{K}}$? This problem has been studied quite extensively over the last few years [@sb:emfde2; @haf:eredd; @haf:eredd2; @sb:emfde; @mv:otsms].
More generally, we might also want to estimate other coarse grained quantities such as the effective drift, or we might want to consider a space dependent eddy diffusivity. This is a challenging problem in statistical inference: data sampled from is only consistent with at sufficiently large scales. In other words, the difficulty stems from the fact that the model used for fitting the data is the wrong model, apart from the large scale part of the data. Furthermore, we do not know *a priori* the length and time scales on which the coarse grained model is valid. On the other hand, we can perform statistical inference in a fully parametric setting for , since only the eddy diffusivity needs to be estimated; statistical inference for would require the non-parametric estimation of the velocity field $v(x,t)$ [@CDRS09]. Parameter estimation for diffusion processes under misspecified or incorrect models has been studied in the statistics literature [@Kut04 Sec 2.6.1].
The problem of parameter estimation for a model that is incompatible with the available data at small scales was studied in [@PavlSt06; @PapPavSt08; @PavlPokStu08] for a class of fast-slow systems of SDEs for which the existence of a coarse grained equation for the slow variables can be proved rigorously. In these papers, parameter estimation for the [**averaging problem**]{}
\[e:av\_intro\] $$\begin{aligned}
\label{e:av_intro_x}
\frac{dx}{dt} &=& f_1(x,y)+\alpha_0(x,y)\frac{dU}{dt}+\alpha_1(x,y)\frac{dV}{dt}, \\
\frac{dy}{dt} &=& {\frac{1}{{\epsilon}}}g_0(x,y)+{\frac{1}{{\sqrt{\epsilon}}}}\beta(x,y)\frac{dV}{dt};\end{aligned}$$
as well as for the [**homogenization problem**]{}
\[e:hom\_intro\] $$\begin{aligned}
\label{e:hom_intro_x}
\frac{dx}{dt} &=& {\frac{1}{{\epsilon}}}f_0(x,y)+f_1(x,y)+\alpha_0(x,y)\frac{dU}{dt}+\alpha_1(x,y)\frac{dV}{dt}, \\
\frac{dy}{dt} & =& {\frac{1}{{\epsilon}^2}}g_0(x,y)+{\frac{1}{{\epsilon}}}g_1(x,y)+{\frac{1}{{\epsilon}}}\beta(x,y)\frac{dV}{dt}.\end{aligned}$$
was studied. In both cases the goal was to fit data obtained from or to the coarse grained equation $$\label{e:coarse_intro}
\frac{d X}{d t} = F(X; \theta) + K(X) \frac{d W}{d t},$$ which describes the dynamics of the slow variable $x(t)$ in the limit as ${\epsilon}\rightarrow 0$. In the aforementioned papers, it was assumed that the vector field $F(X;\theta)$ depends on a set of parameters $\theta$ that we want to estimate using data taken from either the averaging or the homogenization problem. For the homogenization problem it was shown in [@PapPavSt08] that the maximum likelihood estimator is asymptotically biased. In particular, it is necessary to subsample at an appropriate rate in order to estimate the parameters $\theta$ accurately. Similar issues were investigated for the thermal motion of a particle in a multiscale potential [@PavlSt06]. It was shown that subsampling is necessary for the accurate estimation of the drift and diffusion coefficients.
Related issues have been studied in the field of econometrics. In this context, the question is how to accurately estimate the integrated stochastic volatility when market microstructure noise (*i.e.* additive white noise) is present. It was shown in [@AitMykZha05a; @AitMykZha05b] that subsampling reduces the bias in the estimator. It was also shown that subsampling combined with averaging and an appropriate de-biasing step can lead to an accurate and efficient estimator for the integrated stochastic volatility.
In this paper we will study the problem of estimating the eddy diffusivity from noisy Lagrangian observations: $$y_{t_j} = x_{t_j} + \theta {\epsilon}_{t_j}, \quad j=1, \dots N,$$ where $\{x_0,x_1,\ldots,x_N\}$ is a set of samples from a trajectory consistent with equation , ${\epsilon}_{t_j}$ are independent $\mathcal{N}(0,1)$ random variables modelling the observation error, and $\theta$ measures the strength of the observation error.
We will consider time-independent spatially-periodic incompressible velocity fields as well as spatially-periodic velocity fields that are modulated in time by a time-periodic function, or a Gaussian process. In all of these cases, the rescaled trajectory converges weakly to a Brownian motion (see [@PavlSt08 Ch. 13]). The eddy diffusivity depends in a highly nonlinear way on the properties of the velocity field $v(x,t)$. It can be shown (for the class of velocity fields considered in this paper) that the eddy diffusivity ${\mathcal{K}}$ satisfies the upper and lower bounds (we use the notation ${\mathcal{K}}^{\xi} = \langle \xi, {\mathcal{K}}\xi \rangle$ where $\xi$ is an arbitrary vector in ${\mathbb{R}}^d$) [@AvelMajda91] $$\label{e:bounds}
\kappa \leq {\mathcal{K}}^{\xi} \leq \frac{C}{\kappa},$$ for $\kappa$ sufficiently small and some positive constant $C$. We will consider the physically interesting regime $\kappa \ll 1$.
As an eddy diffusivity estimator we will use the quadratic variation $$\label{e:quad_variation}
{\mathcal{K}}_{N,\delta} = \frac{1}{2 N \delta} \sum_{n=0}^{N-1} \big( x_{n+1}- x_n
\big) \otimes \big( x_{n+1}- x_n \big),$$ where $N$ is the number of observations which we assume to be equidistant, with the distance between two subsequent observations being $\delta$, and $T = N \delta$. It is well known [@BasRao80] that, for an SDE of the form , we have the convergence result $$\label{e:kappa_lim}
\lim_{N \rightarrow \infty} \sum_{j=0}^{N-1} \big( x_{(j+1) T 2^{-N}}
- x_{j T 2^{-N}} \big) \otimes \big( x_{(j+1) T 2^{-N}}
- x_{j T 2^{-N}} \big) = 2 \kappa I T, \quad \mbox{a.s.}$$ where $I$ denotes the unit matrix. If we write equation with $x_i$ replaced by $y_i$, the quadratic variation diverges in the limit as $N\to \infty$ due to the observation error. In view of the bounds , it becomes clear that that the estimator ${\mathcal{K}}_{N,\delta}$ underestimates the value of the eddy diffusivity in this limit. In particular, when the eddy diffusivity scales like $\kappa^{-1}$ the estimator can underestimate the eddy diffusivity by several orders of magnitude.
The above suggests that in order to be able to estimate the eddy diffusivity from Lagrangian data, subsampling at an appropriate rate is necessary. However, it is not clear *a priori* what the sampling rate should be. Roughly speaking, we need to look at the data at the scale for which the coarse grained description is valid. The estimation of this time scale is a difficult dynamical question that has been addressed only partially [@Fann01; @HP07]. The [*diffusive time*]{}, the time that it takes for the Lagrangian particle to reach the asymptotic regime described by a Brownian motion with diffusion matrix ${\mathcal{K}}$ depends crucially on the streamline topology and is related to the scaling of the eddy diffusivity with $\kappa$. Clearly we have two sources of error: measurement error, and the error in the estimation of parameters from reduced models using data from the full dynamics which we refer to as the [**multiscale error**]{}. The multiscale error is precisely due to the fact that the reduced model is incompatible with the data at small scales.
In this paper we study the small $\kappa$ asymptotics of the quadratic variation . We show, by means of rigorous analysis and numerical experiments, that, unless we subsample at an appropriate rate, we cannot estimate the eddy diffusivity from the quadratic variation, due to the multiscale error. Additionally, we show that for smooth time-independent spatially periodic velocity fields, the scaling of the optimal sampling rate with $\kappa$ depends on the detailed properties of the velocity field. Our analysis is based on standard limit theorems for stochastic processes, together with careful study of a Poisson equation posed on the unit torus.
From the point of view of statistics, it is clearly not optimal to simply ignore most of the available data by subsampling[^3]. It is natural, therefore, to try to use all data through averaging. We experiment with two different types of averaging: box averaging (computing the quadratic variation using local averages), and shift averaging (which is related to the moving averaging method of statistics). We show by means of numerical experiments, that shift averaging significantly reduces the effects of observation error, but only marginally reduces the multiscale error. On the other hand, box averaging increases the bias of the estimator.
We emphasize that the setting in which we are working is related to but different from the problems studied in [@PavlSt06; @PapPavSt08; @PavlPokStu08]. In particular, we do not assume *a priori* that we have scale separation and that we know the value of the parameter ${\epsilon}$ which measures the degree of scale separation. Rather, the scale separation is induced by the dynamics of . The time scale at which the coarse grained description is valid is essentially what we need to estimate, since this provides us with information about the appropriate sampling rate. For completeness, we also consider the rescaled problem below. The rescaled problem and the original one are equivalent under space-time rescaling for time-independent velocity fields. However, from the point of view of estimating the eddy diffusivity they lead to different problems. When using the quadratic variation to estimate the eddy diffusivity in equation we actually study the small $\kappa$ limit, whereas for the rescaled problem we study the limit of infinite scale separation while keeping $\kappa$ fixed.
The rest of the paper is organized as follows. In Section \[sec:resc\] we we study the problem of estimating the eddy diffusivity using Lagrangian observations from the rescaled equation. In Section \[sec:small\_kappa\] we study the same problem for the unscaled equation , and we also study the effect of observation error on the estimator. In Section \[sec:numerics\] we develop estimators for the eddy diffusivity which are based on a combination of subsampling with averaging and present numerical results for various types of two-dimensional velocity fields. Summary and conclusions are presented in Section \[sec:conclusions\]. Some technical results are included in the appendices.
The Rescaled Problem {#sec:resc}
====================
We consider the equation for the rescaled process $$x^{\epsilon}(t) = {\epsilon}x(t/{\epsilon}^2),$$ given by $$\label{e:rescaled}
\frac{d x}{d t} = \frac{1}{{\epsilon}} v \left(\frac{x}{{\epsilon}} \right) + \sqrt{2
\kappa} \dot{W},$$ where we have dropped the superscript ${\epsilon}$ for notational simplicity.. Our goal is to estimate the eddy diffusivity using data from , in the parameter regime ${\epsilon}\ll 1$ and for $\kappa$ fixed. In particular, we want to find how the sampling rate should scale with ${\epsilon}$ for the accurate estimation of the eddy diffusivity using the quadratic variation estimator. The main result of this section is that, provided that the sampling rate is in between the two characteristic time scales $1$ and ${\epsilon}$ of the problem, then the estimator is asymptotically unbiased, in the limit as $\epsilon\to 0.$
We assume that the velocity field is smooth, divergence-free, mean zero and $1-$periodic, *i.e.* periodic with period $1$ in each Cartesian direction. Under these assumptions, the solution to converges weakly on $C([0,T]; {\mathbb{R}}^d)$ to $X$, as ${\epsilon}\rightarrow 0$, the solution of $$\frac{d X}{d t} =\sqrt{2 {\mathcal{K}}} \frac{d W}{d t},$$ as $\epsilon\to 0$. The eddy diffusivity is given by the formula $$\label{e:eddy_diff}
{\mathcal{K}}= \kappa I + \kappa \int_{{\mathbb{T}}^d} \nabla_z \chi(z) \otimes \nabla_z \chi(z) \, dz$$ where the vector field $\chi(z)$ is the solution of the PDE $$\label{e:cell}
-{\mathcal{L}}_0 \chi(z) = v(z)$$ on ${\mathbb{T}}^d$ with periodic boundary conditions, and where ${\mathcal{L}}_0$ is the generator of the Markov process $z$ on ${\mathbb{T}}^d$: $$\label{e:z_process}
\frac{d z}{d t} = v(z) + \sqrt{2 \kappa} \frac{d W}{d t},$$ *i.e.* $${\mathcal{L}}_0 = v(z) \cdot \nabla_z + \kappa \Delta_z,$$ with periodic boundary conditions. We refer to [@PavlSt08] for the derivation of this result.
Now let ${\mathcal{K}}^\xi = \xi \cdot {\mathcal{K}}\xi$ where $\xi \in {\mathbb{R}}^d$ arbitrary. From it easily follows that $$\label{e:deff_xi}
{\mathcal{K}}^\xi = \kappa \int_{{\mathbb{T}}^d} |\xi + \nabla_z \chi^\xi|^2 \, dz,$$ where $\chi^\xi = \chi \cdot \xi$. Let ${\mathcal{K}}^{\xi}_{N, \delta}$ be the quadratic variation along the direction $\xi$: $$\label{e:qv_xi}
{\mathcal{K}}^{\xi}_{N,
\delta} = \frac{1}{2 N \delta} \sum_{n=0}^{N-1} \big( x_{n+1}^{\xi}
- x^{\xi}_n \big)^2,$$ where $x^{\xi}_n = x (n \delta) \cdot \xi$.
Our goal is to find how the sampling rate $\delta$ should be chosen so that we can estimate the component of the eddy diffusivity [(\[e:eddy\_diff\])]{} along the direction $\xi$ using . The following theorem states that the estimator converges in $L^2$ to the eddy diffusivity in the limit $\epsilon \to
0$, $N\to \infty$, with $T$ fixed.
\[thm:rescaled\] Let $v(z)$ be a smooth, divergence-free, mean zero, 1-periodic vector field and assume that the process $z$ defined in is stationary. Then $${\mathbb{E}}|{\mathcal{K}}_{N, \delta}^\xi - {\mathcal{K}}^\xi |^2 \leq \frac{C}{N} + C \big({\epsilon}^4 \delta^{-2} + {\epsilon}^3 \delta^{-3/2}+ {\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big).$$ In particular, when $\delta = {\epsilon}^{\alpha}, \, \alpha \in (0,2)$, we have $$\lim_{N \rightarrow +\infty}\lim_{{\epsilon}\rightarrow 0} {\mathbb{E}}|
{\mathcal{K}}^{\xi}_{N, \delta} - {\mathcal{K}}^{\xi} |^2 =0,$$ for $N\delta = T$ fixed (i.e. $N \sim {\epsilon}^{-\alpha}$).
The scaling of the optimal sampling rate with ${\epsilon}$, $\delta \sim {\epsilon}^{\alpha},
$ with $\alpha \in (0,2)$ appears to be sharp and it is expected on intuitive grounds, since one would expect that the optimal sampling rate should be in between the two characteristic time scales of the problem $1$ and ${\epsilon}^2$.
The stationarity assumption on $z$ can be removed since even when $z$ starts with arbitrary initial conditions its law converges exponentially fast to the invariant measure of the process which is the Lebesgue measure on ${\mathbb{T}}^d$. We refer to [@bhatta_2] for the details.
For the proof of this theorem we will need the following lemma.
\[lem:rescaled\] Let $v(z)$ be a smooth, divergence-free, mean zero, 1-periodic vector field and assume that the process $z$ defined in is stationary. Then $$|{\mathbb{E}}{\mathcal{K}}^{\xi}_{N, \delta} - {\mathcal{K}}^{\xi} | \leq C \big({\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big).$$ In particular, when $\delta = {\epsilon}^{\alpha}, \, \alpha \in (0,2)$ we have $$\lim_{{\epsilon}\rightarrow 0}|{\mathbb{E}}{\mathcal{K}}^{\xi}_{N, \delta} - {\mathcal{K}}^{\xi} | =0.$$
Notice that in order for the expectation of the quadratic variation to converge to the eddy diffusivity it is not necessary to take the limit $N \rightarrow \infty$. Of course, in order to keep $N \delta = T$ fixed we need to take $N \sim \delta^{-1} =
\kappa^{-\alpha}$.
*Proof of Lemma \[lem:rescaled\]* With the help of the auxiliary process $z = x/{\epsilon}\in {\mathbb{T}}^d$, equation can be rewritten as a system of SDEs:
\[e:rescaled\_system\] $$\frac{d x}{d t} = \frac{1}{{\epsilon}} v \left(z \right) + \sqrt{2
\kappa} \dot{W}.$$ $$\frac{d z}{d t} = \frac{1}{{\epsilon}^2} v \left(z \right) + \sqrt{\frac{2
\kappa}{{\epsilon}^2}} \dot{W}.$$
The generator of the Markov process $\{x(t), \, z(t) \}$ is $$\begin{aligned}
{\mathcal{L}}^{\epsilon}& =& \frac{1}{{\epsilon}^2} \big(v(x)\cdot \nabla_z + \kappa \Delta_z \big)
+ \frac{1}{{\epsilon}} \big(v(x)\cdot \nabla_z + 2 \kappa \nabla_x \cdot \nabla_z \big) + \kappa \Delta_x \\ & =: & \frac{1}{{\epsilon}^2} {\mathcal{L}}_0
+ \frac{1}{{\epsilon}} {\mathcal{L}}_1 + {\mathcal{L}}_2.\end{aligned}$$ Let $\chi^\xi(z)$ denote the solution of the Poisson equation $$-{\mathcal{L}}_0 \chi^\xi = v \cdot \xi =:v^\xi(z)$$ on ${\mathbb{T}}^d$ with periodic boundary conditions. From standard elliptic PDE theory we have that $\xi^{\xi} \in C^{\infty}({\mathbb{T}}^d)$. Hence, we can apply Itô’s formula to $\chi^\xi$ and use the fact that $\chi^\xi$ is independent of $x$ to obtain $$\begin{aligned}
d \chi^\xi &=& \left( \frac{1}{{\epsilon}^2} {\mathcal{L}}_0 \chi^\xi + \frac{1}{{\epsilon}} {\mathcal{L}}_1 \chi^\xi + {\mathcal{L}}_2 \chi^\xi \right)\, dt + \frac{\sqrt{2 \kappa}}{{\epsilon}} \nabla_y
\chi^\xi \cdot d W \\ & = & -\frac{1}{{\epsilon}^2} v^\xi (z)\, dt + \frac{\sqrt{2 \kappa}}{{\epsilon}} \nabla_z \chi^\xi \cdot d W.\end{aligned}$$ Consequently: $$\frac{1}{{\epsilon}} \int_{n \delta}^{(n+1)\delta} v^\xi (z_s) \, ds = - {\epsilon}\big(
\chi^\xi(z_{n+1}) - \chi^\xi(z_{n}) \big) + \sqrt{2 \kappa}\int_{n \delta}^{(n+1)\delta} \nabla_z \chi^\xi \cdot d W.$$ Thus: $$\begin{aligned}
x^\xi_{n+1} - x^\xi_n &=& - {\epsilon}\big(
\chi^\xi(z_{n+1}) - \chi^\xi(z_{n}) \big) + \sqrt{2 \kappa}\int_{n \delta}^{(n+1)\delta} \big(\nabla_z \chi^\xi + \xi \big) \cdot d W \\ & =: & {\epsilon}R_n + \sqrt{2} M_n.\end{aligned}$$ The quadratic variation becomes $$\begin{aligned}
{\mathcal{K}}_{N, \delta}^\xi &=& \frac{1}{2 N \delta} \sum_{n=0}^{N-1} \left( {\epsilon}^2 R_n^2 + 2 \sqrt{2} {\epsilon}R_n M_n + 2 M_n^2 \right).\end{aligned}$$ Since we have assumed that $z(t)$ is stationary, we have that $$\label{e:Mn}
{\mathbb{E}}|M_n|^2 = \kappa \|\xi + \nabla_z \chi^\xi \|^2_{L^2({\mathbb{T}}^d)} \delta = {\mathcal{K}}^\xi \delta$$ from which it follows that $${\mathbb{E}}\left( \frac{1}{ N \delta} \sum_{n=0}^{N-1} M_n^2 \right) = K^\xi.$$ Furthermore, the maximum principle for elliptic PDEs implies that $${\mathbb{E}}|R_n|^2 \leq C.$$ We use now the above calculations and Cauchy-Schwarz to obtain $$\begin{aligned}
{\mathbb{E}}{\mathcal{K}}_{N, \delta}^\xi - {\mathcal{K}}^\xi & = & \frac{1}{2 N \delta} \sum_{n=0}^{N-1} {\epsilon}^2 {\mathbb{E}}R_n^2 + 2 \sqrt{2} {\epsilon}{\mathbb{E}}\big( R_n M_n \big) \\ & \leq & C \big( {\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big).\end{aligned}$$ $\square$
[*Proof of Theorem \[thm:rescaled\]*]{}
From Lemma \[lem:rescaled\] we have that $$\begin{aligned}
{\mathbb{E}}|{\mathcal{K}}_{N, \delta}^\xi - {\mathcal{K}}^\xi |^2 & = & {\mathbb{E}}| {\mathcal{K}}_{N, \delta}^\xi |^2 - |{\mathcal{K}}^\xi|^2 + 2 {\mathcal{K}}^\xi \big({\mathcal{K}}^{\xi} - {\mathcal{K}}^{\xi}_{N,\delta} \big) \\ & \leq & {\mathbb{E}}| {\mathcal{K}}_{N, \delta}^\xi |^2 - |{\mathcal{K}}^\xi|^2 + C \big( {\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big).\end{aligned}$$ Hence, it is sufficient to estimate the difference $E | {\mathcal{K}}_{N, \delta}^\xi |^2 - |K^\xi|^2$. Using the notation introduced in the proof of Lemma \[lem:rescaled\] we can write $$\begin{aligned}
|K^{\xi}_{N, \delta}|^2 & = & \frac{1}{4 N^2 \delta^2} \sum_{n=0}^{N-1}\sum_{\ell=0}^{N-1} \left( {\epsilon}^2 R_n^2 + 2 \sqrt{2} {\epsilon}R_n M_n + 2 M_n^2 \right) \left( {\epsilon}^2 R_{\ell}^2 + 2 \sqrt{2} {\epsilon}R_{\ell} M_{\ell} + 2 M_{\ell}^2 \right) \nonumber \\ & = & \frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1}\sum_{\ell=0}^{N-1} M_n^2 M_{\ell}^2 + R, \label{e:kxi}\end{aligned}$$ where $$\begin{aligned}
R & = & \frac{1}{4 N^2 \delta^2} \sum_{n=0}^{N-1}\sum_{\ell=0}^{N-1} \Big( {\epsilon}^4 R_n^2 R_{\ell}^2 + 4 \sqrt{2} {\epsilon}^3 R_n^2 R_{\ell} M_{\ell}
\\ &&+ 4 {\epsilon}^2 R_n M_{\ell}^2 + 8 {\epsilon}^2 R_n R_{\ell} M_n M_{\ell} +8 \sqrt{2} {\epsilon}M_n M_{\ell}^2 R_n \Big) \\ & =:& I + II + III + IV + V.\end{aligned}$$ The uniform bound on $\chi^{\xi}$ and its derivatives, bounds on moments of stochastic integrals [@KSh91] and the Cauchy-Schwarz inequality yield the bounds $$\begin{aligned}
{\mathbb{E}}\, I \leq C {\epsilon}^4 \delta^{-2}, \quad {\mathbb{E}}\, II \leq C {\epsilon}^3 \delta^{-3/2}, \quad {\mathbb{E}}\, III \leq C {\epsilon}^2 \delta^{-1}, \quad {\mathbb{E}}\, IV \leq C {\epsilon}^2 \delta^{-1}, \quad {\mathbb{E}}\, V \leq C {\epsilon}\delta^{-1/2}.\end{aligned}$$ From the above bounds we deduce that $$\label{e:r_estim}
{\mathbb{E}}\, R \leq C \big({\epsilon}^4 \delta^{-2} + {\epsilon}^3 \delta^{-3/2}+ {\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big).$$ Now we use bounds on moments of stochastic integrals, together with the fact that ${\mathbb{E}}(M_n M_\ell) = 0$ for $n \neq \ell$ to calculate $$\begin{aligned}
{\mathbb{E}}\left(\frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1}\sum_{\ell=0}^{N-1} M_n^2 M_{\ell}^2 \right) & = & \frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1} {\mathbb{E}}( M_n^4 ) + \frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1}\sum_{\ell \neq n} {\mathbb{E}}( M_n^2 ) {\mathbb{E}}( M_{\ell}^2) \\ & \leq & \frac{C}{N} + \frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1}\sum_{\ell \neq n} {\mathbb{E}}( M_n^2 ) {\mathbb{E}}( M_{\ell}^2).\end{aligned}$$ On the other hand, from Equation we deduce that $$\begin{aligned}
{\mathbb{E}}\left|K^\xi_{N,\delta} \right|^2 & = & \frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1}\sum_{\ell =0}^{N-1} {\mathbb{E}}( M_n^2 ) {\mathbb{E}}( M_{\ell}^2) \nonumber
\\ & = &
\frac{1}{ N^2 \delta^2}\sum_{n=0}^{N-1}\sum_{\ell \neq n} {\mathbb{E}}( M_n^2 ) {\mathbb{E}}( M_{\ell}^2) + \mathcal{O} \left(\frac{1}{N} \right). \label{e:kxi_delta}\end{aligned}$$ We combine the above estimates to obtain $$\begin{aligned}
{\mathbb{E}}|{\mathcal{K}}_{N, \delta}^\xi - {\mathcal{K}}^\xi |^2 & \leq & {\mathbb{E}}| {\mathcal{K}}_{N, \delta}^\xi |^2 - |K^\xi|^2 + C \big( {\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big) \\ & \leq & \frac{C}{N} + C \big({\epsilon}^4 \delta^{-2} + {\epsilon}^3 \delta^{-3/2}+ {\epsilon}^2 \delta^{-1} + {\epsilon}\delta^{-1/2} \big).\end{aligned}$$
$\square$
Small $\kappa$ Asymptotics for the Quadratic Variation {#sec:small_kappa}
======================================================
In this section we consider the original problem $$\label{e:sde}
\frac{d x}{d t} = v(x) + \sqrt{2 \kappa} \frac{d W}{d t}.$$ Our goal is to estimate the eddy diffusivity using data from , in the parameter regime $\kappa \ll 1$. In particular, we want to find how the sampling rate should scale with $\kappa$ for the accurate estimation of the eddy diffusivity using the quadratic variation estimator. The main result of this section is that in order for the estimator to be asymptotically unbiased in the limit as $\kappa \rightarrow 0$, it is necessary that the sampling rate (as well as the number of observations, and hence the time interval of observation) must scale with $\kappa$ in an appropriate way, which depends on the detailed properties of the velocity field. In particular, the optimal sampling rate might become unbounded in the limit as $\kappa
\rightarrow 0$ for flows for which the eddy diffusivity also becomes unbounded in this limit. Furthermore, our results are not sharp and detailed analysis is required for each particular flow. In contrast with the rescaled problem that was studied in the previous section, there doesn’t seem to be a simple intuitive argument to explain the scaling of the optimal sampling rate with $\kappa$, since the longest characteristic time scale of the problem (the diffusive time scale) needs to be estimated, as a function of $\kappa$.
As in the previous section we are interested in analyzing the quadratic variation along an arbitrary direction $\xi$ and to calculate the optimal sampling rate in order to be able to estimate the eddy diffusivity from observations. Let ${\mathcal{K}}^{\xi}_{N,
\delta}$ be the quadratic variation along the direction $\xi$ is given by Equation $$\label{e:qv_unresc}
{\mathcal{K}}^{\xi}_{N,
\delta} = \frac{1}{2 N \delta} \sum_{n=0}^{N-1} \big( x_{n+1}^{\xi} - x^{\xi}_n \big)^2$$ where $x^{\xi}_n = x (n \delta) \cdot \xi$. The eddy diffusivity along the direction $\xi$ is given by Equation $${\mathcal{K}}^\xi = \kappa \int_{{\mathbb{T}}^d} |\xi + \nabla_z \chi^\xi|^2 \, dz$$ where $\chi^\xi = \chi \cdot \xi$ is the unique mean zero solution of the elliptic PDE $$\label{e:cell_kappa}
- ( v(z) \cdot \nabla_z + \kappa \Delta_z ) \chi^{\xi} = v^\xi$$ with periodic boundary conditions on the unit torus. In order to study the small $\kappa$ asymptotics of the quadratic variation ${\mathcal{K}}_{N, \delta}^\xi$ we need information on the small $\kappa$ asymptotics of $\chi^\xi$, the solution of . From the PDE and Poincaré’s inequality we deduce the bounds $$\|\chi^\xi \|_{L^2} \leq C \|\nabla_z \chi^\xi \|_{L^2} \leq \frac{C}{\kappa}.$$ The precise asymptotic behavior of $\chi^\xi$ in the small $\kappa$ regime depends on the detailed properties of the velocity field $v(z)$. This difficult problem has been studied quite extensively [@ConstKiselRyzhZl06; @bhatta_1; @Fann01; @MajMcL93]. In this paper we will assume that the solution of the cell problem satisfies the following small-$\kappa$ scaling $$\label{e:kappa_scaling}
\|\chi^\xi \|_{L^p} \sim \|\nabla_z \chi^\xi \|_{L^p} \sim \kappa^{\alpha},
\quad \alpha \in [-1,0], \quad \kappa \ll 1,$$ for $p=2, \, 4$. The notation $ f \sim \kappa^\alpha$ means that there exists constants $C_+, \, C_-$ so that $$C_- \kappa^\alpha \leq f \leq C_+ \kappa^\alpha, \quad \mbox{for} \;\; \kappa \ll 1.$$ Some examples of flows for which the scaling of $\chi^\xi$, the solution of , with $\kappa$ is known are:
1. The two-dimensional shear flow ${\bf v}({\bf x}) = (0, \sin(x))$ [@kramer; @MajMcL93]. For this flow we can solve the Poisson equation explicitly: $$\chi_1(x,y) = 0, \quad \chi_2 (x,y) = -\kappa^{-1} \sin(x)$$ and, consequently, for all $\kappa > 0$, $$\|\chi_2 \|_{L^p} \sim \|\nabla \chi_2 \|_{L^p} \sim \kappa^{-1}.$$
2. The Taylor-Green flow $${\bf v}(x,y) = \nabla^{\bot} \psi_{TG}(x,y), \, \phi_{TG}(x,y) = \sin(x) \sin(y).$$ In this case it is not possible to solve . However it is possible to obtain sharp estimates on the solution of the Poisson equation: $$\|\nabla \chi^\xi \|_{L^2} \sim \kappa^{-1/2}, \quad \kappa \ll 1$$ for all vectors $\xi \in {\mathbb{R}}^2$. See [@Heinz03] for details. On the other hand, by the maximum principle we have that $\|\chi \|_{L^2} \leq C$, uniformly in $\kappa$ [@Fann01].
3. The Childress-Soward flow $${\bf v}(x,y) = \nabla^{\bot} \psi_{CS}(x,y), \, \phi_{CS}(x,y) = \sin(x) \sin(y) +
\lambda\cos(x) \cos(y),$$ where $\lambda\in[0,1]$. This flow interpolates between the Taylor-Green flow (for $\delta=0$) and a shear flow (for $\delta=1$).
In this case we have that $$\|\chi^{\xi_1} \|_{L^2} \sim \kappa^{-1}, \quad \|\chi^{\xi_2} \|_{L^2} \sim 1,$$ where $\xi_1 = 1/\sqrt{2} (1,1), \, \xi_2 = 1/\sqrt{2} (-1,1)$. See [@childress1; @Fann01] for details.
More examples of flows for which the small-$\kappa$ asymptotics of $\chi^\xi$ can be calculated will be presented in Section \[sec:numerics\].
Notice that the above scaling leads to $$\label{e:deff_scaling}
{\mathcal{K}}^{\xi} \sim \kappa^{2 \alpha +1},$$ which is consistent with , since $2 \alpha +1 \in [-1,1]$.
Of course, the exponent $\alpha$ in in general depends in the direction $\xi$ as well as the $L^p$-space, $\alpha =\alpha(\xi,p)$. For simplicity we will assume that $\alpha$ is independent of $p$. The analysis presented below can be easily extended to cover the case where $\alpha =\alpha(p)$.
Convergence results
-------------------
In this section we prove the following.
\[thm:l2\_unrescaled\] Let $v(z)$ be a smooth, divergence-free smooth vector field on ${\mathbb{T}}^d$. Assume that the scaling \[e:kappa\_scaling\] with $p=4$ holds. Then the following estimate holds $$\begin{aligned}
{\mathbb{E}}|{\mathcal{K}}^\xi_{N,\delta} - {\mathcal{K}}^\xi|^2
\leq C \Big(\frac{1}{N} \kappa^{4 \alpha + 2}
+\kappa^{4 \alpha + 1} \delta^{-1} + \kappa^{4 \alpha} \delta^{-2}
+ \kappa^{4 \alpha + \frac{3}{2}} \delta^{-\frac{1}{2}}
+ \kappa^{4 \alpha +\frac{1}{2}} \delta^{-\frac{3}{2}}
\Big). \label{e:l2_bound}\end{aligned}$$ In particular, if $N \sim \kappa^{\zeta}$ with $\zeta > 4 \alpha +2$ and $\delta \sim \kappa^{\gamma}$ with $\gamma < \min(4\alpha +1, 2 \alpha, 8 \alpha+1, \frac{8 \alpha}{3} + \frac{1}{3})$. Then $$\label{e:l2_convergence}
\lim_{\kappa \rightarrow 0} {\mathbb{E}}|{\mathcal{K}}_{N,\delta}^\xi - {\mathcal{K}}^\xi|^2 =0.$$
Estimate is not sharp. See the examples of the steady and modulated in time shear flows in the next section.
Notice that $T = N\delta \to \infty$ as $\kappa$ goes to $\infty$, and notice that the sampling rate may also have to go to $\infty$ depending on the value of $\alpha$. This is in constrast to the rescaled problem, for which convergence occurs as $\epsilon \to 0$ with $T$ fixed.
We first prove the following weak convergence result.
\[lem:expect\_unresc\] Let $v(z)$ be a smooth, divergence-free smooth vector field on ${\mathbb{T}}^d$. Assume that the scaling \[e:kappa\_scaling\] with $p=2$ holds $$\big| {\mathbb{E}}{\mathcal{K}}^{\xi}_{N,\delta} - {\mathcal{K}}^{\xi} \big| \leq C \big(
\kappa^{2 \alpha + \frac{1}{2}} \delta^{-\frac{1}{2}} + \kappa^{2 \alpha} \delta^{-1} \big).$$ In particular, if $\delta = \kappa^{\gamma}$ with $\gamma < \min(2 \alpha, 4 \alpha+1)$ then $$\lim_{\kappa \rightarrow 0} \big| {\mathbb{E}}{\mathcal{K}}^{\xi}_{N,\delta} - {\mathcal{K}}^{\xi} \big| = 0.$$
We apply Itô’s formula to $\chi^\xi$ to write the increment of the process $x^\xi$ as $$\begin{aligned}
x^\xi_{n+1} - x^{\xi}_n & = & \sqrt{2 \kappa} \int_{n \delta}^{(n+1) \delta}
\big( \nabla_z \chi^\xi + \xi \big) \cdot d W - (\chi^\xi(z_{n+1} ) - \chi^\xi(z_{n} ))
\nonumber \\ & =: & \sqrt{2} M_n + R_n, \label{e:decompose}\end{aligned}$$ where $$\langle M_n \rangle = \kappa \int_{n \delta}^{(n+1) \delta} |\nabla_z \chi^\xi + \xi|^2
\, dz \quad \mbox{and} \quad {\mathbb{E}}\langle M_n \rangle = \delta {\mathcal{K}}^\xi.$$
Upon combining and and taking the expectation we obtain $$\begin{aligned}
{\mathbb{E}}{\mathcal{K}}^{\xi}_{N,\delta} = {\mathcal{K}}^{\xi} + \frac{\sqrt{2}}{N \delta} \sum_{n=0}^{N-1} {\mathbb{E}}\left( M_n \, R_n \right) + \frac{1}{2 N \delta} \sum_{n=0}^{N-1} {\mathbb{E}}R_n^2.\end{aligned}$$ We use now and to deduce that $$\begin{aligned}
{\mathbb{E}}{\mathcal{K}}^{\xi}_{N,\delta} - {\mathcal{K}}^{\xi} & \leq & \frac{\sqrt{2}}{N \delta} \sum_{n=0}^{N-1}
({\mathbb{E}}M_n^2)^{1/2} \, ( {\mathbb{E}}R_n^2)^{1/2} + \frac{1}{2 N \delta} \sum_{n=0}^{N-1} {\mathbb{E}}R_n^2 \nonumber \\
& \leq & C \kappa^{2 \alpha + \frac{1}{2}} \delta^{-\frac{1}{2}} + C \kappa^{2 \alpha}
\delta^{-1}. \label{e:estimate_l1}\end{aligned}$$
[*Proof of Theorem \[thm:l2\_unrescaled\].*]{}
From Lemma \[lem:expect\_unresc\] we have that $$\label{e:R_defn}
{\mathbb{E}}{\mathcal{K}}^\xi_{N,\delta} = {\mathcal{K}}^\xi + R$$ with $$\label{e:R_estimate}
|R| \leq C \big( \kappa^{2 \alpha + \frac{1}{2}} \delta^{-\frac{1}{2}} +
\kappa^{2 \alpha} \delta^{-1} \big).$$ We can write $$\label{e:kxi2}
{\mathbb{E}}|{\mathcal{K}}^\xi_{N,\delta} - {\mathcal{K}}^\xi|^2 = {\mathbb{E}}\big| {\mathcal{K}}^\xi_{N,\delta} \big|^2 -
({\mathcal{K}}^\xi)^2 - 2 R {\mathcal{K}}^\xi.$$ We introduce the notation $$\big| {\mathcal{K}}^\xi_{N, \delta} \big|^2 = I^2 + II^2 + III^2 + 2 I \, II + 2 I \,
III + 2 II \, III$$ with $$\begin{aligned}
I = \frac{1}{N \delta} \sum_{n=0}^{N-1} M_n^2, \quad II = \frac{\sqrt{2}}{N \delta}
\sum_{n=0}^{N-1} M_n R_n, \quad III = \frac{1}{ 2 N \delta} \sum_{n=0}^{N-1} R_n^2.\end{aligned}$$ We use to deduce that $${\mathbb{E}}I^2 = \frac{N-1}{N} |K^\xi|^2 + \frac{1}{N^2 \delta^2} \sum_{n=0}^{N-1} {\mathbb{E}}M_n^4.$$ Furthermore, $$\begin{aligned}
{\mathbb{E}}M_n^4 & = & {\mathbb{E}}\left( \sqrt{\kappa} \int_{n \delta}^{(n+1) \delta} (\xi +
\nabla_z \chi^\xi) \, dW \right)^4 \\ & \leq & C \kappa^2 \delta^2 \|\xi +
\nabla_z \chi^\xi \|^4_{L^2({\mathbb{T}}^d)}\end{aligned}$$ Scaling \[e:kappa\_scaling\] together with bounds on moments of stochastic integrals implies that $${\mathbb{E}}M_n^4 \leq C \kappa^{4 \alpha+2} \delta^2.$$ We conclude that $${\mathbb{E}}I^2 \leq |{\mathcal{K}}^\xi|^2 + C \frac{1}{N} \kappa^{4 \alpha +2}.$$ Consequently $${\mathbb{E}}I^2 \leq C \left(1 + \frac{1}{N} \right) \kappa^{4 \alpha +2}.$$
From Assumption we get $$\left( {\mathbb{E}}|R_n|^p \right)^{1/p} \leq C \kappa^\alpha.$$ Now we have $$\begin{aligned}
{\mathbb{E}}II^2 & = & {\mathbb{E}}\left(\frac{2}{N^2 \delta^2} \sum_{n=0}^{N-1} \sum_{k=0}^{N-1}
R_n M_n R_k M_k \right) \\ & \leq & \frac{2}{N^2 \delta^2} \sum_{n=0}^{N-1}
\sum_{k=0}^{N-1} ({\mathbb{E}}|R_n|^4)^{1/4} ({\mathbb{E}}|M_n|^4)^{1/4} ({\mathbb{E}}|R_k|^4)^{1/4}
({\mathbb{E}}|M_k|^4)^{1/4} \\ & \leq & C \kappa^{4\alpha +1} \delta^{-1}.\end{aligned}$$ Similarly, $$\begin{aligned}
{\mathbb{E}}III^2 & = & {\mathbb{E}}\left(\frac{1}{4 N^2 \delta^2} \sum_{n=0}^{N-1} \sum_{k=0}^{N-1}
R_n^2 R_k^2 \right) \\ & \leq & C \kappa^{4 \alpha} \delta^{-2}.\end{aligned}$$ We use now the Cauchy-Schwarz inequality to obtain the estimates (we use the fact that $N \geq 1$)
$$\begin{aligned}
{\mathbb{E}}(I \, II) & \leq & C \kappa^{4 \alpha +\frac{3}{2}} \delta^{-\frac{1}{2}}, \\
{\mathbb{E}}(I \, III) & \leq & C \kappa^{4 \alpha + 1} \delta^{-1} , \\
{\mathbb{E}}(II \, III) & \leq & C \kappa^{4 \alpha + \frac{1}{2}} \delta^{-\frac{3}{2}}.\end{aligned}$$
We use all of the above estimates, together with and estimate , to obtain estimate .
$\square$
![\[shearbars\]Figure showing statistics for estimators of the eddy diffusivity for the shear flow. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=5.1$, and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](shearbars){width="12cm"}
![\[modbars\]Figure showing statistics for estimators of the eddy diffusivity for the periodically-modulated shear flow with modulation frequency $\omega=1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.125$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](modbars){width="12cm"}
![\[OUbars\]Figure showing statistics for estimators of the eddy diffusivity for the OU-modulated shear flow with parameters $\alpha=1$, $\sigma=0.1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.145$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](OUbars){width="12cm"}
![\[TGbars\]Figure showing statistics for estimators of the eddy diffusivity for the Taylor-Green flow. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.342$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](TGbars){width="12cm"}
![ \[eps01shearbars\] Figure showing statistics for estimators of the eddy diffusivity for the shear flow, applied to the rescaled problem with $\epsilon=0.1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (center ) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=5.1$, and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines. ](eps0_1shearbars){width="12cm"}
![\[eps01modbars\]Figure showing statistics for estimators of the eddy diffusivity for the periodically-modulated shear flow with modulation frequency $\omega=1$, applied to the rescaled problem with $\epsilon=0.1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.125$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](eps0_1modbars){width="12cm"}
![\[eps01OUbars\]Figure showing statistics for estimators of the eddy diffusivity for the OU-modulated shear flow with parameters $\alpha=1$, $\sigma=0.1$, applied to the rescaled problem with $\epsilon=0.1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.145$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](eps0_1OUbars){width="12cm"}
![\[eps01TGbars\]Figure showing statistics for estimators of the eddy diffusivity for the Taylor-Green flow, applied to the rescaled problem with $\epsilon=0.1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.342$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](eps0_1TGbars){width="12cm"}
![ \[shearnoise\] Figure showing statistics for estimators of the eddy diffusivity for the shear flow, where $\mathcal{N}(0,0.1)$ observation noise has been added. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (center ) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=5.1$, and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines. ](shear_noise0_1){width="12cm"}
![\[modnoise\] Figure showing statistics for estimators of the eddy diffusivity for the periodically-modulated shear flow with modulation frequency $\omega=1$, where $\mathcal{N}(0,0.1)$ observation noise has been added. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.125$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](shearmod_noise0_1){width="12cm"}
![\[OUnoise\] Figure showing statistics for estimators of the eddy diffusivity for the OU-modulated shear flow with parameters $\alpha=1$, $\sigma=0.1$, where $\mathcal{N}(0,0.1)$ observation noise has been added. with $\epsilon=0.1$. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.145$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](shearOU_noise0_1){width="12cm"}
![\[TGnoise\] Figure showing statistics for estimators of the eddy diffusivity for the Taylor-Green flow, where $\mathcal{N}(0,0.1)$ observation noise has been added. The plots show results for various values of the subsampling interval $\delta$ from (left) the maximum likelihood estimator [(\[e:quad\_variation\])]{}, (centre) the shift-averaged estimator [(\[shift averaging\])]{}, and (right) the box-averaged estimator [(\[box averaging\])]{}. The plots indicate the mean value of the estimators (circular dots), as well as the standard deviation (bars) with statistics computed from 1000 realisations of the Lagrangian trajectory. The correct value ${\mathcal{K}}=0.342$ (3 d.p.), and the value of the small-scale diffusivity $\kappa=0.1$ are both indicated as horizontal lines.](TG_noise0_1){width="12cm"}
The Effect of Observation Error
-------------------------------
In this subsection we study the small $\kappa$ asymptotics of the quadratic variation in the presence of observation error. More specifically, we assume that the observed process (along the direction $\xi$) is $$\label{e:noise}
Y^{\xi}_{t_j} = X^{\xi}_{t_j} + \theta {\epsilon}^{\xi}_{t_j}, \quad j=1, \dots N.$$ The parameter $\theta >0$ measures the strength of the measurement noise which we model through a collection of i.i.d $\mathcal{N}(0,1)$ random variables ${\epsilon}^{\xi}_{t_j}$, which are independent from the Brownian motion driving the Lagrangian dynamics. Since the two sources of noise that appear in the problem are assumed to be independent, the analysis presented in this section also applies to Equation . In particular, we have that $${\mathbb{E}}{\mathcal{K}}^{\xi}_{N,\delta} (Y_t) = {\mathbb{E}}\left( {\mathcal{K}}^{\xi}_{N,\delta} (X_t)\right) + \frac{\theta^2}{\delta}.$$ In view of estimate , we have that $$\left| {\mathbb{E}}\left( {\mathcal{K}}^{\xi}_{N,\delta} (Y_t) \right) - {\mathcal{K}}^{\xi} \right| \leq C \kappa^{2 \alpha + \frac{1}{2}} \delta^{-\frac{1}{2}} + C \kappa^{2 \alpha}
\delta^{-1} + \theta^2 \delta^{-1}.$$ In particular, if $\delta = \kappa^{\gamma}$ with $\gamma < \min(2 \alpha, 4 \alpha+1, 0)$ then $$\lim_{\kappa \rightarrow 0} \big| {\mathbb{E}}{\mathcal{K}}^{\xi}_{N,\delta} - {\mathcal{K}}^{\xi} \big| = 0.$$ We remark that the exponent $\gamma$ is different to the one that appears in the statement of Lemma \[lem:expect\_unresc\], in that it must be negative, irrespective of the scaling of the eddy diffusivity with $\kappa$.
Similarly, in the presence of measurement error, estimate has to be modified. It becomes $$\label{e:unresc_noise}
{\mathbb{E}}\left|{\mathcal{K}}^\xi_{N,\delta} (Y_t) - {\mathcal{K}}^\xi \right|^2 =
{\mathbb{E}}\left|{\mathcal{K}}^\xi_{N,\delta} (X_t) - {\mathcal{K}}^\xi \right|^2 + 3 \frac{\theta^4}{\delta^2} + 2 \theta^2 \left(\frac{1}{\delta} + \frac{2}{N
\delta} \right) ({\mathcal{K}}^\xi + R),$$ where $R$ is defined in equation and estimated in . We can then use Theorem \[thm:l2\_unrescaled\] to bound the first term on the right hand side of equation . Clearly, we require that $\delta \rightarrow \infty$ for the additional terms (which are due to the measurement error) to vanish.
The Two-Dimensional Shear Flow
------------------------------
In this section we present some results for a particular class of flows for which we can compute the quadratic variation explicitly. The purpose of this is to show that the results obtained in Theorem \[thm:l2\_unrescaled\] are not sharp.
For two-dimensional flows of the form $$\label{e:shear_gen}
v(x,y,t) = (0, \eta(t) \sin(x)),$$ where $\eta (t)$ can be either a constant, a periodic function or a stochastic process, we can calculate explicitly the statistics of the quadratic variation of the Lagrangian trajectories [@ma:mmert; @McL98; @kramer]. In the appendix it is shown that for $\eta(t) \equiv 1$, the quadratic variation along the direction of the shear is $$\begin{aligned}
\label{e:quad_shear}
\mathbb{E} {\mathcal{K}}_{N, \delta}
& = & {\mathcal{K}}+ \frac{1}{2\kappa^2\delta}(e^{-\kappa\delta}
-1) + \frac{1}{4\kappa^2T}
\left(
\frac{2}{3}e^{-\kappa\delta} - \frac{1}{6}e^{-4\kappa\delta}-\frac{1}{2}
\right)\frac{1-e^{-4\kappa T}}{1-e^{-4\kappa\delta}},\end{aligned}$$ where $T=N\delta$ and the effective diffusivity is $${\mathcal{K}}= \kappa + \frac{1}{2 \kappa}.$$ From the above formula we immediately deduce that $$\begin{aligned}
\lim_{\kappa \rightarrow 0}\mathbb{E} \left[ {\mathcal{K}}_{N, \delta}-{\mathcal{K}}\right] = 0 \end{aligned}$$ provided that $$\label{e:scaling_shear}
\delta=\kappa^{-2-\epsilon}.$$ for ${\epsilon}>0$, arbitrary. Furthermore, when holds, we have that $$\label{e:conv_rate}
\lim_{\kappa \rightarrow 0} \kappa^{-{\epsilon}} \left( \mathbb{E} {\mathcal{K}}_{N, \delta}-{\mathcal{K}}\right) = -\frac{1}{2} - \frac{1}{8 N},$$ the convergence being exponential in $\kappa$.
It is also possible to calculate ${\mathbb{E}}|{\mathcal{K}}_{N, \delta} - {\mathcal{K}}|^2$. In particular, we have that $$\begin{aligned}
\nonumber
\mathbb{E}
\left| {\mathcal{K}}_{N,\delta}- {\mathcal{K}}\right|^2 & = &
\frac{1}{N \delta^2}\Bigg(
c_1 \frac{1}{\kappa^4} + c_2 \delta \frac{1}{\kappa^3}
+c_3 {\delta}^{2} \frac{1}{\kappa^2} + c_4\delta^2 + \\
& & \qquad\qquad c_5 \frac{\delta}{\kappa}
+
c_6 \kappa^2\delta^2 + c(\delta\kappa)\Bigg)
\nonumber \\
\nonumber && +
\frac{1}{N^2 \delta^2}\Bigg(
d_1 \frac{1}{\kappa^4} + d_2 \delta \frac{1}{\kappa^3}
+ d_3 {\delta}^{2} \frac{1}{\kappa^2} +
\\
& & \qquad \qquad
d_4\delta^2 + d_5 \frac{\delta}{\kappa}
+
d_6 \kappa^2\delta^2 + d(\delta\kappa)\Bigg), \label{e:variance_shear}\end{aligned}$$ where the constants $\{c_i, \, d_i ; i=1, \dots 6\}$ can be calculated explicitly and $c(\delta \kappa), \, d( \delta \kappa)$ converge to a constant exponentially quickly in the limit as $\delta \kappa
\rightarrow + \infty$. From the above formula we immediately deduce that $$\begin{aligned}
\lim_{\kappa \rightarrow 0}\mathbb{E} \left| {\mathcal{K}}_{N, \delta}-{\mathcal{K}}\right|^2 = 0 \end{aligned}$$ provided that holds, together with $N \sim
\kappa^{-2 - {\epsilon}}$, ${\epsilon}>0$. Furthermore, under these assumptions on $\delta$ and $N$ we have that $$\label{e:conv_rate_2}
\lim_{\kappa \rightarrow 0} \kappa^{-{\epsilon}} \mathbb{E} \left| {\mathcal{K}}_{N, \delta}-{\mathcal{K}}\right|^2 = \mbox{const}.$$ This example shows that Theorem \[thm:l2\_unrescaled\] is not sharp. Some details of the calculation of the first two moments of the quadratic variation for the time independent two-dimensional shear flow are presented in Appendix \[sec:shear\].
Numerical experiments {#sec:numerics}
=====================
In this section we illustrate the results of the previous sections with some numerical experiments, and we investigate some modifications to the eddy diffusivity estimator which we shall describe below. The purpose of the numerical experiments that we have performed is to investigate the following issues:
1. The performance of the estimator for the eddy diffusivity as a function of the sampling rate for flows with different streamline topologies.
2. Whether an appropriate averaging procedure can reduce the variance of the estimator.
3. The performance of the estimator for the eddy diffusivity as a function of the sampling rate for the rescaled problem.
4. The performance of the estimator in the presence of measurement noise.
The main conclusions from our numerical experiments can be summarised as follows:
1. The variance of the estimator as well as the optimal sampling rate depend crucially on the streamline topology of the velocity field.
2. Shift averaging (see below) marginally reduces the variance due to multiscale error of the estimator, whereas box averaging (also see below) introduces extra bias into the estimator.
3. There is an optimal sampling rate for the estimator applied to the rescaled problem, but even when using the optimal sampling rate the variance of the estimator can be very large.
4. When the data is subject to measurement noise then subsampling is necessary, even in the absence of multiscale error. Appropriate averaging can reduce the variance due to measurement error.
The Estimators
--------------
We are given a time series of Lagrangian observations of length $T$, sampled at a constant rate $\Delta t$. The number of observations is $N = T/\Delta t$. Our goal is to estimate the eddy diffusivity using the quadratic variation $$\label{e:quad}
{\mathcal{K}}_{N,\delta} = \frac{1}{2 N \delta} \sum_{n=0}^{N-1} \big( x_{n+1}- x_n
\big) \otimes \big( x_{n+1}- x_n \big),$$ We will consider both the unrescaled as well as the rescaled problems . The results presented in Sections and suggest that subsampling at an appropriate rate is necessary in order to estimate the eddy diffusivity correctly, using Lagrangian observations. In the numerical experiments presented in this section we will take the sampling rate to scale either with $\kappa$ (for the unrescaled problem) or with $\epsilon$ (for the rescaled problem), according to the results presented in Theorems \[thm:l2\_unrescaled\] and : $$\delta \sim \kappa^\alpha, \quad \mbox{or} \quad \delta \sim {\epsilon}^{\alpha},$$ for some appropriate exponent $\alpha$.
Even if we use with $\delta$ chosen optimally, the resulting estimator is clearly not optimal since we are using only a very small portion of the available data. Furthermore, the variance of with subsampled data can be enormous, in particular when $\kappa \ll 1$ or ${\epsilon}\ll 1$. One may attempt to reduce the bias and variance in the estimator by making use of all the data. In particular, it is reasonable to expect that subsampling combined with averaging over the data might lead to a more efficient estimator of the eddy diffusivity with reduced bias in comparison to the estimator . This methodology was applied in [@AitMykZha05a; @AitMykZha05b] in order to estimate the integrated stochastic volatility in the presence of market microstructure noise (observation error).
The most natural way of averaging over the data is by splitting the data into $N_B$ bins of size $\delta$ with $\delta N_B = N$ and to perform a local averaging over each bin. We use the notation $$x^j_n :=x ((n-1) \delta + (j-1) \Delta t), \quad n=1, \dots N_B, \;
j=1, \dots J, \quad JN_B = N,$$ for the $j$-th observation in the $n$-th bin. $J=\delta/\Delta t$ is the number of observations in each bin. The maximum likelihood estimator [(\[e:quad\_variation\])]{} is then computed using the averaged values $$\bar{x}_n = \frac{1}{J}\sum_{j=1}^J x^j_n,$$ leading to the [**box-averaged**]{} estimator: $$\label{box averaging}
{\mathcal{K}}_{N_B,\delta}^b = \frac{1}{2 N_B \delta} \sum_{n=0}^{N_B-1} \left(
\frac{1}{J}\sum_{j=1}^J x^j_{n+1}
-
\frac{1}{J}\sum_{j=1}^J x^j_n
\right) \otimes \left(
\frac{1}{J}\sum_{j=1}^J x^j_{n+1}
-
\frac{1}{J}\sum_{j=1}^J x^j_n
\right).$$ A second averaging technique, proposed in [@AitMykZha05b; @AitMykZha05a] to remove the effects of market microstructure noise, is to compute a series of estimators, each using a different observation from each bin, and then to compute the average. This is the [**shift-averaged**]{} estimator: $$\label{shift averaging}
{\mathcal{K}}_{N_B,\delta}^s =
\frac{1}{J}\sum_{j=1}^J
\frac{1}{2 N_B \delta}
\sum_{n=0}^{N_B-1} \left(
x^j_{n+1}
-
x^j_n
\right) \otimes \left(
x^j_{n+1}
-
x^j_n
\right).$$ In all of the tests the box-averaged and shift-averaged estimators were obtained using values from every single timestep. Throughout this section, we only consider the component of the eddy diffusivity along the direction of the shear, since only that component is modified by the flow.
The Velocity Fields
-------------------
The numerical experiments were performed using the following four different idealized divergence-free velocity fields in two dimensions:
1. [ The two-dimensional shear flow]{}: $$\label{e:shear}
{\bf v}({\bf x}) = (0,\sin(x)),$$ for which the eddy diffusivity is is [@kramer] $${\mathcal{K}}= \kappa + \frac{1}{2\kappa}.$$
2. [ The periodically-modulated two-dimensional shear flow]{}: $$\label{e:shear_t}
{\bf v}({\bf x},t) = (0,\sin(x)\sin(\omega t)),$$ with $\omega>0$, for which the eddy diffusivity [@wiggins] is $${\mathcal{K}}= \kappa + \frac{1}{4(\omega + \kappa^2)}.$$
3. [ The stochastically-modulated two-dimensional shear flow]{}: $$\label{e:ou_shear}
{\bf v}({\bf x},t) = (0,\eta(t)\sin(x)),$$ where $\eta(t)$ is an Ornstein-Uhlenbeck process obtained from the equation $$\dot{\eta}(t) = -\alpha \eta(t) + \sqrt{2\sigma}\dot{\beta},$$ and where $\beta$ is a one-dimensional Brownian motion. The eddy diffusivity is $$\label{e:Deff_ou_shear}
{\mathcal{K}}= \kappa + \frac{\sigma}{2(\kappa+\alpha)\alpha}.$$ The calculation of the eddy diffusivity for this velocity field is presented in Appendix \[sec:deff\_ou\_shear\].
4. [ The Taylor-Green flow]{}: $$\label{e:TG}
v(x,t) = \nabla^{\bot} \psi_{TG}(x,y), \quad \psi_{TG}(x,y) = \sin(x)\sin(y).$$ There is no closed formula for the eddy diffusivity for this flow, but it is well known [@childress; @childress1; @childress2; @Fann01; @Kor04] that the eddy diffusivity is isotropic and that $${\mathcal{K}}= c^*\kappa^{1/2}, \quad \kappa \ll 1$$ with a formula for the prefactor $c^*$. For this case we obtain a numerical approximation to the eddy diffusivity ${\mathcal{K}}$ using the spectral method described in [@MajMcL93; @thesis].
We remark that, whereas in the case of the time independent shear flow the eddy diffusivity becomes singular as $\kappa \rightarrow 0$, in all other examples the eddy diffusivity vanishes in the zero molecular diffusion limit. The rate of convergence of ${\mathcal{K}}$ to $0$ is different for the velocity fields , and the Taylor-Green flow . From Theorem \[thm:l2\_unrescaled\] we expect that the different scaling of the eddy diffusivity with $\kappa$ should manifest itself in the scaling of the optimal subsampling rate with $\kappa$.[^4]
Results
-------
Numerical solutions to [(\[e:lagrange\])]{} were obtained for each of these cases using the Euler-Maruyama method with a very small timestep to remove the effects of numerical discretisation error. The estimator [(\[e:quad\_variation\])]{} was then computed for each numerical trajectory and compared with the correct value. In the case of the averaged estimators we used all the data in each bin to compute the averages. These calculations were repeated for 1000 realisations of the trajectory with different Brownian motions, and mean and standard deviations for the estimator values were computed.
### The Unrescaled Process {#s:unrescaled_nums}
Figure \[shearbars\] shows the results of the three estimators applied to the shear flow for various values of $\delta$ with an interval width $T=1000$, from which the number of bins $N_B=T/\delta$ for the averaged estimators can be computed. As is consistent with equation [(\[e:kappa\_lim\])]{}, the maximum likelihood estimator [(\[e:quad\_variation\])]{} underestimates the eddy diffusivity, and converges to the small-scale diffusivity $\kappa$ for small $\delta$. For larger $\delta$, the mean value of the maximum likelihood estimator approaches the correct value of the eddy diffusivity, but the standard deviation of the estimator becomes large, indicating a large variance which means that the probability of accurately estimating the correct value is small. In comparison, the shift-averaged estimator does not improve the bias by much and the variance is only reduced slightly. The box-averaged estimator increases the bias in the estimator in the sense that it substantially underestimates the eddy diffusivity.
Figure \[modbars\] shows the same information for the periodically-modulated shear flow with modulation frequency $\omega$. The small $\delta$ limit is again consistent with equation [(\[e:kappa\_lim\])]{}, and the mean of the estimator increases to a maximum which is well above the correct value, before decreasing again, with increasing standard-deviations for large values of $\delta$. The shift-averaging again shows very little improvement in either the bias or the variance; the box-averaging reduces the mean towards zero in all cases.
Figure \[OUbars\] shows the same information for the OU-modulated shear flow with parameters $\alpha=1$, $\sigma=0.1$. The results for the maximum likelihood estimator indicate an optimum value for $\delta$ which corresponds with a maximum of the mean, however the standard deviation increases monotonically with $\delta$. There is a small improvement in the bias and standard deviation for the shift-averaging, and the box-averaging produces a mean which is less than the small-scale diffusivity $\kappa$ for all values of $\delta$.
Figure \[TGbars\] shows the same information for the Taylor-Green flow. We observe, as is consistent with our theory, that there does seem to be an optimum sampling rate, but the variance is large near the optimal rate, similar to the other cases.
### The Rescaled Problem {#the-rescaled-problem}
We then repeated all of these computations for the rescaled problem [(\[e:rescaled\])]{} with $\epsilon=0.1$. Figures \[eps01shearbars\], \[eps01modbars\], \[eps01OUbars\], and \[eps01TGbars\] show the results for the shear flow, the periodically-modulated shear flow, the OU-modulated shear flow and the Taylor-Green flow respectively. Each of these flows showed that there is an optimal sampling rate for which the mean of the maximum likelihood estimator is close to the correct value, and that the standard deviation is not too large at this sampling rate, although the standard deviation increases for large sampling rates. This illustrates the result of theorem \[thm:rescaled\]: the mean of the maximum likelihood estimator converges to the correct value as $\epsilon\to 0$ and the variance converges to zero as the subsampling rate $\delta$ converges to zero.
### The Effect of Observation Noise
In this section we consider the combined effect of the multiscale structure and of measurement noise; measurement noise is included using equation . The experiments of section \[s:unrescaled\_nums\] were repeated, with $\theta=0.1$. Figures \[eps01shearbars\], \[eps01modbars\], \[eps01OUbars\], and \[eps01TGbars\] show the results for the shear flow, the periodically-modulated shear flow, the OU-modulated shear flow and the Taylor-Green flow respectively. These results confirm equation in showing that the expectation of the estimators tends to infinity as $\delta$ tends to 0 for non-zero $\theta$. This means that it becomes necessary to subsample even if there is no multiscale error. The results also show that for $\theta=0.1$, the multiscale error dominates the variance of the estimator when subsampling is applied. The shift-averaging technique is effective at removing the variance due to measurement error, but not the variance due to multiscale error.
Conclusions {#sec:conclusions}
===========
The problem of estimating the eddy diffusivity from noisy Lagrangian observations was studied in this paper. Apart from the direct relevance of our findings to the problem of the accurate parameterisation of the effects of small scales in oceanic models, we believe that this work is also a step towards the development of efficient methods for data-driven coarse graining. Problems similar to the ones considered in this paper have been studied in the context of data assimilation. For example, one might fit data from the full dynamics (i.e. the primitive equations) to the quasi-geostrophic equation which is a reduced model which is obtained from the full dynamics after averaging, in the limit as the Rossby number $Ro$ goes to $0$. Our results suggest that great care has to be taken when fitting data to a reduced model which is not compatible with the data at all scales. This is particularly the case when the reduced model is obtained through a singular limit such as $Ro
\rightarrow 0$.
In this paper, we considered this problem for a class of velocity fields (divergence-free, smooth, periodic in space and either steady or modulated in time) for which it can be shown rigorously that a parameterisation of the Lagrangian trajectories exists, in terms of an eddy diffusivity tensor. For this class of flows, it was shown, by means of analysis and numerical experiments, that subsampling is necessary in order to be able to estimate the eddy diffusivity from Lagrangian observations. It was also shown that the optimal sampling rate depends on the topological properties of the velocity field.
Parameter estimation methods that combine subsampling with averaging of the data (defined as shift averaging and box averaging) were also proposed. It was shown that shift averaging is very efficient in reducing the effects of observation error, but only slightly reduces the variance of the estimator. It appears that the shift-averaging technique is only useful for removing measurement error (or microstructure noise in the case of econometrics) and not for reducing the multiscale error, as defined in the introduction. On the other hand, box averaging leads to a biased estimator, even when the optimal sampling rate is used. This should not be surprising, since in the trivial case where the velocity field vanishes (*i.e.* pure Brownian motion with diffusivity $\kappa$), the expectation of the box averaged estimator is $\kappa/J$ where $J$ is the number of points per bin. On the other hand, for the same problem, the expectation of the shift averaged estimator is $\kappa$.
For efficient accurate coarse graining it is necessary to develop estimators which can deal with the multiscale error more efficiently. Appropriate averaging over the data appears to be an important ingredient of such an estimator. An alternative method has been proposed in [@CVE06a] based on the reconstruction of the generator of the observed Markov process; methods that combine subsampling and averaging with this approach are currently being developed.
We believe that our conclusions extend to more general types of velocity fields. For example, one can carry out the analysis and numerical experiments presented in this paper using the class of incompressible Gaussian random velocity fields that were considered in [@CC99]. This appears to be a general class of models to consider since one can obtain velocity fields with any chosen energy spectrum. The regularity of such velocity fields should definitely play an important role in the statistical inference procedure.
Clearly the calculation of the optimal sampling rate from the data is crucial for our approach. It appears that frequency domain techniques are more suitable for addressing this issue, and this will be investigated in subsequent publications.
[**Acknowledgements.**]{} The authors are particularly grateful to A.M. Stuart and P.R. Kramer for their very careful reading of an earlier draft of the paper and for many useful suggestions and comments.
Derivation of Formula {#sec:deff_ou_shear}
======================
In this appendix we derive the formula for the effective diffusivity for the OU-modulated shear flow . Homogenization problems for Gaussian incompressible velocity fields that are given in terms of an Ornstein-Uhlenbeck process have been considered in [@carmona; @PavlStuZyg07]. The results presented in these papers imply that $$\lim_{{\epsilon}\rightarrow 0} {\epsilon}y(t/{\epsilon}^2) = \sqrt{2 {\mathcal{K}}} W(t),$$ weakly on $C([0,T];{\mathbb{R}})$ where $W(t)$ is a standard one-dimensional Brownian motion and $$\label{e:Deff_ou_shear1}
{\mathcal{K}}= \kappa + \kappa \|\partial_x \phi \|^2_{L^2(X; \rho)} +
\sigma \|\partial_\eta \phi \|^2_{L^2(X; \rho)}.$$ We have used the notation $X:= (2 \pi {\mathbb{T}})^2 \times {\mathbb{R}}$, and $\phi$ and $\rho$ are the unique solutions of the equations
\[e:phi\_rho\] $$\label{e:phi}
-{\mathcal{L}}\phi = \eta \sin(x), \quad \int_{X} \phi \rho \, dX = 0,$$ $$\label{e:rho}
-{\mathcal{L}}^* \rho = 0, \quad \int_{X} \rho \, dX = 1.$$
We have used the notation $d X = dx dy d \eta$ and ${\mathcal{L}}$ is the generator of the Markov process restricted on $X$: $${\mathcal{L}}= \eta \sin(x) \partial_y + \kappa \partial_x^2 + \kappa \partial_y^2 -\alpha \eta
\partial_\eta + \sigma \partial_\eta^2.$$ ${\mathcal{L}}^*$ denotes the $L^2(X)$-adjoint, i.e. the Fokker-Planck operator. We can easily solve equations and to obtain $$\rho \, dx dy d \eta = \frac{1}{Z} e^{-\frac{\alpha \eta^2}{2 \sigma^2}} \, dx dy d \eta,
\quad Z = 4 \pi^2 \sqrt{\frac{2 \pi \sigma}{\alpha}}$$ and $$\phi(x,y,\eta) = \frac{1}{\kappa +\alpha} \eta \sin(x).$$ Consequently: $$\begin{aligned}
\|\partial_x \phi \|_{(L^2;\rho)}^2 & = & \frac{1}{(\kappa +\alpha)^2} Z^{-1} \int_X \eta^2
(\cos(x))^2 \rho \, dX \\ & = & \frac{\sigma}{2 \alpha} \frac{1}{(\kappa +\alpha)^2}\end{aligned}$$ and $$\begin{aligned}
\|\partial_\eta \phi \|_{(L^2;\rho)}^2 & = & \frac{1}{(\kappa +\alpha)^2} Z^{-1}
\int_X (\sin(x))^2 \rho \, dX \\ & = & \frac{1}{2(\kappa +\alpha)^2}.\end{aligned}$$ Upon inserting the above two formulas in we obtain .
The two-dimensional shear flow {#sec:shear}
==============================
In this appendix we study in more detail the problem of estimating the eddy diffusivity from Lagrangian observations for a class of two-dimensional shear flows. Throughout this appendix we only consider the eddy diffusivity along the direction of the shear. The flows that we will consider are of the form $$\label{e:shear_appendix}
v(x,y,t) = (0, \eta(t) f(x)),$$ where $f(x)$ is a smooth periodic function and $\eta(t)$ is either a constant, a smooth periodic function of time or a stochastic process, e.g. the Onrstein-Uhlenbeck process $$\frac{d \eta}{d t} = - \alpha \eta + \sqrt{2 \sigma} \frac{d W}{d t}.$$ As it has already been noted in [@ma:mmert; @McL98; @kramer], for this class of velocity fields the Lagrangian equations can be solved explicitly. In particular, we have that $$\label{e:y_shear}
y(t) = y(0) + \int_0^t \eta(s) f(x(0) + \sqrt{2 \kappa } W_1(s)) \, ds
+ \sqrt{2\kappa} \, W_2(t),$$ where $W_1(t)$ and $W_2(t)$ are one dimensional independent Brownian motions. Hence, the formula for the quadratic variation becomes $$\label{estimator}
{\mathcal{K}}_{N,\delta} = \frac{1}{2N\delta}\sum_{n=0}^{N-1}\left(\int_{n \delta}^{(n+1)
\delta} \eta(s) f(x(0) + \sqrt{2 \kappa } W_1(s)) \, ds
+ \sqrt{2\kappa} \Delta W_2(n\delta) \right)^2,$$ where $\Delta W_2(n\delta) = W_2((n+1)\delta) - W_2(n\delta)$. Since $f(x)$ is a periodic function, the calculation of the statistics of the quadratic variation can be accomplished by calculating the statistics of integrals of trigonometric functions of the Brownian motion. This calculation can be done by using properties of integrals of symmetric functions, that is functions $f:[n \delta, (n+1)\delta]^d \mapsto {\mathbb{R}}$ for which $f(t_{\sigma_1}, t_{\sigma_2},
\dots t_{\sigma_d}) = f(t_1, t_2, \dots t_d)$ for all permutations $\sigma$ of $(1,2, \dots d)$. In this way, we can calculate the quadratic variation as a function of $\kappa$ and $\delta$ in an explicit form. For simplicity we will consider the case $\eta(t) \equiv 1$, $f(x) = \sin(x)$ and $x(0)= y(0) = 0$. The general case can be treated similarly.
For the velocity field $$v(x,y) = (0, \sin(x))$$ We can calculate the expectation of the 22-component of the quadratic variation equation , and hence prove . Since $W_1(t)$ and $W_2(t)$ are independent, we immediately deduce that $$\begin{aligned}
\mathbb{E}\left[(y_{n+1}-y_n)^2
\right]
& = & \int_{n\delta}^{(n+1)\delta} \int_{n\delta}^{(n+1)\delta}
\mathbb{E}\left[
\sin(\sqrt{2\kappa}W_1(s_1))
\sin(\sqrt{2\kappa}W_1(s_2))\right]
{\diffs_2}{\diffs_1} + 2\kappa\delta.\end{aligned}$$ In order to calculate the integral on the right hand side of the above equation (which we denote by $S$), we use trigonometric identities together with the formula for the expectation of the characteristic function of a Gaussian random variable to obtain $$\begin{aligned}
S & = & -\frac{1}{4}
\sum_{\MM{a}\in I}a_1a_2
\int_{n\delta}^{(n+1)\delta} \int_{n\delta}^{(n+1)\delta}
\mathbb{E}\left[e^{i\sqrt{2\kappa}\left(a_1W_1(s_1)+a_2W_1(s_2)\right)}
\right]{\diffs_2}{\diffs_1}
\\ &=& -\frac{1}{4}
\sum_{\MM{a}\in I}a_1a_2
\int_{n\delta}^{(n+1)\delta} \int_{n\delta}^{(n+1)\delta}
e^
{-2\kappa\delta\sum_{i,j=1}^2a_ia_j\min(s_i,s_j)}
{\diffs_2}{\diffs_1},\end{aligned}$$ where $\MM{a}=(a_1,a_2)$ and $I$ is the index set $\{-1,1\}\times\{-1,1\} =\{-1,1\}^2$. The integrand is symmetric in $s_1$ and $s_2$, and, using properties of multiple integrals of symmetric functions, we can write the above integral in the form $$\begin{aligned}
S &=& -\frac{1}{8}
\sum_{\MM{a}\in I}a_1a_2
\int_{n\delta}^{(n+1)\delta} \int_{s_1}^{(n+1)\delta}
e^
{-2\kappa\delta\sum_{i=1}^2s_i(a_i^2 + \sum_{i<j}a_ia_j)}
{\diffs_2}{\diffs_1}.\end{aligned}$$ Evaluating this formula using Maple gives $$\label{e:shear_exp}
S = 2\kappa\delta + \frac{\delta}{\kappa} +
\frac{1}{2\kappa^2}
\Bigg(-\frac{1}{6}e^{-4\kappa(n+1)\delta}
-\frac{1}{2}e^{-4\kappa n\delta}
+
\frac{2}{3}e^{-\kappa(4n+1)\delta}
+2e^{-\kappa\delta}-2
\Bigg),$$ from which follows upon summation.
We can also calculate $\mathbb{E} \left| {\mathcal{K}}_{N, \delta}- {\mathcal{K}}\right|^2$, leading to equation , and hence . We have $$\label{total variance}
\mathbb{E}\left| {\mathcal{K}}_{N,\delta}-{\mathcal{K}}\right|^2
=
\mathbb{E}\left| {\mathcal{K}}_{N, \delta}\right|^2
-2\mathbb{E}\left({\mathcal{K}}_{N, \delta}\right)\left(\kappa+\frac{1}{2\kappa}\right)
+ \left(\kappa+\frac{1}{2\kappa}\right)^2.$$ We have already calculated the expectation of the quadratic variation, and it remains to compute the second moment. We have $$\begin{aligned}
\mathbb{E}
\left|{\mathcal{K}}_{\delta}\right|^2
&=& \frac{1}{4N^2\delta^2}
\sum_{n=1}^N\sum_{m=1}^N
\mathbb{E}\left((y^n-y^{n-1})^2(y^m-y^{m-1})^2\right) \\
&=& \frac{1}{4N^2\delta^2}
\sum_{n=1}^N\underbrace{\mathbb{E}\left[(y^n-y^{n-1})^4\right]}_{=S_1^n}
+ \frac{1}{2N^2\delta^2}
\sum_{n=1}^N\sum_{m<n}
\underbrace{\mathbb{E}\left[(y^n-y^{n-1})^2(y^m-y^{m-1})^2\right]}_{=S_2^{nm}}.\end{aligned}$$ We shall separately compute these two types of terms, namely the diagonal terms $S_1^n$ and the off-diagonal terms $S_2^{nm}$.
First we compute $S_1^n$. $$\begin{aligned}
S_1^n & = & \underbrace{\mathbb{E}\left[
\left(\int_{n\delta}^{(n+1)\delta}
\sin(\sqrt{2\kappa}W_1(s))\operatorname{d}{s}\right)^4
\right]}_{S_{11}^n} \\
& & + 12\kappa\delta\left(\frac{\delta}{\kappa} +
\frac{1}{2\kappa^2}
\Bigg(-\frac{1}{6}e^{-4\kappa(n+1)\delta}
-\frac{1}{2}e^{-4\kappa n\delta}
+
\frac{2}{3}e^{-\kappa(4n+1)\delta}
+2e^{-\kappa\delta}-2
\Bigg)\right) \\ && + 12 \kappa^2 \delta^2,\end{aligned}$$ where has been used. For the calculation of $S_{11}^n$ we use trigonometric identities, together with the formula for the expectation of the characteristic function of a Gaussian random variable to obtain $$\begin{aligned}
S_{11}^n
& = &
\frac{1}{16} \sum_{\MM{a}\in I} \prod_{k=1}^4a_k
\int_{s_k=n\delta}^{(n+1)\delta}
e^{-2\kappa
\sum_{i,j=1}^4a_ia_j\min(s_i,s_j)
}\operatorname{d}{s}_k ,\end{aligned}$$ where $\MM{a}=(a_1,a_2,a_3,a_4)$ and $I$ is the indexing set $\{-1,1\}^4$. The integrand in this multiple integral is a symmetric function, and hence we may write $$\begin{aligned}
S_{11}^n & = & \frac{3}{2} \sum_{\MM{a}\in I} \prod_{k=1}^4a_k
\int_{s_1=n\delta}^{(n+1)\delta}
\int_{s_2=s_1}^{(n+1)\delta}
\int_{s_3=s_2}^{(n+1)\delta}
\int_{s_4=s_3}^{(n+1)\delta}
e^{-2\kappa
\sum_{i,j=1}^4a_ia_j\min(s_i,s_j)
}\operatorname{d}{s}_1\operatorname{d}{s}_2\operatorname{d}{s}_3\operatorname{d}{s}_4. \end{aligned}$$ This can be computed using Maple: $$\begin{aligned}
S_{11}^n &=& {\frac {1}{26880}}\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,n\delta
}} \right) ^{16} \left( {e^{\kappa\,\delta}} \right) ^{16}}}
\\ &&
\; +{\frac {
261}{64}}\,{\kappa}^{-4}+{\frac {1}{960}}\,{\frac {1}{ \left( {e^{
\kappa\,n\delta}} \right) ^{16}{\kappa}^{4} \left( {e^{\kappa\,\delta}
} \right) ^{4}}}-{\frac {1}{3360}}\,{\frac {1}{ \left( {e^{\kappa\,n
\delta}} \right) ^{16}{\kappa}^{4} \left( {e^{\kappa\,\delta}}
\right) ^{9}}}-{\frac {45}{16}}\,{\frac {\delta}{{\kappa}^{3}}}
\\ && \;
-{
\frac {49}{12}}\,{\frac {1}{{\kappa}^{4}{e^{\kappa\,\delta}}}}-{\frac
{1}{2400}}\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,n\delta}}
\right) ^{4} \left( {e^{\kappa\,\delta}} \right) ^{9}}}+{\frac {1}{
120}}\,{\frac {\delta}{{\kappa}^{3} \left( {e^{\kappa\,n\delta}}
\right) ^{4} \left( {e^{\kappa\,\delta}} \right) ^{4}}}
\\ && \;
+{\frac {19}{
24}}\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,n\delta}} \right) ^{4
}}}-{\frac {1}{480}}\,{\frac {1}{ \left( {e^{\kappa\,n\delta}}
\right) ^{16}{\kappa}^{4}{e^{\kappa\,\delta}}}}-{\frac {229}{288}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,n\delta}} \right) ^{4}{e^{
\kappa\,\delta}}}}
\\ && \;
-5/4\,{\frac {\delta}{{\kappa}^{3}{e^{\kappa\,\delta
}}}}+3/4\,{\frac {{\delta}^{2}}{{\kappa}^{2}}}+{\frac {1}{192}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,\delta}} \right) ^{4}}}
\\ && \;
-{
\frac {5}{12}}\,{\frac {\delta}{{\kappa}^{3} \left( {e^{\kappa\,n
\delta}} \right) ^{4}{e^{\kappa\,\delta}}}}-3/8\,{\frac {\delta}{{
\kappa}^{3} \left( {e^{\kappa\,n\delta}} \right) ^{4}}}+{\frac {7}{
1800}}\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,n\delta}} \right) ^
{4} \left( {e^{\kappa\,\delta}} \right) ^{4}}}+{\frac {1}{768}}\,{
\frac {1}{ \left( {e^{\kappa\,n\delta}} \right) ^{16}{\kappa}^{4}}}\end{aligned}$$
After summation, all the terms containing exponentials given rise to terms which converge to a constant divided by $\delta^2N^2$ faster than any polynomial power of $\delta\kappa$ as $\kappa\delta\to \infty$.
Next we compute $S_2$. Since $n < m$, the term inside the sum is $$\begin{aligned}
\mathbb{E}\left[
(y^{n+1}-y^n)^2(y^{m+1}-y^m)^2
\right]
&=& \mathbb{E}\Bigg[
\left(\int_{n\delta}^{(n+1)\delta}
\sin(\sqrt{2\kappa}W_1(s))\operatorname{d}{s}
+ \sqrt{2\kappa}\int_{n\delta}^{(n+1)\delta}
\operatorname{d}{W_2}(s)\right)^2 \\
& & \quad \times
\left(\int_{m\delta}^{(m+1)\delta}
\sin(\sqrt{2\kappa}W_1(s))\operatorname{d}{s}
+ \sqrt{2\kappa}\int_{m\delta}^{(m+1)\delta}
\operatorname{d}{W_2}(s)\right)^2
\Bigg]
\\
& = & \underbrace{\mathbb{E}\Bigg[
\left(\int_{n\delta}^{(n+1)\delta}
\sin(\sqrt{2\kappa}W_1(s))\operatorname{d}{s}\right)^2
\left(\int_{m\delta}^{(m+1)\delta}
\sin(\sqrt{2\kappa}W_1(s))\operatorname{d}{s}\right)^2
\Bigg]}_{S_{21}^{nm}} \\
& & +
2\kappa\delta \Bigg(
2\kappa\delta + \frac{\delta}{\kappa} +
\frac{1}{2\kappa^2}
\Bigg(-\frac{1}{6}e^{-4\kappa(m+1)\delta}
-\frac{1}{2}e^{-4\kappa m\delta}
+ \\
& & \qquad\qquad\qquad\qquad\qquad\qquad\qquad
\frac{2}{3}e^{-\kappa(4m+1)\delta}
+2e^{-\kappa\delta}-2
\Bigg) \Bigg) \\
& & +
2\kappa\delta \Bigg(
2\kappa\delta + \frac{\delta}{\kappa} +
\frac{1}{2\kappa^2}
\Bigg(-\frac{1}{6}e^{-4\kappa(n+1)\delta}
-\frac{1}{2}e^{-4\kappa n\delta}
+ \\
& & \qquad\qquad\qquad\qquad\qquad\qquad\qquad
\frac{2}{3}e^{-\kappa(4n+1)\delta}
+2e^{-\kappa\delta}-2
\Bigg) \Bigg) \\
& & \quad +4\kappa^2\delta^2,\end{aligned}$$ where has been used again. A similar calculation to that for $S_{11}^n$, making use of the fact that $m>n$ gives $$\begin{aligned}
S_{21}^{nm}
& = &
\int_{s_1=n\delta}^{(n+1)\delta}
\int_{s_2=n\delta}^{(n+1)\delta}
\frac{1}{16}
\sum_{\MM{a}\in I}\left(\prod_{k=1}^4a_k\right)
e^{-2\kappa
\left(\sum_{i,j=1}^2a_ia_j\min(s_i,s_j) + 2\sum_{i=1}^2\sum_{j=3}^4a_ia_js_i\right)}
\\
& & \qquad \times
\int_{s_3=m\delta}^{(m+1)\delta}
\int_{s_4=m\delta}^{(m+1)\delta}
e^{-2\kappa
\sum_{i,j=3}^4a_ia_j\min(s_i,s_j)
}\operatorname{d}{s}_1\operatorname{d}{s}_2\operatorname{d}{s}_3\operatorname{d}{s}_4.\end{aligned}$$ The integrand for the two inner integrals, and the integrand for the two outer integrals are both symmetric functions, and we obtain $$\begin{aligned}
S_{21}^{nm}
& = & \frac{1}{4}
\int_{s_1=n\delta}^{(n+1)\delta}
\int_{s_2=s_1}^{(n+1)\delta}
\sum_{\MM{a}\in I}\left(\prod_{k=1}^4a_k\right)
e^{-2\kappa
\left(\sum_{i=1}^2s_i\left(a_i^2+ 2\sum_{i<j<3}a_ia_j\right)
+ 2\sum_{i=1}^2\sum_{j=3}^4a_ia_js_i\right)}
\\
& & \qquad \times\int_{s_3=m\delta}^{(m+1)\delta}
\int_{s_4=s3}^{(m+1)\delta}
e^{-2\kappa
\sum_{i=3}^4s_i\left(a_i^2+ 2\sum_{i<j}a_ia_j\right)
}\operatorname{d}{s}_1\operatorname{d}{s}_2\operatorname{d}{s}_3\operatorname{d}{s}_4.\end{aligned}$$ This can be computing using Maple: $$\begin{aligned}
S_{21}^{n m} &=& -2\,{\frac {1}{{\kappa}^{4}{e^{\delta\,\kappa}}}}+{\frac {1}{2016}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}
\left( {e^{\delta\,\kappa}} \right) ^{16} \left( {e^{\kappa\,n\delta}
} \right) ^{12}}}+{\frac {1}{210}}\,{\frac {1}{{\kappa}^{4} \left( {e^
{\kappa\,m\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{
6} \left( {e^{\kappa\,n\delta}} \right) ^{12}}}
\\ & & \;
-{\frac {1}{504}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}
\left( {e^{\delta\,\kappa}} \right) ^{13} \left( {e^{\kappa\,n\delta}
} \right) ^{12}}}
\\ & & \;
+{\frac {1}{1440}}\,{\frac {1}{{\kappa}^{4} \left( {e
^{\kappa\,m\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^
{4} \left( {e^{\kappa\,n\delta}} \right) ^{12}}}-{\frac {1}{840}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}
\left( {e^{\delta\,\kappa}} \right) ^{9} \left( {e^{\kappa\,n\delta}}
\right) ^{12}}}-1/12\,{\frac {1}{{\kappa}^{4} \left( {e^{\delta\,
\kappa}} \right) ^{5} \left( {e^{\kappa\,n\delta}} \right) ^{4}}}
\\ & & \;
+{
\frac {1}{60}}\,{\frac {\delta}{{\kappa}^{3} \left( {e^{\kappa\,m
\delta}} \right) ^{4}}}-1/45\,{\frac {\delta}{{\kappa}^{3} \left( {e^{
\kappa\,m\delta}} \right) ^{4}{e^{\delta\,\kappa}}}}+{\frac {1}{180}}
\,{\frac {\delta}{{\kappa}^{3} \left( {e^{\kappa\,m\delta}} \right) ^{
4} \left( {e^{\delta\,\kappa}} \right) ^{4}}}+1/48\,{\frac { \left( {e
^{\kappa\,n\delta}} \right) ^{4}}{{\kappa}^{4} \left( {e^{\kappa\,m
\delta}} \right) ^{4}}}+
\\ & & \;
{\frac {17}{2700}}\,{\frac {1}{{\kappa}^{4}
\left( {e^{\kappa\,m\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}}
\right) ^{4}}}-{\frac {1}{600}}\,{\frac {1}{{\kappa}^{4} \left( {e^{
\kappa\,m\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{9
}}}+1/4\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,n\delta}} \right)
^{4}}}-{\frac {7}{12}}\,{\frac {1}{{\kappa}^{4}{e^{\delta\,\kappa}}
\left( {e^{\kappa\,n\delta}} \right) ^{4}}}
\\ & & \;
-1/12\,{\frac { \left( {e^
{\kappa\,n\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{
3}}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}}}+
\\ & & \;
1/32\,{
\frac { \left( {e^{\kappa\,n\delta}} \right) ^{4} \left( {e^{\delta\,
\kappa}} \right) ^{4}}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}}
\right) ^{4}}}-1/36\,{\frac { \left( {e^{\kappa\,n\delta}} \right) ^{
4}}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}{e^{\delta\,
\kappa}}}}+1/18\,{\frac { \left( {e^{\delta\,\kappa}} \right) ^{2}
\left( {e^{\kappa\,n\delta}} \right) ^{4}}{{\kappa}^{4} \left( {e^{
\kappa\,m\delta}} \right) ^{4}}}
\\ & & \;
+{\frac {1}{288}}\,{\frac { \left( {e^
{\kappa\,n\delta}} \right) ^{4}}{{\kappa}^{4} \left( {e^{\kappa\,m
\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{4}}}-1/4\,
{\frac {\delta}{{\kappa}^{3} \left( {e^{\kappa\,n\delta}} \right) ^{4}
}}+1/3\,{\frac {\delta}{{\kappa}^{3}{e^{\delta\,\kappa}} \left( {e^{
\kappa\,n\delta}} \right) ^{4}}}-1/12\,{\frac {\delta}{{\kappa}^{3}
\left( {e^{\delta\,\kappa}} \right) ^{4} \left( {e^{\kappa\,n\delta}}
\right) ^{4}}}
\\ & & \;
+1/12\,{\frac {1}{{\kappa}^{4} \left( {e^{\delta\,
\kappa}} \right) ^{4} \left( {e^{\kappa\,n\delta}} \right) ^{4}}}+1/3
\,{\frac {1}{{\kappa}^{4} \left( {e^{\delta\,\kappa}} \right) ^{2}
\left( {e^{\kappa\,n\delta}} \right) ^{4}}}+{\frac {1}{{\kappa}^{4}
\left( {e^{\delta\,\kappa}} \right) ^{2}}}+2\,{\frac {\delta}{{\kappa
}^{3}{e^{\delta\,\kappa}}}}
\\ & & \;
+{\frac {{\delta}^{2}}{{\kappa}^{2}}}-2\,{
\frac {\delta}{{\kappa}^{3}}}+{\kappa}^{-4}-{\frac {1}{360}}\,{\frac {
1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}{e^{\delta\,
\kappa}} \left( {e^{\kappa\,n\delta}} \right) ^{12}}}
\\ & & \;
+{\frac {1}{480}}
\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}
\left( {e^{\kappa\,n\delta}} \right) ^{12}}}-{\frac {1}{280}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}
\left( {e^{\delta\,\kappa}} \right) ^{5} \left( {e^{\kappa\,n\delta}}
\right) ^{12}}}
\\ & & \;
+{\frac {1}{672}}\,{\frac {1}{{\kappa}^{4} \left( {e^{
\kappa\,m\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{
12} \left( {e^{\kappa\,n\delta}} \right) ^{12}}}+{\frac {17}{900}}\,{
\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}}}
\\ & & \;
-{
\frac {1}{72}}\,{\frac { \left( {e^{\delta\,\kappa}} \right) ^{3}}{{
\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^{4}}}-{\frac {161}{
5400}}\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}} \right) ^
{4}{e^{\delta\,\kappa}}}}+{\frac {1}{54}}\,{\frac { \left( {e^{\delta
\,\kappa}} \right) ^{2}}{{\kappa}^{4} \left( {e^{\kappa\,m\delta}}
\right) ^{4}}}
\\ & & \;
-{\frac {1}{200}}\,{\frac {1}{{\kappa}^{4} \left( {e^{
\kappa\,m\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{5
}}}+{\frac {1}{150}}\,{\frac {1}{{\kappa}^{4} \left( {e^{\kappa\,m
\delta}} \right) ^{4} \left( {e^{\delta\,\kappa}} \right) ^{6}}}.\end{aligned}$$
After the double summation, all the terms containing exponentials given rise to terms which converge to a constant multiplied by $(N-1)/\delta^2N^2$ faster than any polynomial power of $\delta\kappa$ as $\kappa\delta\to \infty$.
Collecting terms $$\begin{aligned}
\mathbb{E}
\left|{\mathcal{K}}_{N,\delta}\right|^2
& = &
\frac{1}{4 \kappa^2 \delta^2} - \frac{1}{2 \kappa^3 \delta} + \frac{1}{4
\kappa^2} +1 - \frac{1}{\kappa \delta} + \kappa^2 \\ && +
\frac{1}{N \delta^2}\left(
c_1 \frac{1}{\kappa^4} + c_2 \delta \frac{1}{\kappa^3}
+c_3 {\delta}^{2} \frac{1}{\kappa^2} + c_4\delta^2 + c_5 \frac{\delta}{\kappa}
+
c_6 \kappa^2\delta^2 + c(\delta\kappa)\right)
\\ && +
\frac{1}{N^2 \delta^2}\left(
d_1 \frac{1}{\kappa^4} + d_2 \delta \frac{1}{\kappa^3}
+d_3 {\delta}^{2} \frac{1}{\kappa^2} + d_4\delta^2 + d_5 \frac{\delta}{\kappa}
+
d_6 \kappa^2\delta^2 + d(\delta\kappa)\right),\end{aligned}$$ where the constants $\{c_i, \, d_i ; i=1, \dots 6\}$ can be read from the above formulas and $c(\delta \kappa), \, d( \delta \kappa)$ converge exponentially fast to a constant in the limit $\delta \kappa \rightarrow
+ \infty$.
Upon computing the remaining terms in equation we notice that all leading order terms are cancelled and we end up with $$\begin{aligned}
\mathbb{E}
\left| {\mathcal{K}}_{N,\delta}- {\mathcal{K}}\right|^2 & = &
\frac{1}{N \delta^2}\left(
c_1 \frac{1}{\kappa^4} + c_2 \delta \frac{1}{\kappa^3}
+c_3 {\delta}^{2} \frac{1}{\kappa^2} + c_4\delta^2 + c_5 \frac{\delta}{\kappa}
+
c_6 \kappa^2\delta^2 + c(\delta\kappa)\right)
\\ && +
\frac{1}{N^2 \delta^2}\left(
d_1 \frac{1}{\kappa^4} + d_2 \delta \frac{1}{\kappa^3}
+d_3 {\delta}^{2} \frac{1}{\kappa^2} + d_4\delta^2 + d_5 \frac{\delta}{\kappa}
+
d_6 \kappa^2\delta^2 + d(\delta\kappa)\right),\end{aligned}$$ which is precisely equation .
[^1]: Department of Aeronautics, Imperial College London, London SW7 2AZ, UK (colin.cotter@imperial.ac.uk).
[^2]: Department of Mathematics, Imperial College London, London SW7 2AZ, UK (g.pavliotis@imperial.ac.uk).
[^3]: We remark, however, that the small scale data that we ignore are highly correlated and it is not clear how much additional information they contain about the eddy diffusivity.
[^4]: The analysis presented in Section \[sec:small\_kappa\] applies only to time-independent velocity fields, but can be easily generalized to cover the case of time dependent velocity fields. In fact, for the velocity fields and we can analyze directly the quadratic variation without appeal to a general theory. See Appendix .
|
---
abstract: 'Over the last two decades, scanning tunnelling microscopy (STM) has become one of the most important ways to investigate the structure of crystal surfaces. STM has helped achieve remarkable successes in surface science such as finding the atomic structure of Si(111) and Si(001). For high-index Si surfaces the information about the local density of states obtained by scanning does not translate directly into knowledge about the positions of atoms at the surface. A commonly accepted strategy for identifying the atomic structure is to propose several possible models and analyze their corresponding [*simulated*]{} STM images for a match with the experimental ones. However, the number of good candidates for the lowest-energy structure is very large for high-index surfaces, and heuristic approaches are not likely to cover all the relevant structural models. In this article, we take the view that finding the atomic structure of a surface is a problem of stochastic optimization, and we address it as such. We design a general technique for predicting the reconstruction of silicon surfaces with arbitrary orientation, which is based on parallel-tempering Monte Carlo simulations combined with an exponential cooling. The advantages of the method are illustrated using the Si(105) surface as example, with two main results: (a) the correct single-step rebonded structure \[e.g., Fujikawa [*et al.*]{}, Phys. Rev. Lett. 88, 176101 (2002)\] is obtained even when starting from the paired-dimer model \[Mo [*et al.*]{}, Phys. Rev. Lett. 65, 1020 (1990)\] that was assumed to be correct for many years, and (b) we have found several double-step reconstructions that have lower surface energies than any previously proposed double-step models.'
author:
- 'Cristian V. Ciobanu$^{1}$ and Cristian Predescu$^{2}$'
title: '**Reconstruction of silicon surfaces: a stochastic optimization problem**'
---
Introduction
============
Silicon surfaces are the most intensely studied crystal surfaces since they constitute the foundation of the billion-dollar semiconductor industry. Traditionally, the low-index surfaces such as Si(001) are the widely used substrates for electronic device fabrication. With the advent of nanotechnology, the stable high-index surfaces of silicon have now become increasingly important for the fabrication of quantum devices at length scales where lithographic techniques are not applicable. Owing to their grooved or faceted morphology, some high-index surfaces can be used as templates for the growth of self-assembled nanowires. Understanding the self-organization of adatoms on these surfaces, as well as their properties as substrates for thin film growth, requires atomic-level knowledge of the surface structure. Whether the surface unit cells are small \[e.g., Si(113)\] or large \[such as Si(5 5 12)\], in general the atomic-scale models that were first proposed were subsequently contested:[@knal113; @ranke113; @dabrowski; @5512baski; @5512suzuki; @5512takeguchi; @mo; @kds] the potential importance of stable Si surfaces with certain high-index orientations sparked many independent investigations, which led to different proposals in terms of surface structure.
One of the most puzzling cases has been the (105) surface, which appears on the side-facets of the pyramidal quantum dots obtained in the strained layer epitaxy of Ge or Si$_{1-x}$Ge$_x$ ($x>0.2$) on Si(001). Using STM imaging, Mo and coworkers proposed the first model for this surface,[@mo] which was based on unrebonded monatomic steps separated by small (two-dimer wide) Si(001)-2$\times$1 terraces. Subsequently, Khor and Das Sarma reported another possible (105) structure with a lower density of dangling bonds.[@kds] However, the relative surface energy of the two different reconstructions[@mo; @kds] was not computed, and the structure proposed in Ref. had not, at the time, replaced the widely accepted model[@mo] of Mo [*et al*]{}. Only very recently it has been shown [@jap-prl; @jap-ss; @italy-prl; @apl; @susc105] that the actual (105) structure is made of single-height rebonded steps (SR), which are strongly stabilized by the compressive strains present in the Ge films deposited on Si(001) [@italy-prl; @apl] or Si(105). [@jap-prl; @jap-ss; @susc105] Other high-index surfaces such as Si(113) and Si(5 5 12) have sagas of their own,[@ranke113; @knal113; @dabrowski; @5512baski; @5512suzuki; @5512takeguchi] and only in the former case there is now consensus[@dabrowski] about the atomic structure.
The difficulty of finding the atomic structure of a surface is not related to the resolution of the STM techniques, or to understanding of the images obtained. After all, it is well-known that STM gives information about the local density of states at the surfaces and not necessarily about atomic coordinates. [@luth] A common procedure for finding the reconstructions of silicon surfaces consists in a combination of STM imaging and electronic structure calculations as follows. Starting from the bulk truncated surface and taking cues from the experimental data, one proposes several atomic models for the surface reconstructions. The proposed models are then relaxed using density-functional or tight-binding methods, and STM images are simulated in each case. At the end of the relaxation, the surface energies of the structural models are also calculated. A match with the experimental STM data is identified based on the relaxed lowest-energy structures and their simulated STM images. This procedure has long become standard and has been used for many high-index orientations.[@knal113; @dabrowski; @jap-prl; @jap-ss; @italy-prl; @5512baski; @si114] As described, the procedure is heuristic, since one needs to rely heavily on physical intuition when proposing good candidates for the lowest energy structures. In the case of stable high-index Si surfaces, the number of possible good candidates is rather large, and may not be exhausted heuristically; thus, worst-case scenarios in which the most stable models are not included in the set of “good candidates” are very likely. On one hand, it has been recognized[@5512baski] that the minimization of surface energy for semiconductor surfaces is not controlled solely by the reduction of the dangling bond density, but also by the amount of surface stress caused in the process. On the other hand, intuitive reasoning can tackle (at best) the problem of lowering the number of dangling bonds, but cannot account for the increase in surface stress or for the possible nanoscale faceting of certain surfaces.[@baski001-111] For this reason, we adopt the view that finding the structure of high-index Si surfaces is a problem of stochastic optimization, in which the competition between the saturation of surface bonds and the increase in surface stress is intrinsically considered.
To our knowledge, a truly general and robust way of predicting the atomic structure of semiconductor surfaces –understood as finding the atomic configuration of a surface of any arbitrary crystallographic orientation without experimental input, has not been reported. It is not clear that such robust atomic-scale predictions about semiconductor surfaces can even be ventured, since theoretical efforts have been hampered by the lack of empirical or semiempirical potentials that are [*both fast and transferable*]{} for surface calculations. However, the long process which lead to the discovery of the reconstruction of the (105) surface[@mo; @kds; @jap-prl; @jap-ss; @italy-prl; @apl; @susc105] indicates a clear need for a search methodology that does not rely on human intuition. The goal of this article is to present a strategy for finding the lowest-energy reconstructions for an elemental crystal surface. While we hope that this strategy will become a useful tool for many surface scientists, the extent of its applicability remains to be explored. Our initial efforts will be focused on the surfaces of silicon because of their utmost fundamental and technological importance; nonetheless, the same strategy could be applied for any other material surfaces provided suitable models for atomic interactions are available.
The Monte Carlo method
======================
General considerations
----------------------
In choosing a methodology that can help predict the surface reconstructions, we have taken into account the following considerations. First, the number of atoms in the simulation slab is large because it includes several subsurface layers in addition to the surface ones. Moreover, the number of local minima of the potential energy surface is also large, as it scales roughly exponentially[@Sti83; @Sti99] with the number of atoms involved in the reconstruction; by itself, such scaling requires the use of fast stochastic search methods. Secondly, the calculation of interatomic forces is very expensive, so the method should be based on Monte-Carlo algorithms rather than molecular dynamics. Lastly, methods that are based on the modification of the potential energy surface (PES) (such as the basin-hoping algorithm [@basinhopping]), although very powerful in predicting global minima, have been avoided as our future studies are aimed at predicting not only the correct lowest-energy reconstructions, but also the thermodynamics of the surface. These considerations prompted us to choose the parallel-tempering Monte Carlo (PTMC) algorithm [@Gey95; @Huk96] for this study. Before describing in detail the computational procedure and its advantages, we pause to discuss the computational cell and the empirical potential used.
![Schematic computational cell: the “hot” atoms (gray) are allowed to move, while the bottom ones (black) are kept fixed at their bulk locations. Different maximum displacements $\Delta_s$ and $\Delta_b$ are allowed for the atoms that are closer to the surface and deeper in the bulk, respectively.[]{data-label="figgeom"}](geom.eps){width="6cm"}
The simulation cell has a single-face slab geometry with periodic boundary conditions applied in the plane of the surface (denoted $xy$), and no periodicity in the direction ($z$) normal to the surface (refer to Fig. \[figgeom\]). The “hot” atoms from the top part of the slab (corresponding to a thickness of 10–15 Å) are allowed to move, while the bottom layers of atoms are kept fixed to simulate the underlying bulk crystal. Though highly unlikely during the finite time of the simulation performed, the evaporation of atoms is prevented by using a wall of infinite energy that is parallel to the surface and situated 10 Å above it; an identical wall is placed at the level of the lowest fixed atoms to prevent the (theoretically possible) diffusion of the hot atoms through the bottom of the slab. The area of the simulation cell in the $xy$-plane and the number of atoms in the cell are kept fixed during each simulation; as we shall discuss in section IV, these assumptions are not restrictive as long as we consider all the relevant values of the number of atoms per area. Under these conditions, the problem of finding the most stable reconstruction reduces to the global minimization of the total potential energy $V(\mathbf{x})$ of the atoms in the simulation cell (here $\mathbf{x}$ denotes the set of atomic positions). In order to sort through the numerous local minima of the potential $V(\mathbf{x})$, a stochastic search is necessary. The general strategy of such search (as illustrated, for example, by the simulated annealing technique[@Kir83; @Kir84]) is to sample the canonical Boltzmann distribution $ \exp\left[- V(\mathbf{x})/(k_B
T)\right] $ for decreasing values of the temperature $T$ and look for the low-energy configurations that are generated.
In terms of atomic interactions, we are constrained to use empirical potentials because the highly accurate ab-initio or tight-binding methods are prohibitive. Since this work is aimed at finding the [*lowest*]{} energy reconstructions for arbitrary surfaces, the choice of the empirical potential is crucial, as different interaction models can give different energetic ordering of the possible reconstructions. Furthermore, the true structure of the surface may not even be a local minimum of the potential chosen to describe the interactions: it is the case, for example, of the adatom-interstitial reconstructions[@dabrowski] of Si(113), which are not local minima of the Stillinger-Weber potential.[@stillweb] The work of Nurminen *et al.*[@landau-swt3-test] indicates that the most popular empirical potentials for silicon[@stillweb; @tersoff3] are not suitable for finite-temperature simulations of surfaces. After thorough numerical experimentation with several empirical potentials, we have chosen to use the highly optimized empirical potential (HOEP) recently developed by Lenosky *et al.*[@hoep] HOEP is fitted to a database of ab-initio calculations that includes structural and energetic information about small Si clusters, which leads to a superior transferability to the different bonding environments present at the surface.[@hoep]
Advantages of the parallel tempering algorithm as a global optimizer
--------------------------------------------------------------------
The parallel tempering Monte Carlo method (also known as the replica-exchange Monte-Carlo method) consists in running parallel canonical simulations of many statistically independent replicas of the system, each at a different temperature $T_1 < T_2 < \ldots
< T_N$. The set of $N$ temperatures $\{T_i,\ i=1,2,...N \}$ is called a [*temperature schedule*]{}, or [*schedule*]{} for short. The probability distributions of the individual replicas are sampled with the Metropolis algorithm,[@Met53] although any other ergodic strategy can be utilized. The key feature of the parallel tempering method is that swaps between replicas of neighboring temperatures $T_i$ and $T_j$ ($j = i \pm 1$) are proposed and allowed with the conditional probability[@Gey95; @Huk96] given by $$\label{eq:PTMCacc} \min\left\{1, e^{(1/T_j -
1/T_i)\left[V(\mathbf{x}_j)-V(\mathbf{x}_i)\right]/k_B}\right\},$$ where $V(\mathbf{x}_i)$ represents the energy of the replica $i$ and $k_B$ is the Boltzmann constant. The conditional probability (\[eq:PTMCacc\]) ensures that the detailed balance condition is satisfied and that the equilibrium distributions are the Boltzmann ones for each temperature.
In the standard Metropolis sampling[@Met53] of Boltzmann distributions, the probability that the Monte Carlo walker escapes from a given local minimum decreases exponentially as the temperature is lowered. In turn, the average number of Monte Carlo steps needed for the walker to escape from the trapping local minimum increases exponentially with the decrease of the temperature, a scaling that makes the search for a global minimum inefficient at low temperatures. To cope with this problem, the parallel tempering algorithm takes advantage of the fact that the Metropolis walkers running at higher temperatures have larger probabilities of jumping over energy barriers. Parallel tempering significantly decreases the time taken for the walker to escape from local minima by providing an additional mechanism for jumping between basins, namely the swapping of configurations between replicas running at neighboring temperatures. Therefore, if a given (low-temperature) replica of the system is stuck in a local minimum, the configuration swaps with walkers at higher temperatures can provide that replica with states associated with other basins (wells on the potential energy surface), ultimately driving it into the global minimum.
Because of this swapping mechanism, parallel tempering enjoys certain advantages (as a global optimizer) over the more popular simulated annealing algorithm (SA).[@Kir83; @Kir84] In order for SA to be convergent (i.e. to reach the global optimum as the temperature is lowered) the cooling schedule must be of the form[@Gem84; @Haj88] $$\label{eq:SAlaw} T_i = \frac{T_{0}}{\log(i + i_0)}, \quad i \geq
1,$$ where $T_{0}$ and $i_0$ are sufficiently large constants. Such a logarithmic schedule is too slow for practical applications, and faster schedules are routinely utilized. Common SA cooling schedules, such as the geometric or the linear ones,[@Kir83] make SA non-convergent: the Monte Carlo walker has a non-zero probability of getting trapped into minima other than the global one.
The cooling schedule implied by Eq. (\[eq:SAlaw\]) is, of course, asymptotically valid in the limit of low temperatures. In the same limit, the PT algorithm allows for a geometric temperature schedule.[@Sug00; @Pre03] When the temperature drops to zero, the system is well approximated by a multidimensional harmonic oscillator and the acceptance probability for swaps attempted between two replicas with temperatures $T < T'$ is given by the incomplete beta function law[@Pre03] $$\label{eq:AcTT} Ac(T,T') \simeq \frac{2}{B(d/2,d/2)} \int_0^{1/(1
+ R)} \theta^{d/2 - 1}(1 - \theta)^{d/2 -1}d \theta \ ,$$ where $d$ denotes the number of degrees of freedom of the system, $B$ is the Euler beta function, and $R \equiv T' / T$. Since it depends only on the temperature ratio $R$, the acceptance probability (\[eq:AcTT\]) has the same value for any arbitrary replica running at a temperature $T_i$, provided that its neighboring upper temperature $T_{i+1}$ is given by $T_{i+1}=RT_{i}$. The value of $R$ is determined such that the acceptance probability given by Eq. (\[eq:AcTT\]) attains a prescribed value $p$, usually chosen greater that 0.5. Thus, the (optimal) schedule that ensures a constant probability $p$ for swaps between neighboring temperatures is a geometric progression: $$T_i = R^{i-1}T_{min},\quad 1 \leq i \leq N,
\label{eq:schedule}$$ where $T_{min}=T_1$ is the minimum temperature of the schedule. Though more research is required in order to better quantify the relative efficiency of the two different algorithms SA and PT, it is apparent from Eqs. (\[eq:SAlaw\]) and (\[eq:schedule\]) that the parallel tempering algorithm is a global optimizer superior to SA because it allows for a faster cooling schedule. Direct numerical comparisons of the two methods have confirmed that parallel tempering is the superior optimization technique.[@Mor03] The ideas of parallel tempering and simulated annealing are not mutually exclusive, and in fact they can be used together to design even more efficient stochastic optimizers. As shown below, such a strategy that combines parallel tempering and simulated annealing is employed for the present simulations.
![Heat capacity of a Si(105) slab plotted as a function of temperature. The peak is located at 1550K; in order to avoid recalculation of the heat capacity for systems with different numbers of atoms and surface orientations, we set $T_{max}=1600$K as the upper limit of the temperatures range used in the PTMC simulations.[]{data-label="figheatcapacity"}](heatcapacity.eps){width="8cm"}
Description of the algorithm
----------------------------
The typical Monte Carlo simulation done in this work consists of two main parts that are equal in terms of computational effort. In the first stage of the computation, we perform a parallel tempering run for a range of temperatures $[T_{min},\ T_{max}]$. The configurations of minimum energy are retained for each replica, and used as starting configurations for the second part of the simulation, in which each replica is cooled down exponentially until the largest temperature drops below a prescribed value. As a key feature of the procedure, the parallel tempering swaps are not turned off during the cooling stage. Thus, we are using a combination of parallel tempering and simulated annealing, rather than a simple cooling. Finally, the annealed replicas are relaxed to the nearest minima using a conjugate-gradient algorithm. We now describe in detail the stochastic minimization procedure. We shall focus, in turn, on discussing the Monte Carlo moves, the choice of the temperature range $[T_{min},\ T_{max}]$, and the total number of replicas $N$.
The moves of the hot atoms consist in small random displacements with the $x,\ y,\ z$ components given by $$\Delta (2u_{\alpha}-1) \$$ where $u_{\alpha}$ $(\alpha = x, y, z)$ are independent random variables [@Mat98] uniformly distributed in the interval $[0,1]$, and $\Delta$ is the maximum absolute value of the displacement. We update the positions of the individual hot atoms one at a time in a cyclic fashion. Each attempted move is accepted or rejected according to the Metropolis logic.[@Met53] A complete cycle consisting in attempted moves for all hot particles is called a *pass* (or sweep) and constitutes the basic computational unit in this work. We have computed distinct acceptance probabilities for the hot atoms that are closer to the surface (situated within a distance of 5 Å below the surface) and for the deeper atoms, the movements of which are essentially small oscillations around the equilibrium bulk positions. Consequently, as shown in Fig. \[figgeom\], we have employed two different maximal displacements, $\Delta_s$ for the surface atoms, and $\Delta_b$ for the bulk-like atoms lying in the deeper subsurface layers. The displacements $\Delta_s$ and $\Delta_b$ are tuned in the equilibration phase of the simulation in such a way that the Monte Carlo moves are accepted with a rate of 40% to 60%. This tuning of the maximal displacements has been performed automatically by dividing the equilibration phase into several blocks, computing acceptance probabilities for each block, and increasing or decreasing the size of the displacements $\Delta_{s,b}$ until the acceptance probabilities reached values between 40% and 60%. The automatization is necessary because the optimal displacements computed for replicas running at different temperatures have different values. The maximal displacement $\Delta_s$ for the surface atoms is found to be larger than the maximal displacement for the bulk-like atoms. Though expected in view of the larger mobility of the surface atoms, the difference between $\Delta_s$ and $\Delta_b$ is not substantial and the reader may safely employ a single maximal displacement for all hot atoms at a given temperature.
Parallel tempering configuration swaps are attempted between replicas running at neighboring temperatures at every 10 passes in an alternating manner, first with the closest lower temperature then with the closest higher temperature. Exception make the two replicas that run at end temperatures $T_1=T_{min}$ and $T_N=T_{max}$, which are involved in swaps every 20 passes. The range of temperatures $[T_{min}, T_{max}]$ and the temperature schedule $T_1 < T_2 < \cdots < T_N $ have been chosen as described below.
The maximum temperature $T_{max}$ must be high enough to ensure that the corresponding random walker has good probability of escaping from various local minima. However, as the temperature is raised, increasingly more thermodynamic weight is placed on local minima that have high energies compared to the global minimum. Stillinger and Weber[@Sti83; @Sti99] have argued that the number of local minima increases exponentially with the dimensionality of the system. As such, the probability that the walker visits the basin of the global minimum significantly decreases with the increase of temperature. A very strong decrease occurs at the melting point, beyond which most of the configurations visited are associated with the liquid phase. The basins of these configurations are unlikely to contain the global minimum or, in fact, any of the low-energy local minima associated with meaningful surface reconstructions. Therefore, the high-temperature end must be set equal to the melting temperature.
The melting temperature of the surface slab can be determined from a separate parallel tempering simulation by identifying the peak of the heat capacity plotted as a function of temperature. As Fig. \[figheatcapacity\] shows, the melting temperature of a Si(105) sample slab with $70$ hot atoms is about $1550~\text{K}$. Rather than determining a melting temperature for each individual system studied, we have employed a fixed value of $T_{max}=
1600~\text{K}$. The melting temperature of the slab determined here (Fig. \[figheatcapacity\]) is different from the value of 1250K reported for the bulk crystal:[@hoep] the discrepancy is due to surface effects, finite-size effects, as well as to the fact that the hot atoms are always in contact with the rigid atoms from the bottom of the slab. Though we use $T_{max}=1600$K for all simulations, we note that differences of 100K–200K in the melting temperature of the slab do not significantly affect the quality of the Monte Carlo sampling. For most surfaces and system sizes of practical importance, the value of $1600~\text{K}$ is in fact un upper bound for the melting temperature; this may sometimes cause the one or two walkers that run at the highest temperatures to be uncoupled from the rest of the simulation, since they might sample amorphous or liquid states. However, this loss in computational resources is very small compared to the additional effort that would be required by a separate determination of the heat capacity for each surface slab used.
![Exponential cooling of the $N=32$ Monte Carlo walkers (replicas of the surface slab) used in the simulation. For clarity, only eight walkers are shown (every fourth walker). The cooling is performed in 18 steps: at each step the temperature is modified by the same factor $\alpha=0.85$ for all walkers, Eq. (\[eq:cooling\]). For every cooling step $k$, we have a different parallel tempering schedule where each replica is coupled to the walkers running at neighboring temperatures via configuration swaps \[Eq. (\[eq:schedule\]) with $R=4^{1/31}$\]. This coupling is symbolized by the double-arrow lines in the inset.[]{data-label="figschedule"}](expschedule.eps){width="8.7cm"}
In theory, the lowest temperature $T_{min}$ should be set so low that the walker associated with this temperature is virtually localized in the basin associated with the global minima. Nevertheless, obstacles concerning the efficient use of computational resources prevent us from doing so. Numerical experimentation has shown that a temperature of $T_{min} =
400~\text{K}$ is low enough that only local minima associated with realistic surface reconstructions are frequently visited. A further selection among these local minima is performed in the second part of the Monte Carlo simulation, when all temperatures of the initial schedule $\{T_i, \ i=1,2,...N \}$ are gradually lowered to values below $100~\text{K}$; as it turns out, this combination of parallel tempering and simulated annealing makes optimal use of computational resources. Below the melting point the heat capacity of the surface slab is almost constant and well approximated by the capacity of a multidimensional harmonic oscillator (refer to Fig. \[figheatcapacity\]). In these conditions, the acceptance probability for swaps between neighboring temperatures $T$ and $T'$ is given by Eq. (\[eq:AcTT\]) (see also Ref. ). It follows that the optimal temperature schedule on the interval $[T_{min},
T_{max}]$ is the geometric progression (\[eq:schedule\]), where $$R = (T_{max}/T_{min})^{1/[N(d,p)-1]}.$$ We have written $N\equiv N(d,p)$ to denote the smallest number of replicas that guarantees a swap acceptance probability of at least $p$ for a system with $d$ degrees of freedom. Since the best way to run PTMC calculations is to use one processor for each replica of the system, the feasibility of our simulations hinges on values of $N(d,p)$ that translate directly into available processors. The number of walkers $N(d,p)$ can be estimated[@Pre03] by $$\label{eq:minimumN} N(d,p) = \left[d^{1/2}
\frac{\sqrt{2}\ln(T_{max}/T_{min})}{4\text{erf}^{-1}(1-p)}\right]
+ 2,$$ where $[x]$ denotes the largest integer smaller than $x$, and erf$^{-1}$ is the inverse error function. Based on Eq. (\[eq:minimumN\]), we have used $N=32$ walkers for all simulations, which ensures a swap acceptance ratio greater than $p=0.5$ for any system with less than 300 hot atoms, $d<900$. The first part of all Monte Carlo simulations performed in the present article consists of a number of $36\times 10^{4}$ passes for each replica, preceded by $9\times 10^4$ passes allowed for the equilibration phase. When we retained the configurations of minimum energy, the equilibration passes have been discarded so that any memory of the starting configuration is lost.
We now describe the second part of the Monte Carlo simulation, which consists of a combination of simulated annealing and parallel tempering. At the $k$-th cooling step, each temperature from the initial temperature schedule $\{ T_i, i=1,2,..N \}$ is decreased by a factor which is independent of the index $i$ of the replica, $T_{i}^{(k)} = \alpha_{k} T_{i}^{(k-1)}.$ Because the parallel tempering swaps are not turned off, we require that at any cooling step $k$ all $N$ temperatures must be modified by the same factor $\alpha_{k}$ in order to preserve the original swap acceptance probabilities. The specific way in which $\alpha_k$ depends on the cooling step index $k$ is determined by the kind of annealing being sought. In this work we have used a cooling schedule of the form $$\label{eq:cooling} T_{i}^{(k)} = \alpha T_{i}^{(k-1)}=\alpha
^{k-1}T_i \ \ \ \ (k \geq 1),$$ where $T_i\equiv T_i^{(1)}$ and $\alpha$ is determined such that the temperature intervals $[T_{1}^{(k-1)}, T_{N}^{(k-1)}]$ and $[T_{1}^{(k)},T_{N}^{(k)}]$ spanned by the parallel tempering schedules before and after the $k$-th cooling step overlap by $80\%$. This yields a value for $\alpha$ given by $(0.2 T_{min}+
0.8 T_{max})/{T_{max}} = 0.85. $ We have also tested values of $\alpha$ larger than 0.85, and did not find any significant improvements in the quality of the sampling.
The reader may argue that the use of an exponential annealing \[Eq. (\[eq:cooling\])\] is not the best option for attaining the global energy minimum of the system. Apart from the theoretical considerations discussed in the preceding subsection that only a logarithmic cooling schedule would ensure convergence to the ground state,[@Gem84; @Haj88]it is known that the best annealing schedules for a given computational effort oftentimes involve several cooling-heating cycles. We emphasize that in the present simulations, the most difficult part of the sampling is taken care of by the initial PTMC run. In addition, since the configuration swaps are not turned off during cooling (refer to Fig. \[figschedule\]), the Monte-Carlo walkers are subjected to cooling-heating cycles through the parallel tempering algorithm.
The purpose of the annealing (second part of the simulation) is to cool down the best configurations determined by the initial parallel tempering in a way that is more robust than the mere relaxation into the nearest local minimum. If the initial PTMC run is responsible for placing the system in the correct funnels (groups of local minima separated by very large energy barriers), the annealing part of the simulation takes care of jumps between local minima separated by small barriers within a certain funnel. For this reason, the annealing is started from the configurations of minimum energy determined during the first part. The cooling is stopped when the largest temperature in the parallel tempering schedule drops below $100$K. This criterion yields a total of 18 cooling steps, with $2\times 10^4$ MC passes per replica performed at every such step.
Each cooling step is preceded by $5\times 10^3$ equilibration passes, which are also used for the calculation of new maximal displacements $\Delta_s$ and $\Delta_b$, as these displacements depend on temperature and must be recomputed. In fact, each cooling step is a small-scale version of the first part of the simulation. The only difference is that the cooling steps are *not* started from the configurations of minimum energy determined at the preceding cooling steps. (Otherwise, because the number of passes for a given step is quite small, the walkers might not have time to escape from some spurious local minima and we would end up restarting them over again from the respective minima.)
The third and final part of the minimization procedure is a conjugate-gradient optimization of the last configurations attained by each replica. The relaxation is necessary because we aim to classify the reconstructions in a way that does not depend on temperature, so we compute the surface energy at zero Kelvin for the relaxed slabs $i,\ i=1,2,...N$. The surface energy $\gamma$ is defined as the excess energy (with respect to the ideal bulk configuration) introduced by the presence of the surface: $$\gamma =\frac{E_m-n_m e_b}{A}$$ where $E_m$ is the potential energy of the $n_m$ atoms that are allowed to move, $e_b=4.6124$eV is the bulk cohesion energy given by HOEP, and $A$ is the surface area of the slab.
Results for the Si(105) surface
===============================
We have tested the method for a variety of surface orientations, such as (113), (105) and (5 5 12). In this section we are presenting results for Si(105), a choice that was determined by the ubiquity of the (105) orientation on the side facets of the pyramidal quantum dots obtained in the heteroepitaxial deposition of Ge and Si-Ge alloys on Si(001). Recent experimental and theoretical work on the atomic structure of (105) surfaces [@jap-prl; @jap-ss; @italy-prl; @apl; @susc105] provides a strong testing ground for the current investigations. In order to assess the versatility of the method and to provide a direct comparison with a previous heuristic study [@susc105] of the (105) reconstructions, we start our PTMC simulations from each of the structures found in Ref. .
{width="15.00cm"}
{width="13cm"}
To establish the nomenclature for the discussion to follow, we recall that the structures were labelled by SU, SR, DU, DU1, DR, DR1, and DR2, where the first letter denotes the height of the steps (single S, or double D), the second letter indicates whether the step is rebonded (R) or unrebonded (U) and the digit distinguishes between different structures that have the same broad topological features.[@susc105] These reconstructions have different numbers of atoms and different linear dimensions of the periodic cell. The dimensions of the cell are chosen $2a\times
a\sqrt{6.5}$ ($a=5.431$Å is the bulk lattice constant of Si) for all the models considered except DR2, whose topology requires a periodic cell of $2a\times 2a\sqrt{6.5}$. The thickness of the slab corresponds to two unit cells in the $z$ direction, with a maximum of 208 atoms, of which only about half are allowed to move.
The results of the PTMC simulations for the Si(105) surface are plotted in Fig. \[figharvest\], which shows the total energy for each of the $N=32$ replicas at the end of the cooling procedure (circles) and after the conjugate-gradient relaxation (triangles). Figs. \[figharvest\](a), (b), (c) and (d) show the total energies of the reconstructions obtained starting from the SU, DU, DR and DR2 models, respectively. In each case, we have obtained at least two structures with lower surface energies than the starting configurations, which we discuss in turn.
{width="15.00cm"}
Fig. \[figharvest\](a) shows that the (starting) SU structure[@mo] is found only by the two replicas running at the highest temperatures, while colder walkers find a novel double-stepped structure, termed here “transitional” (DT). At even lower temperatures, the double-steps of the DT reconstruction unbunch into single-height rebonded (SR) steps; the three different configurations that correspond to the energies plotted in Fig. \[figharvest\](a) are shown in Fig. \[figSUDTSR\]. Therefore, the correct SR structure [@jap-prl; @jap-ss; @italy-prl; @apl; @susc105] is retrieved even when starting from the topologically different SU model. The usefulness of this PTMC simulation becomes apparent if we recall that the SU structure was widely believed to be correct for more than a decade after its publication. As we shall see, the ground state obtained in our stochastic search is independent of the initial configuration. The only condition for finding the reconstruction with the lowest surface energy is to prescribe the correct number of atoms and the correct dimensions for the simulation slab. We will address these practical aspects in the next section; for now, we continue to describe the results obtained for different numbers of atoms in the computational slab.
The simulation that starts from the DU model finds two distinct rebonded structures, denoted by DX1 and DX2 in Fig. \[figharvest\](b). Both these structures are characterized by the presence of single dimers at the location of steps (see Fig. \[figDTRX\]), which reduces the number of dangling bonds per unit area from 6$db/a^2\sqrt{6.5}$ (starting structure DU) to 5$db/a^2\sqrt{6.5}$. The DX1 reconstruction is the most stable, and it is obtained in all but three replicas of the system.
![Double-step reconstructions of Si(105) with periodic cells (rectangles shown) of $2a\times 2a\sqrt{6.5}$. Although the starting structure \[the DR2 model[@susc105] shown in (d)\] has a reasonably low dangling bond density ($5 db/a^2\sqrt{6.5}$), the Monte Carlo simulation has retrieved three more reconstructions, all having smaller surface energies (refer to Table \[table\_gamma\_smcell\]). These novel structures \[shown in figs. (a)–(c)\] are labelled by DR2$\alpha$, DR2$\beta$, DR2$\gamma$. The atoms are rainbow-colored as indicated in Fig. \[figSUDTSR\].[]{data-label="figDR2"}](DR2.eps){width="8.70cm"}
Although it has a small density of dangling bonds, the DR structure has large surface energy due to the $\sqrt{2} \times 1$ terrace reconstruction.[@susc105] Since the density of dangling bonds is the lowest possible (4$db/a^2\sqrt{6.5}$), the minimization of surface energy is dictated by the reduction of surface stress. Unlike the case of SU and DU structures (described above), not a single replica have retained the starting model DR. Instead, the DT and SR structures are retrieved (refer to Fig. \[figharvest\](c)). When starting from the DR2 structure we obtain at least three low energy structures denoted by DR2$\alpha$, DR2$\beta$ and DR2$\gamma$ (Fig. \[figDR2\]), which have not been previously proposed in Refs. , , or elsewhere. Owing to a larger area of the slab, portions of the newly reconstructed unit cells have patches that resemble the models obtained in prior simulations. In particular, the atomic scale features of the steps on DR2$\alpha$ are very similar to those of the SR structure, a similarity that reflects in the very small relative surface energy of the two models ($\approx 1.6$meV/Å$^2$ ).
We note that the simulations described have a total number of atoms that is between $n=202$ and $n=206$ (Fig. \[figharvest\](a) and (c)) per $2a^2\sqrt{6.5}$ area. To cover all the possibilities for intermediate numbers of atoms, we also perform a simulation with $n=205$; this value of $n$ does not correspond to any of the models reported in Ref. , and the parallel tempering run is started from a bulk-truncated configuration. In this case two new structures are found; these structures are named DY1 and DY2 and shown in Fig. \[figDTRX\]. \[The letters X and Y appearing in DX1, DX2, DY1, DY2 (all denoting double-stepped rebonded structures, Fig. \[figDTRX\]) do not stand for particular words, they are simply intended to unambiguously label the structures in a way that does not complicate the notation.\] While for the DY1 model the rebonding is realized via bridging bonds,[@susc105] in the case of DY2 we find unexpected topological features such as fully saturated surface atoms and over-coordinated bulk atoms. Even though these structural units (seen in the DY2 panel of Fig. \[figDTRX\]) reduce the number of dangling bonds, they also create high atomic-level stresses which make the DY2 reconstruction relatively unfavorable.
We have also performed PTMC simulations with SR, DR1 and DU1 as initial configurations, but have not obtained any other novel reconstructions. We found that SR and DR1 are the global energy minima corresponding to 206 atoms and 203 atoms, respectively. The DU1 structure[@china-Si105] (202 atoms) has lead to the same reconstructions as the SU model (206 atoms). This result indicates a periodic behavior of the surface energy as a function of the total number of atoms, which will be discussed next.
![Atomic structure of the bulk truncated Si(105) surface, viewed from the side (a) and from the top (b). The rectangle of dimensions $2a\times a\sqrt{6.5}$ marks the periodic cell used in most of the simulations, and contains two unit cells of the bulk-truncated surface. For clarity, only a single subsurface (001) layer is shown. In this picture (unlike in Figs. \[figSUDTSR\], \[figDTRX\] and \[figDR2\]) atoms are colored according the their number of dangling bonds ($db$) before reconstruction: red$ = 2db$, green$ = 1db$ and blue$ = 0db$.[]{data-label="figbulktruncated"}](bulktruncated.eps){width="7cm"}
Discussion
==========
To further test that the lowest energy states for given number of atoms are independent of the initial configurations, we have repeated all the calculations using bulk-truncated surface slabs (Fig. \[figbulktruncated\]) instead of reconstructed ones. We have varied the number of atoms $n$ in the simulation cell between 196 and 208, where the latter corresponds to four bulk unit cells of dimensions $a\sqrt{6.5}\times a \times a\sqrt{6.5}$ stacked two by two in the \[010\] and \[105\] directions. For the cases with $n<208$, we have started the PTMC simulations from structures obtained by taking out a prescribed number atoms from random surface sites, and have found the same ground state irrespective of the locations of the removed atoms. For values of $n$ equal to 202, 203, 204 and 206, the ground states (global minima) are also the same as the ones obtained from the reconstructed models DR, DR1, DU, and SU, respectively. Furthermore, we have tested that even when removing arbitrary [*subsurface*]{} atoms the simulation retrieves the same ground states without increasing the computational effort. This finding speaks for the quality of the Monte Carlo sampling and gives confidence in the predictive capabilities of the method described in section II. The lowest surface energies obtained at the end of the numerical procedure are shown in Fig. \[figperiodicsurfene\] as a function of the number of atoms in the simulation cell.
![Surface energy of the global minimum structure plotted versus the total number of atoms $n$ in the simulation slab. Even though there are twelve under-coordinated atoms in each bulk-truncated periodic cell (refer to Fig. \[figbulktruncated\]), the values of the surface energy repeat at intervals of $\Delta n =4$. The underlying bulk structure reduces the number of distinct global minima to four.[]{data-label="figperiodicsurfene"}](periodicsurfene.eps){width="8.7cm"}
As illustrated in Fig. \[figperiodicsurfene\], the simulation finds the same ground states at periodic intervals of $\Delta n =
4$. At first sight, this is somewhat surprising given that the number of under-coordinated surface atoms in a bulk-truncated cell of dimensions $2a \times a\sqrt{6.5}$ is twelve (refer to Fig. \[figbulktruncated\]). The reduced periodicity of the surface energy with the number of atoms in the supercell is due to the underlying crystal structure, which lowers the number of symmetry-distinct global minima to only four. Thus, we have considered all possibilities in terms of numbers of atoms in a simulation slab of area $2a^2\sqrt{6.5}$. The surface energies of the optimal reconstructions for relevant values of $n$, as well as those of some higher-energy structures, are collected in Table \[table\_gamma\_smcell\]. As shown in the table, the global minimum of the surface energy of Si(105) is obtained for the single-height rebonded structure SR. While this finding is in agreement with recent reports,[@jap-prl; @jap-ss; @italy-prl; @apl; @susc105] it is the result of an exhaustive search rather than a comparison between two[@jap-prl; @jap-ss; @italy-prl; @apl] or more[@susc105] heuristically proposed structures.
-------------- ------------- ---------------------- ------------- -------------
$n$ Structure Bond counting HOEP TB
($db/a^2\sqrt{6.5}$) (meV/Å$^2$) (meV/Å$^2$)
206 SR 4 82.20 82.78
DT 4 85.12
SU 6 88.35 83.54
205 DY1 5 86.73
DY2 4.5 88.61
204 DX1 5 84.90
DX2 5 86.04
DU 6 90.18 84.84
203 DR1 5 86.52 85.22
2$\times$203 DR2$\alpha$ 4.5 83.77
2$\times$203 DR2$\beta$ 4.5 84.64
2$\times$203 DR2$\gamma$ 4.5 86.15
2$\times$203 DR2 5 86.34 83.48
-------------- ------------- ---------------------- ------------- -------------
: Surface energies of different Si(105) reconstructions, calculated using the HOEP interatomic potential.[@hoep] The structures are grouped according to the number of atoms $n$ in the simulation cell. Atomic configurations of selected reconstructions are shown in Figs. \[figSUDTSR\], \[figDTRX\] and \[figDR2\]. The third column shows the number of dangling bonds ($db$) per unit area, expressed in units of $a^2 \sqrt{6.5}$. The last column indicates the tight-binding[@tbmd](TB) values reported in Ref. .[]{data-label="table_gamma_smcell"}
From Table \[table\_gamma\_smcell\] we also note that the SR and the DR2$\alpha$ structures have surface energies that are within 1.6meV/Å$^2$ from one another. This gap in the surface energy of the two models (SR and DR2) is smaller than the expected accuracy of relative surface energies determined by an empirical potential. Therefore, it is very likely that these two reconstructions can both be present on the same surface under laboratory conditions. As recently pointed out,[@susc105] the coexistence of several configurations with different topological features but similar surface energies gives rise to the atomically rough and disordered aspect[@tomitori; @china-Si105] of the Si(105) surface. The surface energies computed using HOEP for various rebonded structures (Table \[table\_gamma\_smcell\]) are close to the values obtained previously[@susc105] at the tight-binding level.[@tbmd] For the unrebonded structures (SU and DU), the differences between the HOEP values and the tight-binding ones are larger: this discrepancy is caused by the inability of the HOEP interaction model to capture the tilting of the surface dimers, which is an important mechanism for the relaxation of these unrebonded configurations. Despite this shortcoming, we have found that the HOEP potential is accurate enough to predict the correct bonding topology of the global minimum reconstructions for a variety of surface orientations. If a comparison with experimental STM images is desired, further geometry optimizations are necessary at the level of electronic structure methods: these calculations would have to consider different tiltings of the surface bonds, and in each case the simulated image is to be compared with the experimental one. Thus, even for surfaces where dimer tilting is important, the Monte Carlo simulation based on the HOEP interaction model[@hoep] can still serve as a very efficient tool to find good candidates for the lowest energy structures.
Two practical issues have to be addressed when using PTMC simulations for surface structure prediction. First consideration is related to the size of the computational cell. If a periodic surface pattern exists, the lengths and directions of the surface unit vectors can be determined accurately through experimental means (e.g., STM). In those cases, the periodic lengths of the simulation slab should simply be chosen the same as the ones found in experiments. On the other hand, when the surface does not have two-dimensional periodicity (as it is the case of unstrained Si(105) surface[@tomitori; @china-Si105]), or when experimental data is not available, one should systematically test computational cells with periodic vectors that are low-integer multiples of the unit vectors of the bulk truncated surface; the latter unit vectors can be easily computed from the knowledge of crystal structure and surface orientation. Secondly, the number of atoms in the simulation cell is not [*a priori*]{} known, and there is no simple criterion to find the set of numbers that yield the lowest surface energy for a slab with arbitrary orientation. Adapting the algorithm presented in section II for a grand-canonical ensemble is somewhat cumbersome, as one would have to consider efficiently the combination of two different types of Monte Carlo moves: the small random displacements of the atoms (continuous) and the discrete processes of adding or removing atoms from the simulation slab. The problem of finding the correct number of atoms in the computational cell is not new, as it also appears, for example, in classic algorithms for predicting the bulk crystal structure.[@jap-pr] As shown above for the case of Si(105), a successful way to deal with this problem is to simply repeat the simulation for systems with consecutive numbers of atoms, and look for a periodic behavior of the surface energy of coldest replicas as a function of the number of particles in the computational cell. Note that if the thickness of the slab is sufficiently large, such periodicity of the lowest surface energy with respect to the number of atoms in the supercell is [*guaranteed*]{} to exist: in the worst case, the periodicity will appear when an entire atomic layer has been removed from the simulation cell.
Concluding Remarks
==================
In conclusion, we have developed and tested a stochastic method for predicting the atomic configuration of silicon surfaces. If suitable empirical models for atomic interactions are available, this method can be straightforwardly applied for the determination of the structure of any crystallographic surface of any other material. Using the example of Si(105), we have shown that the PTMC search is superior to heuristic approaches because it ensures that the topology corresponding to the lowest surface energy is considered in the set of good possible structural models. We have performed an exhaustive search for different numbers of atoms in the simulation cell and have found that the global minimum of the (105) surface energy is the single-height rebonded model SR, in agreement with recent studies.[@jap-prl; @jap-ss; @italy-prl; @apl; @susc105] The experiments of Zhao [*et al.*]{}[@china-Si105] indicated that double-stepped structures are present on the unstrained Si(105) surface: our simulations indeed have found double-stepped models with surface energies that are close to the surface energy of the optimal SR reconstruction. In addition, these double-stepped models (termed DR2$\alpha$, DR2$\beta$, and DR2$\gamma$) are energetically more favorable than the double-stepped structures proposed in Refs. and .
We would like to comment on the key role played by the empirical potential used in the present simulations. A highly transferable interatomic potential is required for a satisfactory energetic ordering of different reconstructions. While we would not expect any empirical potential to accurately reproduce the relative surface energies of all the reconstructions found, we can at least expect that the chosen potential correctly predicts the bonding topology for well-known surface reconstructions. In this respect, the HOEP model[@hoep] proved superior to the most widely used interatomic potentials.[@stillweb; @tersoff3] Given this comparison, the results presented here would represent a validation of the work[@hoep] towards more transferable potentials for silicon. We also hope that these results would stimulate further developments of interatomic potentials for other semiconductors.[@feigao]
With the exception of Si(105), Si(113)[@dabrowski] and (likely) Si(114),[@si114] the atomic structure of other stable high-index silicon surfaces has not been elucidated, although a substantial body of STM images has accumulated to date.[@si-highindex] A similar situation exists for Ge surfaces as well.[@ge-highindex] The methodology presented in this article can be used (either directly or in combination with the STM images[@si-highindex]) to determine the configuration of all high-index Si surfaces, as long as the HOEP potential remains satisfactory for all orientations to be investigated. Furthermore, with certain modifications related to the implementation of empirical potentials for systems with [*two atomic species*]{}, the PTMC method could help bring important advances in terms of finding the thermodynamically stable intermixing composition of various nanostructures obtained by heteroepitaxial deposition of thin films on silicon substrates. Though such studies have recently been reported,[@landau-sige] only the intermixing at a given atomic bonding topology has been investigated. The interplay between reconstruction and intermixing is another challenging and important problem that can be tackled via PTMC simulations. Lastly, the method presented in this article may also be used for studying the decomposition of unstable orientations into nanofacets, as well as for predicting the thermodynamics of surfaces in the presence of adsorbates or applied strain.
[[**Acknowledgements.**]{}]{} We gratefully acknowledge funding from National Science Foundation through the Brown MRSEC program (DMR-0079964), and Grants No. CHE-0095053 and CHE-0131114. The simulations were performed at the Center for Advanced Scientific Computation and Visualization at Brown University. We thank Professor J.D. Doll for generous support, Professor M.C. Tringides for useful discussions on the structure of high-index silicon surfaces, and Professor S.J. Singer for valuable comments on the manuscript.
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|
---
abstract: 'There have now been three supernova-associated $\gamma$-ray bursts (GRBs) at redshift $z<0.17$, namely 980425, 030329, and 031203, but the nearby and under-luminous GRBs 980425 and 031203 are distinctly different from the ‘classical’ or standard GRBs. It has been suggested that they could be classical GRBs observed away from their jet axes, or they might belong to a population of under-energetic GRBs. Recent radio observations of the afterglow of GRB 980425 suggest that different engines may be responsible for the observed diversity of cosmic explosions. Given this assumption, a crude constraint on a luminosity function for faint GRBs with a mean luminosity similar to that of GRB 980425 and an upper limit on the rate density of 980425-type events, we simulate the redshift distribution of under-luminous GRBs assuming BATSE and *Swift* sensitivities. A local rate density of about 0.6% of the local supernova Type Ib/c rate yields simulated probabilities for under-luminous events to occur at rates comparable to the BATSE GRB low-redshift distribution. In this scenario the probability of BATSE/HETE detecting at least one GRB at $z<0.05$ is 0.78 over 4.5 years, a result that is comparable with observation. [*Swift*]{} has the potential to detect 1–5 under-luminous GRBs during one year of observation.'
author:
- |
D. M. Coward[^1]\
School of Physics, University of Western Australia, M013, Crawly WA 6009, Australia
title: 'Simulating a faint gamma-ray burst population'
---
\[firstpage\]
cosmology: observations – gamma-rays: bursts – supernovae: general
Introduction
============
Discovery of GRB 980425, associated with the very nearby supernova (SN) 1998bw (at $z=0.0085$, corresponding to about 40 Mpc), heralded a new era in understanding the origin of GRBs [@Galama1; @Galama2]. Recently, the detection of spectroscopic features in the light curve of GRB 030329, similar to those seen in SN 1998bw [@stanek03; @Hjorth03], has strengthened the SN 1998bw/GRB 980425 association. These observations support the ‘collapsar’ model [@zang03] in which a Wolf-Rayet progenitor, possibly in a binary system, undergoes core collapse, producing a compact object surrounded by an accretion disk, which injects energy into the system and thus acts as a ‘central engine’. The energy extracted from the system gives rise to a Type Ib/c SN explosion and drives collimated jets along the progenitor rotation axis, producing a prompt GRB and afterglow emission–see the review by Zhang & Mészáros (2004).
Discovery of the GRB-SN association was an important breakthrough, but GRB 980425 had an unusually low luminosity—it was under-luminous in $\gamma$-rays by three orders of magnitude compared to ‘classical’ GRBs. It was suggested that it could be a very rare event and not a member of the classical GRB population. This explanation seems unlikely given the discovery of the under-luminous and nearby GRB 031203 ($z=0.105$), the first analogue of GRB 980425.
There are presently three GRBs (980425, 030329, and 031203) definitely associated with extremely energetic Type Ib/c SNe [@Proch04; @mal04], all occurring at $z<0.17$. GRB 030329 is classified as classical in the context of energy emission, but was relatively close at $z=0.17$. However, it is difficult to reconcile the under-luminous GRBs 980425 and 031203 with the classical population. One simple explanation is that under-luminous bursts are GRBs observed away from the jet axis.
[@g04] argue that a unified picture can only be obtained by using a luminosity function (LF) that includes all luminosities down to that of GRB 980425, so that the probability of observing the three low-$z$ events is non-negligible. They show that for GRBs 980425, 030329, and 031203 to belong to the classical burst population, the LF must be a broken power-law. This is an attractive proposal in that GRBs 980425 and 031203 can be explained within the bounds of currently popular GRB progenitor models by extending the LF to accommodate GRB 980425. They calculate that if this is the case, no bright burst within $z=0.17$ should be observed by a HETE–like instrument within the next $\sim 20$ yr.
Anomalous GRBs
==============
Evidence for intrinsically sub-energetic events
-----------------------------------------------
[@sod04] argue that if GRB 031203 was observed ‘off axis’, then the radio afterglow should brighten as the ejecta slows down, but they did not observe any re-brightening. They find that the afterglow is faint, indicating that the explosion was under-energetic. Similarly, there is no evidence of re-brightening for GRB 980425, despite radio calorimetry since 1998. [@sod04] conclude that GRBs 980425 and 031203 were intrinsically sub-energetic events.
If SN 1998bw was a rare and unusually sub-energetic SN distinct from local SNe and GRBs, [@sod04] claim the characteristics of SN 1998bw/GRB 980425 are not a result of the observer’s viewing angle but of the properties of its central engine. SN 1998bw was an engine-driven explosion [@li99], in which 99.5% of the kinetic energy ($\sim 10^{50}$ ergs) was coupled to relativistic ejecta of Lorentz factor 2 [@kulk98], while a mere 0.5% went into the ultra-relativistic flow. In contrast, ‘classical’ GRBs couple most of their energy into $\gamma$-rays. [@berg03b] claim that the observed diversity of cosmic explosions (SNe, x-ray flashes (XRF), and GRBs) could be explained with a standard energy source but with a varying fraction of that energy injected into relativistic ejecta. Different engines may be responsible for the observed diversity of cosmic explosions, implying that classical GRBs represent one class of event, one in which $\gamma$-rays channel most of the energy away from a central engine.
It is evident that SN 1998bw could be a member of a distinct class of SN explosions. But how rare is SN 1998bw in the context of Type Ib/c SNe and classical GRBs? [@berg03a] carried out a systematic program of radio observations of Type Ib/c SNe using the Very Large Array to place the first constraint on the rate density of SN 1998bw type events. Of the 33 SNe observed from late 1999 to the end of 2002, they conclude that the fraction of events similar to SN 1998bw is at most 3%. Furthermore they find, by comparison of the SN radio emission to that of GRB afterglows, that none of the observed SNe could have resulted from a classical GRB.
Evidence for an off-axis model for GRB 031203
----------------------------------------------
The evidence that GRB 031203 was an intrinsically faint and nearly spherical explosion is not widely accepted. [@Rameriz04] disagree with this interpretation and argue, from models fitted to the observed x-ray light curve and the radio afterglow, that GRB 031203 was a classical GRB viewed off axis. They find that most spherical models under-predict the x-ray flux at late times by at least two orders of magnitude, and prefer to interpret GRB 031203 as a highly collimated GRB viewed off-axis.
For the case of an observer located outside the jet aperture, $\theta_\mathrm{obs}>\theta_0$, the prompt GRB emission and its early afterglow are considerably weaker than for on axis, $\theta_\mathrm{obs}<\theta_0$. An observer at $\theta_\mathrm{obs}>\theta_0$ sees a rising afterglow light curve at early times, which approaches that seen by an on-axis observer at late times. The emission remains low until the cone of the beam intersects the observer’s line of sight.
In the off-axis jet scenario, with viewing angle $\theta_\mathrm
{obs}\sim 2\theta_0$, GRB 031203 can be modelled as a GRB viewed a few degrees outside of a conical jet. [@Rameriz04] show that if GRB 031203 is modelled as an off-axis observation, its energy emission in $\gamma$-rays is about 10$^{53}$ ergs, consistent with classical GRBs.
The off-axis model of [@Rameriz04] used to explain the weak afterglow of GRB 031203 casts some doubt on the claim that it is an intrinsically weak event. Furthermore, because GRBs 031203 and 980425 share a common deficit in $\gamma$-ray emission, it is possible that these ‘outliers’ are a result of off-axis observations. The possibility that GRBs may represent a class of cosmic explosions with a broad range of energies provides the basis for the following simulation.
A population of under-luminous GRBs
-----------------------------------
Given the potential of the *Swift* satellite, a multi-wavelength GRB observatory (http://swift.gsfc.nasa.gov/) launched on 2004 November 20, to localize hundreds of GRBs, observations of events similar to GRB 980425 will provide new insight into GRB progenitor populations. This is strong motivation to constrain the probability of detecting similar GRB-SNe assuming a simple, if highly uncertain, model. The first upper limits on the rate density of Type Ic SNe associated with GRBs are being made, based on the presence of jetted emissions (e.g. Berger et al. 2003a). Given such upper limits, one can at least provide further constraints on GRB progenitor populations using continued satellite observations. Furthermore, we provide an example to demonstrate how an under-luminous GRB population would manifest during the BATSE observation period and the era of *Swift*.
Firstly, we assume GRB 980425 is a ‘typical’ member of a class of relatively rare GRB (compared to classical GRBs). Based on the luminosity of GRB 980425, the mean luminosity of this population is about 3 orders of magnitude less than that of the classical population. Secondly, the local rate density must be less than 3% of the Type Ib/c SN rate. Another observational constraint is based on the observed rate of GRB 980425 type events out to $z=0.0085$ assuming a 4.5 yr (BATSE) observation period; the probability of occurrence of this very nearby GRB must be compatible with the BATSE GRB distribution. Finally, we assume the $\gamma$-ray emissions are isotropic (not beamed) based on the radio observations of the afterglows of GRBs 980425 and 031203 [@sod04]. With these constraints, we simulate the observed GRB distribution for BATSE and *Swift* sensitivities.
The GRB luminosity function
===========================
The GRB LF, together with the flux sensitivity threshold of an instrument, determine the fraction of all GRBs potentially detectable with that instrument: $$\label{grbrate}
\psi_{\rm GRB}(z)= S_d\int_{L_{\rm lim}(z)}^{\infty}p(L) {\rm d}L
\;,$$ where $\psi_{\rm GRB}(z)$ is the GRB rate scaling function, $S_d$ is the fraction of sky that the detector scans, and $p(L)$ is the GRB LF with $L$ the intrinsic luminosity in units of photons s$^{-1}$. With $f_{\rm lim}$ denoting the instrumental flux sensitivity threshold, in photons s$^{-1}$ m$^{-2}$, the minimum detectable luminosity can be expressed as a function of redshift by $L_{\rm lim}(z)= 4 \pi {D_{\mathrm L}}^{2}(z) f_{\rm lim}$, with $D_{\mathrm L}(z)$ the luminosity distance.
Most models for the classical GRB LF are based on the luminosity–redshift relation [@schaef01]. But the models are biased by the redshift sample and the sensitivity limit related to the redshift estimate. This limit has to be lowered at least by an order of magnitude to encompass the complete range of luminosities. [@firmani04] show that by jointly fitting to the observed differential peak-flux and redshift distributions, the best fit for the LF takes a form that evolves weakly with redshift. However, there is no consensus on the form of LF for classical GRBs; possibilities include a single power law, a double power law and a log-normal distribution.
For an under-luminous population of GRBs modelled on the single GRB 980425, the choice of LF is so uncertain that the form of the function is somewhat arbitrary. Nonetheless, for definiteness and for comparison, we use log-normal distributions for the classical GRBs and an under-luminous population, both with observation-based statistical moments that fit the data: $$\label{PL}
p(L)= \frac{{\rm e}^{-\sigma^{2}/2}}{\sqrt{2\pi \sigma^{2}}} \exp
\bigg\{-\frac{[\ln(L/L_{0})]^{2}}{2\sigma^{2}}\bigg\}\frac{1}{L_{0}}
\;,$$ where $\sigma$ and $L_{0}$ are the width and average luminosity, respectively. We take $\sigma=2$ and $L_{0}=2\times 10^{56}$ s$^{-1}$ for the classical GRBs, with $f_{\rm lim}=0.2$ and 0.04 photons s$^{-1}$ cm$^{-2}$ for BATSE (classical GRBs) and [*Swift*]{} respectively, and take about 0.1 for $S_d$ [@g04]. Assuming that GRB 980425 is representative of an under-luminous population, we take $L_{0}=2\times 10^{53}$ s$^{-1}$, three orders of magnitude less than for the classical GRBs.
GRB rates and detection probability
===================================
One can express the differential GRB rate in the redshift shell $z$ to $z+{\mathrm d}z$ as $$\label{drdz}
\mathrm{d}R = \psi(z) \frac{\mathrm{d}V}{\mathrm{d}z}\frac{r_0
e(z)}{1+z} \mathrm{d}z \,,$$ where $\mathrm{d}V$ is the cosmology-dependent co-moving volume element and $R(z)$ is the GRB event rate, as observed in our local frame, for sources out to redshift $z$. Source rate density evolution is accounted for by the dimensionless evolution factor $e(z)$, which is normalized to unity in the present-epoch universe $(z=0)$, and $r_0$ is the $z=0$ rate density. The $(1 +
z)$ factor accounts for the time dilation of the observed rate by cosmic expansion.
We assume a ‘flat-$\Lambda$’ cosmology with $\Omega_{\mathrm
m}=0.3$ and $\Omega_{\mathrm \Lambda}=0.7$ for the present-epoch density parameters, and take for the Hubble parameter at $z=0$. The star formation rate (SFR) model SF2 of [@mad01] is re-scaled to this cosmology and converted to a normalized evolution factor $e(z)$. This SFR model levels off to an essentially constant rate density, of order 10 times the $z=0$ value, at $z>2$; a star formation cutoff at $z=10$ is assumed.
For the classical GRBs we use $r_0 = 0.9$ yr$^{-1}$ Gpc$^{-3}$, obtained by scaling the all-sky Universal rate to 692 yr$^{-1}$, the value implied by the GUSBAD catalogue. This is comparable to the value of 1.1 yr$^{-1}$ Gpc$^{-3}$ from [@g04], obtained using a different SFR and a broken power-law luminosity function.
As GRBs are independent of each other, their distribution is a Poisson process in time: the probability for at least one event to occur in the volume out to redshift $z$ during observation time $T$ at a mean rate $R(z)$ is given by an exponential distribution: $$\label{prob1}
p(n\ge1;R(z),T) = 1 - e^{-R(z) T}\,.$$ This formula can be used to define a ‘probability event horizon’—as observation time increases, how often will rarer, more local events, be observed? See [@cow041] for a description. Based on equation (\[prob1\]), the probability of at least one GRB occurring in $z<0.17$ during 4.5 yr is about 0.5, implying that the observed GRBs in this volume need not be considered anomalous. But the probability of a GRB occurring in $z<0.01$ over 4.5 yr is 0.00015, implying that GRB 980425 ($z=0.0085$) is either an extreme outlier or a member of a different GRB population. We model the GRB redshift distribution under the latter assumption, with the constraint that the probabilities and rates are consistent with the observed BATSE GRB distribution.
Simulating an under-luminous GRB redshift distribution
======================================================
A GRB distribution comprised of two populations, with different local rate densities and mean luminosities can be expressed as: $$\label{drdz2}
\mathrm{d}R =
\frac{\mathrm{d}V}{\mathrm{d}z}\frac{e(z)}{1+z} \Big[\psi_{\rm
c}(z)r^{\rm c}_0+\psi_{\rm u}(z)r^{\rm u}_0\Big] \mathrm{d}z \,,$$ where $\psi_{\rm c}(z)r^{\rm c}_0$ and $\psi_{\rm u}(z)r^{\rm
u}_0$ are the scaling functions and local rate densities of the classical and under-luminous GRBs respectively. It is assumed that both rates follow the SFR density so that $e(z)$ is the same for both populations.
Figure 1 plots $\psi_{\rm c}(z)$ and $\psi_{\rm u}(z)$ for BATSE and [*Swift*]{} sensitivities. It is evident that at $z>0.1$, the flux-limit of the detectors severely limits the potential detectability of the under-luminous GRB population (mean luminosity $L_{0}=2\times 10^{53}$ s$^{-1}$). For $z<0.1$ the scaling function is non-negligible even though events will be rare inside such a relatively small volume. The sensitivity of [*Swift*]{} implies that it could potentially detect most under-luminous events occurring inside a volume bounded by $z=0.01$.
![GRB scaling functions $\psi_
{\rm c}$ for the classical population, and $\psi_{\rm u}$ for an under-luminous population, shown using BATSE and [*Swift*]{} sensitivities. $\psi_{\rm u}$ is calculated using the assumption that GRB 980425 is a typical event from an under-luminous population of mean luminosity $L_{0}=2\times 10^{53}$ s$^{-1}$, three orders of magnitude less than the mean luminosity of the classical GRBs. A value of 0.1 for the scaling function corresponds to all GRBs in the field of view of the detector (0.1 str) being potentially detectable.](fig1.eps)
Clearly the observed redshift distribution of GRBs and expected rates for very energetic Type Ib/c SNe do impose constraints on the local rate density $r_0^{\rm u}$. Importantly, GRBs 980425 and 031203 show no evidence for jets, implying that there is no geometric rate enhancement factor required to account for unseen bursts of similar type. As a first approximation to $r^{\rm u}_0$, we take the classical rate $r^{\rm c}_0$ increased by the beaming rate enhancement factor for classical bursts, using a value of 250 from [@frail04]. This gives $r^{\rm u}_0\approx 220$ yr$^{-1}$ Gpc$^{-3}$, which is about 0.6% of the local SN Type Ib/c rate, $r_0^{\mathrm {SNIbc}}\approx 3.7\times 10^{4}$ yr$^{-1}$ Gpc$^{-3}$ [@Izzard04]—a result that supports the view that SN1998bw was a relatively rare and unusually energetic SN. We note that the quoted SN Type Ib/c rates may be underestimated because many core-collapse SN are lost to extinction in most surveys to date.
The mean of the luminosity function needs to be reduced by 3-4 orders of magnitude and the variance reduced from 2 to 1.5 to crudely fit the resulting probability distribution with the observed rates of GRBs at small redshift; that is, the probability of occurrence of the very nearby GRB 980425 $(z=0.0085)$ must be compatible with present observations. For this condition to be satisfied we find that $\sigma$ must be smaller than that used for modelling the classical population ($\sigma=2$), otherwise the rate of events at small $z$ would be too large. If a broader luminosity distribution (larger variance) is employed, for example $\sigma=2.5$, the mean luminosity must be reduced to 4 orders of magnitude less than the mean classical GRB luminosity, to yield observationally consistent probabilities. For definiteness and consistency, we assume $\sigma=1.5$ and $L_{0}=2\times 10^{53}$ s$^{-1}$ in all calculations
$p(z<0.0085)$ $p(z<0.05)$ $p(z<0.17)$
---------------------- --------------- ------------- -------------
Classical (BATSE) 0.00015 0.02 0.5
Double (BATSE) 0.04 0.78 0.99
Double ([*Swift*]{}) 0.04 0.96 0.99
: The probability of observing at least one GRB inside volumes bounded by $z=$ 0.0085, 0.05, and 0.17 during 4.5 yr of observation using a single GRB distribution (classical) with BATSE sensitivity and a double distribution (classical + under-luminous) with BATSE and [*Swift*]{} sensitivities, (labelled as Double (BATSE) and Double ([*Swift*]{}) respectively.\[tbl-1\]
Table 1, using the same parameters as figure 1, shows the probabilities for detecting at least one GRB in volumes bounded by $z=$ 0.0085, 0.05, and 0.17 during 4.5 yrs observation, assuming the sensitivities of both BATSE/HETE and [*Swift*]{}. Equation (\[drdz\]) is used to calculate the rates based on a classical distribution and equation (\[drdz2\]) for a distribution comprised of both classical and under-luminous bursts. The double distribution model increases the detection probability of the very nearby GRB 980425 ($z=0.0085$) to a still small (0.04), but significant, level compared with the extremely small probability (0.00015) from the classical distribution alone.
![As for Figure 2 but plotting the cumulative number of GRBs as a function of redshift for an observation time of 1 yr. A comparison of the double and classical models for BATSE shows that the under-luminous population causes a significant increase in numbers at $z<0.1$, resulting in a higher probability of observing small-$z$ GRBs. [*Swift*]{} could potentially detect over 100 GRBs during one year with up to 5 in $z<0.1$](fig3b.eps)
Figures 2 and 3 plot the number distribution and cumulative number of GRBs observed over a 1 yr period for the classical and double distributions assuming BATSE and [*Swift*]{} sensitivities. For BATSE sensitivities it is evident that the under-luminous GRBs have no effect on the observed distribution at $z>0.2$. They do contribute to the cumulative number at $z<0.2$, a result that is compatible with the detection of GRBs 980425 and 031203. The cumulative number increases from about 1 at $z=0.1$, for a BATSE sensitivity, to about 4 or 5 for a [*Swift*]{} sensitivity.
Conclusions
===========
We have shown that a population comprised of classical and under-luminous GRBs is compatible with the currently observed GRB redshift distribution that includes the nearby and under-luminous GRBs 980425 and 031203. We find that a local rate density for an under-luminous GRB population of $r^{\rm u}_0\approx 220$ yr$^{-1}$ Gpc$^{-3}$, which is about 0.6% of the local SN Type Ib/c rate, $r_0^{\mathrm {SNIb/c}}\approx 6.3\times 10^{4}$ yr$^{-1}$ Gpc$^{-3}$, fits the observed low-redshift GRB distribution. Assuming that GRB 980425 is typical in luminosity for an under-luminous population, we make a first crude constraint on such a population by taking a mean luminosity of $L_{0}=2\times
10^{53}$ s$^{-1}$— about 3–4 orders of magnitude less than the mean luminosity of the classical GRBs. These two constraints yield a probability of BATSE/HETE detecting at least one GRB at $z<0.05$ to be 0.78 over 4.5 years, a result that is compatible with the presently observed low-redshift GRB distribution.
GRBs 980425 and 031203 may only appear faint and anomalous because of the sensitivity limit of BATSE/HETE. If such an under-luminous population is present, the increased sensitivity of [*Swift*]{} should enable it to detect and localize 5 times more GRBs than BATSE at redshifts $z<0.1$.
It seems reasonable that the observed broad distribution of observed GRB luminosities may represent related, but different classes of engine-driven emissions powered by rotating massive compact stellar remnants. The definite association of some nearby GRBs with Type Ib/c SNe–a SN type that exhibits considerable diversity–supports the idea that there could be a diverse class of inner engines driving at least a fraction of GRBs. These classes may even form a continuum that encompasses Type Ib/c SNe, XRF, faint GRBs and classical GRBs. Hence the distribution of sources in redshift in this scenario would consist of the sum of the individual populations with different inner engines and local rate densities. There is most likely overlap between the emission characteristics of the various sub-populations, so much so that they may form a continuum of luminosities ranging from XRF to hard GRBs.
If the anomalous GRBs are all shown to be off-axis observations of classical GRBs, then the evidence for different classes of cosmic explosions related to GRBs will become more tenuous. [*Swift*]{} should provide new information on the diversity of inner engines and on the progenitors that produce them, providing a wealth of data to help solve the cosmic riddle of identifying GRB progenitor populations.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank the referee for useful suggestions that have led to improving the clarity of this work. I also thank B. P. Schmidt for discussing the idea of multiple populations of GRBs and R. R. Burman and D. G. Blair for providing useful comments. D. M. Coward is supported by an Australian Research Council fellowship and grant DP0346344.
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[^1]: E-mail: coward@physics.uwa.edu.au
|
---
abstract: 'We present numerical results on two- (2D) and three-dimensional (3D) hydrodynamic core-collapse simulations of an 11.2$M_\odot$ star. By changing numerical resolutions and seed perturbations systematically, we study how the postbounce dynamics is different in 2D and 3D. The calculations were performed with an energy-dependent treatment of the neutrino transport based on the isotropic diffusion source approximation scheme, which we have updated to achieve a very high computational efficiency. All the computed models in this work including nine 3D models and fifteen 2D models exhibit the revival of the stalled bounce shock, leading to the possibility of explosion. All of them are driven by the neutrino-heating mechanism, which is fostered by neutrino-driven convection and the standing-accretion-shock instability (SASI). Reflecting the stochastic nature of multi-dimensional (multi-D) neutrino-driven explosions, the blast morphology changes from models to models. However, we find that the final fate of the multi-D models whether an explosion is obtained or not, is little affected by the explosion stochasticity. In agreement with some previous studies, higher numerical resolutions lead to slower onset of the shock revival in both 3D and 2D. Based on the self-consistent supernova models leading to the possibility of explosions, our results systematically show that the revived shock expands more energetically in 2D than in 3D.'
author:
- 'Tomoya Takiwaki, Kei Kotake, and Yudai Suwa'
bibliography:
- 'ms.bib'
title: 'A Comparison of Two- and Three-dimensional Neutrino-hydrodynamics simulations of Core-collapse Supernovae'
---
Introduction
============
Ever since the first numerical simulation of core-collapse supernovae (CCSNe) [@colgate], the neutrino-driven mechanism has been the leading candidate of the explosion mechanism for more than four decades. In the long history, a very important lesson we have learned from @rampp00 [@lieb01; @thom03; @sumi05] near the Millennium, is that the spherically-symmetric (1D) form of this mechanism fails to explode canonical massive stars. Supported by accumulating supernova observations of the blast morphology (e.g., @wang08, and references therein), a number of multi-dimensional (multi-D) hydrodynamic simulations have been reported so far, which gives us a confidence that hydrodynamic motions associated with neutrino-driven convection (e.g., @herant [@burr95; @jankamueller96; @frye02] and see collective references in @burrows12 [@murphy13; @couch13]) and the SASI (e.g., @blon03 [@Scheck04; @scheck06; @Ohnishi06; @ohnishi07; @ott_multi; @murphy08; @thierry; @thierry07; @thierry12; @iwakami1; @iwakami2; @iwakami13; @endeve10; @endeve12; @rodrigo09_2; @rodrigo09; @rodrigo10; @hanke12]) can help the onset of neutrino-driven explosions.
In fact, a growing number of neutrino-driven explosions have been recently obtained in the state-of-the-art two-dimensional (2D) simulations, in which spectral neutrino transport is solved with different levels of sophistication (e.g., @buras06 [@marek; @bernhard12; @bernhard13; @bruenn13; @suwa10; @suwa11; @suwa13], @janka12 for a review). This success is, however, accompanying new questions. Among them[^1], three-dimensional (3D) effects on the neutrino-driven mechanism are attracting a paramount attention (e.g., @burrows13 [@kotake12] for a review). Unfortunately, however, experimental 3D models that employed a light-bulb scheme (e.g., @murphy08), have provided divergent results so far. The basic result of @nordhaus who were the first to point out that 3D leads to easier explosions than 2D, has been supported by the follow-up studies [@burrows12; @dolence12], but not by @hanke12 [@couch13]. On top of the urgent task to make a detailed comparison between these idealized models, self-consistent 3D simulations should be done in order to have the final word on the 3D effects.
At present, 3D CCSN simulations including spectral neutrino transport are only few [@hanke13; @takiwaki12]. Very recently, @hanke13 succeeded in performing 3D simulations with detailed neutrino transport for a 27 $M_{\sun}$ star. In addition to the first discovery regarding the violent SASI activity in self-consistent 3D models, their results illuminate the importance to go beyond the prevalent light-bulb scheme, only by doing so, the non-linear couplings such as between core-contraction of the proto-neutron star (PNS), the accretion neutrino luminosity, and the multi-D hydrodynamic feedback of neutrino-driven convection and the SASI, can be self-consistently determined. On the other hand, the very high computational cost allowed @hanke13 to focus on a single (self-consistent) 3D model, and it has not been clarified yet whether 3D helps or harms the onset of neutrino-driven explosions compared to 2D.
To address this question, we investigate in this paper how the explosion dynamics will differ from 3D to 2D by systematically changing numerical resolutions and initial seed perturbations in multi-D radiation-hydrodynamic simulations. For the multi-group neutrino transport, the isotropic diffusion source approximation (IDSA) scheme [@idsa] is implemented in a ray-by-ray manner (e.g., @takiwaki12 for more details), which we have updated to achieve a very high computational efficiency. As in @takiwaki12, we here focus on the evolution of an $11.2 M_{\odot}$ star of @woos02. We choose this lighter progenitor because the shock revival occurs relatively earlier after bounce (e.g., @buras06) compared to more massive progenitor models as employed in @hanke13. The updated transport scheme together with the employed earlier-to-explode progenitor allow us to conduct a systematic numerical study, for the first time, in both 2D and 3D (to the best of our knowledge) in the context of self-consistent neutrino-driven supernova models.
![Three dimensional plots of entropy per baryon (top panel), $\tau_{\rm res}/\tau_{\rm heat}$ (bottom left panel) that is the ratio of the residency to the neutrino heating timescale (see the text for details), and the net neutrino heating rate (bottom right panel, in unit of ${\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$) for three snapshots (top and bottom left: $t=230$ ms, and bottom right: $t=150$ ms measured after bounce ($t\equiv 0$) of our model 3D-H-1). The contours on the cross sections in the $x = 0$ (back right), $y = 0$ (back bottom), and $z = 0$ (back left) planes are, respectively, projected on the sidewalls of the graphs. For each snapshot, the length of white line is indicated at right bottom text.[]{data-label="fig1"}](f1c.eps "fig:"){width=".9\linewidth"}\
![Three dimensional plots of entropy per baryon (top panel), $\tau_{\rm res}/\tau_{\rm heat}$ (bottom left panel) that is the ratio of the residency to the neutrino heating timescale (see the text for details), and the net neutrino heating rate (bottom right panel, in unit of ${\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$) for three snapshots (top and bottom left: $t=230$ ms, and bottom right: $t=150$ ms measured after bounce ($t\equiv 0$) of our model 3D-H-1). The contours on the cross sections in the $x = 0$ (back right), $y = 0$ (back bottom), and $z = 0$ (back left) planes are, respectively, projected on the sidewalls of the graphs. For each snapshot, the length of white line is indicated at right bottom text.[]{data-label="fig1"}](f1d.eps "fig:"){width=".45\linewidth"} ![Three dimensional plots of entropy per baryon (top panel), $\tau_{\rm res}/\tau_{\rm heat}$ (bottom left panel) that is the ratio of the residency to the neutrino heating timescale (see the text for details), and the net neutrino heating rate (bottom right panel, in unit of ${\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$) for three snapshots (top and bottom left: $t=230$ ms, and bottom right: $t=150$ ms measured after bounce ($t\equiv 0$) of our model 3D-H-1). The contours on the cross sections in the $x = 0$ (back right), $y = 0$ (back bottom), and $z = 0$ (back left) planes are, respectively, projected on the sidewalls of the graphs. For each snapshot, the length of white line is indicated at right bottom text.[]{data-label="fig1"}](f1b.eps "fig:"){width=".45\linewidth"}
{width=".28\linewidth"} {width=".28\linewidth"} {width=".4\linewidth"}
Numerical Methods and Models
============================
Here we briefly summarize several major updates of the code that we have implemented after our previous work [@takiwaki12] in which the spectral neutrino transport scheme IDSA [@idsa] was implemented in the ZEUS-MP code [@hayes].
In the original IDSA scheme, a steady-state approximation ($\partial {f^{\rm s}}/(\partial t) = 0$) is assumed. Here $f^{\rm s}$ represents the [*streaming*]{} part of the neutrino distribution function (e.g., @idsa). Then one should deal with a Poisson-type equation to find the solution of $f^s$ (e.g., Eq.(10) in @idsa). This is computationally expensive, because a collective data-communication is required on the MPI routines for all the processors (along the given radial direction in the ray-by-ray approximation) to solve the Cauchy problem.
To get around the problem, one needs to solve the evolution of $f^{\rm s}$ as, $$\begin{aligned}
\frac{\partial \mathcal{E}^{\rm s}}{c\partial {t}} &+&
\frac{1}{r^2}\frac{\partial}{\partial r}r^2 \mathcal{F}^{\rm s}
= \mathcal{S}[{j}, {\chi}, {\Sigma}],\label{eq:fs-evol}\\
\mathcal{E}^{\rm s} &\equiv& \frac{1}{2}\int d{\mu}\,{f}^{{\rm s}},\\
\mathcal{F}^{\rm s} &\equiv& \frac{1}{2}\int \mu d{\mu}\,{f}^{{\rm s}},\\
\mathcal{S} &\equiv& -\left( \hat{j}+\hat{\chi}\right)\mathcal{E}^{\rm s}+\Sigma,\end{aligned}$$ where $\mathcal{E}^{\rm s}$ and $\mathcal{F}^{\rm s}$ corresponds to the radiation energy and flux of the streaming particle, and $\mathcal
S$ represents the source term that is a functional of the neutrino emissivity ($j$), absorptivity ($\chi$), and the isotropic diffusion term ($\Sigma$) all defined in the laboratory frame, respectively. From local hydrodynamic quantities (density, $Y_e$, entropy), the source term of Eq.(1) ($\mathcal{S}[{j}, {\chi}, {\Sigma}]$) can be determined. For closure, we use a prescribed relation between the radiation energy and flux as ($\mathcal{F}^{\rm s}/\mathcal{E}^{\rm s}= \frac{1}{2}(1 +\sqrt{1-
[{R_{\nu}}/\max\left(r,R_\nu\right)]^2})$ with $R_\nu$ being the radius of an energy-dependent scattering sphere (see Eq.(11) in @idsa). Since the cell-centered value of the flux, $\mathcal{F}^{\rm s}$, is obtained by the prescribed relation, the cell-interface value is estimated by the first-order upwind scheme assuming that the flux is out-going along the radial direction. With the numerical flux, the transport equation of $\mathcal{E}^{\rm s}$ (Eq.(1)) now expressed in a hyperbolic form is numerically solved. This modification does not produce any significant changes in the numerical results (see Takiwaki et al. in preparation for more details), however, the computational cost becomes more than 10 times smaller than that in the previous treatment. The velocity dependent terms ($O(v/c)$) are only included (up to the leading order) in the trapped part of the distribution function (Eq. (15) in @idsa). Concerning heavy-lepton neutrinos ($\nu_x =
\nu_{\mu},\nu_{\tau},\bar{\nu}_{\mu}, \bar{\nu}_{\tau}$), we employ a leakage scheme to include the $\nu_x$ cooling via pair-, photo, plasma processes (e.g., @ross03 [@itoh89]). We apply the so-called ray-by-ray approach in which the neutrino transport is solved along a given radial direction assuming that the hydrodynamic medium for the direction is spherically symmetric. To improve the accuracy of total energy conservation, we follow the prescription proposed by @mueller. For the calculations presented here, self-gravity is computed by a Newtonian monopole approximation. We use the equation of state (EOS) by @latt91 with a compressibility modulus of $K = 180$ MeV (LS180). Our fiducial 3D models are computed on a spherical polar grid with a resolution of $n_r \times n_{\theta} \times
n_{\phi}$ = $320 \times 64 \times 128 $, in which non-equally spacial radial zones covers from the center to an outer boundary of 5000 km. The radial grid is chosen such that the resolution $\Delta
r$ is better than 2km in the PNS interior and typically better than 5km outside the PNS. For the spectral transport, we use 20 logarithmically spaced energy bins ranging from 3 to 300 MeV and we take a ray-by-ray approximation (e.g., @buras06 [@bruenn13]), in which a ray is cast for every angular zone. In all the multi-D runs, the innermost 5 km is computed in spherical symmetry to avoid excessive time-step limitations. Seed perturbations for aspherical instabilities are imposed by hand at 10 ms after bounce by introducing random perturbations of $1\%$ in velocity behind the stalled shock.
To test the sensitivity of the supernova dynamics to numerical resolutions, we compute 3D model-series with lower angular resolutions, namely half or quarter of the (equidistant) mesh numbers in the azimuthal direction ($n_r \times n_{\theta} \times
n_{\phi}$ = $320 \times 64 \times 64$ and $n_r \times n_{\theta} \times n_{\phi}$ = $320 \times 64 \times 32$). In 2D simulations we vary the mesh numbers in the lateral direction as $n_r \times n_{\theta} $ = $320 \times 64$, $n_r \times n_{\theta} $ = $320 \times 128$ and $n_r \times n_{\theta} $ = $320 \times 256$, respectively (see Table 1). In the table, model 3D-H-1 differs from model 3D-H-2 (and 3D-H-3 etc) only in the random seed perturbations (with the perturbation amplitudes being the same in all cases). Note that the lowest-resolution 3D model in this work corresponds to the best-resolution model in @takiwaki12. By using the fastest [*K*]{} computer in Japan, it took typically 1.3 months (equivalently $\sim$ 4 million core-hour computing-time) for each of our 3D fiducial models.
[lccccccccccccc]{} 3D-H-1 & $ 320 \times 64 \times 128 $ & $ 2.8^{\circ} \times 2.8^{\circ} $ & 223 & & & 284 & 550 & 0.15 & 1,0 & 0.02\
3D-H-2 & $ 320 \times 64 \times 128 $ & $ 2.8^{\circ} \times 2.8^{\circ} $ & 216 & & & 369 & 850 & 0.25 & 2,0 & 0.02\
3D-H-3 & $ 320 \times 64 \times 128 $ & $ 2.8^{\circ} \times 2.8^{\circ} $ & 269 & & & 269 & 400 & 0.15 & 2,0 & 0.02\
3D-M-1 & $ 320 \times 64 \times 64 $ & $ 2.8^{\circ} \times 5.6^{\circ} $ & 192 & & & 269 & 600 & 0.25 & 1,0 & 0.01\
3D-M-2 & $ 320 \times 64 \times 64 $ & $ 2.8^{\circ} \times 5.6^{\circ} $ & 194 & & & 319 & 700 & 0.30 & 1,0 & 0.02\
3D-M-3 & $ 320 \times 64 \times 64 $ & $ 2.8^{\circ} \times 5.6^{\circ} $ & 195 & & & 279 & 700 & 0.27 & 1,0 & 0.01\
3D-L-1 & $ 320 \times 64 \times 32 $ & $ 2.8^{\circ} \times 11.3^{\circ} $ & 193 & & & 314 & 1000 & 0.60 & 2,1 & 0.009\
3D-L-2 & $ 320 \times 64 \times 32 $ & $ 2.8^{\circ} \times 11.3^{\circ} $ & 188 & & & 273 & 800 & 0.45 & 2,1 & 0.009\
3D-L-3 & $ 320 \times 64 \times 32 $ & $ 2.8^{\circ} \times 11.3^{\circ} $ & 183 & & & 273 & 700 & 0.35 & 2,1 & 0.01\
2D-H-1 & $ 320 \times 256 $ & $ 0.7^{\circ} $ & 138 & & & 300 & 1100 & 0.65 & 1,0 & 0.05\
2D-H-2 & $ 320 \times 256 $ & $ 0.7^{\circ} $ & 154 & & & 329 & 1200 & 0.45 & 2,0 & 0.08\
2D-H-3 & $ 320 \times 256 $ & $ 0.7^{\circ} $ & 159 & & & 311 & 1200 & 0.76 & 2,0 & 0.1\
2D-H-4 & $ 320 \times 256 $ & $ 0.7^{\circ} $ & 156 & & & 368 & 1500 & 0.6 & 2,0 & 0.05\
2D-H-5 & $ 320 \times 256 $ & $ 0.7^{\circ} $ & 150 & & & 345 & 1200 & 0.7 & 2,0 & 0.1\
2D-M-1 & $ 320 \times 128 $ & $ 1.4^{\circ} $ & 140 & & & 369 & 1700 & 0.5 & 1,0 & 0.1\
2D-M-2 & $ 320 \times 128 $ & $ 1.4^{\circ} $ & 125 & & & 319 & 1800 & 0.8 & 2,0 & 0.06\
2D-M-3 & $ 320 \times 128 $ & $ 1.4^{\circ} $ & 151 & & & 400 & 1700 & 0.85 & 1,0 & 0.1\
2D-M-4 & $ 320 \times 128 $ & $ 1.4^{\circ} $ & 144 & & & 469 & 2300 & 1.0 & 1,0 & 0.1\
2D-M-5 & $ 320 \times 128 $ & $ 1.4^{\circ} $ & 152 & & & 400 & 1800 & 1.0 & 2,0 & 0.05\
2D-L-1 & $ 320 \times 64 $ & $ 2.8^{\circ} $ & 137 & & & 387 & 2100 & 1.1 & 2,0 & 0.08\
2D-L-2 & $ 320 \times 64 $ & $ 2.8^{\circ} $ & 137 & & & 395 & 2200 & 1.26 & 2,0 & 0.1\
2D-L-3 & $ 320 \times 64 $ & $ 2.8^{\circ} $ & 126 & & & 483 & 2800 & 1.3 & 1,0 & 0.09\
2D-L-4 & $ 320 \times 64 $ & $ 2.8^{\circ} $ & 140 & & & 559 & 2400 & 1.3 & 1,0 & 0.09\
2D-L-5 & $ 320 \times 64 $ & $ 2.8^{\circ} $ & 125 & & & 569 & 2500 & 1.3 & 2,0 & 0.05\
\[tab:models\]
Result
======
As summarized in Table 1, all the computed models including nine 3D models and fifteen 2D models exhibit shock revival, leading to the possibility of explosion. Before going into detail how the explosion dynamics and stochasticity are different in 2D and 3D, we briefly outline the hydrodynamics features taking model 3D-H-1 as an example.
The top panel of Figure \[fig1\] shows the blast morphology of model 3D-H-1 at $t_{\rm pb} = 230$ ms (postbounce) when the revived shock is reaching an angle-averaged radius of 400 km (e.g., red dashed line in the right panel of Figure. \[fig2\]). As seen from the side wall panels, a bipolar explosion is obtained for this model. The bottom left panel (red regions) shows that the ratio of the residency timescale to the neutrino heating timescale (e.g., Equations (6) and (7) in @takiwaki12) exceeds unity behind the shock, which presents evidence that the shock revival is driven by the neutrino-heating mechanism. The bottom right panel of Figure \[fig1\] depicts spacial distribution of the net neutrino heating rate at $t_{\rm pb} = 150$ ms. Small scale inhomogeneities (colored as red or yellow) are seen, which predominantly comes from neutrino-driven convection and anisotropies of the accretion flow, but the shape of the gain region is very close to be spherical before the onset of an explosion. This suggests that the bipolar geometry of the shock is produced not by the global anisotropy of the neutrino heating in the vicinity of the neutrino sphere, but by multi-D effects such as by neutrino-driven convection and the SASI in the gain regions after the explosion (gradually) sets in.
Reflecting the stochastic nature of the multi-D neutrino-driven explosions, the blast morphology changes from models to models. The left and middle panels of Figure \[fig2\] show that a stronger explosion is obtained toward the north direction (model 3D-H-2) and the south pole (model 3D-H-3), which is only different from model 3D-H-1 (e.g., Table 1) in terms of the imposed initial random perturbations. Note that due to the use of the spherical coordinates, we cannot omit the possibility that the polar axis still gives a special direction in our 3D simulations. But more importantly, our results show that the final fate of the 3D and 2D models whether an explosion is obtained or not, is little affected by the stochasticity of the explosion geometry.
In fact, the right panel of Figure \[fig2\] shows the evolution of the average shock radius for our 1D (blue line), 2D (green lines), and 3D (red lines) models, respectively. Before the onset of shock revival (before 100 ms after bounce), the evolution of the shock is all similar to that of the 1D model (blue line). After that, our results show that the shock expansion is systematically more energetic in 2D (green lines) than in 3D (red lines). This feature is qualitatively consistent with @takiwaki12, and also with @hanke13 who recently reported 2D vs. 3D comparison based on a single 3D model but employing more detailed neutrino transport than ours. Due to our lack of necessary computational resources, our 3D models should be terminated typically before $t_{\rm end} \lesssim 300$ ms postbounce (e.g., in Table 1), but we expect them to produce explosions subsequently, seeing a continuous shock expansion out to a radius of 500 km in 3D (and 700 km in 2D). Given the same numerical resolution (e.g., model series 3D-L in Table 1), the average shock radii in this study is smaller than those in @takiwaki12, in which the cooling by heavy-lepton neutrinos was not taken into account. Due to the inclusion of the $\nu_x$ cooling, the (angle-averaged) $\bar{\nu}_e$ luminosity decreases more quickly after bounce (compare Figure \[fig3\] and Figure 14 of [@takiwaki12]), which leads to the less energetic shock expansion in this study. It should be mentioned that by comparing our $\nu_x$ luminosity estimated by the leakage scheme with that obtained by the work by @buras06 with detailed neutrino transport, the peak luminosity is more than 20 % smaller in our case. Such underestimation of cooling by heavy-lepton neutrinos should lead to artificially larger maximum shock extent ($R_{\rm max} \sim 260$ km, blue line in the right panel of Figure \[fig2\]) compared to $R_{\rm max} \sim170$ km in @buras06. We have to emphasize that the use of the leakage scheme together with the omission of inelastic neutrino scattering on electrons and GR effects in the present scheme is likely to facilitate artificially easier explosions. Regarding our 2D models, the relatively earlier shock revival ($\sim 100$ ms postbounce) coincides with the decline of the mass accretion rate onto the central PNS. This could be the reason that the timescale is similar to that in @bernhard12 who reported 2D (GR) models for the same progenitor model with detailed neutrino transport.
As seen from Figure \[fig3\], the angle average neutrino luminosity ($\langle L_{\nu} \rangle$) and the mean neutrino energy ($\langle \epsilon_{\nu} \rangle = \int E^3 \mathcal{F}^{\rm s}dE /\int E^2 \mathcal{F}^{\rm s}dE$, where $E$ is neutrino energy) is barely affected by the imposed initial perturbations (presumably at few-percent levels in amplitudes). This again supports our finding that the explosion stochasticity is very influential to determine the blast morphology but not the working of the neutrino-heating mechanism.
From the bottom panel of Figure \[fig3\], it can be seen that overall trend in the neutrino luminosities and the mean energies is similar between our 3D and 2D model. The neutrino luminosities in the 2D model (green lines) show a short-time variability (with periods of milliseconds to $\gtrsim 10$ ms) after around 100 ms postbounce. Such fast variations in the postbounce luminosity evolution have been already found in previous 2D studies (e.g., @ott_multi [@marek_gw]). This is caused by the modulation of the mass accretion rate due to convective plumes and downflows hitting onto the PNS surface (see also @lund12 [@tamborra13] about the detectability of these neutrino signals). It is interesting to note that such fast variability is less pronounced in our 3D model (red lines in the bottom panel). This is qualitatively consistent with @lund12 who analyzed the neutrino signals from 2D and 3D models, in which an approximate neutrino transport was solved [@annop10] as in @scheck06.
![Top panel shows time evolution of neutrino luminosities and mean energies of electron ($\nu_e$), anti-electron ($\bar{\nu}_e$), or heavy-lepton ($\nu_X$) neutrinos for models 3D-H-2 (green line) and 3D-H-3 (red line), respectively. Bottom panel is the same as the top panel but for the comparison between 2D and 3D (for models 3D-H-1 and 2D-H-1). These quantities are estimated at 500 km.[]{data-label="fig3"}](f3.eps "fig:"){width=".8\linewidth"} ![Top panel shows time evolution of neutrino luminosities and mean energies of electron ($\nu_e$), anti-electron ($\bar{\nu}_e$), or heavy-lepton ($\nu_X$) neutrinos for models 3D-H-2 (green line) and 3D-H-3 (red line), respectively. Bottom panel is the same as the top panel but for the comparison between 2D and 3D (for models 3D-H-1 and 2D-H-1). These quantities are estimated at 500 km.[]{data-label="fig3"}](lumi2D3D.eps "fig:"){width=".8\linewidth"}
Figure \[pns\] shows evolution of the average PNS radius for the 1D (blue line), 2D (green line), and 3D models (red line), respectively, that are defined by a fiducial density of $10^{11}~ {\rm g}~{\rm cm}^{-3}$. The PNS contraction is similar among the 1D, 2D, and 3D models. Although the PNS contraction potentially affect the evolution of the shock [@hanke13; @suwa13], in our cases that are unchanged by the difference of the dimension and that are not main agent to explain the divergence of the shock evolution in 1D, 2D and 3D. The PNS contraction is slightly stronger in the later postbounce phase in 1D ($\gtrsim 150$ ms postbounce, compare blue with green and red lines)) compared to 2D and 3D because no shock revival was obtained in the 1D model and heavier PNS and slightly deeper gravitational potential are obtained compared to that of the multi-D models. In the figure, three more lines (solid, dashed, dotted gray lines) are plotted, in which we estimate the evolution of the PNS radius based on the fitting formula (equation (1) of @scheck06) by changing a final radius of PNS $R_{\rm f}$ for a given set of an exponential timescale of $t_{\rm ib} = 1$ s and an initial radius of PNS $R_{\rm i} =$ 85 km. As can be seen, the dashed gray line ($R_{\rm f} = 12$ km) can most closely reproduce our results, which is just between the slow and fast contraction investigated in the work by @hanke13.
![Average PNS radius (defined by a fiducial density of $10^{11}~ {\rm g}~{\rm cm}^{-3}$) for the 1D (blue line), 2D (green line), and 3D models (red line), respectively. The gray lines labeled as “fit” corresponds to the prescribed PNS evolution with three different values of $R_{\rm f} = 10, 12, $ and 15 km (see text for more details). []{data-label="pns"}](pns.eps){width=".8\linewidth"}
![Top panel shows radial profiles of angle-averaged entropy at 100 ms postbounce for the 1D, 2D-H-1, 3D-H-1 model, respectively. Bottom panel shows spectra of the turbulent kinetic energy as a function of wavenumber ($k$) between our 2D (green line) and 3D (red line) model, respectively. []{data-label="fig:r-s"}](r-ent.eps "fig:"){width=".8\linewidth"}\
![Top panel shows radial profiles of angle-averaged entropy at 100 ms postbounce for the 1D, 2D-H-1, 3D-H-1 model, respectively. Bottom panel shows spectra of the turbulent kinetic energy as a function of wavenumber ($k$) between our 2D (green line) and 3D (red line) model, respectively. []{data-label="fig:r-s"}](sed100ms.eps "fig:"){width=".8\linewidth"}\
Top panel of Figure \[fig:r-s\] shows angle-averaged entropy profile at 100 ms postbounce, after when the difference of the subsequent shock evolution between our 1D, 2D, and 3D models becomes remarkable (e.g., right panel in Figure \[fig2\]). As has been studied in detail since 1990’s (e.g., @herant94 [@burrows95; @jankamueller96]), buoyancy-driven convection supported by turbulence (e.g., @murphy13) transports heat radially outward, leading to a more extended entropy profile in the 2D (green line in the panel) and 3D (red line) model compared to the 1D model (blue line) (see also @hanke12).
Bottom panel of Figure 5 compares the turbulent energy spectra of the anisotropic velocity ([@takiwaki12]) as a function of wavenumber ($k$) between our 2D (green line) and 3D (red line) model, respectively. The green and red line crosses at around $k_{\rm cross}$ = 0.02/km (corresponding to $\sim$ 50 km in spacial scale), above which the amplitude for the 3D model (red line) dominates over that for the 2D model (green), below which the amplitude for the 2D model does for the 3D model. This is qualitatively consistent with the previous results using the light-bulb method (e.g., @hanke12 [@burrows12; @dolence12; @couch13; @fernandez13]. By comparing the 2D and 3D curve with spectral slopes labeled by the corresponding exponent (-5/3:dotted blue line, -3:dotted black line), the power-low dependence ($\propto k^{-5/3}$) approximately holds for $k_{\rm int} \lesssim 0.02$ /km in the 2D model (green line) presumably as a result of the inverse energy cascade [@kraichnan]. Above $k_{\rm cross}$, the slope of the 3D model (red line) become more closer to $k^{-5/3}$ (blue line), while the spectrum slope of the 2D model drops much steeply with the wave number as $k^{-3}$ (gray line). These features are again in accord with the previous studies mentioned above (e.g., @hanke12 [@burrows12; @dolence12; @couch13; @fernandez13]).
As shown in the middle panel of Figure \[fig2\], model 3D-H-3 produces one-sided explosion towards the south pole during the simulation time. Reflecting the unipolarity, the average shock radius is smaller than for the other 3D models (compare dotted line (in red) with solid and dashed red (in red) in the right panel of Figure \[fig2\]). To quantify the vigor of the shock expansion , $t_{\mathrm{400}}$ is a useful quantity that was defined in [@hanke12] as the moment of time when the shock reaches an average radius of 400 km. In fact, as seen from Table 1, $t_{\mathrm{400}}$ of model 3D-H-3 ($t_{\mathrm{400}}
\approx 270~\mathrm{ms}$) is delayed about 50 ms compared to those of models 3D-H-1 and 3D-H-2 ($t_{\mathrm{400}}
\approx 220~\mathrm{ms}$).
The model average of $t_{400}$ (e.g., $t_{400,{\rm av}}$ in Table 1) clearly shows that models with higher numerical resolutions lead to slower onset of the shock revival in both our 3D and 2D models. This feature is qualitatively consistent with the 2D self-consistent models by [@marek] and with the 3D idealized models by [@hanke13; @couch13], respectively.
$\sigma_{400}$ in Table 1 represents the model dispersion of $t_{400}$, which varies much more stochastically in 3D models with different numerical resolution (from 1.2 to 24 ms) than those in 2D (from 6 to 10 ms). For model 3D-H-3, the shock revival is most delayed (e.g., Table 1) and the shock expansion is weakest among the computed models (dotted red line in Figure \[fig1\]). @nagakura13 proposed that shock revival is very sensitive to the imposed seed perturbations near the stalled shock. Following the hypothesis, we speculate that the influence of seed perturbations is seen most remarkably in our weakest explosion model.
$E_{\rm diag}$ in Table 1 denotes the diagnostic energy defined as the total energy (internal plus kinetic plus gravitational), integrated over all matter where the sum of the corresponding specific energies is positive. We include recombination energy in internal energy([@bruenn13]). Reflecting the earlier shock revival, the diagnostic energy is systematically bigger in 2D than in 3D. These diagnostic energies when we terminated the simulation were typically on the order of $\sim 10^{49}$ erg and $\sim 10^{50}$ erg for our 3D and 2D models, respectively. It should be noted that this quantity is estimated at the end of simulation,$t_{\rm end}$. In order to compare the [*diagnostic*]{} energy with the observed kinetic explosion energy ($\sim 10^{51}$ erg), a much longer-term simulation including improved microphysics, general relativity, and nuclear burning would be needed.
{width=".4\linewidth"} {width=".4\linewidth"}
Recently it is enthusiastically discussed which one between neutrino-driven convection and the SASI plays a more crucial role in facilitating neutrino-driven explosions [@thierry; @dolence12; @murphy13; @burrows12; @hanke12; @hanke13]. The left panel of Figure \[fig5\] shows the time evolution of the Foglizzo parameter $\chi$ [@thierry] As seen, $\chi$ continuously exceeds the critical number of $3$ [@thierry] rather shortly after bounce ($\sim 40$ ms), marking the transition to the non-linear phase. The earlier onset of the shock revival and the absence of clear features of the SASI in the linear phase (see discussion below) could indicate that neutrino-driven convection dominates over the SASI when the explosion sets in (e.g., at $t_{400}$ in Table 1).
The right panel of Figure \[fig5\] shows the evolution of normalized coefficient of spherical harmonics of the shock surface for model 3D-H-1. No clear feature of the linear growth of the SASI ($\lesssim 40$ ms postbounce) was obtained for the 11.2 $M_{\odot}$ star explored in this work. In both our 3D and 2D models, the qualitative behaviors of the harmonic modes seen in Figure 6 are less sensitive to the employed numerical resolution and seed random perturbations (see appendix A). The dominant channels are of low modes (e.g., $(\ell,
m)_{\rm max}$ in Table 1), the amplitudes of which (e.g., $|c_{\ell,m}|_{\rm max}$ in the table) are systematically larger in 2D than in 3D. This is qualitatively consistent with previous 3D simulations employing different numerical setups (e.g., @nordhaus [@hanke12; @burrows12; @hanke13]).
After the linear phase comes to an end at around 100 ms postbounce, the trajectory of the revived shock shows a wider diversity (e.g., Figure \[fig1\]) depending on the employed numerical resolution and seed perturbations.
Summary and Discussion
======================
Studying an 11.2 $M_{\odot}$ and changing numerical resolutions and seed perturbations systematically in the multi-D simulations employing the updated IDSA scheme, we studied how the postbounce dynamics is different in 2D and 3D. All the computed models exhibit the neutrino-driven revival of the stalled bounce shock, leading to the possibility of an explosion. Though the blast morphology changes from models to models reflecting the stochastic nature of multi-D neutrino-driven explosions, it was found that the final fate of these multi-D models whether an explosion is obtained or not, is little affected by the explosion stochasticity at least in the current investigated progenitor model. In line with some previous studies, higher numerical resolutions lead to slower onset of the shock revival in both 3D and 2D. Our results systematically showed that the revived shock expands more energetically in 2D than in 3D.
The caveats of our 3D models include the ray-by-ray approximation, the use of the softer EOS, and the omission of detailed neutrino reactions and general relativity (e.g., @kuroda12 [@ott13]). Keeping our efforts to improve them, it is important to study the dependence of progenitors (e.g., @buras06 [@bruenn13]) and EOS (e.g., @marek [@suwa13]) on the neutrino-driven mechanism in 3D computations. A number of exciting issues also remain to be investigated, such as gravitational-wave signatures (e.g., @kotake13), neutrino emission and its detectability (e.g., @lund12), and the possibility of 3D SASI flows generating pulsar kicks and spins (e.g., @annop13). Shifting from individuals to populations of 3D models, a rush of 3D explorations with increasing sophistication is now going to shed light on these fascinating riddles (hopefully not in the distant future) with increasing supercomputing resources on our side.
[ We are thankful to K. Sato and S. Yamada for continuing encouragements. Numerical computations were carried on in part on the K computer at the RIKEN Advanced Institute for Computational Science (Proposal number hp120285), XC30/XT4 at NAOJ, and on SR16000 at YITP in Kyoto University. This study was supported in part by Grants-in-Aid for Scientific Research (Nos. 23540323, 23340069, 24244036, and 25103511) and by HPCI Strategic Program of Japanese MEXT.]{}
Dependence on the amplitude of seed perturbations
=================================================
In this Appendix we briefly report on the dependence on the amplitudes of the initial seed perturbations. We have added a radial velocity perturbation, $\delta v_r(r,\theta,\phi)$ to the profile obtained by 1D simulation,$v_r^{1D}(r)$, according to the equation $\delta v_r = p_{\rm amp} {\rm rand} \times v_r^{1D}$ where rand is pseudorandom number that takes the value from $-1$ to $1$ and $p_{\rm amp}$ represents the absolute amplitude of the seed perturbations that we take as $1 \%$ in the models discussed in the main section. Let us firstly note that the amplitude of seed perturbations $a_{\ell,m}$ assumed in this study scales as $p_{\rm amp}/N^{1/2}$, where $a_{\ell,m}$ denotes the amplitude of the $(\ell,m)$ component of the seed perturbations at a given radius (i.e. $a_{\ell,m} = \int d \phi \int \sin \theta d \theta \left(v_r^{1D}+\delta v_r \right)Y_{\ell,m}^{*}$, where $Y_{\ell,m}^{*}$ is conjugate of spherical harmonic function) and $N = n_{\theta} \times n_{\phi}$ represents the total angular mesh number where $n_{\theta}, n_{\phi}$ being the mesh number in the $\theta$ and $\phi$ direction, respectively(e.g. in our 3D model with highest resolution, $n_{\theta} = 64, n_{\phi} = 128$). Therefore the seed amplitudes assumed in this work depend on numerical resolution (i.e., $N$).
If we keep the seed amplitudes ($a_{\ell,m}$) constant for models with different resolutions, will our results change drastically? Since we cannot rerun 3D models due to the limited computational resources, we compute a number of 2D models to answer to this question. First of all, let us discuss 2D models with different seed amplitudes. Top panel of Figure \[app:fig1\] shows the evolution of $c_{1,0}$ (the normalized harmonic amplitudes of shock position) for 2D models with $p_{\rm amp} = 1\%$ (red lines) or $0.176 \% (= 1\%/\sqrt{32})$ (green lines) with five different realizations of initial random perturbation. The reduced amplitude is determined by the ratio of total grid number of 2D-H models and 3D-H models (i.e. $1/\sqrt{32} = \sqrt{
n_\theta |_{2D-H}/(n_\theta n_\phi)|_{3D-H}}$). One might have a guess that $c_{\ell,m}(t)$ could evolve as $a_{\ell,m} \exp(t/t_0)$ with $t_0$ representing the duration of the linear growth rate of hydrodynamic instabilities including the SASI or neutrino-driven convection. If this could be the case, the linear growth amplitude (at before 35 ms postbounce in the top panel) should be higher about $\sqrt{32}$ times for red lines compared to green lines. But as it is shown, this is not the case[^2]. And after the early rising phase (about 40 ms postbounce in the panel), the saturation amplitudes are shown to be insensitive to the initial perturbation amplitudes (for the initial strength employed above). Remembering that neutrino-driven convection is likely to dominate over the SASI in the non-linear regime for the 11.2 $M_{\odot}$ progenitor employed in this work, the delay of $t_{400}$ (in the non-linear phase) for models with higher numerical resolutions cannot be also simply ascribed to the difference of the seed amplitudes.
Bottom panels of Figure \[app:fig1\] show evolution of $c_{1,0}$ for different resolution (H, M, and L) for our 2D and 3D models, respectively. For a given numerical resolution, $t_{400}$ of the chosen models in these plots is close to a median of $t_{400}$ for each model series. Note that in these models the initial seed amplitudes are dependent on both resolution and dimension. As can be seen, in both 2D and 3D, the amplitudes in the linear phase ($<40$ms postbounce) are comparable for models with different resolutions. And in the non-linear regime, no monotonic dependence of the non-linear evolution on the initial seed amplitudes (between H, M, L models) can be found. These results show that the findings in this work are less sensitive to the assumed initial seed perturbations.
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[^1]: Note that general relativity (GR, e.g., @bernhard12 [@kuroda12; @ott13]) and detailed weak interactions (e.g., @langanke08 [@furusawa13]) are considered as important as 3D effects.
[^2]: Note that if the initial seed perturbations would be more bigger than those assumed in this study, they should affect the postbounce hydrodynamics (see @couch_ott13)
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---
abstract: 'We prove a conjecture by F. Ferrari. Let $X$ be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that $X$ has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.'
author:
- |
U. Bruzzo and A. Ricco\
International School for Advanced Studies (SISSA/ISAS),\
Via Beirut 2, 34014 Trieste, Italy\
Istituto Nazionale di Fisica Nucleare, Sezione di Trieste
date: 16 November 2005
title: |
Normal bundles to Laufer rational curves\
in local Calabi-Yau threefolds
---
*MSC:* 14D15, 14H45, 83E30
*PACS:* 02.10.De, 02.40Tt, 11.25.Mj
*Keywords:* Open strings, geometric transitions, Laufer curves, superpotentials
#### Introduction.
In this paper we consider particular embeddings of smooth rational curves in local Calabi-Yau threefolds, called Laufer curves [@laufer; @Katz:2000ab]. (We use the definition of the physics community, calling Calabi-Yau a quasi-projective threefold with trivial canonical bundle; the term “local” refers to non-compactness.) These geometries have shown to be very useful to understand several features of string theories and supersymmetric gauge theories. In particular they are relevant for brane dynamics and geometric transition/large $N$ dualities. Geometric transition interprets the resummation of the open string sector of an open-closed string theory as a transition in the target space geometry, connecting two different components of a moduli space of Calabi-Yau threefolds. The local Calabi-Yau that we consider represents the open string side of conjectured geometric transitions. In particular, open topological B-type strings in these geometry reduces to matrix models in which the parameters of the complex structure are the coupling constants.
This is directly connected (via F-terms) to the possibility of geometrically engineering supersymmetric gauge theories in Type IIB string theory. Let $\mathbb{R}^4 \times X$ be the target space of the theory, where $X$ is a Calabi-Yau threefold, and $\mathcal{C}$ a rational curve in $X$, with normal bundle $V$ and $N$ D5 branes wrapped on it. The effective field theory is a $\mathcal{N}=1$ supersymmetric gauge theory with gauge group $U(N)$. The space of vacua of this gauge theory, given by the critical points of the effective superpotential, is locally described by the versal deformation space of the curve in $X$. For a given vacua, there are $h^0$ massless chiral superfields in the adjoint representation of the gauge group, where $h^0 := \dim H^0(\mathcal{C}, V)$. On the other hand, the number of massless chiral multiplets is equal to the corank of the Hessian of the superpotential at this vacuum. This relation led [@Ferrari:2003vp] to conjecture the result expressed in our Proposition \[prop:ferrari\].
For an account of these aspects, see also [@Curto; @Mazzucato:2005fe; @Bonelli:2005dc] and references therein.
From a strictly mathematical viewpoint, the problem is the following. Let $V$ be a rank-2 holomorphic vector bundle on a rational curve ${{\mathcal{C}}}$ such that its total space has trivial canonical bundle, and assume that $V$ has a global section. We deform $V$ to a nonlinear fibration $X$ in such a way that $X$ still has trivial canonical bundle and the fibration has sections. The normal bundle to such a section of course splits as a direct sum of two line bundles in view of Grothendieck’s classification of vector bundles on curves of genus zero [@Gro]. The problem is to compute these line bundles. The solution is obtained in terms of a superpotential $W$ than one associates with the deformations of $V$: the sections of $X$ are given by the critical points of $W$, and the degrees of the above mentioned line bundles are given, in accordance with a conjecture by Ferrari [@Ferrari:2003vp], by the rank of the Hessian of $W$ at those critical points.
#### Definition of $X$.
Let ${{\mathcal{C}}}\simeq{{\mathbb{P}}}^1$ be a smooth rational curve and $V \to {{\mathcal{C}}}$ a rank-2 holomorphic vector bundle on ${{\mathcal{C}}}$, with $\det V \simeq K_{{{\mathcal{C}}}} \simeq {{\mathcal{O}}}(-2)$, so that the total space of the bundle $V$ has trivial canonical bundle. Then $V \simeq {{\mathcal{O}}}(-n-2) \oplus {{\mathcal{O}}}(n)$ for some $n$. We consider deformations of $V$ given in terms of transition functions in the standard atlas $\mathcal{U} = \{ U_0, U_1 \}$ of $\mathbb{P}^1$ as $$\label{eq:fibration}
\left\{\begin{array}{rcl}
z' &=& 1/z \\[3pt]
\omega_1' &=& z^{-n} \omega_1 \\[3pt]
\omega_2' &=& z^{n+2} \left( \omega_2 + \partial_{\omega_1} B(z, \omega_1) \right)\ .
\end{array}\right.$$ Note that the complex manifold $X$ defined as the total space of this fibration has again trivial canonical bundle. The term $B(z, \omega)$ is a holomorphic function on $(U_0 \cap U_1) \times \mathbb{C}$ and is called the *geometric potential*. If we expand the function $B$ in its second variable $$B(z,\omega_1)=\sum_{d=1}^\infty \sigma_d(z)\,\omega_1^d$$ each coefficient $\sigma_d$ may be regarded as a cocycle defining an element in the group $$H^1({{\mathbb{P}}}^1,{{\mathcal{O}}}(-2-dn))\simeq H^0({{\mathbb{P}}}^1,{{\mathcal{O}}}(nd))^\ast\,.$$
#### The superpotential.
If we consider ${{\mathcal{C}}}$ as embedded in $X$ as its zero section, and consider the problem of deforming the pair $(X\,{{\mathcal{C}}})$, the space of versal deformations can be conveniently described by a superpotential [@Katz:2000ab]. In the case at hand the superpotential $W$ can be defined as the function of $n+1$ complex variables given by $$\label{eq:superpotential} W(x_0, \dots, x_n) = \frac{1}{2 \pi \mathrm{i}}
\oint_{{{\mathcal{C}}}_0} B\left(z,\omega_1(z) \right) \,{\mathrm{d}}z
$$ where $z$ and $z'$ are local coordinates on $U_0$ and $U_1$, and the parameters $x_0, \dots, x_n$ define sections of the line bundle ${{\mathcal{O}}}(n)$ by letting $$\begin{aligned}
\label{eq:section1}
\omega_1(z)= \sum_{i=0}^{n} x_i z^i \ , \qquad \omega_1'(z') = \sum_{i=0}^{n} x_i (z')^{n-i}\,.\end{aligned}$$ One should note that the superpotential $W$ can be obtained by applying to the function $B$, regarded as an element in $H^0({{\mathbb{P}}}^1,{{\mathcal{O}}}(nd))^\ast$, the dual of the multiplication morphism $$H^0({{\mathbb{P}}}^1,{{\mathcal{O}}}(n))^{\otimes d} \to H^0({{\mathbb{P}}}^1,{{\mathcal{O}}}(nd))$$ (here one should regard the dual of $H^0({{\mathbb{P}}}^1,{{\mathcal{O}}}(nd))$ as a space of Laurent tails).
The key to the result we want to prove is the relationship occuring between the superpotential $W$ and the sections of the fibration $X\to {{\mathcal{C}}}$ (cf. [@Katz:2000ab; @Ferrari:2003vp]).
\[lemma:sect\] The holomorphic sections of the fibration $X\to {{\mathcal{C}}}$ are in a one-to-one correspondence with the critical points of the superpotential, *i.e.*, with the solutions of the equations $$\begin{aligned}
\frac{\partial W}{\partial x_i} = 0 \ , \quad i=0, \dots, n \ .\end{aligned}$$
This can be verified by explicit calculations [@Ferrari:2003vp] after representing the sections of $X$ as $$\label{eq:section2}\begin{array}{rcl}
\omega_2(z) &=& \displaystyle - \frac{1}{2 {\mathrm{i}}\pi}\oint_{C_z} \frac{\partial_\omega B (u,\omega_1(u))}{u-z} \, {\mathrm{d}}u \\[12pt]
\omega_2'(z') &=& \displaystyle \frac{1}{2 {\mathrm{i}}\pi} \oint_{C_{z'}}
\frac{\partial_\omega B (1/u,\omega_1(1/u))}{u^{n+2}(u-z)} \,{\mathrm{d}}u
\end{array}$$ where the contour $C_z$ (resp. $C_{z'}$) encircles the points $0$ and $z$ (resp $z'$). So (\[eq:section1\]) and (\[eq:section2\]) yield a rational curve $\Sigma \subset X$ for each critical point $(x_0, \dots, x_n)$ of $W$.
#### Ferrari’s Conjecture.
Now we state and prove Ferrari’s conjecture.
\[prop:ferrari\] The normal bundle to the section $\Sigma$ of $X$ determined by a critical point $(x_0, \dots, x_n)$ of $W$ is ${{\mathcal{O}}}_\Sigma(-r-1) \oplus {{\mathcal{O}}}_\Sigma(r-1)$ where $r$ is the corank of the Hessian of $W$ at that point.
To calculate the normal bundle to $\Sigma$ we first need to linearize the transition functions around the given section. Defining new coordinates $\delta_i = \omega_i - \omega_i(z)$, $\delta_i' = \omega_i' - \omega_i'(z)$, we obtain $$\begin{aligned}
\delta_2' = z^{n+2} \left(\delta_2 + h(z) \delta_1 + g(z) \right)\end{aligned}$$ where $$\begin{aligned}
g(z) = \partial_\omega B(z, \omega_1(z)) \ , \qquad
h(z) = \partial^2_\omega B(z, \omega_1(z)) \end{aligned}$$ and at a critical point of $W$ we have $g(z) = 0$ using relation (\[eq:derivatives\]) in the appendix. Furthermore, again from (\[eq:derivatives\]), for $h(z)$ we have $$\begin{aligned}
\label{eq:cocycleW}
h(z) = \sum_{i \leq j =0}^{n} \partial_i \partial_j W^{(k)}_{d} z^{-(i+j)-1}\end{aligned}$$ up to terms that can be can be readsorbed by holomorphic change of coordinates (see the Appendix).
Now we need the following. Let us consider an extension of vector bundles on ${{\mathbb{P}}}^1$ of the form $$\begin{aligned}
0 \longrightarrow {{\mathcal{O}}}_{{{\mathbb{P}}}^1}(-n-2) \longrightarrow \Phi
\longrightarrow {{\mathcal{O}}}_{{{\mathbb{P}}}^1}(n) \longrightarrow 0\end{aligned}$$ parametrized by a cocycle $\sigma \in H^1 ({{\mathbb{P}}}^1, {{\mathcal{O}}}_{{{\mathbb{P}}}^1}(-2n-2))$. With respect to the two standard charts $U_0, U_1$ and in the coordinate $z$ of $U_0$, $\sigma$ can be written as $$\begin{aligned}
\sigma(z) = \sum_{k=0}^{2n} \widetilde{t}_k z^{-k-1} \ .\end{aligned}$$ Let us define a quadratic form (quadratic superpotential) on the global sections of the line bundle ${{\mathcal{O}}}_{{{\mathbb{P}}}^1}(n)$: $$\begin{aligned}
H(x_0, \dots, x_n) = \sum_{k=0}^{2n} \widetilde{t}_k \sum_{i,j=0 \atop i+j=k }^{n} x_i x_{j} = \sum_{i,j=0}^{n} H_{ij} x_i x_j \ .\end{aligned}$$
The vector bundle $\Phi$ is ${{\mathcal{O}}}_{\mathbb{P}^1}(r-1) \oplus {{\mathcal{O}}}_{{{\mathbb{P}}}^1}(-r-1)$, where $r$ is the corank of the quadratic form $H$.
By Lemma \[lemma:sect\] the sections of the bundle $\Phi$ correspond to the critical points of $H$, *i.e.,* to the solutions of the linear system $$\begin{aligned}
\sum_{j=0}^{n} H_{ij} x_j = 0 \ .\end{aligned}$$ The dimension of this space is $r$, the corank of $H$. The only rank two vector bundle over $\mathbb{P}^1$ with determinant ${{\mathcal{O}}}_{{{\mathbb{P}}}^1}(-2)$ and $r$ linearly indipendent holomorphic sections is ${{\mathcal{O}}}_{{{\mathbb{P}}}^1}(r-1) \oplus {{\mathcal{O}}}_{\mathbb{P}^1}(-r-1)$.
The proof of Proposition \[prop:ferrari\] is now complete: in fact, by (\[eq:cocycleW\]) the quadratic form $H$ corresponds to the Hessian of the superpotential $W$ at its critical points.
Some formulas for the potentials
================================
We group here some formulas that turn out to be useful in checking the computations involved in the results presented in this paper.
#### The geometric potential.
The geometric potential (deformation term) $B(z, \omega_1)$ is holomorphic on $\mathbb{C}^* \times \mathbb{C}$ and can be cast in the form $$\begin{aligned}
\label{eq:expansionB}
B(z, \omega) = \sum_{d=0}^{\infty} \sum_{k=0}^{dn} t^{(k)}_d B^{(k)}_d (z, \omega)\end{aligned}$$ where $$\begin{aligned}
B^{(k)}_d (z, \omega) = z^{-k-1} \omega^d \qquad k= 0, \dots, dn \ .\end{aligned}$$ The terms with $k < 0$ or $k > dn$ can be readsorbed by a holomorphic change of coordinates. For $l := - k -1 \geq 0$, we define $\widetilde{\omega}_2 := \omega_2 + d z^{l} \omega_1^{d-1}$, and for $m := k - dn -1 \geq 0$, we define $$\begin{aligned}
\widetilde{\omega}_2':= \omega_2' - (z')^m (\omega_1')^{d-1}\ .\end{aligned}$$
#### The superpotential
The superpotential that corresponds to $B^{(k)}_d$, given by (\[eq:superpotential\]), is $$\begin{aligned}
\label{eq:expansionW}
W^{(k)}_{d}(x_0, \dots, x_n) = \sum_{i_1, \dots, i_d=0 \atop i_1+ \dots + i_d=k }^n
x_{i_1} \dots x_{i_d} \ .\end{aligned}$$ We can obtain simple relations for the derivatives of these polynomials:$$\begin{aligned}
\frac{\partial W^{(k)}_{d}}{\partial x_j} &=& \sum_{i_1, \dots, i_d=0 \atop i_1+ \dots + i_d=k }^n
d \left( \frac{\partial x_{i_1}}{\partial x_j} x_{i_2} \dots x_{i_d} \right) \nonumber \\
&=& d \sum_{i_1, \dots, i_{d-1}=0 \atop i_1+ \dots + i_{d-1}=k-j }^n x_{i_1}
\dots x_{i_{d-1}} \ = \ d W^{(k-j)}_{d-1}\end{aligned}$$ and in general we have $$\begin{aligned}
\frac{\partial}{\partial x_{j_1}}\dots \frac{\partial}{\partial x_{j_l}} W^{(k)}_{d} = d (d-1)\dots (d-l+1) W^{(k -j_1\dots- j_l)}_{d-l}\end{aligned}$$
#### Relations between the derivatives of the potentials
Given a section $\omega_1(z)$, we have $$\begin{aligned}
\label{eq:derivatives}
\partial_\omega B(z, \omega_1(z)) = \sum_{j=0}^n \frac{\partial W}{\partial x_j} z^{-j-1} + \mathrm{trivial\ terms} \nonumber \\
\partial^2_\omega B(z, \omega_1(z)) = \sum_{i \leq j =0}^{n} \partial_i \partial_j W z^{-(i+j)-1} + \mathrm{trivial\ terms}\end{aligned}$$ where the “trivial terms” can be readsorbed by a holomorphic change of coordinates. We can obtain these results from (\[eq:expansionB\]) and (\[eq:expansionW\]). We have $$\begin{aligned}
\partial_\omega B^{(k)}_d (z, \omega_1(z)) =
d \sum _{i_1, \dots, i_{d-1} = 0}^{n} x_{i_1} \dots x_{i_{d-1}} z^{i_1 + \dots + i_{d-1}-k-1}\end{aligned}$$ and the only non-trivial terms are such that $0 \leq - (i_1 + \dots + i_{d-1}-k) \leq n$. In the same way, for the second derivatives we have $$\begin{aligned}
\partial^2_\omega B^{(k)}_{d}(z, \omega_1(z)) =
d (d-1) \sum _{i_1, \dots, i_{d-2} = 0}^{n} x_{i_1} \dots x_{i_{d-2}} z^{i_1 + \dots + i_{d-2}-k-1}\end{aligned}$$ The relevant terms are those with $0 \leq - (i_1 + \dots + i_{d-1}-k) \leq 2 n$.
[99]{}
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S. Katz, “Versal deformations and superpotentials for rational curves in smooth threefolds”, \[arXiv: math.ag/0010289\].
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|
---
abstract: 'Many real-world reinforcement learning problems have a hierarchical nature, and often exhibit some degree of partial observability. While hierarchy and partial observability are usually tackled separately (for instance by combining recurrent neural networks and options), we show that addressing both problems simultaneously is simpler and more efficient in many cases. More specifically, we make the initiation set of options conditional on the previously-executed option, and show that options with such Option-Observation Initiation Sets ([OOIs]{}) are at least as expressive as Finite State Controllers (FSCs), a state-of-the-art approach for learning in POMDPs. [OOIs]{}are easy to design based on an intuitive description of the task, lead to explainable policies and keep the top-level and option policies memoryless. Our experiments show that [OOIs]{}allow agents to learn optimal policies in challenging POMDPs, while being much more sample-efficient than a recurrent neural network over options.'
author:
- |
Denis Steckelmacher\
[**Diederik M. Roijers**]{}\
[**Anna Harutyunyan**]{}\
[**Peter Vrancx**]{}\
[**Hélène Plisnier**]{}\
[**Ann Nowé**]{}\
Artificial Intelligence Lab, Vrije Universiteit Brussel, Belgium
bibliography:
- 'biblio.bib'
title: 'Reinforcement Learning in POMDPs with Memoryless Options and Option-Observation Initiation Sets'
---
Introduction
============
Real-world applications of reinforcement learning (RL) face two main challenges: complex long-running tasks and partial observability. Options, the particular instance of Hierarchical RL we focus on, addresses the first challenge by factoring a complex task into simpler sub-tasks [@Barto2003; @Roy2006; @Tessler2016]. Instead of learning what action to perform depending on an observation, the agent learns a top-level policy that repeatedly selects options, that in turn execute sequences of actions before returning [@Sutton1999]. The second challenge, partial observability, is addressed by maintaining a belief of what the agent thinks the full state is [@Kaelbling1998; @Cassandra1994], reasoning about possible future observations [@Littman2001; @Boots2009], storing information in an external memory for later reuse [@Peshkin2001; @Zaremba2015; @Graves2016], or using recurrent neural networks (RNNs) to allow information to flow between time-steps [@Bakker2001; @Mnih2016].
Combined solutions to the above two challenges have recently been designed for planning [@He2011], but solutions for learning algorithms are not yet ideal. HQ-Learning decomposes a task into a sequence of fully-observable subtasks [@Wiering1997], which precludes cyclic tasks from being solved. Using recurrent neural networks in options and for the top-level policy [@Sridharan2010] addresses both challenges, but brings in the design complexity of RNNs [@Jozefowicz15; @Angeline94; @Mikolov14]. RNNs also have limitations regarding long time horizons, as their memory decays over time [@Hochreiter1997].
In her PhD thesis, Precup ([-@Precup2000], page 126) suggests that options may already be close to addressing partial observability, thus removing the need for more complicated solutions. In this paper, we prove this intuition correct by:
1. Showing that standard options do not suffice in POMDPs;
2. Introducing Option-Observation Initiation Sets ([OOIs]{}), that make the initiation sets of options conditional on the previously-executed option;
3. Proving that [OOIs]{}make options at least as expressive as Finite State Controllers (Section \[sec:fsc\]), thus able to tackle challenging POMDPs.
In contrast to existing HRL algorithms for POMDPs [@Wiering1997; @Theocharous2002; @Sridharan2010], [OOIs]{}handle repetitive tasks, do not restrict the action set available to sub-tasks, and keep the top-level and option policies memoryless. A wide range of robotic and simulated experiments in Section \[sec:experiments\] confirm that [OOIs]{}allow partially observable tasks to be solved optimally, demonstrate that [OOIs]{}are much more sample-efficient than a recurrent neural network over options, and illustrate the flexibility of [OOIs]{}regarding the amount of domain knowledge available at design time. In Section \[sec:treemaze\], we demonstrate the robustness of [OOIs]{}to sub-optimal option sets. While it is generally accepted that the designer provides the options and their initiation sets, we show in Section \[sec:duplicatedinput\] that random initiation sets, combined with learned option policies and termination functions, allow [OOIs]{}to be used without any domain knowledge.
Motivating Example {#sec:motivation}
------------------
![Robotic object gathering task. a) Khepera III, the two-wheeled robot used in the experiments. b) The robot has to gather objects from two terminals separated by a wall, and to bring them to the root.[]{data-label="fig:khepera"}](figures/khepera_terminals)
[OOIs]{}are designed to solve complex partially-observable tasks that can be decomposed into a set of fully-observable sub-tasks. For instance, a robot with first-person sensors may be able to avoid obstacles, open doors or manipulate objects even if its precise location in the building is not observed. We now introduce such an environment, on which our robotic experiments of Section \[sec:khepera\] are based.
A Khepera III robothas to gather objects from two terminals separated by a wall, and to bring them to the root (see Figure \[fig:khepera\]). Objects have to be gathered one by one from a terminal until it becomes empty, which requires many journeys between the root and a terminal. When a terminal is emptied, the other one is automatically refilled. The robot therefore has to alternatively gather objects from both terminals, and the episode finishes after the terminals have been emptied some random number of times. The root is colored in red and marked by a paper QR-code encoding `1`. Each terminal has a screen displaying its color and a dynamic QR-code (`1` when full, `2` when empty). Because the robot cannot read QR-codes from far away, the state of a terminal cannot be observed from the root, where the agent has to decide to which terminal it will go. This makes the environment partially observable, and requires the robot to remember which terminal was last visited, and whether it was full or empty.
The robot is able to control the speed of its two wheels. A wireless camera mounted on top of the robot detects bright color blobs in its field of view, and can read nearby QR-codes. Such low-level actions and observations, combined with a complicated task, motivate the use of hierarchical reinforcement learning. Fixed options allow the robot to move towards the largest red, green or blue blob in its field of view. The options terminate as soon as a QR-code is in front of the camera and close enough to be read. The robot has to learn a policy over options that solves the task.
The robot may have to gather a large number of objects, alternating between terminals several times. The repetitive nature of this task is incompatible with HQ-Learning [@Wiering1997]. Options with standard initiation sets are not able to solve this task, as the top-level policy is memoryless [@Sutton1999] and cannot remember from which terminal the robot arrives at the root, and whether that terminal was full or empty. Because the terminals are a dozen feet away from the root, almost a hundred primitive actions have to be executed to complete any root/terminal journey. Without options, this represents a time horizon much larger than usually handled by recurrent neural networks [@Bakker2001] or finite history windows [@Lin1993].
[OOIs]{}allow each option to be selected conditionally on the previously executed one (see Section \[sec:theory\]), which is much simpler than combining options and recurrent neural networks [@Sridharan2010]. The ability of [OOIs]{}to solve complex POMDPs builds on the time abstraction capabilities and expressiveness of options. Section \[sec:khepera\] shows that [OOIs]{}allow a policy for our robotic task to be learned to expert level. Additional experiments demonstrate that both the top-level and option policies can be learned by the agent (see Section \[sec:duplicatedinput\]), and that [OOIs]{}lead to substantial gains over standard initiation sets even if the option set is reduced or unsuited to the task (see Section \[sec:treemaze\]).
![Observations of the Khepera robot. a) Color image from the camera. b) Color blobs detected by the vision system, as observed by the robot. QR-codes can only be decoded when the robot is a couple of inches away from them.[]{data-label="fig:khepera_observations"}](figures/khepera_observations)
Background
==========
This section formally introduces Markov Decision Processes (MDPs), Options, Partially Observable MDPs (POMDPs) and Finite State Controllers, before presenting our main contribution in Section \[sec:ooi\].
Markov Decision Processes
-------------------------
A discrete-time Markov Decision Process (MDP) $\langle S, A, R, T, \gamma \rangle$ with discrete actions is defined by a possibly-infinite set $S$ of states, a finite set $A$ of actions, a reward function $R(s_t, a_t, s_{t+1}) \in \mathcal{R}$, that provides a scalar reward $r_t$ for each state transition, a transition function $T(s_t, a_t, s_{t+1}) \in [0, 1]$, that outputs a probability distribution over new states $s_{t+1}$ given a $(s_t, a_t)$ state-action pair, and $0 \le \gamma < 1$ the discount factor, that defines how sensitive the agent should be to future rewards.
A stochastic memoryless policy $\pi(s_t, a_t) \in [0, 1]$ maps a state to a probability distribution over actions. The goal of the agent is to find a policy $\pi^*$ that maximizes the expected cumulative discounted reward $E_{\pi^*}[\sum_t \gamma^t r_t]$ obtainable by following that policy.
Options
-------
The options framework, defined in the context of MDPs [@Sutton1999], consists of a set of options $O$ where each option $\omega \in O$ is a tuple $\langle \pi_\omega, I_\omega, \beta_\omega \rangle$, with $\pi_\omega (s_t, a_t) \in [0, 1]$ the memoryless option policy, $\beta_\omega(s_t) \in [0, 1]$ the termination function that gives the probability for the option $\omega$ to terminate in state $s_t$, and $I_\omega \subseteq S$ the initiation set that defines in which states $\omega$ can be started [@Sutton1999].
The memoryless top-level policy $\mu(s_t, \omega_t) \in [0, 1]$ maps states to a distribution over options and allows to choose which option to start in a given state. When an option $\omega$ is started, it executes until termination (due to $\beta_\omega$), at which point $\mu$ selects a new option based on the now current state.
Partially Observable MDPs
-------------------------
Most real-world problems are not completely captured by MDPs, and exhibit at least some degree of partial observability. A Partially Observable MDP (POMDP) $\langle \Omega, S, A, R, T, W, \gamma \rangle$ is an MDP extended with two components: the possibly-infinite set $\Omega$ of observations, and the $W : S \rightarrow \Omega$ function that produces observations $x$ based on the unobservable state $s$ of the process. Two different states, requiring two different optimal actions, may produce the same observation. This makes POMDPs remarkably challenging for reinforcement learning algorithms, as memoryless policies, that select actions or options based only on the current observation, typically no longer suffice.
Finite State Controllers
------------------------
Finite State Controllers (FSCs) are commonly used in POMDPs. An FSC $\langle {\ensuremath{\mathcal{N}}}, \psi, \eta, \eta^0 \rangle$ is defined by a finite set [$\mathcal{N}$]{} of nodes, an action function $\psi (n_t, a_t) \in [0, 1]$ that maps nodes to a probability distribution over actions, a successor function $\eta (n_{t-1}, x_t, n_t) \in [0, 1]$ that maps nodes and observations to a probability distribution over next nodes, and an initial function $\eta^0 (x_1, n_1) \in [0, 1]$ that maps initial observations to nodes [@Meuleau1999].
At the first time-step, the agent observes $x_1$ and activates a node $n_1$ by sampling from $\eta^0 (x_1, \cdot)$. An action is performed by sampling from $\psi (n_1, \cdot)$. At each time-step $t$, a node $n_t$ is sampled from $\eta (n_{t-1}, x_t, \cdot)$, then an action $a_t$ is sampled from $\psi (n_t, \cdot)$. FSCs allow the agent to select actions according to the entire history of past observations [@Meuleau1999], which has been shown to be one of the best approaches for POMDPs [@Lin1992]. [OOIs]{}, our main contribution, make options at least as expressive and as relevant to POMDPs as FSCs, while being able to leverage the hierarchical structure of the problem.
Option-Observation Initiation Sets {#sec:ooi}
==================================
Our main contribution, Option-Observation Initiation Sets ([OOIs]{}), make the initiation sets of options conditional on the option that has just terminated. We prove that [OOIs]{}make options at least as expressive as FSCs (thus suited to POMDPs, see Section \[sec:fsc\]), even if the top-level and option policies are memoryless, while options without [OOIs]{}are strictly less expressive than FSCs (see Section \[sec:vanilla\]). In Section \[sec:experiments\], we show on one robotic and two simulated tasks that [OOIs]{}allow challenging POMDPs to be solved optimally.
Conditioning on Previous Option {#sec:theory}
-------------------------------
Descriptions of partially observable tasks in natural language often contain allusions at sub-tasks that must be sequenced or cycled through, possibly with branches. This is easily mapped to a policy over options (learned by the agent) and sets of options that may or may not follow each other.
A good memory-based policy for our motivating example, where the agent has to bring objects from two terminals to the root, can be described as “go to the green terminal, then go to the root, then go back to the green terminal if it was full, to the blue terminal otherwise”, and symmetrically so for the blue terminal. This sequence of sub-tasks, that contains a condition, is easily translated to a set of options. Two options, $\omega_{GF}$ and $\omega_{GE}$, sharing a single policy, go from the green terminal to the root (using low-level motor actions). $\omega_{GF}$ is executed when the terminal is full, $\omega_{GE}$ when it is empty. At the root, the option that goes back to the green terminal can only follow $\omega_{GF}$, not $\omega_{GE}$. When the green terminal is empty, going back to it is therefore forbidden, which forces the agent to switch to the blue terminal when the green one is empty.
We now formally define our main contribution, Option-Observation Initiation Sets ([OOIs]{}), that allow to describe which options may follow which ones. We define the initiation set $I_\omega$ of option $\omega$ so that the set [$\mathcal{O}_t$]{} of options available at time $t$ depends on the observation $x_t$ and previously-executed option $\omega_{t-1}$:
$$\begin{aligned}
I_\omega &\subseteq \Omega \times (O \cup \{ \emptyset \}) \\
{\ensuremath{\mathcal{O}_t}}&\equiv \{ \omega \in O : (x_t, \omega_{t-1}) \in I_\omega \}
\end{aligned}$$
with $\omega_0 = \emptyset$, $\Omega$ the set of observations and $O$ the set of options. [$\mathcal{O}_t$]{} allows the agent to condition the option selected at time $t$ on the one that has just terminated, even if the top-level policy does not observe $\omega_{t-1}$. The top-level and option policies remain memoryless. Not having to observe $\omega_{t-1}$ keeps the observation space of the top-level policy small, instead of extending it to $\Omega \times O$, without impairing the representational power of [OOIs]{}, as shown in the next sub-section.
[OOIs]{}Make Options as Expressive as FSCs {#sec:fsc}
------------------------------------------
Finite State Controllers are state-of-the-art in policies applicable to POMDPs [@Meuleau1999]. By proving that options with [OOIs]{}are as expressive as FSCs, we provide a lower bound on the expressiveness of [OOIs]{}and ensure that they are applicable to a wide range of POMDPs.
\[thm:fsc\] [OOIs]{}allow options to represent any policy that can be expressed using a Finite State Controller.
The reduction from any FSC to options requires one option [$\langle n'_{t-1}, n_t \rangle$]{} per ordered pair of nodes in the FSC, and one option [$\langle \emptyset, n_1 \rangle$]{} per node in the FSC. Assuming that $n_0 = \emptyset$ and $\eta (\emptyset, x_1, \cdot) = \eta^0 (x_1, \cdot)$, the options are defined by:
$$\begin{aligned}
\label{eq:fsc_beta}
\beta_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} (x_t) &= 1 \\
\label{eq:fsc_pio}
\pi_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} (x_t, a_t) &= \psi(n_t, a_t) \\
\label{eq:fsc_pi}
\mu (x_t, {\ensuremath{\langle n'_{t-1}, n_t \rangle}}) &= \eta (n'_{t-1}, x_t, n_t) \\
\nonumber
I_{{\ensuremath{\langle \emptyset, n_1 \rangle}}} &= \Omega \times \{ \emptyset \} \\
\nonumber
I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} &= \Omega \times \{{\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}: n'_{t-1} = n_{t-1} \}
\end{aligned}$$
Each option corresponds to an edge of the FSC. Equation \[eq:fsc\_beta\] ensures that every option stops after having emitted a single action, as the FSC takes one transition every time-step. Equation \[eq:fsc\_pio\] maps the current option to the action emitted by the destination node of its corresponding FSC edge. We show that $\mu$ and $I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}}$ implement $\eta (n_{t-1}, x_t, n_t)$, with $\omega_{t-1} = {\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}$, by:
$$\begin{aligned}
\mu &(x_t, {\ensuremath{\langle n'_{t-1}, n_t \rangle}}) = \\ &\begin{cases}
\eta (n_{t-1}, x_t, n_t) &\begin{array}{l}
{\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}\in I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} \\
\Leftrightarrow n'_{t-1} = n_{t-1}
\end{array} \\
0 &\begin{array}{l}
{\ensuremath{\langle n'_{t-2}, n_{t-1} \rangle}}\notin I_{{\ensuremath{\langle n'_{t-1}, n_t \rangle}}} \\
\Leftrightarrow n'_{t-1} \ne n_{t-1}
\end{array}
\end{cases}
\end{aligned}$$
Because $\eta$ maps nodes to nodes and $\mu$ selects options representing pairs of nodes, $\mu$ is extremely sparse and returns a value different from zero, $\eta (n_{t-1}, x_t, n_t)$, only when [$\langle n'_{t-2}, n_{t-1} \rangle$]{} and [$\langle n'_{t-1}, n_t \rangle$]{} agree on $n_{t-1}$.
Our reduction uses options with trivial policies, that execute for a single time-step, which leads to a large amount of options to compensate. In practice, we expect to be able to express policies for real-world POMDPs with much less options than the number of states an FSC would require, as shown in our simulated (Section \[sec:duplicatedinput\], 2 options) and robotic experiments (Section \[sec:khepera\], 12 options). In addition to being sufficient, the next sub-section proves that [OOIs]{}are necessary for options to be as expressive as FSCs.
Original Options are not as Expressive as FSCs {#sec:vanilla}
----------------------------------------------
![Two-nodes Finite State Controller that emits an infinite sequence ABAB... based on an uninformative observation $x_{\emptyset}$. This FSC cannot be expressed using options without [OOIs]{}.[]{data-label="fig:fsc"}](figures/hrlnext_fsc_bw)
While options with regular initiation sets are able to express some memory-based policies [@Sutton1999 page 7], the tiny but valid Finite State Controller presented in Figure \[fig:fsc\] cannot be mapped to a set of options and a policy over options (without [OOIs]{}). This proves that options without [OOIs]{}are strictly less expressive than FSCs.
Options without [OOIs]{}are not as expressive as Finite State Controllers.
Figure \[fig:fsc\] shows a Finite State Controller that emits a sequence of alternating A’s and B’s, based on a constant uninformative observation $x_{\emptyset}$. This task requires memory because the observation does not provide any information about what was the last letter to be emitted, or which one must now be emitted. Options having memoryless policies, options executing for multiple time-steps are unable to represent the FSC exactly. A combination of options that execute for a single time-step cannot represent the FSC either, as the options framework is unable to represent memory-based policies with single-time-step options [@Sutton1999].
Experiments {#sec:experiments}
===========
The experiments in this section illustrate how [OOIs]{}allow agents to perform optimally in environments where options without [OOIs]{}fail. Section \[sec:khepera\] shows that [OOIs]{}allow the agent to learn an expert-level policy for our motivating example (Section \[sec:motivation\]). Section \[sec:duplicatedinput\] shows that the top-level and option policies required by a repetitive task can be learned, and that learning option policies allow the agent to leverage random [OOIs]{}, thereby removing the need for designing them. In Section \[sec:treemaze\], we progressively reduce the amount of options available to the agent, and demonstrate how [OOIs]{}still allow good memory-based policies to emerge when a sub-optimal amount of options are used.
All our results are averaged over 20 runs, with standard deviation represented by the light regions in the figures. The source code, raw experimental data, run scripts, and plotting scripts of our experiments, along with a detailed description of our robotic setup, are available as supplementary material. A video detailing our robotic experiment is available at <http://steckdenis.be/oois_demo.mp4>.
Learning Algorithm {#sec:nnet}
------------------
All our agents learn their top-level and option policies (if not provided) using a single feed-forward neural network, with one hidden layer of 100 neurons, trained using Policy Gradient [@Sutton2000] and the Adam optimizer [@kingma2014adam]. Our neural network $\pi$ takes three inputs and produces one output. The inputs are problem-specific observation features $\mathbf{x}$, the one-hot encoded current option $\boldsymbol{\omega}$ ($\boldsymbol{\omega} = \mathbf{0}$ when executing the top-level policy), and a mask, $\mathbf{mask}$. The output $\mathbf{y}$ is the joint probability distribution over selecting actions or options (so that the same network can be used for the top-level and option policies), while terminating or continuing the current option:
$$\begin{aligned}
\mathbf{h_1} &= \tanh(\mathbf{W}_1 [\mathbf{x}^T \boldsymbol{\omega}^T]^T + \mathbf{b}_1), \\
\mathbf{\hat{y}} &= \sigma(\mathbf{W}_2 \mathbf{h_1} + \mathbf{b}_2) \circ \mathbf{mask}, \\
\mathbf{y} &= \frac{\mathbf{\hat{y}}}{\mathbf{1}^T \mathbf{\hat{y}}},
\end{aligned}$$
with $W_i$ and $b_i$ the trainable weights and biases of layer $i$, $\sigma$ the sigmoid function, and $\circ$ the element-wise product of two vectors. The fraction ensures that a valid probability distribution is produced by the network. The initiation sets of options are implemented using the $\mathbf{mask}$ input of the neural network, a vector of $2 \times (|A| + |O|)$ integers, the same dimension as the $\mathbf{y}$ output. When executing the top-level policy ($\boldsymbol{\omega} = \mathbf{0}$), the mask forces the probability of primitive actions to zero, preserves option $\omega_i$ according to $I_{\omega_i}$, and prevents the top-level policy from terminating. When executing an option policy ($\boldsymbol{\omega} \ne \mathbf{0}$), the mask only allows primitive actions to be executed. For instance, if there are two options and three actions, $\mathbf{mask} = \begin{smallmatrix}end \\ cont\end{smallmatrix} (
\begin{smallmatrix}
0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 1
\end{smallmatrix}
)$ when executing any of the options. When executing the top-level policy, $\mathbf{mask} = \begin{smallmatrix}end \\ cont\end{smallmatrix} (
\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 \\
a & b & 0 & 0 & 0
\end{smallmatrix}
)$, with $a = 1$ if and only if the option that has just finished is in the initiation set of the first option, and $b = 1$ according to the same rule but for the second option. The neural network $\pi$ is trained using Policy Gradient, with the following loss:
$$\begin{aligned}
\mathcal{L}(\pi) &= -\sum\limits_{t=0}^{T} (\mathcal{R}_t - V(x_t, \omega_t)) \log (\pi(x_t, \omega_t, a_t))
\end{aligned}$$
with $a_t \sim \pi(x_t, \omega_t, \cdot)$ the action executed at time $t$. The return $\mathcal{R}_t = \sum_{\tau=t}^{T} \gamma^{\tau} r_{\tau}$, with $r_{\tau} = R(s_{\tau}, a_{\tau}, s_{\tau+1})$, is a simple discounted sum of future rewards, and ignores changes of current option. This gives the agent information about the complete outcome of an action or option, by directly evaluating its flattened policy. A baseline $V(x_t, \omega_t)$ is used to reduce the variance of the $\mathcal{L}$ estimate [@Sutton2000]. $V(x_t, \omega_t)$ predicts the expected cumulative reward obtainable from $x_t$ in option $\omega_t$ using a separate neural network, trained on the monte-carlo return obtained from $x_t$ in $\omega_t$.
Comparison with LSTM over Options {#sec:lstm}
---------------------------------
In order to provide a complete evaluation of [OOIs]{}, a variant of the $\pi$ and $V$ networks of Section \[sec:nnet\], where the hidden layer is replaced with a layer of 20 LSTM units [@Hochreiter1997; @Sridharan2010], is also evaluated on every task. We use 20 units as this leads to the best results in our experiments, which ensures a fair comparison of LSTM against OOIs. In all experiments, the LSTM agents are provided the same set of options as the agent with [OOIs]{}. Not providing any option, or less options, leads to worse results. Options allow the LSTM network to focus on important observations, and reduces the time horizon to be considered. Shorter time horizons have been shown to be beneficial to LSTM [@Bakker2001].
Despite our efforts, LSTM over options only manages to learn good policies in our robotic experiment (see Section \[sec:khepera\]), and requires more than twice the amount of episodes as [OOIs]{}to do so. In our repetitive task, dozens of repetitions seem to confuse the network, that quickly diverges from any good policy it may learn (see Section \[sec:duplicatedinput\]). On TreeMaze, a much more complex version of the T-maze task, originally used to benchmark reinforcement learning LSTM agents [@Bakker2001], the LSTM agent learns the optimal policy after more than 100K episodes (not shown on the figures). These results illustrate how learning with recurrent neural networks is sometimes difficult, and how [OOIs]{}allow to reliably obtain good results, with minimal engineering effort.
Object Gathering {#sec:khepera}
----------------
The first experiment illustrates how [OOIs]{}allow an expert-level policy to be learned for a complex robotic partially-observable repetitive task. The experiment takes place in the environment described in Section \[sec:motivation\]. A robot has to gather objects one by one from two terminals, green and blue, and bring them back to the root location. Because our actual robot has no effector, it navigates between the root and the terminals, but only pretends to move objects. The agent receives a reward of +2 when it reaches a full terminal, -2 when the terminal is empty. At the beginning of the episode, each terminal contains 2 to 4 objects, this amount being selected randomly for each terminal. When the agent goes to an empty terminal, the other one is re-filled with 2 to 4 objects. The episode ends after 2 or 3 emptyings (combined across both terminals). Whether a terminal is full or empty is observed by the agent only when it is at the terminal. The agent therefore has to remember information acquired at terminals in order to properly choose, at the root, to which terminal it will go.
![Cumulative reward per episode obtained on our object gathering task, with [OOIs]{}, without [OOIs]{}, and using an LSTM over options. [OOIs]{}learns an expert-level policy much quicker than an LSTM over options. The LSTM curve flattens-out (with high variance) after about 30K episodes.[]{data-label="fig:kresults"}](terminals-results/plot)
The agent has access to 12 memoryless options that go to red ($\omega_{R1..R4}$), green ($\omega_{G1..G4}$) or blue objects ($\omega_{B1..B4}$), and terminate when the agent is close enough to them to read a QR-code displayed on them. The initiation set of $\omega_{R1,R2}$ is $\omega_{G1..G4}$, of $\omega_{R3,R4}$ is $\omega_{B1..B4}$, and of $\omega_{G_i,B_i}$ is $\omega_{R_i} ~ \forall i = 1..4$. This description of the options and their [OOIs]{}is purposefully uninformative, and illustrates how little information the agent has about the task. The option set used in this experiment is also richer than the simple example of Section \[sec:theory\], so that the solution of the problem, not going back to an empty terminal, is not encoded in [OOIs]{}but must be learned by the agent.
Agents with and without [OOIs]{}learn top-level policies over these options. We compare them to a *fixed* agent, using an expert top-level policy that interprets the options as follows: $\omega_{R1..R4}$ go to the root from a full/empty green/blue terminal (and are selected accordingly at the terminals depending on the QR-code displayed on them), while $\omega_{G1..G4,B1..B4}$ go to the green/blue terminal from the root when the previous terminal was full/empty and green/blue. At the root, [OOIs]{}ensure that only one option amongst *go to green after a full green*, *go to green after an empty blue*, *go to blue after a full blue* and *go to blue after an empty green* is selected by the top-level policy: the one that corresponds to what color the last terminal was and whether it was full or empty. The agent goes to a terminal until it is empty, then switches to the other terminal, leading to an average reward of 10.
When the top-level policy is learned, [OOIs]{}allow the task to be solved, as shown in Figure \[fig:kresults\], while standard initiation sets do not allow the task to be learned. Because experiments on a robot are slow, we developed a small simulator for this task, and used it to produce Figure \[fig:kresults\] after having successfully asserted its accuracy using two 1000-episodes runs on the actual robot. The agent learns to properly select options at the terminals, depending on the QR-code, and to output a proper distribution over options at the root, thereby matching our expert policy. The LSTM agent learns the policy too, but requires more than twice the amount of episodes to do so. The high variance displayed in Figure \[fig:kresults\] comes from the varying amounts of objects in the terminals, and the random selection of how many times they have to be emptied.
Because fixed option policies are not always available, we now show that [OOIs]{}allow them to be learned at the same time as the top-level policy.
Modified DuplicatedInput {#sec:duplicatedinput}
------------------------
In some cases, a hierarchical reinforcement learning agent may not have been provided policies for several or any of its options. In this case, [OOIs]{}allow the agent to learn its top-level policy, the option policies and their termination functions. In this experiment, the agent has to learn its top-level and option policies to copy characters from an input tape to an output tape, removing duplicate B’s and D’s (mapping ABBCCEDD to ABCCED for instance; B’s and D’s always appear in pairs). The agent only observes a single input character at a time, and can write at most one character to the output tape per time-step.
The input tape is a sequence of $N$ symbols $x \in \Omega$, with $\Omega = \{A, B, C, D, E\}$ and $N$ a random number between 20 and 30. The agent observes a single symbol $x_t \in \Omega$, read from the $i$-th position in the input sequence, and does not observe $i$. When $t = 1$, $i = 0$. There are 20 actions ($5 \times 2 \times 2$), each of them representing a symbol (5), whether it must be pushed onto the output tape (2), and whether $i$ should be incremented or decremented (2). A reward of 1 is given for each correct symbol written to the output tape. The episode finishes with a reward of -0.5 when an incorrect symbol is written.
![Cumulative reward per episode obtained on modified DuplicatedInput, with random or designed [OOIs]{}, without [OOIs]{}and using an LSTM over options. Despite our efforts, an LSTM over options repeatedly learns then forgets optimal policies, as shown by the high variance of its line.[]{data-label="fig:diresults"}](modifiedduplicatedinput-results/plot)
The agent has access to two options, $\omega_1$ and $\omega_2$. [OOIs]{}are designed so that $\omega_2$ cannot follow itself, with no such restriction on $\omega_1$. No reward shaping or hint about what each option should do is provided. The agent automatically discovers that $\omega_1$ must copy the current character to the output, and that $\omega_2$ must skip the character without copying it. It also learns the top-level policy, that selects $\omega_2$ (skip) when observing B or D and $\omega_2$ is allowed, $\omega_1$ otherwise (copy).
Figure \[fig:diresults\] shows that an agent with two options and [OOIs]{}learns the optimal policy for this task, while an agent with two options and only standard initiation sets ($I_\omega = \Omega ~ \forall \omega$) fails to do so. The agent without [OOIs]{}only learns to copy characters and never skips any (having two options does not help it). This shows that [OOIs]{}are necessary for learning this task, and allow to learn top-level and option policies suited to our repetitive partially observable task.
When the option policies are learned, the agent becomes able to adapt itself to random [OOIs]{}, thereby removing the need for designing [OOIs]{}. For an agent with $N$ options, each option has $\frac{N}{2}$ randomly-selected options in its initiation set, with the initiation sets re-sampled for each run. The agents learn how to leverage their option set, and achieve good results on average (16 options used in Figure \[fig:diresults\], more options lead to better results). When looking at individual runs, random [OOIs]{}allow optimal policies to be learned, but several runs require more time than others to do so. This explains the high variance and noticeable steps shown in Figure \[fig:diresults\].
The next section shows that an improperly-defined set of human-provided options, as may happen in design phase, still allows the agent to perform reasonably well. Combined with our results with random [OOIs]{}, this shows that [OOIs]{}can be tailored to the exact amount of domain knowledge available for a particular task.
TreeMaze {#sec:treemaze}
--------
The optimal set of options and [OOIs]{}may be difficult to design. When the agent learns the option policies, the previous section demonstrates that random [OOIs]{}suffice. This experiment focuses on human-provided option policies, and shows that a sub-optimal set of options, arising from a mis-specification of the environment or normal trial-and-error in design phase, does not prevent agents with [OOIs]{}from learning reasonably good policies.
TreeMaze is our generalization of the T-maze environment [@Bakker2001] to arbitrary heights. The agent starts at the root of the tree-like maze depicted in Figure \[fig:treemaze\], and has to reach the extremity of one of the 8 leaves. The leaf to be reached (the goal) is chosen uniformly randomly before each episode, and is indicated to the agent using 3 bits, observed one at a time during the first 3 time-steps. The agent receives no bit afterwards, and has to remember them in order to navigate to the goal. The agent observes its position in the current corridor (0 to 4) and the number of T junctions it has already crossed (0 to 3). A reward of -0.1 is given each time-step, +10 when reaching the goal. The episode finishes when the agent reaches any of the leaves. The optimal reward is 8.2.
![TreeMaze environment. The agent starts at $x_1$ and must go to one of the leaves. The leaf to be reached is indicated by 3 bits observed at time-steps 1, 2 and 3.[]{data-label="fig:treemaze"}](figures/hrlnext_treemaze_bw)
We consider 14 options with predefined memoryless policies, several of them sharing the same policy, but encoding distinct states (among 14) of a 3-bit memory where some bits may be unknown. 6 partial-knowledge options $\omega_{0{{-}}{{-}}}$, $\omega_{1{{-}}{{-}}}$, $\omega_{00{{-}}}$, ..., $\omega_{11{{-}}}$ go right then terminate. 8 full-knowledge options $\omega_{000}$, $\omega_{001}$, ..., $\omega_{111}$ go to their corresponding leaf. [OOIs]{}are defined so that any option may only be followed by itself, or one that represents a memory state where a single 0 or - has been flipped to 1. Five agents have to learn their top-level policy, which requires them to learn how to use the available options to remember to which leaf to go. The agents do not know the name or meaning of the options. Three agents have access to all 14 options (with, without [OOIs]{}, and LSTM). The agent with [OOIs]{}(8) only has access to full-knowledge options, and therefore cannot disambiguate unknown and 0 bits. The agent with [OOIs]{}(4) is restricted to options $\omega_{000}$, $\omega_{010}$, $\omega_{100}$ and $\omega_{110}$ and therefore cannot reach odd-numbered goals. The options of the (8) and (4) agents terminate in the first two cells of the first corridor, to allow the top-level policy to observe the second and third bits.
Figure \[fig:treemazeresults\] shows that the agent with [OOIs]{}(14) consistently learns the optimal policy for this task. When the number of options is reduced, the quality of the resulting policies decreases, while still remaining above the agent without [OOIs]{}. Even the agent with 4 options, that cannot reach half the goals, performs better than the agent without [OOIs]{}but 14 options. This experiment demonstrates that [OOIs]{}provide measurable benefits over standard initiation sets, even if the option set is largely reduced.
Combined, our three experiments demonstrate that [OOIs]{}lead to optimal policies in challenging POMDPs, consistently outperform LSTM over options, allow the option policies to be learned, and can still be used when reduced or no domain knowledge is available.
![Cumulative reward per episode obtained on TreeMaze, using 14, 8 or 4 options. Even with an insufficient amount of options (8 or 4), [OOIs]{}lead to better performance than no [OOIs]{}but 14 options. LSTM over options learns the task after more than 100K episodes.[]{data-label="fig:treemazeresults"}](treemaze-results/plot)
Conclusion and Future Work {#sec:conclusion}
==========================
This paper proposes [OOIs]{}, an extension of the initiation sets of options so that they restrict which options are allowed to be executed after one terminates. This makes options as expressive as Finite State Controllers. Experimental results confirm that challenging partially observable tasks, simulated or on physical robots, one of them requiring exact information storage for hundreds of time-steps, can now be solved using options. Our experiments also illustrate how [OOIs]{}lead to reasonably good policies when the option set is improperly defined, and that learning the option policies allow random [OOIs]{}to be used, thereby providing a turnkey solution to partial observability.
Options with [OOIs]{}also perform surprisingly well compared to an LSTM network over options. While LSTM over options does not require the design of [OOIs]{}, their ability to learn without any a-priori knowledge comes at the cost of sample efficiency and explainability. Furthermore, random [OOIs]{}are as easy to use as an LSTM and lead to superior results (see Section \[sec:duplicatedinput\]). [OOIs]{}therefore provide a compelling alternative to recurrent neural networks over options, applicable to a wide range of problems.
Finally, the compatibility between [OOIs]{}and a large variety of reinforcement learning algorithms leads to many future research opportunities. For instance, we have obtained very encouraging results in continuous action spaces, using CACLA [@VanHasselt2007] to implement parametric options, that take continuous arguments when executed, in continuous-action hierarchical POMDPs.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author is “Aspirant” with the Science Foundation of Flanders (FWO, Belgium), grant number 1129317N. The second author is “Postdoctoral Fellow” with the FWO, grant number 12J0617N.
Thanks to Finn Lattimore, who *gave* a computer to the first author, so that he could finish this paper while attending the UAI 2017 conference in Sydney, after his own computer unexpectedly fried. Thanks to Joris Scharpff for his very helpful input on this paper.
|
---
abstract: 'Since the complexity of the practical environment, many distributed networked systems can not be illustrated with the integer-order dynamics and only be described as the fractional-order dynamics. Suppose multi-agent systems will show the individual diversity with difference agents, where the heterogeneous (integer-order and fractional-order) dynamics are used to illustrate the agent systems and compose integer-fractional compounded-order systems. Applying Laplace transform and frequency domain theory of the fractional-order operator, consensus of delayed multi-agent systems with directed weighted topologies is studied. Since integer-order model is the special case of fractional-order model, the results in this paper can be extend to the systems with integer-order models. Finally, numerical examples are used to verify our results.'
address: |
1. National Key Laboratory on Aircraft Control Technology, Beihang University, Beijing 100191, P.R. China; e-mail: $\{$hyyang$\_$ld, lguo66$\}$@yahoo.com.cn.\
2. School of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, China; e-mail: hntjxx@163.com.\
3. School of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210046, China; e-mail: caokc@njupt.edu.cn.
author:
- 'Hong-yong Yang$^1$, Lei Guo$^1$, Xun-lin Zhu$^2$, Ke-cai Cao$^3$'
title: 'Coordination Control of Heterogeneous Compounded-Order Multi-Agent Systems with Communication Delays'
---
Coordination control; multi-agent systems; heterogeneous dynamics; compounded-order; communication delays.\
[**Pacs**]{}: 89.75.F6; 05.30.Pr
Introduction
============
Nowadays, the past three decades have witnessed significant progress on fractional calculus, because the applications of fractional calculus are found in more and more scientific fields, covering mechanics, physics, engineering, informatics, and materials. The list of such applications is long, for instance, it includes viscoelasticity, colored noise, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, control engineering and so on[@podlubny; @hilfer; @ladaci; @hwanga; @ahn; @li; @sabat; @lu]. In fact, real word processes generally or most likely are fractional-order systems[@podlubny; @hilfer]. Furthermore, fractional order controllers have so far been implemented to enhance the robustness and the performance of the closed loop control systems, the stability problem of fractional-order systems has been investigated both from an algebraic and an analytic point of view[@ladaci; @hwanga; @ahn; @li; @sabat; @lu].
On the other hand, as one of the most basic problem of coordination control for networked systems, consensus of multi-agent systems has been widespread concerned as an important research topic in the field of systems control. Consensus of multi-agent systems means that several distributed agents achieve the same state or output through local mutual coupling effect among the individuals, where the centralized control is not used. Based on the computer model proposed by Reynolds which imitates animals’ flocking [@5], Vicsek et al. [@6] firstly proposed a non-equilibrium multi-agent system model from the point of view of statistical mechanics, simulation shows that all individuals in the system can run in accordance with the same direction under certain conditions. Recently, since the wide application of multi-agent systems in various fields, many scholars have devoted themselves to study the consensus of multi-agent systems [@olfati; @ren07; @lishihua; @lin; @yujun; @yang; @tian]. When the information transferring, communication delays will occur in networked control system and affect the features of the system. The effects of communication delays on movement consensus of multi-agent systems have been concerned by many scholars [@lin; @yujun; @yang; @tian], and the stability of delay system also become a hot topic in the multiple agents field.
The important results of the above literatures pay attention to the consensus problem of integer-order multi-agent systems. In the complex environment, many dynamic characteristics of natural phenomena can not be described in the form integer-order equation, but only be described in the dynamics of fractional-order (non-integer order) behavior, for example: flocking movement and food searching by means of the individual secretions and microbial, submarine underwater robots in the bottom of the sea with a large number of microorganisms and viscous substances, unmanned aerial vehicles running in the complex space environment [@ren2011]. Cao and Ren [@cao1; @cao2] studied distribution coordination of fractional-order multi-agent systems firstly, and gave the relationship between the number of individuals and the fractional order in the stable multi-agent systems. However, to the best of authors’ knowledge, there are few researches done on the coordination control of fractional-order multi-agent systems with communication delays.
In this paper, we suppose agents work in the complex environment, the heterogeneous dynamics with fractional-order and integer-order is presented for multi-agent systems. The main innovation of this paper lies in the study on consensus of compound-order (fractional-order and integer-order) distributed multi-agent systems with different time delays. This paper is organized as follows. In Section 2, some necessary definitions and notations are given on fractional calculus. In Section 3, the compound-order dynamics of the fractional order system and the integer order system is presented. The consensus of integer-fractional compounded-order multi-agent systems with communication delays is studied in Section 4. The corresponding simulation results are provided in Section 5 to demonstrate the effectiveness of the proposed conditions. Finally, the conclusions are drawn in Section 6.
Fractional Calculus
===================
Fractional calculus plays an important role in modern science. There are mainly two widely used fractional operators: Caputo and Riemann-Liouville (R-L) fractional operators[@podlubny]. In physical systems, Caputo fractional operator is more practical than R-L fractional operator because R-L fractional operator has initial value problems. Therefore, in this paper we will use Caputo fractional operator to model the system dynamics and analyze the stability of the proposed coordination algorithms. Generally, Caputo fractional operator includes Caputo integral and Caputo derivative. Caputo integral is defined as $$\begin{aligned}
^{C}_{a}D^{-p}_{t}f(t)=\frac{1}{\Gamma(p)}\int^{t}_{a}\frac{f(\theta)}{(t-\theta)^{1-p}}d \theta,\end{aligned}$$ where the integral order $p\in (0,1]$, $\Gamma(.)$ is the Gamma function, and a is an arbitrary real number. Based on the Caputo integral, for a nonnegative real number $\alpha$, Caputo derivative is defined as $$^{C}_{a}D^{\alpha}_{t}f(t)=^{C}_{a}D^{-p}_{t}[\frac{d^{[\alpha]+1}}{dt^{[\alpha]+1}}f(t)],
\label{e1}$$ where $p=[\alpha]+1-\alpha\in (0,1]$ and $[\alpha]$ is the integral part of $\alpha$. If $\alpha$ is an integer, then $p=1$ and the Caputo derivative is equivalent to the integer-order derivative. In this paper, a simple notation $f^{(\alpha)}(t)$ is used to replace $^{C}_{a}D^{\alpha}_{t}f(t)$. Let $\mathfrak{L}()$ denote the Laplace transform of a function, the Laplace transform of Caputo derivative is shown as $$\mathfrak{L}(f^{(\alpha)}(t))=s^{\alpha}F(s)-\sum^{[\alpha]+1}_{k=1}s^{\alpha-1}f^{(k-1)}(0),
\label{e2}$$ where $F(s)=\mathfrak{L}(f(t))=\int^\infty_{0-}e^{-st}f(t)dt$ is the Laplace transform of function $f(t)$, $f^{(k)}(0)=\lim_{\xi\rightarrow 0-}f^{(k)}(\xi)$ and $f^{(0)}(0)=f(0)=\lim_{\xi\rightarrow 0-}f(\xi)$.
Problem statement
=================
Assume that multi-agent systems consist of $n$ autonomous agents, connected relations among the agents constitute a network topology $\mathcal{G}$. Let $\mathcal{G}=\{V,E,A\}$ represent a directed weighted graph, in which $V=\{v_1,v_2,...,v_n\}$ represents a collection of $n$ nodes, and its set of edges is $E\subseteq V\times V$. The node indexes belong to a finite index set $I=\{1,2,...,n\}$, adjacency matrix $A=[a_{ij}]\in R^{n\times n}$ with weighted adjacency elements $a_{ij}\geq 0$. An edge of the weighted diagraph $\mathcal{G}$ is denoted by $e_{ij}=(v_{i},v_{j})\in E$. We assume that the adjacency element $a_{ij}> 0$ when $e_{ij}\in E$, otherwise, $a_{ij}= 0$. The set of neighbors of node $i$ is denoted by $N_{i}=\{j\in I:
a_{ij}>0\}$.
Let $\mathcal{G}$ be a weighted digraph without self-loops, i.e., $a_{ii}=0$, and matrix $D=\mathrm{diag}\{d_1,d_2,...,d_n\}$ be the diagonal matrix with the diagonal elements $d_{i}=\sum_{j=1}^{n}a_{ij}$ representing the sum of the elements in the $i$-th row of matrix $A$. The Laplacian matrix of the weighted digraph $\mathcal{G}$ is defined as $L=D-A$. For two nodes $i$ and $k$, there is subscript set $\{k_{1},
k_{2}, ... k_{l}\}$ satisfying $a_{ik_{1}}>0$, $a_{k_{1}k_{2}}>0$, ..., $a_{k_{l}k}>0$, then there is a directed linked path between node $i$ and node $k$ which is used for the information transmission, also we can say node $i$ can receive the information from node $k$. If node $i$ can find a path to reach any node of the graph, then node $i$ is globally reachable from every other node in the digraph. For any two nodes in the graph, there are at least one directed linked path, then G is strongly connected.
[**Lemma 1**]{}[@ren07]. $0$ is a simple eigenvalue of Laplacian matrix $L$, and $X_{0}=C[1, 1, ..., 1]^{T}$ is corresponding right eigenvector, i.e., $LX_{0}=0$, if and only if the digraph $\mathcal{G}=(V, E,
A)$ has a globally reachable node.
Assume that there are individual differences in the complex environment of multi-agent systems; there are two groups of multi-agent systems with integer-order dynamics and fractional-order dynamics. The compounded-order dynamical equations are described as follows:
$$\begin{array}{lc}
\dot{x}_{i}(t)=u_i(t), i=1,...,m,\\
x_{l}^{(\alpha)}(t)=u_l(t), l=m+1,...,n,
\end{array}\label{e3}$$
where $x_i(t)\in R$ and $u_i(t)\in R$ represent the $i$-th agent’s state and control input respectively, $\dot{x}_{i}(t)$ represents the first-order derivative for the state $x_i(t)$, $x_{l}^{(\alpha)}$ represents the $\alpha $ order Caputo derivative, and $\alpha\in (0, 1]$. Assume the following control protocols are used in multi-agent systems: $$u_{i}(t)=-\gamma\sum_{k\in N_i}a_{ik}[x_{i}(t)-x_{k}(t)], i=1,...,m,m+1,...,n.
\label{e4}$$ where $a_{ik}$ represents the $(i, k)$ elements of adjacency matrix $A$, $\gamma>0$ is control gain, $N_i$ represents the neighbors collection of the $i$-th agent.
This article assumes that there are communication delays in the dynamical systems, and consensus of the integer-fractional-compounded-order agent systems with communication delays is studied. Under the influence of communication delays, we can get the following algorithm: $$\begin{array}{lc}
\dot{x}_{i}(t)=u_i(t-\tau_i), i=1,...,m,\\
x_{l}^{(\alpha)}(t)=u_l(t-\tau_l), l=m+1,...,n,
\end{array}
\label{e5}$$ where $\tau_i>0$ is the communication delay of agent $i$. Through a simple change we can get $$[\dot{X}_1(t),X_2^{(\alpha)}]^T=-\gamma[\Sigma_{i=1}^{m}L_{i}X^T(t-\tau_i), \Sigma_{i=m+1}^{n}L_{i}X^T(t-\tau_i)]^T
\label{e6}$$ where $L_i=E(i)L$, $E(i)$ represents matrix whose element of $(i, i)$ is 1 and the rest are 0s, $X_1(t)=[x_1(t),...,x_m(t)]$, $X_2(t)=[x_{m+1}(t),...,x_n(t)]$, $X(t)=[x_1(t),x_2(t),...,x_n(t)]$, and $L=\Sigma_{i=1}^{n}L_{i}$. Suppose that for any initial value of the system, the states of autonomous agents meet $\lim_{t\rightarrow\infty}(x_i(t)-x_k(t))=0$, for $i,k\in I$, then we call multi-agent systems asymptotically reach consensus. In this paper, we apply Laplace transform and frequency domain theory of the fractional-order operator to study the consensus of delayed compounded-order multi-agent systems with directed weighted topologies.
Coordination control for compounded-order multi-agent systems with communication-delays
=======================================================================================
[**Theorem 1**]{} Suppose that multi-agent systems are composed of $n$ independent agents whose connected network topology is directed and has a globally reachable node. Then, compounded-order multi-agent system (\[e6\]) with time delays can asymptotically reach consensus, if $$\tau_i<\frac{\pi}{2(2\gamma \bar{d})^{1/\alpha}}
\label{e7}$$ where $\bar{d}=\max\{d_i, i=1,...,n\}$, $d_i=\Sigma_{k=1}^{n}a_{ik}$.
[**Proof.**]{} Applying Laplace transformation to system (\[e6\]), the characteristic equation can be gotten $$\det(\left(\begin{array}{cc}sI_m\ \ & \\ & s^\alpha I_{n-m} \end{array}\right)+\gamma E(s)L)=0,
\label{e8}$$ where $I_m$ represents a unit matrix with $m$-dimensions, $E(s)=\mathrm{diag}\{e^{-\tau_1 s},...,e^{-\tau_n s}\}$. 0 is a single eigenvalue of the Laplacian matrix $L$ because the system has a globally reachable node from Lemma 1. Due to $\alpha>0$, the characteristic equation has a characteristic root $s=0$.
When $s\neq 0$, let $$\begin{aligned}
F(s)=(I_n+\gamma \left(\begin{array}{cc}s^{-1}I_m\ \ & \\ & s^{-\alpha} I_{n-m} \end{array}\right) E(s)L),\end{aligned}$$ the characteristic equation (\[e8\]) is equivalent to $F(s)=0$. Nextly, we will prove that all zero points of $F(s)=0$ have negative real parts. Let $$\begin{aligned}
G(s)=\gamma \left(\begin{array}{cc}s^{-1}I_m\ \ & \\ & s^{-\alpha} I_{n-m} \end{array}\right)E(s)L,\end{aligned}$$ according to the generalized Nyquist criterion [@desoer], if for $s=j\omega$, where $j$ is complex number unit, point $-1+j0$ is not surrounded by the Nyquist curve of $G(j\omega)$’s eigenvalues, then all zero points of $F(s)$ have negative real parts. Let $s=j\omega$, we can get $$G(j\omega)=\gamma \left(\begin{array}{cc}\omega^{-1} e^{-j\pi/2}I_m\ \ & \\ & \omega^{-\alpha}e^{-j\alpha\pi/2}I_{n-m} \end{array}\right)\ E(j\omega)L.
\label{e9}$$ In the following proof, Gerschgorin’s disc theorem will be applied to estimate the eigenvalues $\lambda(G(j\omega))$ of the matrix $G(j\omega)$. According to the Gerschgorin’s disc theorem, we have $$\lambda(G(j\omega))\in \bigcup_{i\in I} G_i,
\label{e10}$$
where for $i=1,...,m$, $$\begin{aligned}
\begin{array}{cl}
G_i=& \{\zeta\in C, |\zeta-\omega^{-1}\gamma d_{i}e^{-j(\omega\tau_i+\pi/2)}|\\
& \leq \omega^{-1}\sum_{k=1,k\neq i}^{n}|\gamma a_{ik}e^{-j(\omega\tau_i+\pi/2)}|\},
\end{array}\end{aligned}$$ and for $i=m+1,...,n$, $$\begin{aligned}
\begin{array}{cl}
G_i=& \{\zeta\in C, |\zeta-\omega^{-\alpha}\gamma d_{i}e^{-j(\omega\tau_i+\alpha\pi/2)}|\\
& \leq \omega^{-\alpha}\sum_{k=1,k\neq i}^{n}|\gamma a_{ik}e^{-j(\omega\tau_i+\alpha\pi/2)}|\},
\end{array}\end{aligned}$$ where $d_{i}=\sum_{k=1}^{n}a_{ik}$. After simple sorting it can be $$\begin{aligned}
\begin{array}{cl}
G_i=& \{\zeta\in C, |\zeta-\omega^{-1}\gamma d_{i}e^{-j(\omega\tau_i+\pi/2)}|\leq \omega^{-1}\gamma d_{i}\}, i=1,...,m,
\end{array}\end{aligned}$$ and $$\begin{aligned}
\begin{array}{cl}
G_i=& \{\zeta\in C, |\zeta-\omega^{-\alpha}\gamma d_{i}e^{-j(\omega\tau_i+\alpha\pi/2)}|\leq \omega^{-\alpha} \gamma d_{i}\}, i=m+1,...,n,
\end{array}\end{aligned}$$
When the Nyquist curve of the origin of the disc $G_i$ changing, the disc changes along with it. Next, we will prove that point $-a+j0 (a\geq 1)$ is not in every disc $G_i $.
For $i=1,...,m$, the changes of the disc $G_i$ is to be studied in the following. The origin of the disc $G_i$ is $\omega^{-1}\gamma d_{i}e^{-j(\omega\tau_i+\pi/2)}$, the radius of the disc $G_i$ is $\omega^{-1}\gamma d_{i}$. Let $$\begin{aligned}
\Delta=|-a+j0-\omega^{-1}\gamma d_{i}e^{-j(\omega\tau_i+\pi/2)}|^2-(\omega^{-1}\gamma d_{i})^2,\end{aligned}$$ there is $$\begin{aligned}
\Delta=a(a+2\omega^{-1}\gamma d_{i}\cos(\omega\tau_i+\pi/2)).\end{aligned}$$ When $\omega_c \tau_i+\pi/2=\pi$, there is $\cos(\omega_c\tau_i+\pi/2)=-1$. We can get $$\begin{aligned}
\Delta\geq a(a-2\omega_c^{-1}\gamma d_{i}),\end{aligned}$$ where $$\begin{aligned}
2\omega_c^{-1}\gamma d_{i}=2(\pi/(2\tau_i))^{-1}\gamma d_i.\end{aligned}$$ According to the conditions of the theorem $$\begin{aligned}
2(\pi/(2\tau_i))^{-1}\gamma d_i\leq 2((\pi/(2\tau_i))^{-\alpha}\gamma \bar{d}<1,\end{aligned}$$ from the hypothesis $a\geq 1$, we can get $$\begin{aligned}
\Delta>0.\end{aligned}$$ Then, it gets, for $i=1, ..., m$, $$\begin{aligned}
|-a+j0-\omega^{-1}\gamma d_{i}e^{-j(\omega\tau_i+\pi/2)}|>\omega^{-1}\gamma d_{i}.\end{aligned}$$
By means of the same deduction, we can get, for $i=m+1, ..., n$, $$\begin{aligned}
|-a+j0-\omega^{-\alpha}\gamma d_{i}e^{-j(\omega\tau_i+\alpha\pi/2)}|>\omega^{-\alpha}\gamma d_{i}.\end{aligned}$$
When $a\geq 1$, the point $-a+j0$ is not in disc $G_i$. Thus the point $-1+j0$ is not surrounded by curves of eigenvalue $\lambda(G(j\omega))$ of matrix $G(j\omega)$. Therefore, all zero points of $F(s)=0$ have negative real parts.
Due to the equilibrium point of the system meeting $LX^*=0$, then $X^*=C[1,...,1]^T$ (where $C$ is a constant) is the eigenvector with the corresponding eigenvalue 0 of the Laplacian matrix $L$. Therefore, $\lim_{t\rightarrow\infty}x_{i}(t)=C$, and the system asymptotically reaches consensus.
[**Corollary 1.**]{} Suppose multi-agent systems are composed of $n$ independent agents, whose connection network topology is directed and symmetrical, and there is a global reachable node. Then compounded-order multi-agent system (\[e6\]) with time delays can asymptotically reach consensus, if $$\tau_i<\frac{\pi}{2(\gamma \rho_{L})^{1/\alpha}},
\label{e11}$$ where $\rho_L$ represents the spectral radius of matrix $L$ with $\rho_L=\max\{|\lambda_i|,i=1,...,n\}$ and $\lambda_i$ is the eigenvalue of the Laplacian matrix $L$.
[**Proof.**]{} According to theorem 1, the characteristic equation of the system is $$\begin{aligned}
\det(\left(\begin{array}{cc}sI_m & \\ & s^\alpha I_{n-m} \end{array}\right)+\gamma E(s)L)=0.\end{aligned}$$ Since the Laplacian matrix $L$ is symmetrical, there is orthogonal matrix $P$ satisfying $L=P\Lambda P^{-1}$, where $\Lambda=diag\{ \lambda_1,...,\lambda_n\}$. Because there is a global reachable point, we can know $Rank(L)=n-1$ and 0 is a single eigenvalue of matrix from Lemma 1. Therefore, the characteristic equation has a root $s=0$.
When $s\neq 0$, let $F(s)$ and $G(s)$ be same as the proof of Theorem 1, and let $$H(s)=\left(\begin{array}{cc}s^{-1}I_m & \\ & s^{-\alpha} I_{n-m} \end{array}\right)\mathrm{diag}\{e^{-\tau_is}, i=1,...,n\}.
\label{e12}$$
Let $s=j\omega$, we can get $$G(j\omega)=H(j\omega)\gamma L,
\label{e13}$$ where $$\begin{aligned}
\begin{array}{cl}
H(j\omega)&=\mathrm{diag}\{H_i(j\omega),i=1,...,n\}\\
&=\left(\begin{array}{cc}\omega^{-1}e^{-j\pi/2} & \\ & \omega^{-\alpha}e^{-j\alpha\pi/2} I_{n-m} \end{array}\right)\mathrm{diag}\{e^{-j\omega\tau_i}, i=1,...,n\}.
\end{array}\end{aligned}$$
Let $M=\mathrm{diag}\{M_i, i=1,...n\}$ where $M_i=\pi/(2\tau_i)$ (for $i=1,...,m$) and $M_l=((2-\alpha)\pi/(2\tau_l))^\alpha$ (for $l=m+1,...,n$). Matrix $MH(j\omega)=\mathrm{diag}\{M_iH_i(j\omega),\ i=1,...,n\}$ is a diagonal matrix where the Nyquist curve of its diagonal elements passes over point $-1+j0$. Suppose $\lambda(G(j\omega))$ is the eigenvalue of matrix $G(j\omega)$, we have $$\begin{aligned}
\begin{array}{cl}
\lambda(G(j\omega))& =\lambda(MH(j\omega)\gamma M^{-1/2}LM^{1/2})\\
& \in \rho(\gamma M^{-1/2}LM^{1/2}) \times Co(0\cup \{M_iH_i(j\omega),i=1,...,n\}),
\end{array}\end{aligned}$$ Where $\rho()$ represents the spectral radius of matrix, $Co(\xi)$ represents the convex hull of $\xi$. Because of $M_iH_i(j\omega)$ will passes over point $-1+j0$, point $-1+j0$ is included in convex hull $Co(0\cup \{M_iH_i(j\omega), i=1,...,n\})$. Since $$\begin{aligned}
\begin{array}{cl}
\rho(\gamma M^{-1/2}LM^{1/2})&= \rho(\gamma M^{-1/2}\Lambda M^{1/2})\\
& =\max\{|\gamma M_i^{-1}\lambda_i|,i=1,...,n\},
\end{array}\end{aligned}$$ according to the hypothesis condition $\tau_i<\frac{\pi}{2(\gamma \rho_{L})^{1/\alpha}}$, we can get $$\begin{aligned}
\begin{array}{cl}
\rho(\gamma M^{-1/2}LM^{1/2})&<1.
\end{array}\end{aligned}$$ Therefore, point $-1+j0$ is not included in $\rho(\gamma M^{-1/2}LM^{1/2}) \times Co(0\cup \{M_iH_i(j\omega),i=1,...,n\})$. That is, point $-1+j0$ is not included in the Nyquist curve of the eigenvalue of $G(j\omega)$. According to generalized Nyquist theorem [@desoer], the zero points of the characteristic equation have negative real parts. Therefore the multi-agent systems can asymptotically reach consensus, and $\lim_{t\rightarrow\infty}x_{i}(t)=C$.
[**Corollary 2.**]{} Suppose multi-agent systems are composed of $n$ independent agents, whose connection network topology is directed and symmetrical, and there is a global reachable node. Then compounded-order multi-agent system (\[e6\]) with time delays can asymptotically reach consensus with $\alpha=1$, if $$\gamma\tau_i<\pi/(2\lambda_{max}),
\label{e14}$$ where $\lambda_{max}$ is the maximum eigenvalue of matrix $L$.
[**Corollary 3.**]{} Suppose multi-agents are system composed of $n$ independent agents, whose connection network topology is directed and symmetrical, and there is a global reachable node. Then compounded-order multi-agent system (\[e6\]) can asymptotically reach consensus when $\alpha=1$ and time delays $\tau_i=\tau$, if $$2\gamma\tau<\pi/\rho,
\label{e15}$$ where $\rho$ represents Spectral radius of matrix $L$.
[**Remark 1.**]{} The consensus result in Corollary 3 for $\gamma=1$ is in accord with that in [@olfati].
Examples Simulations
====================
Suppose the system is composed of four dynamical agents (Fig. 1) with two integer-order agent systems (agnet1 and agent2) and two fractional-order agent systems (agent3 and agent4). The connection weights between individuals are $a_{21} = 0.7$, $a_{42} = 0.8$, $a_{31} = 0.9$, $a_{14} = 1$. The order of the fractional multi-agent dynamics is $\alpha=0.9$, through the topology of the system,we can get the adjacency matrix $$\begin{aligned}
A=\left( \begin{array}{cccc}
0& 0& 0& 1\\
0.7 & 0 & 0 & 0\\
0.9& 0& 0 & 0\\
0& 0.8 & 0& 0
\end{array}
\right).\end{aligned}$$
\[0.25\][]{}
According to Theorem 1, we can get the relationship between the system control gain and the upper bound of communication delays (Fig. 2). With the help of Fig.2, we can select the control gain according to the communication delay of the system, or decide the upper bound of communication delays by means of the control gain of the system, to make the system meet the condition of reaching consensus. Suppose the communication delay is $\tau=0.6s$, the system control gain should be selected as $\gamma \leq 1.19$ from Fig. 2; suppose the system control gain $\gamma=1$, we can obtain that the upper bound of communication delays is $0.7s$ from Fig.2.
Assume the communication delay of multi-agent systems is 0.6s and the system control gain $\gamma=1$ in simulation, we set the expect objective at 0.5, consensus can be asymptotically reached (Fig. 3) through compounded-order coordination algorithm.
{height="3in" width="4in"}
[Fig. 2: Relationship between the control gain and the upper bound of communication delays. ]{}
{height="3in" width="4in"}
[Fig. 3: Movement trajectories of the multi-agent systems with delay 0.6s. ]{}
Assume the communication delay of multi-agent systems is 0.7s and the system control gain $\gamma=1$ in simulation, consensus of compounded-order multi-agent systems can not be reached (Fig. 4).
{height="3in" width="4in"}
[Fig. 4: Movement trajectories of the multi-agent systems with delay 0.7s. ]{}
Conclusions
===========
This paper studies distributed coordination of integer-fractional-compounded-order multi-agent systems with communication delays. Consensus of multi-agent systems with directed network topology is studied through the stability theory of frequency domain, and the consensus conditions for compounded-order delayed multi-agent systems are presented. The relationship between the control gain of multi-agent systems and the upper bound of time delays is derived. Suppose the orders of the fractional dynamical systems are all 1, the extended conclusion in this paper is the same with ordinary integer order system. In the following work, research of the robust stability of integer-fractional-compounded-order multi-agent systems will be carried out.
Acknowledgements {#acknowledgements .unnumbered}
================
This research is supported in part by the State Key Development Program for Basic Research of China (Grant No. 2012CB720003), the National Natural Science Foundation of China (Grant No. 91016004, 61273152, 61127007) and the Natural Science Foundation of Shandong Province of China (No. ZR2011FM017).
[99]{} I. Podlubny, Fractional Differential Equations, Academie Press, New York, 1999. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2001.
S. Ladaci, J.J. Loiseau, A. Charef, Fractional order adaptive high-gain controllers for a class of linear systems, Communications in Nonlinear Science and Numerical Simulation 13, 707-714, 2008.
C. Hwanga, Y.C. Cheng, A numerical algorithm for stability testing of fractional delay systems, Automatica 42, 825-831, 2006. H.S. Ahn, Y.Q. Chen, Necessary and sufficient stability condition of fractional-order interval linear systems, Automatica 44(11):2985-2988, 2008.
Y. Li, Y.Q. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mitta-Leffler stability, Computers and Mathematics with Applications 24, 1429-1468, 2009. J. Sabatier, M. Moze, C. Farges, LMI stability conditions for fractional order systems, Computers and Mathematics with Applications 59, 1594-1609, 2010. J.G. Lu, Y.Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order: the case $0<\alpha<1$, IEEE Transactions on Automatic Control 55(1):152-159,2010.
C. W. Reynolds, Flocks, herds, and schools: a distributed behavioral model, Computer Graphics, 21(4): 25-34, 1987. T. Vicsek, A. Cziroo’k, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letter, 75(6): 1226-1229, 1995. R Olfati-Saber, R M Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49(9): 1520-1533, 2004. W Ren, R W Beard, E M Atkins. Information consensus in multivehicle cooperative control: Collective group behavior through local interaction. IEEE Control Systems Magazine, 27(2): 71-82, 2007. Shihua Li, Haibo Du, Xiangze Lin. Finite-time consensus algorithm for multi-agent with double-integrator dynamics, Automatica, 47(8): 1706-1712, 2011. Peng Lin, Yingmin Jia. Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies. Automatica, 45(9): 2154-2158, 2009. Junyan Yu, Long Wang. Group consensus in multi-agent systems with switching topologies and communication delays. Systems $\&$ Control Letters, 59(6): 340-348, 2010. Hongyong Yang, Zhenxing Zhang, Siying Zhang. Consensus of Second-Order Multi-Agent Systems with Exogenous Disturbances, International Journal of Robust and Nonlinear Control, 2011, 21(9): 945-956. Yu-ping Tian, Chen-Lin Liu. Consensus of multi-agent systems with diverse input and communication delays, IEEE Trans. on Automatic Control, 2008, 53(9): 2122-2128.
Wei Ren, Yongcan Cao. Distributed coordination of multi-agent networks, Springer-Verlag, London, 2011. Yongcan Cao, Yan Li, Wei Ren, YangQuan Chen. Distributed coordination of networked fractional-order systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40(2): 362-370, 2010. Yongcan Cao, Wei Ren. Distributed coordination for fractional-order systems: dynamic interaction and absolute/relative damping, Systems $\&$ Control Letters, 43(3-4): 233-240, 2010. C.A. Desoer, Y.T Wang. On the generalized Nyquiststability criterion. IEEE Transactionson Automatic Control, 25, 187-196, 1980.
|
---
abstract: 'Traffic Jam has been a daily problem for people in Jakarta which is one of the busiest city in Indonesia up until now. Even though the official government has tried to reduce the impact of traffic issues by developing a new public transportation which takes up a lot of resources and time, it failed to diminish the problem. The actual concern to this problem actually lies in how people move between places in Jakarta where they always using their own vehicle like cars, and motorcycles that fill most of the street in Jakarta. Among much other public transportations that roams the street of Jakarta, Buses is believed to be an efficient transportation that can move many people at once. However, the location of the bus stop is now have moved to the middle of the main road, and it’s too far for the nearby residence to access to it. This paper proposes an optimal location of optimal bus stops in West Jakarta that is experimentally proven to have a maximal distance of 350 m. The optimal location is estimated by means of mean shift clustering method while the optimal routes are calculated using Ant Colony algorithm. The bus stops locations rate of error is 0.07% with overall route area of 32 km. Based on our experiments, we believe our proposed bus stop plan can be an interesting alternative to reduce traffic congestion in West Jakarta.'
address:
- '*System Information Dept.*, Multimedia Nusantara University, Banten, Indonesia'
- '*System Information Dept.*, Multimedia Nusantara University, Banten, Indonesia'
author:
- Kenny Supangat
- Yustinus Eko Soelistio
title: 'Bus Stop’s Location and Bus Route Planning Using Mean Shift Clustering and Ant Colony in West Jakarta'
---
Introduction
============
According to the new start-stop index created by Castrol motor oil company \[1\], Jakarta has become number one city with the worst traffic in the world. The fact that Jakarta already has quite a variety of transportation like trains, angkot (public car established by private company), public bus (runs by a private company), ojek (a motorcycle runs by an individual person with low rate of fare), and the newest addition to it is the Busway (BRT-Bus Rapid Transit established by the Official Government of Jakarta) are still not proven to be an optimal solution for solving the traffic problem. Moreover, the actual issue that shows a huge contribution to the traffic problems was the increasing number of a private vehicle in Jakarta streets \[2\].The survey presented by the Statistic Department of Central Jakarta illustrate the number of a private two-wheel motorcycle is filling most of the roads with roughly 13.084.372 units or 74.67% on the road. In contrast, other vehicles especially buses only take 2.07% part of the population, while the other like private cars, trucks, buses, and governments vehicles are each takes the percentage of 18.64%, 3.84%, and 0.79% respectively \[2\].
The private vehicle owners mostly absorb the total usage of public transportation, especially the private car and motorcycle owner. The bus is one of a few public transportation that get the least attention from people due to some reason presented in the early survey. The survey asked 46 respondents (24 male and 22 female) about the transportation they usually use with their own reason. The survey shows that 63% (29 people) of the respondents don’t mainly use bus because “the bus is too far away from my home”, the other reasons are “there are no bus stop nearby my residence” with 27%, and just 10% of the respondent felt awfully uncomfortable with the bus condition.
According to the data found during the survey, this research will be focusing on finding the optimal location for bus stops that will be located relatively near to the residential area, and the shortest route for the newly discovered bus stops location in this research.
Data Acquisition
================
The residential area located in West Jakarta holds the largest number of houses in Jakarta region \[3\]. Moreover, the region includes several well-known residential districts in the center of West Jakarta area (i.g. Green Garden, Kembangan, Kedoya district, and more), along with more than five thousand houses outside these districts. A large portion of West Jakarta have been chosen as the designated area for this experiment, one of the areas is shown in Figure 1 where the houses serve as the main data is marked as red pins in the figure.
![\[label\]Example of part of West Jakarta on Google Maps. Red marks are annotated houses.](Capture.jpg)
The file containing the whole data will later serve as data input to finding the optimal bus stop locations by using mean shift algorithm implemented in python version 2.7.11.
Methodology
===========
Bus Stop’s Location Estimation
------------------------------
We apply mean shift \[4-6\] \[11\] (Equation 1) clustering algorithm to estimate the optimal location of the bus stops. The location of the bus stops are represented by the cluster center calculated by the mean shift, and the radius of the bus stop’s service areas are the bandwidth of the clusters.
$$\label{eq:meanShift}
m(x)=\frac{\sum_{s\in S}K(s - x)s}{\sum_{s\in S}K(s - x)}$$
Where the difference of m(x) - x is called as mean shift in \[4\], x is initial centroid estimates (the longitude and latitude of inputs), S is the longitude and longitude of labeled houses, and K is the kernel used in the method (Gaussian kernel with $K(s-x)=e^{-c||s-x||^2}$). We also using a bandwidth or radius that represent how far is the service area of the new bus stop will be, in this case, we will set an arbitrary range of bandwidth of 500m as the default bandwidth for the first attempt in this experiment.
Bus Stops Routes Optimization
-----------------------------
Previous studies have shown that ant colony can be used to solve optimal route problem (Bedi, 2007 \[13\]; Alves, 2010 \[14\]).The bus stops routes are optimized using ant colony algorithm \[7\] \[8\] \[12\] following:
$$P_{ij}=\frac{(\tau_{ij}^\alpha)(\eta_{ij}^\beta)}{\sum(\tau_{ij}^\alpha)(\eta_{ij}^\beta)}$$
where ,$P-ij$. is the probability of choosing the state, $T-ij$. is the intensity of the pheromone trail for each interstate, ,$N-ij$. is the visibility of a solution that would be selected by the ants. For the pheromone part, $\alpha$ is a parameter controlling the intensity of the pheromone trail where $\alpha\geq0$, and $\beta$ is a parameter controlling the visibility where $\beta\geq0$ [@Haryanto].
The new pheromone trail $(T-ij)$ is recalculated in each state by:
$$\tau_{ij}=(1 - \rho)\tau_{ij} + \rho \Delta \tau_{ij}$$
where $\rho$ is the constant evaporation of pheromone trail $0 >\rho>1$ [@Haryanto].
The ant colony algorithm is implemented in 2013 version of Matlab. In our implementation we use $\alpha=4$ and $\beta=1$ as the main parameter of both the intensity of the pheromone trail and ants visibility. To calculate the pheromone trail, we set the $\rho=0.15$.
Result and Analysis
===================
Bus Stops Location
------------------
We estimate the location of bus stops using an arbitrary 500 meters bandwidth on Equation \[eq:meanShift\]. It produces eight cluster centers that will be used as the new location for the new bus stop location (Figure \[fig:bandwidth500\]. Unfortunately, the result is not quite reliable, because in the real situation every road in the street is full of curves and not just a straight road.
To find the amount of error produced by the first result, a customized google maps API code is made to calculate how many houses miss the actual bus stop service radius or we can say further than 500m from house to bus stop. There are 594 houses or 7.72% out of 7962 total houses that miss the bus stop service radius, and also the other error information is the maximum, minimum, and average range of error of each house.
After seeing the error result, we decide to use various numbers of arbitrary bandwidth radius ranging from 450m, 400m, 350m, 300m, and 250m. From each of the result that can be seen in Table \[table:meanShift\], and we choose one result that has the least error as the final solution.
Bandwidth 500 m 450 m 400 m 350 m 300 m 250 m
------------------- --------- --------- --------- --------- --------- ---------
Total Error 594 22 57 6 10 7
Error Percentages 7.72% 0.28% 0.74% 0.07% 0.13% 0.09%
Max Error 0.4 km 0.07 km 0.08 km 0.09 km 0.11 km 0.1 km
Min Error 0.01 km 0.01 km 0.01 km 0.01 km 0.01 km 0.01 km
Median Error 0.09 km 0.03 km 0.02 km 0.02 km 0.03 km 0.07 km
Bus Stops Spawned 9 14 14 19 22 33
: \[table:meanShift\]Summary of mean shift result. The best fit radius is 350 m indicated by the lowest error percentage.
From the result illustrated by the Table \[table:meanShift\], shows a various result from different bandwidth (radius). We pick the 350 m bandwidth because the result produces least error percentage with 0.07%, i.g. six houses miss the boundary of 350 m.
From the result using 350 m bandwidth in Figure \[fig:bandwidth350\], we can see that there is 19 bus stop generated in the map, compare with Figure \[fig:bandwidth500\] that use 500 m bandwidth which only generates eight bus stops with higher bandwidth radius. This means the result of 350 m bandwidth can reach out smaller area than the result from 500 m bandwidth.
The placement of the new bus stop and the existing bus stops can be seen in Figure \[fig:newButStops\]. The blue box represents the new bus stop, and the red box represents the old bus stop. We can see there are only 4 bus stops marked with a red box that currently operating and located in the middle of the main road far from the residential area . In the other hand, there are 19 new bus stops marked with a blue box scattered around the places so that many people can access the bus stop more easily across the residential area.
Bus Route
---------
From the chosen result before, the geographical location of the new bus stop locations will be inserted into the source code as the input by using Matlab programming tools. There are several main inputs used in the like the number of ants, a number of cities (in here will be served as bus stop geographical location) and a number of iteration or repetition because the ant colony works in random and has many different kinds of possible outputs [@Dorigo3].
As presented in Figure \[fig:finalRoutes\] are the final result of ant colony method where the route leads from bus stop from number 1 through 19 and back to bus stop number 1. The visualization of the route is created using Google Maps feature.
The 26.44 km route comes from several run attempts of changing the parameter of ant colony method. From Table 2, 3, 4, and 5 are the attempt to find the shortest by changing some parameter in ant colony method which is number of ants, and number loop or attempt that the ants going to do to find the shortest path in that attempt.
Conclusion
==========
This study has successfully determined the optimum solution for bus stop locations and its sub-optimal route. We prove that mean shift and ant colony algorithms can handle this particular problem, and should the case where the area is expanded further. And based on this experiments, we believe our proposed bus stop plan can be an interesting alternative to reduce traffic congestion in West Jakarta. Next we should expand the area of interest to cover wider Jakarta area and calculate more optimize bandwidth to find best fit radius.
References {#references .unnumbered}
==========
[9]{}
Toppa, Sabrina, (2015, February). These Cities Have The Worst Traffic in the World, Says a New Index. TIME, Retrieved from http://time.com/3695068/worst-cities-traffic-jams/?iid=sr-link1. Statistic Department Center of DKI Jakarta Province, “Transportation Statistic of DKI Jakarta 2015”, DKI Jakarta BPS Province, 2015 StreetDirectory, “Komplek Perumahan in Jakarta Barat”. Retrieved from http://www.streetdirectory.co.id/indonesia/jakarta/landmark/zone/jakarta+barat/komplek+perumahan/. Fukunaga, Keinosuke, and Larry D. Hostetler. “The estimation of the gradient of a density function, with applications in pattern recognition.”Information Theory, IEEE Transactions on 21.1 (1975): 32-40. Cheng, Yizong. “Mean shift, mode seeking, and clustering.” Pattern Analysis and Machine Intelligence, IEEE Transactions on 17.8 (1995): 790-799. Cheng, Yizong, and King-Sun Fu. “Conceptual clustering in knowledge organization.” Pattern Analysis and Machine Intelligence, IEEE Transactions on 5 (1985): 592-598. Dorigo, Marco, et al., eds. Ant Colony Optimization and Swarm Intelligence: 6th International Conference, ANTS 2008, Brussels, Belgium, September 22-24, 2008, Proceedings. Vol. 5217. Springer, 2008. Dorigo, Marco, and Thomas Stützle. “Ant colony optimization: overview and recent advances.” Techreport, IRIDIA, Universite Libre de Bruxelles (2009). Dorigo, Marco, and Luca Maria Gambardella. “Ant colonies for the travelling salesman problem.” BioSystems 43.2 (1997): 73-81. Haryanto, Ardy Wibowo, Adhi Kusnadi, and Yustinus Eko Soelistio. “Warehouse Layout Method Based on Ant Colony and Backtracking Algorithm.” arXiv preprint arXiv:1508.04872 (2015). Comaniciu, Dorin, and Peter Meer. “Mean shift: A robust approach toward feature space analysis.” Pattern Analysis and Machine Intelligence, IEEE Transactions on 24.5 (2002): 603-619. Dorigo, Marco, and Luca Maria Gambardella. “Ant colony system: a cooperative learning approach to the traveling salesman problem.”Evolutionary Computation, IEEE Transactions on 1.1 (1997): 53-66. Bedi, Punam, et al. “Avoiding traffic jam using ant colony optimization-a novel approach.” Conference on Computational Intelligence and Multimedia Applications, 2007. International Conference on. Vol. 1. IEEE, 2007. Alves, Diogo, et al. “Ant colony optimization for traffic dispersion routing.” Intelligent Transportation Systems (ITSC), 2010 13th International IEEE Conference on. IEEE, 2010.
|
---
abstract: 'The heating of the intergalactic medium in the early, metal-poor Universe may have been partly due to radiation from high mass X-ray binaries (HMXBs). Previous investigations on the effect of metallicity have used galaxies of different types. To isolate the effects of metallicity on the production of HMXBs, we study a sample consisting only of 46 blue compact dwarf galaxies (BCDs) covering metallicity in the range 12+log(O/H) of 7.15 to 8.66. To test the hypothesis of metallicity dependence in the X-ray luminosity function (XLF), we fix the XLF form to that found for near-solar metallicity galaxies and use a Bayesian method to constrain the XLF normalization as a function of star formation rate (SFR) for three different metallicity ranges in our sample. We find an increase by a factor of 4.45 $\pm$ 2.04 in the XLF normalization between the metallicity ranges 7.1-7.7 and 8.2-8.66 at a statistical significance of 99.79 per cent. Our results suggest that HMXB production is enhanced at low metallicity, and consequently that HMXBs may have contributed significantly to the reheating of the early Universe.'
author:
- |
S. Ponnada$^{1}$[^1], M. Brorby$^{1}$[^2], P. Kaaret$^{1}$\
$^{1}$Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242\
bibliography:
- 'MyRefs2.bib'
title: 'Effects of Metallicity on High Mass X-ray Binary Formation'
---
\[firstpage\]
galaxies: dwarf galaxies, starburst — X-rays: galaxies
Introduction {#sect:intro}
============
Around 300,000 years after the Big Bang, the mostly hydrogen plasma that permeated all of space had cooled enough to combine and form neutral atoms. As neutral matter fell into the primordial dark matter gravitational wells, the first stars and galaxies formed and began to heat and ionize the surrounding baryonic matter [@Barkana2001]. @Mirabel2011 proposed that high-mass X-ray binaries (HMXBs) in star-forming galaxies were the main source of X-rays in the early universe. Early X-ray binaries heated the intergalactic medium, producing a strong impact on the 21-cm hydrogen recombination signals that serve as primary diagnostics of the physics of the epoch of reionization, as well as impacting the formation and structure of galaxies in the early universe [@Artale2015; @Fialkov2017; @Madau2017].
HMXBs have been found to have populations which depend on the star formation rate (SFR) [@Griffiths1990; @David1992; @Ranalli2003; @Grimm2003; @Kaaret2008; @Mineo2012a], but these samples consist of well-evolved galaxies with near-solar metallicities. The stars and galaxies during the time of reionization had low metallicities. However, direct studies of the sources that produced X-rays in the early universe are currently impractical due to their high redshifts. [@Kunth2000] have suggested that blue compact dwarf (BCD) galaxies may be local analogs of the X-ray producing, metal-deficient galaxies found in the early universe.
Models [e.g., @Dray2007] and studies [@Mapelli2010; @Kaaret2011; @Prestwich2013; @Brorby2014] have shown enhanced X-ray binary formation in these low metallicity galaxies. There is now growing evidence which shows that the production of X-ray sources, HMXBs in particular, increases as metallicity decreases [e.g., @Mapelli2010; @Kaaret2011; @Prestwich2013; @Basu-Zych2013; @Brorby2014; @Brorby2016; @Douna2015]. Given that galaxies at very high redshifts have lower metal content, one would expect a larger number of HMXBs in early galaxies than would be predicted using the relation found for nearby, near-solar metallicity galaxies.
Using a sample consisting predominantly of BCDs, [@Prestwich2013] showed that the number of the brightest HMXBs, or ultraluminous X-ray sources (ULX), in star-forming galaxies is enhanced for these extremely metal-poor galaxies by a factor of 7 $\pm$ 3. In a comparative sample of near and sub-solar metallicity galaxies, all of which were spiral galaxies, they found no significant increase in the number of ULXs relative to SFR. A similar study by [@Douna2015] also measured HMXB populations across a large metallicity range and found enhancement at lower metallicities, but they too used different galaxy types at low and high metallicities (BCDs and spirals, respectively). For a review, see @Kaaret2017.
These observational results match predictions from simulations done by [@Linden2010] who showed a dramatic increase in bright HMXBs below 20 percent solar metallicity. [@Linden2010] explain that this bright HMXB population increase is due to a relative enhancement in the number of binaries accreting through Roche lobe overflow (RLO) rather than wind accretion systems. RLO is the dominant accretion mechanism for close binaries and produces brighter HMXBs than wind accretion, which is more common in binaries with longer orbital periods. A separate set of simulations, done by [@Fragos2013a], showed that the $L_X$/SFR increased by an order of magnitude going from solar metallicity to 10 percent the solar value.
Although many attribute the evolution of the X-ray luminosity function (XLF) to metallicity effects, there may also be effects correlated to the total stellar mass of the galaxy, star formation history, or gas mass fraction. However, no studies have attempted to test the evolution of the total luminosity and number of HMXB across a broad range of metallicity while maintaining the uniformity of a single galaxy type. Here, we use a single galaxy type (BCDs) across a large range of metallicities to test the metallicity dependence more strictly by eliminating most of these other variables. We study a large sample of BCDs, varying in metallicity by almost two orders of magnitude, to test for metallicity dependence of the XLF and $L_X$/SFR relations. In Section 2, we discuss our sample selection criteria. Section 3 describes our X-ray image analysis methods, and Section 4 details our procedures for analyzing UV and IR images to determine star formation rates. In Section 5, we discuss the Bayesian method we use for testing XLF evolution over metallicity, our analysis of HMXB formation evolution, and the evolution of $L_X$/SFR over metallicity. We conclude with a summary and discussion in Section 6.
The Sample
==========
Defined by optically blue continuua in their spectra, BCDs exhibit strong recent star formation, indicating similar star formation histories. Their recent star formation activity would also indicate that their X-ray binary populations are dominated by HMXBs [@Colbert2004; @Fragos2013a]. Being dwarf galaxies, they also have low stellar masses, with the originally defined magnitude cut-off being M$_{B}$ $>$ -18, thereby providing the expectation that BCDs do not range drastically in mass [@Thuan1981]. The gas mass fraction of BCDs is typically higher than 30 per cent [@Zhao2013]; however, [@Thuan2016] found that the gas mass fraction appears to decrease with metallicity. @Thuan2016’s findings should be taken into consideration for our sample, however, we note that their sample did not consist of strictly BCDs.
We expand a set of 25 BCDs studied in [@Brorby2014], hereafter B14, by selecting BCDs over all metallicities up to a distance of 60 Mpc. Cross-referencing the BCD population defined by [@Izotov2007] and the NED catalogue with the *Chandra* archive, we find a total of 21 additional BCDs with *Chandra* observations, *GALEX* observations, and published metallicities. We add this to our sample for an overall sample size of 46 BCDs.
For the final sample, we determine star formation rates using *GALEX* (UV) data following the methods used in previous studies, B14 and @Brorby2016, hereafter B16. We obtain metallicities from published values, all of them obtained through the direct-temperature method, with an exception. The distance and metallicity measurements from reported values in the literature are listed in Table \[tab:sample\].
Analysis {#sect:Analysis}
========
\[subsect:observations\] Following B14, we analyze observations carried out with ACIS-S3, one of the back-illuminated chips aboard *Chandra*, where target galaxies are close to the chip’s aimpoint. Reprocessing level 1 event files using the latest versions of CIAO (4.9) and CALDB (4.7.7), we locate X-ray sources within the 0.5$-$8 keV band using the CIAO tool `wavdetect`. Using a Python 2.7 script and the `ciao.contrib.runtool`, we automate the data pipeline process. Running the CIAO tool `mkpsfmap` with an enclosed counts fraction (ECF) of 90 per cent, we use the resulting PSF map for the `wavdetect` tool, with pixel scales set to the $\sqrt{2}$ series from 1 to 8. The significance threshold (`sigthresh`) is set to $10^{-6}$, resulting in less than one false detection per image. We use `maxiter` = 10, `iterstop` $= 0.00001$, and `bkgsigthresh` = 0.0001.
The source list generated by `wavdetect` is used in the background-deflaring procedure using the CIAO tool `deflare`, which first excludes source regions and proceeds to scan the event file for time periods where the background is at the mean count value. From this, good time intervals (GTIs) are extracted to be used in filtering the event file when calculating fluxes.
To determine whether the sources found using `wavdetect` are within the target galaxy, we construct $D_{25}$ ellipses using positions and dimensions in the HyperLeda[^3] data base and the CIAO tool `dmmakereg`, which takes position parameters and dimensions to create region files. Before creating the region file, we increase the dimensions of our ellipses to cover an extra 200 pc to account for binaries moving from their formation regions, and an additional 1.18 arcsec to account for positional error from HyperLeda and *Chandra*’s astrometric error – following the procedure of B14.
We calculate fluxes using the CIAO tool `srcflux`, which converts source counts to fluxes and uses point spread function models to correct source and background regions. The input parameters for this tool are the source list, the neutral hydrogen column density (found using `Colden`[^4]), the photon index ($\Gamma = 1.7$), point spread function method (’arfcorr’), and the energy range ($0.5 - 8$ keV). Using `srcflux`, we calculate unabsorbed fluxes for the given energy interval with an absorbed power law model. The fluxes are later converted to luminosities using the distances to the target galaxies. X-ray source photometry information is listed in Table \[tab:Xray\].
We follow the procedure of B14, to determine the minimum number of counts in the 0.5-8 keV range for a 95 per cent probability threshold of source detection. We refer to this as the completeness number. From this, we find completeness luminosities which are the luminosities corresponding to the completeness number for the aforementioned likelihood of source detection. This involves using relations given in @Zezas2007 for the probability of detecting a source with a certain number of counts over a background with a given number of background counts per pixel. From this calculation, we find that the completeness number is 9. Using this minimum number, we utilize the `ChandraPIMMS`[^5] tool and $\log N - \log S$ curves of @Georgakakis2008 to determine the minimum fluxes for source detection and the expected number of background sources respectively. We note that in Equation 2 of @Georgakakis2008, the terms with a coefficient of K$^{'}$ must be adjusted to have minus signs. We convert these minimum fluxes to ’completeness’ luminosities, and hereafter refer to the completeness luminosity as L$_\text{min}$.
Star Formation Rates from *GALEX* and *Spitzer*
===============================================
To determine star-formation rates with UV measurements, we obtain images corresponding to each galaxy from the *GALEX*[[^6]]{} archive in the far-ultraviolet (FUV) and near-ultraviolet (NUV). To find the respective SFRs, we follow the steps taken in B14 and B16, first extracting count rates for each galaxy. By using the CIAO tool `dmmakereg`, we use our previous method of creating $D_{25}$ ellipses and adjust position and angles individually for the *GALEX* images, using them as source regions. We extract background-subtracted FUV and NUV count rates for the galaxies by using source-centered annular ellipses with areas eight times that of the respective $D_{25}$ region and the CIAO tool `dmstat`. We then follow the relations given in B14 between count rates, fluxes, and luminosities to subsequently determine the star-formation rates in each UV band.
As the SFR determined by the NUV component does not account for reprocessing of light by dust, we also determine IR components of the SFR for each galaxy using *Spitzer* archive[^7] images and following the procedure of B14.
In Section 3.3.3 of B14, a correlation was found between the FUV SFR measure of @Hunter2010 and the NUV+IR method used in @Mineo2012a with a slope of 1.23 $\pm$ 0.11. The correlation included galaxies with FUV SFRs up to 0.1 .
We analyze this correlation between the two SFR methods for our sample. We find that all of the galaxies lacking IR observations lie within a FUV SFR of 0.1 . Using 0.1 as a cutoff FUV star formation rate for the linear regression for the galaxies with IR observations, we use this new correlation to determine NUV+IR SFRs for the remaining 18 galaxies. We remove two galaxies, Mrk 996 and NGC 5253, from the regression as they were clear outliers to the observed trend. The new correlation, plotted in red in Figure \[fig:SFRreanalysed\], has a slope of 1.39 $\pm$ 0.33, which is consistent with the slope of 1.23 $\pm$ 0.11 found in B14. Our UV and IR measurements and corresponding SFRs are detailed in Table \[tab:sfr\].
Results {#sect::Results}
=======
![ vs for the sample galaxies with Spitzer images (black circles). The correlation found for galaxies with < 0.1 (red line) has a slope of 1.39 $\pm$ 0.33. Shaded regions represent uncertainties in slope values.[]{data-label="fig:SFRreanalysed"}](SFRcorr_corrected_0909.png){width="48.00000%"}
XLF Normalization {#subsect::XLF}
-----------------
For this study, we aimed to determine how the XLF changed with metallicity. In order to do so, we binned our galaxies by metallicity, with a cutoff of $12+\log_{10}({\rm O/H})$ = 7.7 for low metallicity, 7.7 < $12+\log_{10}({\rm O/H})$ < 8.2 for intermediate metallicity, and $12+\log_{10}({\rm O/H})$ $\geq$ 8.2 for high metallicity. Our limit for low metallicity is consistent within rounding for the limit of 7.65 used by @Prestwich2013 for extremely metal-poor galaxies. We chose to bin the intermediate and high metallicity galaxies with the specified cutoff values to maintain a similar number of galaxies with source detections as the low metallicity bin. The bins and their compositions are detailed in Table \[tab:binsxlf\].
To determine XLF normalization evolution with metallicity, we employ Bayesian methods described in B14. The XLF has the form $$\frac{dN}{dL_{38}} = q s L{ \begingroup
\settowidth{\@tempdima}{\textsubscript{38}} \settowidth{\@tempdimb}{\textsuperscript{-$\alpha$}} \ifdim\@tempdima<\@tempdimb
\setlength{\@tempdima}{\@tempdimb} \fi
\makebox[\@tempdima][l]{ \rlap{\textsubscript{38}}\textsuperscript{-$\alpha$}} \endgroup}$$ where q is the normalization constant, s is the SFR (M$_\odot$ yr$^{-1}$), $L_{38}$ is the $L_X$/10$^{38}$, and $\alpha$ is the power-law index [@Grimm2003]. We take the gamma distribution as the conjugate prior of the Poisson distribution. The gamma distribution is defined by $$\label{gammaDist}
\text{GAMMA}(q;X,B) = \frac{B^{X} q^X e^{-qB}}{(X-1)!},$$ where $X = \sum x_i$, $x_i$ is the number of observed sources in the $i^\text{th}$ galaxy, and $B = \sum \frac{s_i}{1-\alpha}\left(L_\text{cut}^{1-\alpha} - L_\text{min,\ i}^{1-\alpha}\right)$, where $s_i$ is the SFR (M$_\odot$ yr$^{-1}$) of the i$^{th}$ galaxy. The gamma distribution is the posterior probability distribution, , for the XLF normalization q given the data D, where the likelihood function is given by the Poisson distribution *P(D|q)*. L$_\text{cut}$ is the cut-off luminosity for the XLF, 110 $\times$ 10$^{38}$ erg s$^{-1}$ [@Mineo2012a]. D represents the number of sources, and the minimum luminosity values and SFRs for the binned galaxies are hyperparameters of the prior.
As in B14, GAMMA(q;0,0) is initially taken as a placeholder since it is the “know-nothing” prior. The case of choosing a uniform prior could bias our result as this would be proportional to a prior of GAMMA(q;1,0), which is equivalent to adding a galaxy with a SFR of 0 containing an X-ray source. To this end, we simulate a random sample of 46 galaxies over various SFR ranges and perform the Bayesian calculation using a uniform prior as well as the “know-nothing” prior. We find that at low SFRs, the “know-nothing” prior better matches the data, and high SFRs, the results of both calculations converge as expected due to more data.
In this method, the XLF normalization parameter and its error are given by the mean (*X/B*) and standard deviation $\sqrt{(\textit{$X/B^{2}$})}$ of the gamma distribution. These values are calculated for each metallicity range in the final sample, which is described in Table \[tab:binsxlf\]. The quoted uncertainties do not take into account uncertainty on star formation rate.
Our results for the posterior probability distributions for each metallicity bin are shown in Figure \[fig:xlf\]. We see a significant increase in the XLF normalization of the low metallicity range, which has a mean metallicity of $12+\log_{10}({\rm O/H})$ = 7.48, relative to the high metallicity ranges, which have mean metallicities of $12+\log_{10}({\rm O/H})$ = 7.97 and 8.37 respectively, indicating evolution with metallicity. Our XLF normalization value for the low metallicity bin of 10.38 $\pm$ 3.83 also is considerably increased with respect to the @Mineo2012a value of 1.49 $\pm$ 0.07 for near-solar metallicity galaxies.
We analyze the significance of these differences by simulating a distribution of a random variable defined as the difference between two random variables drawn from the two gamma distributions, i.e., we test the null hypothesis of the higher metallicity bin having a greater “q” value than the low-metallicity bin. To this end, the test statistic is constructed by subtracting the higher metallicity parameter from the low-metallicity parameter. Since we are testing only whether the higher metallicity value is greater, we opt for the one-sided test. The fraction of values that fall below zero corresponds to the probability that the difference between the two normalization parameters is less than zero. We find the probability of q$_{int}$ and q$_{high}$, the normalization parameters of the intermediate and high-metallicity bins, being greater than the low-metallicty bin to be true at confidences of 2.2 $\times$ 10$^{-4}$ and 2.1 $\times$ 10$^{-3}$ respectively. Alternatively, this can be stated as a 99.98 and 99.79 per cent confidence that the null hypotheses can be rejected. Our normalization value for the low metallicity galaxies is greater than the value found by @Mineo2012a for near-solar metallicity galaxies at a confidence of 99.994 per cent.
The normalization parameters for the intermediate and high metallicity bins do not differ from the Mineo value significantly. This suggests that HMXB formation is not influenced by the nature of the host galaxy. Futhermore, @Prestwich2013 found a break in the number of ULXs normalized to SFR and @Douna2015 found a break in the number of HMXBs normalized to SFR and in the XLF at $12+\log_{10}({\rm O/H})$ of $\approx$ 8.0. In conjunction with our result, this indicates that the XLF enhancement seems to diminish in the metallicity range 7.7-8.0.
------------ --------------------------- ------------------------------------ -------------------------- --------------------------------------------------
Bin Number $12+\log_{10}({\rm O/H})$ Number of galaxies with detections Number of non-detections XLF Normalization *q* ($M_\odot^{-1} \text{yr}$)
1 7.1-7.7 8 18 10.38 $\pm$ 3.83
2 7.7-8.2 8 6 1.69 $\pm$ 0.32
3 8.2-8.66 5 1 2.33 $\pm$ 0.63
------------ --------------------------- ------------------------------------ -------------------------- --------------------------------------------------
\
![Normalization parameters of high metallicity BCDs (solid line), intermediate metallicity BCDs (dot-dashed line), and low metallicity BCDs (dashed line).[]{data-label="fig:xlf"}](xlf_comp_08_21.png){width="50.00000%"}
$L_X$/SFR - Metallicity relation {#subsect::reg}
--------------------------------
In addition to studying XLF normalization, we analyze the relationship between the total HMXB luminosity, SFR, and metallicity. @Douna2015 and B16 found correlations between $L_X$/SFR and metallicity with galaxy samples containing various types over a similar range of metallicity to ours. However, these studies ignored upper-limits in the linear regression procedure.
To take the upper-limits in our sample into account, we follow survival analysis techniques described in @Schmitt1985 [@Isobe1986]. The method we employ is the Buckley-James regression method, which is conveniently packaged in the Space Telescope Science Data Analysis System (STSDAS). The Buckley-James method uses the non-parametric Kaplan-Meier estimator to modify the regression in the presence of upper-limits by giving large residuals lower weighting [@Isobe1986]. Applied statistics literature on the Buckley-James method suggests that the regression method should not be used in cases with over 20 per cent censoring. If we include all the upper-limits in our sample, our censoring fraction would exceed 50 per cent. To see how the regression results change with the number of upper-limits included, we perform the analysis three different ways, including only the 4 upper-limits which constrain the @Mineo2012a $L_X$-SFR relation, including only the 16 upper-limits which constrain the B16 $L_X$-SFR-metallicity relation, and including no upper-limits. We run the `buckleyjames` routine within STSDAS with default parameters.
We apply this analysis to our sample by fitting an equation of the form $$\text{log($\frac{L_X}{ergs^{-1}}$)} = a \text{log($\frac{SFR}{M_{\odot} yr^{-1}}$)} + b \text{log($\frac{(O/H)}{(O/H)_{\odot}}$)} + c$$ with *a* set to unity. We plot this relation in Figure \[fig:lxsfr\] along with B16’s findings and the relevant model found in @Fragos2013a. The survival analysis results are tabulated in Table \[tab:lxsfr\].
Our measured slopes and their uncertainties vary as a function of the number of upper-limits included. When including the 16 upper-limits that constrain the B16 relation, the survival analysis seems to break down due to the abundance of censored data points. In the low-censoring case, including only the 4 constraining @Mineo2012a, the result indicates a negative correlation between the total resolved point source luminosity normalized to SFR and metallicity with slope of -0.954 $\pm$ 0.37, consistent with @Douna2015’s value of -1.01 and B16’s value of -0.59 $\pm$ 0.13. Analysing the trend when excluding all upper-limits, as done in @Douna2015 and B16, yields similar results to those studies, with an observed slope of -0.996 $\pm$ 0.38.
![L$_{X}$/SFR - metallicity relation for our sample. The black points represent the total source luminosity normalized to SFR for a given galaxy while upper limits given by arrows represent completeness luminosities. The solid, black line represents the model derived from stellar population synthesis simulations of @Fragos2013a, and the red, dashed line represents the multi-sample correlation found in B16. Our survival analysis correlation for the BCDs is represented by the solid blue line for the 3 upper-limit case. The 0 upper limit case is shown with the purple dot-dashed line. We list regression coefficients in Table \[tab:lxsfr\].[]{data-label="fig:lxsfr"}](lx-sfr-08-03.png){width="50.00000%"}
------------------------ ------------------- ------------------ ----------------------------------
Number of upper-limits Slope Intercept Standard deviation of regression
0 -0.996 $\pm$ 0.38 39.14 $\pm$ 0.34 0.359
4 -0.954 $\pm$ 0.37 39.20 0.654
16 -0.446 $\pm$ 0.38 39.28 0.649
------------------------ ------------------- ------------------ ----------------------------------
\
**Notes.** STSDAS results for Buckley-James regression in the case of censoring and linear regression results for 0 censoring case.
Summary and Discussion
======================
Previous studies have shown enhanced production of HMXBs in samples of low metallicity galaxies [e.g., @Mapelli2010; @Kaaret2011; @Prestwich2013], as well as B14, however none have studied the evolution of HMXB populations with a galaxy sample of uniform type over a broad range of metal content. From our study of catalogued BCDs over a wide metallicity range, we observe a significant increase in the XLF normalization at low metallicity relative to high by a factor of 4.45 $\pm$ 2.04 at a statistical significance of 99.79 per cent, with no significant distinction between high and intermediate metallicity. Future studies on this subject may choose to focus on a large sample of galaxies of the same type with $12+\log_{10}({\rm O/H})$ between 7.7 and 8.0 to elucidate the mechanism behind the observed decrease in the XLF. To probe potential changes in the XLF shape parameter due to variations in the dominant accretion mechanism, a larger sample of HMXB detections, especially at lower metallicities, would also be required.
By utilizing survival analysis regression methods, we build upon previous correlations found between $L_X$/SFR and metallicity for different censoring compositions. We find agreement with previous relations in the low-censoring and no censoring cases, and unconstraining results in the high-censoring case.
Eliminating potentially confounding variables due to galaxy type in our study of solely BCDs, we find enhanced production of HMXBs at low metallicity and a correlation between $L_X$/SFR and decreasing metallicity. Our results further indicate that in the low-metallicity environment of the early Universe, the production of HMXBs would have been enhanced and thus HMXBs could be more important than previously thought in determining the thermal conditions of the epoch of reionization.
While we assume elimination of other factors not discussed in this study such as stellar masses, dust fractions, gas mass fractions, and star formation histories due to maintaining a uniform galaxy type, future studies may probe the effects of these variables on the evolution of HMXBs. Since we cannot disentangle the potentially changing gas mass fraction with metallicity in this study, future studies with gas mass measurements may wish to explore potential correlations with HMXB formation. The results from this study can further be used to evaluate effects on the early Universe 21-cm hydrogen recombination signals.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
The authors thank the referee, Dr. Leonardo J. Pellizza, for the constructive feedback provided on the manuscript. M.B. thanks Hai Fu for insightful discussions which helped to improve the quality of the paper. STSDAS is a product of the Space Telescope Science Institute, which is operated by AURA for NASA. The scientific results reported in this article are based on joint observations made by the Chandra X-ray Observatory and the NASA/ESA Hubble Space Telescope. Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award No. AR7-18005X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics and Space Administration under contract NAS8-03060.
Tables
======
[lrccccc]{} Name &ObsID &$12+\log_{10}({\rm O/H})$ &RA &DEC &Exposure (ks) &Distance (Mpc)\
DDO 68 & 11271 & $7.15^a$ & 09 56 45.7 & +28 49 33.6 & 10.0 & 5.9\
I Zw 18 & 805 & $7.18^a$ & 09 34 02.4 & +55 14 26.4 & 40.0 & 18.2\
SBS 1129+576 & 11283 & $7.41^a$ & 11 32 02.42 & +57 22 45.4 & 14.75 & 26.3\
SBS 0940+544 & 11288 & $7.48^a$ & 09 44 17.15 & +54 11 28.5 & 16.83 & 22.1\
RC2 A1116+51 & 11287 & $7.51^a$ & 11 19 34.36 & +51 30 12.3 & 11.65 & 20.8\
.. & .. & .. &.. &.. &.. & ..\
\
**Notes.** The full table is available in the online version.\
The table includes the name, Chandra ObsID, metallicity, RA and DEC (J2000), Chandra exposure time, and distance for each galaxy. References – @Brorby2014$^a$, @Thuan2016$^b$, @Izotov2014$^c$, @Engelbracht2008$^d$, @Zhao2013$^e$, $^f$, $^g$ , @Guseva2000$^h$, @Kreckel2015$^i$, @Pilyugin2007$^j$.
[l c c l r c c c c]{} Name & ${n_H}^a$ & & Angle$^b$ & ${N(>L_\text{min})}^c$ & ${N_\text{bkg}}^d$ & ${L_\text{min}}^e$\
& $(10^{20} \text{cm}^{-2})$ & & (deg) & & & $\left(10^{38} \text{erg s}^{-1}\right)$\
DDO 68 & 1.97 & 66.49 & 32.94 & 77.7 & 2 & 0.3096 & 0.286\
I Zw 18 & 1.99 & 11.16 & 8.86 & -55.0 & 1 & 0.0449 & 0.548\
SBS 1129+576 & 0.87 & 23.81 & 9.6 & -72.0 & 1 & 0.0452 & 3.767\
SBS 0940+544 & 1.34 & 22.69 & 12.58 & -63.5 & 1 & 0.0614 & 2.354\
RC2 A1116+51 & 1.19 & 16.38 & 11.44 & -0.7 & 1 & 0.0308 & 3.003\
... & ... & ... &... &... & ... &... &...\
**Notes.**\
The full table is available in the online version.\
$^a$ Column densities were found using the `Colden`tool.\
$^b$ Properties of the D$_{25}$ ellipses were taken from the HyperLedadatabase and modified as described in Section \[subsect:observations\]. These represent 0.5 $\times$ D$_{25}$ dimensions.\
$^c$ Number of observed sources within the D$_{25}$ ellipse with luminosity greater than L$_\text{min}$.\
$^d$ Number of expected background sources within the D$_{25}$ ellipse, as determined by $\log N - \log S$ curves of @Georgakakis2008.\
$^e$ Luminosities limits calculated for the $0.5-8.0$ keV energy band.\
$^*$ Denotes galaxies without source detections constraining the Mineo relation.
[l c c c c c c c]{} Name & $E(B-V)^a$ & Count Rate$_\text{FUV}^b$ & $L_\text{FUV}^c$ & $\text{SFR}_{\text{FUV}}^d$ & Count Rate$_\text{NUV}^e$ & $\text{SFR}_{\text{NUV},0}^f$ & $\text{SFR}_{\text{IR}}^f$\
& (mag) & (cps) & $\left(10^{26} \text{erg s}^{-1} \text{Hz}^{-1}\right)$ & $\left(10^{-3} M_{\odot}\ \text{yr}^{-1}\right)$ & (cps) & $\left(10^{-3} M_{\odot}\ \text{yr}^{-1}\right)$ & $\left(10^{-3} M_{\odot}\ \text{yr}^{-1}\right)$\
DDO 68 & 0.0158 & 20.572 & 1.055 & 13.395 & 65.883 & 15.686 & 0.884\
I Zw 18 & 0.0292 & 9.918 & 5.362 & 68.1 & 27.894 & 63.197 & 2.025\
SBS 1129+576 & 0.0115 & 2.849 & 2.809 & 35.67 & 10.229 & 48.395 & 6.465\
SBS 0940+544 & 0.0114 & 1.605 & 1.116 & 14.175 & 5.639 & 18.837 & 3.196\
RC2 A1116+51 & 0.0128 & 4.325 & 2.693 & 34.203 & 13.776 & 40.764 & 0.013\
... & ... & ... & ... & ... & ... & ... & ...\
\
**Notes.**\
The full table is available in the online version.\
$^a$ Extinctions were found using the Infrared Science Archive (IRSA) tool DUST.\
$^b$ Count rate in the FUV band of GALEX observations.\
$^c$ Luminosity in the FUV band, $1350-1750$Å.\
$^d$ Star-formation rate as determined by the @Hunter2010 method.\
$^e$ Count Rate in the NUV band of GALEX, $1750-2800$Å.\
$^f$ Star-formation rate in the NUV. $^e$ Star-formation rate in the IR via Spitzer images [@Brorby2014].\
[^1]: E-mail: sam-ponnada@uiowa.edu
[^2]: E-mail: matthew-brorby@uiowa.edu
[^3]: <http://leda.univ-lyon1.fr/>
[^4]: <http://cxc.harvard.edu/toolkit/colden.jsp>
[^5]: <http://cxc.harvard.edu/toolkit/pimms.jsp>
[^6]: [](http://galexgi.gsfc.nasa.gov/docs/galex/FAQ/counts_background.html)
[^7]: <http://sha.ipac.caltech.edu/applications/Spitzer/SHA/>
|
---
abstract: 'We have studied the low-lying excitations of a chain of coupled circuit-QED systems in the ultrastrong coupling regime, and report several intriguing properties of its two nearly degenerate ground states. The ground states are Schrödinger cat states at a truly large scale, involving maximal entanglement between the resonators and the qubits, and are mathematically equivalent to Majorana bound states. With a suitable design of physical qubits, they are protected against local fluctuations and constitute a non-local qubit. Further, they can be probed and manipulated coherently by attaching an empty resonator to one end of the circuit-QED chain.'
author:
- 'Myung-Joong Hwang'
- 'Mahn-Soo Choi'
bibliography:
- 'paper.bib'
title: 'Large-Scale Schrödinger-Cat States and Majorana Bound States in Coupled Circuit-QED Systems'
---
Confronted with formidable difficulties in solving strongly interacting many-body systems, it has been desired to find good quantum simulators. It may seem natural to simulate a many-body system with another tunable system of massive particles such as ultracold atomic gases [@Bloch:2012jy]. In fact, any controllable quantum system, notably quantum computer if ever practical, can simulate efficiently many-body systems [@Feynman:1982gn]. Indeed it has been recognized that photons confined in coupled-cavities simulate closely the quantum behaviors of strongly-correlated many-body systems [@Hartmann:2006kv; @*Greentree:2006jg; @Angelakis:2007ho]. Subsequent studies have revealed that Bose-Hubbard model [@Rossini:2007fx; @*Irish:2008ci; @*Makin:2008hr; @*Koch:2009hh; @*Schmidt:2009cs; @*Schmidt:2010kl], interacting spin models [@Angelakis:2007ho; @Hartmann:2007db; @*Kay:2008ip; @*Cho:2008et], and other exotic quantum phases [@Cho:2008ct; @*Carusotto:2009dr; @*Koch:2010eu] can be simulated efficiently using the coupled-cavities. Further, recent advances in solid-state devices such as circuit-QED systems [@Wallraff:2004dy; @Schoelkopf:2008cs] and micro-cavities [@Aoki:2006gq; @Hennessy:2007hi] and ongoing efforts to fabricate large-scale cavity arrays [@Houck:2012iq; @*Underwood:2012vq] make the array of coupled cavities a promising candidate for an efficient quantum simulator.
Meanwhile, the ultrastrong coupling regime of the cavity-QED system, where the light-matter coupling energy is comparable to or even higher than the energy of the cavity field, has been envisioned [@Devoret:2007gi; @*Bourassa:2009gy; @*Bourassa:2012tv] and experimentally demonstrated [@Niemczyk:2010gv; @*FornDiaz:2010by; @*Gunter:2009gc]. The ultrastrong coupling brings about fundamentally different physics deeply connected to the high degree of entanglement between the “matter” and the photon [@Ashhab:2010eh; @Hwang:2010jn; @Nataf:2010dy; @Nataf:2011ff; @Braak:2011hc; @*Irish:2007bo; @*Hausinger:2010eb; @*Hausinger:2011bc; @*Casanova:2010kd; @*Zueco:2009it]. However, the effect of ultrastrong coupling on the low-energy excitations of an array of coupled cavity-QED systems remains unclear, and is our main concern in this work.
In this paper, we investigate the low-lying excitations of a one-dimensional (1D) array of circuit-QED systems (cQEDs), with each cQED being in the ultrastrong coupling regime; see Fig. \[fig:1\]. It turns out that the array permits two nearly degenerate ground states separated by a finite energy gap from the continuum of higher-energy states. We find several intriguing properties of the two ground states: (i) They are Schrödinger cat states at a truly large scale, and involve maximal entanglement between the resonators and the qubits. (ii) With a suitable design of physical qubits, the two ground states are protected against local fluctuations and constitute a non-local qubit [@Tserkovnyak:2011dl]. (iii) They are mathematically equivalent to the long-searched Majorana bound states . (iv) They can be probed and manipulated coherently by attaching an empty resonator to one end of the circuit-QED chain. Such configuration turns the total system (the circuit-QED chain plus the empty resonator) into another effective circuit-QED system. There are many promising types of superconducting qubits, among which we focus on Fluxonium [@Manucharyan:2009fo; @*Koch:2009ec]. As we illustrate below, its strong inductive coupling with the superconducting resonator [@Nataf:2010dy] and its anisotropic noise characteristics [@Nataf:2011ff] are well suited for our purpose.
![Schematic of 1D circuit-QED arrays. The red dots indicate qubits placed inside of superconducting resonator. The $N$th resonator is coupled to the detection resonator. The circuit-QED array realizes the transverse field Ising model (TFIM), and the detection resonator can measure and control the degenerate ground state of the TFIM.[]{data-label="fig:1"}](fig1){width="8cm"}
*System: a circuit-QED chain* — We consider a 1D array of cQEDs; see Fig. \[fig:1\]. Each cQED consists of the “resonator”, a superconducting microwave transmission line, and the “qubit”, a superconducting quantum bit (two-level system) [@Blais:2004kn; @Wallraff:2004dy], and is theoretically described by the Rabi Hamiltonian $$\label{paper::eq:2}
H^\mathrm{cQED}_i = \omega_0 a_i^\dagger a_i
-\lambda(a_i+a_i^\dagger)\sigma_i^x+\frac{\Omega}{2}\sigma_i^z$$ where $a_i$ and $a_i^\dag$ are the field operators of the resonator with frequency $\omega_0$, $\sigma_i^x$ and $\sigma_i^z$ Pauli operators of the qubit with energy splitting $\Omega$, and $\lambda$ the resonator-qubit coupling energy in the $i$th cQED. The resonators of neighboring cQEDs are coupled capacitively to each other, and photons hop from one resonator to nearby ones. The Hamiltonian of the whole chain is thus given by $$\label{paper::eq:1}
H=\sum_i^N H^\mathrm{cQED}_i -J\sum_{i}^{N-1}
(a^\dagger_i a_{i+1}+ a_{i}a^\dagger_{i+1})$$ where $J$ is the photon hopping amplitude and $N$ is the number of cQEDs in the chain.
Before discussing the energy levels and associated wavefunctions of the whole chain, we first briefly review the properties of the low-lying states of a single cQED in the ultrastrong coupling regime ($\lambda\gtrsim\omega_0$). The strong coupling disables the standard rotating wave approximation, which reduces Eq. (\[paper::eq:2\]) to the Jaynes-Cummings Hamiltonian. As a consequence, the ground state of Eq. (\[paper::eq:2\]) is not a simple vacuum anymore as in the Jaynes-Cumming model. Instead, it contains finite average photon numbers, and shows non-classical properties such as squeezing and entanglement [@Ashhab:2010eh; @Hwang:2010jn]. To see this, let us examine the ground-state wavefunction more closely: Approximate expressions for the nearly-degenerate ground states have been derived in Ref. (see also Ref. ). Here we take a different approach and explore the parity symmetry in the Rabi Hamiltonian, which is important to understand the effect of photon hopping. The Hamiltonian (\[paper::eq:2\]) commutes with the “parity” operator $\Pi_i=\exp(-i\pi a_i^\dagger
a_i)\sigma_i^z$, and thus the Hilbert space is classified into subspaces $\mathcal{E}_i^\pm$ of $\pm$ parity. Within each subspace $\mathcal{E}_i^\pm$, the Hamiltonian can be described in effect by a single bosonic operator, $b_i=a_i\sigma^x_i$: $H_i^\mathrm{cQED}\to H^\pm_i=H_i^0\pm H_i^1$ with $H^0_i =\omega_0(b_i^\dagger-\lambda/\omega_0)
(b_i-\lambda/\omega_0)-\lambda^2/\omega_0$ and $H^1_i=\frac{\Omega}{2}\cos(\pi b_i^\dagger
b_i)$ [@Hwang:2010jn]. $H^0_i$ is simply a displaced harmonic oscillator and the ground state is a coherent state ${\left|\textstyle{\lambda/\omega_0}\right\rangle}_{b_i}^\pm$. For $\lambda/\omega_0\gg1$ (regardless of $\Omega$), $H^1_i$ can be treated perturbatively and shifts the energies of ${\left|\textstyle{\lambda/\omega_0}\right\rangle}_{b_i}^\pm$ relatively by an exponentially small amount $\Delta=\frac{\Omega}{2}
e^{-2(\lambda/\omega_0)^2}$. Now, back in the $\{a_i,\sigma_i^z\}$-basis, the nearly degenerate ground states ${\left|\textstyle{\lambda/\omega_0}\right\rangle}_{b_i}^\pm$ are expressed as
\[paper::eq:3\] $$\begin{aligned}
\label{paper::eq:3a}
{\left|\textstyle{0}\right\rangle}_i&\equiv \frac{1}{\sqrt{2}}\left({\left|\textstyle{\lambda/\omega_0}\right\rangle}_i{\left|\textstyle{+}\right\rangle}_i-{\left|\textstyle{-\lambda/\omega_0}\right\rangle}_i{\left|\textstyle{-}\right\rangle}_i\right), \\
\label{paper::eq:3b}
{\left|\textstyle{1}\right\rangle}_i&\equiv \frac{1}{\sqrt{2}}\left({\left|\textstyle{\lambda/\omega_0}\right\rangle}_i{\left|\textstyle{+}\right\rangle}_i+{\left|\textstyle{-\lambda/\omega_0}\right\rangle}_i{\left|\textstyle{-}\right\rangle}_i\right),\end{aligned}$$
where ${\left|\textstyle{\alpha}\right\rangle}_i$ ($\alpha\in\mathbb{C}$) is the eigenstate (coherent state) of $a_i$ and ${\left|\textstyle{\pm}\right\rangle}_i$ are the eigenstates of $\sigma_i^x$. In short, these two ground states, ${\left|\textstyle{0}\right\rangle}_i$ and ${\left|\textstyle{1}\right\rangle}_i$, residing in distinct parity subspaces are nearly degenerate with an energy splitting of $2\Delta$, separated far from higher-energy states by an energy gap $\omega_0$.
*Effective model: a transverse-field Ising chain* — Let us now investigate the whole chain described by the Hamiltonian (\[paper::eq:1\]). Typically $J\ll\omega_0$, and we are mainly interested in the low-lying excitations, well below $\omega_0$. In this limit, each cQED remains within the subspace spanned by the states ${\left|\textstyle{0}\right\rangle}_i$ and ${\left|\textstyle{1}\right\rangle}_i$ in Eq. (\[paper::eq:3\]) and can be regarded as a *pseudo-spin*: $$\label{paper::eq:4}
\sum_i^N H^\mathrm{cQED}_i= -\Delta\sum_i^N \tau^z_i$$ where $\tau^z_i={\left|\textstyle{0}\right\rangle}_i{\left\langle\textstyle{0}\right|}-{\left|\textstyle{1}\right\rangle}_i{\left\langle\textstyle{1}\right|}$ and the energy splitting $\Delta$ plays the role of Zeeman field. Hopping of a photon into or out of a cavity changes the parity of its state, or more explicitly $a_i{\left|\textstyle{0}\right\rangle}_i=\lambda/\omega_0{\left|\textstyle{1}\right\rangle}_i$ and $a_i{\left|\textstyle{1}\right\rangle}_i=\lambda/\omega_0{\left|\textstyle{0}\right\rangle}_i$. Based on these observation, we can identify $a_i$ and $a^\dagger_i$ as a pseudo-spin-flip operator $\tau^x_i$ and $a_ia^\dagger_{i+1}$ as Ising interaction $\tau^x_i\tau^x_{i+1}$. That is, the photon-hopping part of the Hamiltonian becomes $$\label{paper::eq:6}
J\sum_{i}^{N-1} (a^\dagger_i a_{i+1}+ a_{i}a^\dagger_{i+1})= J_\mathrm{eff}\sum_i^{N-1} \tau^x_i\tau^x_{i+1}$$ with $J_\mathrm{eff}=2J(\lambda/\omega_0)^2$. The effective Ising interaction strength, $J_\mathrm{eff}$, is renormalized with respect to $J$ by the factor $(\lambda/\omega_0)^2$ because the field part of the pseudo-spin states in Eq. (\[paper::eq:3\]) is a coherent state with amplitudes $\lambda/\omega_0$ and the field-field interaction between resonators is proportional to the amplitudes of the resonator fields.
Putting both terms in Eqs. (\[paper::eq:4\]) and (\[paper::eq:6\]) together, the low-energy effective Hamiltonian for the cQED chain becomes the so-called transverse-field Ising model (TFIM), $$\label{paper::eq:7}
H_\mathrm{Ising}
= -\Delta\sum_i^N \tau^z_i-J_\mathrm{eff}\sum_i ^{N-1}\tau^x_i\tau^x_{i+1}.$$ The TFIM exhibits a quantum phase transition between the *magnetically ordered* phase for $\Delta<J_\mathrm{eff}$ and the *quantum paramagnet* phase for $\Delta>J_\mathrm{eff}$ [@Sachdev:2011uj]. The former is particularly interesting for our purposes. For $\Delta=0$, $H_\mathrm{Ising}$ has two degenerate ground states, ${\left|\textstyle{\Rightarrow}\right\rangle}\equiv\prod_i{\left|\textstyle{\rightarrow}\right\rangle}_i$ and ${\left|\textstyle{\Leftarrow}\right\rangle}\equiv\prod_i{\left|\textstyle{\leftarrow}\right\rangle}_i$, where ${\left|\textstyle{\rightarrow}\right\rangle}_i$ and ${\left|\textstyle{\leftarrow}\right\rangle}_i$ are eigenstates of $\tau_i^x$. For $\Delta>0$ (yet $\Delta<J_\mathrm{eff}$), $\tau_i^z$ tends to flip the pseudo-spins, ${\left|\textstyle{\rightarrow}\right\rangle}_i\leftrightarrow{\left|\textstyle{\leftarrow}\right\rangle}_i$. It causes tunneling between ${\left|\textstyle{\Rightarrow}\right\rangle}$ and ${\left|\textstyle{\Leftarrow}\right\rangle}$ via soliton propagation, and hence the true eigenstates become $$\label{paper::eq:10}
{\left|\textstyle{\Psi_0}\right\rangle}
= \frac{1}{\sqrt{2}}\left({\left|\textstyle{\Rightarrow}\right\rangle}+{\left|\textstyle{\Leftarrow}\right\rangle}\right)
\,,\quad
{\left|\textstyle{\Psi_1}\right\rangle}
= \frac{1}{\sqrt{2}}\left({\left|\textstyle{\Rightarrow}\right\rangle}-{\left|\textstyle{\Leftarrow}\right\rangle}\right)$$ However, as the tunneling involves $N$ spins, the tunneling amplitude is exponentially suppressed with the system size $N$. In other words, ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ are nearly degenerate with energy splitting, $\delta\sim\exp(-N/\xi)$ with $\xi$ being the correlation length of the Ising chain, exponentially small in system size $N$. Both are separated from the continuum of excitations by the energy gap $J_\mathrm{eff}$.
The two states ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ in Eq. (\[paper::eq:10\]) have non-local combinations of many pseudo-spins and are widely known as Greenberger-Horne-Zeilinger (GHZ) states [@Greenberger89a]. Moreover, by expressing them in the original $\{a_i,\sigma_i^x\}$-basis $$\label{paper::eq:12}
{\left|\textstyle{\Psi_s}\right\rangle} =
\frac{1}{\sqrt{2}}
\left[\prod_i^N{\left|\textstyle{\lambda/\omega_0}\right\rangle}_i{\left|\textstyle{+}\right\rangle}_i
+(-1)^s\prod_i^N{\left|\textstyle{-\lambda/\omega_0}\right\rangle}_i{\left|\textstyle{-}\right\rangle}_i\right]$$ with $s=0$ or $1$, one can see that they involve high degree of non-local entanglement between cavity fields and qubits. They are thus *Schrödinger cat states* at a truly large scale while many theoretically proposed or experimentally demonstrated Schrödinger cat states [@Yurke:1986cm; @Ourjoumtsev:2007cw] contain merely a single radiation field. Below we illustrate that the two states in (\[paper::eq:12\]) are protected against local fluctuations and constitute a non-local qubit [@Tserkovnyak:2011dl].
*Effective model: a Majorana chain* — 1D TFIM discussed above is equivalent to a chain of Majorana fermions [@Kitaev:2001up; @Kitaev:2009ut]. The latter has attracted great interest because it permits localized Majorana modes that can be used for topologically protected quantum computation [@Kitaev:2001up; @Kitaev:2009ut; @Kitaev:2006ik]. A very recent experiment [@Mourik:2012je] suggests that the Majorana chain can be realized in a solid-state system, and intensive efforts are made in this direction .
Here we re-express the two nearly degenerate states in Eq. (\[paper::eq:10\]) or (\[paper::eq:12\]) in terms of localized Majorana fermions, and later discuss an experimentally feasible way of probing such Majorana fermions. The equivalence between the TFIM and the Majorana chain can be seen through a Jordan-Wigner transformation [@Lieb:1961dn]: $c^\dagger_i=\tau_i^+\prod^{i-1}_{j=1}(-\tau^z_j)$ with $\tau^+_i={\frac{1}{2}}(\tau^x_i+i \tau^y_i)$. The operators $c_i$ and $c_i^\dag$ describe Dirac fermions and satisfy $\{c_i,c^\dagger_j\}=\delta_{ij}$ and $\{c_i,c_j\}=0$. The Dirac fermion operators are further represented with self-conjugate Majorana operators, $\gamma_{2i-1}=c^\dagger_i+c_i$ and $\gamma_{2i}=i(c^\dagger_i-c_i)$. The TFIM (\[paper::eq:7\]) is then reduced to $$\label{paper::eq:8}
H_\mathrm{Majorana} = \frac{i}{2}\left[
\Delta\sum_{i=1}^N\gamma_{2i-1}\gamma_{2i}+
J_\mathrm{eff}\sum_i^{N-1}\gamma_{2i}\gamma_{2i+1}\right]$$ At $\Delta=0$, the Majoranas at the two ends, $\gamma_1$ and $\gamma_{2N}$, in the chain does not appear in the Hamiltonian, which implies the existence of two degenerate ground states. These are nothing but ${\left|\textstyle{\Rightarrow}\right\rangle}$ and ${\left|\textstyle{\Leftarrow}\right\rangle}$ in Eq. (\[paper::eq:10\]). For finite $\Delta$, the two states ${\left|\textstyle{\Rightarrow}\right\rangle}$ and ${\left|\textstyle{\Leftarrow}\right\rangle}$ are mixed linearly into ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ in Eq. (\[paper::eq:10\]) due to the tunneling between two Majorana modes $\gamma_1$ and $\gamma_{2N}$, and the degeneracy is lifted. Since the tunneling is through the whole chain, the energy splitting $\delta$ is exponentially small (as long as $\Delta<J_\mathrm{eff}$). One can check that $(\gamma_{1}+i \gamma_{2N}){\left|\textstyle{\Psi_0}\right\rangle} = 0$ and $(\gamma_{1}+i \gamma_{2N}){\left|\textstyle{\Psi_1}\right\rangle} = 2{\left|\textstyle{\Psi_0}\right\rangle}$, which means that ${\left|\textstyle{\Psi_1}\right\rangle}$ has one more fermion than ${\left|\textstyle{\Psi_0}\right\rangle}$ or equivalently that ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ have different fermion parities.
Here we emphasize that the two Majoranas localized at the ends of the Majorana chain are actually non-local in the physical chain, i.e., the cQED chain or the Ising chain [@Bardyn:2012td]: The Majorana operators is represented in terms of $\tau_j^x$ and $\tau_j^z$ as $$\label{paper::eq:9}
\gamma_{1}=\tau_1^x,\quad
\gamma_{2N}=i\tau^x_N \prod_{j=1}^{N}(-\tau^z_j),$$ and $\gamma_{2N}$ involves the *string* operator $\prod_{j=1}^{N}(-\tau^z_j)$. This implies that the two nearly-degenerate ground states ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ are not protected topologically against local noise even though mathematically they correspond to two distinct Majorana modes. It is in stark contrast to the case where the two Majorana modes at the ends of a $p$-wave superconducting wire are topologically protected. However, we will see below that the two states ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ are vulnerable only to a certain type of local noise and there exist realistic systems with such type of local noise significantly suppressed.
*Noise* — It is evident from the expression in Eq. (\[paper::eq:12\]) that the non-local spin qubits are prone to the local noise in $\sigma_i^x$ of the physical qubits and the one in $a_i+a_i^\dagger$ of the resonators. The states are intrinsically robust against the $\sigma_i^y$ and $\sigma_i^z$ noise since ${}_i{{\left\langle\textstyle{\Psi_1}\right|}}\sigma_i^{y,z}{\left|\textstyle{\Psi_0}\right\rangle}_i\sim
e^{-(\lambda/\omega_0)^2}$, which is reminiscent of the Franck-Condon effect. The $a_i+a_i^\dag$ noise affects only the resonator at the end of the chain (which is usually connected external microwave environment for measurement), and can be easily avoided by replacing it by a high-Q resonator. The problem with $\sigma_i^x$ noise can be circumvented, for example, by using Fluxonium for qubits. Fluxonium is known to have anisotropic noise characteristics with $\sigma_i^{y,z}$ being the dominant noises and the $\sigma_i^x$ noise ignorable [@Nataf:2011ff].
What about the effect of inhomogeneity in system parameters? Above we have assumed $\omega_0$, $\Omega$, $\lambda$ and $J$ of each circuit-QED to be homogeneous. Deviations in $\omega_0$, $\Omega$, $\lambda$ lead to fluctuations in $\Delta$. The inter-cavity coupling strength, $J$, can also be varied from cavity to cavity, which leads to inhomogeneous TFIM, $$-\sum_i\Delta_i\tau_i^z -\sum_i J_i^\mathrm{eff}\tau_i^x\tau_{i+1}^x.$$ This Hamiltonian still conserves the parity symmetry, $\mathcal{P}=\prod^N_{i=1}\tau_z^i$ which is respected by the degenerate ground states. Therefore, the ground states will be robust to small fluctuations in $\Delta_i$ and $J_i$. We thus conclude that the nearly degenerate ground states ${\left|\textstyle{\Psi_0}\right\rangle}$ and ${\left|\textstyle{\Psi_1}\right\rangle}$ can be kept well protected by a careful design of the physical qubits in the system.
![(a) Energy diagram for the circuit-QED Hamiltonian (\[paper::eq:2\]) as a function of $\lambda/\omega_0$. (b) Plot of $\frac{\omega_0}{\lambda} {}_i{{\left\langle\textstyle{1}\right|}}a_i{\left|\textstyle{0}\right\rangle}_i$. We can conclude that $\lambda>2\omega_0$ is required for our model to be valid because the transverse field $\Delta$ almost vanishes and the identification of photon annihilation operator as a spin flip operator, $\frac{\omega_0}{\lambda}
a_i=\tau_i^x$, is justified.[]{data-label="fig:2"}](fig2a "fig:"){width="8cm"} ![(a) Energy diagram for the circuit-QED Hamiltonian (\[paper::eq:2\]) as a function of $\lambda/\omega_0$. (b) Plot of $\frac{\omega_0}{\lambda} {}_i{{\left\langle\textstyle{1}\right|}}a_i{\left|\textstyle{0}\right\rangle}_i$. We can conclude that $\lambda>2\omega_0$ is required for our model to be valid because the transverse field $\Delta$ almost vanishes and the identification of photon annihilation operator as a spin flip operator, $\frac{\omega_0}{\lambda}
a_i=\tau_i^x$, is justified.[]{data-label="fig:2"}](fig2b "fig:"){width="8cm"}
*Detection and control* — In this section, we suggest a scheme to control and measure the non-local spin qubit. It can be also interpreted as detecting the Majorana bound states. Our proposal consists only of an additional empty resonator coupled to the resonator at the end of the circuit-QED chain. Consider a resonator with a frequency, $\omega_d$, capacitively coupled to $N$th cavity, so that we have $$\label{paper::eq:11}
H_d=J_d(a_N^\dagger a_d +a_N a_d^\dagger)+\omega_da^\dagger_d a_d$$ where $a_N$ represents the field operator of $N$th cavity, and $a_d$ the field operator of the detection cavity. As shown earlier, the $N$th cavity’s creation and annihilation operators are equivalent to $\lambda/\omega_0\tau_N^x$ for the $N$th effective spin. Moreover, for the non-local spin qubits, $\tau_i^x$ is equivalent to $S^x={\left|\textstyle{\Psi_0}\right\rangle}{\left\langle\textstyle{\Psi_1}\right|}+{\left|\textstyle{\Psi_1}\right\rangle}{\left\langle\textstyle{\Psi_0}\right|}$ for any $i$ as $\tau_i^x{\left|\textstyle{\Psi_s}\right\rangle}={\left|\textstyle{\Psi_{1- s}}\right\rangle}$ ($s=0,1$). Therefore, assuming that $\tilde{J}_d\equiv J_d\lambda/\omega_0\ll J_\mathrm{eff}$, the low-energy effective Hamiltonian (\[paper::eq:7\]) combined with the detection Hamiltonian (\[paper::eq:11\]) leads again to the Rabi Hamiltonian $$\label{paper::eq:13}
H_\mathrm{Rabi}
= \frac{\delta}{2}S^z+\tilde J_dS^x(a_d +a_d^\dagger)+\omega_da^\dagger_d a_d$$ Here we can make the rotating wave approximation, then the Hamiltonian reduces to the Jaynes-Cummings Hamiltonian. Therefore, by just adding an empty resonator at one end of the circuit-QED array, we can realize a circuit-QED Hamiltonian for the non-local spin qubit. It allows us to tap into the standard techniques available for the circuit-QED to control and measure the non-local spin qubit. For example, since the detuning between the detection cavity frequency and the non-local spin qubit splitting, $\Delta_d=\omega_d-\delta$ is large compared to $\tilde J_d$, it is in the dispersive regime where the cavity frequency pulling by the non-local spin qubit is $ \delta\omega_d={2J_d^2}/{\Delta_d}$ [@Blais:2004kn]. This can be experimentally measured since one can have $\tilde J_d\sim10^{-4}\omega_0$ which achieves the standard strong-coupling regime for the circuit QED [@Wallraff:2004dy].
*Experimental feasibility* — Finally we examine the experimental feasibility of the ideas explained above, estimating possible values of physical parameters of the system. Two requirements must be satisfied: First, the two ground states of each cQED in the system must be nearly degenerate and well separated from higher excitations. In Fig. (\[fig:2\]) (a) are plotted the energies of individual circuit-QED Hamiltonian (\[paper::eq:2\]) in the resonant case ($\omega_0=\Omega$). Figure \[fig:2\] (b) plots $\frac{\omega_0}{\lambda} {}_i{{\left\langle\textstyle{1}\right|}}a_i{\left|\textstyle{0}\right\rangle}_i$ to illustrate how good (its value close to 1) the approximation $a_i=\lambda/\omega_0\tau^x_i$ is. One can see that $\lambda\sim2\omega_0$ suffices for the requirement. Second, the system should be in the magnetically ordered phase (in terms of the effective TFIM), $\Delta<J_\mathrm{eff}$ or equivalently $\Omega\exp\left[-2(\lambda/\omega_0)^2\right] < 4J(\lambda/\omega_0)^2.$ This requirement is satisfied provided that $J>10^{-5}\omega_0$. The desired coupling strength, $\lambda>2\omega_0$, seems achievable for the Fluxonium coupled inductively to the superconducting resonator [@Nataf:2011ff]. Moreover, $J>10^{-5}\omega_0$ is also realistic for the superconducting resonators, with $J$ in the range of a few MHz.
*Conclusion* – We have found several intriguing properties of the two nearly degenerate ground states of a chain of coupled circuit-QED systems in the ultrastrong coupling regime. The ground states are Schrödinger cat states at a truly large scale, and are mathematically equivalent to Majorana bound states. With a suitable design of the system, they are protected against local fluctuations, and may be probed and manipulated coherently by attaching an extra empty resonator.
Finishing this work, we have noticed a closely related preprint [@1205.3083S]. While they focus on the phase transition of the circuit-QED chain, we are mainly concerned about the quantum properties of the nearly degenerated ground states on one side of the phase transition. In this respect, both works are complementary to each other.
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bibliography:
- 'literature.bib'
- 'literature2.bib'
- 'mazin3.bib'
---
Astro2020 Science White Paper
Directly Imaging Rocky Planets from the Ground
**Thematic Areas:** Planetary Systems $\square$ Star and Planet Formation $\square$ Formation and Evolution of Compact Objects $\square$ Cosmology and Fundamental Physics $\square$ Stars and Stellar Evolution $\square$ Resolved Stellar Populations and their Environments $\square$ Galaxy Evolution $\square$ Multi-Messenger Astronomy and Astrophysics
**Principal Author:**
Name: B. Mazin Institution: University of California Santa Barbara Email: bmazin@ucsb.edu Phone: (805)893-3344
**Co-authors:** (names and institutions) É. Artigau, Université de Montréal\
V. Bailey, California Institute of Technology/JPL\
C. Baranec, University of Hawaii\
C. Beichman. California Institute of Technology/JPL\
B. Benneke, Université de Montréal\
J. Birkby, University of Amsterdam\
T. Brandt, University of California Santa Barbara\
J. Chilcote, University of Notre Dame\
M. Chun, University of Hawaii\
L. Close, University of Arizona\
T. Currie, NASA-Ames Research Center\
I. Crossfield, Massachusetts Institute of Technology\
R. Dekany, California Institute of Technology\
J.R. Delorme, California Institute of Technology/JPL\
C. Dong, Princeton University\
R. Dong, University of Victoria\
R. Doyon, Université de Montréal\
C. Dressing, University of California Berkeley\
M. Fitzgerald, University of California Los Angeles\
J. Fortney, University of California Santa Cruz\
R. Frazin, University of Michigan\
E. Gaidos, University of Hawai‘i\
O. Guyon, University of Arizona/Subaru Telescope\
J. Hashimoto, Astrobiology Center of NINS\
L. Hillenbrand, California Institute of Technology\
A. Howard, California Institute of Technology\
R. Jensen-Clem, University of California Berkeley\
N. Jovanovic, California Institute of Technology\
T. Kotani, Astrobiology Center of NINS\
H. Kawahara, University of Tokyo\
Q. Konopacky, University of California San Diego\
H. Knutson, , California Institute of Technology\
M. Liu, University of Hawaii\
J. Lu, University of California Berkeley\
J. Lozi, Subaru Telescope\
B. Macintosh, Stanford University\
J. Males, University of Arizona\
M. Marley, NASA Ames\
C. Marois, University of Victoria\
D. Mawet, California Institute of Technology/JPL\
S. Meeker, California Institute of Technology/JPL\
M. Millar-Blanchaer, California Institute of Technology/JPL\
S. Mondal, S. N. Bose National Centre for Basic Sciences\
N. Murakami, Hokkaido University\
R. Murray-Clay University of California Santa Cruz\
N. Narita, National Astronomical Observatory of Japan\
T.S. Pyo, National Astronomical Observatory of Japan\
L. Roberts, California Institute of Technology/JPL\
G. Ruane, California Institute of Technology/JPL\
G. Serabyn, California Institute of Technology/JPL\
A. Shields, University of California Irvine\
A. Skemer, University of California Santa Cruz\
L. Simard, Herzberg Astronomy and Astrophysics Research Centre\
D. Stelter, University of California Santa Cruz\
M. Tamura, University of Tokyo, Astrobiology Center of NINS\
M. Troy, California Institute of Technology/JPL\
G. Vasisht, California Institute of Technology/JPL\
J. K. Wallace, California Institute of Technology/JPL\
J. Wang, The Ohio State University\
J. Wang, California Institute of Technology\
S. Wright, University of California San Diego\
Introduction
============
Over the past three decades instruments on the ground and in space have discovered thousands of planets outside the solar system. These observations have given rise to an astonishingly detailed picture of the demographics of short-period planets ($P\lesssim 30$ days), but are incomplete at longer periods where both the sensitivity of transit surveys and radial velocity signals plummet. Even more glaring is that the spectra of planets discovered with these indirect methods are either inaccessible (radial velocity detections) or only available for a small subclass of transiting planets with thick, clear atmospheres [@2015ApJ...815..110M].
Direct detection can be used to discover and characterize the atmospheres of planets at intermediate and wide separations, including non-transiting exoplanets. Today, a small number of exoplanets have been directly imaged, but they represent only a rare class of young, self-luminous super-Jovian-mass objects orbiting tens to hundreds of AU from their host stars. Atmospheric characterization of planets in the $<$5 AU regime, where radial velocity (RV) surveys have revealed an abundance of other worlds, is technically feasible with 30-m class apertures in combination with an advanced AO system, coronagraph, and suite of spectrometers and imagers.
There is a vast range of unexplored science accessible through astrometry, photometry, and spectroscopy of rocky planets, ice giants, and gas giants. In this whitepaper we will focus on one of the most ambitious science goals — detecting for the first time habitable-zone rocky ($<1.6 R_\Earth$, @2017AJ....154..109F) exoplanets in reflected light around nearby M-dwarfs[^1]. Other whitepapers will address the second potential way to detect rocky exoplanets by looking at thermal (10 ) emission around a handful of the nearest Sun-like stars.
To truly understand exoplanets we need to go beyond detection and demographics and begin to characterize these exoplanet and their atmospheres. High-resolution spectroscopic capabilities will not only illuminate the physics and chemistry of exo-atmospheres, but may also probe these rocky, temperate worlds for signs of life in the form of atmospheric biomarkers (combination of water, oxygen, methane and other molecular species). The Astro 2020 White Paper “Detecting Earth-like Biosignatures on Rocky Exoplanets around Nearby Stars with Ground-based Extremely Large Telescopes” (Lopez-Morales et al.) includes more details on potential biosignatures.
Landscape and Demographics
==========================
Led by *Kepler*, transit surveys of the last decade have revealed that about half of all solar-type stars host planets with sizes between Earth and Neptune (1–4 $R_\Earth$) and orbital periods less than a year, and that these planets are frequently found in closely packed multiple systems. The occurrence rate of such small planets declines at the shortest orbital periods ($P<$10days), but is roughly flat in $\log P$ for periods between a month and a year . Jovian worlds are known to be less common than terrestrial planets [@howard_etal10; @mayor_etal11], but their higher detectability means that they comprise the bulk of the presently detected population of long-period planets ($P > 1$year; NASA Exoplanet Archive). At the longest periods, current imaging surveys of young systems have found that very massive planets are rare (0.6% for 5–13 $M_\textrm{Jup}$ and $a = $30–300AU; @bowler_etal16), but true Jupiter analogs are beyond the reach of existing instruments. Even with instrumentation that reaches fundamental limits (AO correction limited by photon shot noise, near perfect coronagraphs, etc.) the science return of 8–10m telescopes is fundamentally limited by inner working angle (IWA) and sensitivity, but we do expect future surveys on existing telescopes to reveal both cooler and less massive (perhaps sub-Jovian at the youngest ages) planets at moderate separations from their host stars.
While *Kepler* has probed the landscape and demographics of small exoplanets on short period orbits, and RV surveys have given valuable information for wider orbits, the census of small planets in the solar neighborhood is highly incomplete. *Kepler* has also not provided information on small planets with intermediate periods (above 200 days). A large ground-based telescope, like those being considered in the US ELT program, could fill in this important gap in our knowledge, potentially in conjunction with upcoming high precision RV surveys.
The large increase in aperture going from current telescopes to 30-m class telescopes both decreases the inner working angle, probes smaller separations, and increases the achievable contrast (even with equal wavefront control performance, as the fractional stellar flux per $\lambda/D$ element scales as $1/D^2$). As our understanding of the challenges of direct imaging and our technology continues to improve, we expect direct imaging instruments for ELTs to approach photon-noise limited final post-processed contrasts of around $10^{-8}$ [@males_guyon_2018; @2018SPIE10703E..0ZG].
The IWA of 30-m telescopes will enable access to the habitable zones of hundreds of cool dwarfs. This synergizes well with future NASA missions like HabEx and LUVOIR that plan to search for earth-like planets around Sun-like stars (see also Astro2020 White Paper “The Critical, Strategic Importance of Adaptive Optics-Assisted Ground-Based Telescopes for the Success of Future NASA Exoplanet Direct Imaging Missions” by Currie et al.). Together, ground-based 30-m telescopes working in the visible and NIR and future visible space-based planet finders will provide a complete picture of habitable zone planets in the solar neighborhood.
Outside M-dwarf habitable zones, these large ground-based telescopes will also have the ability to greatly contribute to the demographics of planets from 0.5 to beyond 5AU, especially around stellar types that are not amenable to RV measurements and for face-on systems.
{width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"}
Planetary Characteristics
=========================
Even more crucial than detecting planets with a variety of masses and separations is probing their atmospheric properties with spectroscopy and their bulk properties through combined mass and radius measurements. The classic “core accretion” model of planet formation is supported by the enhanced metallicities and detailed compositions of the giant planets in our solar system and their correlation with planetary mass and semimajor axis. When spectroscopy is combined with atmospheric models it allows us to infer the atmospheric compositions of exoplanets. Ongoing studies of giant exoplanet metallicity are beginning to explore whether core accretion is the dominant formation pathway for most exoplanetary systems. To date, however, observations of water or carbon abundance have been practical only for transiting hot-Jupiter planets and for self-luminous young giant planets discovered through direct imaging.
Over the next decade, the spectroscopic exploration of transiting planets will advance rapidly with the combination of *TESS* and *JWST*, including a potential sample of $\sim$100 giant planets and many more small planets (albeit biased towards planets with high equilibrium temperatures and/or planets orbiting low-mass stars; @sullivan_etal15). ELTs, however, will measure cooler planets and those orbiting earlier-type stars. By probing different regimes of atmospheric chemistry than transit observations, we increase the parameter space spanned by our physical models, helping to identify their biases. Together, transit and direct methods will provide the spectroscopy of the diverse array of planetary targets that are needed to significantly further our understanding of planet formation.
In addition to composition, measurements at high spectral resolution can determine planets’ rotation rates and cloud through Doppler imaging [@crossfield14; @2014Natur.509...63S]. Photometry and polarimetry as a function of orbital phase can also be used to constrain clouds, hazes, and surface features . These science cases are enabled by these telescope’s unprecedented angular resolution, light-collecting power, and instrumentation. Moreover, these instruments have the potential opportunity to detect biosignatures using a technique called high-dispersion coronagraphy . HDC takes advantage of high-resolution spectroscopy ($R>$50k) to mitigate previous contrast limits from the ground and in principle reach the sensitivities needed to characterize planets that are ${\sim}10^8$ times fainter than their host stars.
Rocky Planets
-------------
Characterising the atmosphere of planets in the habitable zone of nearby stars to search for biosignature gases is perhaps one of the most compelling science goals in all of astronomy. This importance is reflected, for example, in Strategic Objective 1.1 of the 2018 NASA Strategic Plan which states: *“Are we alone?” is a central research question that involves biological research and research in the habitability of locations in the solar system such as Mars, the moons of outer planets, or thousands of potentially habitable worlds around other stars. This research supports a fundamental science topic at the interface of physics, chemistry, and biology.*
Around the nearest M dwarfs, ELTs will be able to detect starlight reflected by rocky, habitable-zone exoplanets in addition to ice and gas giants. The low luminosities of M dwarfs means that planets must orbit close to the star to receive Earth-like radiance levels. While a boon for transit surveys due to the increased probability of transit, the proximity of M dwarf habitable zones to the star poses a challenge for direct imaging because the habitable zone typically lies well inside the IWA of 8-m telescopes. The small IWA of ELTs makes them ideal for characterizing planets in the habitable zones of M dwarfs. Only a small fraction of nearby planets will transit, and we await an understanding of the success rates of NIR-optimized RV surveys in discovering low-mass planets. Despite the uncertainties, Proxima Cen b, the TRAPPIST-1 planets, LHS 1140 b, and Gliese 411 (see Table \[table:planets\]) were all discovered in the last two years, and several more surveys targeting M dwarfs have just started or will begin in the near future. ELTs will be powerful instruments not only in obtaining more complete and less biased statistics on planetary demographics through surveys that image planets orbiting low-mass stars, but also in characterizing these discoveries.
**Name** **Telescope** **Primary** **Dist (pc)** **Radius** **Temp (K)** **Sep $\lambda/D$ (mas)** **Contrast**
------------- --------------- ------------- --------------- ----------------- -------------- --------------------------- ----------------------
Prox. Cen b GMT M5.5 1.3 1.1 R$_\Earth$ 235 4.5 (37) $3.5 \times 10^{-8}$
Gliese 411 TMT M1.9 2.5 1.5 R$_\Earth$ 350 4.5 (31) $2.5 \times 10^{-8}$
Ross 128 b Both M4 3.4 1.1 R$_\Earth $ 300 2.2 (15) $3.3 \times 10^{-8}$
YZ Cet d Both M4.5 3.6 1.1 R$_\Earth $ 260–370 1.1 (7.7) $9.8 \times 10^{-8}$
GJ 273 b Both M3.5 3.8 1.5 R$_\Earth $ 260 3.5 (24) $2.1 \times 10^{-8}$
Wolf 1061 c Both M3.5 4.3 1.7 R$_\Earth $ 225 3.0 (21) $2.4 \times 10^{-8}$
: The most promising currently known rocky exoplanet candidates with equilibrium temperatures below 350 K from the NASA Exoplanet Archive. Contrasts are calculated with an albedo of 0.3, phase correction of 50%, and with radii of the RV planets estimated from models of rocky exoplanets [@2016ApJ...819..127Z]. Separation in $\lambda/D$ at 1 is calculated with the most favorable telescope if both TMT and GMT can see the star. All of these candidates have separations and contrasts that are likely to be accessible with TMT and GMT. Many more planets will be discovered before ELT first light through near-IR RV surveys.[]{data-label="table:planets"}
These observations will be among the first opportunities to detect biosignatures in the atmospheres of other worlds. While 10 $\mu$m observations may allow detection of rocky planets around a small sample of solar-type stars [@2015IJAsB..14..279Q], planets around faint M-type stars are the most favorable targets for spectroscopic follow up at shorter wavelengths with ELTs and *JWST* [in the case of transiting planets; @kasting_etal14]. There are abundant lines from biosignature gases, O$_2$, H$_2$O, CH$_4$, and CO$_2$, in the near infrared (i.e., $\sim$1 to 4), where the HDC technique is expected to reach optimal performance. For example, a simultaneous detection of oxygen (O$_2$) and methane (CH$_4$) would be highly suggestive of life [@desmarais02]. It is important to note that the very precise characterization of molecular spectra in the laboratory are essential for these detections and should be pursued in support of ELT success. A detection of CH$_4$, however, is out of reach for *JWST* given the low concentration of CH4 and the relatively high mean molecular weight / small scale height of an Earth-like atmosphere. Such measurements with ELTs can potentially provide a strong case for life activities on nearby worlds: there are $>$30 M dwarfs within 5 pc that are observable by TMT and GMT, and there is at least one rocky planet per M dwarf . Given the many nascent instruments that will undertake M-dwarf planet surveys in the near term, ELTs will likely have access to a full census of characterizable planets in these nearest systems.
Planetary System Architectures
==============================
Significant advances in understanding planet formation are possible by investigating the correlation of planet properties with location in circumstellar disks. ELTs will be able to place individual planets into formation context in systems that are still forming. The smaller IWA will allow them to peer into the inner regions of other systems that are inaccessible to today’s telescopes. This work is discussed in detail in a companion whitepaper, “Planet Formation Science with US ELT Direct Imaging” by Dr. Steph Sallum *et al*.
[^1]: cf. whitepaper to the Exoplanet Science Strategy committee of the U.S. National Academy of Science: <https://goo.gl/nELRGX>
|
---
abstract: 'The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$ with $d=2,$ $1\leq \alpha\leq 2$ and $d\geq 3,$ $0< \alpha\le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [@ST6]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [@Kim].'
address:
- 'Bradley University, Department of Mathematics, Peoria, IL 61625 USA'
- 'Bradley University, Department of Mathematics, Peoria, IL 61625 USA'
author:
- 'Selma Y[i]{}ld[i]{}r[i]{}m Yolcu'
- Türkay Yolcu
nocite: '[@*]'
title: Refined Eigenvalue Bounds on the Dirichlet Fractional Laplacian
---
introduction
============
Fractional Laplacian operators are usually considered as the prototype of the non-local operators [@ChenKimSong] and have recently garnered much attention in many applications in mathematics and physics. Related problems usually lie at the interface of probability, stochastic processes, partial differential equations and spectral theory such as [@BanYol; @ChenKimSong; @ChenSong]. For some applications, we refer the reader to graphene models [@HajjMeh], obstacle problems [@Silv], non-local minimal surfaces [@CafRoSav].
In this article, we study eigenvalues of the fractional Laplacian operator $(-\Delta)^{\alpha/2}$ defined by $$\label{cpp}
\begin{split}
(-\Delta)^{\alpha/2}\,\phi_j &=\lambda_j^{(\alpha)}\,\phi_j \quad \hbox{in} \,\,\Omega,\\
\phi_j &=0 \quad\hbox{in} \,\,{\mathbb R}^d\backslash \Omega
\end{split}$$ where $\Omega$ is a bounded connected domain with smooth boundary in ${\mathbb R}^d$, for either $d=2,$ $1\le \alpha\le 2$ or $d\ge 3$ and $0< \alpha \le 2.$ Since $\Omega$ is bounded, the spectrum of the fractional Laplacian is discrete and eigenvalues $\{\lambda_j^{(\alpha)}\}_{j=1}^{\infty}$ (including multiplicities) can be sorted in an increasing order.
Unlike Laplacian, fractional Laplacian is a non-local operator such that for suitable test functions, including all functions in $u\in C_0^{\infty}({\mathbb R}^d)$, it is defined as $$\label{fracdef1}
(-\Delta)^{\alpha/2}u(x)={\mathcal A}_{d,\alpha}\lim_{\epsilon\to0^+}\int_{\{|y|>\epsilon\}}\frac{u(x+y)-u(x)}{|y|^{d+\alpha}}dy,$$ where ${\mathcal A}_{d,\alpha}$ is a well-known positive normalizing constant. From a probabilistic point of view, $(-\Delta)^{\alpha/2}$ can be considered as the infinitesimal generator of the semigroup of the symmetric $\alpha-$stable process, denoted by $X_t$, with the characteristic function $$\label{probdefn}
e^{-t|\mu|^{\alpha}}=E(e^{i\mu\cdot X_t})= \int_{{\mathbb R}^d}e^{i\mu\cdot y}p_t^{(\alpha)}(y)dy,\qquad t>0, \quad \mu\in{\mathbb R}^d,$$ where $p_t^{(\alpha)}(x,y)=p_t^{(\alpha)}(x-y)$ is called the transition density of the stable process (or the heat kernel of the fractional Laplacian). While explicit formulae for the transition density of symmetric $\alpha$-stable processes are only available for the Cauchy process ($\alpha=1$) and the Brownian motion ($\alpha=2$), these processes share many of the basic properties of the Brownian motion. Another process of importance is the Holtsmark distribution ($\alpha=3/2$) which is used to model gravitational fields of stars (See e.g., [@Zolo]). Stable processes do not have continuous paths which is related to non-locality of the fractional Laplacian operator [@BanYol; @Bogetal].
When fractional Laplacians involved, some of the known methods fail because of the fractional power and non-locality of such operators. This drawback can be evaded by using the Fourier transform definition. Recall that the Fourier transform and its inverse are defined as $${\mathcal F}[u](\mu) =\hat{u}(\mu)=c_d\int_{{\mathbb R}^d}e^{-i\mu\cdot z}\,u(z)\,dz, \quad{\mathcal F}^{-1}[u](z)=c_d\int_{{\mathbb R}^d}e^{i\mu\cdot z}\,u(\mu)\,d\mu,$$ where $c_d=(2\pi)^{-\frac{d}{2}}$ is the normalizing constant. Interestingly, the fractional Laplacian operator on $\Omega\subset{\mathbb R}^d$ can be defined as a pseudo-differential operator as $$\label{defnbilapfourier}
(-\Delta)^{\alpha/2}|_{\Omega}u = {\mathcal F}^{-1}\left[|\mu|^{\alpha }{\mathcal F}[u]\right],\qquad 0<\alpha \le2, \quad u\in H_0^{\alpha/2}(\Omega).$$ Here, $H_0^{\alpha/2}(\Omega)$ denotes the Sobolev space of order $\alpha/2$. When $\Omega={\mathbb R}^d$, one can look at Proposition 3.3. [@Vald] for the proof of the equivalence between the definitions in and .
There is an extensive literature devoted to the inequalities involving the eigenvalues of the Dirichlet Laplacian operator, which can be regarded as the fractional Laplacian when $\alpha=2$. One may consult the articles [@AshBen; @AshBenLaug; @AshLaug; @FraGei; @Henrot; @KovVuWei; @LapGeiWei; @LapWei; @Weidl] and references therein for a through literature review. It is worth pausing here for a moment to consider the Dirichlet Laplacian results relevant to our main result. The first such result is the Li-Yau inequality that provides a lower bound for the sums of eigenvalues, sharp in the sense of Weyl asymptotics [@LiYau]. The authors generalized this result in [@ST2] by obtaining the following Li-Yau type inequality for the eigenvalues of the Diriclet fractional Laplacian operator: $$\sum_{j=1}^k\lambda_j^{(\alpha)}
\ge (4\pi)^{\frac{\alpha}{2}}\,\frac{d}{ \alpha+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{\alpha }{d}}k^{1+\frac{\alpha }{d}},\label{BLY}$$ where $|\Omega|$ represents the volume of $\Omega$ and $\Gamma(x)$ denotes the Gamma function $\Gamma(x)=\int_0^{\infty} t^{x-1}e^{-t}\,dt$ for $x>0.$ One may also look at [@HarYil] for a Li-Yau type inequality involving the eigenvalues of the (massless) Klein–Gordon square root operators $(-\Delta)^{1/2}|_{\Omega}$, (i.e., the case $\alpha=1$). To look at this inequality from a different perspective, one may take the Legendre transform of the following result by Laptev [@Laptev2] and obtain : $$\sum_{j}(z-\lambda_j^{(\alpha)})_+\leq (4\pi)^{-\frac{d}{2}} \frac{\alpha}{\alpha+d}\frac{|\Omega|}
{\Gamma\left(1+\frac{d}{2}\right)} z^{1+\frac{d}{\alpha}}.\label{AL}$$ When we set $\alpha=2$ in , we recover an earlier result by Berezin [@Bere], which supplies the Li-Yau inequality after an application of the Legendre transform. Thus, in what follows, we call as the Berezin-Li-Yau inequality.
In [@Melas], Melas proved the following improvement to the Berezin-Li-Yau inequality ($\alpha=2$): $$\label{melasimp}
\sum_{j=1}^k\lambda_j^{(2)}
\ge 4\pi\,\frac{d}{ 2+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{2 }{d}}k^{1+\frac{2}{d}}+\frac{1}
{24(2+d )}\frac{|\Omega|}{{\mathcal I}(\Omega)}\,k,$$ where ${\mathcal I}(\Omega)$, the moment of inertia, is defined by $${\mathcal I}(\Omega)=\min_{y\in{\mathbb R}^d}\int_{\Omega} |z-y|^2\,dz.$$ By a translation of the origin and a rotation of axes if necessary, in the sequel, we assume that the origin is the center of mass of $\Omega$ and that $${\mathcal I}(\Omega)=\int_{\Omega} |z|^2\,dz.\label{ID}$$ Melas type bounds and their many variants and extensions have recently attracted a lot of attention, see for instance [@Ilyin; @KovWei; @Selma; @ST4; @ST2; @ST5; @ST3; @ST6; @Turkay]. In particular, in [@ST2], the authors obtained a refinement of , stating that $$\begin{split}\sum_{j=1}^k\lambda_j^{(\alpha)}
& \ge (4\pi)^{\frac{\alpha}{2}}\,\frac{d}{ \alpha+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{\alpha }{d}}k^{1+\frac{\alpha }{d}}
\\ &\quad +
\frac{\alpha}
{48( \alpha+d )}\frac{|\Omega|^{1-\frac{\alpha-2}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha-2}{d}}}{(4\pi)^{1-\frac{\alpha}{2}}\,{\mathcal I}(\Omega)}\,k^{1+\frac{\alpha-2}{d}}.
\end{split}\label{ccmbiz}$$ Remark that $\alpha=2$ in recovers Melas’ bound in [@Melas] for the Dirichlet Laplacian eigenvalues. Unfortunately, it is not easy to take Legendre transform of to find a similar improved Berezin type bound in the case of the fractional Laplacian.
For $0<\ell\leq 1$, authors also proved in [@ST6] that $$\label{imbound2e}
\begin{split}
\sum_{j=1}^k \left(\lambda_j^{(\alpha)}\right)^\ell
&\ge (4\pi)^{\frac{\alpha\ell}{2}}\,\frac{d}{ \alpha\ell+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{\alpha\ell}{d}} k^{1+\frac{\alpha\ell }{d}} \\
&\quad+\frac{\alpha\ell}
{16(\alpha\ell+d )} \frac{|\Omega|^{1-\frac{\alpha\ell-2}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha\ell-2}{d}}}{(4\pi)^{1-\frac{\alpha\ell}{2}}\,{\mathcal I}(\Omega)}\,k^{1+\frac{\alpha\ell-2}{d}}
\\& \quad - \frac{\alpha\ell}{640(\alpha\ell+d )}
\frac{|\Omega|^{2-\frac{\alpha\ell -4}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha\ell -4}{d}}}{ {(4\pi)^{2-\frac{\alpha\ell}{2}}\,\mathcal I}(\Omega)^{2}}\,k^{1+\frac{\alpha\ell -4}{d}}.
\end{split}$$ The main purpose of this article is to obtain analogous but sharper bounds with Dirichlet Laplacian replaced by Dirichlet fractional Laplacian. The first step, inspired by results in [@Selma; @ST4; @ST2; @ST6; @Turkay], is establishing the following result:
\[impBLY\] For $k\ge 1$, and either $1\le\alpha\le 2$ and $d=2$ or $0<\alpha\le 2$ and $d\ge 3,$ the eigenvalues $\{\lambda_j^{(\alpha)}\}_{j=1}^{\infty}$ of the fractional Laplacian operator defined on $\Omega \subset {\mathbb R}^d$ satisfy $$\label{imbound1}
\begin{split}
\sum_{j=1}^k\lambda_j^{(\alpha)}
&\ge(4\pi)^{\frac{\alpha}{2}}\,\frac{d}{ \alpha+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{\alpha }{d}}k^{1+\frac{\alpha }{d}}
\\& \quad+\frac{\alpha}
{2( \alpha+d)} \frac{|\Omega|^{\frac{1}{2}-\frac{\alpha-1}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha-1}{d}}}{(4\pi)^{\frac{1}{2}-\frac{\alpha}{2}}\,{\mathcal I}(\Omega)^{\frac{1}{2}}}\,k^{1+\frac{\alpha-1}{d}}
\\& \quad - \frac{5\alpha}
{16( \alpha+d )}\frac{|\Omega|^{1-\frac{\alpha-2}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha-2}{d}}}{(4\pi)^{1-\frac{\alpha}{2}}\,{\mathcal I}(\Omega)}\,k^{1+\frac{\alpha-2}{d}}
\\& \quad+\frac{\alpha}{16( \alpha+d )}
\frac{|\Omega|^{\frac{3}{2}-\frac{\alpha -3}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha -3}{d}}}{ {(4\pi)^{\frac{3}{2}-\frac{\alpha}{2}}\,\mathcal I}(\Omega)^{\frac{3}{2}}}\,k^{1+\frac{\alpha -3}{d}}.
\end{split}$$
Note that the constants in the leading terms on the right side of , which is a fractional version of Weyl’s law, are optimal due to a classical result of Blumenthal and Getoor [@BluGet]. While the constant in the second term in is still sub-optimal, the estimate in is substantially stronger than previous known results in [@ST2; @ST6; @Turkay]. In addition, we recover the lower bounds in the case of the Dirichlet Laplacian [@ST4] when we set $\alpha=2$ in .
In [@ChenSong], Chen and Song obtained that $$\lambda_j^{(\alpha\ell)}\le \left(\lambda_j^{(\alpha)}\right)^\ell\label{CS}$$ for each $j$ and any constant $\ell\in(0,1]$. Thus, Proposition \[impBLY\] along with an application of leads to our principal result:
\[impBLYT\] For $k\ge 1$, $0<\ell\le 1$ and either $1\le\alpha\le 2$ and $d=2$ or $0<\alpha\le 2$ and $d\ge 3,$ the eigenvalues $\{\lambda_j^{(\alpha)}\}_{j=1}^{\infty}$ of the fractional Laplacian operator defined on $\Omega \subset {\mathbb R}^d$ satisfy $$\label{imbound21}
\begin{split}
\sum_{j=1}^k\left(\lambda_j^{(\alpha)}\right)^{\ell}
&\ge(4\pi)^{\frac{\alpha\ell}{2}}\,\frac{d}{ \alpha\ell+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{\alpha\ell}{d}}k^{1+\frac{\alpha\ell }{d}}
\\& \quad+\frac{\alpha\ell}
{2( \alpha\ell+d)} \frac{|\Omega|^{\frac{1}{2}-\frac{\alpha\ell-1}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha\ell-1}{d}}}{(4\pi)^{\frac{1}{2}-\frac{\alpha\ell}{2}}\,{\mathcal I}(\Omega)^{\frac{1}{2}}}\,k^{1+\frac{\alpha\ell-1}{d}}
\\& \quad - \frac{5\alpha\ell}
{16( \alpha\ell+d )}\frac{|\Omega|^{1-\frac{\alpha\ell-2}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha\ell-2}{d}}}{(4\pi)^{1-\frac{\alpha\ell}{2}}\,{\mathcal I}(\Omega)}\,k^{1+\frac{\alpha\ell-2}{d}}
\\& \quad+\frac{\alpha\ell}{16( \alpha\ell+d )}
\frac{|\Omega|^{\frac{3}{2}-\frac{\alpha\ell -3}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha\ell -3}{d}}}{ {(4\pi)^{\frac{3}{2}-\frac{\alpha\ell}{2}}\,\mathcal I}(\Omega)^{\frac{3}{2}}}\,k^{1+\frac{\alpha\ell -3}{d}}.
\end{split}$$
On a side note, the proof of Proposition \[impBLY\] consists of a very delicate application of Steffensen’s type inequalities, which is mainly about the comparison of the integrals on the subsets of interval $[0,\infty)$.
The article is structured as follows: In Section \[sec:BLY2\], we revisit the relevant facts about the eigenvalues and eigenfunctions of the fractional Laplacian operator. After providing the proof of an auxilliary lemma that plays a crucial role in proving Proposition \[impBLY\], we present the proof of our main results in Section \[sec:impBLY\]. Finally, we end Section \[sec:impBLY\] with a remark which extends the main result even further for some elliptic operators studied in [@Kim; @SV]. Please see Remark \[rmkgen\] for details.
Preliminaries {#sec:BLY2}
=============
In this section, we give an overview of the definitions and tools that are essential to establish the estimates in . Even though, these were previously studied in [@Selma; @ST2; @ST6; @Turkay], we include them so that the article is self-contained. By using Plancherel’s theorem, one can show that the set of Fourier transforms $\{\hat{\phi}_j\}_{j=1}^{\infty}$ of $\{\phi_j\}_{j=1}^{\infty}$ forms an orthonormal set in $L^2({\mathbb R}^d)$ since the set of eigenfunctions $\{\phi_j\}_{j=1}^{\infty}$ is an orthonormal set in $L^2(\Omega).$ To ease the notation, we set $$\label{Uk}
\Phi_k(\mu):=\sum_{j=1}^k|\hat{\phi}_j(\mu)|^2=\frac{1}{(2\pi)^{d}}\sum_{j=1}^k\left|\int_{\Omega}
e^{-i z\cdot\mu}\phi_j(z)\,dz\right|^2 \ge 0.$$ Because the support of $\phi_j$ is $\Omega,$ the integral is taken over $\Omega$ instead of $\mathbb{R}^d$. Interchanging the sum and integral and using $\|\hat{\phi}_j\|_2=1,$ we derive $$\label{Fintk}
\int_{{\mathbb R}^d}\Phi_k(\mu)\, d\mu=k.$$ Observe that $$\label{Uksumlambda}
\begin{split}
\sum_{j=1}^k \lambda_j^{(\alpha)}&=\sum_{j=1}^k \langle \phi_j,\lambda_j^{(\alpha)} \phi_j\rangle =\sum_{j=1}^k \langle \phi_j,(-\Delta)^{\alpha/2}\phi_j\rangle \\ & =\sum_{j=1}^k \langle \phi_j,\mathcal{F}^{-1}[|\mu|^{\alpha }\mathcal{F}[\phi_j]]\rangle =\sum_{j=1}^k \int_{{\mathbb R}^d}|\mu|^{\alpha }\,|\hat{\phi}_j(\mu)|^2\,d\mu \\ & =\int_{\mathbb{R}^d} |\mu|^{\alpha }\, \Phi_k(\mu)\, d\mu.
\end{split}$$ Application of Bessel’s inequality gives an upper bound for $\Phi_k$: $$\Phi_k(\mu)\le \frac{1}{(2\pi)^{d}}\int_{\Omega}\left|e^{-iz\cdot\mu}\right|^2\,dz = \frac{|\Omega|}{(2\pi)^{d}}:=\Omega_d.\label{fksiless}$$ Next, we find an estimate for $|\nabla \Phi_k|$. Note that $$\sum_{j=1}^{k}|\nabla \hat{\phi}_j(\mu)|^2\leq\frac{1}{(2\pi)^{d}}\int_{\Omega}\left|iz e^{-iz\cdot\mu}\right|^2\,dz
=\frac{{\mathcal I}(\Omega)}{(2\pi)^{d}}.\label{bugrad}$$ In view of H[" o]{}lder’s inequality and utilizing and , we arrive at the following uniform bound: $$\label{gradUk}
\begin{split}
|\nabla \Phi_k(\mu)|&\le 2\left(\sum_{j=1}^{k}|\hat{\phi}_j(\mu)|^2\right)^{1/2}
\left(\sum_{j=1}^{k}|\nabla \hat{\phi}_j(\mu)|^2\right)^{1/2}\\ & \le 2(2\pi)^{-d}\sqrt{|\Omega|\, {\mathcal I}(\Omega)}:=\beta.
\end{split}$$ Let $B_R(z):=\{y\in\mathbb{R}^d : |y-z|\le R\}$ designate the ball of radius $R$ centered at $z$ in $\mathbb{R}^d$ and $\omega_d$ denotes the volume of $d$ dimensional unit ball $B_1(z)$ in $\mathbb{R}^d$ given by $$\omega_d=\dfrac{\pi^{\frac{d}{2}}}{\Gamma\left(1+\frac{d}{2}\right)}.\label{wd}$$ Now assume that $R$ is such that $|\Omega|=\omega_dR^d$. That is, $B_R(0)$ is the symmetric rearrangement of $\Omega.$ Note that $${\mathcal I}(\Omega)\geq \int_{B_R(0)}|z|^2\,dz=\frac{d\omega_d}{d+2}R^{d+2}
=\frac{d}{d+2}\omega_d^{-\frac{2}{d}}|\Omega|^{\frac{d+2}{d}},\label{IDineq}$$ roughly giving $$\beta
\geq (2\pi)^{-d}\,\omega_d^{-\frac{1}{d}}\,|\Omega|^{\frac{d+1}{d}}.\label{boundform}$$ Let $\Phi_k^*(\mu)$ denote the decreasing radial rearrangement of $\Phi_k(\mu).$ There exists a real valued absolutely continuous function $\varphi_k:[0,\infty)\to [0,\Omega_d]$ such that $$\Phi_k^*(\mu)=\varphi_k(|\mu|).\label{zetak}$$ By using P[ó]{}lya-Szeg[ö]{} inequality, one can show that $$0\le -\varphi_k'(t)\le \beta.\label{bdv}$$ For more details, see for example [@ST2].
Proof of Proposition \[impBLY\] {#sec:impBLY}
===============================
Before diving into the proof of the main results, we present the following surprising sharper inequality which will be the main ingredient in the proof of the sharper lower bound in . Our method of proof has been previously explored in several articles, for instance [@Melas; @Selma; @ST4; @ST5; @ST6], with crucial differences.
\[lem:ineq\] For either $d=2$ and $1\le\alpha\le 2$ or $3\le d\in \mathbb{N}$ and $0<\alpha\le 2,$ $a>0,$ $b>0,$ we have the following inequality $$a^{d+\alpha}\ge \frac{d+\alpha}{d}a^{d} b^{\alpha}-\frac{\alpha}{d}b^{d+\alpha}+\frac{\alpha}{d} \,b^{d+\alpha-3}\,\left(2a+b\right) (a-b)^2.
\label{keyineq}$$
A direct but lengthy proof of this lemma is given in [@ST6]. Here, we shall give a more intuitive and rigorous proof using convexity.
First, let us show that $$h(x):=dx^{d+\alpha}-(d+\alpha)x^d+\alpha-\alpha(2x+1)(x-1)^2\ge 0.\label{key1}$$ for $x\ge 0,$ either $d=2$ and $1\le\alpha\le 2$ or $3\le d\in \mathbb{N}$ and $0<\alpha\le 2.$
**Case 1:** Assume that $d\geq 3$ and $0< \alpha \leq 2$. Observe that $h$ can be written as $h(x)=x^2g(x)$ where $$g(x)=dx^{d+\alpha-2}-(d+\alpha)x^{d-2}-2\alpha x+3\alpha.$$ Differentiating we get $$g'(x)=d(d+\alpha-2)x^{d+\alpha-3}-(d+\alpha)(d-2)x^{d-3}-2\alpha,$$ $$g''(x) = x^{d-4}d(d+\alpha-2)(d+\alpha-3) \left(x^{\alpha}-\frac{(d+\alpha)(d-2)(d-3)}{d(d+\alpha-2)(d+\alpha-3)}\right).$$ Note that $g''(x_{d,\alpha})=0$ where $$x_{d,\alpha}:=\left(\frac{(d+\alpha)(d-2)(d-3)}{d(d+\alpha-2)(d+\alpha-3)}\right)^{1/\alpha} < 1.$$ When $x\ge x_{d,\alpha},$ we have $g''(x)\ge 0$, implying that $g$ is convex on $[x_{d,\alpha},\infty)$. Thus, $$g(x)\geq g(1)+g'(1)(x-1)=0,$$ since $g(1)=0$ and $g'(1)=0$. That is, $g(x)\geq 0$ on $[x_{d,\alpha},\infty)$. In particular, $g(x_{d,\alpha})\ge 0.$ On the other hand, when $0\le x\le x_{d,\alpha},$ we have $g''(x)\le 0$ yielding that $g'$ is decreasing on $[0, x_{d,\alpha}]$. This implies that $g'(x)\le g'(0)=-2\alpha<0,$ meaning that $g$ is decreasing on $[0, x_{d,\alpha}].$ This leads to $g(x)\ge g(x_{d,\alpha})\ge 0$ for $x\in[0, x_{d,\alpha}].$ Hence, $g(x)\geq 0$ for $x\in[0,\infty)$. Therefore, we deduce that $h(x)=x^2g(x)\geq 0$ for $x\in[0,\infty).$
 
**Case 2:** Now, assume that $d=2$ and $1\leq \alpha\leq 2$. Then $h$ becomes $$h(x)=2x^{2+\alpha}-(2+\alpha)x^2+\alpha-\alpha(2x+1)(x-1)^2.$$ As before, we can write $h(x)=x^2g(x)$ where $g(x)=2x^{\alpha}-2\alpha x +(2\alpha-2).$ Note that if $\alpha=1,$ then $g(x)=0.$ Differenting again, we obtain $$g'(x)=2\alpha x^{\alpha-1}-2\alpha\quad\mbox{and}\quad
g''(x)=2\alpha(\alpha-1)x^{\alpha-2}.$$ Notice that for $1\leq \alpha\leq 2$, $g''(x)\geq 0,$ meaning that $g$ is convex. Since $g(1)=0$ and $g'(1)=0$, $g(x)\geq g(1)+g'(1)(x-1)=0$ implies that $g(x)\geq 0$ and, therefore, $h(x)\geq 0.$
Setting $x=a/b$ in , multiplying through by $b^{d+\alpha}/d$ and rearranging the terms, we conclude , as desired.
When $0<\alpha<1$ and $d=2,$ the inequality above fails to hold, therefore, we do not resolve this case in this manuscript.
With Lemma \[lem:ineq\] in hand, we are now ready to prove Proposition \[impBLY\].
Assume that - hold and $d\geq 2.$ Consider the decreasing, absolutely continuous function $\varphi_k :[0,\infty)\to [0,\infty)$ defined by . We know that $0\leq-\varphi_k'(t)\leq \beta$ for $t\ge 0$ where $\beta>0$ is given by . Since $\varphi_k (0)>0$ due to let us first define $$\label{trk}\Theta_k(a):=\frac{1}{\varphi_k (0)}\varphi_k \left(\frac{\varphi_k (0)}{\beta}a\right).$$ Note that $\Theta_k$ is positive, $\Theta_k(0)=1$ and $0\le -\Theta_k'(a)\le 1$. To simplify the notation, we also set $\theta_k(a):=-\Theta_k'(a)$ for $t\geq 0$. Hence, $0\le \theta_k(a)\le 1$ for $t\ge 0$ and $$\displaystyle{\int_{0}^{\infty}}\theta_k(a)\,da=\Theta_k(0)=1.$$ Now, set $$\label{defint}
\zeta_k=\int_0^{\infty} a^{d-1}\,\Theta_k(a)\,da\qquad\mbox{and}\qquad \eta_k=\int_0^{\infty} a^{ \alpha+d -1}\,\Theta_k(a)\,da.$$ Using we get $$\label{eqnkA}
k=\int_{{\mathbb R}^d} \Phi_k(\mu)\,d\mu
=\int_{{\mathbb R}^d} \Phi_k^*(\mu)\,d\mu
=d\omega_d\int_{0}^{\infty}a^{d-1}\varphi_k(a)\,da.$$ Moreover, since the map $\mu\mapsto|\mu|^{\alpha }$ is radial and increasing, by , we obtain that $$\label{sumxiB}
\begin{split}
\sum_{j=1}^k\lambda_j^{(\alpha)}&=\int_{{\mathbb R}^d} |\mu|^{\alpha }\,\Phi_k(\mu)d\mu
\\ &\ge \int_{{\mathbb R}^d} |\mu|^{\alpha }\,\Phi_k^*(\mu)d\mu\\ &=d\omega_d\int_{0}^{\infty} a^{ \alpha+d-1} \varphi_k(a)\,da. \end{split}$$ Substitution of into yields $$\label{imp01}\begin{split}
\zeta_k &=
\frac{\beta^{d}}{\varphi_k(0)^{d+1}}\int_0^{\infty} a^{d-1}\varphi_k(a)\,da=\frac{\beta^{d}k}{d\,\omega_d\,\varphi_k(0)^{d+1}},\\
\eta_k&=
\frac{\beta^{ \alpha+d }}{\varphi_k(0)^{ \alpha+d +1}}\int_0^{\infty} a^{ \alpha+d -1}\varphi_k(a)\,da\le \frac{\beta^{ \alpha+d }\,\sum_{j=1}^k\lambda_j^{(\alpha)}}{d\,\omega_d\,\varphi_k(0)^{ \alpha+d +1}}
\end{split}$$ Observe that application of Fubini’s theorem together with $$\Theta_k(b)=\int_b^{\infty} \theta_k(a)\, da$$ leads to $$\begin{aligned}
\frac{1}{t+d}\int_0^{\infty}a^{t+d}\,\theta_k(a)\,da&=& \int_0^{\infty}\left(\int_0^a b^{t+d-1}\, db\right)\theta_k(a)\,da\\ &=& \int_0^{\infty}b^{t+d-1}\left(\int_b^{\infty} \theta_k(a)\, da\right)db\\ &=& \int_0^{\infty}b^{t+d-1} \Theta_k(b)\, db,\end{aligned}$$ which together with $y=0$ and $y=\alpha$ respectively yields $$\label{crucial1}
\int_0^{\infty}a^{d}\,\theta_k(a)\,da=\zeta_k\, d
\qquad\mbox{and}\qquad
\int_0^{\infty}a^{ \alpha+d }\,\theta_k(a)\,da=\eta_k\,( \alpha+d ).$$ Notice that $$\big(a^{d}-1\big)\big(\theta_k(a)-\mathds{1}_{[0,1]}(a)\big)\ge 0, \qquad a\in[0,\infty).
\label{ineq:00}$$ Integrating from $0$ to $\infty$ gives $$\int_0^{\infty}a^{d}\theta_k(a)\,da \ge \frac{1}{d+1}=\gamma_d(0),$$ where $\gamma_d:[0,\infty)\to(0,\infty)$ is defined by $$\gamma_d(x)=\int_x^{x+1}a^{d}\,da.$$ Since $\gamma_d$ is continuous and non-decreasing and $\gamma_d(x)\to \infty$ as $x\to \infty,$ the Intermediate Value Theorem provides us with the existence of $\tau\ge 0$ such that $$\gamma_d(\tau)=
\int_\tau^{\tau+1}a^{d}\,da = \int_0^{\infty}a^{d}\,\theta_k(a)\,da,$$ which, by , concludes that $$\label{dk}
\int_{\tau}^{\tau+1}a^{d}\,da=d\,\zeta_k.$$ Now consider the polynomial $$T(x)=x^{ \alpha+d }-\nu_1x^{d}+\nu_2=x^d(x^{\alpha }-\nu_1)+\nu_2$$ where $$\nu_1=\frac{(\tau+1)^{ \alpha+d }-\tau^{ \alpha+d }}{(\tau+1)^{d}-\tau^{d}}>0,\qquad \nu_2=\frac{(\tau+1)^{ \alpha+d }-\tau^{ \alpha+d }}{(\tau+1)^{d}-\tau^{d}}\tau^d-\tau^{ \alpha+d }\ge 0$$ are chosen so that $T(\tau)=0$ and $T(\tau+1)=0$ and $T$ remains negative on $(\tau,\tau+1)$ and positive on $[0,\infty)\backslash [\tau,\tau+1].$ It is immediate to observe that $$T(a)\left(\mathds{1}_{[\tau,\tau+1]}(a)-\theta_k(a)\right)\le 0\quad\mbox{on}\quad [0,\infty).
\label{ineq:01}$$ Integration of on $[0,\infty)$ leads to $$\int_{\tau}^{\tau+1}a^{ \alpha+d }\,da\le \int_0^{\infty} a^{ \alpha+d }\,\theta_k(a)\,da-\nu_1\left( \int_0^{\infty} a^{d}\,\theta_k(a)\,da-\int_{\tau}^{\tau+1}a^{d}\,da\right),$$ simplifying to $$\int_{\tau}^{\tau+1} a^{ \alpha+d }\,da\le \int_0^{\infty} a^{ \alpha+d }\,\theta_k(a)\,da.
\label{eqn:kv}$$ Using , we infer that $$\label{do}
\int_{\tau}^{\tau+1}a^{ \alpha+d }\,da\leq\eta_k \,( \alpha+d ).$$ Observe that $$\label{dkJ}
d\,\zeta_k=\int_{\tau}^{\tau+1}a^{d}\,da\ge \int_{0}^{1}a^d\,da= \frac{1}{d+1}.$$ Notice that gives the key inequality in the proof of this lemma. Indeed, integrating in $p$ from $\tau$ to $\tau+1$ we obtain $$\label{m1}\begin{split}
\int_{\tau}^{\tau+1}a^{ \alpha+d }\,da&\ge \frac{ \alpha+d }{d}b^{\alpha }\int_{\tau}^{\tau+1}a^{d}\,da-\frac{\alpha }{d}b^{ \alpha+d }\\ &\quad +\frac{\alpha }{d} b^{ \alpha+d -3}\int_{\tau}^{\tau+1}(2a+b)(a-b)^2\,da.
\end{split}$$ Note that [@ST4] $$\label{m31}
\int_{\tau}^{\tau+1}(a-b)^2\,da\ge\min_{\tau\ge 0,\; b\ge 1/2}\int_{\tau}^{\tau+1}(a-b)^2\,da=\frac{1}{12}.$$ $$\label{m03}
\int_{\tau}^{\tau+1}a\,(a-b)^2\,da
\ge \min_{\tau\ge 0,\; b\ge 1/2} \int_{\tau}^{\tau+1}a\,(a-b)^2\,da
\ge \frac{1}{2}b^2-\frac{2}{3}b+\frac{1}{4}$$ and so, we have $$\label{m3}
\int_{\tau}^{\tau+1}(2a+b)\,(a-b)^2\,da\ge b^2-\frac{5}{4}b+\frac{1}{2}$$ for any $b\ge1/2$ and $\tau\ge 0.$ Since $(\zeta_k d)^{1/d}\ge (d+1)^{-1/d}\ge 3^{-1/2}\ge 1/2$ due to , setting $b=(\zeta_k d)^{1/d}$ and using and , we deduce that yields to $$\label{m4}
\begin{split}
\eta_k &\ge \frac{1}{ \alpha+d }(\zeta_k d)^{1+\frac{\alpha }{d}}+\frac{\alpha}{d( \alpha+d )}(\zeta_k d)^{1+\frac{\alpha -1}{d}}\\ &\quad -\frac{5\alpha}{4d( \alpha+d )}(\zeta_k d)^{1+\frac{\alpha -2}{d}}+\frac{\alpha}{2d( \alpha+d )}(\zeta_k d)^{1+\frac{\alpha -3}{d}}.
\end{split}$$ Equations in combined with and turn into $$\label{zeta}
\begin{split}
\sum_{j=1}^{k}\lambda_j^{(\alpha)}&\geq \frac{d}{ \alpha+d }\,\omega_d^{-\frac{\alpha }{d}} \,\varphi_k(0)^{-\frac{\alpha }{d}}
\,k^{1+\frac{\alpha }{d}}
\\
&\quad +\frac{\alpha}{( \alpha+d )}\beta^{-1}\,\omega_d^{-\frac{\alpha -1}{d}} \,\varphi_k(0)^{1-\frac{\alpha -1}{d}}
\,k^{1+\frac{\alpha -1}{d}} \\
&\quad -\frac{5\alpha}{4(\alpha+d )}\beta^{-2}\,\omega_d^{-\frac{\alpha -2}{d}} \,\varphi_k(0)^{2-\frac{\alpha -2}{d}}
\,k^{1+\frac{\alpha -2}{d}}\\
&\quad +\frac{\alpha}{2(\alpha+d )}\beta^{-3}\,\omega_d^{-\frac{\alpha -3}{d}} \,\varphi_k(0)^{3-\frac{\alpha -3}{d}}
\,k^{1+\frac{\alpha -3}{d}}.
\end{split}$$ To finish the proof of Proposition \[impBLY\] we shall minimize the right side of over $\varphi_k(0).$ To prove , we show that the function, which we call $\vartheta(x)$, on the right-hand side of with $x:=\varphi_k(0)>0$ decreases on $(0,\Omega_d].$ By we know that $0< x\le \Omega_d.$ First, define $$\label{Sen}
\vartheta(x)= \vartheta_1(x)+\vartheta_2(x)$$ where $$\label{psi1}
\vartheta_1(x)=\frac{d\,
k^{1+\frac{\alpha }{d}}}{( \alpha+d )\,\omega_d^{\frac{\alpha }{d}}} x^{-\frac{\alpha}{d}}
+\frac{\alpha
\,k^{1+\frac{\alpha -1}{d}}}{( \alpha+d )\beta\,\omega_d^{\frac{\alpha -1}{d}}} x^{1-\frac{\alpha-1}{d}}$$ and $$\label{psi2}
\vartheta_2(x)=\frac{\alpha
\,k^{1+\frac{\alpha -3}{d}}}{2( \alpha+d )\beta^{3}\,\omega_d^{\frac{\alpha -3}{d}}} x^{3-\frac{\alpha-3}{d}}
-\frac{5\alpha
\,k^{1+\frac{\alpha -2}{d}}}{4(\alpha+d )\beta^2\,\omega_d^{\frac{\alpha -2}{d}}} x^{2-\frac{\alpha-2}{d}}$$ Thus, it is enough to show that $\vartheta_1$ and $\vartheta_2$ defined by and are also decreasing on the interval $(0,\Omega_d]$. Differentiating $\vartheta_1$ and $\vartheta_2,$ we observe that $\vartheta_1(x)$ and $\vartheta_2(x)$ are decreasing on the interval $(0,x_1)$ and $(0,x_2),$ respectively, where $$x_1=\left(\frac{d\, \beta \,k^{\frac{1}{d}}}{2(d-\alpha+1)\,
\omega_d^{\frac{1}{d}}}\right)^{\frac{d}{d+1}},\qquad x_2= \left(\frac{5(2d+2-\alpha)\, \beta \,k^{\frac{1}{d}}}{2(3d+3-\alpha)\,
\omega_d^{\frac{1}{d}}}\right)^{\frac{d}{d+1}}.$$ Hence, we particularly obtain that $\vartheta$ is decreasing on $(0,\Omega_d]$ when we have $\Omega_d \le
\min\left\{x_1,x_2
\right\}
$ for any $k\ge 1.$ Since $x\mapsto \Gamma(x)$ is increasing for $x\ge 2$ we obtain that $$\Gamma\left(1+\frac{d}{2}\right)\ge \Gamma(2)=1.\label{gamsiz}$$ In view of , $\beta\ge (2\pi)^{-d}\,\omega_d^{-\frac{1}{d}}|\Omega|^{\frac{d+1}{d}}$ and definition of $\omega_d$ given in we observe that $$x_1\ge \left(\frac{d\, |\Omega|^{\frac{d+1}{d}} \,k^{ \frac{1}{d}}}{(d-\alpha+1)\,(2\pi)^{d}\,
\omega_d^{\frac{2}{d}}}\right)^{\frac{d}{d+1}} \ge\left(\frac{\left[\Gamma\left(1+\frac{d}{2}\right)\right]^{\frac{2}{d}} \,|\Omega|^{\frac{d+1}{d}} k^{\frac{1}{d}}}{(2\pi)^{d+1}
}\right)^{\frac{d}{d+1}}\ge \Omega_d,$$ as $k\ge 1$ and $d-\alpha+1\le 2d.$ Similarly, we obtain that $$x_2\ge \left(\frac{5(2d+2-\alpha)\,|\Omega|^{\frac{d+1}{d}} \,k^{\frac{1}{d}}}{2(3d+3-\alpha)\,(2\pi)^{d}\,
\omega_d^{\frac{2}{d}}}\right)^{\frac{d}{d+1}}\ge
\left(\frac{\left[\Gamma\left(1+\frac{d}{2}\right)\right]^{\frac{2}{d}}\,|\Omega|^{\frac{d+1}{d}} \,k^{\frac{1}{d}}}{(2\pi)^{d+1}\,
}\right)^{\frac{d}{d+1}}\ge \Omega_d,$$ as $k\ge 1$ and $5(2d+2-\alpha)\ge (3d+3-\alpha).$ In conclusion, we obtain that $\vartheta(x)\ge \vartheta(\Omega_d)$ as $\vartheta$ is decreasing on $(0,\Omega_d]$. Substitution of $\beta=2(2\pi)^{-d}\,\sqrt{|\Omega|\,{\mathcal I}(\Omega)}$ given in together with $\varphi_k(0)=\Omega_d$ turns into .
\[rmkgen\] In view of the recent work [@Kim; @SV], it is worth noting that one can easily extend this for elliptic operators ${\mathcal E}_f$ defined by a kernel $f$ $$\label{elliptic}
{\mathcal E}_f u(x)=\lim_{\epsilon\to0^+}\int_{\{|y|>\epsilon\}}
(u(x+y)-u(x))\,f(y)\,dy,$$ where $f$ satisfies $$f(y)\ge \sigma \frac{{\mathcal A}_{d,\alpha}}{|y|^{d+\alpha}},\label{ker}$$ ${\mathcal A}_{d,\alpha}$ is the normalizing constant in the fractional Laplacian definition and $\sigma>0.$ To this end, let us consider the eigenvalue problem defined by $$\label{genel}
\begin{split}
-{\mathcal E}_f\,\phi_j &=\lambda_j\,\phi_j \quad \hbox{in} \,\,\Omega,\\
\phi_j &=0 \quad\hbox{in} \,\,{\mathbb R}^d\backslash \Omega
\end{split}$$ It is shown in [@SV] that the spectrum of ${\mathcal E}_f$ is also discrete and the eigenvalues $\{\lambda_j\}_{j=1}^{\infty}$ (including multiplicities) can be sorted in an increasing order. Also, the set of Fourier transforms $\{\hat{\phi}_j\}_{j=1}^{\infty}$ of $\{\phi_j\}_{j=1}^{\infty}$ forms an orthonormal set in $L^2({\mathbb R}^d)$ since the set of eigenfunctions $\{\phi_j\}_{j=1}^{\infty}$ is an orthonormal set in $L^2(\Omega).$ Note that we use the same notation for eigenvalues and eigenfunctions to illuminate the striking similarities though they might be different for each ${\mathcal E}_f.$ Defining $\Phi_k$ as in , we obtain immediately. However, needs to be re-written as the following inequality $$\label{Nsumlambda}
\begin{split}
\sum_{j=1}^k \lambda_j&=\sum_{j=1}^k \langle \phi_j,\lambda_j \phi_j\rangle =\sum_{j=1}^k \langle \phi_j,-{\mathcal E}_f\phi_j\rangle =\sum_{j=1}^k \int_{{\mathbb R}^d} \varrho_{\alpha}(\mu)\,|\hat{\phi}_j(\mu)|^2\,d\mu \\ & \ge \sigma\int_{\mathbb{R}^d} |\mu|^{\alpha }\, \Phi_k(\mu)\, d\mu.
\end{split}$$ where we used the fact (Proposition 3.3. in [@Vald]) that $$\varrho_{\alpha}(\mu)=\int_{{\mathbb R}^d} (1-\cos(y\cdot \mu))\,f(y)\,dy\ge \sigma\,{\mathcal A}_{d,\alpha}\int_{{\mathbb R}^d} \frac{1-\cos(y\cdot \mu)}{|y|^{d+\alpha}}\,dy =\sigma |\mu|^{\alpha}.$$ Having in hand, changes as follows: $$\label{NsumxiB}
\begin{split}
\sum_{j=1}^k\lambda_j&\ge \sigma\int_{{\mathbb R}^d} |\mu|^{\alpha }\,\Phi_k(\mu)d\mu
\\ &\ge \sigma \int_{{\mathbb R}^d} |\mu|^{\alpha }\,\Phi_k^*(\mu)d\mu\\ &=\sigma d\omega_d\int_{0}^{\infty} a^{ \alpha+d-1} \varphi_k(a)\,da. \end{split}$$ and proceeding exactly as before using in place of , we immediately arrive at the following remarkable estimate:
\[CimpBLY\] For $k\ge 1$, and either $1\le\alpha\le 2$ and $d=2$ or $0<\alpha\le 2$ and $d\ge 3,$ the eigenvalues $\{\lambda_j\}_{j=1}^{\infty}$ of defined on $\Omega \subset {\mathbb R}^d$ satisfy $$\label{Cimbound1}
\begin{split}
\sum_{j=1}^k\lambda_j
&\ge(4\pi)^{\frac{\alpha}{2}}\,\frac{\sigma\,d}{ \alpha+d }
\left(\frac{\Gamma\left(1+\frac{d}{2}\right)}{ |\Omega|}\right)^{\frac{\alpha }{d}}k^{1+\frac{\alpha }{d}}
\\& \quad+\frac{\sigma\alpha}
{2( \alpha+d)} \frac{|\Omega|^{\frac{1}{2}-\frac{\alpha-1}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha-1}{d}}}{(4\pi)^{\frac{1}{2}-\frac{\alpha}{2}}\,{\mathcal I}(\Omega)^{\frac{1}{2}}}\,k^{1+\frac{\alpha-1}{d}}
\\& \quad - \frac{5\sigma\alpha}
{16( \alpha+d )}\frac{|\Omega|^{1-\frac{\alpha-2}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha-2}{d}}}{(4\pi)^{1-\frac{\alpha}{2}}\,{\mathcal I}(\Omega)}\,k^{1+\frac{\alpha-2}{d}}
\\& \quad+\frac{\sigma\alpha}{16( \alpha+d )}
\frac{|\Omega|^{\frac{3}{2}-\frac{\alpha -3}{d}}\,\Gamma\left(1+\frac{d}{2}\right)^{\frac{\alpha -3}{d}}}{ {(4\pi)^{\frac{3}{2}-\frac{\alpha}{2}}\,\mathcal I}(\Omega)^{\frac{3}{2}}}\,k^{1+\frac{\alpha -3}{d}}.
\end{split}$$
Note that this estimate also improves the main result of [@Kim], which is simply the multiple of the lower bound stated in [@ST2] by $\sigma$.
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abstract: |
Robert C. Elston was born on February 4, 1932, in London, England. He went to Cambridge University to study natural science from 1952–1956 and obtained B.A., M.A. and Diploma in Agriculture (Dip Ag). He came to the US at age 24 to study animal breeding at Cornell University and received his Ph.D. in 1959. From 1959–1960, he was a post-doctoral fellow in biostatistics at University of North Carolina (UNC), Chapel Hill, where he studied mathematical statistics. He then rose through the academic ranks in the department of biostatistics at UNC, becoming a full professor in 1969. From 1979–1995, he was a professor and head of the Department of Biometry and Genetics at Louisiana State University Medical Center in New Orleans. In 1995, he moved to Case Western Reserve University where he is a professor of epidemiology and biostatistics and served as chairman from 2008 to 2014. Between 1966 and 2013, he directed 42 Ph.D. students and mentored over 40 post-doctoral fellows. If one regards him as a founder of a pedigree in research in genetic epidemiology, it was estimated in 2007 that there were more than 500 progeny. Among his many honors are a NIH Research Career Development Award (1966–1976), the Leadership Award from International Society of Human Genetics (1995), William Allan Award from American Society of Human Genetics (1996), NIH MERIT Award (1998) and the Marvin Zelen Leadership Award, Harvard University (2004). He is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics as well as a Fellow of the Ohio Academy of Science. A leader in research in genetic epidemiology for over 40 years, he has published over 600 research articles in biostatistics, genetic epidemiology and applications. He has also coauthored and edited 9 books in biostatistics, population genetics and methods for the analysis of genetic data.
The original conversation took place on August 4, 2009, during the Joint Statistical Meetings (JSM) in Washington, DC by GZ and ZL. NLG had dinner with RCE during the 2013 JSM in Montreal, Canada, and added supplementary material and edited the conversation. RCE updated and clarified certain points.
address:
- 'Gang Zheng was a Mathematical Statistician, Office of Biostatistics Research, National Heart, Lung and Blood Institute, Bethesda, Maryland 20892-7913, USA. He passed away on January 9, 2014, without completing this.'
- 'Zhaohai Li is a Professor of Statistics and Biostatistics and the Chair, Department of Statistics, George Washington University, Rome Hall, 5th Floor, 801 22nd St. NW, Washington, DC 20052, USA .'
- 'Nancy L. Geller is the Director of Office of Biostatistics Research, National Heart, Lung and Blood Institute, 6701 Rockledge Drive, Bethesda, Maryland 20892-7913, USA .'
author:
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title: 'A Conversation with Robert C. Elston'
---
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Early Education
===============
**Gang and Zhaohai:** Robert, it is a great pleasure to have this opportunity to talk with you about your life, research, career, mentorship and some of your views of genetic epidemiology.
Can you begin by telling us about your early years?
**Robert:** I was born in London, and I was 7 years old when World War II broke out (1939). My brothers and I were evacuated to a little village, Lea Green in Hertfordshire, about 30 miles from London. That’s where I first loved farming and thought I’d be a farmer when I grew up. In 1941, my father arranged for us to live in Hertford, where Battersea Grammar school had been evacuated from London. I don’t know how he got me into that grammar school since I was really too young. They put me in the lowest form (grade). I eventually took what was called the school certificate at 14 while most took it when they were at 16.
**Zhaohai:** What did you study in high school?
**Robert:** In addition to the usual subjects, we studied French and a year later got to choose Latin or German. I did Latin mainly because my elder brothers had done Latin, and because I knew I needed Latin to go to Oxford or Cambridge. The following year, the class master who taught Latin chose the two or three best students and said: Okay, you will do Greek. My brothers had done Greek but they had to give up physics to do Greek and I was not going to give up physics! I said, if you want me to do Greek, I’ll need to eliminate history or geography or both. So they agreed!
When World War II ended, Battersea Grammar school moved back to London, so I had one year at Hertford Grammar school. In 1946, we returned to London as a family. Then I went to University College School, which was “a public school,” meaning it was open to anyone who was willing to pay (laughs). Although I studied Latin and Greek and the classics for two years, I also wanted to do science. As I was young, I could stay there in the sixth form for four years, mixing 2 years of classics (Latin, Greek, a little French and ancient history) with 2 years of biology, physics and chemistry. I never studied calculus, highly unusual in the US, but not so unusual at that time for a science student in England. When I got to the states, I estimated that what we did in England in the sixth form was equivalent to one or two years of undergraduate work in the states.
Cambridge University
====================
**Gang:** How did you get to Cambridge University?
**Robert:** In those days, the way you got to a university was either you were rich or royalty, or you sat for a scholarship examination. I applied to both Reading and Cambridge Universities and had to take a scholarship examination at each. The scholarship wasn’t much in terms of the money, but if you passed the scholarship exam, the local government would pay for your education.
**Zhaohai:** So you got the scholarship?
**Robert:** No, I actually failed. For Reading University, the three subjects I chose for the exam were Greek, French and chemistry. They thought that combination quite useless. At Cambridge, there were six of us competing for one scholarship. At that time, the School of Agriculture at Cambridge had a three-year Bachelor’s degree in agriculture. I had already decided to do something in agriculture. But I had taken a special scholarship exam at Magdalene College for people who would spend four years. The first two years would be Part I of the natural science tripos. \[A tripos is the course system at the University of Cambridge.\] Then at the end of the second year, I would have a choice, either a two year diploma in agriculture or continue with Part II of the natural science tripos followed by a one year diploma in agricultural science. Although I didn’t get that scholarship, I did well in the exam, and they said they would accept me into Magdalene College for the four-year program in two years’ time, with the government giving me some support. Why two years? Because they “knew” I would have to serve two years in the military. So I had those two years to spend. Since I was going to study agriculture, I was able to get a deferment from the military to work on a farm, but I remained eligible for military service and could be called up later. So I worked on a farm for a year and then spent my next year in France where I perfected my French (my mother’s mother tongue).
**Gang:** What happened after those two years?
**Robert:** I returned to England and went to Cambridge. For the natural science tripos, I spent two years doing work for Part I. I had to have three science subjects. My original idea was to do botany, zoology and chemistry (organic and biochemistry). After one year, I really didn’t like botany and decided I wanted to do mathematics. So I changed from full subject botany to half subject botany and half subject mathematics. I had to teach myself calculus, which I did with a little book called *Calculus Made Easy* by Silvanus P. Thompson ([-@Tho46]). Clearly, I needed private tutoring for mathematics. Wally Smith (Walter Laws Smith), whom I knew from my extracurricular activity on the stage (we were both members of the Pentacle club, which was a magic club), became my mathematics supervisor. He told me I wasn’t very good at mathematics, but I stuck it out! I know mathematics lowered my exam result at Cambridge!
**Zhaohai:** What did you do next? Did you get to do genetics at Cambridge?
**Robert:** My first choice for a Part 2 tripos was biochemistry, which I really enjoyed, but between the lectures and the labs, the hours were too long, so I went to my tutor for advice. I said I was thinking about Part 2 genetics. I remembered his words so well, “You know this program in genetics here is new. And this man \[R. A.\] Fisher is considered eccentric by some, and it may not stand you in good stead in later life for it to be known you worked with him.” That is why I did not do a year with Fisher! So I ended up doing the two-year diploma in agriculture.
**Gang:** Who taught you statistics at Cambridge?
**Robert:** I had lectures from four people. Dennis Lindley gave us three weeks on statistics as part of the half-subject mathematics. He taught me significance testing. He did not believe in it but he taught it! In the same year, I did have lectures from Fisher because those who did zoology could do an optional series of lectures with Fisher on genetics. So I did do genetics with Fisher. I’ll tell you a joke he told (translated into American). I did not know what he was doing with adding, subtracting and dividing for a 2 by 2 table. He came up with this number. He said, “Now, I am going to call this number chi-squared. Don’t be alarmed. I know you are all biologists. It is no worse than calling a dog ‘Lassie’!” He said, “If this number is greater than 4, perhaps there is something going on.” This was in a lecture hall which could hold 200 people. At the first lecture, there might have been 150 people; second lecture, 50; third, about 15! He made us all sit in the front row. He could sense his audience. I don’t remember exactly what he said, but it was something like: if I say something is always transmitted from mother to daughter, then clearly it is never transmitted from mother to son. Then he said, “I do hope I’m not making a mistake in logic. Do stop me if I make a mistake in logic, won’t you?” He could sense that we were all thinking about the truth of what he had said!
While I was spending a year in France, I had read Fisher’s book for research workers ([-@Fis50]) and his book on experimental design ([-@Fis51]). So one day after lecture I said to Fisher, I read your book. What is the difference between a standard deviation and a standard error? He looked me up and down and said: “Your height is a deviation from the mean. It is not an error.”
Then I also had lectures with Anscombe and R. C. Campbell, because they were teaching agriculture students. There we learned about experimental and split-plot design, and basic statistics. We also knew how to calculate F-statistics using a hand calculator. And I had lectures from Wishart. Wishart used a little book he had written with Sanders called *Principles and Practice of Field Experimentation* ([-@WisSan55]). He taught us how to lay out plots in the field and for agricultural experimental designs. We were just agriculture people! He was in the School of Agriculture.
**Zhaohai:** And you still wanted to be a farmer after you got your diploma in agriculture?
**Robert:** What else was I going to do? I learned how to run a farm in England and knew quite a lot of animal physiology, plant physiology, soil science, how to work out the feeding and animal nutrition, but I had no capital. To be a farmer, you needed capital for the land and the machinery. I couldn’t afford a farm.
Coming to the US for a Ph.D.
============================
**Zhaohai:** How did you end up coming to US for your Ph.D. in animal breeding?
**Robert:** I got the B.A. in 1955. The way it worked was that two years after you got your B.A., you could pay 10 guineas and you got an M.A. So I had an M.A. My mother wanted me to take an academic job. I saw a notice: Fellowships to America. All right, I thought, I’ll just go to America for just one year. These were King George VI memorial fellowships from the English Speaking Union of the US and they were giving about 25 scholarships a year. You could have up to three choices of where you wanted to go, but you had to sign that you would go wherever they sent you. My choices were UC Davis where Michael Pease was doing chicken breeding or Ames, Iowa, which was known for dairy cow breeding. I left the third choice blank. They sent me to Cornell, where there was a department of animal husbandry. There I was sent to Chuck (Charles Roy) Henderson, who said all his students minored in biometry, and suggested that I go to see Professor Federer. So I went to see Walter Federer, who asked me why I did not stay for a Ph.D. I said I had only money for a year. He told me that they would find me money.
I had to return to England at the end of the year because I was called for military service; and if I passed my 26th birthday outside of England, I could have been called up to age 36. Again I avoided military service, this time by working on a pig farm. I was able to leave England before my 26th birthday because the farm owner was willing to say I was still there. I actually spent my 26th birthday on the high seas en route back to Cornell. I did this so I could get to Cornell when the semester began.
So I returned to study animal breeding for a Ph.D. with Chuck Henderson with minors in biometry and mathematics. My thesis was in mixed model nonorthogonal ANOVA. We had one of the first computers, an IBM 650. I spent three months with punch cards to invert a $79\times 79$ matrix!
**Zhaohai:** Who were your contemporaries in graduate school at that time?
**Robert:** I was exactly contemporaneous with a student of Chuck Henderson’s, Shayle Searle. He had a degree in mathematics and a diploma in statistics. Chuck Henderson had just spent a year at the New Zealand Dairy Board with Shayle Searle and recruited him to be a graduate student at Cornell. I learned a lot of from both of them. Chuck Henderson told me he was an animal scientist, not a statistician. The reason he was doing BLUP (Best Linear Unbiased Prediction) was because statisticians wouldn’t do it for him. He never considered himself as a statistician at all.
From Cornell to University of North Carolina at Chapel Hill (UNC)
=================================================================
**Gang:** How did you choose a post-doctoral fellowship in statistics after finishing up your Ph.D. in animal breeding?
**Robert:** I was going to finish my Ph.D. in the summer of 1959. I didn’t know what to do next. Walter Federer advised me to do a post-doctoral fellowship in statistics. He said I should apply to Princeton, where there were fellowships in statistics for biologists. They paid \$5000 a year, tax free. So I applied. Before I heard from Princeton, I drove to Miami for an international student conference. On my way back, I stopped at Chapel Hill to see my old friend Wally Smith, who had moved to the Department of Statistics there. He told me they could offer me \$4,800 a year as a tax-free fellowship in the Department of Biostatistics, as Bernard Greenberg, the chairman, had some money. So I went to UNC at Chapel Hill. I wrote to Princeton that I was no longer interested, and received the nicest letter back from Sam Wilks. From that experience, I learned that when you are trying to recruit students or post-docs, write a nice letter.
**Zhaohai:** How long was your post-doc at Chapel Hill?
**Robert:** Just one year. Bernie Greenberg insisted that teaching was part of the training for all students, pre- and post-docs, so I became a teaching assistant for Statistics 101 for public health. My affiliation was with biostatistics, but I had my office in the statistics department. In addition, I took 5 theoretical statistics courses in each of the two semesters, although I did not do all the homework. I took courses in multivariate analysis from S. N. Roy and Norman Johnson, response surface designs from R. C. Bose, experimental design from Indra Chakravarti and David Duncan, nonparametric statistics from Wassily Hoeffding (U-statistics) and mathematics for statistics from Wally Smith. During that time, James Durbin, Maurice Kendall and E. J. Hannan (time series) were visitors. I also published work from my dissertation, my first paper in *Biometrics* ([-@Els61]).
**Gang:** You spent most of 1959 to 1979 at UNC Chapel Hill. Since you came as a post-doctoral fellow, how did you manage to stay?
**Robert:** In order for me to stay at Chapel Hill, Bernie Greenberg suggested a job as a Research Assistant Professor of Pathology to work on a project for the blood bank, which he thought must be related to genetics, my interest. I worked on a project to estimate the amount of blood that the blood bank should keep on hand (Elston, [-@Els62]; Elston and Pickrel, [-@ElsPic63]). I simulated blood units being purchased by the blood bank and being sent out for use (Elston, [-@Els62]).
Computers and statistics had not been used before in blood banking. Analyzing six months of the blood bank’s records, I found I could fit a negative binomial distribution to the number of units that came into the bank for seven of the eight major blood types, but not for the O negative blood data. That was because O negative blood can be transfused into anyone and so O negative donors were often requested to donate blood, rather than donating simply at random. The blood bank director was impressed with that finding purely by statistical analysis. I ended up writing several other papers on the blood bank project (Elston and Pickrel, [-@ElsPic65]; Elston, [-@Els66; -@Els68; -@Els70]).
Toward the end of that second year, Bernie Greenberg said he needed someone to teach bioassay the next year, and did I know anything about it? I said I knew a little, and yes, I could teach that, but in truth, I was one chapter ahead of the students most of the time.
**Gang:** But then there was this two year gap, 1962–64 when you went to Aberdeen. Why was that?
**Robert:** My third year at Chapel Hill was the sixth year I had been in the US. I had a J-1 visa and US law required that you had to return to your home country for at least two years. Exceptions to stay in the US were only by an act of Congress. David Finney contacted me for a permanent position as a Senior Biometric Fellow in Aberdeen Scotland and, since I couldn’t stay in the US, I accepted. My wife and I didn’t like the idea of going to Aberdeen very much, but this was a permanent position. (I had just got married in Chapel Hill and my wife came from Gloucestershire, 100 miles west of London, so we were both from southern England.) After being in Aberdeen for about six months, I put down a deposit on a house, and the next day I got a letter from Bernie Greenberg asking me to come back to Biostatistics as an Associate Professor after the required two years outside the US. He asked me what salary I would want and when I named the largest salary I dared, he offered me 25% more. I was trapped! This time I came to the US with a green card. We returned to Chapel Hill with a nine-month old daughter, and for her to get her visa, I had to sign on her behalf that she wasn’t coming into the US for the purpose of becoming a prostitute; she remarked recently that she kept her half of the bargain!
**Gang:** Describe the Biostatistics Department on your return.
**Robert:** I was the 6th or 7th faculty member of the department. The department grew with the help of federal grant support. In the mid-sixties, we were tremendously successful. In 1966, I managed to get a five-year Career Development Award and then a five-year renewal. There was an interdepartmental training grant in genetics and biostatistics had its own training grant, but there was no Ph.D. in biostatistics. At that time, the Ph.D. students funded by the departmental training grant took either a Ph.D. in Experimental Statistics at Raleigh or a Ph.D. in Statistics at Chapel Hill, with a minor in Public Health. I wanted a Ph.D. program in biostatistics with a minor in genetics to have students funded by the interdepartmental training grant, so I wrote the Ph.D. proposal. Greenberg was told it wasn’t broad enough, so I rewrote it allowing for minors in genetics, demography and other fields as well. The Ph.D. program in Biostatistics officially began in 1968 with Rose Gaines-Das being the first to get a Ph.D. in biostatistics, with a thesis in statistical genetics.
Initially, it was difficult for my students to get positions. That’s why Joe Haseman went to the National Institute of Environmental Health Sciences; there were no academic positions in statistical genetics. Haseman could not get a job in statistical genetics despite the fact that Haseman and Elston (*Behavior Genetics*, [-@HasEls72], from Haseman’s dissertation) became the most cited paper ever published in *Behavior Genetics*.
By now, many of my Ph.D. students funded on that training genetics grant are retired. I can’t imagine why!
**Gang:** While you were at UNC, you did a lot of traveling. How did you manage that?
**Robert:** With the Career Development Award, my position didn’t cost the university and Bernie Greenberg said I could do whatever I wanted because it didn’t cost him anything. This allowed me to visit the University of Hawaii to work with Newton Morton and D. C. Rao for one year and, during summers I had further trips to Hawaii and England (the Galton Laboratory in London and the University of Cambridge). When I visited the Galton laboratory (1967), I met John Stewart, who was a graduate student at Cambridge. We ended up writing a paper together, which appeared in *Human Heredity* ([-@ElsSte71]), a minor journal at that time. We computed the likelihood of the model for the observed phenotype data in a given pedigree. We could handle large pedigrees and relatively few markers. I didn’t then know I was using Bayes’ theorem recursively to compute the likelihood. Stewart’s contribution was to apply the result to linkage. In the discussion, Stewart wrote that this paper answered a fundamental question in human genetics, that is, is some phenotype polygenic or is there a major gene? It was Ken Lange who named this “the Elston–Stewart algorithm.” It was overall a most productive time.
From UNC to Louisiana State University Medical Center (LSUMC)
=============================================================
**Gang:** Why did you leave UNC for LSU in 1979?
**Robert:** I moved to LSU for two major reasons. I went to New Orleans for the ENAR meeting and they wanted me to come there to be chair of the Department of Biometry in the LSU Medical Center. They offered me a hard money position. That was the first reason: all positions at UNC were soft money and by then I had four children, all within six years of age, to put through college. The second reason was that at UNC I had gotten a grant which allowed me to purchase my own computer and the university would only let me house it in the computer center. At that time, nobody in the School of Public Health was permitted to have his own computer. So those were the primary reasons that I left.
**Zhaohai:** Tell us about your years at LSU.
**Robert:** Even though I had a hard money position, I kept writing grants. Because I had all of these federal grants, I was able to start a Ph.D. program in statistical genetics and expand the faculty. I wrote four proposals for Ph.D.s and masters’ degrees in Biometry and Genetics. Alec Wilson, Joan Bailey-Wilson and George Bonney became part of my faculty.
**Gang:** What kind of training did you give at LSU?
**Robert:** I trained several post-docs there. I especially like to train statisticians to do genetics. At LSU, I had a training grant from NHLBI which was initially only to train post-docs. From 1992–1993, Dan Schaid of the Mayo Clinic was my post-doc. I remember the year because Hurricane Andrew hit Louisiana that year. It was supposed to hit New Orleans. We boarded up the windows and left for our eldest daughter’s wedding in Ann Arbor. We thought we might not have a house when we got back. But the hurricane missed New Orleans. Dan Schaid completed his year with me and went back to the Mayo Clinic and was able to analyze the genetic data that they had been collecting. He is now a leader in the field of genetic epidemiology.
**Zhaohai:** Why did you leave LSU?
**Robert**: My faculty was good and I wanted to raise their salaries. The administration said there were no faculty raises and “no exceptions.” Of course, there were exceptions! That’s why I’m a lousy administrator: I refuse to lie! In one of the following years, the Chancellor wrote a letter to the department heads saying that again there would be no faculty raises and noted that good people would leave and “this should be taken as an opportunity.” I was fed up with being a department head anyway and had \$1,000,000 in grant money. That and the climate were the reasons I decided to leave!
From LSU to Case Western Reserve University (CWRU)
==================================================
**Zhaohai**: You have been at CWRU since 1995. Why did you choose CWRU?
**Robert:** My wife hates the heat, so staying in the south was out of the question. She wanted to go to Maine or Vermont, so Cleveland was a compromise. I accepted a full professorship at CWRU without administrative responsibilities so I could get some work done!
**Gang:** Tell us about the department when you arrived.
**Robert:** I was hired by an epidemiologist, Alfred Rimm. He wanted me to have my own division, so we called it Genetic and Molecular Epidemiology. The department is really a mini-school of Public Health. Aside from my division, it had divisions of Epidemiology, Biostatistics, Health Services Research and Public Health. The names have changed over the years, but with the exception that there are no longer formal divisions, the structure is the same.
When I moved, only two people from LSU came with me, Xiuqing Guo, a graduate student, and Hemant Tiwari, a post-doc. I was also able to take my training grant in biometric genetic analysis because nobody remained at LSU who could do the work. Joan Bailey-Wilson and Alec Wilson moved to the National Human Genome Research Institute (NHGRI) of NIH because they had family nearby, in Baltimore. I was also able to take my computers.
Al Rimm asked me to do genetics only, not biostatistics and I did that for over ten years. One project was S.A.G.E. (Statistical Analysis forGenetic Epidemiology), which I had started in New Orleans, funded by an NIH Resource Grant. I also took that with me. The Resource Grant required collaborations, providing a service for which you had to charge (the S.A.G.E. software), training and dissemination (S.A.G.E. courses). Initially, there was a charge for S.A.G.E. because the grant required that. Beginning in 2005, we were able to distribute S.A.G.E. for free (see Elston and Gray-McGuire, [-@ElsGra04] and <https://code.google.com/p/opensage/>). Version 6.2 was meant to be web based so people could use other programs with it, but funding to complete this project never materialized.
**Gang:** How come you became department chair at CWRU?
**Robert:** In 2008, Al Rimm resigned as chair and they asked me to be interim department chair and I agreed. They needed a real chair to apply for stimulus money, so in 2009, they took the “interim” away. They continued to advertise for a real chair and it took several years—until now (2014)—to fill the position. Call it a second childhood!
![Robert Elston at Case Western Reserve University, 2007.[]{data-label="f4"}](497f04.eps)
**Zhaohai:** How did you arrange your time as chair on administration, research and mentoring graduate students and post-docs?
**Robert:** When I first became chair, we needed to reorganize our Ph.D. program. Under the previous chair, the department was acting as though each division had a separate Ph.D. program. The different divisions found it hard to agree on one Ph.D., but the graduate school did not recognize multiple Ph.D. programs in one department. So I put a lot of effort into establishing the one Ph.D. program in Epidemiology and Biostatistics, with several different concentrations. When that was done, with several faculty committees to make sure that the program ran smoothly, a lot of the administration was taken care of. And I have always considered my research and mentoring of students, both pre- and post-docs, to be all part of one and the same thing.
These days I spend a lot of time writing grants. It’s getting harder to get them. Renewal of my research grants and the training grant is taking more time than I wish. It’s hard to know how much detail the reviewers want. Sometimes the projects described in the grant proposal get published before the grant gets funded! And of course, if we don’t get funding, we won’t be able to support Ph.D. students to do research.
I also spend a lot of time helping others. For junior faculty, I tell them not to put my name on their paper as an author because when they come up for tenure, people may think the paper was my idea and not theirs. If my name is on a paper, you can be sure I really contributed something. I don’t notice all the authors when I am reading a paper, but I find that people notice if my name is on a paper. So I have to be sure that every sentence is accurate. My purpose is to be pedagogical as well as do research, and this makes me very fussy about proper wording and clarity. I read the galley proofs personally and I have my secretary read them, too. I learned that at Aberdeen from Finney. He had a sign up in the tea room which said, “No paper leaves this department without the Professor’s permission.”
I still work on family studies, although that has become less fashionable than case-control GWAS. My recent work is still a mixture of theoretical and applied and I still enjoy writing and publishing with students.
**Gang:** Do you ever plan to retire?
**Robert:** I don’t know when I shall retire—probably when I am no longer able to get grants to fund my research. I have four children and ten grandchildren. By the way, three of my children are university professors in mathematics/health sciences, and the one who isn’t decided to be creative and studied acting. She puts on the high school play every year (and her older son is majoring in mathematics at college). I look forward to spending more time with my family when I do .
Summing Up
==========
**Gang**: You have directed 40 Ph.D. theses and had 45 post-docs. By now they too have had trainees. What does your “research pedigree” look like?
**Robert:** My research pedigree has more than 500 progeny The International Genetic Epidemiology Society had a special tribute for me on my 70th birthday and someone drew it out. At that time, half of the field of genetic epidemiology was in my pedigree. There are at least four generations.
**Gang:** How would you sum up your career?
**Robert:** Like that of many other academics, my career path was an accident. When I talk to others in academia, most of the time they had no idea what field they would end up in. In my case, I didn’t even expect to go to academia. But once I decided to apply statistics to genetics, I think I made a happy choice; and I’m glad I decided early on to make all of my students collaborators. I only hope they learned as much from me as I from them.
**Gang:** What is your advice to a young statistical geneticist starting out today?
**Robert:** My advice is quite simple. First, make sure you keep learning as much statistics as you can and second, keep up to date with computing technology. Statistical genetics may go out of fashion, but there will always be a need for statisticians who can compute.
**Zhaohai:** Do you have any closing comments?
**Robert:** It is really nice to have this conversation appear in *Statistical Science*. I actually never considered myself to be a statistician. I was a geneticist among the statisticians and a statistician among the geneticists!
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---
abstract: 'We revisit the Chiral Magnetic Effect (CME) using the chiral Lagrangian. We demonstrate that the electric-current formula of the CME is derived immediately from the contact part of the Wess-Zumino-Witten action. This implies that the CME could be, if observed, a signature for the local parity violation, but a direct evidence for neither quark deconfinement nor chiral restoration. We also discuss the reverse Chiral Magnetic Primakoff Effect, i.e. the real photon production through the vertex associated with the CME, which is kinematically possible for space-time inhomogeneous configurations of magnetic fields and the strong $\theta$ angle. We make a qualitative estimate for the photon yield to find that it comparable to the thermal photon.'
author:
- Kenji Fukushima
- Kazuya Mameda
bibliography:
- 'photon.bib'
title: 'Wess-Zumino-Witten action and photons from the Chiral Magnetic Effect'
---
The vacuum structure in Quantum Chromodynamics (QCD) has been an important subject investigated in theory for a long time. It has been well-known that gauge configurations with topologically non-trivial winding such as the instanton, the magnetic monopole, etc should play a crucial role in the spontaneous breaking of chiral symmetry [@Schafer:1996wv], color confinement [@Diakonov:2009jq], the mass of $\eta'$ meson [@'tHooft:1986nc], and the strong $\theta$ angle [@Kim:2008hd].
Among others the problem of the strong $\theta$ angle is still posing a theoretical challenge. There is no consensus on the unnatural smallness of $\theta$ and thus the absence of [ $\mathcal{P}$ ]{}and [ $\mathcal{CP}$ ]{}violation in the strong interaction. Recently, more and more researchers in the field of the relativistic heavy-ion collision are getting interested in the possibility of fluctuating $\theta$ in a transient state of QCD matter and searching for a signature to detect the local [ $\mathcal{P}$ ]{}violation (LPV) experimentally [@Morley:1983wr; @*Kharzeev:1998kz; @*Fugleberg:1998kk; @*Buckley:1999mv; @Voloshin:2000xf; @*Voloshin:2004vk; @*Finch:2001hs; @*Abelev:2009ac].
In this context the discovery of the Chiral Magnetic Effect (CME) [@Kharzeev:2007tn; @*Kharzeev:2007jp; @Fukushima:2008xe] has triggered constructive discussions and lots of works have been devoted to the interplay between the topological effects and the external magnetic field ${\boldsymbol{B}}$ [@Son:2007ny; @Mizher:2008hf; @*Mizher:2010zb; @Asakawa:2010bu], while the strong-${\boldsymbol{B}}$ effect itself on nuclear or QCD matter had been [@Rafelski:1975rf] and are still [@D'Elia:2010nq; @*Bali:2011qj] attracting theoretical interest. (See Ref. [@Vilenkin:1980fu; @*Metlitski:2005pr] for earlier works related to the CME.) If $\theta$ temporarily takes a non-zero value in hot and dense QCD matter, its time derivative induces an excess of either left-handed or right-handed quarks. Because of the alignment of the spin and the momentum directions of left-handed and right-handed quarks, ${\boldsymbol{B}}$ would generate a net electric current parallel to ${\boldsymbol{B}}$, which may be in principle probed by the fluctuations of $\mathcal{P}$-odd observables in the heavy-ion collision [@Voloshin:2000xf; @*Voloshin:2004vk; @*Finch:2001hs; @*Abelev:2009ac].
It should be an urgent problem of paramount importance, we believe, to sort out the proper physics interpretation of the CME and the LPV in general since the LPV is under intensive investigations in ongoing experiments at present. It is also under active discussions whether the Chiral Magnetic Wave (CMW) should account for the discrepancy between the elliptic flows of positively and negatively charged hadrons [@Kharzeev:2010gd; @*Burnier:2011bf]. It is often said that the CME could be a signature for quark deconfinement and chiral symmetry restoration, as stated also by one of the present authors in Ref. [@Fukushima:2008xe]. This was conjectured because the intuitive explanation for the CME seemed to require almost massless $u$ and $d$ quarks. One should be, however, careful of the physics interpretation of anomalous phenomena which sometimes look counter-intuitive. The first half of our discussions is devoted to considerations on the implication of the CME in terms of the chiral Lagrangian. We conclude that the CME is insensitive to whether the fundamental degrees of freedom are quarks or hadrons, so that it could be seen without deconfinement. Chiral symmetry restoration is, on the other hand, necessary to realize the hadronic LPV in the same manner as in the case of the disoriented chiral condensate (DCC) [@Anselm:1989pk; @*Anselm:1991pi; @*Blaizot:1992at; @*Rajagopal:1993ah].
In the last half of our discussions, as an application of the chiral Lagrangian, we address the real photon production through the process that we call the reverse Chiral Magnetic Primakoff Effect. The typical process in the ordinary Primakoff effect is the $\pi^0$ (or some neutral meson generally) production from a single photon picking up another photon from the external electromagnetic field [@Primakoff:1951pj]. In the relativistic heavy-ion collision the neutral pseudo-scalar field $\theta(x)$ can couple to a photon in ${\boldsymbol{B}}$ leading to a single photon emission, i.e. $\theta + B \to \gamma$, which can be viewed as a reverse process of the Primakoff effect. Such a mechanism for the photon production can be traced back to the old idea to detect the axion via the Primakoff effect [@Sikivie:1983ip; @*Raffelt:1987im], and is similar to the recent idea on a novel source of photons from the conformal anomaly [@Basar:2012bp]. In short, a crucial difference between our idea and that in Ref. [@Basar:2012bp] lies in the neutral meson involved in the process — $\sigma$ meson (which turns to a hydrodynamic mode) in the conformal anomaly case and $\theta$ or $\eta_0$ in our case of the CME vertex (see also Ref. [@Fukushima:2002mp] for the diphoton emission from the $\sigma$ meson). From this point of view, it would be very natural to think of photons as a signature of the CME; instead of the axion [@Sikivie:1983ip], CME requires a *background* $\theta(x)$, which may cause the same process of the single photon production as the axion detection.
The interesting point in our arguments for the photon production is that the real photon emission is attributed to exactly the same vertex as to describe the electric-current generation in the CME. As long as ${\boldsymbol{B}}$ and $\theta$ are spatially homogeneous, as often assumed for simplicity, the real particle production is prohibited kinematically, but once ${\boldsymbol{B}}$ and $\theta$ are space-time dependent (and they are indeed so in the heavy-ion collision!), the energy-momentum conservation is satisfied, so that the real photon can come out.
Here, one might have wondered how the physical constant $\theta$ can be lifted up in hot and dense matter and treated as if it were a particle. In other words, what is the origin of the chiral chemical potential $\mu_5$ in the hadronic environment? This is an important question and related to the physical mechanism to cause the LPV. At extremely high energy the Color Glass Condensate and the Glasma initial condition [@Kharzeev:2001ev] may be the most relevant and its characteristic scale is then given by the saturation scale $Q_s$. In this case the role of $\theta$ in the pure Yang-Mills dynamics is more non-trivial [@D'Elia:2012vv] than full QCD with dynamical quarks where $\theta$ can be regarded as the $\mathrm{U(1)_A}$ rotation angle. In the hadronic phase at low energy, the chiral Lagrangian provides us with a clear picture, which consists of three parts, $${\mathcal{L}}_{\text{eff}} = {\mathcal{L}}_\chi + {\mathcal{L}}_{\text{WZW}} + {\mathcal{L}}_{\text{P}}\;,$$ where the first one is the usual chiral Lagrangian that is given by [@DiVecchia:1980ve; @*Witten:1980sp; @*Leutwyler:1992yt; @Kaiser:2000gs; @*Kaiser:2000ck] $$\begin{split}
{\mathcal{L}}_\chi &= \frac{f_\pi^2}{4} {\text{tr}}\bigl[ D_\mu U^\dagger
D^\mu U + 2\chi (MU^\dagger + UM ) \bigr] \\
&\qquad\qquad -\frac{{N_{\text{f}}}\chi_{\text{top}}}{2}
\Bigl[\theta-\frac{i}{2}{\text{tr}}(\ln U - \ln U^\dagger)
\Bigr]^2\;,
\end{split}$$ in the lowest order including the topological terms that break $\mathrm{U(1)_A}$ symmetry. Here, $\chi_{\text{top}}$ represents the pure topological susceptibility, the covariant derivative involves the vector and the axial-vector fields as $D_\mu U \equiv \partial_\mu U - i r_\mu U + i U l_\mu +
\frac{i}{2}(\partial_\mu \theta + 2{\text{tr}}(a_\mu)) U$ with $r_\mu \equiv v_\mu + a_\mu$ and $l_\mu = v_\mu - a_\mu$, and $\chi\equiv -\langle\bar{q}q\rangle/f_\pi^2$ from the Gell-Mann-Oakes-Renner relation. It is obvious that, as discussed in Ref. [@DiVecchia:1980ve; @*Witten:1980sp; @*Leutwyler:1992yt], the $\theta$-dependence is to be absorbed in the phase of $U$ if the current quark mass matrix $M$ has a zero component. Then, one can understand that $\theta$ and the phase of $U$ or $\eta_0/f_{\eta_0}$ are simply identifiable apart from the mass terms proportional to $\chi$ and $M$. This means that, if the system has the DCC in the iso-singlet channel $\eta_0(x)$ and if $\chi M\simeq 0$ at high enough $T$, we can interpret this $\eta_0(x)$ as an effective $\theta(x)$ in a transient state (this reinterpretation exactly corresponds to the normalization condition for $U$ in Ref. [@Kaiser:2000ck]). We note that in the whole argument this is the only place where (partial) chiral symmetry restoration is required in the hadronic picture of the CME. Hence, in the hadronic phase, the DCC of $\eta_0$ is the source for $\mu_5(x)$. Its strength and distribution could be in principle figured out in numerical simulations as in Ref. [@Ikezi:2003kn].
The anomalous processes such as $\pi^0\to \gamma\gamma$ and $\gamma \pi^0\to\pi^+ \pi^-$ are described by the Wess-Zumino-Witten (WZW) part that can be written in a concise way in the two-flavor case [@Kaiser:2000ck] as $$\begin{split}
& {\mathcal{L}}_{\text{WZW}} = -\frac{{N_{\text{c}}}}{32\pi^2}\epsilon^{\mu\nu\rho\sigma}
\Biggl[ {\text{tr}}\Bigl\{ U^\dagger\hat{r}_\mu U\hat{l}_\nu
- \hat{r}_\mu\hat{l}_\nu \\
& + i\Sigma_\mu (U^\dagger\hat{r}_\nu U \!+\! \hat{l}_\nu) \Bigr\}
{\text{tr}}(v_{\rho\sigma}) \!+\! \frac{2}{3} {\text{tr}}\bigl( \Sigma_\mu \Sigma_\nu
\Sigma_\rho \bigr)\, {\text{tr}}(v_\sigma) \Biggr]
\end{split}
\label{eq:WZW}$$ with $v_{\mu\nu}\equiv\partial_\mu v_\nu-\partial_\nu v_\mu-i[v_\mu,v_\nu]$, and $\Sigma_\mu \equiv U^\dagger\partial_\mu U$. A hat symbol indicates the traceless part, i.e.$\hat{r}_\mu \equiv r_\mu - \tfrac{1}{2}{\text{tr}}(r_\mu)$ and $\hat{l}_\mu \equiv l_\mu - \tfrac{1}{2}{\text{tr}}(l_\mu)$. There is one more part that has no dynamics of chiral field $U$ and thus is called the contact part; $$\begin{aligned}
&{\mathcal{L}}_{\text{P}} = \frac{{N_{\text{c}}}}{8{N_{\text{f}}}\,\pi^2}\epsilon^{\mu\nu\rho\sigma}
\Biggl\{ {\text{tr}}\Bigl[ v_\mu \Bigl( \partial_\nu v_\rho
-\frac{2i}{3}v_\nu v_\rho \Bigr)\Bigr] \partial_\sigma\theta \\
& + {\text{tr}}\bigl( a_\mu D^v_\nu a_\rho \bigr) \Bigl( \frac{4}{3}
{\text{tr}}(a_\sigma) + \partial_\sigma \theta \Bigr)
\!-\!\frac{2}{3{N_{\text{f}}}}{\text{tr}}(a_\mu) {\text{tr}}(\partial_\nu a_\rho)
\partial_\sigma \theta \Biggr\}\;, \notag\end{aligned}$$ where $D^v_\mu a_\nu\equiv\partial_\mu a_\nu-i v_\mu a_\nu-ia_\mu v_\nu$.
Now that we have the chiral effective Lagrangian that should encompass the anomalous processes, it is straightforward to read the current in the presence of space-time dependent $\theta(x)$ and the electromagnetic field $A_\mu$. To this end, in the two-flavor case, the vector and the axial-vector fields are set to be $$v_\mu = eQA_\mu\;, \qquad
a_\mu = 0$$ with the electric-charge matrix, $Q=\text{diag}(2/3, -1/3)=1/6+\tau_3$.
Let us first simplify ${\mathcal{L}}_{\text{WZW}}$ and ${\mathcal{L}}_{\text{P}}$, respectively, which are of our central interest. It should be mentioned that the quadratic terms of $A_\mu$ vanish due to the anti-symmetric tensor, $\epsilon^{\mu\nu\rho\sigma}$. Then, the first term in Eq. vanishes and the rest takes the following form, $$\begin{aligned}
{\mathcal{L}}_{\text{WZW}} &= -\frac{{N_{\text{c}}}{\text{tr}}(Q)}{32\pi^2}
\epsilon^{\mu\nu\rho\sigma}
\Bigl\{ ie^2 {\text{tr}}\Big[ \bigl(\Sigma_\mu + \tilde{\Sigma}_\mu\bigr)
\tau_3 \Bigr] A_\nu\partial_\rho A_\sigma \notag\\
&\qquad\qquad\qquad\qquad\quad -\frac{2e}{3} {\text{tr}}\bigl(
\Sigma_\mu \Sigma_\nu \Sigma_\rho\bigr) A_\sigma \Bigr\} \;,\end{aligned}$$ where we defined $\tilde{\Sigma}_\mu = (\partial_\mu U) U^\dagger$. Similarly the contact term can become as simple as $${\mathcal{L}}_{\text{P}} = \frac{{N_{\text{c}}}e^2 {\text{tr}}(Q^2)}{8{N_{\text{f}}}\,\pi^2}
\epsilon^{\mu\nu\rho\sigma} A_\mu(\partial_\nu A_\rho)\,
\partial_\sigma \theta\;.
\label{eq:P}$$
Now, we are ready to confirm that we can reproduce the electric current corresponding to the CME in the hadronic phase. We shall next compute the electric current by taking the differentiation of the effective action with respect to the gauge field coupled to it, that is, $$j^\mu(x) = \frac{\delta}{\delta A_\mu(x)}
\int d^4x\,{\mathcal{L}}_{\text{eff}}\;.$$ The current from the usual chiral Lagrangian ${\mathcal{L}}_\chi$ at the lowest order results in $$\begin{split}
j^\mu_\chi &= -i\frac{e f^2_\pi}{4} {\text{tr}}\bigl[\bigl(\Sigma^\mu -
\tilde{\Sigma}^\mu\bigr)\tau^3\bigr] \\
&\simeq e \bigl( \pi^- i\partial^\mu \pi^+
-\pi^+ i\partial^\mu \pi^- \bigr) + \cdots \;,
\end{split}
\label{eq:jchi}$$ which represents the electric current carried by the flow of charged pions, $\pi^\pm$, which is clear from the expanded expression. There appears no term involving $\partial_\mu\theta$ in this part. More non-trivial and interesting is the current associated with the WZW terms, leading to $$\begin{aligned}
& j_{\text{WZW}}^\mu = -\frac{{N_{\text{c}}}{\text{tr}}(Q)}{32\pi^2}
\epsilon^{\mu\nu\rho\sigma} \Bigr\{ 2ie^2 {\text{tr}}\big[
(\Sigma_\nu + \tilde{\Sigma}_\nu)\tau_3 \big] \partial_\rho A_\sigma
\notag\\
& + e^2 {\text{tr}}\big[ \partial_\rho(\Sigma_\nu + \tilde{\Sigma}_\nu)
\tau_3 \big] A_\sigma - \frac{2e}{3}
{\text{tr}}(\Sigma_\nu \Sigma_\rho \Sigma_\sigma) \Bigl\} \;,
\label{eq:jwzw}\end{aligned}$$ The physical meaning of this current will be transparent in the expanded form using $U\sim 1
+i\boldsymbol{\pi}\cdot \boldsymbol{\tau}/f_\pi+\cdots$. Then we find that the first term in Eq. is written as, $$j_{\text{WZW}}^\mu = \frac{{N_{\text{c}}}{\text{tr}}(Q) e^2}{8\pi^2 f_\pi}\,
\epsilon^{\mu\nu\rho\sigma}(\partial_\nu \pi^0) F_{\rho\sigma}\;.
\label{eq:jpi0}$$ The second term in Eq. is vanishing and the last term represents a topological current purely from the entanglement of all $\pi^0$ and $\pi^\pm$. The physics implication of Eq. has been discussed with the $\pi^0$-domain wall [@Son:2007ny] and the pion profile in the Skyrmion [@Eto:2011id]. Finally we can reproduce the CME current from the contact interaction as $$j_{\text{P}}^\mu = \frac{{N_{\text{c}}}\, e^2\, {\text{tr}}(Q^2)}{4{N_{\text{f}}}\,\pi^2}
\epsilon^{\mu\nu\rho\sigma} (\partial_\nu A_\rho)\,
\partial_\sigma \theta \;.
\label{eq:jp}$$ We can rewrite the above expression in a more familiar form using $\mu_5 = \partial_0\theta/(2{N_{\text{f}}})$ and $B^i = \epsilon^{ijk} \partial_j A_k$ to reach, $$\boldsymbol{j}_{\text{P}} = \frac{{N_{\text{c}}}\, e^2\, {\text{tr}}(Q^2)}{2\pi^2} \mu_5
\boldsymbol{B} \;.
\label{eq:CME}$$ It should be noted that $\epsilon_{0123}=+1$ in our convention.
This derivation of the CME is quite suggestive on its own and worth several remarks.
First, it is known that the contact term ${\mathcal{L}}_{\text{P}}$ is not renormalization-group invariant [@Kaiser:2000ck]. This means that ${\mathcal{L}}_{\text{P}}$ and thus $j_{\text{P}}$ are scale dependent like the running coupling constant. It is often said that $j_{\text{P}}$ is an exact result from the quantum anomaly, but it may be a little misleading. The functional form itself could be protected (though there is no rigourous proof) but ${\boldsymbol{B}}$ and $\mu_5$ in Eq. should be renormalized ones. Indeed it has been pointed out that interaction vertices in the (axial) vector channels result in the dielectric correction to ${\boldsymbol{B}}$ [@Gorbar:2009bm; @*Fukushima:2010zza]. The knowledge on the chiral Lagrangian strongly supports the results of Ref. [@Gorbar:2009bm].
Second, to find Eq. , we do not need quark degrees of freedom explicitly but only hadronic variables. This is naturally so because the idea of the WZW action is to capture the anomalous effects from the ultraviolet regime in terms of infrared variables. It is clear from the above derivation, therefore, that the CME occurs without massless quarks in the quark-gluon plasma. (See also Ref. [@Sadofyev:2010pr; @*Sadofyev:2010is; @*Gao:2012ix; @*Son:2012wh] for another derivations of the CME without referring to quarks explicitly.) Then, a conceptual confusion might arise; what really flows that contributes to an electric current in the hadronic phase? One may have thought that it is $\pi^\pm$, but such a current is rather given by $j_\chi^\mu$, and the CME current $j_{\text{P}}^\mu$ originates from the contact part that is decoupled from $U$. The same question is applied to Eq. if the system has a $\pi^0$ condensation.
In a sense these currents associated with the $\theta(x)$ or $\pi^0(x)$ backgrounds are reminiscent of the Josephson current in superconductivity. Suppose that we have a $\pi^0$ condensate, then such a coherent state behaves like a macroscopic wave-function of $\pi^0$ field. Then, a microscopic current inside of the wave-function $\pi^0$ could be a macroscopic current in the whole system since the wave-function spreads over the whole system. In the case of the CME, $\theta(x)$ or $\eta_0(x)$ plays the same role as $\pi^0(x)$. In this way, strictly speaking, it is a high-momentum component of quarks and anti-quarks in the wave-function of $\pi^0$ or $\eta_0$ that really flow to make a current, though these quarks do not have to get deconfined.
This sort of confusing interpretation of the CME current arises from the assumption that $\theta(x)$ and ${\boldsymbol{B}}(x)$ are spatially homogeneous. Once this assumption is relaxed, as we discuss in what follows, an interesting new possibility opens, which may be more relevant to experiments.
![Schematic figure for the single photon production as a consequence of the axial anomaly and the external magnetic field. The angular distribution of the emitted photons is proportional to $(q_z^2+q_x^2)/(q_x^2+q_y^2+q_z^3)$ where $q_y$ is in the direction parallel to ${\boldsymbol{B}}$ and $q_z$ and $q_x$ are perpendicular to ${\boldsymbol{B}}$.[]{data-label="fig:photon"}](photon.eps){width="0.6\columnwidth"}
From now on, let us revisit Eq. from a different point of view. If we literally interpret Eq. as usual in the quantum field theory, it should describe a vertex of the processes involving two photons and the $\theta$ field such as $\theta \to
\gamma\gamma$ and $\theta + B \to \gamma$ in the magnetic field. The latter process can be viewed as the reverse of the Primakoff effect involving the $\theta(x)$ background instead of neutral mesons. It is a very intriguing question how much photon can be produced from this reverse Primakoff effect. For this purpose we shall decompose the vector potential into the background $\bar{A}_\mu$ (corresponding to $B$) and the fluctuation ${\mathcal{A}}_\mu$ (corresponding to photon). Then, Eq. turns into $${\mathcal{L}}_{\text{P}} = \frac{{N_{\text{c}}}\, e^2\, {\text{tr}}(Q^2)}{8{N_{\text{f}}}\,\pi^2}\,
\epsilon^{\mu\nu\rho\sigma} \bigl[
{\mathcal{A}}_\mu(\partial_\nu {\mathcal{A}}_\rho) + {\mathcal{A}}_\mu \bar{F}_{\nu\rho}
\bigr] \partial_\sigma\theta \;,
\label{eq:vertex}$$ where the first term represents the two-photon production process $\theta\to \gamma\gamma$ similar to $\pi^0\to \gamma\gamma$, and the second represents the reverse Primakoff effect ($\theta + B \to \gamma$) involving the background field strength $\bar{F}_{\mu\nu}=\partial_\mu\bar{A}_\nu-\partial_\nu\bar{A}_\mu$. Here we are interested only in the situation that the background field is so strong that we can neglect the contribution from the first term.
Even when $|eB|\sim {\Lambda_{\text{QCD}}}$ in the heavy-ion collision, we can still utilize the perturbative expansion in terms of the electromagnetic coupling constant. In the leading order, from the LSZ reduction formula, the $S$-matrix element for the single-photon production with the momentum $q=(|{\boldsymbol{q}}|,{\boldsymbol{q}})$ and the polarization $\varepsilon^{(i)}({\boldsymbol{q}})$ is deduced immediately from the vertex , $$\begin{split}
&i{\mathcal{M}}(i;{\boldsymbol{q}}) = \langle \varepsilon^{(i)}({\boldsymbol{q}})|\Omega\rangle
= i \frac{{N_{\text{c}}}\, e^2\, {\text{tr}}(Q^2)}
{8{N_{\text{f}}}\,\pi^2\sqrt{(2\pi)^3 2q_0}} \\
&\qquad\times
\epsilon^{\mu\nu\rho\sigma} \varepsilon^{(i)\mu}({\boldsymbol{q}})
\int d^4 x\,e^{-iq\cdot x} \bar{F}_{\nu\rho}(x)\,
\partial_\sigma \theta(x) \;,
\end{split}$$ where $q_0 = |\boldsymbol{q}|$. This expression becomes very simple when the background field has only the magnetic field in the $y$ direction, i.e. $B=\bar{F}_{zx}$ and the rest is just vanishing. Thus, we have, $$\begin{split}
&\epsilon^{\mu\nu\rho\sigma} \varepsilon^{(i)\mu}({\boldsymbol{q}})
\int d^4 x\,e^{-iq\cdot x} \bar{F}_{\nu\rho}(x)\,
\partial_\sigma \theta(x) \\
&\qquad\qquad = -2\varepsilon^{(i) y}({\boldsymbol{q}})\int d^4x\, e^{-iq\cdot x}
B(x)\, \partial_0 \theta(x)\;,
\end{split}$$ and replacing $\partial_0\theta$ by the chiral chemical potential $\mu_5$ by $\mu_5=\partial_0\theta/(2{N_{\text{f}}})$ and using $\sum_i \varepsilon^{(i)j}({\boldsymbol{q}})\,\varepsilon^{(i)k}({\boldsymbol{q}})
=\delta^{jk}-q^j q^k/{\boldsymbol{q}}^2$ with ${\boldsymbol{q}}^2=q_x^2+q_y^2+q_z^2$, we can finally arrive at $$\begin{aligned}
&q_0 \frac{dN_\gamma}{d^3q} = q_0 \sum_i |{\mathcal{M}}(i;{\boldsymbol{q}})|^2 \notag\\
&= \frac{1-(q_y)^2/{\boldsymbol{q}}^2}{2 (2\pi)^3}
\biggl( \frac{{N_{\text{c}}}\, e^2\, {\text{tr}}(Q^2)}{2\pi^2}\int d^4x\, e^{-iq\cdot x}
B(x)\mu_5(x)\biggr)^2 \notag\\
&= \frac{q_z^2+q_x^2}{2 (2\pi)^3 {\boldsymbol{q}}^2}
\cdot \frac{25\,\alpha_e\, \zeta({\boldsymbol{q}})}{9\pi^3} \;,\end{aligned}$$ where we used ${N_{\text{c}}}=3$ and ${\text{tr}}(Q^2)=5/9$ for the two-flavor case in the last line and $\alpha_e\equiv e^2/(4\pi)\simeq 1/137$ is the fine structure constant. In the above we defined, $$\zeta({\boldsymbol{q}})\equiv \biggl|\int d^4x\,e^{-iq\cdot x} eB(x)\,
\mu_5(x)\biggr|^2 \;.$$
It is quite interesting to see that the final expression is proportional to the momenta $q_z^2+q_x^2$ which are perpendicular to the ${\boldsymbol{B}}$ direction. This could be another source for the elliptic flow $v_2$ of the direct photon in a similar mechanism as pointed out in Ref. [@Basar:2012bp].
Because there is no reliable model to predict $\mu_5(x)$, it is difficult to calculate $\zeta({\boldsymbol{q}})$ as a function of the momentum. For a first attempt, therefore, let us make a qualitative order estimate. The strength of the magnetic field is as large as ${\Lambda_{\text{QCD}}}^2$ or even bigger at initial time. A natural scale for $\mu_5$ is also given by ${\Lambda_{\text{QCD}}}$, or if the origin of the LPV is the color flux-tube structure in the Glasma [@Kharzeev:2001ev], the typical scale is the saturation momentum $Q_s\sim 2\;\text{GeV}$ for the RHIC energy. The space-time integration picks up the volume factor $\sim \tau_0^2
A_\perp$ with $\tau_0$ being the life time of the magnetic field, i.e.$\tau_0\simeq 0.01\sim 0.1\;\text{fm/c}$, and $A_\perp$ the transverse area $\sim 150\;\text{fm}^2$ for the Au-Au collision. Then, $\zeta
\simeq 0.1\sim 10^{3}$, where the smallest estimate for $\tau_0=0.01\;\text{fm/c}$ and $\mu_5\sim{\Lambda_{\text{QCD}}}$ and the largest one for $\tau_0=0.1\;\text{fm/c}$ and $\mu_5\sim Q_s$. Then, the photon yield is expected to be $q_0 (dN_\gamma/d^3q) \simeq (10^{-7} \sim 10^{-3})\text{GeV}^{-2}$. This is a rather conservative estimate, for the magnetic field may live longer with backreactions and is of detectable level of the photon yield as compared to the conventional photon production from the thermal medium [@Alam:2000bu; @*Srivastava:2001hz; @*Rasanen:2002qe; @*Gelis:2004ep; @*Turbide:2003si; @*Turbide:2007mi].
One may think that not only the polarization but also $\zeta({\boldsymbol{q}})$ has strong asymmetry because of the presence of ${\boldsymbol{B}}$. The typical domain size of the LPV should be, however, much smaller than the impact factor $b\sim$ a few fm at least, and thus the asymmetry effect turns out only negligible. In reality, depending on the spatial position, there are not only $B_y$, but $B_x$ and $B_z$ and also the electric fields $E_x$, $E_y$, and $E_z$. We are now performing full numerical calculations including all those fields and the LPV based on the Glasma flux-tube picture. Since such model buildings postulate lots of arguments on assumptions and justifications, we will leave them to a separate publication under preparation.
In summary, we have formulated the CME in terms of the chiral Lagrangian with the WZW terms, which provides us with the physics picture to understand the CME in the hadronic phase. We derived the current of the CME correctly from the contact term that is not RG invariant. We established how the CME could be realized through $\eta_0(x)$ as a result of the DCC in the iso-singlet channel. Then, the key observation in view of the chiral Lagrangian is that the vertex responsible for the CME also describes the single photon production for space-time inhomogeneous $\theta(x)$ and ${\boldsymbol{B}}(x)$. We have given the expression for the photon yield to find that its angular distribution is asymmetric with the direction perpendicular to ${\boldsymbol{B}}$ more preferred. We made a qualitative estimate for the yield and found it comparable to the thermal photon contribution. Unlike the thermal photon the $p_t$ distribution should reflect the domain size of the LPV. Electromagnetic probes as a signature for the LPV (see Ref. [@Andrianov:2010ah; @*Andrianov:2012hq] for the dilepton production) deserve further investigations and we believe that this work would shed light on future developments in this direction.
We thank Dima Kharzeev, Naoki Yamamoto, and Qun Wang for useful discussions. K. F. is grateful for discussions in a workshop “Heavy Ion Pub” at Hiroshima University, which inspired him. K. F. is supported by Grant-in-Aid for Young Scientists B (24740169).
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abstract: 'These are the written discussions of the paper “Bayesian measures of model complexity and fit" by D. Spiegelhalter et al. (2002), following the discussions given at the Annual Meeting of the Royal Statistical Society in Newcastle-upon-Tyne on September 3rd, 2013.'
address: ' $^1$Universidad de Granada, $^2$Universidad de Las Palmas de Gran Canaria, $^3$TiDES Institute, Spain, $^4$University of Warwick, UK, and $^{5}$Université Paris-Dauphine, France'
author:
- 'Elías Moreno$^{1}$, Francisco–José Vázquez–Polo$^{2,3}$, and Christian P. Robert$^{4,5}$'
title: 'Two discussions of the paper “Bayesian measures of model complexity and fit" by D. Spiegelhalter et al., Read before The Royal Statistical Society at a meeting organized by the Research Section on Wednesday, March 13th, 2002'
---
Discussion by E. Moreno and F.-J. Vázquez–Polo
==============================================
This is an interesting paper, in which a new dimension correction to penalise over-fit models is presented. It has given rise to considerable discussion; here, we focus on the DIC model selection procedure defined in the paper.
Eleven years later, model selection for complex models remains an open problem. The weak link of the Bayesian model selection approach is the elicitation of the prior over models and over the model parameters to be used in the procedure. Several priors have been proposed for interesting model selection problems, such as variable selection in high dimensional regression, clustering, change points and classification, but none of them satisfy all reasonable requirements. Thus, we fully agree with the authors’ claim in justifying DIC that “*full elicitation of informative priors and utilities is simply not feasible in most situations*”. However, this does not imply that in model selection we can avoid the use of priors in a coherent way (Berger and Pericchi, 2001).
Does the DIC have a justification from a decision theory viewpoint?
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In model selection we have a sample $\mathbf{y}_{n}$ of size $n$, a discrete class of $k$ competing sampling models $\mathfrak{M}$, the sampling density of model $M_{i}$ is $f(\mathbf{y}_{n}|\theta _{i},M_{i})$, and a prior for models and model parameters $\pi (\theta _{i},M_{i})=\pi (\theta _{i}|M_{i})$ $\pi (M_{i})$, where $\theta _{i}\in $ $\Theta _{i}\,$. The parameter spaces are typically continuous.
In model selection the quantity of interest is the model, and therefore the decision space is $\mathfrak{D}=\big\{d_{j},$ $j=1,...,k \big\}$, where $d_{j}$ is the decision to choose model $M_{j}$, and the states of nature is the class of models $\mathfrak{M}$. Given a loss function $\mathfrak{L}(d_{i},M_{j}), \; \mathfrak{L}:\mathfrak{D\times M} \longrightarrow \mathbb{R}^{+}$, the optimal Bayesian decision is to choose the model $M^{\pi }$ such that
$$M^{\pi }=\arg \min_{i=1,...,k}\sum_{j=1}^{k}\mathfrak{L}(d_{i},M_{j})%
\pi (M_{j}|\mathbf{y}_{n}),$$
where $$\displaystyle
\pi (M_{j}|\mathbf{y}_{n})=\frac{m_{j}(\mathbf{y}_{n})\pi (M_{j})}{%
\sum_{j=1}^{k}m_{j}(\mathbf{y}_{n})\pi (M_{j})},$$ and the marginal $\displaystyle m_{i}(\mathbf{y}_{n})=\int_{\Theta _{i}}f(\mathbf{y}%
_{n}|\theta _{i},M_{i})\pi (\theta _{i}|M_{i})d\theta _{i},$ is the likelihood of model $M_{i}, i=1,...,k$. This means that whatever loss function $\mathfrak{L}(d_{i},M_{j})$ we use, the optimal decision depends on the posterior model probabilities; that is, the decision formulation takes into account the uncertainty of the model. However, the DIC does not depend on $\pi(M_j|\mathbf{y}_{n}), \; j=1,...,k.$
Does the DIC correspond to a Bayesian procedure?
------------------------------------------------
The Bayesian procedures automatically penalise model complexity without any adjustment (Dawid, 2002), and this is a good reason to require a model selection procedure to be Bayesian. Another reason is that the competing models can be averaged, with the weights being the model posterior probabilities. On the other hand, for Schwarz’s Bayesian information criterion (BIC), to compare model $M_{i}$ with $M_{j}$, $$-2\log BIC_{ij}(\mathbf{y}_{n})=-2\log \frac{f(\mathbf{y}_{n}|\hat{%
\theta}_{i}(\mathbf{y}_{n}),M_{i})}{f(\mathbf{y}_{n}|\hat{\theta}_{j}(%
\mathbf{y}_{n}),M_{j})}+(d_{i}-d_{j})\log n,$$where $d_{i},d_{j}$ are the dimensions of $\Theta _{i}$ and $\Theta _{j}$, there is a Bayes factor $B_{ij}$ such that $|-2\log~B_{ij} - 2 \log BIC_{ij}|=O_{P}(n^{-1/2})$ (Kass and Wasserman, 1995), and thus the $BIC$ asymptotically corresponds to a Bayes factor, we do not see that a similar correspondence can be established with the $$DIC_{ij}(\mathbf{y}_{n})=-2\log \frac{f(\mathbf{y}_{n}|\bar{\theta}%
_{i}(\mathbf{y}_{n}),M_{i})}{f(\mathbf{y}_{n}|\bar{\theta}_{j}(\mathbf{y}%
_{n}),M_{j})}+ \mbox{Correction}_{ij}$$where $\bar{\theta}_{i}(\mathbf{y}_{n})=E_{\theta _{i}|\mathbf{y}_{n}}\theta
_{i}$, and$$\displaystyle
\mbox{Correction}_{ij} = 4 \; \Big\{E_{\theta _{i}|\mathbf{y}_{n}}\log f(%
\mathbf{y}_{n}|\theta _{i},M_{i})-E_{\theta _{j}|\mathbf{y}_{n}}\log
f(\mathbf{y}_{n}|\theta _{j},M_{i}) \Big\}
+ \; 4\log \frac{f(\mathbf{y}_{n}|\tilde{\theta}_{i}(\mathbf{y}%
_{n}),M_{i})}{f(\mathbf{y}_{n}|\tilde{\theta}_{j}(\mathbf{y}_{n}),M_{j})}.$$
We note that under mild conditions ${\Large |}\hat{\theta}(\mathbf{y}_{n})-%
\bar{\theta}(\mathbf{y}_{n}){\Large |}=O_{P}(n^{-1})$, and hence the main difference between BIC and DIC comes from the correction term. As a result of this term, the DIC does not correspond to a Bayesian procedure.
Asymptotic.
-----------
The DIC is not a consistent model selection procedure and although it is a negative property of the procedure, this does not seem to worry the authors, who argue that “*we neither believe in a true model nor would expect the list of models being considered*”. This implies that the probability of a model has no meaning, as no model space is considered. However, the point is that if we applied the DIC to a case in which the class of models were known, we would have consistency.
On the other hand, some statisticians, for instance Fraser (2011), have suggested that the sampling properties of the Bayesian methods should be studied. In this respect, Wasserman (2011) asserts that “we must be vigilant and pay careful attention to the sampling properties of procedures". We agree with both these views. Moreover, consistency is a very useful sampling property that allows us to compare the behaviour of alternative Bayesian model selection procedures for complex models.
Consistency in a model selection procedure for a given class of models $\mathfrak{M}$ means that when sampling from a model in $\mathfrak{M},$ the posterior probability of this model tends to one as the sample size tends to infinity. Bayesian procedures for model selection are typically consistent when the dimension of the models is small compared with the sample size (David, 1992; Casella et al., 2009). Furthermore, when the model from which we are sampling is not in the class $\mathfrak{M}$, the Bayesian procedure asymptotically chooses a model in $\mathfrak{M}$ that is as close as possible to the true one, in the Kullback–Leibler distance.
On the other hand, consistent Bayesian procedures for low dimensional models are not necessarily consistent for high dimensional models. For example: (a) Schwarz’s approximation to the Bayes factor BIC is not necessarily consistent in high dimensional settings (Berger, 2003; Moreno et al., 2010). $\,$ (b) When the number of models increases with the sample size, as occurs in clustering, change point or classification problems, consistency of the Bayesian model selection procedure depends not only on the prior over the model parameters but also on the prior over the models. In fact, default priors commonly used for discrete spaces may give an inconsistent Bayesian model selection procedure, as occurs in clustering when using the uniform prior over the models (Casella et al., 2012). $\,$ (c) In variable selection in regression when the number of regressors $p$ increases with the sample size, i.e., $p=O(n^{b}), 0\leq b\leq 1$, some priors that are commonly used over the model parameters and over the model space make the Bayesian procedures inconsistent. For instance, the $g-$priors (Zellner, 1986) with $g=n$ produce an inconsistent Bayesian procedure. The mixture of $g-$priors with respect to the InverseGamma$(g|1/2, n/2),$ or the intrinsic priors (Moreno et al., 1998) over the model parameters when combined with the independent Bernoulli prior on the model space (George and McCulloch, 1997; Raftery et al., 1997) may also provide an inconsistent Bayesian procedure.
These results show that consistency can be a very useful property for the difficult task of selecting priors for model selection in complex models.
Discussion by C.P. Robert
=========================
The main issue with DIC undoubtedly is the question of its worth for (or within) Bayesian decision analysis (since I doubt there exist many proponents of DIC outside the Bayesian community). The appeal of DIC is, I presume, to deliver a [*single*]{} summary per model for all models under comparison and to allow therefore for a complete ranking of those models. I however object at the worth of simplicity for simplicity’s sake: models are complex (albeit less than reality) and their usages are complex as well. To consider that model A is to be preferred upon model B just because $DIC(A)=1228 < DIC(B)=1237$ is a mimicry of the complex mechanisms at play behind model choice, especially given the wealth of information provided by a Bayesian framework. (Non-Bayesian paradigms may be more familiar with procedures based on a single estimator value.) And to abstain from accounting for the significance of the difference between $DIC(A)$ and $DIC(B)$ clearly makes matters worse.
This is not even discussing the stylised setting where one model is considered as “true" and where procedures are compared by their ability to recover the “truth". David Spiegelhalter repeatedly mentioned during his talk that he was not interested in this. This stance inevitably brings another objection, though, namely that models–as tools instead of approximations to reality–can only be compared against their predictive abilities, which DIC seems unable to capture. Once again, what is needed in this approach to model comparison is a multi-factor and all-encompassing criterion that evaluates the predictive models in terms of their recovery of some features of the phenomenon under study. Or of the process being conducted. (Even stooping down to a one-dimensional loss function that is supposed to summarise the purpose of the model comparison does not produce anything close to the DIC function, unless one agrees to massive approximations.)
Obviously, considering that asymptotic consistency is of no importance whatsoever (as repeated by David Spiegelhalter in his presentation) easily avoids some embarrassing questions, except the (still embarrassing) one about the true purpose of statistical models and procedures. How can those be compared if no model is true and if accumulating data from a given model is not meaningful? How can simulation be conducted in such a barren landscape? I find this minimalist attitude the more difficult to accept that models are truly used as if they were or could be true, at several stages in the process. It also prevents the study of the criterion under model misspecification, which would clearly be of interest.
Another point worth discussing, already exposed in Celeux et al. ([-@celeux:forbes:robert:titterington:2006])), is that there is no unique driving principle for constructing DICs. In that paper inspired from the discussion by De Iorio and Robert ([-@deiorio:robert:2002]), we examined eight different and natural versions of DIC for mixture models, resulting in highly diverging values for DIC and the effective dimension of the parameter, I believe that such a lack of focus is bound to reappear in any multi-modal setting and fear that the answer about (eight) different focus on what matters in the model is too cursory and lacks direction for the hapless practitioner.
My final and critical remark about DIC is that the criterion shares very much the same perspective as Murray Aitkin’s integrated likelihood, as already stressed in [@robert:titterington:2002]. Both Aitkin ([-@aitkin:1991], [-@aitkin:2010]) and Spiegelhalter et al. ([-@spiegbestcarl]) consider a posterior distribution on the likelihood function, taken as a function of the parameter but omitting the delicate fact that it also depends on the observable and hence does not exist a priori. See Gelman et al. ([-@gelman:robert:rousseau:2013])) for a detailed review of Aitkin’s (2010) book, since most of the criticisms therein equally apply to DIC, and I will not reproduce them here, except for pointing out that DIC escapes the Bayesian framework (and thus requires even more its own justifications).
[7]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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|
---
author:
- 'V. Bosch-Ramon'
- 'D. Khangulyan'
- 'F. A. Aharonian'
title: 'Non-thermal emission from secondary pairs in close TeV binary systems'
---
Introduction
============
Recently, several compact TeV emitters harboring an OB type star have been found in our Galaxy: PSR B1259$-$63 is a massive binary system containing a young non-accreting pulsar detected by HESS (Aharonian et al. [@aharonian05a]); LS 5039 is likely a high-mass microquasar also detected by HESS (Aharonian et al. [@aharonian05b]); LS I +61 303 is a high-mass X-ray binary detected first by MAGIC (Albert et al. [@albert06]) and recently also by VERITAS (Maier [@maier07]); is a high-mass microquasar harboring a black-hole detected by MAGIC (Albert et al. [@albert07]). In this type of objects, a significant fraction of the energy radiated above 100 GeV can be absorbed via photon-photon interactions (e.g. Ford [@ford84]; Protheroe & Stanev [@protheroe87]; Moskalenko & Karakula [@moskalenko94], Bednarek [@bednarek97]; Boettcher & Dermer [@boettcher05]; Dubus [@dubus06b]; Khangulyan et al. [@khangulyan07]; Reynoso et al. [@reynoso08]) producing secondary pairs that, under the strong radiation and magnetic fields present close to the massive star, can radiate efficiently via IC or synchrotron processes. In the case in which IC is the dominant cooling channel, EM cascades can develop (see, e.g., Aharonian et al. [@aharonian06b]; Bednarek [@bednarek06]; Khangulyan et al. [@khangulyan07]; Orellana et al. [@orellana07]). If synchrotron emission is the dominant cooling channel, the absorbed radiation is reemitted at lower energies (e.g. Khangulyan et al. [@khangulyan07]). We note that the scenario described here could take place not only in microquasars or non-accreting pulsar systems but in any source harboring a gamma-ray emitter embedded in a dense photon field.
In this work, we study the radiation of the secondary pairs created by photon-photon absorption of gamma-rays in the surroundings of a hot massive star. We focus mainly on the case when the ambient magnetic field is large enough to suppress effectively EM cascading. The structure of the paper goes as follows: in Sect. \[mod\], the adopted model is described; in Sect. \[res\], the results are presented and discussed, and our conclusions are given in Sect. \[conc\].
A model for pair creation and secondary pair emission {#mod}
=====================================================
The general picture
-------------------
We describe here the physical system formed by an OB star and a gamma-ray emitter separated by a short distance. The star is approximated as a point-like source of (monoenergetic) photons of energy $\epsilon_0$ and luminosity $L_*$. The emitter, located at distance $R=d_*$ from the star and assumed to be point-like, isotropically produces gamma-rays with an injection power $L_{\rm \gamma~inj}$, following a power-law distribution of index $\Gamma$. We consider here only the primary radiation above the minimum threshold energy for secondary pair production, i.e. $\epsilon_{\rm min~th}=1/\epsilon_0$ (in $m_{\rm e} c^2$ units), since we focus on the study of the production of secondary pairs and their emission. The maximum energy of the primary photons is fixed at 100 TeV. Throughout most of this work, we characterize the star/VHE emitter as follows: $L_*=10^{39}$ erg/s, $d_*=3\times 10^{12}$ cm s$^{-1}$, and $\epsilon_0=2\times 10^{-5}$ (corresponding to a star temperature $\approx 40000$ k); we take two values for $\Gamma$, 2 and 3, to account either for hard or soft primary gamma-ray spectra.
Important ingredients of the model are the stellar wind and the ambient magnetic field. OB stars present fast winds with mass loss rates $\dot{M}_{\rm w}\sim 10^{-7}-10^{-5}$ M$_{\odot}$ yr$^{-1}$ and velocities $V_{\rm w}\sim (1-3)\times
10^8$ cm $s^{-1}$ (e.g. Puls et al. [@puls06]). We adopt here $\dot{M}_{\rm w}=10^{-6}$ M$_{\odot}$ yr$^{-1}$ and $V_{\rm w}=2\times 10^8$ cm s$^{-1}$. Moreover, the star generates a magnetic field in its surroundings with values in the stellar surface that might be as high as $B_0\sim 1000$ G (e.g. Donati et al. [@donati02]; Hubrig et al. [@hubrig07]). We assume here that a significant fraction of the magnetic field is disordered. A sketch of our scenario is presented in Fig. \[picture\].
secondary pairs in the system
-----------------------------
To compute the secondary pair injection spectra in the different regions of the binary system, we use the anisotropic differential pair production cross section given by eq. (15) in Böttcher & Schlickeiser ([@boettcher97]). The secondary pair injection spectrum presents a slope similar to that of the primary gamma-rays, with the low-energy cutoff similar to the pair creation one, which depends on the angle between the two incoming photons: . The point-like approximation for the star fails for $R\la 2R_*\sim 20\,R_{\odot}$ from the stellar center. For our choice of values of $d_*$, $R_*$, $L_*$, and $\epsilon_0$, only photons with energy $>$ few TeV emitted at an angle $\la 30^\circ$ with respect to the star-emitter line can reach the region where the finite size of the star comes into play (Dubus [@dubus06b]). The monoenergetic photon target approximation is roughly valid provided we deal with a broad distribution of primary gamma-rays. The deviation from the real case is small and will be neglected here.
We assume that diffusion takes place in the Bohm regime. Once secondary pairs are injected in different parts of the system, they isotropize and suffer wind advection, which is the dominant transport mechanism in the system. Ionization losses in the wind, synchrotron emission in the ambient magnetic field, and IC scattering of the stellar photons, are the relevant cooling processes of secondary pairs ($\dot{\gamma}$). Relativistic Bremsstrahlung of the secondary pairs in the stellar wind is not considered since the timescales for this process are much longer than ionization, other radiative, and advective timescales. All dependencies on $R$ of the relevant physical quantities (the wind density, and the energy densities of the ambient magnetic and stellar radiation fields) are taken as $\propto 1/R^2$. We consider that the TeV emitter produces radiation long enough for the formation of a steady state distribution of secondary pairs in the system ($n(\gamma,R)$), which can be obtained from the following differential equation: $$V_{\rm w}{\partial
n(\gamma,R)\over \partial R}+ {\partial \dot{\gamma}(\gamma,R)n(\gamma,R)\over \partial
\gamma}=q(\gamma,R)\,, \label{eqdif}$$ where $\gamma$ is the particle Lorentz factor and $q(\gamma,R)$ is the secondary pair injection rate as a function of energy and distance (derived in the Appendix).
A restriction on $n(\gamma,R)$ is that EM cascades above must be effectively suppressed by the ambient $B$, which must therefore be above a certain critical value, $B_{\rm c}$ (see Khangulyan et al. [@khangulyan07]): $$B_{\rm c}\approx 70\left(L_*\over 10^{39} {\rm erg/s}\right)^{1/2}\left(R\over R_*\right)^{-1}{\rm G}\ .
\label{eq:b_crit}$$ Eq. (\[eq:b\_crit\]), together with the $R$-dependence assumed here, shows that EM cascades are already suppressed for a $\approx 20$ G magnetic field at $R=d_*$. This value appears quite moderate when looking at the possible stellar surface magnetic field values given above. However, even in the case of weaker magnetic fields, $n(\gamma,R)$ does not change much if EM cascades are not accounted for $\Gamma\ga 2.5$, since the amount of EM cascade reprocessed energy will be relatively small.
For the adopted parameter values in our model, most of the energy of the primary VHE radiation is absorbed for distances $\la 10^{12}$ cm from the emitter and $\sim d_*$ from the star. Depending on $\Gamma$, radiative cooling leads to an electron distribution producing synchrotron emission that peaks at either sharply X-ray energies or smoothly in the range X- to soft gamma-rays. Along with this a fraction of energy is radiated in the GeV range via IC scattering. The fact that the minimum energy of the injected secondary pairs is $\sim \epsilon_{\rm min~th}m_{\rm e}c^2$ implies that, even under strong synchrotron or Thomson IC cooling, the emission of particles with energies below this value will not dominate the total radiative output. Ionization losses and wind advection lead to steady state secondary pairs radiating very little radio emission for $R\la d_*$ from the star. However, the wind transports secondary pair energy to larger $R$, where this energy can be still efficiently radiated. In these farther regions, synchrotron radio emission can eventually become the dominant radiative channel.
Results {#res}
=======
Injection and evolution of secondary pairs
------------------------------------------
In this section we discuss the spectrum of the secondary pairs injected in the system, $Q_{\rm int}(\gamma)$ $(=\int q(\gamma,R)dR)$, and the secondary pair energy distribution once the steady state is reached, $N_{\rm int}(\gamma)$ $(=\int n(\gamma,R)dR)$. For illustrative purposes, we perform our calculations using $L_{\rm \gamma~inj}=3\times 10^{35}$ erg s$^{-1}$, which is similar to the value inferred in Khangulyan et al. ([@khangulyan07]) for LS 5039. We remark that the secondary pair emission luminosity scales linearly with $L_{\rm \gamma~inj}$.
In Figs. \[pairs1\] and \[pairs2\], upper panels, we show $Q_{\rm int}\,\gamma^2$ for the whole volume, adopting $\Gamma=2$ and 3, respectively. In the same figures, lower panels, $N_{\rm int}\,\gamma^2$ for the whole region is also shown. In all the plots, the contributions to the total $Q_{\rm int}\,\gamma^2$ and $N_{\rm int}\,\gamma^2$ from $R< d_*$ and $>d_*$ are presented as well.
As seen in the figures, $Q_{\rm int}(\gamma)$ has a similar shape to that of the primary gamma-rays $\propto
\epsilon_{\gamma}^{-\Gamma}$. As mentioned in Sect. \[mod\], the minimum energy of the injected secondary pairs is the pair creation threshold energy, i.e. $\epsilon_{\rm th}m_{\rm e}c^2$, which changes with the angle between the two incoming photons. As a consequence, both $q(\gamma,R)$ and $n(\gamma,R)$ depend strongly on $R$ as well as on $\Gamma$. In addition, the shape of $N_{\rm int}(\gamma)$ is related to the dominance of different cooling mechanisms and the impact of wind advection.
For particle energies $> \epsilon_{\rm min~th}m_{\rm e}c^2$, depending on $\gamma$, $N_{\rm int}(\gamma)$ is: cooled by synchrotron losses ($\propto \gamma^{-\Gamma-1}$; for high $B$ in both $R< d_*$ and $> d_*$); uncooled (injection spectrum slope; for low $B$ and $R< d_*$; because of the fast advection of the particles from this relatively small region); or cooled by KN IC ($\propto \gamma^{-\Gamma+1}$; mainly for low $B$ and $R> d_*$; because there particles have time to lose energy). For particle energies $< \epsilon_{\rm min~th}m_{\rm e}c^2$, $N_{\rm int}(\gamma)$ is: cooled by synchrotron/Thomson IC ($\propto \gamma^{-2}$; $R< d_*$); advection dominated ($\propto
\gamma^{-2}$; $R> d_*$; because of advection of cooled particles created at $R<d_*$); or cooled by ionization losses ($\propto$ constant; in both $R< d_*$ and $> d_*$). In the range $\gamma\sim 10-100$, the $1/R^2$-dependence of losses leads to a strong pileup of particles at $R\gg d_*$, giving a spike feature in the particle energy distribution. The cooling mechanisms in different $\gamma$ ranges for $\Gamma=2$ are pointed to in Fig. \[pairs1\], lower panel; for $\Gamma=3$, despite the softer slope,the pattern is the same.
In Fig. \[pairdistr\], we present the total energy ($E_{\rm int}$ in Fig. \[pairdistr\]) accumulated in secondary pairs up to distance $R$, and the total energy rate ($dE_{\rm int}/dt$ in Fig. \[pairdistr\]) from pair creation accumulated up to $R$. The $dE_{\rm int}/dt$ curve is shown up to the distance at which injection becomes negligeable, i.e. $R_{\rm inj}\sim $ AU. As seen in Fig. \[pairdistr\], a significant amount of energy, $\sim 10^{35}$ erg, is transported up to several AU. Despite this energy budget, the efficiency of the emission from large $R$ will be limited by the low magnetic and radiation energy densities and the ionization loss dominance.
Emission from secondary pairs
-----------------------------
The calculated SEDs of the synchrotron and IC emission produced by the injected secondary pairs are shown in Figs. \[sed1\] and \[sed2\]. In Fig. \[sed1\], we have adopted a $B=100$ G, and $\Gamma=2$ (upper panel) and 3 (lower panel). The SEDs of the emission generated at $R< d_*$ and $> d_*$ from the star are also shown. In Fig. \[sed2\], the SEDs have been computed adopting $B=1$ (dotted line), 10 (long-dashed line) and 100 G (solid line), and $\Gamma=2$ (upper panel) and 3 (lower panel).
For energies of photons produced by secondary pairs with energies $> \epsilon_{\rm min~th}m_{\rm e}c^2$, the spectrum becomes softer for larger $B$ due to the dominating synchrotron channel. For $B=1$ G and $\Gamma=2$, there is a clear hardening in the synchrotron and IC spectra produced by KN IC cooling (although a proper treatment requires EM cascading). In this photon energy range, the larger $\Gamma$ is, the softer the synchrotron and IC spectra are.
For energies of photons produced by secondary pairs with energies $< \epsilon_{\rm min~th}m_{\rm e}c^2$, the emission is produced either by synchrotron/Thomson IC cooled secondary pairs ($R\la d_*$), or by secondary pairs affected already by wind advection ($R>d_*$). The spectral hardening at low energies in the synchrotron and IC emission for $R\la d_*$ is produced by ionization losses. The particle pileup around $\gamma\sim 10$ impacts on the low energy radiation from $R> d_*$.
The inhomogeneity of $q(\gamma,R)$, and the sensitivity of $n(\gamma,R)$ to the cooling and transport conditions, imply that the produced emission changes strongly with location in the system. Further complexities like a magnetic field with angular dependences, or a strongly inhomogeneous diffusion coefficient, would yield globally and locally different SEDs.
### LS 5039 and
Adopting $B_0\sim 100$ G, our model predicts radio fluxes and X-ray luminosities of several mJy and several 10$^{33}$ erg s$^{-1}$, respectively. The parameter values adopted in this work are similar to those of LS 5039 and , making worthy a discussion of the results in the context of these two sources.
The obtained radio fluxes and X-ray luminosities are not far from those found in LS 5039 (e.g. Paredes et al. [@paredes00]; Bosch-Ramon et al. [@bosch07]), implying that the secondary pair contribution in this source may be comparable, if not dominant, to that of any intrinsic radio and X-ray component linked to the TeV emitter itself. We remark that, especially in the radio band, a slightly higher $B$-field would yield significantly larger radio fluxes due to the $B^2$-dependence of the synchrotron emission. The moderate X-ray luminosity in LS 5039 permits a look for such a secondary pair component. In the GeV regime, the secondary pair IC fluxes are far from those found by EGRET (Paredes et al. [@paredes00]), suggesting that the latter are due to the combination of both primary and secondary pair IC radiation.
In the case of Cygnus X-1, adopting the same $L_{\rm \gamma~inj}$ as for LS 5039, the predicted radio flux from secondary pairs is a substantial part of the total radio flux of the source (e.g. Stirling et al. [@stirling01]). In X-rays, explained as thermal emission from accretion disk/corona-like regions, the high luminosity of makes it impossible observationally to disentangle a possible secondary pair component.
The lack of evidence of accretion in LS 5039 (Bosch-Ramon et al. [@bosch07] and references therein) is an important difference between this source and , which shows clear X-ray accretion features (Albert et al. [@albert07] and references therein). Nevertheless, as noted, e.g. by Bogovalov & Kelner ([@bogovalov05]), the fact that in some sources the accretion disk luminosity is undetectable does not imply lack of accretion. This may explain why LS 5039 does not show accretion signatures in its X-ray spectrum.
### LS I +61 303
For a compact system with the properties listed in Sect. \[mod\], only VLBI interferometers with angular resolution $\la
0.1$ milliarcseconds would be able to resolve the radio emission. Nonetheless, within the constraints of our model, we have explored the possibility of explaining the extended radio emission of a less compact system, LS I +61 303, which presents a peculiar extended radio structure with changing morphology along the orbit (Dhawan et al. [@dhawan06]). We have computed the radio emission produced at the spatial scales of the observed extended radio structures. Adopting[^1] $B_0=100$ G, we have obtained the SEDs for the synchrotron radio emission originated in different regions: $R< 1$, $> 1$, $> 2$, and $>
3$ AU. The radio and broadband (synchrotron plus IC) SEDs are presented in Fig. \[sed3\] (lower -left- and upper panels, respectively); the spatial distribution of the corresponding emitting particles around LS I +61 303 after an injection time of one orbital period is also shown (lower -right- panel).
To implement these calculations, we adopt the orbital distance corresponding to the phases when the source was detected by MAGIC (Albert et al. [@albert06]): $\sim 6\times 10^{12}$ cm. $\Gamma$ is taken 2.6 (Albert et al. [@albert06]), and $L_{\rm \gamma~inj}=3\times 10^{35}$ erg s$^{-1}$, the same as the one taken for LS 5039, enough to explain observations. We note that LS I +61 303 is not detected by MAGIC above a few hundred GeV when outside the phase range $\sim
0.5-0.7$. Nevertheless, primary gamma-rays with softer spectra may still be injecting a significant amount of secondary pairs in the system despite being barely detectable above few hundreds GeV. For the spatial distribution of particles, we followed individually their energy and spatial evolution accounting for advection and (Bohm) diffusion in the wind using a simple Monte-Carlo simulation (Bosch-Ramon et al., in preparation). The orbital parameters of LS I +61 303, like the eccentricity of the system ($e=0.72$; see Casares et al. [@casares05] for the system parameters), were considered.
From Fig. \[sed3\], it is seen that radiation flux levels of $\sim 20$ mJy (8 GHz) are reached. In this source, most of the secondary pair radio emission would appear extended, pointing in the direction opposite to the star and bending due to orbital motion, as shown in the secondary pair spatial distribution in Fig. \[sed3\]. This kind of behavior is similar to that found by Dhawan et al. ([@dhawan06]) in LS I 61 +303. These authors associated the radio morphology of this source to a particular non-accreting pulsar scenario (see Dubus [@dubus06a]), although recent hydrodynamical simulations of stellar/pulsar colliding winds predict quite different morphologies (see Romero et al. [@romero07] and Bogovalov et al. [@bogovalov07]).
The predicted X-ray luminosities are similar to those observed in this source (e.g. Sidoli et al. [@sidoli06]), which, as in LS 5039, is small enough to allow the study of a secondary pair component. Despite the fact that the complexity of the X-ray emission cannot be completely explained by our model, the processes considered here may be behind a significant fraction of the observed X-ray emission. Like LS 5039, LS I +61 303 was also proposed to be a (variable) GeV source (Tavani et al. [@tavani98]), and the secondary pair contribution to this energy range may be significant (see also Bednarek [@bednarek06]).
{width="68.00000%"}
Conclusions {#conc}
===========
We conclude that the presence of a powerful VHE emitter near a massive hot star leads unavoidably to non-thermal emission in the stellar wind. The question whether this secondary pair radiation is detectable depends on the magnetic and radiation field strength in the system.
In the scenario explored here, we predict moderately hard spectra, and fluxes of several mJy for typical galactic distances, not far from the fluxes detected in some microquasars. We remark that radio emission is produced mainly in the regions with $R> d_*$ and may be dominant over any other radio component (e.g. linked to the TeV emitter itself), and the radio morphology, flux and spectrum are strongly sensitive to the geometry and $R$-dependence of the magnetic field. Interestingly, the predicted radio morphology could resemble that shown by the VHE emitting X-ray binary LS I +61 303.
The fluxes of the (likely non-thermal) X-ray emission of several TeV emitting X-ray binaries (e.g. LS I +61 303, LS 5039), typically around 10$^{33}-10^{34}$ erg s$^{-1}$, are similar to the values predicted here. It implies that the secondary pair X-ray radiation could be comparable to, or even dominate over, any intrinsic component linked to the TeV emitter itself. It is worth noting that due to their moderate X-ray luminosities LS 5039 and LS I +61 303 are good candidates to look for a secondary pair contribution in this energy range.
We show here that the secondary pair emission at $\sim$ GeV energies, if not strongly dominated by an intrinsic GeV emitter, could be revealed by GLAST. Above $\epsilon_{\rm min~th}\sim 10$ GeV, soft primary spectra and/or moderate-to-high magnetic fields would imply low fluxes. In the latter case, significant synchrotron energy losses would suppress EM cascading.
Our calculations show that even with simple assumptions on the system geometry, primary gamma-ray injected spectra, and $B$-field and wind structure, the resulting situation is quite complex. Thus a detailed characterization of the secondary pair non-thermal emission in a particular source is a difficult task for which stellar wind physics, primary VHE emission modeling, and high quality data are required.
The authors thank the anonymous referee for constructive comments. The authors are grateful to Andrew Taylor for a thorough reading of the manuscript. V.B-R. gratefully acknowledges support from the Alexander von Humboldt Foundation. V.B-R. acknowledges support by DGI of MEC under grant AYA2007-68034-C03-01, as well as partial support by the European Regional Development Fund (ERDF/FEDER).
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Appendix
========
The secondary pair injection function for a monoenergetic point-like source of target photons is determined by the following integral: $$\begin{aligned}
\nonumber
q(\gamma,R)={L_*\over4\pi m_{\rm e}c^3\epsilon_0}\times\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\\
\quad\quad\quad\quad\int\limits_{-1}^1{\rm d}
\cos\theta\,{(1-\cos\alpha(r))\over r^2}\int{\rm d} \epsilon\, {{\rm
d}N_\gamma(\epsilon)\over {\rm d}\epsilon{\rm d}t}\,{\rm e}^{-\tau}{{\rm
d}\sigma_{\rm p}\over {\rm d} \gamma}\, ,\quad\quad \end{aligned}$$ where ${\rm d}N_\gamma(\epsilon)/{\rm d}\epsilon{\rm d}t$ is the primary gamma-ray injection spectrum per time unit, $r$/$\tau$ is the distance/optical depth from the gamma-ray emitter to the secondary pair creation location, and $\alpha(r)$ is the interaction angle at this location. A sketch of the situation is presented in Fig. \[opac\]. The cross-section ${\rm
d}\sigma_{\rm p}/ {\rm d} \gamma$ is given be the eq. (15) from Böttcher & Schlickeiser (1997) and the kinematics constraints: $${\epsilon\left(1-\sqrt{1-{2\over\epsilon\epsilon_0
(1-\cos\alpha(r))}}\right)\over2}<\gamma<{\epsilon\left(1+\sqrt{1-{2\over\epsilon\epsilon_0
(1-\cos\alpha(r))}}\right)\over2}\,.$$ The distance and the interaction angle are defined as the following: $$r^2=R^2+d_*^2-2d_*R\cos\theta\,,\quad \cos\alpha(x)={\rho^2+x^2-d_*^2\over2\rho x}\,,$$ where $\rho=\sqrt{x^2+d_*^2+x(R^2-d_*^2-r^2)/r}$ is the distance from the optical star to the gamma-ray absorption point. Finally, the optical depth $\tau$ is calculated like $$\tau(r,R)={L_*\over 4\pi m_{\rm e}c^3\epsilon_0}
\int\limits_0^r{\rm d}x\,
{(1-\cos\alpha(x))\over\rho^2}\sigma_{\rm p}\,,$$ where $\sigma_{\rm p}$ is the total pair production cross-section (see e.g. Berestetskii et al. ([@berestetskii82])).
![Geometry of the photon-photon interaction.[]{data-label="opac"}](secondary.eps){width="34.00000%"}
[^1]: The relevant parameters characterizing the system properties can be found in Bosch-Ramon et al. ([@bosch06]).
|
---
abstract: 'In this paper we prove that the weighted linear combination of products of the $k$-subsets of an $n$-set of positive real numbers with weight being the harmonic mean of their reciprocal sets is less than or equal to uniformly weighted sum of products of the $k$-subsets with weight being the harmonic mean of the whole reciprocal set.'
address: 'Stat Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore-560059, India'
author:
- 'C.P. Anil Kumar'
title: An Inequality
---
Introduction
============
There is a version of this inequality for $2$-subsets of an $n$-set which appears in [@OMPDSL2006] page $327$, Problem $4$ as a short-listed problem for the Forty Seventh $IMO\ 2006$ held in Ljubljana, Slovenia. This inequality has an interesting generalization as stated in the Main Theorem \[theorem:SumProductSymmetricFunctionInequality\] below.
The Main Inequality
===================
Let $A=\{a_1,a_2,\ldots,a_n\}$ be an $n$-set of positive real numbers. Here we allow the numbers to repeat. i.e. $a_i=a_j$ for some $1 \leq i \neq j \leq n$. Let $S=\{i_1,i_2,\ldots,i_k\} {\subset}\{1,2,3,\ldots,n\}$ be a $k$-subset for some $k \leq n$. Let $B_S=\{a_{i_1},a_{i_2},a_{i_3},\ldots, a_{i_k}\} {\subset}A$ be its corresponding set. The reciprocal set denoted by $B_S^{-1}$ of the set $B_S$ is defined to be the set $B_S^{-1}=\{\frac{1}{a_{i_1}},\frac{1}{a_{i_2}},\frac{1}{a_{i_3}},\ldots,
\frac{1}{a_{i_k}}\}$.
Now we state the main theorem.
\[theorem:SumProductSymmetricFunctionInequality\] Let $[{\underline}{n}]
= \{1,2,3,\ldots,n\}$ denote the set of first $n$ natural numbers. Let $A=\{a_i: i=1, \ldots,n\}$ be an $n$-set of positive real numbers. For any subset $S {\subset}[{\underline}{n}]$, let $\prod a_S$ denote ${\underset}{i \in S}{\prod} a_i$ and $\sum a_S$ denote ${\underset}{i \in S}{\sum}
a_i$. Then $${\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum}\bigg(\frac{\prod a_S}{\sum
a_S}\bigg) \leq \frac{n}{k} \bigg(\frac{{\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum} \prod a_S}{\sum a_{[{\underline}{n}]}}\bigg)$$ i.e. The weighted linear combination of products of the $k$-sets with weight as harmonic mean of their reciprocal sets is less than or equal to uniformly weighted sum of products of the $k$- sets with weight the harmonic mean of the whole reciprocal set.
We also observe that the equality occurs if and only if $a_1=a_2=\ldots=a_n$.
Some Simple Cases of the General Inequality
===========================================
We prove a few lemmas.
\[lemma:ReciprocalTriples\] Let $a_1,a_2,a_3$ be three positive real numbers. Then the sum of the reciprocals of $a_i$ is greater than or equal to sum of the reciprocals of their pairwise averages. $$\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3} \geq
\frac{1}{\frac{a_1+a_2}{2}} + \frac{1}{\frac{a_2+a_3}{2}} +
\frac{1}{\frac{a_3+a_1}{2}}$$
We have from $AM-HM$ inequality applying to the reciprocals $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3}$ we get [$$\begin{aligned}
\frac{\frac{1}{a_1}+\frac{1}{a_2}}{2} &\geq \frac{2}{a_1+a_2}\\
\frac{\frac{1}{a_2}+\frac{1}{a_3}}{2} &\geq \frac{2}{a_2+a_3}\\
\frac{\frac{1}{a_3}+\frac{1}{a_1}}{2} &\geq \frac{2}{a_3+a_1}
\end{aligned}$$ ]{} and adding these inequalities we have $$\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3} \geq
\frac{1}{\frac{a_1+a_2}{2}} + \frac{1}{\frac{a_2+a_3}{2}} +
\frac{1}{\frac{a_3+a_1}{2}}$$ Hence the Lemma \[lemma:ReciprocalTriples\] follows.
\[lemma:MainThree\] Let $a_1,a_2,a_3$ be three positive real numbers. Then $$\frac{a_1a_2}{a_1+a_2}+\frac{a_2a_3}{a_2+a_3}+\frac{a_3a_1}{a_1+a_3}
\leq \frac{3(a_1a_2+a_2a_3+a_3a_1)}{2(a_1+a_2+a_3)}$$
In order to prove Lemma \[lemma:MainThree\] first we make a simplification by assuming without loss of generality that $a_1+a_2+a_3=1$. This can be done by normalizing with $a_1+a_2+a_3$. Now [$$\begin{aligned}
&2\frac{a_1a_2}{a_1+a_2}+2\frac{a_2a_3}{a_2+a_3}+2\frac{a_3a_1}{a_1+a_3}-2(a_1a_2+a_2a_3+a_3a_1)\\
&=a_1a_2a_3\bigg(\frac{1}{\frac{a_1+a_2}{2}} +
\frac{1}{\frac{a_2+a_3}{2}}
+ \frac{1}{\frac{a_3+a_1}{2}}\bigg)\\
&\leq a_1a_2a_3\bigg(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}\bigg)
\text{ Using Lemma~\ref{lemma:ReciprocalTriples} }\\
&=a_1a_2+a_2a_3+a_3a_1
\end{aligned}$$ ]{} Hence the Lemma \[lemma:MainThree\] follows.
\[lemma:Reciprocal\] Let $a_1,a_2,a_3,\ldots,a_n$ be $n$ positive real numbers. Then the sum of the reciprocals of $a_i$ is greater than or equal to the sum of the reciprocals of their $(n-1)$-wise averages. $$\label{eq:Reciprocal}
\begin{aligned}
\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+ \ldots + \frac{1}{a_n} & \geq \frac{1}{\frac{a_1+a_2+\ldots+a_{n-1}+a_n-a_n}{n-1}}+ \frac{1}{\frac{a_1+a_2+a_3+\ldots+a_n-a_{n-1}}{n-1}}\\
& + \frac{1}{\frac{a_1+a_2+a_3+ \ldots + a_n-a_{n-2}}{n-1}} + \ldots
+ \frac{1}{\frac{a_1+a_2+a_3+\ldots+a_n-a_{1}}{n-1}}
\end{aligned}$$
This is a generalization of Lemma \[lemma:ReciprocalTriples\] to $n$-positive real numbers $a_1,a_2,\ldots,a_n$. We have from $AM-HM$ inequality applying to the reciprocals $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\ldots,\frac{1}{a_n}$ we get the following set of inequalities. For every $1 \leq j \leq n$, we get [$$\frac{1}{(n-1)}\bigg({\underset}{i \neq j,1=1}{{\overset}{n}{\sum}}
\frac{1}{a_i}\bigg) \geq \frac{(n-1)}{{\underset}{i \neq j,
i=1}{{\overset}{n}{\sum}} a_i}$$ ]{} and adding these inequalities we have $$\begin{aligned}
\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+ \ldots + \frac{1}{a_n} & \geq \frac{1}{\frac{a_1+a_2+\ldots+a_{n-1}+a_n-a_n}{n-1}}+ \frac{1}{\frac{a_1+a_2+a_3+\ldots+a_n-a_{n-1}}{n-1}}\\
& + \frac{1}{\frac{a_1+a_2+a_3+ \ldots + a_n-a_{n-2}}{n-1}} + \ldots
+ \frac{1}{\frac{a_1+a_2+a_3+\ldots+a_n-a_{1}}{n-1}}
\end{aligned}$$ Hence the Lemma \[lemma:Reciprocal\] follows.
Let $[{\underline}{n}] = \{1,2,3,\ldots,n\}$ denote the set of first $n$ natural numbers. Let $A=\{a_i: i=1,\ldots, n\}$ be a set of $n$ positive real numbers. Then $${\underset}{(n-1)-subset\ S {\subset}[{\underline}{n}]}{\sum}\bigg(\frac{\prod a_S}{\sum
a_S}\bigg) \leq \frac{n}{(n-1)} \bigg(\frac{{\underset}{(n-1)-subset\ S
{\subset}[{\underline}{n}]}{\sum} \prod a_S}{\sum a_{[{\underline}{n}]}}\bigg)$$
This is a generalization of the above Lemma \[lemma:MainThree\] to the case of $(n-1)$-subsets of an $n$-set. The proof is similar to the proof of Lemma \[lemma:MainThree\] except here we use Lemma \[lemma:Reciprocal\] instead of Lemma \[lemma:ReciprocalTriples\].
\[lemma:MainTwoSets\] Let $A=\{a_i: i=1,\ldots,n\}$ be a set of $n$-positive real numbers. Then $${\underset}{i<j}{\sum}\frac{a_ia_j}{a_i+a_j} \leq
\frac{n}{2(a_1+a_2+\ldots+a_n)} {\underset}{i<j}{\sum}a_ia_j$$
The proof is as follows. Again by normalizing with ${\underset}{i=1}{{\overset}{n}{\sum}} a_i$ we can assume that ${\underset}{i=1}{{\overset}{n}{\sum}} a_i=1$ and it is enough to prove that $${\underset}{i<j}{\sum}\frac{a_ia_j}{a_i+a_j} \leq \frac{n}{2}
{\underset}{i<j}{\sum}a_ia_j$$ So consider [$$\begin{aligned}
&2{\underset}{i<j}{\sum}\frac{a_ia_j}{a_i+a_j}-2{\underset}{i<j}{\sum}a_ia_j
={\underset}{i<j}{\sum}\frac{2}{a_i+a_j}\bigg({\underset}{k \neq i,k \neq
j}{\sum}a_ia_ja_k\bigg)\\
&=\bigg({\underset}{3-subset\ S {\subset}[{\underline}{n}]}{\sum} \bigg({\underset}{2-subset\ T
{\subset}S}{\sum} \frac{2\prod a_S}{\sum a_T}\bigg)\bigg)\\
&\text{ Now using Lemma~\ref{lemma:ReciprocalTriples} for all } 3
\text{ subsets of }\{1,2,3,\ldots,n\} \text{ we get } \\
&\leq (n-2)\bigg({\underset}{2-subset\ S {\subset}[{\underline}{n}]}{\sum} \prod
a_S\bigg)
\end{aligned}$$ ]{} Hence the lemma follows.
Proof of the Main Theorem
=========================
Here we prove the Main Theorem \[theorem:SumProductSymmetricFunctionInequality\]
Now we generalize to the case given in the Theorem \[theorem:SumProductSymmetricFunctionInequality\] by first normalizing the inequality with $\sum a_i$ so that we can assume that $\sum a_i=1$. And we have to proof the following inequality $${\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum}\bigg(\frac{\prod
a_S}{\frac{\sum a_S}{k}}\bigg) \leq n\bigg({\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum} \prod a_S\bigg)$$ We have $$\begin{aligned}
&{\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum}\bigg(k\frac{\prod a_S}{\sum a_S}\bigg)-k\bigg({\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum} \prod a_S\bigg)\\
&= k\bigg({\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum} \frac{\prod a_S(1-\sum a_S)}{\sum a_S}\bigg)\\
&=k\bigg({\underset}{(k+1)-subset\ S {\subset}[{\underline}{n}]}{\sum} \bigg({\underset}{k-subset\ T {\subset}S}{\sum} \frac{\prod a_S}{\sum a_T}\bigg)\bigg)\\
&\leq(n-k)\bigg({\underset}{k-subset\ S {\subset}[{\underline}{n}]}{\sum} \prod
a_S\bigg) \text{ Using inequality~\ref{eq:Reciprocal} in
Lemma~\ref{lemma:Reciprocal}}
\end{aligned}$$
The equality occurs when if and only all the AM-HM inequalities involved give equality which holds if and only if $a_1=a_2=\ldots=a_n$. Hence the Main Theorem \[theorem:SumProductSymmetricFunctionInequality\] follows.
Acknowledgements
================
The author likes to thank Prof. C.R. Pranesachar, HBCSE, TIFR, Mumbai for mentioning this problem on inequalities given in Lemma \[lemma:MainTwoSets\] which the author was able to suitably generalize to the Main Theorem \[theorem:SumProductSymmetricFunctionInequality\] in this article.
[1]{} D. Dujukic,V. Janokovic,I. Matic,N. Petrovic. [The IMO Compendium]{}, Problem Books in Mathematics.
|
---
abstract: 'Motivated by stringy considerations, Randall & Sundrum have proposed a model where all the fields and particles of physics, save gravity, are confined on a 4-dimensional brane embedded in 5-dimensional anti-deSitter space. Their scenario features a stable bound state of bulk gravity waves and the brane that reproduces standard general relativity. We demonstrate that in addition to this zero-mode, there is also a discrete set of metastable bound states that behave like massive 4-dimensional gravitons which decay by tunneling into the bulk. These are resonances of the bulk-brane system akin to black hole quasinormal modes—as such, they give rise to the dominant corrections to 4-dimensional gravity. The phenomenology of braneworld perturbations is greatly simplified when these resonant modes are taken into account, which is illustrated by considering gravitational radiation emitted from nearby sources and early universe physics.'
author:
- |
Sanjeev S. Seahra[^1]\
Institute of Cosmology & Gravitation\
University of Portsmouth\
Portsmouth, PO1 2EG, UK
title: |
**Metastable massive gravitons from\
an infinite extra dimension[^2]**
---
String theory, one of the leading candidates for the ‘theory of everything,’ tells us that the universe has more than four dimensions, but everyday experience seems to suggest otherwise. Over the years, there have been several attempts to explain why extra dimensions are hidden, including the conventional assumption that they have a compact topology with radii on the order of the Planck length. But recently, certain developments in non-perturbative string theory have provided alternatives to this ‘Kaluza-Klein’ compactifaction. The key ingredient is so-called $d$-branes, which are $(d+1)$-dimensional hypersurfaces on which the standard model particles and fields can be consistently confined. This raises the possibility that the observable universe is a 3-brane, i.e., we are living on a ‘braneworld’ [@Maartens:2003tw review].
But in these scenarios, gravity is free to propagate in the full higher-dimensional ‘bulk’ manifold. This is potentially worrisome, since the force of gravity we are familiar with behaves in an entirely 4-dimensional manner. For example, in a higher-dimensional universe we would expect deviations from Newton’s inverse square law at large distances, and such deviations have never been measured. We could again invoke compactification to rescue the model, but there is a different intriguing strategy: What if we were able to find scenarios where the bulk graviton was, in some sense, *dynamically* bound to the brane? Could we then allow the extra dimensions to be infinite?
In 1999, Randall & Sundrum [@Randall:1999vf] proposed a phenomenological realization of this idea in 5-dimensional anti-deSitter space. Their model has the line element $$ds^2 = e^{-2|y|/\ell} \eta_{\alpha\beta} dx^\alpha dx^\beta + dy^2,
\quad G^{(5)}_{ab} = (6/\ell^2) g^{(5)}_{ab},$$ where $\ell$ is the anti-deSitter length scale. The brane is identified with the intrinsically flat geometric defect at $y =
0$, and the Israel junction conditions imply that it is actually a thin sheet of vacuum energy. Fluctuations of this model can be written as $\eta_{\alpha\beta} \rightarrow \eta_{\alpha\beta} +
h_{\alpha\beta}$ with $$h_{\alpha\beta} = e^{-|y|/2\ell}
\psi_k(t,y) e^{-i\mathbf{k} \cdot \mathbf{x}} \epsilon_{\alpha\beta},
\quad \epsilon_{\alpha\beta} = \text{constant},$$ where $\psi_k$ satisfies a one-dimensional wave equation $$\label{wave equation}
-\ell^2 {\partial}_t^2 \psi_k = -{\partial}_z^2 \psi_k + V_k(z) \psi_k, \quad z = e^{y/\ell} \ge 1.$$ Here, we have restricted attention to one half of the bulk. The potential $V_k$ is shown in Figure \[fig:potential\]. The delta function enforces the boundary condition ${\partial}_z (z^{3/2} \psi_k) =
0$ at $z = 1$, which ensures that the matter content of the brane is unaltered by the perturbation. The attractive nature of the delta function allows for a normalizable, stable bound state of the brane and the 5-dimensional graviton $$\psi^{(0)}_k = e^{i\omega t} z^{-3/2}, \quad \omega = k.$$ From the point of view of 4-dimensional brane observers, the fluctuation behaves exactly like a massless spin-2 field propagating on a flat background. Hence, this so-called ‘zero-mode’ reproduces 4-dimensional weak field gravity on the brane, and shows how one can recover standard general relativity with an infinite extra dimension.
![The potential governing gravity waves in the Randall-Sundrum braneworld[]{data-label="fig:potential"}](potential.eps)
However, $\psi^{(0)}_k$ is not the only solution to (\[wave equation\]). There is a whole spectrum of modes parameterized by a separation constant $m$: $$\label{massive mode solution}
\psi^{(m)}_k = e^{i \omega t} \sqrt{z} \left[ {\text{H}^{(1)}_{1}}(m\ell)
{\text{H}^{(2)}_{2}}(m\ell z) - {\text{H}^{(2)}_{1}}(m\ell) {\text{H}^{(1)}_{2}}(m\ell z) \right],
\quad \omega^2 = k^2 + m^2,$$ where ${\text{H}^{(n)}_{\nu}}$ are Hankel functions. The dispersion relation on the right implies that each mode looks like a massive spin-2 field to a brane observer. So, in addition to the bound state solution that reproduces general relativity, there is a continuum of massive graviton ‘Kaluza-Klein’ modes that predict deviations from it. A complete description of brane gravity must include both.
What is the essential behaviour of the massive modes on the brane, and under what circumstances are they important relative to the zero-mode? To be sure, a direct attack on this problem with Green’s functions can be mounted, but we pursue a more physical approach by asking a different, more subtle question: Does the potential shown in Figure \[fig:potential\] treat each value of $m$ democratically, or do certain masses tend to dominate the gravity wave spectrum? That is, are any *resonant* massive modes within the Kaluza-Klein continuum? Resonant phenomena play a prominent role in many branches of physics, and can often be used to simply characterize the most important features of a given system’s dynamics. There are at least two examples of this that are useful to highlight: In black hole perturbation theory, the master equations governing gravity wave propagation support resonant solutions known as *quasinormal modes* [@Nollert], which tend to dominate the late time behaviour of scattered gravity waves, irrespective of the initial configuration of the perturbation. A second example comes from accelerator physics, where resonances in scattering cross sections allows one to identify bound states of elementary particles *without* having a complete theoretical grasp on the interactions between them. In both cases, we do not need to actually solve the equations, subject to a given source, to predict the behaviour of the system—the resonant effects are predominant. These model problems suggest that the key to gaining physical intuition about the massive modes in the braneworld is to find the resonant solutions of (\[wave equation\]).
Asymptotically far from the brane, the exact solution (\[massive mode solution\]) reduces to a superposition of travelling waves, and the ratio of the coefficients of the outgoing and incoming contributions defines the scattering matrix. As in the two example problems cited above, the resonant modes of our system are defined by the poles of this scattering matrix.[^3] A simple calculation [@Seahra:2005wk] yields that these correspond to a discrete set of *complex* masses $$\label{resonant masses}
\{m_j\ell\} = \{\mu_j\} = \{0.419 + 0.577\,i,\, 3.832 +
0.355\,i,\, 7.016 + 0.350 \,i\, \cdots\}.$$ If $k$ is real, the dispersion relation $\omega^2 = k^2 + m^2$ implies that these modes have $\text{Im}\,\omega > 0$, i.e., they are exponentially damped in time. In this sense, they are exactly analogous to the quasinormal modes of perturbed black holes. Because these solutions decay in time, it is sensible to call them the metastable bound states of the brane and the bulk graviton. A useful interpretation comes from recalling Gamov’s classic 1928 treatment of the radioactive alpha decay [@Gasiorowicz], where the $\alpha$-particle is thought to be trapped in a potential well surrounding a much heavier partner. In that problem, the metastable bound states represent solutions where the $\alpha$-particle is mostly localized near the daughter nucleus, but there is a finite probability of it tunnelling through the potential barrier and out to infinity. In this spirit, we see that the resonant masses (\[resonant masses\]) represent a spectrum of massive 4-dimensional gravitons mostly localized on the brane, but subject to decay by tunnelling into the bulk.
This opens up a whole new way of looking at perturbative gravity in the Randall-Sundrum scenario. Before, one thought of a zero-mass graviton accompanied by continuum of massive spin-2 fields. Now, we realize that the essential behaviour is that of a stable massless graviton augmented by a discrete family of quasi-bound massive cousins. To gain a better understanding of the phenomenology of these ‘particles,’ we consider a wavepacket of gravitational radiation on the brane. We assume motion in the $x$-direction, and a momentum space profile $\alpha(k)$. The pulse’s evolution on the brane will dominated by contributions from the zero-mode and resonant masses: $$\label{pulses}
\delta h_{\alpha\beta} \sim \epsilon_{\alpha\beta} \int dk \, \alpha(k) \left[
\exp{ik(t-x)} + \sum_j c_j \exp{ik\left(\frac{t}{n_j}-x\right)} \right].$$ Here, $j$ runs over the resonances, the $c_j$ expansion coefficients are determined from the initial extra-dimensional pulse profile, and $n_j$ is the complex reflective index [@Jackson] $$n_j = n_j(k) = \frac{k}{\omega_j(k)} = \frac{k\ell}{\sqrt{(k\ell)^2+\mu_j^2}}.$$ Hence, $h_{\alpha\beta}$ is a superposition of a discrete pulses corresponding to the zero-mode and massive gravitons. Since $n_j$ has a nonzero real and imaginary parts, each of the massive pulses behaves like it is travelling in an absorptive medium, slower than the speed of light. On the other hand, the zero mode acts like it is propagating in a vacuum.
If $\alpha(k)$ is peaked about some $k = k_0$, we can sensibly ask: How fast do the massive pulses travel? And how far do they get before decaying? We define the group velocity $v_j$ and lifetime $\tau_j$ of each of the modes in the usual way: $$v_j = \text{Re} \, \omega_j'(k_0) = \text{Re}\,n_j(k_0),
\quad \tau_j = \frac{1}
{\text{Im} \, \omega_j(k_0)}
= \frac{\text{Im}\,n_j(k_0)}{k_0}.$$ Together, these give an attenuation length $d_j$, which is the distance a given massive mode travels before its amplitude decreases by a factor of $e$: $$\frac{d_j}{\ell} = \frac{v_j \tau_j}{\ell} =
\frac{k_0\ell}{\text{Re}\,\mu_j\,\text{Im}\,\mu_j}.$$ The denominator on the right is of order unity or larger, so we see that modes with $k_0 \ell \gg 1$ can travel for large distances, while modes with $k_0 \ell \ll 1$ have very short streaming lengths on the brane.
Now, tabletop experiments of Newton’s law limit $\ell \lesssim 0.1
\text{ mm}$, which immediately dashes any hopes of seeing any bulk effects in the gravity waves emitted from nearby sources. The reasoning is as follows: Astrophysical systems have sizes much larger than 0.1 mm, which means that any gravitational radiation will primarily be composed of partial waves with $k\ell \ll 1$. Thus, the signal from the massive modes will only propagate for a minuscule distance $d \ll \ell$ along the brane before decaying away, making their direct detection impossible.[^4]
The situation is quite different when we consider cosmological braneworld scenarios, where the motion of the brane in the extra dimension accounts for the expansion of the universe. In the early universe epoch $H\ell \sim 1$ of such a model, all sub-horizon modes will have $k\ell \gtrsim 1$. Hence, the attenuation length of the massive gravitons will be of order the horizon size or larger, implying that they should have an important effect of the gravity wave background. However, our calculations to this point have been for a static brane, and it is unclear how general brane motion affects the quasi-particle excitations. But some (heuristic) progress can be made by considering the small-scale fluctuations, which have $$\label{dispersion}
k\ell \gg 1 \quad \Rightarrow \quad \omega_j(k)
\approx k \left[ 1 + \frac{\mu_j^2}{2(k\ell)^2} \right]
\quad \Rightarrow \quad |\omega_j(k)| \gg H.$$ The last inequality means that the typical oscillation timescale is much less than the speed of cosmological expansion, so it is safe to say that these modes ‘see’ the brane to be stationary, and are still valid resonant solutions. Hence, in the early universe the high frequency component of the gravity wave background will effectively be governed by a discrete spectrum of spin-2 fields obeying the above dispersion relation. As the universe expands the ‘fingerprint’ of these fields expands with it, eventually decoupling from the massive modes when its size grows larger than $\ell$. Therefore, this ‘back of the envelope’ calculation predicts that the relic gravity wave background carries within it primordial signatures of the extra dimension, encoded in $\omega_j(k)$ and thereby the discrete set of complex numbers $\{\mu_j\}$.
So, what has our knowledge of the resonances between bulk gravity waves and the brane achieved for us? We have seen that the late time behaviour of metric fluctuations is dominated by a discrete set of spin-2 fields with complex masses plus the zero-mode. Kinematically, the resonant massive modes behave like gravitons travelling in an absorptive medium—the dissipation is due to the tunnelling of gravitational radiation into the extra dimension. For a given AdS length scale $\ell$ in the bulk, we have calculated the lifetime and streaming-length of these particles on the brane, which are of order $\ell$ and $\ell/c$ respectively. With $\ell \lesssim 0.1$ mm, we see that the massive part of the spectrum cannot play a large role astrophysical processes in the nearby universe, but may be significant in earlier epochs where $H\ell \sim 1$. This highlights the ephemeral nature of bulk effects on the brane in the Randall-Sundrum scenario, and why such a model is credible description of the *real* world. By and large, we see that Einstein’s theory of gravitation can—in principle—live peacefully with infinite extra dimensions, and the regions of conflict are neatly parameterized by discrete spectrum of massive decaying gravitons.
I would like to thank Chris Clarkson, Roy Maartens, and David Wands for helpful discussions and encouragement, and <span style="font-variant:small-caps;">NSERC</span> for financial support.
[1]{}
Roy Maartens. “Brane-world gravity.” , 7:7, 2004. gr-qc/0312059.
Lisa Randall and Raman Sundrum. “An alternative to compactification.” , 83:4690–4693, 1999. hep-th/9906064.
Hans-Peter Nollert. “Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars.” , 16:R159, 1999.
John R. Taylor. . Wiley, New York, 1972.
L. D. Landau and E. M. Lifshitz. , volume 3 of [ *Course of Theoretical Physics*]{}. Elsevier, London, 3rd edition, 1977.
Sanjeev S. Seahra. “Ringing the [R]{}andall-[S]{}undrum braneworld.” 2005. hep-th/0501175.
Stephen Gasiorowicz. . Wiley, New York, 2nd edition, 1996.
John David Jackson. . Wiley, New York, 3rd edition, 1999.
[^1]: Email: sanjeev.seahra@port.ac.uk
[^2]: This essay received an honourable mention in the 2005 Gravity Research Foundation essay competition
[^3]: For a comprehensive introduction to the theory of one-dimensional scattering and the associated resonant phenomena, see the textbooks by Taylor [@Taylor] or Landau and Lifshitz [@LL].
[^4]: A complimentary result can be derived by integrating over real values of $\omega$ in (\[pulses\]) and assuming that $k$ is complex. Then, one finds that the attenuation length becomes long only when the source involves frequencies with $\omega\ell \gg 1$, i.e., with $\omega \gg 10^{12} \text{ sec}^{-1}$.
|
---
author:
- |
Kanishka Perera$^{a}$, Cyril Tintarev$^{b}$, Jun Wang$^{c}$, Zhitao Zhang$^{d}$[^1]\
[$^{a}$Department of Mathematical Sciences, Florida Institute of Technology,]{}\
[Melbourne, FL 32901, USA]{}\
[$^{b}$Department of Mathematics, Uppsala University,75 106 Uppsala, Sweden]{}\
[$^{c}$Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R. China]{}\
[$^{d}$Academy of Mathematics and Systems Science, Chinese]{}\
[Academy of Sciences, Beijing 100190, P.R. China]{}
title: 'Ground and bound state solutions for a Schrödinger system with linear and nonlinear couplings in $\mathbb{R}^N$'
---
> [**Abstract:**]{} We study the existence of ground and bound state solutions for a system of coupled Schrödinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we prove the existence of ground and bound state solutions under suitable assumptions. Our results are new even for the limit system.\
> [**Keywords**]{}: [Coupled Schrödinger systems; positive solutions; variational methods; linear and nonlinear couplings]{}\
> [**AMS Subject Classification (2010):**]{} 35B32, 35B38, 35J50.
Introduction and main results
=============================
In this paper we study the $2$-component coupled nonlinear Schrödinger system $$\label{auto}
\begin{cases}
- \Delta
u+u=(a_0(x)+a(x))|u|^{p-2}u+(\beta_0+\beta(x))|u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}}
+(\kappa_0+\kappa(x))v,\\
- \Delta
v+v=(b_0(x)+b(x))|v|^{p-2}v+(\beta_0+\beta(x))|u|^{\frac{p}{2}}
|v|^{\frac{p}{2}-2}v+(\kappa_0+\kappa(x))u,\\
(u,v) \in H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N),
\end{cases}$$ where $2<p<\frac{2N}{N-2}$ if $N\geq3$, and $2<p<\infty$ if $N=1,2$. We assume that the following condition holds:
- $a_0,\, b_0 \in L^\infty(\mathbb{R}^N)$ are positive ${\mathbb{Z}}^N$-periodic functions, $\beta_0, \kappa_0\in\mathbb{R}$, $a, b, \beta, \kappa
\in L^\infty(\mathbb{R}^N)$ go to zero as $|x| \to \infty$, and $$\label{a-2}
\begin{split}
&\inf_{x \in \mathbb{R}^N}\, (a_0(x) + a(x))>0,\quad \inf_{x \in \mathbb{R}^N}\, (b_0(x) + b(x))>0,\\
&0<\kappa_0+\inf_{x \in \mathbb{R}^N}\, \kappa(x)\leq
\kappa_0+\sup_{x \in \mathbb{R}^N}\, \kappa(x)<1.
\end{split}$$
This system of equations is related to the following important Schrödinger system with linear and nonlinear couplings arising in Bose-Einstein condensates (see [@DKNF]): $$\label{original}
\left\{ \begin{aligned}
&-i\frac{\partial \Phi}{\partial t}=\Delta\Phi-V(x)\Phi+\mu_1|\Phi|^2\Phi+\beta|\Psi|^2\Phi+\kappa\Psi,~t>0,x\in \Omega, \\
&-i\frac{\partial \Psi}{\partial t}=\Delta\Psi-V(x)\Psi+\mu_2|\Psi|^2\Psi+\beta|\Phi|^2\Psi+\kappa\Phi,~t>0,x\in \Omega,
\end{aligned} \right.$$ where $\Omega$ is a smooth domain in $\mathbb{R}^N$, $V$ is the relevant potential, typically consisting of a magnetic trap and/or an optical lattice, and $\Phi$ and $\Psi$ are the (complex-valued) condensate wave functions. The intra- and interspecies interactions are characterized by the coefficients $\mu_1, \mu_2 > 0$ and $\beta$, respectively, while $\kappa$ denotes the strength of the radio-frequency (or electric-field) coupling. This system also arises in the study of fiber optics, where the solution $(\Phi,\Psi)$ is two coupled electric-field envelopes of the same wavelength, but of different polarizations, and the linear coupling is generated either by a twist applied to the fiber in the case of two linear polarizations, or by an elliptic deformation of the fiber’s core in the case of circular polarizations. Looking for solitary wave solutions of the form $\Phi(x,t)=e^{i\lambda t}u(x),\, \Psi(x,t)=e^{i\lambda t}v(x)$, where $\lambda > 0$ is a constant, leads to the following elliptic system for $u$ and $v$: $$\label{li-zhang1.1}
\left\{ \begin{aligned}
&-\Delta u+(\lambda +V(x))u=\mu_1 u^3+\beta u v^2+\kappa v & ~~in &~~~~ \Omega, \\
&-\Delta v+(\lambda+V(x)) v=\mu_2 v^3+\beta u^2 v+\kappa u & ~~in & ~~~~ \Omega,\\
&u =v=0~~\mbox{on}~~\partial \Omega~ (\hbox{or}~ u,v\in H^1(\mathbb{R}^N)~ \hbox{if}~
\Omega=\mathbb{R}^N),
\end{aligned} \right.$$ which has received considerable attention in recent years.
Interesting existence and multiplicity results in various domains for system with $\kappa = 0$ have been obtained in [@AC; @ACR; @BDW; @Dancer-Wei-2009-TRMS; @DW2; @DWZ1; @DWZ2; @DWZ3; @DWZ4; @NTTV; @TT; @TV]. In particular, bifurcation results were obtained in [@BDW], and larger systems and their limiting equations were considered in [@DWZ1; @DWZ2; @DWZ3; @DWZ4]. It was shown in Ambrosetti-Colorado [@AC] that when $\kappa = 0$ and $V(x) \equiv 0$ there exist constants $\beta_2 > \beta_1 > 0$ such that this sytem has a positive ground state solution for $\beta > \beta_2$ and no positive ground state solution for $\beta < \beta_1$. A solution is called a ground state solution (or positive ground state solution) if its energy is minimal among all the nontrivial solutions (or all the positive solutions) of or .
Existence and asymptotic behavior of multi-bump solitons of system with $\kappa\neq 0,\beta = 0$ and $V(x) \equiv 0$ were studied in Ambrosetti-Cerami-Ruiz [@ACR] using perturbation methods. Ambrosetti-Cerami-Ruiz [@ACR-1] studied positive ground and bound state solutions of the system $$\label{ACR1}
\left\{ \begin{aligned}
&-\Delta u+ u =(1+a(x)) |u|^{p-1}u+\kappa v, \\
&-\Delta v+ v=(1+b(x)) |v|^{p-1}v+\kappa u,\\
&(u,v) \in H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)
\end{aligned} \right.$$ when $\kappa>0$, $N\geq 2, 1<p<2^*-1,$ where $$2^*=\left\{\begin{array}{lll}\frac{2N}{N-2}, ~\hbox{if}~ N\geq 3,\\
+\infty,~\hbox{if}~ N=2.\end{array}\right.$$ Under the general assumptions $a(x), b(x)\in L^{\infty}(\mathbb{R}^N)$, $\lim\limits_{|x|\rightarrow+\infty} a(x)=\lim\limits_{|x|\rightarrow+\infty} b(x)=0$, $\inf\limits_{x\in\mathbb{R}^N}(1+a(x))>0,\, \inf\limits_{x\in\mathbb{R}^N}(1+b(x))>0$, and additional suitable hypotheses, they used concentration compactness type arguments to obtain some interesting results about the existence of positive ground and bound state solutions.
System with both $\kappa \ne 0$ and $\beta \ne
0$ has been much less studied. Topological methods were used in Beitia-Garca-Torres [@BPT] to obtain a positive bound state solution, and variational methods and index theory were used in Li-Zhang [@Li-Kui-Zhangzhitao-2015-Preprint] to obtain a ground state solution and infinitely many positive bound state solutions. Some existence results when $V(x) \equiv 0$ were obtained in Tian-Zhang [@tian-zhang] using variational and bifurcation arguments. In the present paper we consider the system , which generalizes , with $\kappa_0 \ne 0$ and $\beta_0
\ne 0$. Our results seem to be new even for the limit system when $a(x)=b(x)=\beta(x)=\kappa(x)\equiv0$.
Our first results is Theorem \[th1.1\] below, which is concerned with the existence of a positive ground state solution of .
\[th1.1\] Suppose that $(A_0)$ holds, $0 < \kappa_0 < 1$, and $$\label{a-6}
\kappa(x) \ge 0, \quad a(x) \ge 0, \quad b(x) \ge 0, \quad \beta(x) \ge 0,$$ with at least one of the inequalities in strict on a set of positive measure. Then has a ground state solution. For the case $u=v$, if $$\label{a-7}
\kappa(x)\ge 0, \quad a(x)+b(x)+2\beta(x)\ge 0,$$ with at least one of these inequalities strict on a set of positive measure, then has a ground state solution.
We give conditions more general than and that guarantee the existence of a ground state solution of (see Theorem \[zh0127\] in Section 6), which are not stated here in order to simplify notation.
Next we study when the functions $a, b,
\kappa$ and $\beta$ are non-positive. In this case there exists no ground state solution (see Lemma \[Lemma 7.1\]), and bound states must be sought at higher levels. We assume that $a_0$ and $b_0$ are positive constants and $p=4$ in . Then we have the following result.
\[th1.2\] Let $a_0, b_0>0$, $0 < \kappa_0 < 1$, $p=4$, and assume that and one of the following conditions holds:
- $\beta_0\geq3$;
- $1\leq\beta_0\leq3$ and $w(0)\leq
\sqrt{\frac{2\kappa_0(1+\beta_0)}{(3-\beta_0)(1-\kappa_0)}}$, where $w$ is the unique positive solution of the scalar equation $-\Delta
u+u=u^{3},\ u\in H^1(\mathbb{R}^{N})$;
- $-1<\beta_0<1$, $w(0)\leq
\sqrt{\frac{2\kappa_0(1+\beta_0)}{(3-\beta_0)(1-\kappa_0)}}$, and
- if $N=1$, then $\kappa_0$ or $\beta_0-1$ is sufficiently small,
- if $N=2,3$, then $\beta_0, \kappa_0>0$ are sufficiently small, or $|\beta_0|$ is sufficiently small and $\kappa_0$ is close to 1.
If $\kappa(x), a(x), b(x), \beta(x) \le 0$, with at least one of the inequalities strict on a set of positive measure, then system has a positive bound state solution provided that $$R_0:=\left(1 + \frac{|\kappa|_\infty}{1 - \kappa_0}\right)^2\left(1
- \max
\left\{\frac{|a|_\infty}{a_0},\frac{|b|_\infty}{b_0},\frac{|\beta|_\infty}{\beta_0}\right\}\right)^{-1}$$ is sufficiently small.
- We give a more precise assumption on $R_0$ (see Lemma \[Lemma 7.5\]), which is not stated here in order to simplify notation.
- A key step in the proof of Theorem \[th1.2\] is showing the uniqueness and nondegeneracy of the positive solution of the limit system $$\label{a-8}
\begin{cases}
- \Delta
u+u=a_0|u|^{p-2}u+\beta_0|u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}}
+\kappa_0v,\\
- \Delta v+v=b_0|v|^{p-2}v+\beta_0|u|^{\frac{p}{2}}
|v|^{\frac{p}{2}-2}v+\kappa_0u,\\
(u,v) \in H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N).
\end{cases}$$ Compared to the paper [@ACR-1], we have more coupled terms here, namely $|u|^{\frac{p}{2}} |v|^{\frac{p}{2}-2}v$ and $|u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}}$. These terms present new difficulties for proving the uniqueness and nondegeneracy of the positive solution of for general $p$. Using some ideas from [@Dancer-Wei-2009-TRMS; @WeiYao2012-CPAA; @Ikoma-2009-Nodea], we prove this for the case $p=4$ here (see Lemma \[lem-3.5\]). The proof for general $p$ is an interesting open problem.
The main results are Theorems \[th1.1\] and \[th1.2\]. For the limit system used to prove the main results, Lemmas \[lem-3.1\] – \[lem-3.5\] are new and of independent interest.
The paper is organized as follows. In Section 2, we give some notations and preliminaries. In section 3, we study the existence and asymptotic behavior of the positive solution of the limit system , and consider the uniqueness and nondegeneracy of the positive solution of when $p=4$ and $a_0, b_0$ are positive constants. In section 4, we give a concentration compactness result. In section 5, we study existence of ground state solutions for a functional-analytic model of our problem. In section 6, we give several sufficient conditions for the existence of ground state solutions. In section 7, we prove the existence of bound state solutions of system when $p = 4$ and $a_0, b_0$ are positive constants.
Preliminaries
=============
We will use the following notations:
- for a positive function or constant $M$, $\|\cdot\|_M$ is the equivalent norm on $H^1(\mathbb{R}^N)$ defined by ${\displaystyle}\|u\|_{M}^{2}=\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+M|u|^{2})$;
- $\|(u,v)\|_E=(\|u\|^{2}+\|v\|^{2})^{1/2}$ is the norm of $E=H^1(\mathbb{R}^{N})\times H^1(\mathbb{R}^N)$, where $\|\cdot\|$ is a norm on $H^1(\mathbb{R}^N)$;
- for $1\le p<\infty$, $|\cdot|_p$ is the usual norm of $L^p(\mathbb{R}^N)$ defined by $|u|_p={\displaystyle}\left(\int_{\mathbb{R}^N}|u|^p\right)^{1/p}$;
- $2^*$ is the critical Sobolev exponent given by $2^{*}=\frac{2N}{N-2}$ if $N\geq3$, and $2^{*}=\infty$ if $N=1,\, 2$;
- $C_i,\, i=1,2,\dots$ denote positive constants.
The energy functional associated with the system is given by $$\label{b-1}
\begin{split}
\Phi(u,v)&=\frac{1}{2}(\|u\|^{2}+\|v\|^2)-\frac{1}{p}\int_{\mathbb{R}^{N}}\left[(a_0(x)+a(x))|u|^{p}
+(b_0(x)+b(x))|v|^{p}\right]\\
&\quad-\frac{2}{p}\int_{\mathbb{R}^{N}}(\beta_0+\beta(x))|u|^{\frac{p}{2}}|v|^{\frac{p}{2}}-\int_{\mathbb{R}^{N}}
(\kappa_0+\kappa(x))uv,~ \forall(u,v) \in H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N).
\end{split}$$ To obtain nontrivial solutions of , we use the associated Nehari manifold $$\label{b-2}
\mathscr{N}=\{z=(u,v)\in E\setminus\{(0,0)\}: \Phi'(u,v)(u,v)=0\}.$$ Clearly, $\Phi\in C^{2}(E,\mathbb{R})$ and all nontrivial critical points of $\Phi$ are on $\mathscr{N}$. For $(u,v)\in\mathscr{N}$, $$\label{b-3}
\begin{split}
\|u\|^{2}+\|v\|^2&=\int_{\mathbb{R}^{N}}\left[(a_0(x)+a(x))|u|^{p}
+(b_0(x)+b(x))|v|^{p}\right]\\
&\quad+2\int_{\mathbb{R}^{N}}(\beta_0+\beta(x))|u|^{\frac{p}{2}}|v|^{\frac{p}{2}}+2\int_{\mathbb{R}^{N}}
(\kappa_0+\kappa(x))uv\\
&\leq
C\, (\|u\|^p+\|v\|^p)+(\kappa_0+\sup\kappa(x))(\|u\|^2+\|v\|^2)
\end{split}$$ by the Sobolev inequality. Since $\kappa_0+\sup\kappa(x)<1$ and $p > 2$, it follows from this that $\|u\|+\|v\|\geq\sigma>0$ for all $(u,v)\in\mathscr{N}$, so $\mathscr{N}$ is uniformly bounded away from the origin in $E$.
Set $$c=\inf_{(u,v)\in\mathscr{N}}\, \Phi(u,v).$$ A pair of functions $(u,v)\in\mathscr{N}$ such that $\Phi(u,v)=c$ will be called a ground state solution of . We have $$\label{b-5}
\begin{split}
\Phi|_{\mathscr{N}}(u,v)&=\left(\frac{1}{2}-\frac{1}{p}\right)\bigg[\int_{\mathbb{R}^{N}}\left[(a_0(x)+a(x))|u|^{p}
+(b_0(x)+b(x))|v|^{p}\right]\\
&\quad+2\int_{\mathbb{R}^{N}}(\beta_0+\beta(x))|u|^{\frac{p}{2}}|v|^{\frac{p}{2}}\bigg]\\
&=\left(\frac{1}{2}-\frac{1}{p}\right)\left[\|u\|^{2}+\|v\|^{2}-2\int_{\mathbb{R}^{N}}(\kappa_0+\kappa(x))uv\right]\\
&\geq\left(\frac{1}{2}-\frac{1}{p}\right)\left[\|u\|^{2}+\|v\|^{2}-2\, (\kappa_0+\sup \kappa(x))\|u\|\|v\|\right]\\
&\geq\left(\frac{1}{2}-\frac{1}{p}\right)[1-(\kappa_0+\sup \kappa(x))]\left[\|u\|^{2}+\|v\|^{2}\right].
\end{split}$$ Since $\kappa_0+\sup \kappa(x)<1$, it follows that $c>0$. First we have the following lemma regarding the role of $c$.
\[lem-2.1\] If $c$ is attained at $z\in\mathscr{N}$, then $z$ is a solution of .
Assume that $z_{0}=(u_{0},v_{0})\in\mathscr{N}$ is such that $\Phi(u_{0},v_{0})=c$. According to [@ChangKungChing2005 Theorem 4.1.1], $\mathscr{N}$ is a locally differentiable manifold and so there exists a Lagrange multiplier $\ell\in\mathbb{R}$ such that $$\label{b-6}
\Phi'(u_{0},v_{0})=\ell G'(u_{0},v_{0}),$$ where $G(u,v)=\Phi'(u,v)(u,v)$. We infer from $z_{0}=(u_{0},v_{0})\in\mathscr{N}$ and $\kappa_0+\sup \kappa(x)<1$ that $$\label{b-7}
\begin{split}
G'(u_{0},v_{0})(u_{0},v_{0})&=2(\|u_{0}\|^{2}+\|v_{0}\|^{2})
-p\int_{\mathbb{R}^{N}}\left[(a_0(x)+a(x))|u_0|^{p}
+(b_0(x)+b(x))|v_0|^{p}\right]\\
&\quad-2p\int_{\mathbb{R}^{N}}(\beta_0+\beta(x))|u_0|^{\frac{p}{2}}|v_0|^{\frac{p}{2}}-4\int_{\mathbb{R}^{N}}
(\kappa_0+\kappa(x))u_0v_0\\
&=(2-p)\left[(\|u_0\|^2+\|v_0\|^2)-2\int_{\mathbb{R}^{N}}
(\kappa_0+\kappa(x))u_0v_0\right]<0.
\end{split}$$ Testing the equation with $(u_{0},v_{0})$, it follows from that $\ell=0$. Thus, we have $\Phi'(u_{0},v_{0})=0$, i.e., $z_{0}$ is a critical point of $\Phi$.
The limit equations
===================
Ground state solution
---------------------
In this subsection we study the existence and asymptotic behavior of the positive solution of the limit system $$\label{c-1}
\begin{cases}
- \Delta
u+u=a_0(x)|u|^{p-2}u+\beta_0|u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}}
+\kappa_0v,\quad x\in\mathbb{R}^{N},\\
- \Delta v+v=b_0(x)|v|^{p-2}v+\beta_0|u|^{\frac{p}{2}}
|v|^{\frac{p}{2}-2}v+\kappa_0u,\quad x\in\mathbb{R}^{N},
\end{cases}$$ where $2<p<2^*$, $a_0(x)$ and $b_0(x)$ are $1$-periodic positive functions. The energy functional corresponding to is defined by $$\label{c-2}
\begin{split}
\Phi_0(u,v)&=\frac{1}{2}(\|u\|^{2}+\|v\|^2)-\frac{1}{p}\int_{\mathbb{R}^{N}}\left[a_0(x)|u|^{p}
+b_0(x)|v|^{p}\right]\\
&\quad-\frac{2}{p}\int_{\mathbb{R}^{N}}\beta_0|u|^{\frac{p}{2}}|v|^{\frac{p}{2}}-\int_{\mathbb{R}^{N}}
\kappa_0uv.
\end{split}$$ The corresponding Nehari manifold is $$\label{c-3}
\mathscr{N}_0=\{z=(u,v)\in E\setminus\{(0,0)\}:
\Phi'_0(u,v)(u,v)=0\}.$$ Clearly, $\Phi_0\in C^{2}(E,\mathbb{R})$ and all nontrivial solutions are contained in $\mathscr{N}_0$. Set $$\label{c-4}
c_0=\inf_{(u,v)\in\mathscr{N}_0}\, \Phi_0(u,v).$$ As in -, one can show that if $0<\kappa_0<1$, $\mathscr{N}_0$ is uniformly bounded away from the origin $(0,0)$. Moreover, if we replace $\mathscr{N}$ and $c$ by $\mathscr{N}_0$ and $c_0$, respectively, the conclusion of Lemma \[lem-2.1\] remains true for $0<\kappa_0<1$.
Now we are ready to prove the existence of a ground state solution of .
\[lem-3.1\] If $0<\kappa_0<1$, the periodic system has a positive ground state solution $(u,v)\in\mathscr{N}_0$.
Let $w_0$ denote the positive solution of $-\Delta
u+u=a_0(x)|u|^{p-2}u, u\in H^{1}(\mathbb{R}^{N})$. Then $(w_0,0)\in\mathscr{N}_0$, and $\mathscr{N}_0\neq\emptyset$. Let $\{(u_{n},v_{n})\}\subset\mathscr{N}_0$ be a minimizing sequences. By using the Ekeland’s variational principle type arguments(see [@wangjun2014pre Lemma 3.10] or [@Willem1996book]), we can assume that there exists a subsequence of $\{(u_{n},v_{n})\}\subset\mathscr{N}_0$(still denote by $(u_{n},v_{n})$) such that $$\label{c-5}
\Phi_0(u_{n},v_{n})\rightarrow c_{0},\quad
\Phi_0'|_{\mathscr{N}_0}(u_{n},v_{n})\rightarrow0.$$ Similar to and , it follows from $\kappa_0+\sup \kappa(x)<1$ that $0<\sigma\leq\|u_{n}\|+\|v_{n}\|\leq C_1$. We claim that $\Phi_0'(u_{n},v_{n})\rightarrow0$ as $n\rightarrow\infty$. Indeed, it is clear that $$\label{c-6}
o(1)=\Phi_0|_{\mathscr{N}_0}'(u_{n},v_{n})=\Phi_0'(u_{n},v_{n})-\ell_{n}G'_{0}(u_{n},v_{n}),$$ where $\ell_{n}\in\mathbb{R}$ and $G_0(u,v)=\Phi_0'(u,v)(u,v)$. As in Lemma \[lem-2.1\], one can check that $G_0(u_n,v_n)(u_n,v_n)\leq-C_2<0$. So, we know that $\ell_{n} \to 0$ in . Thus, it follows that $$\label{c-7}
\Phi_0(u_{n},v_{n})\rightarrow c_{0},\quad
\Phi_0'(u_{n},v_{n})\rightarrow0.$$ From the boundedness of $\{(u_n,v_n)\}$, without loss of generality we assume that $u_{n}\rightharpoonup u_{0}$, $v_{n}\rightharpoonup
v_{0}$ in $H^{1}(\mathbb{R}^{N})$, $u_{n}\rightarrow u_{0}$ and $v_{n}\rightarrow v_{0}$ in $L_{loc}^{p}(\mathbb{R}^{N})$, $\forall
p\in(2,2^{*})$.
We claim that $\{(u_{n},v_{n})\}$ is nonvanishing, i.e., there exists $R>0$ such that $$\label{c-8}
\liminf_{n\rightarrow\infty}\int_{B_{R}(y_{n})}(u_{n}^{2}+v_{n}^{2})\geq\delta>0,$$ where $y_{n}\in\mathbb{R}^{N}$ and $B_{R}(y_{n})=\{y\in\mathbb{R}^{N}: |y-y_{n}|\leq R\}$. Arguing by contradiction, if is not satisfied, then $\{(u_{n},v_{n})\}$ is vanishing, i.e., $$\label{c-9}
\lim_{n\rightarrow\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{r}(y)}(u_{n}^{2}+v_{n}^{2})=0,\
\text{for\ all}\ r>0.$$ According to Lions’s concentration compactness lemma(see [@Willem1996book Lemma 1.21]) that $u_{n}\rightarrow0$ and $v_{n}\rightarrow0$ in $L^{t}(\mathbb{R}^{N})(\forall
t\in(2,2^{*}))$. So, we infer from $\Phi_0'(u_{n},v_{n})(u_{n},v_{n})=0$ that $$\label{c-10}
\begin{split}
\|u_n\|^{2}+\|v_n\|^2&=\int_{\mathbb{R}^{N}}\left[a_0(x)|u_n|^{p}
+b_0(x)|v_n|^{p}\right]\\
&\quad+2\int_{\mathbb{R}^{N}}\beta_0|u_n|^{\frac{p}{2}}|v_n|^{\frac{p}{2}}+2\int_{\mathbb{R}^{N}}
\kappa_0u_nv_n\rightarrow0,
\end{split}$$ as $n\rightarrow\infty$. This contradicts with $\|u_n\|+\|v_n\|\geq\sigma>0$. Hence, holds. Moreover, there exist $\{k_n\}\subset\mathbb{Z}^{N}$ and $R_0>R>0$ such that $$\label{c-11}
\liminf_{n\rightarrow\infty}\int_{B_{R_0}(k_{n})}(u_{n}^{2}+v_{n}^{2})\geq\frac{\delta}{2}>0.$$
Set $\tilde{u}_{n}=u_{n}(x+k_{n})$ and $\tilde{v}_{n}=v_{n}(x+k_{n})$. Since $a_0(x)$ and $b_0(x)$ are $1$-periodic functions, it follows that the norms and $\Phi$ are invariance under the translations $x\mapsto x+k_n$. Thus, we can assume that $\tilde{u}_{n}\rightharpoonup \tilde{u}_{0}$, $\tilde{v}_{n}\rightharpoonup \tilde{v}_{0}$ in $H^{1}(\mathbb{R}^{N})$, $\tilde{u}_{n}\rightarrow \tilde{u}_{0}$ and $\tilde{v}_{n}\rightarrow \tilde{v}_{0}$ in $L_{loc}^{t}(\mathbb{R}^{N})(\forall t\in(2,2^{*}))$. Moreover, it follows from that $$\label{c-12}
\liminf_{n\rightarrow\infty}\int_{B_{R_0}(0)}(\tilde{u}_{n}^{2}+\tilde{v}_{n}^{2})\geq\frac{\sigma}{2}>0.$$ So, we have $\tilde{u}_{0}\neq0$ or $\tilde{v}_{0}\neq0$. Furthermore, it follows from the weak continuous of $\Phi_0'$ and that $\Phi_0'(\tilde{u}_{0},\tilde{v}_{0})=0$ and $\tilde{z}_{0}=(\tilde{u}_{0},\tilde{v}_{0})\in\mathscr{N}$. As in [@Li-Kui-Zhangzhitao-2015-Preprint], we define the following inner product $$\label{c-13}
\left<(u_1,v_1),(u_2,v_2)\right>=\left<(u_1,v_1),(u_2,v_2)\right>_{1}-
\kappa_0\int_{\mathbb{R}^{N}}\left(u_1v_2+u_2v_1\right),$$ where $\left<(\cdot,\cdot),(\cdot,\cdot)\right>_{1}$ denotes the inner product in $E$. Correspondingly, the induced norm denotes by $\|(\cdot,\cdot)\|_{\kappa_0}$. Furthermore, it follows from $0<\kappa_0<1$ that the norms $\|(\cdot,\cdot)\|_{\kappa_0}$ and $\|(\cdot,\cdot)\|_{E}$ are equivalent in $E$. Hence, we infer from the weak lower semicontinuity of the norm that $$\label{c-14}
\begin{split}
c_0&\leq\Phi_0(\tilde{u}_{0},\tilde{v}_{0})=\left(\frac{1}{2}-\frac{1}{p}\right)\left[\|\tilde{u}_{0}\|^{2}+\|\tilde{v}_{0}\|^{2}
-2\int_{\mathbb{R}^{N}}\kappa_0\tilde{u}_{0}\tilde{v}_{0}\right]\\
&=\left(\frac{1}{2}-\frac{1}{p}\right)\|(\tilde{u}_{0},\tilde{v}_{0})\|_{\kappa_0}^2
\leq\liminf_{n\rightarrow\infty}\left(\frac{1}{2}-\frac{1}{p}\right)\|(\tilde{u}_{n},\tilde{v}_{n})\|_{\kappa_0}^2\\
&=\liminf_{n\rightarrow\infty}\left(\frac{1}{2}-\frac{1}{p}\right)\left[\|\tilde{u}_{n}\|^{2}+\|\tilde{v}_{n}\|^{2}
-2\int_{\mathbb{R}^{N}}\kappa_0\tilde{u}_{n}\tilde{v}_{n}\right]\\
&=\liminf_{n\rightarrow\infty}\Phi_0(\tilde{u}_{n},\tilde{v}_{n})=
\lim_{n\rightarrow\infty}\Phi_0(u_{n},v_{n})=c_0.
\end{split}$$ So, $\tilde{z}_{0}=(\tilde{u}_{0},\tilde{v}_{0})\neq(0,0)$ is a ground state solution of . Finally, we prove that $\tilde{z}_{0}=(\tilde{u}_{0},\tilde{v}_{0})$ is positive. Obviously, there exists unique $t>0$ such that $t(|\tilde{u}_{0}|,|\tilde{v}_{0}|)\in\mathscr{N}_0$. So, one sees that $$\label{c-15}
\begin{split}
t^{p-2}&=\frac{\|\tilde{u}_{0}\|^{2}+\|\tilde{v}_{0}\|^{2}
-2\int_{\mathbb{R}^{N}}\kappa_0|\tilde{u}_{0}||\tilde{v}_{0}|}{\int_{\mathbb{R}^{N}}\left[a_0(x)|u|^{p}
+b_0(x)|v|^{p}+2\beta_0|u|^{\frac{p}{2}}|v|^{\frac{p}{2}}\right]}\\
&\leq\frac{\|\tilde{u}_{0}\|^{2}+\|\tilde{v}_{0}\|^{2}
-2\int_{\mathbb{R}^{N}}\kappa_0\tilde{u}_{0}\tilde{v}_{0}}{\int_{\mathbb{R}^{N}}\left[a_0(x)|u|^{p}
+b_0(x)|v|^{p}+2\beta_0|u|^{\frac{p}{2}}|v|^{\frac{p}{2}}\right]}=1.
\end{split}$$ Furthermore, we infer from $t(|\tilde{u}_{0}|,|\tilde{v}_{0}|)\in\mathscr{N}_0$ that $$\label{c-16}
\begin{split}
c_0\leq\Phi_0(t|\tilde{u}_{0}|,t|\tilde{v}_{0}|)=t^2\left(\frac{1}{2}-\frac{1}{p}\right)\|(|\tilde{u}_{0}|,|\tilde{v}_{0}|)\|_{\kappa_0}^2
\leq\left(\frac{1}{2}-\frac{1}{p}\right)\|(\tilde{u}_{0},\tilde{v}_{0})\|_{\kappa_0}^2=c_0.
\end{split}$$ Thus, one deduces that $t=1$, $(|\tilde{u}_{0}|,|\tilde{v}_{0}|)\in\mathscr{N}_0$ and $\Phi(|\tilde{u}_{0}|,|\tilde{v}_{0}|)=c_0$. Hence, we can assume that $(\tilde{u}_{0},\tilde{v}_{0})$ is a nonnegative solution of . Moreover, by using the maximum principle we know that $(\tilde{u}_{0},\tilde{v}_{0})$ is a positive ground state solution of .
Next we consider the asymptotic behavior of the ground state solution of as the parameter $\kappa_0\rightarrow0$ in the simple case where $p=4$, and $a_0$ and $b_0$ are constants. To emphasize the dependency on $\kappa_0$, in the following we write $c_0^{\kappa_0}$, $\mathscr{N}_0^{\kappa_0}$ and $\Phi_0^{\kappa_0}$ for $c_0$, $\mathscr{N}_0$ and $\Phi_0$, respectively. We have the following result.
\[Pro-3.2\] Let $a_0, b_0 > 0$, $\beta_0 > 0$, $0 < \kappa_0 < 1$, and let $(u_{\kappa_0},v_{\kappa_0})$ be any positive ground state solution of .
- If $N=2$ or $3$, then $(u_{\kappa_0}, v_{\kappa_0})\rightarrow(u_{0}, v_{0})$ in $H^{1}(\mathbb{R}^{N})$ as $\kappa_0\rightarrow0$. Moreover, the following conclusions hold: (i) If $0<\beta_{0}<\min\{a_0,b_0\}$, or $\beta_{0}>\max\{a_0,b_0\}$, or $0<a_0=b_0<1$ and $\beta_0=a_0$, then $(u_0,v_0)$ is a positive radial ground state solution of with $\kappa_0=0$. (ii) If $\min\{a_0,b_0\}\leq\beta_{0}\leq\max\{a_0,b_0\}$ and $a_0\neq b_0$, then one of $u_0$ and $v_0$ is zero, and $(u_0,v_0)$ is a semipositive radial ground state solution of with $\kappa_0=0$. (iii) If $a_0=b_0\geq1$ and $\beta_0=a_0$, then $(u_0,v_0)$ is a nonnegative radial ground state solutions of with $\kappa_0=0$.
- If $N=1$, then $(u_{\kappa_0}, v_{\kappa_0})\rightharpoonup(u_{0}, v_{0})$ in $H^{1}(\mathbb{R}^{N})$ as $\kappa_0\rightarrow0$, and $(u_0, v_0)$ has the same properties as above.
For $a_0, b_0>0$, we take $\kappa_0^{n}>0$ such that $\kappa_0^{n}\rightarrow0$ as $n\rightarrow\infty$. Let $(u_{n},
v_{n}):=(u_{\kappa_0^{n}}, v_{\kappa_0^{n}})$ denote the positive ground state solution of with $\kappa=\kappa_0^{n}$. We first claim that $c_0^{\kappa_0}$ is a decreasing function on $\kappa_0$. In fact, for $0<\kappa_0^{1}\leq\kappa_0^{2}$, we let $(u_{1},v_{1})$ and $(u_{2},v_{2})$ denote the positive ground state solution corresponding to $\kappa_0=\kappa_0^{1}$ and $\kappa_0=\kappa_0^{2}$ respectively. Then there exists unique $t_{1}(\kappa_0^{1})>0$ such that $t_{1}(\kappa_0^{1})(u_{2},v_{2})\in\mathscr{N}_0^{\kappa_0^{1}}$. As in we know that $t_{1}(\kappa_0)$ is a decreasing function on $\kappa_0$. Hence, we infer from $(u_{2},v_{2})\in\mathscr{N}_0^{\kappa_0^{2}}$ that $\hat{t}_{1}:=t_{1}(\kappa_0^{1})\leq t_{1}(\kappa_0^{2})=1$, where $t_{1}(\kappa_0^{2})$ satisfies $t_{1}(\kappa_0^{2})(u_{2},v_{2})\in\mathscr{N}_0^{\kappa_0^{2}}$. So, it follows that $$\label{c-17}
\Phi_0^{\kappa_0^{1}}(u_{1},v_{1})\leq\Phi_0^{\kappa_0^{1}}(\hat{t}_{1}u_{2},\hat{t}_{1}v_{2})=\frac{\hat{t}_{1}^{2}}{6}\|(u_{2},v_2)\|_{\kappa_0}^{2}
\leq\frac{1}{6}\|(u_{2},v_2)\|_{\kappa_0}^{2}=\Phi_0^{\kappa_0^{2}}(u_{2},v_{2}).$$ Thus, the claim holds. Hence, we know that $$\label{c18}
c_0^{\kappa_0^{1}}\leq
c_0^{\kappa_0^{n}}=\left(\frac{1}{2}-\frac{1}{p}\right)\|(u_{n},v_{n})\|_{\kappa_0^{n}}^2
\leq c_0^{\kappa_0^{n+1}}\leq\cdot\cdot\cdot\leq c_0^{0}.$$ This implies that $\{(u_n, v_n)\}$ is bounded in $E$. In addition, since $\beta_0>0$ and $\kappa_0^{n}>0$ close to zero, by the moving plane method (see [@Busca-Sirakov-2000-JDE Theorem 2]), $u_n$ and $v_n$ must be radially symmetric and strictly decreasing functions.
We first consider the case $2\leq N\leq3$. Without loss of generality we assume that $(u_{n},v_{n})\rightharpoonup(u_{0},v_{0})$ in $E_{r}=H_{r}^{1}(\mathbb{R}^{N})\times H_{r}^{1}(\mathbb{R}^{N})$, and $(u_{n},v_{n})\rightarrow(u_{0},v_{0})$ in $L_{r}^{p}(\mathbb{R}^{N})\times L_{r}^{p}(\mathbb{R}^{N})$$(\forall
p\in(2,2^{*}))$. Moreover, $u_{0}, v_{0}\geq0$ in $\mathbb{R}^{N}$. For each $(\varphi,\phi)\in C_{0}^{\infty}(\mathbb{R}^{N})\times
C_{0}^{\infty}(\mathbb{R}^{N})$, one has that $$\label{c19}
\begin{split}
\left(\Phi_{0}^{\kappa_0^{n}}\right)'(u_{n},v_{n})(\varphi,\phi)&=\int_{\mathbb{R}^{N}}
[\nabla u_{n}\nabla\varphi+u_{n}\varphi+\nabla
v_{n}\nabla\phi+v_{n}\phi-3(a_0u_{n}^{2}\varphi+b_0v_{n}^{2}\phi)]\\
&\quad-\beta_0\int_{\mathbb{R}^{N}}(u_{n}v_n^{2}\varphi+v_{n}u_n^{2}\phi)+\kappa_0^{n}\int_{\mathbb{R}^{N}}
(v_{n}\varphi+u_{n}\phi)\\
&=\int_{\mathbb{R}^{N}} [\nabla u\nabla\varphi+u\varphi+\nabla
v\nabla\phi+v\phi-3(a_0u^{2}\varphi+b_0v^{2}\phi)]\\
&\quad-\beta_0\int_{\mathbb{R}^{N}}(uv^{2}\varphi+vu^{2}\phi).
\end{split}$$ Thus, $(u_{0},v_{0})$ satisfies with $\kappa_0=0$. Moreover, as in one infers that $$\label{c20}
c_0^{\kappa_0^{n}}=\Phi_0^{\kappa_0^{n}}(u_n,v_n)\rightarrow
c_0^{0}\quad\text{and}\quad
\left(\Phi_0^{\kappa_0^{n}}\right)'(u_n,v_n)=0.$$ As in Lemma \[lem-3.1\], we infer that $\int_{\mathbb{R}^{N}}(u_{n}^{4}+v_n^4)\geq
\int_{B_{R}(y_n)}(u_{n}^{4}+v_n^4)\geq\delta>0$. Thus, from Brezis-Lieb lemma(see [@Willem1996book]) we infer that $\int_{\mathbb{R}^{N}}(u_{n}^{4}+v_n^4)\rightarrow\int_{\mathbb{R}^{N}}(u_{0}^{4}+v_0^4)$, and $\int_{\mathbb{R}^{N}}(u_{0}^{4}+v_0^4)\geq\delta>0$. Hence, we know that at least one of $u_{0}$ and $v_0$ is not equal to zero. Moreover, we infer from that $(u_0,v_0)$ is a nonnegative ground state solution of with $\kappa_0=0$. To make it clear we divide into the following cases:
- If $\beta_{0}>\max\{a_0,b_0\}$, as in the proof of [@Sirakov-2007-CMP Theorem 1], we know that has positive radial ground state solution with $\kappa_0=0$. Moreover, $C_0^0<\frac{S_{1}^2}{4}\max\{\frac{1}{a_0^2},\frac{1}{b_0^2}\}$. Thus, $u_0\not\equiv0$ and $v_0\not\equiv0$ is a nonnegative radial ground state solution of with $\kappa_0=0$. By maximum principal, one deduces that $(u_0,v_0)$ is a positive radial ground state solution of with $\kappa_0=0$.
- If $0<\beta_{0}<\min\{a_0,b_0\}$, by using similar arguments as in [@Sirakov-2007-CMP Proposition 3.3], we know that $\int_{\mathbb{R}^{N}}u_{n}^{4},
\int_{\mathbb{R}^{N}}v_{n}^{4}\geq\delta_{0}>0$. Hence, $u_0\not\equiv0$ and $v_0\not\equiv0$ is a positive radial ground state solution of with $\kappa_0=0$.
- If $\min\{a_0,b_0\}\leq\beta_{0}\leq\max\{a_0,b_0\}$ and $a_0\neq
b_0$, as in [@Sirakov-2007-CMP Theorem 1], the system does not have a nontrivial solution(both component are not equal to zero) with nonnegative components. So, one of $u_0$ and $v_0$ must be zero.
- If $0<a_0=b_0<1$ and $\beta_0=a_0$, we claim that both $u_0$ and $v_0$ are non zero. In fact, if $u_0=0$, we infer from [@Sirakov-2007-CMP Proposition 3.2] that $$\label{c21}
\frac{S_1^2}{4a_0^2}=c_0^{0}\leq\frac{S_1^2}{4a_0}.$$ This is a contradiction. So, $u_0\not\equiv0$ and $v_0\not\equiv0$ is a positive radial ground state solution of with $\kappa_0=0$ in this case.
Next we consider the case $N=1$. As in , we obtain $(u_{n},v_{n})$ is bounded in $E_{r}$. Without loss of generality we assume that $(u_{n},v_{n})\rightharpoonup(u_{0},v_{0})$ in $H_{r}^{1}(\mathbb{R})\times H_{r}^{1}(\mathbb{R})$, and $(u_{n},v_{n})\rightarrow(u_{0},v_{0})$ in $L_{r,loc}^{p}(\mathbb{R})\times L_{r,loc}^{p}(\mathbb{R})$$(\forall
p\in(2,\infty))$. Moreover, similar to , one deduces that $(u_{0},v_{0})$ is an nonnegative solution of with $\kappa_0=0$. Since $C_0^{\kappa_0}$ is a decreasing function on $\kappa_0$, it follows from that $$\label{c22}
\frac{1}{4}\int_{\mathbb{R}}(a_{0}u_{0}^{4} +b_{0}v_{0}^{4}+2\beta_0
u_{0}^{2}v_{0}^{2})\geq c_0^{0}\geq c_0^{\kappa_{0}^n}\geq
c^{\kappa_{0}^1}\geq\delta>0.$$ So, at least one of $u_{0}$ and $v_0$ is not equal to zero. The rest of the proof is almost the same as the case $2\leq N\leq3$. We omit the details here.
Uniqueness and nondegeneracy of the positive solution
-----------------------------------------------------
In this subsection we consider the uniqueness and nondegeneracy of the positive solution of when $p=4$, and $a_0$ and $b_0$ are positive constants. Then the system is $$\label{auto-2}
\begin{cases}
-\Delta u+u=a_{0}u^{3}+\beta_{0}v^{2}u+\kappa_{0}v,\quad &\text{in}\ \mathbb{R}^{N}\\
-\Delta v+v=b_{0}v^{3}+\beta_{0}u^{2}v+\kappa_{0}u,\quad &\text{in}\
\mathbb{R}^{N}.
\end{cases}$$ We look for a synchronized solution of the form $z=(a_{1}w(a_3x),a_2w(a_3x))$, where $w$ is the unique positive solution of $$\label{c-18}
-\Delta u+u=u^{3},\quad u\in H^1(\mathbb{R}^{N}).$$ Substituting $z$ into gives $$\label{c-19}
\begin{cases}
a_1a_3^2=a_1-\kappa_{0}a_2,\\
a_3^2=a_{0}a_1^2+\beta_{0}a_2^2,\\
a_2a_3^2=a_2-\kappa_{0}a_1,\\
a_3^2=b_{0}a_2^2+\beta_0a_1^2.
\end{cases}$$ Solving gives $$\label{c-20}
\begin{cases}
a_{0}=b_{0}:=\mu>0,\\
a_1=a_2=\pm\sqrt{\frac{1-\kappa_0}{\mu+\beta_0}}, \beta_0>-\mu, 0<\kappa_0<1,\\
a_3=\sqrt{1-\kappa_0},
\end{cases}$$ or $$\label{c-21}
\begin{cases}
a_{0}=b_{0}:=\mu>0,\\
a_1=-a_2=\pm\sqrt{\frac{1-\kappa_0}{\mu+\beta_0}}, \beta_0>-\mu, 0<\kappa_0<1,\\
a_3=\sqrt{1-\kappa_0}.
\end{cases}$$ Without loss of generality we may assume that $a_{0}=b_{0}=\mu=1$. Then the system has the four synchronized solutions $z_{1,2}=\left(\pm\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),
\pm\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x)\right)$ and $z_{3,4}=\left(\pm\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),
\mp\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x)\right)$. We first consider the uniqueness of the positive solution $z_1$.
\[lem-3.2\]
1. If $N=1$ and $\mu=1$, then $z_1$ is the unique positive solution of in the following cases: (i) $1\leq\beta_0$ and $0<\kappa_0<1$, (ii) $-1<\beta_0<1$ and $\kappa_0>0$ is sufficiently small, (iii) $0<\kappa_0<1$ and $\beta_0-1<0$ is sufficiently small.
2. If $N=2$ or $3$, then $z_1$ is the unique positive solution of in the following cases: (i) $\beta_0\geq1$ and $0<\kappa_0<1$, (ii) $\beta_0, \kappa_0>0$ are sufficiently small, (iii) $\beta_0>0$ is sufficiently small and $\kappa_0$ is close to $1$.
3. If $\beta_0<0$ is sufficiently small, and $\kappa_0=0$ or $\kappa_0$ is close to $1$, then $z_1$ is the unique radial positive solution of .
<!-- -->
1. We modify the argument of [@WeiYao2012-CPAA Theorem 1.1]. If $N=1$ and $\mu=1$, system reduces to $$\label{c-22}
\begin{cases}
&-u''+u=u^{3}+\beta_{0}v^{2}u+\kappa_{0}v,\quad \text{in}\ [0,\infty),\\
&-v''+v=v^{3}+\beta_{0}u^{2}v+\kappa_{0}u,\quad \text{in}\
[0,\infty),\\
&u(r)>0, v(r)>0\ \text{in}\ [0,\infty),\\
&u'(0)=v'(0)=0,\ u(r),\ v(r)\rightarrow0\ \text{as}\
r\rightarrow\infty.
\end{cases}$$ Let $(u,v)$ be a positive solution of . Thus, we only need to prove that $v(r)=u(r)$ for all $r\geq0$ by the uniqueness result of the single scalar equation. Multiplying the first and second equations of by $v$ and $u$ respectively, then we have that $$\label{c-23}
(u'v)'-u'v'-uv+u^{3}v+\beta_{0}v^{3}u+\kappa_{0}v^2=0,$$ and $$\label{c-24}
(uv')'-u'v'-uv+v^{3}u+\beta_{0}u^{3}v+\kappa_{0}u^2=0.$$ Subtracting by gives $$\label{c-25}
(u'v-uv')'+(1-\beta_0)uv(u^2-v^2)+\kappa_{0}(v^2-u^2)=0.$$ Integrating over $(0,\infty)$ and using $u'(0)=v'(0)=u(\infty)=v(\infty)$, we have $$\label{c-26}
\int_{0}^{\infty}[(1-\beta_0)uv-\kappa_{0}](u+v)(u-v)=0.$$ We first claim that for $\beta_0>-1$, $u\geq v$ or $u\leq v$. Suppose not, then $g=u-v$ changes sign. It is easy to see that $$\label{c-27}
g''-(1+\kappa_{0})g-(u^2+uv+v^2-\beta uv)g=0\quad \text{in}\
[0,\infty).$$ By using unique continuation property for elliptic equation, we know that $g$ is not equal to zero in any nonempty interval. Furthermore, by Maximum principle we infer that $g(r)=u(r)-v(r)$ changes sign only finite time. Without loss of generality we may assume that $g(r)>0$ for large $r$. Thus there exists $R_1>0$ such that for $r>R_1$ and $\beta_0\neq1$ $$\label{c-28}
u(R_1)v(R_1)<\frac{\kappa_{0}}{|1-\beta_0|},\quad
u(R_1)-v(R_1)=0\quad\text{and}\quad u(r)-v(r)>0.$$ This implies that $$\label{c-29}
u'(R_1)-v'(R_1)\geq0.$$ Integrating over ($R_1,\infty$) we obtain $$\label{c-30}
-(u'v-uv')(R_1)+\int_{R_1}^{\infty}[(1-\beta_0)uv-\kappa_{0}](u+v)(u-v)=0.$$ We infer from - that $$\label{c-31}
\begin{split}
&-(u'v-uv')(R_1)=-u(R_1)[u'(R_1)-v'(R_1)]\leq0 \quad \text{and}\\
&(1-\beta_0)u(r)v(r)-\kappa_{0}\leq|1-\beta_0|u(R_1)v(R_1)-\kappa_{0}<0,\
\forall r\geq R_1.
\end{split}$$ This contradicts with . If $\beta_0=1$, we can also find the contradiction by using the argument of -. So, we prove the claim that for $\beta_0>-1$, $u\geq v$ or $u\leq v$. Finally, we need prove that $u\equiv v$. We divide into the following three cases:
1. If $\beta_0\geq1$, we know that $$\label{c-32}
(1-\beta_0)u(r)v(r)-\kappa_{0}<0,\ \forall r>0.$$ Moreover, we infer from $u\geq v$ or $u\leq v$ that the left hand of the integral is strict less than zero. This is contradiction. Thus, $u\equiv v$ in this case.
2. For each $-1<\beta_0<1$, without loss of generality we assume that $u\geq v$. It is clear that there exists $R_2>0$ large enough such that $$\label{c-33}
\int_{0}^{R_2}(u(r)+v(r))(u(r)-v(r))dr>\int_{R_2}^{\infty}(u(r)+v(r))(u(r)-v(r))dr.$$ Furthermore, we can choose $\kappa_0$ small enough such that $[(1-\beta_0)u(R_2)v(R_2)-\kappa_0]>0$. So, we obtain that $$\label{c-34}
\begin{split}
& \int_{0}^{\infty}[(1-\beta_0)uv-\kappa_{0}](u+v)(u-v)\\
\geq& [(1-\beta_0)u(R_2)v(R_2)-\kappa_{0}]\left(\int_{0}^{R_2}(u^2-v^2)dr-\int_{R_2}^{\infty}(u^2-v^2)dr\right)\\
>& 0.
\end{split}$$ This contradicts with .
3. For each $0<\kappa_0<1$, we should prove that the conclusion holds if $\beta-1$ close to $0^+$. Since $u(0)=\max u(x)$ and $v(0)=\max v(x)$, if $u(0)v(0)<\frac{\kappa_0}{1-\beta_0}$, we have that $$\label{c-35}
\int_{0}^{\infty}[(1-\beta_0)uv-\kappa_{0}](u+v)(u-v)<0.$$ This contradicts with .
2. We first use the idea of [@WeiYao2012-CPAA] to consider the case ($i$). Let $\Gamma_{+}=\{x\in\mathbb{R}^{N}: u(x)-v(x)>0\}$. Then $\Gamma_{+}$ is a piecewise $C^1$ smooth domain. Multiplying the first equation in by $v$ and the second equation in by $u$ and then integrating by parts on $\Gamma_{+}$ and subtracting together, we obtain the following integral identity $$\label{a-21}
\int_{\partial\Gamma_{+}}(v\frac{\partial u}{\partial
n}-u\frac{\partial v}{\partial
n})+\int_{\Gamma_{+}}[(1-\beta_0)uv-\kappa_{0}](u+v)(u-v)=0,$$ where $n$ denotes the unit outward normal to $\Gamma_{+}$. Since $u(x)-v(x)>0$ in $\Gamma_{+}$, $u(x)-v(x)=0$ on $\partial\Gamma_{+}$ and $\lim_{|x|\rightarrow\infty}u(x)=\lim_{|x|\rightarrow\infty}v(x)=0$, it follows that $$\label{a-22}
\int_{\partial\Gamma_{+}}(v\frac{\partial u}{\partial
n}-u\frac{\partial v}{\partial
n})=\int_{\partial\Gamma_{+}}u\frac{\partial (u-v)}{\partial
n}\leq0.$$ On the other hand, one sees that for $\beta\geq1$ and $0<\kappa_0<1$ $$\label{a-23}
\int_{\Gamma_{+}}[(1-\beta_0)uv-\kappa_{0}](u+v)(u-v)<0.$$ Hence, $\Gamma_{+}=\emptyset$. Similarly, we may prove that the set $\Gamma_{-}=\{x\in\mathbb{R}^{N}: u(x)-v(x)>0\}$ is also an empty set. Therefore, $u=v$ and we complete the proof of the case $(i)$.
Second, we consider the case $(ii)$. According to [@WeiYao2012-CPAA Theorem 4.1], if $\beta_0>0$ small and $\kappa_0=0$, we know that $z_1=(\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),
\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$ is a unique positive solution of . Moreover, $z_1$ is nondegenerate in $E_r=H_{r}^{1}(\mathbb{R}^{N})\times
H_{r}^{1}(\mathbb{R}^{N})$ by [@WeiYao2012-CPAA Lemma 2.2]. For each $z=(u,v)\in E_r$ we define $$\label{a-19}
\Phi_{\kappa_0}(u,v)=\frac{1}{2}(\|u\|^{2}+\|v\|^{2})-\frac{1}{4}\int_{\mathbb{R}^{N}}(u^4+v^4+2\beta_0
u^2v^2)-\kappa_0\int_{\mathbb{R}^{N}}uv.$$ Let $\Psi(\kappa_0,u,v)=\Phi'_{\kappa_0}(u,v)$. Obviously, we have that $\Psi(0,z_1)=0$. Moreover, $\Psi_{z}(0,z_1)=\Phi''_{\kappa_0}(z_1)$ is invertible. By the implicit function theorem, there exist $\widetilde{\beta}_0> 0, R_0
>0$ and $\psi:(-\widetilde{\beta}_0,\widetilde{\beta}_0)\rightarrow B_{R_0}(z_1)$ such that for any $\beta_0\in(-\widetilde{\beta}_0,\widetilde{\beta}_0)$, $\Psi(\kappa_0,z)=0$ has a unique solution $z=\psi(\beta_0)$ in $B_{R_0}(z_1)$. Furthermore, by using the same blow up arguments as [@Dancer-Wei-2009-TRMS Lemma 2.4], we know that for each fixed $0<\kappa_0<1$, there exists $C_{\kappa_0}>0$ such that $$\label{a-20}
|u|_{L^{\infty}(\mathbb{R}^{N})}+|v|_{L^{\infty}(\mathbb{R}^{N})}\leq
C_{\kappa_0},$$ where $(u,v)$ is a nonnegative solution of . Thus for $\kappa_0$ sufficiently small, the set of solutions to system is contained in $B_{R_0}(z_1)$.
Finally, we prove the case ($iii$). For $\bar{\beta}_0>0$, we define $$\mathcal {S}_{\bar{\beta}_0}=\{z=(u,v)\in E_{r}: z\ \text{is\ a\
positive\ solution\ of\ \eqref{auto-2}\ with}\
\beta_0\in[0,\bar{\beta}_0]\},$$ where $E_{r}=H_{r}^1(\mathbb{R}^{N})\times H_{r}^1(\mathbb{R}^{N})$. By using a minor modification of the arguments of [@Ikoma-2009-Nodea Corollary 2.4], we know that $\mathcal {S}_{\bar{\beta}_0}$ is compact in $E_r$. Moreover, according to [@ACR-1 Lemma 3.13], we know that the unique positive solution $\tilde{z}_0=(\sqrt{1-\kappa_0}w(\sqrt{1-\kappa_0}x),
\sqrt{1-\kappa_0}w(\sqrt{1-\kappa_0}x))$ with $\beta_0=0$ of is nondegenerate. So, by using the same arguments as in the proof of the case ($i$), we can prove that for $\beta_0>0$ small, has a unique positive solution.
3. Let $\mathcal {S}_{-\bar{\beta}_0}=\left\{z=(u,v)\in E_{r}: z\ \text{is\ a\
positive\ solution\ of\ \eqref{auto-2}\ with}\
\beta_0\in[\bar{\beta}_0,0]\right\}$. We first claim that for any $\bar{\beta}_0>0$, there exists $C_{\bar{\beta}_0}>0$ such that $$|u|_{\infty}+|v|_{\infty}\leq C_{\bar{\beta}_0}.$$ Similarly, we also use the blow up arguments as [@Dancer-Wei-2009-TRMS Lemma 2.4]. Assume that there exist a sequence of positive solutions $\{z_{n}=(u_{n},v_{n})\}$ of with $\beta_{n}\in[-\bar{\beta}_0,0]$ such that $\beta_{n}\rightarrow\tilde{\beta}$ and $|v_{n}|_{\infty}\leq|u_{n}|_{\infty}\rightarrow\infty$ as $n\to\infty$. We set $$\eta_{n}=\frac{1}{|u_{n}|_{\infty}},\quad
(w_{n}(x),h_{n}(x))=(\eta_{n}u_{n}(\sqrt{\eta_{n}}x),\eta_{n}v_{n}(\sqrt{\eta_{n}}x)).$$ Since $u_{n}$ and $v_{n}$ are radially symmetric and decreasing in the radial direction. Hence $|h_{n}|_{\infty}\le
|w_{n}|_{\infty}=w_n(0)=1$. It is easy to verify that $(w_{n},h_{n})$ satisfies $$\begin{cases}
-\Delta
w_{n}+\eta_{n}w_{n}=w_{n}^3+\beta_{n}h_{n}^2w_{n}+\eta_{n}\kappa_0h_{n},\\
-\Delta
h_{n}+\eta_{n}h_{n}=h_{n}^3+\beta_{n}w_{n}^2h_{n}+\eta_{n}\kappa_0w_{n}.
\end{cases}$$ By the standard elliptic argument, we may assume that, subject to a subsequence, $(w_{n},h_{n})\rightarrow(w_{0},h_{0})$ in $C_{loc}^{2}(\mathbb{R}^{3})$ as $n\to\infty$, where $(w_{0},h_{0})$ is a nonnegative solution of $$\label{c-j41}
\begin{cases}
-\Delta
w_{0}=w_{0}^3+\tilde{\beta}h_0^{2}w_{0},\\
-\Delta h_{0}=h_{0}^3+\tilde{\beta}w_0^{2}h_{0}.
\end{cases}$$ Since $w_0(0)=1$ and the maximum principle shows that $w_0(x)>0$ in $\mathbb{R}^N$. However, as in [@Wei-Juncheng-2010-Anals Theorem 2.1], we know that for $\tilde{\beta}>-1$, any nonnegative solution of is zero. This is a contradiction.
By [@WeiYao2012-CPAA Theorems 4.1 and 4.2], we have the following result for $\kappa_0=0$.
\[lem-3.3\]
- If $N=1$ and $0\leq\beta_0\not\in[\min\{a_0,b_0\}, \max\{a_0,b_0\}]$, then $z_0=(\sqrt{\frac{\beta_0-b_0}{\beta_0^2-a_0b_0}}w(x)$, $\sqrt{\frac{\beta_0-a_0}{\beta_0^2-a_0b_0}}w(x))$ is the unique solution of .
- If $N=2$ or $3$, and $\beta_0>\max\{a_0,b_0\}$ or $\beta_0>0$ is sufficiently small, then $z_0$ is the unique positive solution of .
Next we study the nondegeneracy of solutions of the system . Recall that $(U_1,U_2)$ is a nondegenerate solution if the solution set of the linearized system $$\label{a-24}
\begin{cases}
&\Delta \phi_{1}-\phi_{1}+\kappa_0\phi_2+3U_{1}^{2}\phi_1+\beta_{0}U_{2}^{2}\phi_1+2\beta_{0}U_{1}U_{2}\phi_2=0,\\
&\Delta
\phi_{2}-\phi_{2}+\kappa_0\phi_1+3U_{2}^{2}\phi_2+\beta_{0}U_{1}^{2}\phi_2+2\beta_{0}U_{1}U_{2}\phi_1=0,\\
&\phi_{1}=\phi_{1}(r),\ \phi_{2}=\phi_{2}(r)
\end{cases}$$ is $N$-dimensional, i.e., $$\label{a-25}
\phi=\left(\begin{array}{cc}
\phi_{1}\\
\phi_{2} \\
\end{array}
\right)
=\sum_{j=1}^{N}k_j\left(\begin{array}{cc}
\frac{\partial U_1}{\partial x_j}\\
\frac{\partial U_2}{\partial x_j} \\
\end{array}
\right).$$ Set $z_1=
(c_{0}w_{0},c_{0}w_{0}):=(\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),$ $\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$. In the following we study the nondegeneracy of the solution $z_1$. First we have the following result for $\kappa_0=0$ as in [@Dancer-Wei-2009-TRMS lemma 2.2].
\[lem-3.4\] If $0\leq\beta_0\not\in[\min\{a_0,b_0\},\max\{a_0,b_0\}]$, then $\hat{z}_1=(\sqrt{\frac{\beta_0-b_0}{\beta_0^2-a_0b_0}}w(x),$ $\sqrt{\frac{\beta_0-a_0}{\beta_0^2-a_0b_0}}w(x))$ is nondegenerate in the space of radial functions.
For $\kappa_0\neq0$, we have the following result.
\[lem-3.5\] Let $0<\kappa_0<1$. If $\beta_{0}\geq3$, or $-1<\beta_{0}<3$ and $w(0)\leq
\sqrt{\frac{2\kappa_0(1+\beta_0)}{(3-\beta_0)(1-\kappa_0)}}$, then $z_1=
(c_{0}w_{0},c_{0}w_{0}):=(\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),$ $\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$ is nondegenerate in the space of radial functions, where $w(0)=\max w$ and $w$ is the unique positive solution of the scalar equation.
If $\kappa_0\neq0$, we shall prove the nondegenerate of $z_1=
(c_{0}w_{0},c_{0}w_{0}):=(\sqrt{\frac{1-\kappa_0}{1+\beta_0}}$ $w(\sqrt{1-\kappa_0}x),
\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$. The linearized problem of at $z_1$ becomes $$\label{c-43}
\begin{cases}
&\Delta \phi_{1}-\phi_{1}+\kappa_0\phi_2+(3+\beta_{0})c_{0}^{2}w_{0}^{2}\phi_1+2\beta_{0} c_{0}^{2}w_{0}^{2}\phi_2=0,\\
&\Delta
\phi_{2}-\phi_{2}+\kappa_0\phi_1+(3+\beta_{0})c_{0}^{2}w_{0}^{2}\phi_2+2\beta_{0}
c_{0}^{2}w_{0}^{2}\phi_1=0,\\
&\phi_{1}=\phi_{1}(r),\ \phi_{2}=\phi_{2}(r).
\end{cases}$$ By an orthonormal transformation, can be transformed to two single equations $$\label{c-44}
\begin{cases}
&\Delta\Phi_{1}-(1-\kappa_0)\Phi_{1}+3(1-\kappa_{0})w^{2}(\sqrt{1-\kappa_0}x)\Phi_1=0,\\
&\Delta\Phi_{2}-(1-\kappa_0)\Phi_{1}+\left[\frac{(3-\beta_0)(1-\kappa_0)}{1+\beta_0}w^{2}(\sqrt{1-\kappa_0}x)-2\kappa_0\right]\Phi_2=0.
\end{cases}$$ By scaling $x\mapsto\frac{y}{\sqrt{1-\kappa_0}}$, we know that becomes $$\label{c-45}
\begin{cases}
&\Delta\Psi_{1}-\Psi_{1}+3w^{2}(y))\Psi_1=0,\\
&\Delta\Psi_{2}-\Psi_{2}+\left[\frac{(3-\beta_0)}{1+\beta_0}w^{2}(y)-\frac{2\kappa_0}{1-\kappa_0}\right]\Psi_2=0,
\end{cases}$$ where $\Psi_{i}(y)=\Phi_i(\frac{y}{\sqrt{1-\kappa_0}})(i=1,2)$. On the other hand, since the eigenvalues of $$\label{c-46}
\Delta\Psi-\Psi+\lambda w^{2}\Psi=0,\quad \Psi\in
H^{1}(\mathbb{R}^{N})$$ are $$\label{c-47}
\lambda_1=1,\ \lambda_2=\cdot\cdot\cdot=\lambda_{N+1}=3,\
\lambda_{N+2}>3,$$ where the eigenfunction corresponding to $\lambda_1$ is $cw$, and the eigenfunctions corresponding to $\lambda_2$ are spanned by $\frac{\partial w}{\partial x_{j}}(j=1,2,\cdot\cdot\cdot,N)$.
So, the first equation has only zero solution, i.e., $\Psi_1=0$.
If $\beta_{0}\geq3$ and $0<\kappa_0<1$ we know that $K(\beta_0,\kappa_0):=\frac{(3-\beta_0)}{1+\beta_0}w^{2}(y)-\frac{2\kappa_0}{1-\kappa_0}<0$. It follows that $\Psi_2=0$.
If $-1<\beta_{0}<3$, $0<\kappa_0<1$ and $w(0)\leq\sqrt{\frac{2\kappa_0(1+\beta_0)}{(3-\beta_0)(1-\kappa_0)}}$, then $K(\beta_0,\kappa_0)\leq0$. Thus, $\Psi_2=0$.
\[rem-3.7\] Similarly, under the same conditions of Lemma \[lem-3.5\], one can prove that $z_2=
(-c_{0}w_{0},-c_{0}w_{0})=(-\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),-\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$ is also nondegenerate in the space of radial functions.
Concentration compactness lemma
===============================
The following profile decomposition is an immediate consequence of Theorem 3.1, that trivially adapts the reasoning for the scalar case of Corollary 3.2, from [@ccbook] to the Hilbert space $E=H^1({\mathbb{R}}^N) \times H^1({\mathbb{R}}^N)$, equipped with the group $D$ of lattice translations $D=\{(u,v)\mapsto (u(\cdot-y),v(\cdot-y)),\;y\in\mathbb {\mathbb{Z}}^N\}$.
\[profdec\] Let $(u_k, v_k)$ be a bounded sequence in $E$. There exists a renamed subsequence and a sequence $(y_k^{(n)})_k\subset \mathbb {\mathbb{Z}}^N)$, $n\in\mathbb N$, such that $y_k^{(1)}=0$, $$\label{wlim}
U^{(n)}=\stackrel{\rightharpoonup}{\lim} u_k(\cdot +y_k^{(n)});\; V^{(n)}=\stackrel{\rightharpoonup}{\lim} v_k(\cdot +y_k^{(n)});$$ $$|y_k^{(m)}-y_k^{(n)}|\to\infty\text{ for } m\neq n,$$ $$\sum_n \|U^{(n)}\|^2\le \|u_k\|^2+o(1); \; \sum_n \|V^{(n)}\|^2\le \|v_k\|^2+o(1);$$ and for any $p\in(2,2^*)$, $$\label{pd}
\rho_k:=u_k- \sum_n U^{(n)}(\cdot -y_k^{(n)})\to 0 \text{ in } L^p;\;
\tau_k:=v_k- \sum_n V^{(n)}(\cdot -y_k^{(n)})\to 0 \text{ in } L^p;\;$$ and the series in the last relations are convergent unconditionally in $H^1$ and uniformly with respect to $k$.
Note that it is a priori possible that one of the components of $(U^{(n)},V^{(n)})$ is zero.
We will now evaluate the asymptotic value of the functional on a sequence provided by the theorem above.
\[prop:energy\] Let $\Phi$ be the functional and let $\Phi_0$ be the functional . Let $(u_k,v_k)$ be the sequence provided by Theorem \[profdec\]. Then $$\label{eq:energy}
\Phi(u_k,v_k) \ge \Phi(U^{(1)},V^{(1)}) + \sum_{n=2}^\infty \Phi_0(U^{(n)},V^{(n)}).$$ Moreover, if, in addition, $\Phi'(u_k,v_k)\to 0$ and $\Phi(u_k,v_k) \to c \in {\mathbb{R}}$, then $(U^{(1)},V^{(1)})$ is a critical point of the functional $\Phi$, $(U^{(n)},V^{(n)})$ for any $n\ge 2$ is a critical point of the functional $\Phi_0$, and $$\label{eq:energy2}
\Phi(U^{(1)},V^{(1)}) + \sum_{n=2}^\infty \Phi_0(U^{(n)},V^{(n)}) = c.$$
By continuity of $\Phi-\Phi_0$ with respect to the weak convergence it suffices to prove for $\Phi=\Phi_0$, which can be immediately obtained by iteration of the Brezis-Lieb lemma (see [@CwiTi], Appendix B, for the scalar case). Since the map $\Phi'-\Phi'_0$ is continuous with respect to the weak convergence, the conclusion that $(U^{(n)},V^{(n)})$ is a critical point for respective functional is immediate. In order to show , let $\rho_k$ and $\tau_k$ be as in and note that $\Phi'(u,v)=(u,v)+ \varphi'(u,v)$ with continuous $\varphi'(u,v):L^p\times L^p\to E$. From here and from the criticality of points $(U^{(n)},V^{(n)})$ it follows from $(u_k,v_k)+ \varphi'(u_k,v_k)=\Phi'(u_k,v_k)\to 0$ by a standard continuity argument that $(\rho_k,\tau_k)\to 0$ in $E$. Consequently, recalling again that $\Phi-\Phi_0$ is weakly continuous, we have $$\begin{aligned}
c & = &\lim \Phi(u_k,v_k)\\
& = &\lim \Phi(\sum_n U^{(n)}(\cdot -y_k^{(n)}), \sum_n V^{(n)}(\cdot -y_k^{(n)}))\\
&= &\lim \Phi_0(\sum_n U^{(n)}(\cdot -y_k^{(n)}), \sum_n V^{(n)}(\cdot -y_k^{(n)}))+(\Phi-\Phi_0)(U^{(1)},V^{(1)})\\
&= &\sum \Phi_0(U^{(n)},V^{(n)})+(\Phi-\Phi_0)(U^{(1)},V^{(1)}),\end{aligned}$$ which proves .
Ground state solutions - functional-analytic setting
====================================================
In this section we study existence of ground state solutions for a functional-analytic model of our problem. We identify, for a class of functionals defined below, the ground state with the mountain pass solution. We then formulate a sufficient condition for existence of a ground state in terms of comparison with the problem at infinity (which, in these general settings, is not required itself to admit a ground state). Verification of the comparison condition and existence of the ground state for the problem at infinity is a subject of the next section, where more specific properties of the functional are invoked. The number $p>2$ remains fixed throughout the section.
Let $H$ be a Hilbert space. We say that a functional $\Phi\in C^2(H)$ is of class ${\mathcal S}_p$, if it is of the form $\Phi(u)=\frac12\|u\|^2-\frac{1}{p}\psi(u)$, where the functional $\psi\in C^2(H)$ is bounded on bounded sets, homogeneous of degree $p$ and positive except at $u=0$, the norm refers to any of equivalent norms of $H$, and assume that $\Phi'$ is weak-to-weak continuous on $H$. It is to be understood that the norm $\|\cdot\|$ is not fixed, but is one of equivalent norms of $H$, that may vary for different functionals in the class.
Note that the functional , and consequently , are of the class ${\mathcal S}_p$.
\[comp\] Let $\Phi\in {\mathcal S_p}$ and let ${\mathcal N}=\{u\in H\setminus\{0\}: (\Phi'(u),u)=0\}$. Then $w\in H\setminus\{0\}$ minimizes $\Phi$ on $\mathcal N$ if and only if the path $t\mapsto tw$, $0\le t<\infty$ minimizes $$\label{abstr-mpg}
c=\inf_{\eta\in P}\max_{t>0}\Phi(\eta(t)),$$ where $$P=\{\eta\in C([0,\infty);H), \eta(0)=0, \Phi(\eta(+\infty))=-\infty\}.$$ Moreover, $w$ is a critical point of $\Phi$.
1\. First note that $\Phi$ has the classical mountain pass geometry. Note also that since $\psi$ is bounded on bounded sets and homogeneous, $0\le\psi(u)\le C\|u\|^p$ which implies that $\mathcal N$ is bounded away from the origin. Furthermore, by Euler theorem for homogeneous functions, $(\psi{''}(u)u,u)=p(p-1)\psi(u)>0$ unless $u=0$, and therefore a minimizer $w$ of $\Phi$ on $\mathcal N$ is a nonzero critical point of $\Phi$.
2\. Note that every path in $P$ intersects $\mathcal N$, which implies that $c\ge \Phi(w)$. On the other hand, $c\le \max_t \Phi(tw)=\Phi(w)$. It is immediate then that whenever $w$ is a minimizer of $\Phi$ on $\mathcal N$, the path $t\mapsto tw$ minimizes .
3\. Conversely, if $w\in H\setminus\{0\}$ is such that the path $t\mapsto tw$ minimizes , then the maximum of $\Phi$ on the path is necessarily a critical point of $\Phi$, and thus belongs to $\mathcal N$, and consequently is attained at $t=1$, so $w$ is a critical point of $\Phi$. If, however, $w$ is not a minimal point of $\Phi$ on $\mathcal N$, and $w_1$ is such a minimizer, the maximum of $\Phi$ on $tw_1$ will be smaller than $c$, a contradiction.
\[lem:nonzero\] Let $\Phi\in\mathcal{S}_p$. Let $c$ be the minimax value . Then, if $\Phi(w_k)\to c$ and $\Phi'(w_k)\to 0$ in $H$, then $w_k$ is bounded. Moreover, if $w_k\rightharpoonup w\neq 0$, then $w_k\to w$ in the norm of $H$ and $w$ is a ground state of $\Phi$.
The proof of the first assertion of the lemma follows the classical argument of Ambrosetti-Rabinowitz. Multiplication of $\Phi'(w_k)\to 0$ by $w_k/\|w_k\|$ (the case when $w_k=0$ on a subsequence is trivial) gives $$\|w_k\|-\frac{\psi(w_k)}{\|w_k\|}\to 0.$$ This implies that $\psi(w_k)=\|w_k\|(\|w_k\|+o(1))$ and $\Phi(w_k)=(\frac{1}{2}-\frac{1}{p})\|w_k\|^2+o(\|w_k\|)\to c$, which implies that $w_k$ is bounded in norm. Then we also have $\Phi(w_k)=(\frac{1}{2}-\frac{1}{p})\|w_k\|^2+o(1)$.
We now prove the second assertion when $w_k\rightharpoonup w\neq 0$. By weak semicontinuity of the norm, $$\Phi(w_k)=(\frac{1}{2}-\frac{1}{p})\|w_k\|^2+o(1)\ge (\frac{1}{2}-\frac{1}{p})\|w\|^2+o(1)=\Phi(w)+o(1).$$ By weak-to-weak continuity of $\Phi'$, the element $w$ is a (nonzero) critical point of $\Phi$, and thus $w\in\mathcal N$. Evaluation of the functional on the path $t\mapsto tw$ gives $\lim \Phi(w_k)=c\le \Phi(w)$. Together with the previous inequality we have that $\Phi(w_k)\to \Phi(w)$. This implies that $\|w_k\|^2\to \|w\|^2$, which in turn means that $w_k\to w$ in $H$. By Lemma \[comp\], the element $w$ is a ground state of $\Phi$.
Lemma \[lem:nonzero\] shows that the ground state exists as long as the critical sequence at the mountain pass level does not converge weakly to zero. The next lemma introduces a (still implicit) sufficient condition for the latter and thus for the existence of a ground state.
From now on we assume that $H$ is a space of functions ${\mathbb{R}}^N\to {\mathbb{R}}^m$, $m\in{\mathbb{N}}$, such that for every sequence $y_k\in \mathbb{R}^N$, $|y_k|\to\infty $ and every $w\in H$, $w(\cdot-y_k)\rightharpoonup 0$.
Let $\Phi\in\mathcal{S}_p$. One says that a functional $\Phi_0\in\mathcal{S}_p$ is a limit of $\Phi$ at infinity if for every $y\in \mathbb{Z}^N$ and every $w\in H$, $\Phi_0(w(\cdot+y))= \Phi_0(w)$, and the maps $\Phi-\Phi_0$ and $\Phi'-\Phi'_0$ are continuous with respect to weak convergence.
Note that $c\le c_0$, which is easy to show by evaluating $\Phi$ on the paths approximating $c_0$ for $\Phi_0$, translated by $y$ far enough from the origin of ${\mathbb{Z}}^N$ (on which the difference between $\Phi_0$ and $\Phi$ is insignificant).
In what follows we will make the following assumption on the functional $\psi_0$: $$\label{coco}
w_k\in H, y_k\in{\mathbb{Z}}^N, w_k(\cdot-y_k)\rightharpoonup 0 \Longrightarrow \psi_0(w_k)\to 0.$$ Note that our notation is consistent with the notation in the previous sections in the sense that whenever $0<\kappa_0<1$, the functional is the limit at infinity of the functional .
Assume that $\Phi\in\mathcal{S}_p$ has a limit functional $\Phi_0\in\mathcal{S}_p$ at infinity and that is satisfied. Let $c$ and $c_0$ be the mountain pass values for $\Phi$ and $\Phi_0$, respectively. If $c<c_0$, then $\Phi$ has a ground state.
Let $w_k\rightharpoonup 0$ be a critical sequence for $\Phi$ with $\Phi(w_k)\to c$. By definition of the functional at infinity, we have $c=\lim \Phi(w_k)=\lim \Phi_0(w_k)=\lim \Phi_0(w_k(\cdot-y_k))$ for any sequence $y_k\in{\mathbb{Z}}^N$ such that $|y_k|\to\infty $. Note also that we may assume that $w_k(\cdot-y_k)$ has a weak limit $w_0$, which is necessarily a critical point of $\Phi_0$. We may also assume that $w_0\neq 0$, since if it would happen for any sequence $y_k$ with $|y_k|\to\infty $, by we would have $\Phi_0(w_k)\to 0$, and thus $c=\lim\Phi(w_k)= 0$, a contradiction. Then, by weak-to-weak continuity of $\Phi_0'$ (assured by the definition of $\mathcal{S}_p$) we have $w_0\in \mathcal{N}_0$, and therefore $\Phi_0(w_0)\ge c_0$.
Since $\Phi_0(w_k(\cdot-y_k))=(\frac{1}{2}-\frac{1}{p})\|w_k\|_0^2+o(1)$ and the norm is weakly lower semicontinuous, we have $$c+o(1)=\Phi_0(w_k(\cdot-y_k))\ge (\frac{1}{2}-\frac{1}{p})\|w_0\|_0^2=\Phi_0(w_0)\ge c_0,$$ which contradicts the assumption $c<c_0$. We conclude that the critical sequence has to have a subsequence with a nonzero weak limit, which by Lemma \[lem:nonzero\] implies existence of a ground state.
\[lem:less\] Assume that $\Phi\in\mathcal{S}_p$ has a limit functional $\Phi_0\in\mathcal{S}_p$ at infinity. Let $c$ and $c_0$ be the mountain pass values for $\Phi$ and $\Phi_0$, respectively. Assume that $\Phi_0$ has a ground state $w_0$. If $$\label{eq:less}
(\frac{1}{2}-\frac{1}{p})\left(\frac{\|w_0\|^p}{\psi(w_0)}\right)^\frac{2}{p-2}<\Phi_0(w_0),$$ then $c<c_0$.
By defintion, $$c\le\max_{t>0} \Phi(tw_0)=\max_{t>0}\frac{t^2}{2}\|w_0\|^2-\frac{t^p}{p}\psi(w_0).$$ Elementary evaluation of the maximum gives $$c\le (\frac{1}{2}-\frac{1}{p})\left(\frac{\|w_0\|^p}{\psi(w_0)}\right)^\frac{2}{p-2}.$$ Since $c_0=\Phi_0(w_0)$, the inequality above and imply $c<c_0$.
\[cor:less\] Under conditions of Lemma \[lem:less\], if $$\label{eq:less2}
\frac{\|w_0\|^p}{\psi(w_0)}<\frac{\|w_0\|_0^p}{\psi_0(w_0)},$$ then $c<c_0$.
Since $\Phi_0\in\mathcal{S}_p$, we have the representation $\Phi_0=\frac{1}{2}\|\cdot\|_0^2-\frac{1}{p}\psi_0$ which, by repeating calculations in the proof of Lemma \[lem:less\] allows to represent the ground state value of $\Phi_0$ as $$c_0=\max_t\Phi_0(tw_0)=(\frac{1}{2}-\frac{1}{p})\left(\frac{\|w_0\|_0^p}{\psi_0(w_0)}\right)^\frac{2}{p-2}.$$ Thus can be written in the equivalent form as .
Ground state solutions
======================
Now we give several sufficient conditions to have . We will always assume that $0<\kappa_0<1$, so that the functional $\Psi_0$ has a positive ground state, which we denote as $(u,v)$, by Lemma \[lem-3.1\].
\[zh0127\] The functional has a ground state if one of the following conditions - holds. $$\label{01}
\left(\frac{\|(u,v)\|^2_0-\int\kappa(x)uv}{\|z\|^2_0}\right)^{p/2}<\frac{\psi(u,v)}{\psi_0(u,v)};$$ $$\label{02}
\kappa(x)\ge 0, a(x)\ge 0, b(x)\ge 0, \beta(x)\ge 0,$$ provided that at least one of the inequalities is strict on a set of positive measure;
For the case $u=v$ (see sufficient conditions in Section 3), $$\label{03}
(a(x)+b(x)+2\beta(x))u^{p-2}+p\kappa(x)\ge 0;$$ or in particular if $$\label{04}
\kappa(x)\ge 0 \text{ and } a(x)+b(x)+2\beta(x)\ge 0,$$ provided that at least one of the inequalities is strict on a set of positive measure.
Condition is a restatement of . Condition obviously implies . Condition is restated for $u=v=w$, and condition trivially implies .
From the results of Theorem \[zh0127\], we know that Theorem \[th1.1\] hold.
Finally we give another proof for Lemma \[lem-3.1\].
Let $w_k$ be the subsequence given by Theorem \[profdec\] of a critical sequence for $\Phi_0$. Assume that the series contains at least two nonzero terms. Then, using Proposition \[prop:energy\] we have $$\begin{array}{lll}
c_0&=&\Phi_0(w_k)+o(1)=(\frac{1}{2}-\frac{1}{p})\psi_0(w_k)+o(1)\\
& \ge &(\frac{1}{2}-\frac{1}{p})(\psi_0(U^{(1)},V^{(1)})+\psi_0(U^{(2)},V^{(2)}))+o(1)\\
&=& \Phi_0(U^{(1)},V^{(1)})+\Phi_0(U^{(2)},V^{(2)})+o(1).
\end{array}$$
At the same time, considering the functional $\Phi_0$ on the path $t\mapsto t(U^{(1)},V^{(1)})$, we have $c_0\le \Phi_0(U^{(1)},V^{(1)})$, which is a contradiction. We conclude therefore that the critical sequence $\tilde w_k=w_k(\cdot+y_k^{(1)})$ is convergent in $L^p$ to $(U^{(1)},V^{(1)})$, from which one can easily conclude that $(U^{(1)},V^{(1)})$ is a ground state.
Bound state solutions
=====================
Throughout this section we fix $0 < \kappa_0 < 1$, and assume that $\kappa(x) \le 0$, $a(x) \le 0$, $b(x) \le 0$ and $\beta(x) \le 0$, with at least one of the inequalities strict on a set of positive measure.
\[Lemma 7.1\] $c = c_0$ and $c$ is not attained.
The periodic system has a positive ground state solution $(u,v) \in \mathscr{N}_0$ by Lemma \[lem-3.1\]. Take a sequence $y_k \in {\mathbb{Z}}^N$ such that $|y_k| \to \infty$ and set $$(u_k,v_k) = (t_k\, u(\cdot + y_k),t_k\, v(\cdot + y_k)),$$ where $t_k > 0$ is such that $(u_k,v_k) \in \mathscr{N}$, i.e., $$\begin{aligned}
t_k^{p-2} & = & \frac{\|u(\cdot + y_k)\|^2 + \|v(\cdot + y_k)\|^2 - 2 {\displaystyle \int}_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x))\, u(x + y_k)\, v(x + y_k)\, dx}{\begin{split}
{\displaystyle \int}_{{\mathbb{R}}^N} \big[(a_0(x) + a(x))\, u(x + y_k)^p + (b_0(x) + b(x))\, v(x + y_k)^p\\[-10pt]
+ 2\, (\beta_0 + \beta(x))\, u(x + y_k)^{p/2}\, v(x + y_k)^{p/2}\big] dx
\end{split}}\\[10pt]
& = & \frac{\|u\|^2 + \|v\|^2 - 2 {\displaystyle \int}_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x - y_k))\, uv\, dx}{\begin{split}
{\displaystyle \int}_{{\mathbb{R}}^N} \big[(a_0(x) + a(x - y_k))\, u^p + (b_0(x) + b(x - y_k))\, v^p\\[-10pt]
+ 2\, (\beta_0 + \beta(x - y_k))\, u^{p/2}\, v^{p/2}\big] dx
\end{split}}\\[10pt]
& = & \frac{{\displaystyle \int}_{{\mathbb{R}}^N} \big[a_0(x)\, u^p + b_0(x)\, v^p + 2 \beta_0\, u^{p/2}\, v^{p/2}\big] dx
- 2 {\displaystyle \int}_{{\mathbb{R}}^N} \kappa(x - y_k)\, uv\, dx}{\begin{split}
{\displaystyle \int}_{{\mathbb{R}}^N} \big[a_0(x)\, u^p + b_0(x)\, v^p + 2 \beta_0\, u^{p/2}\, v^{p/2}\big] dx
+ {\displaystyle \int}_{{\mathbb{R}}^N} \big[a(x - y_k)\, u^p + b(x - y_k)\, v^p\\[-10pt]
+ 2 \beta(x - y_k)\, u^{p/2}\, v^{p/2}\big] dx
\end{split}}.\end{aligned}$$ Since $|y_k| \to \infty$, $t_k \to 1$ and hence $\Phi(u_k,v_k) \to c_0$, so $c \le c_0$. To see that the reverse inequality holds, for any $(u,v) \in \mathscr{N}$ such that $uv \ge 0$, let $t > 0$ be such that $(tu,tv) \in \mathscr{N}_0$, i.e., $$\begin{aligned}
\label{7.1}
&~& t^{p-2} = \frac{\|u\|^2 + \|v\|^2 - 2 \kappa_0 {\displaystyle \int}_{{\mathbb{R}}^N} uv\, dx}{{\displaystyle \int}_{{\mathbb{R}}^N} \big[a_0(x)\, |u|^p + b_0(x)\, |v|^p + 2 \beta_0\, |u|^{p/2}\, |v|^{p/2}\big] dx} \notag\\[10pt]
& = & \frac{\begin{split}
{\displaystyle \int}_{{\mathbb{R}}^N} \big[(a_0(x) + a(x))\, |u|^p + (b_0(x) + b(x))\, |v|^p + 2\, (\beta_0 + \beta(x))\, |u|^{p/2}\, |v|^{p/2}\big] dx\\[-10pt]
+ 2 {\displaystyle \int}_{{\mathbb{R}}^N} \kappa(x)\, uv\, dx
\end{split}}{{\displaystyle \int}_{{\mathbb{R}}^N} \big[a_0(x)\, |u|^p + b_0(x)\, |v|^p + 2 \beta_0\, |u|^{p/2}\, |v|^{p/2}\big] dx}.\end{aligned}$$ Since $\kappa(x), a(x), b(x), \beta(x) \le 0$ and $uv \ge 0$, $t \le 1$ and hence $$\begin{gathered}
c_0 \le \Phi_0(tu,tv) = t^2 \left(\frac{1}{2} - \frac{1}{p}\right) \left[\|u\|^2 + \|v\|^2 - 2 \kappa_0 \int_{{\mathbb{R}}^N} uv\, dx\right] \le \left(\frac{1}{2} - \frac{1}{p}\right) \bigg[\|u\|^2 + \|v\|^2\\[10pt]
- 2 \kappa_0 \int_{{\mathbb{R}}^N} uv\, dx\bigg]
\le \left(\frac{1}{2} - \frac{1}{p}\right) \left[\|u\|^2 + \|v\|^2 - 2 \int_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x))\, uv\, dx\right] = \Phi(u,v)\end{gathered}$$ by , so $c_0 \le c$. If $\Phi(u,v) = c$, then equality holds throughout and hence $t = 1$ and $(u,v)$ is a ground state solution of system , so an argument similar to that in the proof of Lemma \[lem-3.1\] shows that $uv > 0$. Then implies that $\kappa(x), a(x), b(x), \beta(x) \equiv 0$, which is contrary to assumptions.
By Lemma \[Lemma 7.1\], system has no solutions at the level $c$, so we look for a solution at a higher energy level using the notion of barycenter as in [@ACR-1]. In view of Lemma \[lem-3.5\], we only consider the case where $p = 4$ and $a_0, b_0$ are positive constants, so our system is $$\label{7.2}
\begin{cases}
- \Delta u + u = (a_0 + a(x))\, u^3 + (\beta_0 + \beta(x))\, v^2 u + (\kappa_0 + \kappa(x))\, v \quad & \text{in } {\mathbb{R}}^N,\\
- \Delta v + v = (b_0 + b(x))\, v^3 + (\beta_0 + \beta(x))\, u^2 v + (\kappa_0 + \kappa(x))\, u \quad & \text{in } {\mathbb{R}}^N,
\end{cases}$$ where $N \le 3$, $\beta_0 \in {\mathbb{R}}$, $a, b, \beta, \kappa \in L^\infty(\mathbb{R}^N)$ go to zero as $|x| \to \infty$, and $$\begin{gathered}
\label{7.10}
a_0 + \inf_{x \in {\mathbb{R}}^N}\, a(x) > 0,~ b_0 + \inf_{x \in {\mathbb{R}}^N}\, b(x) > 0,
0 < \kappa_0 + \inf_{x \in {\mathbb{R}}^N}\, \kappa(x) \le \kappa_0 + \sup_{x \in {\mathbb{R}}^N}\, \kappa(x) < 1.\end{gathered}$$ The associated energy functional is $$\begin{split}
\Phi(u,v) & = \frac{1}{2} \left(\|u\|^2 + \|v\|^2\right) - \frac{1}{4} \int_{{\mathbb{R}}^N} \left[(a_0 + a(x))\, u^4 + (b_0 + b(x))\, v^4\right]\\
& \quad - \frac{1}{2} \int_{{\mathbb{R}}^N} (\beta_0 + \beta(x))\, u^2 v^2 - \int_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x))\, uv, \quad (u,v) \in E.
\end{split}$$
For $u \in H^1({\mathbb{R}}^N)$, let $$\mu(u)(x) = \frac{1}{|B_1|} \int_{B_1(x)} |u(y)|\, dy$$ and note that $\mu(u)$ is a bounded continuous function on ${\mathbb{R}}^N$. Then set $$\hat{u}(x) = \left[\mu(u)(x) - \frac{1}{2}\, \max \mu(u)\right]^+,$$ so that $\hat{u} \in C_0({\mathbb{R}}^N)$. The barycenter of a pair $(u,v) \in E \setminus \{(0,0)\}$ was defined in [@ACR-1] by $$\xi(u,v) = \frac{1}{|\hat{u}|_1 + |\hat{v}|_1} \int_{{\mathbb{R}}^N} x \left(\hat{u}(x) + \hat{v}(x)\right) dx$$ (see also [@MR1989833]). Since $\hat{u}$ and $\hat{v}$ have compact supports, $\xi : E \setminus \{(0,0)\} \to {\mathbb{R}}^N$ is a well-defined continuous map. As noted in [@ACR-1], it has the following properties:
1. If $u$ and $v$ are radial functions, then $\xi(u,v) = 0$.
2. For $t \ne 0$, $\xi(tu,tv) = \xi(u,v)$.
3. For all $y \in {\mathbb{R}}^N$, $\xi(u(\cdot + y),v(\cdot + y)) = \xi(u,v) - y$.
Set $$\tilde{c} = \inf_{(u,v) \in \mathscr{N}\!,\, \xi(u,v) = 0}\, \Phi(u,v).$$ Clearly, $\tilde{c} \ge c$. As in [@ACR-1], we have the following lemma.
\[Lemma 7.2\] $\tilde{c} > c$.
Suppose $\tilde{c} = c$. Then there exists a sequence $(u_k,v_k) \in \mathscr{N}$ such that $\xi(u_k,v_k) = 0$ and $\Phi(u_k,v_k) \to c$. By Ekeland’s variational principle (see [@MR0346619]), there exists another sequence $(\tilde{u}_k,\tilde{v}_k) \in \mathscr{N}$ such that
1. $\Phi(\tilde{u}_k,\tilde{v}_k) \to c$,
2. $\Phi|_{\mathscr{N}}'(\tilde{u}_k,\tilde{v}_k) \to 0$,
3. $\|(\tilde{u}_k,\tilde{v}_k) - (u_k,v_k)\| \to 0$.
As in the proof of Lemma \[lem-3.1\], $\Phi'(\tilde{u}_k,\tilde{v}_k) \to 0$. This together with the mean value theorem and (3) above imply that $$\label{7.3}
\Phi'(u_k,v_k) \to 0$$ since $\Phi''$ maps bounded sets onto bounded sets.
Since $\kappa_0 + \sup \kappa(x) < 1$, it follows from that $(u_k, v_k)$ is bounded. We pass to the renamed subsequence provided by Theorem \[profdec\], and note that $$\label{7.4}
(u_k,v_k) - \sum_n\, (U^{(n)}(\cdot - y_k^{(n)}),V^{(n)}(\cdot - y_k^{(n)})) \to 0 \text{ in } E$$ by , , and the continuity of the Sobolev imbedding. By Proposition \[prop:energy\], $$\Phi(U^{(1)},V^{(1)}) + \sum_{n \ge 2}\, \Phi_0(U^{(n)},V^{(n)}) = c,$$ $(U^{(1)},V^{(1)}) \in \mathscr{N} \cup \{(0,0)\}$, and $(U^{(n)},V^{(n)}) \in \mathscr{N}_0 \cup \{(0,0)\}$ for $n \ge 2$. In view of Lemma \[Lemma 7.1\], then $(U^{(1)},V^{(1)})$ trivial and $(U^{(n)},V^{(n)})$ is nontrivial for at most one $n \ge 2$. Since $\mathscr{N}$ is bounded away from the origin, then implies that $(U^{(n)},V^{(n)})$ is nontrivial for exactly one $n \ge 2$, say, for $n_0$, which then is a ground state solution of system . Then $(u_k,v_k) - (U^{(n_0)}(\cdot - y_k^{(n_0)}),V^{(n_0)}(\cdot - y_k^{(n_0)})) \to 0$ and hence $(u_k(\cdot + y_k^{(n_0)}),v_k(\cdot + y_k^{(n_0)})) \to (U^{(n_0)},V^{(n_0)})$ after a translation, so $$\label{7.5}
\xi(u_k(\cdot + y_k^{(n_0)}),v_k(\cdot + y_k^{(n_0)})) \to \xi(U^{(n_0)},V^{(n_0)})$$ by the continuity of the barycenter. However, $$\xi(u_k(\cdot + y_k^{(n_0)}),v_k(\cdot + y_k^{(n_0)})) = \xi(u_k,v_k) - y_k^{(n_0)} = - y_k^{(n_0)}$$ and $|y_k^{(n_0)}| \to \infty$, contradicting .
Let $(u,v)$ be a radially symmetric positive ground state solution of system and consider the continuous map $$\Gamma : {\mathbb{R}}^N \to \mathscr{N}, \quad \Gamma(y) = (t_y\, u(\cdot - y),t_y\, v(\cdot - y)),$$ where $t_y > 0$ is such that $\Gamma(y) \in \mathscr{N}$. We have $$\label{7.11}
\xi(\Gamma(y)) = \xi(u,v) + y = y$$ since $u$ and $v$ are radial functions. Moreover, as in the proof of Lemma \[Lemma 7.1\], $$\label{7.6}
t_y^2 = \frac{\|u\|^2 + \|v\|^2 - 2 {\displaystyle \int}_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x + y))\, uv\, dx}{{\displaystyle \int}_{{\mathbb{R}}^N} \left[(a_0 + a(x + y))\, u^4 + (b_0 + b(x + y))\, v^4 + 2\, (\beta_0 + \beta(x + y))\, u^2 v^2\right] dx}$$ and $$\label{7.8}
\Phi(\Gamma(y)) \to c_0 = c \quad \text{as } |y| \to \infty.$$
\[Lemma 7.3\] Assume that $0<\kappa_0<1$ and one of the following conditions holds:
- $\beta_0\geq3$;
- $1\leq\beta_0\leq3$ and $w(0)\leq
\sqrt{\frac{2\kappa_0(1+\beta_0)}{(3-\beta_0)(1-\kappa_0)}}$, where $w$ is the unique positive solution of the scalar equation ;
- $-1<\beta_0<1$, $w(0)\leq
\sqrt{\frac{2\kappa_0(1+\beta_0)}{(3-\beta_0)(1-\kappa_0)}}$, and
- if $N=1$, then $\kappa_0$ or $\beta_0-1$ is sufficiently small,
- if $N=2,3$, then $\beta_0, \kappa_0>0$ are sufficiently small, or $|\beta_0|$ is sufficiently small and $\kappa_0$ is close to 1.
Then $c_0$ is an isolated critical level of $\Phi_0$.
We use an indirect argument. Suppose that there exists a sequence $(u_n,v_n)\in
E$ such that ($i$) $\Phi'_0(u_n,v_n)=0$; $(ii)$ $\Phi_0(u_n,v_n)>c_0$; ($iii$) $\Phi_0(u_n,v_n)\rightarrow c_0$. As in Proposition \[prop:energy\], there exists $y_n\in\mathbb{R}^{N}$ such that $(\tilde{u}_n(x),\tilde{v}_n(x))=(u_n(x-y_n),v_n(x-y_n))$ converges to some $(u,v) \in E$ strongly. Moreover, $\Phi'_0(u,v)=0$ and $\Phi_0(u,v)=c_0$, i.e., $z=(u,v)$ is a ground state solution of . As in [@ACR-1 Lemma 3.5], then $u,v>0$ or $u,v<0$. By Lemma \[lem-3.2\], $z_1=(\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$ is the unique positive solution. From this and the structure of the system we infer that the solution of with both components negative is also unique. Hence $z_2=(-\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x),-\sqrt{\frac{1-\kappa_0}{1+\beta_0}}w(\sqrt{1-\kappa_0}x))$ is the unique negative solution of . In conclusion, we have $z=z_1$ or $z=z_2$. As in Lemma \[lem-3.5\] and Remark \[rem-3.7\], $z_1$ and $z_2$ are nondegenerate solutions of . Thus, $\Phi_0(u_n,v_n)=c_0$ by [@ACR-1 Lemma 3.7], contradicting $(ii)$.
Henceforth we assume the hypotheses of Lemma \[Lemma 7.3\], so that $c_0$ is an isolated critical level of $\Phi_0$. Let $d_0 = \inf\, \{d > c_0 : d \text{ is a critical level of } \Phi_0\}$ and set $\tilde{c}_0 = \min\, \{d_0, 2c_0\}$. Then $\tilde{c}_0 > c_0$ and we have the following lemma.
\[Lemma 7.4\] $\Phi$ satisfies the [*(PS)*]{}$_d$ condition for all $d \in (c_0,\tilde{c}_0)$.
Let $d \in (c_0,\tilde{c}_0)$ and let $(u_k,v_k) \in E$ be a (PS)$_d$ sequence for $\Phi$, i.e., $\Phi(u_k,v_k) \to d$ and $\Phi'(u_k,v_k) \to 0$. Since $$\Phi(u_k,v_k) - \frac{1}{4}\, \Phi'(u_k,v_k)\, (u_k,v_k) = \frac{1}{4} \left(\|u_k\|^2 + \|v_k\|^2 - 2 \int_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x))\, u_k v_k \right)$$ and $\kappa_0 + \sup \kappa(x) < 1$, it follows as in that $(u_k, v_k)$ is bounded. Passing to the renamed subsequence provided by Theorem \[profdec\] and utilizing Proposition \[prop:energy\], $$\Phi(U^{(1)},V^{(1)}) + \sum_{n \ge 2}\, \Phi_0(U^{(n)},V^{(n)}) = d,$$ $(U^{(1)},V^{(1)}) \in \mathscr{N} \cup \{(0,0)\}$, and $(U^{(n)},V^{(n)}) \in \mathscr{N}_0 \cup \{(0,0)\}$ for $n \ge 2$. Since $c = c_0$ and $d < 2c_0$, then $(U^{(n)},V^{(n)})$ is nontrivial for at most one $n \ge 1$. Since $d > 0$, then $(U^{(n)},V^{(n)})$ is nontrivial for exactly one $n$, say, for $n_0$. Since $c_0 < d < d_0$ and $\Phi_0$ has no critical levels in this interval, $n_0 = 1$. Then $(u_k,v_k) \to (U^{(1)},V^{(1)})$ as in the proof of Lemma \[Lemma 7.2\].
\[Lemma 7.5\] If $$\label{7.7}
\frac{\left(1 + \frac{|\kappa|_\infty}{1 - \kappa_0}\right)^2}{1 - \max \left\{\frac{|a|_\infty}{a_0},\frac{|b|_\infty}{b_0},\frac{|\beta|_\infty}{\beta_0}\right\}} < \frac{\tilde{c}_0}{c_0},$$ then $\Phi(\Gamma(y)) < \tilde{c}_0$ for all $y \in {\mathbb{R}}^N$.
Since $\Gamma(y) \in \mathscr{N}$, $$\Phi(\Gamma(y)) = \frac{t_y^2}{4} \left[\|u\|^2 + \|v\|^2 - 2 \int_{{\mathbb{R}}^N} (\kappa_0 + \kappa(x + y))\, uv\, dx\right]$$ by . Since the numerator in is less than or equal to $$\begin{gathered}
\|u\|^2 + \|v\|^2 - 2\, (\kappa_0 - |\kappa|_\infty) \int_{{\mathbb{R}}^N} uv\, dx = \left(1 + \frac{|\kappa|_\infty}{1 - \kappa_0}\right) \left[\|u\|^2 + \|v\|^2 - 2 \kappa_0 \int_{{\mathbb{R}}^N} uv\, dx\right]\\[10pt]
- \frac{|\kappa|_\infty}{1 - \kappa_0} \left[\|u\|^2 + \|v\|^2 - 2 \int_{{\mathbb{R}}^N} uv\, dx\right] \le 4c_0 \left(1 + \frac{|\kappa|_\infty}{1 - \kappa_0}\right)\end{gathered}$$ and the denominator is greater than or equal to $$\begin{gathered}
\int_{{\mathbb{R}}^N} \left[(a_0 - |a|_\infty)\, u^4 + (b_0 - |b|_\infty)\, v^4 + 2\, (\beta_0 - |\beta|_\infty)\, u^2 v^2\right] dx\\[10pt]
\ge 4c_0 \left(1 - \max \left\{\frac{|a|_\infty}{a_0},\frac{|b|_\infty}{b_0},\frac{|\beta|_\infty}{\beta_0}\right\}\right),\end{gathered}$$ the conclusion follows from .
The main result of this section is the following theorem.
\[zhang\] Let $a_0, b_0 > 0$, $0 < \kappa_0 < 1$, and assume that and the hypotheses of Lemma \[Lemma 7.3\] are satisfied. If $\kappa(x), a(x), b(x), \beta(x) \le 0$, with at least one of the inequalities strict on a set of positive measure, and holds, then the system has a nontrivial bound state solution.
By Lemmas \[Lemma 7.1\] and \[Lemma 7.2\], and , there exists a constant $R > 0$ such that $$\label{7.12}
c_0 < \max_{|y| = R}\, \Phi(\Gamma(y)) < \tilde{c}.$$ Let $$Q = \Gamma(\overline{B_R(0)}), \qquad S = \{(u,v) \in \mathscr{N} : \xi(u,v) = 0\},$$ and note that $\partial Q \cap S = \emptyset$ by . We claim that $\partial Q$ links $S$, i.e., $h(Q) \cap S \ne \emptyset$ for every map $h$ in the family $${\mathcal H} = \{h \in C(Q,\mathscr{N}) : h|_{\partial Q} = {id_{\, }}\}.$$ To see this, let $h \in \mathcal H$ and consider the continuous map $\varphi = \xi \circ h \circ \Gamma : \overline{B_R(0)} \to {\mathbb{R}}^N$. If $|y| = R$, then $\Gamma(y) \in \partial Q$ and hence $h(\Gamma(y)) = \Gamma(y)$, so $\varphi(y) = y$ by . By the Brouwer fixed point theorem, then there exists $y \in B_R(0)$ such that $\varphi(y) = 0$, i.e., $h(\Gamma(y)) \in S$.
We have $$\label{7.13}
\inf_S\, \Phi = \tilde{c} > \max_{\partial Q}\, \Phi > c_0$$ by . Set $$d = \inf_{h \in \mathcal H}\, \max_{(u,v) \in h(Q)}\, \Phi(u,v).$$ Since $\partial Q$ links $S$, $d \ge \tilde{c} > c_0$, and since ${id_{\, }}\in \mathcal H$, $d < \tilde{c}_0$ by Lemma \[Lemma 7.5\], so $\Phi$ satisfies the (PS)$_d$ condition by Lemma \[Lemma 7.4\]. So $d$ is a critical value of $\Phi$ by a standard argument.
From Theorem \[zhang\], we know that the results of Theorem \[th1.2\] hold.
**Acknowledgements:** This work was completed while K. Perera was visiting the Academy of Mathematics and Systems Science at the Chinese Academy of Sciences, and he is grateful for the kind hospitality of the host institution. J. Wang was supported by Natural Science Foundation of China (No: 11571140) and Natural Science Foundation of Jiangsu Province (Nos: BK2012282, BK20150478). Z.-T. Zhang was supported by National Natural Science Foundation of China (Nos: 11325107, 11271353, 11331010).
[1]{}
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[^1]: Corresponding author. E-mail addresses: kperera@fit.edu (K. Perera), tintarev@math.uu.se (C. Tintarev), wangmath2011@126.com (J. Wang), zzt@math.ac.cn (Z. Zhang)
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---
author:
- 'K. M. Punzi, P. Hily-Blant, J. H. Kastner, G.G. Sacco, T. Forveille'
title: 'An Unbiased 1.3 mm Emission Line Survey of the Protoplanetary Disk Orbiting LkCa 15'
---
|
---
abstract: 'We say that a graph $H$ is [*planar unavoidable*]{} if there is a planar graph $G$ such that any red/blue coloring of the edges of $G$ contains a monochromatic copy of $H$, otherwise we say that $H$ is [*planar avoidable*]{}. I.e., $H$ is planar unavoidable if there is a Ramsey graph for $H$ that is planar. It follows from the Four-Color Theorem and a result of Gonçalves that if a graph is planar unavoidable then it is bipartite and outerplanar. We prove that the cycle on $4$ vertices and any path are planar unavoidable. In addition, we prove that all trees of radius at most $2$ are planar unavoidable and there are trees of radius $3$ that are planar avoidable. We also address the planar unavoidable notion in more than two colors.'
author:
- 'M. Axenovich[^1], C. Thomassen[^2], U. Schade[^3], T. Ueckerdt[^4]'
title: Planar Ramsey graphs
---
Introduction
============
Ramsey’s theorem [@R] claims that any graph is Ramsey in the class of all complete graphs, i.e., for any graph $G$ and any number $k$ of colors there is a sufficiently large complete graph such that in any coloring of its edges in $k$ colors there is a monochromatic copy of $G$. In general for graphs $G$ and $H$, we write $G\rightarrow_k H$ and say that $G$ $k$-[*arrows*]{} $H$ if any coloring of the edges of $G$ in $k$ colors contains a monochromatic copy of $H$. We write $G\rightarrow H$ and say that $G$ [*arrows*]{} $H$ if $k=2$. There are classes of graphs that are Ramsey in their own class, meaning that for any graph $H$ in a class ${\mathcal{F}}$ there is a graph $G\in {\mathcal{F}}$ such that $G\rightarrow H$. Examples of such classes include bipartite graphs, graphs with a given clique number, and graphs of a given odd girth, see [@NR1; @NR2]. Here, we are concerned with Ramsey properties of the class of all planar graphs. We say that a planar graph $H$ is $k$-[*planar unavoidable*]{} if there is a planar graph $G$ such that $G\rightarrow_k H$, otherwise we call $H$ $k$-[*planar avoidable*]{}. Similarly, we define [*outerplanar unavoidable*]{} and [*outerplanar avoidable*]{} graphs. When $k=2$, we write [*planar unavoidable*]{} instead of $2$-[*planar unavoidable*]{}, or, if clear from context, simply [*unavoidable*]{}. The complexity of the problem to edge-color planar graphs with a given number of colors so that there is no monochromatic copy of a given graph was addressed by Broersma et al. [@BFKW]. A related problem of bounding local density of Ramsey graphs has been addressed for example in [@RR] and [@MP].\
A result of Gonçalves [@Go] states that any planar graph can be edge-colored in two colors so that each color class is an outerplanar graph. Thus any planar unavoidable graph is outerplanar. The Four Color Theorem [@AH] implies that any planar graph is a union of two bipartite graphs. In general any graph that is $2^k$-colorable for $k\in \mathbb{N}$ is a union of at most $k$ bipartite graphs.\
This shows that any planar unavoidable graph is bipartite and outerplanar and thus gives necessary conditions for planar unavoidability.\
Next we give several sufficient conditions. Here, a [*generalized broom*]{} is a union of a path and a star such that they share only the center of the star.
\[good2\] If $H$ be a path, a cycle on $4$ vertices, a tree of radius at most $2$, or a generalized broom, then $H$ is planar unavoidable. Moreover, if $H$ is a path, then it is outerplanar unavoidable.
The next result shows that not only odd cycles and non-outerplanar graphs are planar avoidable, but also some trees.
\[not-good\] There is a planar avoidable tree of radius $3$ and an outerplanar avoidable tree of radius $2$.
Moreover, any planar avoidable tree has at least $8$ vertices and there is a planar avoidable tree on $106$ vertices.
A result of Hakimi et al. [@Ha], see also [@AA], states that any planar graph can be edge-decomposed into at most five star forests. Thus the $k$-planar unavoidable graphs for $k\geq 5$ are precisely the star forests. Next we summarise our results for $k$-planar unavoidable graphs, for $k=3$ and $4$.
\[more-colors\] If $H$ is $k$-planar unavoidable for $k\geq 3$, then $H$ is a forest. If $H$ is $4$-planar unavoidable, then $H$ is a caterpillar forest. There are $3$- and $4$-planar avoidable trees of radius $2$.
Moreover, there are $3$- and $4$-planar avoidable trees on $10$ and $6$ vertices, respectively.
We provide some definitions in Section \[definitions\]. Sections \[sec:good-2\], \[sec:not-good-1\], and \[sec:more-colors\] contain the proofs of Theorems \[good2\], \[not-good\], and \[more-colors\] respectively. Finally Section \[conclusions\] states some concluding remarks and open questions.
Definitions
===========
We denote a complete graph, a path, and a cycle on $n$ vertices by $K_n, P_n,$ and $ C_n$, respectively. A complete bipartite graph with parts of sizes $m$ and $n$ is denoted by $K_{m,n}$. For an integer $k$, $k\geq 2$, a [*$k$-ary tree*]{} is a rooted tree in which each vertex has at most $k$ children. A [*perfect*]{} $k$-ary tree is a $k$-ary tree in which every non-leaf vertex has $k$ children and all leaf vertices have the same distance from the root. For all other standard graph theoretic definitions, we refer the reader to the book of West [@W].\
[**Iterated Triangulation ${{\rm Tr}}(n)$**]{}:\
An [*iterated triangulation*]{} is a plane graph ${{\rm Tr}}(n)$ defined as follows: ${{\rm Tr}}(0)=K_3$ is a triangle, ${{\rm Tr}}(i) \subseteq {{\rm Tr}}(i+1)$, ${{\rm Tr}}(i+1)$ is obtained from ${{\rm Tr}}(i)$ by inserting a vertex in each of the inner faces of ${{\rm Tr}}(i)$ and connecting this vertex with edges to all the vertices on the boundary of the respective face, see Figure \[Def Tr(n)\]. We see that ${{\rm Tr}}(i)$ is a triangulation and each triangle of ${{\rm Tr}}(i)$ bounds a face of ${{\rm Tr}}(j)$ for some $j \leq i$.\
![The iterative construction of ${{\rm Tr}}(2)$.[]{data-label="Def Tr(n)"}](DefTr-n)
[**Universal outerplanar graph ${{\rm UOP}}(n)$:**]{}\
A [*universal outerplanar graph*]{} ${{\rm UOP}}(n)$ is defined as follows: ${{\rm UOP}}(1)$ is a triangle. An edge on the outer face is called an [*outer*]{} edge. For $k>1$, ${{\rm UOP}}(k)$ is an outerplanar graph that is a supergraph of ${{\rm UOP}}(k-1)$ obtained by introducing, for each outer edge $e=xy$, a new vertex $v_e$ and new edges: $v_e x$ and $v_ey$. Then the set of outeredges of ${{\rm UOP}}(k)$ is $\{v_ex, v_ey: ~ e=xy \mbox{ is an outeredge of } {{\rm UOP}}(k-1)\}.$\
[**Triangulated Grid ${{\rm Gr}}(n)$:**]{}\
Let a [*triangulated grid*]{} be a graph ${{\rm Gr}}(n)=(V,E)$ with $V=[n]\times[n]$ and $(k,j)(k',j')\in E$ if and only if either ($k=k'$ and $|j-j'|=1$) or ($|k-k'|=1$ and $j=j'$) or ($k=k'-1$ and $j=j'-1$) or ($k=k'+1$ and $j=j'+1$). We define left, right, top, and bottom sides of the grid as subsets of vertices $[n]\times \{1\}$, $[n]\times \{n\}$, $\{1\}\times [n]$, and $\{n\}\times [n]$ respectively.\
[**Fish:**]{}\
A graph $G$ is called a [*fish*]{} and denoted $F_{x,y}$ if $V(G)=\{x, y\}\cup S$, where $S\cap \{x,y\} = \emptyset$, $x$ and $y$ are each adjacent to each vertex in $S$, $S$ induces a path in $G$, and $xy$ is an edge. We call $S$ the set of [*spine vertices*]{}, $G[S]$ is called the [*spine*]{}, $xs, ys$ are called [*ribs*]{}, $s\in S$, and the paths $x,s,y$ of length $2$ are called [*double ribs*]{}. Sometimes we say that a fish $F_{x,y}$ [*hangs*]{} on an edge $xy$. In an edge-colored fish, a double rib is called [*bicolored*]{} if there are different colors used on two edges of this double rib. We will call two double ribs $x,s,y$ and $x,s',y$, with $s\neq s'$, $s,s'\in S$, [*identically bicolored*]{}, if the same color is used on both of the edges $xs$ and $xs'$, and a different color is used on both of the edges $sy$ and $s'y$. Note that for any positive integers $m$ and $k$ and for any edge $xy \in E({{\rm Tr}}(m))$, there is a fish on $xy$ in ${{\rm Tr}}(m+k)$ with $k$ spine vertices. Indeed, consider an inner face $xyz$ in ${{\rm Tr}}(m)$. We can pick spine vertices $s_1,\ldots,s_k$ such that $s_i\in V({{\rm Tr}}(m+i)-{{\rm Tr}}(m+i-1))$, $i\in\{1,\dots,k\}$ and such that $s_i$ is inserted in the face $xys_{i-1}$ of ${{\rm Tr}}(m+i-1)$, $i=1,\ldots,k$, $s_0=z$.\
![A fish $F_{x,y}$ with $k$ spine vertices.[]{data-label="DefFish"}](DefFish)
Proof of Theorem \[good2\] {#sec:good-2}
==========================
Theorem \[good2\] follows immediately from the following lemmas.
The following proof closely resembles the Hex-lemma [@Ga].
\[path\] Let $G$ be a near-triangulation with outer cycle $C$, that is, $G$ is a planar graph with outer face boundary $C$ and each other face is bounded by a triangle. Let $a,b,c,d$ be vertices on $C$ in clockwise order dividing the edges of $C$ in four paths $C(a,b),C(b,c),C(c,d),C(d,a)$, respectively. If the edges of $G$ are colored red and blue, then either there is a blue path from $C(a,b)$ to $C(c,d)$ or a red path from $C(b,c)$ to $C(d,a)$ (or both).
Suppose there is no blue path from $C(a,b)$ to $C(c,d)$. Then the red graph contains a minimal edge-cut separating $C(a,b)$ and $C(c,d)$. A minimal edge-cut in $G$ is a cycle $C'$ in the dual graph $G^*$. This cycle $C'$ must contain the vertex $v^*$ corresponding to the outer face of $G$. Since $G$ is a near-triangulation, it follows that any two consecutive edges of $C'$ (except the two edges incident with $v^*$) correspond to two edges in $G$ that are incident with the same vertex. Thus the edges in $G$ corresponding to the edges in $C'$ contain a red path path from $C(b,c)$ to $C(d,a)$.
Any path is planar unavoidable, even in a class of planar graphs of bounded degrees (in fact of maximum degree at most $6$).
If the edges of the triangulated grid ${{\rm Gr}}(k)$ are colored red or blue, then there is a monochromatic $P_k$ by Lemma \[path\], where the paths $C(a,b),C(b,c),C(c,d),C(d,a)$ correspond to the top, right, bottom, and the left sides of the grid.
The above gives planar graphs of bounded maximum degree that arrow arbitrarily long paths, which however have large tree-width. Complementary, we can find planar graphs of tree-width $2$ that also arrow arbitrarily long paths, where however the maximum degree is large.
\[lem:path\] Any path is outerplanar unavoidable. In particular, for any positive integer $n$, ${{\rm UOP}}(n^2) \rightarrow P_n$.
We shall show that ${{\rm UOP}}(n^2)\rightarrow P_n$. Let $G = G_{n^2} = {{\rm UOP}}(n^2)$ and let it be edge-colored red and blue. We see that each edge of $G$ is on the outer face of $G_i={{\rm UOP}}(i)$ for some $i \leq n^2$, where $G_1\subseteq G_2\subseteq \cdots \subseteq G_{n^2}$ as in the definition of the universal outer planar graph. Consider the unique outerplanar embedding of $G$ and for each edge $e$, consider $G_i$ such that $e$ is on the outer face of $G_i$. For a vertex let its rank be the least $i \in \{1,\ldots,n^2\}$ for which it is in the vertex set of $G_i$. For each edge $e$, we define graphs $G({\rm out},e)$ and $G({\rm in}, e)$ such that $G = G({\rm out}, e) \cup G({\rm in}, e)$, where $G({\rm out}, e)$ and $G({\rm in}, e)$ share only the edge $e$ and no vertices except for the endvertices of $e$. We require in addition that $G({\rm in}, e)$ contains $G_1$ as a subgraph, see Figure \[fig:outerplanar-definitions\]. Observe that among vertices of rank $i$ in $G$ there are two at distance $i$ in $G$, $i=1,\ldots,n^2$.
![The universal outerplanar graph ${{\rm UOP}}(5)$. Vertices with the same rank lie on concentric circles. For the thick edge $e$, the vertices in $G({\rm in},e)$ are shown in black. The two vertices $x,y$ in ${{\rm UOP}}(5)$ have distance $5$.[]{data-label="fig:outerplanar-definitions"}](outerplanar-definitions)
For an edge $e$ in $G_i$ with endvertex $v$ of rank $i$, we define the following. Let $R(e)$ be a longest red path in $G_i$ with last edge $e$ and last vertex $v$. Let $B(e)$ be a longest blue path in $G({\rm in},e)$ with last vertex $v$. We shall write $e>e'$ for two edges of $G$ if $|R(e)| > |R(e')|$, or $|R(e)| = |R(e')|$ and $|B(e)| > |B(e')|$.\
Consider the edges in $G_n$. Assume that the endvertices of each such edge belong to the same blue component. Then for any two vertices of rank $n$, there is a blue path joining them. Since there are two such vertices at distance at least $n$ in $G$, we see that there is a blue path on $n$ edges, and we are done. So, assume that $e_n$ is an outer edge of $G_n$ such that its endvertices belong to different blue components of $G$. Assume we constructed a sequence of edges $e_n < e_{n+1}< \cdots < e_{n+i}$ of outer edges in $G_n,\ldots,G_{n+i}$ respectively such that the endvertices of each of these edges belong to different blue components of $G$. Consider $e=e_{n+i}$, we shall construct $e_{n+i+1}$. Let $e=uv$ with $v$ of rank $n+i$ and let $e', e''$ be two adjacent outer edges of $G_{n+i+1}$ that are incident to $u$ and $v$, respectively. Let $e'=uw$ and $e''=vw$ where $w$ has rank $n+i+1$. Note that either ($u$ and $w$) or ($v$ and $w$) are in different blue components in $G$, otherwise $u$ and $v$ would have been in the same blue component. See Figure \[fig:outerplanar-cases\] for illustrations.\
![Illustrations of Case 1 (left) and Case 2 (right) in the proof of Lemma \[lem:path\].[]{data-label="fig:outerplanar-cases"}](outerplanar-cases)
Case 1. $v$ and $w$ are in different blue components. Then $e'' = vw$ is red and the path $R(e) \cup vw$ is a red path in $G_{n+i+1}$ of length $|R(e)|+1$ ending in $e''$ at vertex $w$. Then let $e_{n+i+1} = e''$. We see that $e'' > e$.\
Case 2. $v$ and $w$ are in the same blue component and $u$ and $w$ are in different blue components. Then $e' = uw$ is red and the path $(R(e) - uv) \cup uw$ is a red path of length $|R(e)|$ in $G_{n+i+1}$ ending with $e'$ at vertex $w$. Since $v$ and $w$ are in the same blue component, there is a blue path $P$ of length $q$, $q\geq 1$, between them in $G({\rm out}, e'')$. The union of $P$ and the blue path $B(e)$ ending at $v$ in $G({\rm in}, e)$ forms a blue path ending at $w$ in $G({\rm in}, e')$. Let $e_{n+i+1}= e'$. We see that $e' > e$.\
We can continue in this manner until rank $n^2$, i.e., we create a desired sequence $e_n < \cdots < e_{n^2}$. Note that $|R(e_i)| \geq 1$ and $|B(e_i)| \geq 0$ for $i = n,\ldots,n^2$, and $(|R(e_i)|,|B(e_i)|) \neq (|R(e_j)|,|B(e_j)|)$ whenever $i \neq j$. As there are exactly $n^2-n+1 = (n-1)n + 1$ edges in this sequence, there exists some $i \in \{n,\ldots,n^2\}$ with $|R(e_i)| \geq n$ or $|B(e_i)| \geq n$, proving that there is a red or a blue path of length at least $n$.
\[Broomhd\] Any generalized broom is planar unavoidable.
Let $H$ be a union of $P_{2k+1}$ and $K_{1,k}$ that share only their center vertices. Note that any generalized broom on at most $k$ vertices is a subgraph of $H$. Let $n$ be sufficiently large, say $n\geq 10k^2$. Consider $G={{\rm Tr}}(10n)$ colored red and blue. Since ${{\rm UOP}}(8n)\subseteq {{\rm Tr}}(8n)$, we see that there is monochromatic path $P$ on edges $e_1, \ldots, e_n$ in order in a two-edge colored ${{\rm Tr}}(8n)$, say $P$ is red. Consider a set ${\mathcal{F}}$ of $n-2k$ fishes hanging on $e_{k+1}, e_{k+2}, \ldots, e_{n-k}$ respectively such that the spines of fishes from ${\mathcal{F}}$ are pairwise disjoint and each fish has at least $4k$ spine vertices. If at least one of these fishes contains a red star of size $k$ centered at a vertex of $P$, we have a red $H$. Otherwise, each fish in ${\mathcal{F}}$ contains at least $2k$ blue double ribs. The union of blue subgraphs of fishes from ${\mathcal{F}}$ clearly contains a blue copy of $H$.
\[lem:height-2\] Any tree of radius $2$ is planar unavoidable.
Let $H$ be a perfect $k$-ary tree of radius $2$. Consider ${{\rm Tr}}(19k)$ together with a fixed edge coloring in red and blue.
![Two monochromatic stars of different colors with the same leaf-set.[]{data-label="TCase1"}](TCase1)
If there is a red star $S_r$ and a blue star $S_b$ on $2k$ edges in ${{\rm Tr}}(n)$, $n \leq 18k$, such that the stars have the same leaf-set $L$, then there is a monochromatic copy of $H$ in ${{\rm Tr}}(n+k)$.
Let $x,y$ denote the centers of $S_r$ and $S_b$, respectively. Each vertex $z \in L$ has at least $2k$ neighbors in ${{\rm Tr}}(n+k)$ that are not neighbors of any vertex in $L - z$. Hence, by pigeonhole principle $z$ is the center of a monochromatic star on $k$ edges, whose leaves have distance at least two to $L - z$, see Figure \[TCase1\]. At least $k$ of these monochromatic stars are of the same color that together with either $S_r$ or $S_b$ form a monochromatic copy of $H$. This proves the Claim.
![Part of a fish $F$ with $k$ blue double ribs, part of a fish $F_{x,s}$ with red double ribs, and part of a fish $F_{s,s'}$ between $s$ and $s'$ with monochromatic red double ribs.[]{data-label="TCase2"}](TCase2-new)
Now consider any two adjacent vertices $x,y$ in ${{\rm Tr}}(n)$, $n \leq 12k$, and the set $L$ of their at least $6k$ common neighbors in ${{\rm Tr}}(n+6k)$. By the Claim, we may assume that fewer than $2k$ vertices of $L$ have a red edge to $x$ and a blue edge to $y$, and fewer than $2k$ vertices of $L$ have a blue edge to $x$ and a red edge to $y$. Each of the remaining at least $2k$ vertices in $L$ has its edges to $x$ and $y$ in the same color, and by pigeonhole principle we may assume that for at least $k$ of these vertices this the same color. We let $K(x,y)$ denote this monochromatic copy of $K_{2,k}$ in ${{\rm Tr}}(n+6k)$.
Finally, consider two adjacent vertices $x,y$ in ${{\rm Tr}}(0)$. Say that $K(x,y) \subset {{\rm Tr}}(6k)$ is blue. If for every vertex $z$ in $K(x,y) - \{x,y\}$ we find a monochromatic $K(z,a) \subset {{\rm Tr}}(18k)$ in blue for some $a$, then there is a blue copy of $H$, as desired. So assume that for at least one vertex $z$ in $K(x,y)- \{x,y\}$ all monochromatic $K(z,a)$ for some $a$ are red; see Figure \[TCase2\]. Then in particular $K(z,y) \subseteq {{\rm Tr}}(12k)$ is red with vertices $z,y$ and $w_1,\ldots,w_k$. Moreover, for each $i=1,\ldots,k$ the monochromatic $K(z,w_i) \subset {{\rm Tr}}(18k)$ is red. However, this gives a red copy of $H$ rooted at $y$; see Figure \[TCase2\].
\[C4\] A cycle $C_4$ is planar unavoidable. For $n \geq 16$, ${{\rm Tr}}(n) \rightarrow C_4$.
Consider the graph $G$ consisting of a fish $F_{x,y}$ hanging on edge $xy$ with $15$ spine vertices $s_1,\ldots,s_{15}$, and a vertex of degree three in each face of $F_{x,y}$ bounded by two spine vertices; see the left part of Figure \[fig:C4-full-graph\]. Note that $G \subset {{\rm Tr}}(15)$. Consider any fixed edge-coloring of $G$ in red and blue.
![Left: A planar graph $G$ with $G \rightarrow C_4$. Right: Illustrations for the two cases in the proof of Lemma \[C4\].[]{data-label="fig:C4-full-graph"}](C4-full-graph)
First we claim that $F_{x,y}$ contains a monochromatic $C_4$ or a monochromatic inner face $f$ such that any two vertices $u,v$ of $f$ have a common neighbor $w$ in $F_{x,y}$, not in $f$, such that edges $uw$ and $vw$ have the same color. To this end, consider the spine vertices $s_1,\ldots,s_{15}$ and the corresponding double ribs $x,s_i,y$, $i = 1,\ldots,15$. If $F_{x,y}$ contains no monochromatic $C_4$, at most two double ribs are monochromatic – one red and one blue. Hence there are five consecutive spine vertices $s_i,\ldots,s_{i+4}$ whose double ribs are bicolored. Assume, without loss of generality, that $i=1$. Further assume that the edges $xy$ and $xs_{3}$ are red, so the edge $s_{3}y$ is blue.
Case 1: $xs_{2}$ is red.
: Then $s_{2}y$ is blue. If the spine edge $s_{2}s_{3}$ is blue, then $s_{2},s_{3},y$ bound an inner face $f$ with the desired properties ensured by the vertex $x$ that sends red edges to $f$. So we may assume that $s_{2}s_{3}$ is red. For the same reason, if $xs_1$ is also red, then also $s_1s_{2}$ is red, giving a red $C_4$ with vertices $x,s_1,s_{2},s_{3}$. So we may assume that $xs_1$ is blue and hence $s_1y$ is red. Now if $s_1s_{2}$ is red, there is a red $C_4$ with vertices $x,y,s_1,s_{2}$. So we may assume that $s_1s_{2}$ is blue.
Symmetrically, we may assume that $xs_{4}$ is blue, hence $s_{4}y$ is red, and $s_{3}s_{4}$ is blue. But now $s_{2},s_{3},x$ bound an inner face with the desired properties; see the right part of Figure \[fig:C4-full-graph\].
Case 2: $xs_{2}$ is blue.
: Then $s_{2}y$ is red. Now if $s_{2}s_{3}$ is red, we have a red $C_4$ with vertices $x,y,s_{2},s_{3}$. So we may assume that $s_{2}s_{3}$ is blue. By symmetry we may also assume that $xs_{4}$ is blue, hence $s_{4}y$ is red, and $s_{3}s_{4}$ is blue. But then we have a blue $C_4$ with vertices $x,s_{2},s_{3},s_{4}$; see the right part of Figure \[fig:C4-full-graph\].
This proves the claim that $F_{x,y}$ contains a monochromatic $C_4$ or a monochromatic inner face $f$, say in red, such that any two vertices of $f$ are joined by a blue $P_3$ in $F_{x,y}$. In the former case we are done. In the latter case note that as $f$ is all red, any two vertices of $f$ are also joined by a red $P_3$ in $F_{x,y}$. Now consider the vertex $z$ in $G - F_{x,y}$ whose three neighbors are the vertices of $f$. As two of the three edges incident to $z$ have the same color, there are two vertices in $f$ that are joined by two distinct but identically colored $P_3$’s in $G$. That is, there is a monochromatic copy of $C_4$ in $G$.
Proof of Theorem \[not-good\] {#sec:not-good-1}
=============================
Let $T_1$ be a tree of radius $3$ with root $r$ and all vertices of distance $0,1,2$ to $r$ having degree $5$. See Figure \[fig:trees-and-colorings\]. Let $G$ be a planar graph. Let $V_1, V_2, V_3$ be a partition of $V(G)$ such that each $V_i$ induces a linear forest in $G$, $i=1,2,3$, such a partition exists by a result of Poh [@Poh]. Further, consider an orientation of $G$ with out-degree at most $3$ at each vertex, see [@H]. (This orientation result also follows from [@NW].) For $i = 1,2,3$ color the edges in $G[V_i]$ alternately red and blue along the paths in $G[V_i]$. For each remaining directed edge $uv$ of $G$ we have $u \in V_i$ and $v \in V_j$ for some $i \neq j$. Color $uv$ red if $i < j$ and blue if $i > j$. See Figure \[fig:trees-and-colorings\]. Assume that there is a monochromatic copy of $T_1$, say red. Since the out-degree of each vertex in $G$ is at most $3$, we see that each non-leaf vertex of $T_1$ has at least two incoming edges. Due to the color alternation in each $G[V_i]$, at least one of the two incoming edges has its two endvertices in distinct parts. In particular, the root $r$ is in $V_2$ or in $V_3$. Then at least one vertex at distance $1$ or $2$ from $r$ is in $V_1$. However, the vertices of $V_1$ have in-degree at most $1$ in the red graph, a contradiction.\
Let $T_2$ be a tree of radius $2$ with root $r$ and all vertices of distance $0,1$ to $r$ having degree $4$. See Figure \[fig:trees-and-colorings\]. Similarly, let $G$ be an outerplanar graph. Let $V_1, V_2$ be a partition of $V(G)$ such that each $V_i$ induces a linear forest in $G$, $i=1,2$, such a partition exists by a result of Cowen et al. [@CCW]. Further, consider an orientation of $G$ with out-degree at most $2$ at each vertex, see [@H; @NW]. For $i = 1,2$ color the edges in $G[V_i]$ alternately red and blue along the paths in $G[V_i]$. For each remaining directed edge $uv$ of $G$ we have $u \in V_i$ and $v \in V_j$ for some $i \neq j$. Color $uv$ red if $i < j$ and blue if $i > j$. See Figure \[fig:trees-and-colorings\]. Assume that there is a red copy of $T_2$. Since the out-degree of each vertex in $G$ is at most $2$, each non-leaf vertex of $T_2$ has two incoming edges. Thus the root $r$ is in $V_2$ and at least one of its neighbors is in $V_1$, a contradiction.\
The trees $T_1$ and $T_2$ have $106$ and $21$ vertices, respectively, and are illustrated in Figure \[fig:trees-and-colorings\]. We know that every planar avoidable tree has at least $8$ vertices since it has radius at least three and it is not a generalized broom.
![Illustrations of trees $T_1,\ldots,T_4$ defined in the proofs of Theorem \[not-good\] and \[more-colors\]: $T_1$ is planar avoidable with $2$ colors. $T_2$ is avoidable with $2$ colors in the class of outerplanar graphs. $T_3$ is planar avoidable with $3$ colors. $T_4$ is planar avoidable with $4$ colors. The colorings below illustrate patterns of how to color any planar (outerplanar) graph on basis of a partition $V_1,V_2,V_3$ ($V_1,V_2$) of the vertices, and an orientation of the edges between the parts. []{data-label="fig:trees-and-colorings"}](trees-and-colorings-new)
Proof of Theorem \[more-colors\] {#sec:more-colors}
================================
A result of Nash-Williams [@NW] implies that any planar graph can be edge-decomposed into at most three forests. Thus any graph $H$ that is not a forest is $3$-planar avoidable. Another result of Gonçalves [@Go1] states that any planar graph can be edge-colored in four colors so that each color class is a forest of caterpillars. Thus any graph $H$ that is not a caterpillar forest is $4$-planar avoidable.\
For the remainder of the proof let $G$ be any planar graph. Let $V_1,V_2,V_3$ be a partition of the vertex set $V(G)$ so that $G[V_i]$ is a linear forest [@Poh]. We shall define two colorings $c_3$ and $c_4$ of the edges of $G$ with three and four colors, respectively. To this end, consider the bipartite subgraphs $B_1,B_2,B_3$ of $G$ with partitions $(V_2,V_3)$, $(V_1,V_3)$, $(V_1,V_2)$, and containing all edges of $G$ between respective parts. For each $i=1,2,3$ orient the edges of $B_i$ so that the out-degree at each vertex is no more than $2$. (Such an orientation exists by [@H; @NW] as bipartite $n$-vertex planar graphs have no more than $2n-3$ edges, by Euler’s formula.)
Coloring $c_3$:
: For $i=1,2,3$, color all edges in $G[V_i]$ and all edges of $G$ that are oriented incoming at a vertex of $V_i$ in color $i$.
Coloring $c_4$:
: For $i=1,2,3$, color all edges of $G$ that are oriented incoming at a vertex of $V_i$ in color $i$. Further, color all edges in $G[V_1]$, $G[V_2]$, $G[V_3]$ in color $4$.
Next we show that a tree $T_3$ of radius $2$ with root $r$ and all vertices of distance $0,1$ to $r$ of degree at least $3$ (see Figure \[fig:trees-and-colorings\]) is $3$-planar avoidable. We claim that $c_3$ does not contain a monochromatic copy of $T_3$. In fact, if $v$ is any vertex with at least three incident edges of the same color $i$, then $v$ must be a vertex in $V_i$. However, $G[V_i]$ has maximum degree at most $2$, while the vertices of degree at least $3$ in $T_3$ induce a subgraph of maximum degree at least $3$. Hence there is no monochromatic copy of $T_3$ in $G$ under coloring $c_3$.\
Finally, we show that a symmetric double star $T_4$ on $6$ vertices, i.e, a tree with two adjacent vertices of degree $3$ and four leaves (see Figure \[fig:trees-and-colorings\]) is $4$-planar avoidable. We claim that $c_4$ does not contain a monochromatic copy of $T_4$. First, color $4$ is a disjoint union of paths, and thus there is no copy of $T_4$ in color $4$. For color $i \in \{1,2,3\}$ we see that, as before, only vertices in $V_i$ may have three incident edges of color $i$. However, as $V_i$ is an independent set in the subgraph of color $i$, there is no copy of $T_4$ in that subgraph. Hence there is no monochromatic copy of $T_4$ in $G$ under coloring $c_4$, as desired.\
Let us remark that coloring $c_3$ shows that every graph $H$ in which the vertices of degree at least $4$ induce a subgraph of maximum degree at least $3$ is $3$-planar avoidable. Similarly, coloring $c_4$ shows that every graph $H$ with an odd-length path whose two endvertices have degree at least three each, is $4$-planar avoidable.
Conclusions
===========
In this paper we initiated the study of Ramsey properties of planar graphs. When two colors are considered, only some outerplanar bipartite graphs are unavoidable and even some trees are avoidable. We showed that $C_4$ is unavoidable. The following questions remain open:\
[**1.**]{} Are other even cycles unavoidable?\
[**2.**]{} What is the smallest number of vertices in an avoidable tree?\
All of our positive results, showing that some graphs are unavoidable, use the fact that the iterated triangulation ${{\rm Tr}}(n)$ arrows these graphs.\
[**3.**]{} Is is true that for each planar unavoidable graph $H$ there is $n=n(H)$ such that ${{\rm Tr}}(n) \rightarrow G$?
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Merker, M., Postle, L., [*Bounded diameter arboricity,*]{} Journal of Graph Theory, (2018), 1–13.
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[^1]: Department of Mathematics, Karlsruhe Institute of Technology
[^2]: Department of Applied Mathematics and Computer Science, Technical University of Denmark
[^3]: Department of Mathematics, Karlsruhe Institute of Technology
[^4]: Computer Science Department, Karlsruhe Institute of Technology
|
[ **[ Isomonodromic Deformations with\
an Irregular Singularity and Hyperelliptic Curves ]{}** ]{}
Department of Engineering Science, Niihama National College of Technology,\
7-1 Yakumo-chou, Niihama, Ehime, 792-8580.
By Kazuhide Matsuda
In this paper, we extend the result of Kitaev and Korotkin [@KK] to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $\tau$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $\tau$-function is expressed by the hyperelliptic $\Theta$ function moving the argument ${\mathbf{z}}$ and the period ${\mathbf{B}},$ where $t$ and the positions of regular singularities move $z$ and ${\mathbf{B}},$ respectively.
the $\tau$-function; the $\Theta$ function; monodromy-preserving deformation; irregular singular point; hyperelliptic curves.
Introduction {#introduction .unnumbered}
============
In this paper, we extend the result of Kitaev and Korotkin [@KK] to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $\tau$-function represented by the hyperelliptic $\Theta$ function moving both the argument ${\mathbf{z}}$ and the period ${\mathbf{B}}$. In [@Matsuda], we constructed the $\tau$-function by the elliptic $\Theta$ function moving the argument $z$ and the period $\Omega$.
Miwa, Jimbo and Ueno [@MM1] extended the work of Schlesinger [@Sch] and established a general theory of monodromy-preserving deformation for a first order matrix system of ordinary linear differential equations, $$\frac{dY}{dx}
=
A(x)Y,
\quad
A(x)
=
\sum_{\nu=1}^{n}
\sum_{k=0}^{r_{\nu}}
\frac{A_{\nu,-k}}{(x-a_{\nu})^{k+1}}
-
\sum_{k=1}^{r_{\infty}}
A_{\infty, -k} x^{k-1},$$ having regular or irregular singularities of arbitrary rank.
The monodromy data to be preserved are (i) Stokes multipliers $S_j^{(\nu)} \,\,
(j=1, \ldots , 2r_{\nu}),$ (ii) connection matrices $C^{(\nu)},$ (iii) “exponents of formal monodromy" $T_0^{(\nu)}.$ Miwa, Jimbo and Ueno found a deformation equation as a necessary and sufficient condition for the monodromy data to be independent of deformation parameters and defined the $\tau$-function for the deformation equation.
Let us explain the relationship between the $\tau$-function and the $\Theta$ function. Miwa and Jimbo [@MM2] constructed a monodromy-preserving deformation with irregular singularities and expressed the $\tau$-function with the $\Theta$ function by moving its argument ${\mathbf{z}},$ which $t,$ the parameters of the irregular singularities, move.
Kitaev and Korotkin [@KK] constructed a monodromy-preserving deformation of $2\times2$ Fuchsian systems, whose deformation parameters are the positions of $2g+2$ regular singularities. Its $\tau$-function is expressed by the hyperelliptic $\Theta$ function moving the period ${\mathbf{B}},$ which the positions of regular singularities move. When $g=1,$ the $\tau$-function is equivalent to Picard’s solution of the sixth Painlevé equation by the Bäcklund transformation. The aim of this paper is to unify Miwa and Jimbo’s result and Kitaev and Korotkin’s result in the hyperelliptic case. We note that Deift, Its, Kapaev and Zhou [@DIKZ] also constructed the isomonodromic deformations of $2\times 2$ Fuchsian systems in terms of the hyperelliptic $\Theta$ functions.
This paper is organized as follows. In Section 1, we explain the basic properties of the $\Theta$ function, the prime-form and the canonical bi-meromorphic differential.
In Section 2, we study an ordinary differential equation which is given by $$\label{equ:diff1}
\frac{d\Psi}{d\lambda}
=
\left(
\sum_{j=1}^{2g+2}
\frac{A_{j}}{\lambda-\lambda_{j}}
-B_{-1}
\right)
\Psi(\lambda),$$ whose deformation parameters are $\lambda_1,\lambda_2, \ldots \lambda_{2g+2}$ and diagonal elements of $B_{-1}$. Then, following Miwa, Jimbo and Ueno [@MM1], we introduce the $\tau$-function.
In Section 3, we solve a class of Riemann-Hilbert (inverse monodromy) problem for special parameters. Furthermore, we prove that the solution $\Psi(\lambda)$ has the following special monodromy data: (i) Stokes multipliers around $\lambda=\infty$ $S^{\infty}_1 = S^{\infty}_2 =1,$ (ii) connection matrices around $\lambda=\lambda_{j}$ $C_{j} \,(1 \leq j \leq 2g+2),$ (iii) the exponents of formal monodromy $
T_{\infty,0}=0, T_{j,0}={\qopname\relax o{diag}}(-\frac14, \frac14)
\, (1 \leq j \leq 2g+2).
$
In Section 4, we calculate the monodromy-preserving deformation which the solution $\Psi(\lambda)$ satisfies and compute the coefficients, $B_{-1}, A_{j} \,\,(1 \leq j \leq 2g+2).$
In Section 5, we find the $\tau$-function. Section 5 consists of three subsections. Subsection 5.1 is devoted to $H_t$, the Hamiltonian at an irregular singular point $\lambda=\infty$. Subsection 5.2 is devoted to Fay’s identities and Rauch’s variational formulas. Subsection 5.3 is devoted to $H_{j} \,(1 \leq j \leq 2g+2)$, the Hamiltonians on the deformation parameters $\lambda_{j} \,\,(1 \leq j \leq 2g+2).$ From $H_t$ and $H_j\,\,(1\leq j \leq 2g+2),$ we compute the $\tau$-function and prove Theorem 0.1.
In order to state our main theorem, we define hyperelliptic curves and the $\Theta$ function and introduce the prime-form and the canonical bi-meromorphic differential, following Fay [@Fay]. The hyperelliptic curves $\mathcal{L}$ are $$\omega^2 = \prod_{j=1}^{2g+2} (\lambda-\lambda_j),$$ whose branch points $\lambda_i \in {\mathbb{C}}\,\,(1 \leq i \leq 2g+2)$ are distinct and canonical homological basis $\{a_j,b_j\}_{1 \leq j \leq g}$ are chosen according to Figure \[rs\].
(400,90) (40,50) (40,30)[$\lambda_1$ ]{} (100,50) (100,30)[$\lambda_2$]{} (40,50)[(1,0)[60]{}]{}
(80,50)(130,70)(160,50) (120,57)[$>$]{} (120,65)[$b_1$]{} (80,50)(130,30)(160,50)
(50,50)(190,120)(330,50) (190,83)[$>$]{}
(200,89)[$b_g$]{}
(50,50)(190,0)(330,50)
(140,50) (140,30)[$\lambda_3$ ]{} (200,50) (200,30)[$\lambda_4$]{} (140,50)[(1,0)[60]{}]{}
(170,50)[(80,30)]{} (170,62)[$<$]{} (210,65)[$a_1$]{}
(220,50)[……]{}
(300,50) (300,30)[$\lambda_{2g+1}$]{} (360,50) (360,30)[$\lambda_{2g+2}$]{} (300,50)[(1,0)[60]{}]{}
(330,50)[(80,30)]{} (330,62)[$<$]{} (370,65)[$a_g$]{}
The basic holomorphic one forms are expressed by $$dU^0_k = \frac{\lambda^{k-1} d \lambda}{\omega}, \quad 1 \leq k \leq g.$$ Then, the $g \times g$ matrices of a- and b-periods are given by $$\mathcal{A}_{kj}= \oint_{a_j} dU^0_k, \quad
\mathcal{B}_{kj}= \oint_{b_j} dU^0_k, \quad
1 \leq k,j \leq g.$$ Thus, the normalized holomorphic one forms are defined by $$dU_k=\frac{1}{\omega} \sum_{j=1}^g (\mathcal{A}^{-1})_{kj} \lambda^{j-1}d\lambda
\quad 1 \leq k \leq g,$$ which satisfy $$\oint_{a_j} dU_k = \delta_{jk}.$$ Furthermore, from $\mathcal{A}$ and $\mathcal{B}$, we can construct $$\bf{B}=\mathcal{A}^{-1} \mathcal{B}.$$
Following Mumford [@Mum], we define the $\Theta$ function with characteristic $[{\mathbf{p}},{\mathbf{q}}]\,({\mathbf{p}},{\mathbf{q}}\in {\mathbb{C}}^g)$ by $$\label{eqn:theta}
\Theta [{\mathbf{p}},{\mathbf{q}}] ({\mathbf{z}}| {\mathbf{B}})
=
\sum_{{\mathbf{m}}\in {\mathbb{Z}}^g}
\exp
\{
\pi i
\langle
{\mathbf{B}}({\mathbf{m}}+{\mathbf{p}}),{\mathbf{m}}+{\mathbf{p}}\rangle
+ 2 \pi i
\langle
{\mathbf{z}}+{\mathbf{q}},{\mathbf{m}}+{\mathbf{p}}\rangle
\},$$ where ${\mathbf{z}}\in {\mathbb{C}}^g$ and the sum extends over all integer vectors in ${\mathbb{C}}^g.$
Following Fay [@Fay], we introduce the prime-form $E(P,Q)$ and the canonical bi-meromorphic differential $W(P,Q)$. The prime-form is defined by $$\begin{aligned}
E(P,Q)
&=
\frac{\Theta [\mathbf{p}^{*},\mathbf{q}^{*}] (U(P)-U(Q))}{h_{*}(P)h_{*}(Q)} \\
(h_{*}(P))^2
&=
\sum_{k=1}^g \frac{\partial \Theta [\mathbf{p}^{*},\mathbf{q}^{*}]}{\partial z_k}(0 |\mathbf{B}) {\it dU_k(P)}
\quad
\mathrm{for} \,\,
P,Q \in \mathcal{L},\end{aligned}$$ where $[\mathbf{p}^{*},\mathbf{q}^{*}]$ is an arbitrary odd non-singular half integer characteristic. The canonical bi-meromorphic differential $W(P,Q)$ is given by $$W(P,Q)
=
dx_Pdx_Q
\log E(P,Q),$$ where $dx_P,dx_Q$ are the exterior differentiations with respect to the local parameter of $P,Q,$ respectively. When $P=\infty^1$ and $Q=\infty^2,$ we define the local coordinates of $P,Q$ by $$x_{\infty^1}=\frac{1}{\lambda}, \,\,
x_{\infty^2}=\frac{1}{\lambda},$$ and have $$W(\infty^1,\infty^2)
=
\left(
\left.
\frac{\partial^2}{\partial x_{\infty^1}\partial x_{\infty^2}}
\log E(P,Q)
\right|_{P=\infty^1,Q=\infty^2}
\right)dx_{\infty^1}dx_{\infty^2}.$$
According to Fay [@Fay], the projective connection $S(Q)$ is given by the equation $$W(P,Q)
=
\left(
\frac{1}{(x_{P}-x_{Q})^2}
+
\frac{1}{6}
S(Q)
+
O(x_{P}-x_{Q})
\right)
dx_{P}dx_{Q},$$ where $P$ and $Q$ have local coordinates $x_{P},x_{Q}$ in a neighborhood of $Q\in \mathcal{L}$. When $Q=\infty^1$, we define the local coordinate of $Q$ by $$x_{P}=x_{Q}=\frac{1}{\lambda}
\quad
\mathrm{if} \,\, P \longrightarrow Q=\infty^1.$$
Our main theorem is as follows:
For the monodromy-preserving deformation (\[eqn:mpd\]), the $\tau$-function is $$\begin{aligned}
{3}
\tau(\lambda_1, \ldots, \lambda_{2g+2} ; t)
&=
\Theta[{\mathbf{p}},{\mathbf{q}}]
\left(
\mathbf{v}(t) |{\mathbf{B}}\right) &
&(\det \mathcal A)^{-\frac12}
\prod_{1 \leq j<k \leq 2g+2}
(\lambda_j-\lambda_{k})^{-\frac18} \\
& &
&\times
\exp \Big\{\frac{t^2}{4} \left(\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}\right) \Big\},\end{aligned}$$ where $$\mathbf{v}(t)
=
t \times
\left(
\frac{dU_1}{dx_{\infty^1}}(\infty^1),
\frac{dU_2}{dx_{\infty^1}}(\infty^1),
\ldots,
\frac{dU_{g}}{dx_{\infty^1}}(\infty^1)
\right)$$ and $x_{\infty^1}=\displaystyle\frac{1}{\lambda},$ which is a local coordinate of $\infty^1.$
By setting $t=0$, we obtain Kitaev and Korotkin’s $\tau$-function in [@KK]. We had a branch point at $\infty$ in [@Matsuda], while we do not in this paper.
Hyperelliptic Curves and the $\Theta$ Function
==============================================
In this section, we explain the detailed properties of the $\Theta$ function, the prime-form $E(P,Q)$ and the canonical bi-meromorphic differential $W(P,Q)$.
The $\Theta$ function possesses the following periodicity properties: $$\label{eqn:per1}
\Theta [{\mathbf{p}},{\mathbf{q}}] ({\mathbf{z}}+{\mathbf{e}}_j | {\mathbf{B}})
=
\exp \{ 2 \pi i p_j \}
\Theta [{\mathbf{p}},{\mathbf{q}}] ({\mathbf{z}}| {\mathbf{B}})$$
$$\label{eqn:per2}
\Theta [{\mathbf{p}},{\mathbf{q}}] ({\mathbf{z}}+ {\mathbf{B}}{\mathbf{e}}_j | {\mathbf{B}})
=
\exp \{ -2 \pi i q_j - \pi i {\mathbf{B}}_{jj} -2 \pi i z_j \}
\Theta [{\mathbf{p}},{\mathbf{q}}] ({\mathbf{z}}| {\mathbf{B}}),$$
where $$\mathbf{e}_j ={}^t(0, \ldots, \stackrel{j th}{1}, \ldots, 0).$$
We define the Abel map $U(P) \in {\mathbb{C}}^g \,P\in \mathcal{L}$ by $$\begin{aligned}
U(P)
&=
{}^t
\left(
U_1(P), U_2(P), \ldots, U_g(P)
\right) \\
U_j(P)
&=
\int_{\lambda_1}^P dU_j \quad (1\leq j \leq g).\end{aligned}$$ For the canonical homological basis $\{a_j, b_j \}_{1 \leq j \leq g}$ and the base point $\lambda_1$, the Riemann constants $\mathbf{K} \in {\mathbb{C}}^g$ are as follows: $$\mathbf{K}=
\frac12
{\mathbf{B}}\left(
\mathbf{e}_1+\mathbf{e}_2+\cdots+\mathbf{e}_g
\right)
+
\frac12
\left(
\mathbf{e}_1+2\mathbf{e}_2+\cdots+g\mathbf{e}_g
\right).$$
A characteristic $[\mathbf{p},\mathbf{q}]$ is a $g \times 2$ matrix of complex numbers which is given by $$[\mathbf{p},\mathbf{q}]
=
\begin{bmatrix}
p_1 & q_1 \\
p_2 & q_2 \\
\vdots & \vdots \\
p_g & q_g
\end{bmatrix},$$ where $$\mathbf{p}
=
{}^t
(p_1,p_2, \ldots, p_g),
\mathbf{q}
=
{}^t
(q_1,q_2,\ldots,q_g).$$ We consider $p_i,q_i \,\,(1\leq i \leq g)$ as elements of ${\mathbb{C}}^g/{\mathbb{Z}}^g.$ If all the components are half-integers, $[{\mathbf{p}},{\mathbf{q}}]$ is called a half-integer characteristic. A half-integer characteristic is in one-to-one correspondence with a half-period $\mathbf{B} \mathbf{p} + \mathbf{q}$. If the scalar product $4 \langle \mathbf{p}, \mathbf{q} \rangle$ is odd, then the characteristic is called odd and the related $\Theta$ function is odd with respect to its argument $\mathbf{z}$. If this scalar product is even, then the characteristic is called even and the related $\Theta$ function is even with respect to its argument $\mathbf{z}$.
The odd characteristics which are important for us in the sequel correspond to any subset $$S=
\{ \lambda_{i_1}, \lambda_{i_2},\ldots, \lambda_{i_{g-1}}
\},$$ whose components are arbitrary distinct branch points. The odd half-period corresponding to the subset $S$ is expressed by $$\mathbf{B} \mathbf{p}^S + \mathbf{q}^S
=
U(\lambda_{{\it i_1}})+U(\lambda_{{\it i_2}})+ \cdots +U(\lambda_{{\it i_{g-1}}})-\mathbf{K}.$$
Analogously, we shall be interested in the even half-period corresponding to the subset $$T=
\{
\lambda_{i_1}, \lambda_{i_2},\ldots, \lambda_{i_{g+1}}
\},$$ which consists of arbitrary $g + 1$ branch points. The even half-period is given by $$\mathbf{B} \mathbf{p}^T + \mathbf{q}^T
=
U(\lambda_{{\it i_1}})+U(\lambda_{{\it i_2}})+ \cdots +U(\lambda_{{\it i_{g+1}}})-\mathbf{K}.$$ We fix the choice of $T$ and define $$\{\lambda_{j_1},\lambda_{j_2},\ldots,\lambda_{j_{g+1}} \}
:=
\{\lambda_1,\lambda_2,\ldots,\lambda_{2g+2}\}
\setminus T.$$
We explain the detailed property of $E(P,Q)$. On page 13–14 of Fay [@Fay], we find $$\begin{aligned}
E(P,Q)
&=
\frac{ \Theta [\bf{p}^T,\bf{q}^T]({\it U(P)-U(Q)})}{\Theta [\bf{p}^T,\bf{q}^T](0) {\it m_T(P,Q)}}, \\
m_{T}(P,Q)
&=
\frac{
\omega(Q) \prod_{k=1}^{g+1}(\lambda(P)-\lambda_{i_k})
+
\omega(P) \prod_{k=1}^{g+1}(\lambda(Q)-\lambda_{i_k})
}
{2 (\lambda(Q)-\lambda(P))} \\
&
\hspace{20mm}
\times
\left[
\frac{
d \lambda(P) d \lambda(Q)
}
{
\omega(P)\omega(Q)
\prod_{k=1}^{g+1}(\lambda(P)-\lambda_{i_k})(\lambda(Q)-\lambda_{i_k})
}
\right]^{\frac12},\end{aligned}$$ because $\mathcal{L}$ is hyperelliptic.
The Schlesinger System
======================
We study an ordinary differential equation which is given by $$\label{equ:diff2}
\frac{d \Psi}{d \lambda}
=
\left(
\sum_{j=1}^{2g+2}
\frac{A_j}{\lambda-\lambda_{j}}
-B_{-1}
\right)
\Psi(\lambda),$$ where $A_1,A_2,\ldots,A_{2g+2}, B_{-1} \in sl(2, {\mathbb{C}})$ are independent of $\lambda$.
The monodromy data of are as follows: (i) Stokes multipliers around $\lambda=\infty$ $S^{\infty}_1 = S^{\infty}_2 ;$ (ii) connection matrices around $\lambda_j$ $C_j, \,(j=1,2,\ldots, 2g+2);$ (iii) the exponents of formal monodromy $T_{\infty,0}, T_{j,0} \, (j=1,2,\ldots, 2g+2).$ In the next section, we obtain a convergent series around $\lambda = \infty, \lambda_{j} \,\,(j=1,2,\ldots, 2g+2)$ which are expressed by $$\begin{aligned}
&
\Psi(\lambda)
=
(
1
+
O(\frac{1}{\lambda})
)
\exp
T^{\infty}(\lambda)
=\hat{\Psi}^{\infty}(\lambda)
\exp T^{\infty}(\lambda), \\
&
\Psi(\lambda)
=
G_{j}
(
1
+
O(\lambda-\lambda_{j})
)
\exp T_{j}(\lambda)
=
G_{j}
\hat{\Psi}_{j}(\lambda)
\exp
T_{j,0}(\lambda), \notag \\
&
\hspace{90mm}(j=1,2,\ldots, 2g+2),\end{aligned}$$ where $$\begin{aligned}
T^{\infty}(\lambda)
&=
\left(
\begin{array}{cc}
-\frac{t}{2} & \\
& \frac{t}{2} \\
\end{array}
\right)
\lambda
+
T_{\infty,0}
\log(\frac{1}{\lambda}), \\
T_{j}(\lambda)
&=
T_{j,0}
\log(\lambda-\lambda_{j})
\,\, (j=1,2,\ldots, 2g+2).\end{aligned}$$
For the deformation parameters $t, \lambda_1, \lambda_2, \ldots, \lambda_{2g+2},$ the closed one form is defined by $$\begin{aligned}
\Omega
&=
\omega_{\infty}
+
\omega_{\lambda_1}
+
\omega_{\lambda_2}
+
\cdots
+
\omega_{\lambda_{2g+2}} \\
&=
H_t dt
+
H_1 d \lambda_1
+
H_2 d \lambda_2
+
\cdots
+
H_{2g+2} d \lambda_{2g+2}, \end{aligned}$$ where $$\begin{aligned}
\omega_{\infty}
&=
-
{\textrm{Res}}_{\lambda=\infty} \,\,
{\qopname\relax o{tr}}\hat{\Psi}^{\infty}(\lambda)^{-1}
\frac{\partial \hat{\Psi}^{\lambda}}{\partial \lambda}
(\lambda)
\,
d T^{\infty}(\lambda), \\
\omega_{\lambda_j}
&=
-
{\textrm{Res}}_{\lambda=\lambda_{j}} \,\,
{\qopname\relax o{tr}}\hat{\Psi}_{j}(\lambda)^{-1}
\frac{\partial \hat{\Psi}_{j}}{\partial \lambda}(\lambda)
\,
d T_{j}(\lambda) \,\,(j=1,2,\ldots, 2g+2 ),\end{aligned}$$ and $d$ is the exterior differentiation with respect to the deformation parameters $t, \lambda_1, \lambda_2, \ldots, \lambda_{2g+2}$. Especially, we can write $$\omega_{\lambda_{j}}
=
\left[
{\textrm{Res}}_{\lambda=\lambda_{j}}
\frac12
{\qopname\relax o{tr}}\left(
\frac{d \Psi}{d \lambda}\Psi^{-1}
\right)^2
\right]
d \lambda_{j}.$$ Then, from the closed one form $\Omega$, the $\tau$-function is defined by $$\Omega := d \log \tau (\lambda_1, \lambda_2, \ldots, \lambda_{2g+2},t).$$
The Riemann-Hilbert Problem for Special Parameters
==================================================
In this section, we concretely construct a $2\times2$ matrix valued function $\Psi(\lambda)$, whose monodromy data, (i) Stokes multipliers, (ii) connection matrices, (iii) exponents of formal monodromy, are independent of the deformation parameters $t, \lambda_1, \lambda_2, \ldots, \lambda_{2g+2}$.
(400,150)
(70,70)[(1,1)[120]{}]{} (100,70)[(3,4)[90]{}]{}
(85,70)[(30,55)\[b\]]{} (85,60) (80,50)[$\lambda_1$]{} (81,39)[$>$]{} (82,25)[$l_{1}$]{}
(130,70)[(1,2)[60]{}]{} (160,70)[(1,4)[30]{}]{}
(145,70)[(30,55)\[b\]]{} (145,60) (142,50)[$\lambda_2$]{} (139,39)[$>$]{} (142,25)[$l_2$]{}
(180,39)[$\cdots$]{}
(190,190) (190,195)[$\lambda_0$]{}
(220,70)[(-1,4)[30]{}]{} (250,70)[(-1,2)[60]{}]{}
(235,70)[(30,55)\[b\]]{} (235,60) (232,50)[$\lambda_{2g+1}$]{} (232,39)[$>$]{} (232,25)[$l_{2g+1}$]{}
(280,70)[(-3,4)[90]{}]{} (310,70)[(-1,1)[120]{}]{}
(295,70)[(30,55)\[b\]]{} (295,60) (292,50)[$\lambda_{2g+2}$]{} (292,39)[$>$]{} (292,25)[$l_{2g+2}$]{}
The involution of $\mathcal{L}$ is defined by $$*:
(\lambda,\omega)
\longrightarrow
(\lambda,-\omega).$$ Then, from the $\Theta$ function and $W(P,Q)$, we define the $2 \times 2$ matrix valued function $\Phi(P)$ by $$\Phi (P) =
\left(
\begin{array}{cc}
\varphi(P) & \varphi(P^{*}) \\
\psi(P) & \psi(P^{*})
\end{array}
\right),$$ where $$\begin{aligned}
\varphi(P)
&=
\Theta [{\mathbf{p}},{\mathbf{q}}] (U(P)+U(P_{\varphi})+\mathbf{v}(t))
\Theta [{\mathbf{p}}^S,{\mathbf{q}}^S](U(P)-U(P_{\varphi})) \\
&
\quad
\times
\exp
-\frac{t}{2}
\{
-
\int^P_{\lambda_1}
W(P,\infty^1)
+
\int^P_{\lambda_1}
W(P,\infty^2)
\}, \\
\psi(P)
&=
\Theta [{\mathbf{p}},{\mathbf{q}}] (U(P)+U(P_{\psi})+\mathbf{v}(t))
\Theta [{\mathbf{p}}^S, {\mathbf{q}}^S] (U(P)-U(P_{\psi})) \\
&
\quad
\times
\exp
-\frac{t}{2}
\{
-
\int^P_{\lambda_1}
W(P,\infty^1)
+
\int^P_{\lambda_1}
W(P,\infty^2)
\},
\\
\mathbf{v}(t)
&=
t \times
\left(
\frac{dU_1}{dx_{\infty^1}} (\infty^1),
\frac{dU_2}{dx_{\infty^1}} (\infty^1),
\ldots,
\frac{dU_{g}}{dx_{\infty^1}} (\infty^1)
\right),\end{aligned}$$ where $P_{\varphi}, P_{\psi}$ are arbitrary points of $\mathcal{L}$ and $x_{\infty^1}=\displaystyle\frac{1}{\lambda},$ which is a local coordinate of $\infty^1.$
\[prop:reg\] (1) The function $\Phi(P)$ is invertible outside of the branch points $\lambda_1, \lambda_2, \ldots, \lambda_{2g+2}.$ (2) $\det \Phi(P)$ has zeros at $\lambda_j \notin S$ with the first order and has zeros at $\lambda_j \in S$ with the third order. (3) The function $\Phi(P)$ transforms as follows with respect to the tracing along the canonical homological basis, $a_j, b_j\,(j=1,2,\ldots, g)$: $$\begin{aligned}
\label{eqn:trans1}
T_{a_j}
\big[
\Phi (P)
\big]
&=
\Phi (P)
\left(
\begin{array}{cc}
\exp \{ 2\pi i (p_j+p^S_j) \} & \\
& \exp \{ - 2\pi i (p_j+p^S_j) \}
\end{array}
\right), \\
\label{eqn:trans2}
T_{b_j}
\big[
\Phi (P)
\big]
&=
\Phi (P)
\left(
\begin{array}{cc}
\exp \{ - 2\pi i (q_j+q^S_j) \} & \\
& \exp \{ 2\pi i (q_j+q^S_j) \}
\end{array}
\right)
\exp \{ -2 \pi i {\mathbf{B}}_{jj}-4 \pi i U_j(P)\},\end{aligned}$$ where $T_{l}$ denotes the operator of analytic continuation along the contour $l$.
By using the periodicity (\[eqn:per1\]) and (\[eqn:per2\]), we obtain $$\begin{aligned}
T_{a_j}[\varphi(P)]&=
\exp\{2\pi i (p_j+p_j^S)\} \varphi(P), \\
T_{b_j}[\varphi(P)]&=
\exp\{-2\pi i (q_j+q_j^S)-2 \pi i {\mathbf{B}}_{jj}-4 \pi i U_j(P)\} \varphi(P).\end{aligned}$$ We deduce the same transformation laws for $\psi(P)$.
The actions of the involution $*$ on $\{a_j,b_j\}$ and $dU_j$ are given by $$a_j^{*}=-a_j,\,
b_j^{*}=-b_j,\,
dU_j(P^{*})=-dU_j(P),\,
(j=1,2,\ldots,g),$$ respectively. Therefore, we get $$\begin{aligned}
T_{a_j}[\varphi(P^*)]
&=
\exp \{-2\pi i(p_j+p_j^S)\}
\varphi(P^*), \\
T_{b_j}[\varphi(P^*)]&= \exp\{2\pi i(q_j+q_j^S)-2\pi i {\mathbf{B}}_{jj}-4\pi i U_j(P)\}\varphi(P^*).\end{aligned}$$ We deduce the same transformation laws for $\psi(P^*)$. Then, we complete the proof of (3).
The equations (\[eqn:trans1\]) and (\[eqn:trans2\]) imply that $$\begin{aligned}
T_{a_j}[\det \Phi(P)]&=\det \Phi(P),\\
T_{b_j}[\det \Phi(P)]&=\det \Phi(P)
\exp \{-4\pi i {\mathbf{B}}_{jj}-8\pi i U_j(P)\},\end{aligned}$$ which imply that $$\frac{1}{2 \pi i}
\oint_{\partial \mathcal{L}}
\frac
{
d(\det \Phi(P))
}
{\det \Phi(P)}
=
4g.$$ Thus, it follows that $$3(g-1)+g+3=4g,$$ because $\det \Phi(P)$ has zeros at the branch points $\lambda_j$ and has zeros at $\lambda_j \in S$ of the order of at least three. Therefore, it follows that $\det \Phi(P)$ does not vanish outside of the branch points and that $\det \Phi(P)$ has zeros at $\lambda_j \notin S$ with the first order and has zeros at $\lambda_j \in S$ with the third order, which complete the proof of (1) and (2), respectively.
In order to normalize $\Phi(\lambda)$ near $\lambda=\infty$, we have
\[lem:Fay1\] For $P,Q \in \mathcal{L},$ $$\begin{aligned}
W(P,Q)
&=
\left(
\frac{1}{(x_{P}-x_{Q})^2}
+
\frac{1}{6}
S(Q)
+
O(x_{P}-x_{Q})
\right)
dx_{P}dx_{Q}, \\
\label{eqn:proj2}
\frac16 S(Q)
&=
\frac16 \{\lambda,x_{Q}\}(Q)
+
\frac{1}{16}
\left(
\frac{d}{dx_{Q}}
\log
\prod_{k=1}^{g+1}
\frac{\lambda-\lambda_{i_k}}{\lambda-\lambda_{j_k}}(Q)
\right)^2 \notag \\
& \hspace{30mm}
-\sum_{k,l=1}^g
\frac{\partial^2}{\partial z_k \partial z_l}
\log \Theta [{\mathbf{p}}^T,{\mathbf{q}}^T](0)
\frac{dU_k}{dx_{Q}}(Q)\frac{dU_l}{dx_{Q}}(Q),\end{aligned}$$ where $x_{P},x_{Q}$ are local coordinates of $P,Q \in \mathcal{L}$ and $$\{\lambda,x\}
=
\frac{\lambda^{\prime \prime \prime}}{\lambda^{\prime}}
-\frac32
\left(
\frac{\lambda^{\prime \prime}}{\lambda^{\prime}}
\right)^2.$$
See pp. 20 in [@Fay].
In Lemma \[lem:Fay1\]. we set $$x_{P}=x_{Q}=
\frac{1}{\lambda},
\quad
Q=\infty^1,$$ and take the limit $P \longrightarrow \infty^1.$ Furthermore, we define the constant terms $c_{\infty^1},c_{\infty^2}$ by $$\begin{aligned}
{11}
&
\int_{\lambda_1}^P
W(P,\infty^1)
&
&=
-\lambda &
+&c_{\infty^1} &
&+\frac16 S(\infty^1)
\lambda^{-1} &
&+\cdots & \quad &\mathrm{near} \,\,\lambda=\infty^1, \\
&
\int_{\lambda_1}^P
W(P,\infty^2)
&
&=
&
&c_{\infty^2} &
&+
\frac{
W(\infty^1,\infty^2)
}{dx_{\infty^1}dx_{\infty^2}}\lambda^{-1} &
&+\cdots & &\mathrm{near} \,\,\lambda=\infty^1. \end{aligned}$$ Therefore, $\Phi(\lambda)$ can be developed near $\lambda=\infty$ as
$$\Phi(\lambda)
=
\left(
G^{\infty}
+
O(\frac{1}{\lambda})
\right)
\exp
\left(
\begin{array}{cc}
-
\frac{t}{2} \lambda& \\
& \frac{t}{2}\lambda \\
\end{array}
\right),$$
where $G^{\infty}$ is a $2 \times 2$ matrix whose matrix elements are given by $$\begin{aligned}
(G^{\infty})_{11}
&=
\Theta[{\mathbf{p}},{\mathbf{q}}](U(\infty^1)+U(P_{\varphi})+\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](U(\infty^1)-U(P_{\varphi}))
\exp
\Big\{
\frac{t}{2}
\left(
c_{\infty^1}-c_{\infty^2}
\right)
\Big\}, \\
(G^{\infty})_{21}
&=
\Theta[{\mathbf{p}},{\mathbf{q}}](U(\infty^1) + U(P_{\psi}) +\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](U(\infty^1)-U(P_{\psi}))
\exp
\Big\{
\frac{t}{2}
\left(
c_{\infty^1}-c_{\infty^2}
\right)
\Big\}, \\
(G^{\infty})_{12}
&=
\Theta[{\mathbf{p}},{\mathbf{q}}](U(\infty^2) +U(P_{\varphi}) +\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](U(\infty^2) -U(P_{\varphi}))
\exp
\Big\{
-
\frac{t}{2}
\left(
c_{\infty^1}-c_{\infty^2}
\right)
\Big\}, \\
(G^{\infty})_{22}
&=
\Theta[{\mathbf{p}},{\mathbf{q}}](U(\infty^2) +U(P_{\psi}) +\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](U(\infty^2)-U(P_{\psi}))
\exp
\Big\{
-
\frac{t}{2}
\left(
c_{\infty^1}-c_{\infty^2}
\right)
\Big\}.\end{aligned}$$ Proposition \[prop:reg\] shows that $$\det G^{\infty}
=
\det \Phi (\infty)
\neq
0.$$ We define a matrix valued function $\Psi(\lambda)$ by $$\Psi(P)
=
\frac
{
\sqrt{\det \Phi(\infty)}
}
{
\sqrt{\det \Phi(P)}
}
(G^{\infty})^{-1}
\Phi(P).$$
The expansions of $\Psi(\lambda)$ near $\lambda=\infty$ are as follows:
\[lem:ya\] $$\begin{aligned}
&
\Psi(\lambda)
=
\left(
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right)
+
O(\frac{1}{\lambda})
\right)
\exp
T^{\infty}(\lambda), \\
&
T^{\infty}(\lambda)
=
T^{\infty}_{-1}
\lambda, \,\,
T^{\infty}_{-1}
=
\left(
\begin{array}{cc}
-\frac{t}{2} & \\
& \frac{t}{2}
\end{array}
\right), \end{aligned}$$ where the Taylor series of $\Psi(\lambda) \exp\left\{-T^{\infty}(\lambda)\right\}$ is convergent. Especially, if $P_{\varphi}=\infty^1$ and $ P_{\psi}=\infty^2,$
$$\begin{aligned}
\Psi(\lambda)
&=
\left(
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right)
+
\Psi_{-1}^{\infty}
\lambda^{-1}
+
\cdots
\right)
\exp T^{\infty}(\lambda), \\
\left(\Psi_{-1}^{\infty}\right)_{11}
&=
\sum_{k=1}^g
\frac{\partial}{\partial z_k}
\log \Theta [{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))
\frac{dU_k}{dx_{\infty^1}}(\infty^1)
+
\frac{t}{2}
\left(
\frac16
S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right),
\\
\left(\Psi_{-1}^{\infty}\right)_{21}
&=
\frac{i}{E(\infty^2,\infty^1)}
\frac{\Theta[{\mathbf{p}},{\mathbf{q}}](2U(\infty^1)+\mathbf{v}(t))}{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\exp
\left\{
t
\left(
c_{\infty^1}-c_{\infty^2}
\right)
\right\}, \\
\left(\Psi_{-1}^{\infty}\right)_{12}
&=
\frac{i}{E(\infty^1,\infty^2)}
\frac
{\Theta[{\mathbf{p}},{\mathbf{q}}](2U(\infty^2)+\mathbf{v}(t))}
{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\exp
\left\{
-t
\left(
c_{\infty^1}-c_{\infty^2}
\right)
\right\}, \\
\left(\Psi_{-1}^{\infty}\right)_{22}
&=
\sum_{k=1}^g
\frac{\partial}{\partial z_k}
\log \Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))
\frac{dU_k}{dx_{\infty^2}}(\infty^2)
-
\frac{t}{2}
\left(
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right).\end{aligned}$$
In the following theorem, we determine the monodromy matrices and the Stokes matrices of $\Psi(\lambda)$.
\[thm:mono\] For $\nu = 1, 2, \ldots, 2g+2,$ the monodromy matrix $M_{\nu}$ of $\Psi(\lambda)$ corresponding to the contour $
l_{\nu}
$ is given by $$M_{\nu}
=
\left(
\begin{array}{cc}
0 & m_{\nu} \\
- m_{\nu}^{-1} & 0
\end{array}
\right),$$ where $$\begin{aligned}
&
m_{1} = -i, \,\,
m_{2} = i (-1)^{g+1}\exp \left\{-2 \pi i \sum_{k=1}^g p_k \right\}, \\
&
m_{2j+1} = i (-1)^g \exp \left\{ 2 \pi i q_j-2\pi i\sum_{k=j}^g p_k \right\}, \\
&
m_{2j+2} = i (-1)^{g+1} \exp \left\{ 2\pi i q_j -2\pi i \sum_{k=j+1}^g p_k \right\},\end{aligned}$$ for $j=1,2,\ldots,g$. The Stokes matrices are expressed by $$S_1^{\infty}
=
S_2^{\infty}
=
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right).$$
By the involution $*$, we get $$\Psi(\lambda) M_{1}
=
\Psi(\lambda)
\left(
\begin{array}{cc}
& i \\
i &
\end{array}
\right),
\,
\textrm{or}
\,
=
\Psi(\lambda)
\left(
\begin{array}{cc}
& -i \\
-i &
\end{array}
\right).$$ We define $$\begin{aligned}
M_{1}
=
\left(
\begin{array}{cc}
& m_{1} \\
-m_{1}^{-1} &
\end{array}
\right)
=
\left(
\begin{array}{cc}
& -i \\
-i &
\end{array}
\right).\end{aligned}$$ Proposition \[prop:reg\] implies that $$\begin{aligned}
T_{a_j}
\left[
\Psi(\lambda)
\right]
&=
\Psi(\lambda) M_{2j+2} M_{2j+1} \\
&=
\Psi(\lambda)
\frac{T_{l_{2j+1}\circ l_{2j+2}}[\sqrt{\det \Phi(P)}]}{\sqrt{\det \Phi(P)}}
\exp \{2 \pi i (p_j+p_j^S) \}
\sigma_3,\end{aligned}$$ and $$\begin{aligned}
T_{-b_j+b_{j-1}}
\left[
\Psi(\lambda)
\right]
&=
\Psi(\lambda) M_{2j+1} M_{2j} \\
&=
\Psi(\lambda)
\frac{T_{l_{2j}\circ l_{2j+1}}[\sqrt{\det \Phi(P)}]}{\sqrt{\det \Phi(P)}}
\exp \{2 \pi i (q_j-q_{j-1}+q_j-q_{j-1})\}
\sigma_3,\end{aligned}$$ where $
\sigma_3=
\left(
\begin{array}{cc}
1 & 0 \\
0 &-1
\end{array}
\right).
$ In order to determine the monodromy matrices, we have $$\begin{aligned}
{3}
&U(\lambda_1)=0, &
&U(\lambda_2) \,\, = \,\, \frac12 \sum_{k=1}^g {\mathbf{e}}_k, \\
&U(\lambda_{2j+1})=\frac12 {\mathbf{B}}{\mathbf{e}}_j +\frac12 \sum_{k=j}^g {\mathbf{e}}_k, & \quad
&U(\lambda_{2j+2}) \, = \, \frac12 {\mathbf{B}}{\mathbf{e}}_j +\frac12 \sum_{k=j+1}^g {\mathbf{e}}_k, \,\,
(j=1,2,\ldots,g).\end{aligned}$$ Then, we get $$p_j^S=\frac12
\left(
\delta_{2j+1}+\delta_{2j+2}+1
\right), \,
q_{j+1}-q_{j}
=
\frac12
\left(
\delta_{2j+2}+\delta_{2j+3}+1
\right),$$ where $$\begin{cases}
\delta_j=1 \quad \mathrm{if} \,\, \lambda_j \in S \\
\delta_j=0 \quad \mathrm{if} \,\, \lambda_j \notin S.
\end{cases}$$
The function $\sqrt{\det \Phi(P)}$ transforms with respect to the tracing along the cycles $l_j$ in the following way: $$\label{eqn:root}
\begin{cases}
T_{l_{2j+1}\circ l_{2j+2}}[\sqrt{\det \Phi(P)}]
=
\exp \{\pi i (\delta_{2j+1}+\delta_{2j+2}+1) \}
\sqrt{\det \Phi(P)}, \\
T_{l_{2j}\circ l_{2j+1}}[\sqrt{\det \Phi(P)}]
=
\exp \{\pi i (\delta_{2j+2}+\delta_{2j+3}+1) \}
\sqrt{\det \Phi(P)}.
\end{cases}$$ In order to prove (\[eqn:root\]), we have only to note that if $\lambda_j$ is in $S$, from Proposition \[prop:reg\], $\det \Phi(P)$ has a zero of order one at $\lambda=\lambda_j$ and that if $\lambda_j$ is not in $S$, from Proposition \[prop:reg\], $\det \Phi(P)$ has a zero of order three at $\lambda=\lambda_j$. Therefore, we obtain $$\begin{cases}
M_{2j+2}M_{2j+1}=
\exp \{ 2 \pi i \sigma_3 \} \\
M_{2j+1} M_{2j} =
\exp \{2 \pi i (q_j-q_{j-1}) \sigma_3 \}.
\end{cases}$$ By considering $$M_{2g+2} M_{2g+1}
\cdots
M_{1}
=
I,$$ we get the monodromy matrices.
Because $\Psi(\lambda) \exp\left\{-T^{\infty}(\lambda)\right\}$ can be developed near $\lambda=\infty$ as a convergent series, the Stokes multipliers are $
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right).
$
We can describe the monodromy data of $\Psi(\lambda)$ in the following way.
$\Psi(\lambda)$ has the following monodromy data: $\mathrm{(i)}$ Stokes multipliers $S^{\infty}_1=S^{\infty}_2=
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right)
,$ $\mathrm{(ii)}$ connection matrices $
C_j
=
\frac{1}{\sqrt{2 i m_{j}}}
\left(
\begin{array}{cc}
1 & i m_{j} \\
-1 & i m_{j}
\end{array}
\right)
\,\,
(j=1,2, \ldots, 2g+2),
$ $\mathrm{(iii)}$ exponents of formal monodromy $
T_{j,0}
=
{\qopname\relax o{diag}}(-\frac14, \frac14)
\,\,
(j=1,2,\ldots,2g+2).
$ Especially, the developments of $\Psi(\lambda)$ near $
\lambda=\lambda_{j} \,
(j=1,2,\ldots, 2g+2)
$ are expressed by $$\begin{aligned}
&
\Psi(\lambda)
=
G_{j}
(
1
+
O(\lambda-\lambda_{j})
)
\exp T_{j} (\lambda)
C_{j}, \\
&
T_{j}(\lambda)
=
T_{j,0} \log (\lambda-\lambda_j ).\end{aligned}$$
\(i) is clear. (ii) and (iii) can be obtained by diagonalizing the monodromy matrices $M_{j} \,\, (j=1,2,\ldots,2g+2).$
This corollary means that the monodromy data of $\Psi(\lambda)$ are independent of the deformation parameters $t, \lambda_1, \lambda_2, \ldots, \lambda_{2g+2}.$
Monodromy-Preserving Deformation
================================
In this section, we prove that $\Psi(\lambda)$ satisfies an ordinary differential equation which is expressed by $$\frac{d \Psi}{d \lambda}
=
\left(
\sum_{j=1}^{2g+2}
\frac{A_{j}}{\lambda-\lambda_{j}}
-
B_{-1}
\right)
\Psi(\lambda),$$ and concretely determine the coefficients $B_{-1}, A_{j} \,\, (1 \leq j \leq 2g+2).$
We note that the monodromy matrices and the Stokes matrices are independent of the parameters $P_{\varphi}, P_{\phi}, p_k^S, q_k^S\,\,(1\leq k \leq g)$ in Theorem \[thm:mono\]. Thus, in this section, we set $
P_{\varphi}=\infty^2,
\,
P_{\phi}=\infty^1
$ and take $S_j$ so that $\lambda_j$ is not in $S_j$ because of the uniqueness of a solution of the Riemann-Hilbert problem. Therefore, we get $$\Psi(\lambda)
=
\frac{1}{\sqrt{\det \Phi^{\infty}(\lambda)}}
\Phi^{\infty}(\lambda),$$ where $$\begin{aligned}
&
\Psi(P)
=
\left(
\begin{array}{cc}
\varphi_j^{\infty}(P) & \varphi_j^{\infty} (P^{*}) \\
\psi_j^{\infty}(P) & \psi_j^{\infty} (P^{*})
\end{array}
\right), \\
&
\varphi_j^{\infty}(P)=
\frac
{
\Theta[{\mathbf{p}},{\mathbf{q}}](U(P)+U(\infty^2)+\mathbf{v}(t))
\Theta[{\mathbf{p}}^{S_j},{\mathbf{q}}^{S_j}](U(P)-U(\infty^2))
}
{
\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](-2U(\infty^2))
} \\
&
\hspace{30mm}
\times
\exp \{-\frac{t}{2}(c_{\infty^1}-c_{\infty^2}) \}
\exp \Pi (P),
\\
&
\psi_j^{\infty}(P)=
\frac
{
\Theta[{\mathbf{p}},{\mathbf{q}}](U(P)+U(\infty^1)+\mathbf{v}(t))
\Theta[{\mathbf{p}}^{S_j},{\mathbf{q}}^{S_j}](U(P)-U(\infty^1))
}
{
\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](-2U(\infty^1))
} \\
&
\hspace{30mm}
\times
\exp \{\frac{t}{2}(c_{\infty^1}-c_{\infty^2}) \}
\exp \Pi (P).\end{aligned}$$
\[thm:mpd\] $\Psi(\lambda)$ satisfies the following ordinary differential equation: $$\label{eqn:mpd}
\frac{d \Psi}{d \lambda}
=
\left(
\sum_{j=1}^{2g+2}
\frac{A_{j}}{\lambda-\lambda_{j}}
-
B_{-1}
\right)
\Psi(\lambda),$$ where $$\begin{aligned}
&
B_{-1}
=
{\qopname\relax o{diag}}(\frac{t}{2},-\frac{t}{2} ), \\
&
A_j
=
-\frac14
F_j^{\infty} \sigma_3 (F_j^{\infty})^{-1}, \quad
\sigma_3=
\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right),
\\
&
F_j^{\infty}
=
\left(
\begin{array}{cc}
\varphi_j^{\infty}(\lambda_j) & \frac{d}{dx_j} \varphi_j^{\infty}(\lambda_j) \\
\psi_j^{\infty}(\lambda_j) & \frac{d}{dx_j} \psi_j^{\infty}(\lambda_j)
\end{array}
\right), \end{aligned}$$ and $x_j=\sqrt{\lambda-\lambda_j}.$
From Lemma \[lem:ya\], it follows that $$\Psi_{\lambda} \Psi^{-1}
=
\left(
\begin{array}{cc}
-\frac{t}{2} & \\
& \frac{t}{2}
\end{array}
\right)
+
O(\lambda^{-1})
:=
-B_{-1}+O(\lambda^{-1}).$$
Near $\lambda=\lambda_j,$ we have $$\begin{cases}
\varphi_j^{\infty}(P)
=
\varphi_j(\lambda_j)
+
\sqrt{\lambda-\lambda_j}
\frac{d}{dx_j}\varphi_j(\lambda_j)
+\cdots \\
\psi_j^{\infty}(P)
=
\psi_j(\lambda_j)
+
\sqrt{\lambda-\lambda_j}
\frac{d}{dx_j}\psi_j(\lambda_j)
+\cdots,
\end{cases}$$ which implies that $$\det \Phi^{\infty}(P)
=
-2
\sqrt{\lambda-\lambda_j}
\det F_j^{\infty}
+
O(\lambda-\lambda_j).$$ From the definition of $S_j$, it follows that $\det F^{\infty}_j \neq 0.$
We set $$\Psi(\lambda)
:=
G_j
(1+O(\lambda-\lambda_j))
(\lambda-\lambda_j)^
{
\left(
\begin{array}{cc}
-\frac14 & \\
& \frac14
\end{array}
\right)
}
C_j.$$ Then, we get $$\begin{aligned}
G_j=
\left.
\Psi(\lambda)
C_j^{-1}
(\lambda-\lambda_j)^
{
\left(
\begin{array}{cc}
\frac14 & \\
& -\frac14
\end{array}
\right)
}
\right|_{\lambda=\lambda_j}
&=
(-2\det F^{\infty}_j)^{-\frac12}
(1+O(\sqrt{\lambda-\lambda_j})) \\
&
\hspace{10mm}
\left.
\times
\Phi^{\infty}(P)
C_j^{-1}
(\lambda-\lambda_j)^
{
\left(
\begin{array}{cc}
1 & \\
& -\frac12
\end{array}
\right)
}
\right|_{\lambda=\lambda_j} \\
&=
(-2 \det F^{\infty}_j)^{-\frac12}
\times
2i m_j
F^{\infty}_j.\end{aligned}$$ Thus, we obtain $$\Psi_{\lambda}\Psi^{-1}
=
-\frac14
\frac{1}{\lambda-\lambda_j}
F^{\infty}_j \sigma_3 (F^{\infty}_j)^{-1}
+
\cdots
=
\frac{A_j}{\lambda-\lambda_j}
+\cdots.$$
From Theorem \[thm:mpd\], we can obtain the following deformation equation:
The deformation equation of the monodromy-preserving deformation (\[eqn:mpd\]) is as follows. For $j,k=1,2,\ldots,2g+2,$ $$\begin{aligned}
&
dA_j = [\Theta_j,A_j], \\
&
d F^{\infty}_j = \Theta_j A_j, \\
&
\Theta_j =
\sum_{k \neq j}
A_k
\frac{d\lambda_k-d\lambda_j}{\lambda_k-\lambda_j}
-
\left[
\Psi^{\infty}_{-1},dB_{-1}
\right]
-
d
\left(
\lambda_j B_{-1}
\right),\end{aligned}$$ where $``\,d\,"$ is the exterior differentiation with respect to the deformation parameters, $t, \lambda_1, \lambda_2, \ldots \lambda_{2g+2}$.
See [@MM1].
The $\tau$-Function for the Schlesinger System
==============================================
In this section, we calculate the $\tau$-function for the monodromy-preserving deformation (\[eqn:mpd\]). This section consists of three subsections. Subsection 5.1 is devoted to the Hamiltonian $H_t$. Subsection 5.2, 5.3 is devoted to the Hamiltonian $H_{j} \, (j=1,2,\ldots, 2g+2).$ In subsection 5.2, we quote Fay’s identities and Rauch’s variational formulas in order to compute the $\tau$-function. In subsection 5.3, we calculate $H_{j}$ and the $\tau$-function.
The Hamiltonian at the Irregular Singular Point
-----------------------------------------------
In this subsection, we prove Proposition \[prop:omegaa\], where we compute $\omega_{\infty}$ and the Hamiltonian $H_t$.
\[prop:omegaa\] $$\begin{aligned}
\omega_{\infty}
&=
\left(
\frac{1}{\Theta [{\mathbf{p}},{\mathbf{q}}] ({\it \mathbf{v}}(t))}
\sum_{k=1}^g
\frac{\partial}{\partial z_k}
\{\Theta [{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t)) \}
\frac{dU_k}{dx_{\infty^1}}(\infty^1)
+
\frac{t}{2}
\left(
\frac16 S(\infty^1)-
\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right)
\right) dt \\
&= H_t dt.\end{aligned}$$
We define $$\Pi(P)
=
-\frac{t}{2}
\{
-
\int^P_{\lambda_1}
W(P,\infty^1)
+
\int^P_{\lambda_1}
W(P,\infty^2)
\},$$ and let $\hat{\Pi}(P)$ denote the regular part of $\Pi(P)$ around $\lambda=\infty^1$ which is given by $$\hat{\Pi}(P)
=
-\frac{t}{2}
\{
const \, +
\left(-\frac16 S(\infty^1)+\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}\right)\lambda^{-1}
+\cdots
\}.$$ Furthermore, we set $$\begin{aligned}
\hat{\varphi}(P)
&=
\Theta[{\mathbf{p}},{\mathbf{q}}](U(P)+U(P_{\varphi})+\mathbf{v}(t))
\Theta [{\mathbf{p}}^S,{\mathbf{q}}^S](U(P)-U(P_{\varphi})), \\
\hat{\psi}(P)
&=
\Theta[{\mathbf{p}},{\mathbf{q}}](U(P)+U(P_{\psi})+\mathbf{v}(t))
\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](U(P)-U(P_{\psi})).\end{aligned}$$ Then, we get $$\begin{aligned}
\Psi(P)
&=
\frac
{
\sqrt{\det \Phi(\infty)}
}
{
\sqrt{\det \Phi(P)}
}
(G^{\infty})^{-1}
\Phi(P) \\
&=
\frac
{
\sqrt{\det \Phi(\infty)}
}
{
\sqrt{\det \Phi(P)}
}
(G^{\infty})^{-1}
\left(
\begin{array}{cc}
\hat{\varphi}(P) \exp (\hat{\Pi} (P))
&
\hat{\varphi}(P^{*}) \exp (\hat{\Pi} (P^{*})) \\
\hat{\psi}(P) \exp (\hat{\Pi} (P))
&
\hat{\psi}(P^{*}) \exp (\hat{\Pi} (P^{*}))
\end{array}
\right) \\
&
\hspace{50mm}
\times
{\qopname\relax o{diag}}\left(
\exp
\Big\{
-
\frac{t}{2}
\lambda
\Big\},
\exp
\Big\{
\frac{t}{2}
\lambda
\Big\}
\right) \\
&:=
\hat{\Psi}^{\infty}(\lambda)
\exp T^{\infty} (\lambda).\end{aligned}$$
From the definition of $\omega_{\infty},$ it follows that $$\omega_{\infty}
=
-
{\textrm{Res}}_{\lambda=\infty}
{\qopname\relax o{tr}}\hat{\Psi}^{\infty}(\lambda)^{-1}
\frac{\partial}{\partial \lambda}
\hat{\Psi}^{\infty}(\lambda)
d T^{\infty} (\lambda).$$ In order to compute $\omega_{\infty},$ we set $$A(\lambda)
=
(G^{\infty})^{-1}
\left(
\begin{array}{cc}
\hat{\varphi}(P) \exp \hat{\Pi} (P)
&
\hat{\varphi}(P^*) \exp \hat{\Pi}(P^*) \\
\hat{\psi} (P) \exp \hat{\Pi} (P)
& \hat{\psi}(P^*) \exp \hat{\Pi}(P^*)
\end{array}
\right).$$ Therefore, we get $$-
{\qopname\relax o{tr}}\hat{\Psi}^{\infty}(\lambda)^{-1}
\frac{\partial}{\partial \lambda}
\hat{\Psi}^{\infty} (\lambda)
d T^{\infty} (\lambda)
=
\frac
{d t}
{2}
\lambda
{\qopname\relax o{tr}}A^{-1}(\lambda) A^{\prime}(\lambda)
\left(
\begin{array}{cc}
1 & \\
& -1
\end{array}
\right),$$ and $$\begin{aligned}
\label{eqn:trace1}
&
{\qopname\relax o{tr}}A^{-1}(\lambda) A^{\prime}(\lambda)
\left(
\begin{array}{cc}
1 & \\
& -1
\end{array}
\right) \notag \\
&=
\frac{1}{\det \Phi (P)}
\Big[
\det
\left(
\begin{array}{cc}
\{
\hat{\varphi} (P) \exp \hat{\Pi}(P)
\}^{\prime}
& \hat{\varphi}(P^*) \exp \hat{\Pi}(P^*) \notag \\
\{
\hat{\psi} (P) \exp \hat{\Pi}(P)
\}^{\prime}
&
\hat{\psi}(P^*) \exp \hat{\Pi}(P^*)
\end{array}
\right) \\
&
\hspace{40mm}
-
\det
\left(
\begin{array}{cc}
\hat{\varphi} (P) \exp \hat{\Pi}(P)
&
\{
\hat{\varphi}(P^*) \exp \hat{\Pi}(P^*)
\}^{\prime} \\
\hat{\psi} (P) \exp \hat{\Pi} (P)
&
\{ \hat{\psi}(P^*) \exp \hat{\Pi} (P^*) \}^{\prime}
\end{array}
\right)
\Big],\end{aligned}$$ where $\prime$ means the differentiation with respect to the variable $\lambda$.
We have normalized the matrix function $\Psi(\lambda)$ around $\lambda=\infty$ in Lemma \[lem:ya\] and have proved that the monodromy data of $\Psi(\lambda)$ are independent of $P_{\varphi}, P_{\psi}$ in Theorem \[thm:mono\] and its corollary. Therefore, we can choose the parameters, $P_{\varphi}, P_{\psi}$ at our disposal to simplify the calculation.
Firstly, we multiply both the numerators and the denominators of (\[eqn:trace1\]) by $\displaystyle\frac{1}{\lambda_{\psi}-\lambda_{\varphi}}$. Then, we take the limit $P_{\psi} \rightarrow P_{\varphi}$ and get $$\hat{\psi}(P)
=
\frac
{
\partial \hat{\varphi}(P)
}
{\partial \lambda_{\varphi}}.$$ Next, we multiply both the numerators and the denominators of (\[eqn:trace1\]) by $\displaystyle\frac{1}{\lambda_{\varphi}-\lambda}$. Then, we take the limit $
P_{\varphi} \rightarrow P
$ and obtain $$\begin{aligned}
{\qopname\relax o{tr}}A^{-1}(\lambda) A^{\prime}(\lambda)
\left(
\begin{array}{cc}
1 & \\
& -1
\end{array}
\right)
&=
2
\frac{1}{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\lim_{P_{\varphi} \rightarrow P}
\frac{\partial}{\partial x_{\varphi}}
\Theta[{\mathbf{p}},{\mathbf{q}}](-U(P) + U(P_{\varphi}) + \mathbf{v}(t)) \notag \\
&
\hspace{50mm}
+
2
\frac{\partial}{\partial \lambda}
\{
\hat{\Pi} (P)
\}.\end{aligned}$$ From the definition of $\omega_{\infty},$ it follows that $$\omega_{\infty} =
\frac{dt}{\Theta [{\mathbf{p}},{\mathbf{q}}] (\mathbf{v}(t))}
\sum_{k=1}^g
\frac{\partial}{\partial z_k}
\{\Theta [{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t)) \}
\frac{dU_k}{dx_{\infty^1}}(\infty^1)
+
\frac{t}{2}dt
\left(
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right).$$
Fay’s Identities and Rauch’s Variational Formulas
-------------------------------------------------
In this subsection, we quote Fay’s identities and Rauch’s variational formulas in order to determine the Hamiltonians $H_j\,(j=1,2,\ldots,2g+2).$ Lemma \[lem:Fay2\] and \[lem:Rauch\] are devoted to Fay’s Identities and Rauch’s Variational Formulas, respectively.
\[lem:Fay2\] [*(1) For $P,Q \in \mathcal{L},$ $$\frac{
\Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ]^2(U(P)-U(Q))}
{
\Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ]^2(0)
E^2(P,Q)
}
=W(P,Q)+
\sum_{k,l=1}^g
\frac{\partial^2}{\partial z_k \partial z_l}
\log \Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ](0)
dU_k(P) dU_l(Q).$$ (2) For $P,Q \in \mathcal{L},$ $$\begin{aligned}
&
\frac{\Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ](2(U(P)-U(Q)))}
{
\Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ](0)
E^4(P,Q)
}
-
\frac{\Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ]^4(U(P)-U(Q))}
{
\Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ]^4(0)
E^4(P,Q)
} \\
&
\hspace{30mm}
=
\frac12
\sum_{k,l,m,n=1}^g
\frac{\partial^4}{\partial z_k \partial z_l \partial z_m \partial z_n}
\log \Theta [\mathbf{p}^{T}, \mathbf{q}^{T} ](0)
dU_k(P) dU_l(P)dU_m(Q) dU_n(Q).\end{aligned}$$ (3) For $P,Q\in\mathcal{L},$ $$\begin{aligned}
\frac
{\Theta[\mathbf{p}^T, \mathbf{q}^T](0)\Theta[\mathbf{p},\mathbf{q}]\left(2\left(U(P)-U(Q)\right)\right)}
{\Theta[\mathbf{p}^T,\mathbf{q}^T]\left(U(P)-(Q)\right)^2E(P,Q)^2}
&=
W(P,Q) \\
&\quad\quad
+
\sum_{k,l=1}^g
\frac{\partial^2}{\partial z_k \partial z_l}
\log \Theta[\mathbf{p}^T,\mathbf{q}^T]\left(U(P)-U(Q)\right)dU_k(P)dU_l(Q) \\
&=
dx_{P}dx_{Q}
\log
\frac{E(P,Q)}{\Theta[\mathbf{p}^T,\mathbf{q}^T]\left(U(P)-U(Q)\right)},\end{aligned}$$ where $x_P,x_Q$ are local coordinates of $P,Q,$ respectively. Especially, since $\mathcal{L}$ is hyperelliptic, it follows that $$\frac
{\Theta[\mathbf{p}^T, \mathbf{q}^T](0)\Theta[\mathbf{p},\mathbf{q}]\left(2\left(U(P)-U(Q)\right)\right)}
{\Theta[\mathbf{p}^T,\mathbf{q}^T]\left(U(P)-(Q)\right)^2E(P,Q)^2}
=
dx_{P} dx_{Q}\log \frac{1}{m_T(P,Q)}.$$* ]{}
For (1), (2) and (3), see pp 26, pp 28 and pp 29 in Fay’s book [@Fay], respectively.
Rauch [@Rauch] described the dependence of $dU_k\,(k=1,2,\ldots,g)$ and ${\mathbf{B}}_{kl}\,(k,l=1,2,\ldots,g)$ on the moduli of the Riemann surfaces. The moduli space of hyperelliptic curves can be parameterized by the positions of the branch points $\lambda_j \, (j=1,2,\ldots,2g+2)$. Korotkin [@Korotkin] proved the variational formulas of the following useful form.
\[lem:Rauch\] [*(1) For $P\in \mathcal{L},$ $$\label{eqn:Rauch1}
\frac{\partial}{\partial \lambda_j}
\left\{
\frac{dU_k}{dx_{P}}(P)
\right\}
=
\frac12 \frac{W(P,\lambda_j)}{dx_{P} d x_j}
\frac{dU_k}{dx_j}(\lambda_j),$$ where $x_{P}$ is a local coordinate of $P$ and $x_j=\sqrt{\lambda-\lambda_j},$ which is a local coordinate of the branch point $\lambda_j$ for any $j=1,2,\ldots, 2g+2.$ (2) For the branch points $\lambda_j \,\,(j=1,2,\ldots, 2g+2),$ $$\label{eqn:Rauch2}
\frac{\partial {\mathbf{B}}_{kl}}{\partial \lambda_j}
=
\pi i
\frac{dU_l}{dx_j}(\lambda_j)
\frac{dU_k}{dx_j}(\lambda_j) \quad (k,l=1,2,\ldots, g),$$ where $x_j=\sqrt{\lambda-\lambda_j},$ which is a local coordinate of $\lambda_j.$* ]{}
The $\tau$-Function
-------------------
In this subsection, we compute $\omega_{\lambda_j} (j=1,2,\ldots,2g+2)$ and calculate the $\tau$-function.
\[lem:omeganu\] For $j=1,2,\ldots,2g+2,$ $$\begin{aligned}
H_j
&=
\frac{\partial}{\partial \lambda_j}
\log \Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t) | {\mathbf{B}})
-\frac12
\frac{\partial}{\partial \lambda_j}
\log \det \mathcal{A}
-\frac18
\frac{\partial}{\partial \lambda_j}
\log
\prod_{k<l}
(\lambda_k-\lambda_l) \\
&
\hspace{60mm}
+
\frac{\partial}{\partial \lambda_j}
\left\{
\frac{t^2}{4}\left(\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}\right)
\right\}.\end{aligned}$$
By direct calculation, we obtain $$\frac12
{\qopname\relax o{tr}}\left( \Psi^{\prime}(\lambda)\Psi^{-1}(\lambda) \right)^2
=
-
\frac{\det \left( \Phi_x \right)}{\det \Phi}
+
\frac14
\left(
\frac
{
(\det \Phi)_{\lambda}
}
{\det \Phi}
\right)^2.$$ We calculate $\omega_{\lambda_j}$ in the same way as $\omega_{\infty}$ in Proposition \[prop:omegaa\].
We multiply both the numerators and the denominators of $
\displaystyle
\frac{\det \left( \Phi_{\lambda} \right)}{\det \Phi},
\frac
{
(\det \Phi)_{\lambda}
}
{\det \Phi}
$ by $\displaystyle\frac{1}{\lambda_{\varphi}-\lambda_{\psi}}$. Then, we take the limit $P_{\psi} \rightarrow P_{\varphi}$ and get $$\psi(P)
=
\frac{\partial \varphi(P)}{\partial \lambda_{\varphi}}.$$ Furthermore, we multiply both the numerators and the denominators of $
\displaystyle
\frac{\det \left( \Phi_{\lambda} \right)}{\det \Phi},
\frac
{
(\det \Phi)_{\lambda}
}
{\det \Phi}
$ by $\displaystyle\frac{1}{\lambda_{\varphi}-\lambda}$ and take the limit $P_{\varphi} \rightarrow P$. Then, we obtain $$\begin{aligned}
\frac
{
(\det \Phi)_{\lambda}
}
{\det \Phi}
&=
2
\frac{\partial}{\partial \lambda}
\log \Theta [{\mathbf{p}}^S,{\mathbf{q}}^S](-2U(P)),
\\
\frac{\det \left( \Phi_{\lambda} \right)}{\det \Phi}
&=
\frac{1}{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\lim_{P_{\varphi} \rightarrow P}
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\Theta[{\mathbf{p}},{\mathbf{q}}](-U(P)+U(P_{\varphi})+\mathbf{v}(t)) \\
&+
\frac{2}{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\lim_{P_{\varphi} \rightarrow P}
\frac{\partial }{\partial \lambda}
\Theta[{\mathbf{p}},{\mathbf{q}}] (-U(P)+U(P_{\varphi})+\mathbf{v}(t))
\frac{\partial }{\partial \lambda} \Pi (P) \\
&+
\frac{1}{\Theta[{\mathbf{p}}^S,{\mathbf{q}}^S](-2U(P))}
\lim_{P_{\varphi} \rightarrow P}
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\Theta [{\mathbf{p}}^S,{\mathbf{q}}^S] (-U(P)-U(P_{\varphi})) \\
&-
\left(
\frac{\partial}{\partial \lambda} \Pi (P)
\right)^2,\end{aligned}$$ which implies that $$\begin{aligned}
&
\frac12
{\qopname\relax o{tr}}\left( \Psi^{\prime}(\lambda)\Psi^{-1}(\lambda) \right)^2
=
-
\left.
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\log \Theta [{\mathbf{p}}^S,{\mathbf{q}}^S] (-U(P) -U(P_{\varphi}))
\right|_{P_{\varphi}=P} \\
&
\hspace{25mm}
-
\frac{1}{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\left.
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\Theta[{\mathbf{p}},{\mathbf{q}}] (-U(P) + U(P_{\varphi}) +\mathbf{v}(t))
\right|_{P_{\varphi}=P} \\
&
\hspace{25mm}
-
\left.
\frac{2}{\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\frac{\partial}{\partial \lambda} \Theta[{\mathbf{p}},{\mathbf{q}}](-U(P)+U(P_{\varphi})+\mathbf{v}(t))
\frac{\partial}{\partial \lambda}
\Pi (P)
\right|_{P_{\varphi=P}} \\
&
\hspace{25mm}
+
\left(
\frac{\partial}{\partial \lambda}
\Pi (P)
\right)^2.\end{aligned}$$
Firstly, we calculate the residue of (5.12) at $\lambda=\lambda_j$. For the purpose, we set $$P=\lambda_j, \,x_{P}=x_j:=\sqrt{\lambda-\lambda_j}$$ in (\[eqn:proj2\]) of Lemma \[lem:Fay1\]. Then, we have
$$\begin{aligned}
&
{\textrm{Res}}_{\lambda=\lambda_1}
\left\{
\left.
-
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\log \Theta [{\mathbf{p}}^S,{\mathbf{q}}^S] (-U(P) -U(P_{\varphi}))
\right|_{P_{\varphi}=P}
\right\}
\\
&
\hspace{20mm}
=
\frac18 \sum_{k\neq j} \frac{n_j n_k}{\lambda_j-\lambda_k}
-\frac{1}{4 \Theta [{\mathbf{p}}^T,{\mathbf{q}}^T]}(0)
\sum_{l,k=1}^g \frac{\partial^2 \Theta[{\mathbf{p}}^T,{\mathbf{q}}^T]}{\partial z_l \partial z_k}(0)
\frac{dU_l}{dx_j}(\lambda_j) \frac{dU_k}{dx_j}(\lambda_j),\end{aligned}$$
where $n_k=1$ for $\lambda_k \in T$ and $n_k =- 1$ for $\lambda_k \notin T.$ For the calculation, we use (\[eqn:Rauch2\]) in Lemma \[lem:Rauch\] and the heat equation $$\label{eqn:heat}
\frac{\partial^2 \Theta[{\mathbf{p}},{\mathbf{q}}]({\mathbf{z}}|{\mathbf{B}})}{\partial z_l \partial z_k}
=
4 \pi i
\frac{\partial \Theta[{\mathbf{p}},{\mathbf{q}}]({\mathbf{z}}|{\mathbf{B}})}{\partial {\mathbf{B}}_{lk}}.$$ Then, we get $$\begin{aligned}
&
{\textrm{Res}}_{\lambda=\lambda_j}
\left\{
-
\left.
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\log \Theta [{\mathbf{p}}^S,{\mathbf{q}}^S] (-U(P) -U(P_{\varphi}))
\right|_{P_{\varphi}=P}
\right\} \\
&
\hspace{65mm}
=
\frac18 \sum_{k \neq j}\frac{n_j n_k}{\lambda_j-\lambda_k}
-
\frac{\partial}{\partial \lambda_j}
\log
\Theta [{\mathbf{p}}^T,{\mathbf{q}}^T](0 |{\mathbf{B}}).\end{aligned}$$ By using the Thomae’s formula $$\Theta^4[{\mathbf{p}}^T,{\mathbf{q}}^T](0)
=
\pm
\frac{(\det \mathcal{A})^2}{(2 \pi i)^{2g}}
\prod_{l<k,\,l,k=1}^{g+1} (\lambda_{i_l}-\lambda_{i_k})
\prod_{l<k,\,l,k=1}^{g+1} (\lambda_{j_l}-\lambda_{j_k}),$$ we get $$\begin{aligned}
&
{\textrm{Res}}_{\lambda=\lambda_j}
\left\{
-
\left.
\frac{\partial^2}{\partial \lambda \partial \lambda_{\varphi}}
\log \Theta [{\mathbf{p}}^S,{\mathbf{q}}^S] (-U(P) -U(P_{\varphi}))
\right|_{P_{\varphi}=P}
\right\} \\
&
\hspace{60mm}
=
-\frac12
\frac{\partial}{\partial \lambda_j}
\log \det \mathcal{A}
-\frac18
\frac{\partial}{\partial \lambda_j}
\log
\prod_{k<l}
(\lambda_k-\lambda_l). \end{aligned}$$
Secondly, we calculate the residue of the sum of (5.13) and (5.14) at $\lambda=\lambda_j,$ which is $$\begin{aligned}
\label{eqn:secandthir}
&
\frac{1}{4\Theta[{\mathbf{p}},{\mathbf{q}}]( \mathbf{v}(t) )}
\sum_{k,l=1}^g
\frac{\partial^2 \Theta[{\mathbf{p}},{\mathbf{q}}]}{\partial z_k \partial z_l}
(\mathbf{v}(t))
\frac{dU_k}{dx_j}(\lambda_j) \frac{dU_l}{dx_j}(\lambda_j) \notag \\
&
\hspace{40mm}
+
\frac{1}{2\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t))}
\sum_{k=1}^g
\frac{\partial \Theta[{\mathbf{p}},{\mathbf{q}}]}{\partial z_k}
(\mathbf{v}(t))
\times t
\frac{W(\lambda_1,\infty^1)}{dx_jdx_{\infty^1}}. \end{aligned}$$ From Lemma \[lem:Rauch\], it follows that (\[eqn:secandthir\]) is $$\frac{\partial}{\partial \lambda_j}
\log
\Theta[{\mathbf{p}},{\mathbf{q}}](\mathbf{v}(t)|{\mathbf{B}}).$$
Lastly, we deal with the residue of (5.15) at $\lambda=\lambda_j,$ which is $\displaystyle\frac{t^2}{4}\left(\frac{W(\lambda_j,\infty^1)}{dx_jdx_{\infty^1}}\right)^2.$ We prove that if $\lambda_j \in T,$ $$\frac{\partial}{\partial \lambda_j}
\left\{
\frac{t^2}{4}
\left(
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right)
\right\}
=
\frac{t^2}{4}
\left(
\frac{
W(\lambda_j,\infty^1)
}{dx_jdx_{\infty^1}}
\right)^2.$$ If $\lambda_j \not\in T,$ this formula can be proved in the same way.
From Lemma \[lem:Fay1\] and Lemma \[lem:Fay2\] (1), it follows that $$\begin{aligned}
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1} dx_{\infty^2}}
&=
\frac{1}{16}
\left(
\sum_{k=1}^{g+1}
\lambda_{i_k}-\sum_{k=1}^{g+1}\lambda_{j_k}
\right)^2
-
\frac
{
\Theta[{\mathbf{p}}^T,{\mathbf{q}}^T]^2(U(\infty^1)-U(\infty^2))
}
{
\Theta[{\mathbf{p}}^T,{\mathbf{q}}^T]^2(0)
\{E(\infty^1,\infty^2)\sqrt{dx_{\infty^1}}\sqrt{dx_{\infty^2}}\}^2
} \\
&\hspace{30mm}
-2
\sum_{k,l=1}^g
\frac{\partial^2}{\partial z_k \partial z_l}
\log \Theta [{\mathbf{p}}^T,{\mathbf{q}}^T](0)
\frac{dU_k}{dx_{\infty^1}}(\infty^1)
\frac{dU_l}{dx_{\infty^1}}(\infty^1) \\
&=
\frac18\left(\sum_{k=1}^{g+1} \lambda_{i_k}-\sum_{k=1}^{g+1}\lambda_{j_k}\right)^2 \\
&\hspace{30mm}
-2
\sum_{i,j=1}^g
\frac{\partial^2}{\partial z_i \partial z_j}
\log \Theta [{\mathbf{p}}^T,{\mathbf{q}}^T](0)
\frac{dU_i}{dx_{\infty^1}}(\infty^1)
\frac{dU_j}{dx_{\infty^1}}(\infty^1), \end{aligned}$$ where we have used $$\frac
{
\Theta[{\mathbf{p}}^T,{\mathbf{q}}^T](U(\infty^1)-U(\infty^2))
}
{
\Theta[{\mathbf{p}}^T,{\mathbf{q}}^T](0)
E(\infty^1,\infty^2)
}
=
m_T(\infty^1,\infty^2)
=
\frac{\sqrt{-1}}{4}
\left(\sum_{k=1}^{g+1} \lambda_{i_k}-\sum_{k=1}^{g+1}\lambda_{j_k}\right)\sqrt{dx_{\infty^1}}\sqrt{dx_{\infty^2}}.$$ Then, from Lemma \[lem:Rauch\], it follows that $$\begin{aligned}
&\frac{\partial}{\partial \lambda_j}
\left\{
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right\} \\
&=
\frac14\left(\sum_{k=1}^{g+1} \lambda_{i_k}-\sum_{k=1}^{g+1}\lambda_{j_k}\right) \\
&
-
\frac12
\sum_{k,l,m,n=1}^g
\frac{\partial^4}{\partial z_k \partial z_l \partial z_m \partial z_n}
\log \Theta[\mathbf{p}^T,\mathbf{q}^T](0)
\frac{dU_k}{dx_{\infty^1}}(\infty^1)\frac{dU_l}{dx_{\infty^1}}(\infty^1)\frac{dU_m}{dx_{\infty^1}}(\infty^1)\frac{dU_n}{dx_{\infty^1}}(\infty^1) \\
&
-
\left(
\sum_{k,l=1}^g \frac{\partial^2}{\partial z_k\partial z_l}
\log \Theta[\mathbf{p}^T,\mathbf{q}^T](0)\frac{dU_k}{dx_j}(\lambda_j)\frac{dU_l}{dx_{\infty^1}}(\infty^1)
\right)^2 \\
&
-2
\left(
\sum_{k,l=1}^g \frac{\partial^2}{\partial z_k\partial z_l}
\log \Theta[\mathbf{p}^T,\mathbf{q}^T](0)\frac{dU_k}{dx_j}(\lambda_j)\frac{dU_l}{dx_{\infty^1}}(\infty^1)
\right)
\frac{W(\lambda_j,\infty^1)}{dx_jdx_{\infty^1}}.\end{aligned}$$ Thus, from Lemma \[lem:Fay2\] (1) and (2), we have $$\begin{aligned}
\frac{\partial}{\partial \lambda_1}
\left\{
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right\}
&=
\frac14\left(\sum_{k=1}^{g+1} \lambda_{i_k}-\sum_{k=1}^{g+1}\lambda_{j_k}\right)
-
\frac{\Theta[\mathbf{p}^T,\mathbf{q}^T]\left(2\left(U(\infty^1)-U(\lambda_j)\right)\right)}
{\Theta[\mathbf{p}^T,\mathbf{q}^T](0)\{E(\lambda_j,\infty^1)\sqrt{dx_j}\sqrt{dx_{\infty^1}}\}^4} \\
&+\left(\frac{W(\lambda_j,\infty^1)}{dx_jdx_{\infty^1}}\right)^2.\end{aligned}$$ Therefore, from Lemma \[lem:Fay2\] (3), we obtain $$\frac{\partial}{\partial \lambda_j}
\left\{
\frac16 S(\infty^1)-\frac{W(\infty^1,\infty^2)}{dx_{\infty^1}dx_{\infty^2}}
\right\}
=
\left(
\frac{W(\lambda_j,\infty^1)}{dx_jdx_{\infty^1}}\right)^2,$$ which completes the proof of the proposition.
Since we have calculated the Hamiltonians $H_t, H_1, H_2, \ldots, H_{2g+2}$, we finally obtain Theorem 0.1.
[5]{} P. Deift., A. Its., A., Kapaev. and X. Zhou., [*On the algebro-geometric integration of the Schlesinger equations and on elliptic solutions of the Painlevé 6 equation, Commun. Math. Phys. [**203**]{}.* ]{} (1999) 613-633. J. D. Fay., [*Theta-functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. [**352**]{}, Springer-Verlag, Berlin-New York,* ]{} (1973). H. Farkas., I. Kra., [ *Riemann Surfaces, 2nd ed, Springer.*]{} K. Matsuda., [ *Isomonodromic deformation with an irregular singularity and the elliptic theta functions, J. Phys. A: Math. Theor. [**40**]{}.* ]{} (2007) 11939-11960. M. Jimbo, T. Miwa and K. Ueno, [ *Monodromy preserving deformations of linear differential equations with rational coefficients, Physica* ]{} [**1D**]{} (1980) 80-158. M. Jimbo and T. Miwa, [*Monodromy preserving deformations of linear differential equations with rational coefficients II, Physica* ]{} [**2D**]{} (1981) 407-448. A. V. Kitaev and D. A. Korotkin, [*On solutions of the Schlesinger equations in terms of $\Theta$ functions, Intern. Math. Research Notices* ]{} [**17**]{} (1998) 877-905. D. A. Korotkin, [*Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann.* ]{} [**329,**]{} no. 2, (2004), 335–364. D. Mumford, [*Tata Lectures on Theta I, Progress in Math, v 28, Birkhauser* ]{} (1983). H. E. Rauch., [*Weierstrass points, branch points and moduli of Riemann Surfaces, Comm. Pure. Appl. Math.*]{} [**12**]{} (1959) 543-560. L. Schlesinger., [*Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten, J. Reine u. Angew. Math.* ]{} [**141**]{} (1912) 96-145.
The author thanks Professor Yousuke Ohyama for the careful guidance and the referees for useful comments.
|
---
abstract: 'We study the dynamics of a one-dimensional fluid of orientable hard rectangles with a non-coarse-grained microscopic mechanism of facilitation. The length occupied by a rectangle depends on its orientation, which is coupled to an external field. The equilibrium properties of our model are essentially those of the Tonks gas, but at high densities, the orientational degrees of freedom become effectively frozen due to jamming. This is a simple analytically tractable model of glassy phase. Under a cyclic variation of the pressure, hysteresis is observed. Following a pressure quench, the orientational persistence exhibits a two-stage decay characteristic of glassy systems.'
author:
- 'Jeferson J. Arenzon'
- Deepak Dhar
- Ronald Dickman
title: Glassy dynamics and hysteresis in a linear system of orientable hard rods
---
Introduction
============
Almost all liquids, if cooled sufficiently fast, form a glassy structure; much effort has been directed towards understanding this phenomenon. Monte Carlo simulations have shown that systems with purely hard-core interactions can describe qualitatively much of the observed phenomenology of the glass transition, e.g., the very fast rise of relaxation times, and the absence of an associated latent heat. Theoretical analysis of such models is hampered by an incomplete understanding of the equilibrium (fluid-solid) phase transition. The best studied hard-core model is a system of hard spheres, which has experimental realizations in colloidal, granular and other systems [@PuMe86; @ArTs06; @PrSeWe07]. In many cases size dispersion, or other built-in complexity, is introduced in order to avoid crystallization and thereby observe a glass transition [@SaKr00; @PaZa05; @PaZa06; @Zamponi07b]. For monodisperse systems, there is no transition in one dimension [@Tonks36], while two and three dimensional systems are highly prone to crystallize. In higher dimensions [@SkDoStTo06; @MeFrCh09; @ZaVaSaPoCaPu09; @ChIkMeMi10], nucleation rates are low and the glass state is more easily attained. Indeed, in the limit of very high spatial dimensionality, there seems to be an ideal glass transition [@PaZa06; @ScSc10], although how far this extends down to lower dimensions is still debated [@Tarzia07].
Hard core potentials need not be spherically symmetric. On a lattice, for example, where the symmetry is discrete, the behavior strongly depends on the dimension of the system, the lattice structure, and the exclusion range (see [@Panagiotopoulos05; @FeArLe07] and refs. therein).
Here we study the dynamic properties of a one-dimensional system of classical hard rectangles, with only two orientations allowed, horizontal and vertical. These will be called “rods" in what follows. Classical linear fluids have been extensively studied over the last decades [@Tonks36; @Gursey50; @Baxter65; @LiMa66; @CaRu69; @Bloomfield70; @Percus76; @Percus82; @Marko89; @Davis90; @DuHu03; @FeLeAr07; @TrPa08]; the case of elongated rods with orientational freedom has also been analyzed [@CaRu69; @FrRu73; @FuLo79; @BeKr06; @KaKa09a; @KaKa09b; @GuVa10]. While the equilibrium properties of this class of models are well understood, their dynamic properties, in particular those related to the glass transition, have received somewhat limited attention [@Bowles00; @StLuMe02; @StDe02; @BoSa06; @DhLe10]. The simple, one-dimensional model considered here, reproduces (at least in part) the glassy phenomenology with an explicit (non-coarse-grained) microscopic mechanism of facilitation. In analogy with closely related models in which the constraints are kinetic instead of geometric [@FrAn84; @KoAn93; @RiSo03], due to a spatial and temporal coarse-graining, here there is no thermodynamic transition and the only nontrivial behavior is kinetic. In the present context this represents an advantage, in that the relaxation is not complicated by critical slowing down or metastability.
At short time scales, the glassy phase can be modelled as a metastable phase, in restricted thermal equilibrium. An exact calculation of the properties of such metastable states, including the equation of state, has been made recently for a toy model [@Dhar02; @DhLe10]. In these papers, the ergodicity is explicitly broken, and one assumes that in the glassy phase some transition rates are exactly zero. The simple model considered here provides an extension of the treatment in Ref. [@DhLe10] to include the description of slow evolution of the macroscopic glassy state at longer time scales. In our model, ergodicity is not explicitly broken, and the dynamics is capable of bringing the system to equilibrium, but at high pressures, the relaxation of orientations becomes so slow that these variables are effectively [*frozen*]{} over any reasonable time scale. Under steady increase of pressure, they never fully relax. Macroscopic properties in the frozen regime are found to depend on history, in particular, on the rate at which the pressure is increased.
The structure of the paper is as follows. The next section introduces the model while Sec. \[sec.equilibrium\] reviews its equilibrium properties. We describe the dynamic behavior in section \[sec.dynamics\]. Sec. \[sec.conclusions\] contains some concluding remarks.
Model {#sec.model}
=====
We consider a system of $N$ rigid rods of length $\sigma \geq 1$ and unit width on a line of length $L$. Each rod is described by the variables $(x_i,S_i)$, where $x_i\in [0,L)$ is the position of the center of mass and $S_i$ denotes its orientation ($0$ for horizontal, 1 for vertical). Rods in state $S_i=0$ occupy a length $\sigma$ and those in state $S_i=1$ occupy a unit length (see Fig. \[fig.rods3\]). Since the order of the particles cannot change, we take $x_1 < x_2 < \ldots < x_N$. For convenience, we define $x_0 \equiv 0$ and $x_{N+1}\equiv L$; there are no orientational degrees of freedom associated with these variables. We define inter-rod distances, $y_i = x_{i+1} - x_i$, so that $\sum_{i=0}^N y_i = L$. When all rods are constrained to have the same orientation or, equivalently, $\sigma=1$, the original Tonks gas [@Tonks36] is recovered. Without this restriction, the model is analogous [@Onsager49] to a binary mixture of rods (with conservation of the total particle number, but not the number of each species separately). The particles are subject to an external field $h'$, coupled to the orientations $\{S_i\}$, so that the potential energy takes the form $${\cal H} =
\sum_i \phi_{\scriptstyle\rm hc}(y_i) + h'\sum_i S_i,$$ where the first term denotes hard core interactions and the sums extend over the $N$ rods; for $h'>0$ the horizontal orientation is favored.
The hard-core interaction between the $i$th rod and its right neighbor is $$\phi_{\scriptstyle\rm hc}(y_i) =
\begin{cases}
0, & y_i\geq a_i \\
\infty, & y_i< a_i,
\end{cases}$$ where $a_i$ is the minimum distance between the centers of rods $i$ and $i+1$, given by $$a_i = \sigma + \frac{1}{2}(S_i+S_{i+1})(1-\sigma).
\label{mindist}$$ An important quantity is the mean free volume per particle, $v_f= v-\sigma+(\sigma-1)m$, where $v=\rho^{-1}= L/N$ and $m$ is the fraction of rods in the vertical orientation, or “magnetization”.
![Example of a rod configuration.[]{data-label="fig.rods3"}](rods3.eps){width="6cm"}
We define a local, continuous-time, stochastic dynamics for the positions (or equivalently, the separations $y_i$), and orientations. The time evolution is Markovian, and satisfies the detailed balance condition. Each rod executes unbiased diffusive motion. The wall at $x_{N+1} = L$ also undergoes diffusive motion, but is subject to a bias, with ratio of jump rate that increase $L$ by $\delta L$ to the backward transition is $\exp( - p \delta L)$. The rods can also change their orientation, rotating about its center of mass. While the orientation varies continuously during a transition, we suppose the latter occurs so rapidly that orientations other than vertical or horizontal may be neglected. In addition, the dynamics includes an important constraint on orientational transitions: particle $i$ can only change its orientation if its neighbors to the left and right are sufficiently far away that its rotation (through 90$^{\rm o}$) is not blocked geometrically (see Fig. \[flip\]). In order for particle $i$ to change its orientation, it is necessary that $$y_i>\frac{\sigma+S_{i+1}(1-\sigma)+r}{2}
\label{eq.condition}$$ where $r\equiv \sqrt{1+\sigma^2}$ is the rod diagonal. The analogous relation for $y_{i-1}$ must be satisfied as well. Note that these conditions depend on the states of the neighboring rods ($i\pm 1$), but not on $S_i$ itself. Without the nonoverlapping constraint, $r$ is replaced by $\sigma$ in the above equation. Any violation of the excluded volume condition is rejected. These transitions are accepted in accordance with the Metropolis criterion, that is, with probability $\min [1,\exp(-\beta\Delta {\cal H})]$. One Monte Carlo step (MCS) consists in an attempt to update all degrees of freedom (i.e., all positions and orientations, and the volume).
![Geometrical constraint associated with an orientational transition. Analogous restrictions apply also for the other possible orientations.[]{data-label="flip"}](flip.eps){width="6cm"}
Our Monte Carlo simulations were performed on systems of $N=1000$ rods; and results represent averages over 100 (or more) independent realizations.
Equilibrium Properties {#sec.equilibrium}
======================
The equilibrium properties of the model are indeed very simple. The canonical configurational partition function is given by $$Z_N(h,L) =
\sum_{\{S_i\}} e^{-h \sum_i S_i}
\int_R dy_1 \cdots dy_N \label{Z}$$ where $h=\beta h'$ and the subscript $R$ denotes the restrictions $\sum_{i=1}^N y_i = L$ and $y_i \geq a_i$, where the minimum distances $a_i$ are defined in Eq. (\[mindist\]). The usual factor of $1/N!$ is absent due to the fixed order of the particles on the line. Introducing the variables $z_i = y_i -
a_i$, (with $z_0 \equiv x_1$ and $z_N \equiv L - x_N$), we have, $$\begin{aligned}
Z_N(h,L) &=&
\sum_{\{S_i\}} e^{-h \sum_i S_i}
\int_0 dz_0 \cdots \int_0 dz_N \;\nonumber \\ &&
\times \delta \left[ \sum_{i=0}^N z_i - L +
N\sigma - (\sigma-1)\sum_{i=1}^N S_i \right].
\label{Z1}\end{aligned}$$ We study the system in the constant-pressure ensemble; the partition function is $$%\begin{eqnarray}
Y_N(h,p) = \int dL e^{-p L} Z_N(h,L) ,
%\nonumber \\ &=&
%%\frac{1}{\Lambda^N}
%\sum_{\{S_i\}} \int_0^\infty \! dz_0 \cdots \int_0^\infty \! dz_{N-1} \nonumber \\
%&& \prod_i \exp \left\{-p \left[ z_i + \sigma -
%(\sigma-1+h)S_i \right] \right\}
%\end{eqnarray}$$ where $p$ denotes the pressure divided by $k_B T$. A simple calculation yields $$Y_N (h,p) = %\frac{e^{-p N \sigma}}{\Lambda^N p^N}(1+ \kappa)^N
e^{-p N \sigma} p^{-N}(1+ \kappa)^N$$ with $\kappa \equiv \exp [(\sigma-1)p - h]$. The Gibbs free energy per particle is given by $g = -(N \beta)^{-1}
\ln Y$, where, in the thermodynamic limit, $$\lim_{N \to \infty} \frac{1}{N} \ln Y_N = -\sigma p + \ln (1+\kappa)
-\ln p . %- \ln \Lambda .
\label{tlim}$$ In this limit the volume per particle is, $$v(p,h) = -\frac{1}{N} \frac{\partial \ln Y}{\partial p}
= \sigma - \frac{(\sigma -1) \kappa}{1+\kappa} + \frac{1}{p},
\label{volperpart}$$ while the fraction of vertical rods is, $$m_{eq}(p,h) = -\frac{1}{N} \frac{\partial \ln Y}{\partial h} =
\frac{\kappa}{1+\kappa}. \label{fracvert}$$ Eqs. (\[volperpart\]) and (\[fracvert\]) imply the relation $$%v = \frac{1}{p} + \sigma - (\sigma -1)m_{eq},
v = \frac{1}{p} + m_{eq} + \sigma (1-m_{eq}),
\label{vmp}$$ which implies that the free volume per particle is $v_f = 1/p$, as in the Tonks gas. It is worth noting than in equilibrium, the variables $\{y_i \}$ and $\{S_i \}$ are all mutually independent. The behavior of $m$ as a function of the pressure for several values of $\sigma$ and $h$ is illustrated in Fig. \[fig.mp\].
![Equilibrium fraction of vertical rods as a function of pressure $p$, for $h=0$ and 4, (empty and filled symbols, respectively). Points are simulation results, while the lines correspond to Eq. (\[fracvert\]). When $p$ is large enough, all rods are aligned vertically. []{data-label="fig.mp"}](mp.eps){width="8cm"}
Time-dependent Properties {#sec.dynamics}
=========================
In this section we study the kinetics of the magnetization $m$ and molecular volume $v$. We assume that the diffusive relaxation is much faster than the orientational relaxation. Then the displacement degrees of freedom may be assumed to be in thermal equilibrium. At high pressures, most update attempts for orientation change fail, as they are blocked. Suppose first that the orientations are fixed. In this case, the different sectors in the pico-canonical ensemble are specified by orientation of each rod. Different sectors with the same number of vertical rods are macroscopically equivalent. Within a sector, the displacement degrees of freedom are assumed to be in equilibrium. Then the translational dynamics will bring the system to a constrained equilibrium distribution, in which the distances $z_i$ are independent, exponentially distributed random variables with mean $1/p$, and the mean length of the system is $\langle L \rangle = N[1 + (1-m) \sigma] + (N+1)/p]$, where $m$ is the fraction of vertically oriented particles, not necessarily equal to the equilibrium value $m_{eq}$.
Now, allowing the orientations to fluctuate, an equation of motion for $m(t)$ can be derived if we assume that the translation dynamics is rapid, so that between any pair of successive orientational transitions, the interparticle distances attain the constrained equilibrium distribution mentioned above. Under this hypothesis, the particle orientations are mutually independent, as in equilibrium (we assume as well that the orientations are initially uncorrelated).
Consider, for example, a transition from $S_i=0$ to 1. This would be allowed only if the two gaps on the two sides of the rod are large enough. In the restricted equilibrium ensemble, separations are independent, exponentially distributed random variables, $P(y)=p\exp(-py)$, and the probability that a hole larger than $w=(r-\sigma)/2$ appears at one of its sides is $P(v_f>w)=\exp[-p(r-\sigma)/2]$. Given that such gaps must exist at both sides and that this flip is against the field, we see that the effective transition rate is proportional to $\exp[-p(r-\sigma)-h]$, and that it is independent of the state of the neighboring rods. Thus we may write the transition rate for $S_i=0$ to 1 as $$\gamma_+ = \gamma \exp[- p(r -\sigma) - h],
\label{gammaplus}$$ where $\gamma$ is an arbitrary attempt rate, independent of $p$ and $h$. On the other hand, if the transition is from 1 to 0, $w=(r-1)/2$ and $$\gamma_- = \gamma \exp[- p(r - 1)].
\label{gammamin}$$ Note that only $\gamma_+$ depends on $h$ and that these rates satisfy detailed balance. For small values of the pressure, free space is abundant and the slowest process is a flip against the field, so that the larger time scale involved is given by $\gamma_+^{-1}$. On the other hand, when the pressure is large enough, the production of large enough holes is the dominant slow process and the relevant characteristic time now scales as $\gamma_-^{-1}$.
The evolution of $m(t)$ is governed by $$\frac{dm}{dt} = \gamma_+ (1-m) - \gamma_- m \equiv - \Gamma m + \gamma_+ ,
\label{dmdt}$$ where $\Gamma = \gamma_+ + \gamma_-$. If the rates are time independent, then letting $\phi = m - m_{eq} = m - \gamma_+/\Gamma$, we have $$\frac{d\phi}{dt} = -\Gamma \phi,$$ showing an exponential approach to equilibrium.
The system is driven out of equilibrium if the pressure (or the external field, $h$) is time-dependent. Suppose that $\gamma_+$ and $\gamma_-$ depend on time through the pressure. Then we have $$m(t) = e^{-{\cal G}(t)}
\left( m_0 + \int_0^t ds \; \gamma_+ (s) \; e^{{\cal G}(s)} \right),
\label{mt}$$ where $${\cal G}(t) = \int_0^t dt' \, \Gamma(t').$$
A particularly interesting example is that of a system initially in equilibrium at pressure $p_0$, and subject to a pressure that increases linearly with time, $p(t) = p_0 + \lambda t$ for $t > 0$, where $\lambda$ is the annealing rate. In this case, $$\begin{aligned}
{\cal G}(t) = \frac{\gamma}{\lambda} &
\left[ e^{-h-p_0(r-\sigma)} \frac{1 - e^{-\lambda (r-\sigma)t}}{r-\sigma} \right.\nonumber\\
& \left. + e^{-p_0(r-1)} \frac{1 - e^{-\lambda (r-1)t}}{r-1} \right]\end{aligned}$$ so that $$\lim_{t \to \infty} {\cal G}(t) = \frac{\gamma}{\lambda}
\left[\frac{e^{-h-p_0(r-\sigma)}}{r-\sigma}
+ \frac{e^{-p_0(r-1)}}{r-1} \right],$$ which is finite for $\lambda > 0$. Inserting this result in Eq. (\[mt\]), we see that the initial magnetization $m_0$ is not “forgotten" even when $t \to \infty$. It is easily verified that memory of the initial magnetization persists for a pressure increase of the form $p(t) = p_0 + \lambda t^{\alpha}$, for any positive values of $\lambda$ and $\alpha$.
![Fraction $m$ of vertical rods versus pressure $p$ for $h=5$, $\sigma=2$, $\gamma=1$ and several values of the rate of pressure increase, $\lambda$. From top to bottom, $\lambda=0$ (equilibrium, bold line), $10^{-3}$, $10^{-2}$, and $10^{-1}$. Points: simulation; solid lines: theory, Eq. (\[mt\]).[]{data-label="p_ann_h5_lambda"}](p_ann_h5_lambda.eps){width="8cm"}
![Volume per molecule $v$ versus pressure $p$ for the same parameters as in Fig. \[p\_ann\_h5\_lambda\] during the pressure annealing. Points represent the simulation data while the solid lines are the improved theoretic predictions based on Eq. (\[dvfdt\]) (with $\gamma\simeq 0.3$). The lines for $\lambda=10^{-4}$ and $\lambda=0$ (equilibrium, bold line) are indistinguishable at this scale. Inset: free volume $v_F$. Analogous deviations from equilibrium are again seen for large values of $\lambda$.[]{data-label="p_ann_h5_lambda2"}](p_ann_h5_lambda2.eps){width="8cm"}
Examples of $m(t)$ (for pressure increasing linearly with time) are shown in Fig. \[p\_ann\_h5\_lambda\] for $p_0=1$, $h=5$, $\sigma = 2$, and $\gamma = 1$. For $\lambda = 10^{-4}$, the difference between $m(p)$ and the equilibrium result is small. For larger rates of pressure increase, on the other hand, there are marked differences between the final value of $m$ and the equilibrium result. Although $m(t)$ is well described by Eq. (\[mt\]), as can be seen by the excellent agreement with the simulation, the same does not occur for the molecular volume, $v(p)$, and the free volume, at larger rates of pressure increase $\lambda$. The reason is that in this case the rate of pressure increase is large enough so that we can no longer treat the interparticle distances as in instantaneous thermal equilibrium, on the time scale of the orientational relaxation. This effect can be incorporated in an approximate manner in the theory if we assume that the free volume per molecule follows a relaxational dynamics, that is, $$\frac{d v_F}{dt} = - \Gamma_v \left( v_F - \frac{1}{p} \right)
\label{dvfdt}$$ To evaluate the transition rates $\gamma_+$ and $\gamma_-$ we require the probability density $p(z)$. The simplest hypothesis is that $p(z)$ is exponential, as in equilibrium, but with the mean free volume $v_F$ in place of its equilibrium value, $1/p$, so that $p(z) = v_F^{-1}\exp(-z/v_F)$. Then the evolution of the magnetization is given by Eq. (\[dmdt\]), with transition rates as in Eqs. (\[gammaplus\]) and (\[gammamin\]), but with $p$ replaced by $1/v_F$. With an appropriate choice of the relaxation rate $\Gamma_v$, this simple theory yields reasonable agreement with simulation results at larger quench rates, as is shown in Fig. \[p\_ann\_h5\_lambda2\]. Some deviations between the theoretical prediction and simulations are evident at the highest quench rate, for smaller pressures; this is not surprising given the simplifications introduced.
In addition to the “freezing out” of the orientational degrees of freedom under a steady pressure quench, the system exhibits interesting hysteresis effects under an oscillatory pressure. Hysteresis loops obtained through numerical simulation are illustrated in Fig. \[histerese\], for various values of $\lambda = |dp/dt|$ in triangle-wave cycles of pressure variation. Similar results are obtained via numerical integration of Eq. (\[dmdt\]), assuming rapid equilibration of the free volume. Notice that for larger rates, the system describes a sequence of irreversible loops before entering a reversible one. For even larger rates than those shown in the figure, we find a greater number of irreversible loops, analogous to those obtained in compaction experiments of rods under vibration [@ViLaMuJa00].
![Fraction of vertical rods under cyclic variation of the pressure, showing hysteresis loops. For the highest rate, $\lambda=0.1$, the loop only closes after several cycles. Parameters: $h=5$, $\sigma=2$; the pressure varies between 1 and 20 at rates $\lambda$ as indicated.[]{data-label="histerese"}](histerese.eps){width="8cm"}
It is worth noting that an external field $h$ is not required to observe freezing. Even with $h=0$, the rapid reduction in the transition rate $\gamma_+$, Eq. (\[gammaplus\]), with increasing pressure ensures that the orientational degrees of freedom cannot equilibrate. Inhibition of orientational relaxation is greatest for $\sqrt{1 + \sigma^2} - \sigma$ as large as possible, i.e., for $\sigma$ tending to unity. (Of course this tends to reduce the excess of $v$ over its equilibrium value.) Thus, for $h=0$, $\sigma = 1.1$, and other parameters as above, one finds $m_\infty \equiv \lim_{t \to \infty} m(t) = 0.620$ if the pressure increases at a rate of $\lambda = 1$, and $m_\infty = 0.5409$ for $\lambda = 10$. (As $\lambda$ is increased, $m_\infty $ approaches the initial magnetization, equal to $1/2$ for $h=0$.)
![Persistence $P(t)$ as a function of time as the system pressure is quenched from 1 to $p$ at $t=0$, for $\sigma=2$ and $h=5$. When $p$ is small, the decay is exponential while for larger values, after an initial exponential decay, the persistence develops a plateau whose height is close to the initial magnetization $m_0$ and whose width increases with $p$. The solid lines are from Eq. (\[eq.per\]).[]{data-label="fig.quench_per"}](quench_per.eps){width="9cm"}
If instead of a smooth annealing, the system is suddenly quenched from low to high pressure, a two-step decay typical of glassy behavior is observed in the orientational persistence (the fraction of rods that did not flip since $t=0$). Fig. \[fig.quench\_per\] shows the results for several values of the final pressure after the system is quenched from an equilibrium state at $p_0=1$. Whatever the value of the final pressure, the initial decay of the persistence function is exponential and ruled by the against the field flip of rods that are horizontal at $t=0$, so that $P(t)\simeq \exp(-\gamma_+t)$ in this regime. As the final pressure is increased, the flip from 1 to 0 becomes slower and a plateau develops, after the initial fast decay, whose width increases with the final pressure. The height of this plateau is on the order of the initial magnetization, $m_0$. (Its precise value is slightly smaller than $m_0$ since some of the up rods will have already flipped during the initial fast regime). When the final pressure is large, it takes a long time for a rod to flip, since enough space must be freed, which requires a cooperative rearrangement of the neighboring rods. This happens with a rate proportional to $\gamma_-$. Thus, the data in Fig. \[fig.quench\_per\] are well fit by a sum of exponentials, corresponding to the fast and slow processes in the model: $$P(t) \simeq {\rm e}^{-\gamma_+t} + m_0{\rm e}^{-b\gamma_-t}
\label{eq.per}$$ where, for the parameters of Fig. \[fig.quench\_per\], $b\simeq 0.4$ and $m_0\simeq 0.01$. The coefficient of the second term, 0.01, is the height of the plateau and roughly corresponds to the initial magnetization. Thus, starting with an even smaller initial pressure, and thus a larger magnetization, the plateau can be tuned to higher values. The width of the plateau increases with the final value of the pressure and diverges as $p\to\infty$. This diverging relaxation time, $\gamma_-^{-1}$, is related to the increasing length of the cooperative region [@KaKa09b]. Notice that, at variance with other models for the glass transition, the slow relaxation is not associated with stretched exponentials.
Conclusions {#sec.conclusions}
===========
One-dimensional systems with short-range interactions, such as the model studied here, do not exhibit a phase transition at finite temperature and pressure in equilibrium. A jamming transition involving certain degrees of freedom may nevertheless occur [@KaKa09b], with a diverging length scale, as the control parameter (inverse pressure or temperature) goes to zero. Here we study a simple, geometric model on the line, for which analytical results for both the statics and the dynamics may be obtained and compared with numerical simulations, with excellent agreement. This system of hard rods is subject to geometric constraints that prohibit a rod changing its orientation if the distance from its neighbors is too small (i.e., in the absence of sufficient free volume). In our model, the time spent in transit between the two orientations is assumed to be small, and the orientation degree was taken to be a discrete variable. It is straightforward to extend the discussion to the case where we allow continuous orientations, although no qualitative differences are expected.
In the presence of an external field that disfavors the vertical position, our model may be seen as an in-layer description of a higher-dimensional system. In a dense system, the vertical position will be disfavored due to excluded-volume interactions with the rods in the neighboring layers. Thus the external field may be interpreted as an effective interaction to take into account the remaining dimensions.
Despite its simplicity, the model presents several properties characteristic of the glass transition, such as annealing rate dependence and two-step relaxation. The dynamical behavior can be understood in terms of the two microscopic reorientation processes involved.
The work of JJA and RD is partially supported by the Brazilian agencies FAPERGS, FAPEMIG, CAPES and CNPq. RD was partially supported by CNPq under project 490843/2007-7. DD was partially supported by the Department of Science and Technology, Government of India under the project DST/INT/Brazil/RPO-40/2007.
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---
abstract: 'We consider the defocusing nonlinear Schrödinger equation on $\mathbb T^2$ with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the uniqueness of the solution as limit of solutions to canonically truncated systems. The invariance of [ the]{} Gibbs measure under the global dynamics follows as a consequence.'
address:
- '$^1$ Department of Mathematics, University of Sourthern California, Los Angeles, CA 90089, USA '
- '$^2$ Department of Mathematics, University of Massachusetts, Amherst MA 01003'
- '$^3$Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA'
author:
- Yu Deng$^1$
- 'Andrea R. Nahmod$^2$'
- Haitian Yue$^3$
title: Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two
---
[^1]
[^2]
Introduction
============
In this paper we study the (defocusing) Wick ordered nonlinear Schrödinger equation on the torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$, $$\label{nls}
\left\{
\begin{split}(i\partial_t+\Delta)u&=W^{2r+1}(u),\\
u(0)&=u_{\mathrm{in}},\end{split}
\right.$$ where $r$ is a given positive integer, $W^{2r+1}$ is the Wick ordered power nonlinearity of degree $2r+1$, which will be defined below. We prove that, almost surely with respect to the associated Gibbs measure, the equation (\[nls\]) has a global *strong* solution, which is the unique limit of solutions to the canonical finite dimensional truncations. Moreover, this solution map keeps the Gibbs measure invariant.
For $r=1$ (cubic nonlinearity) this was proved by Bourgain [@Bourgain]; the results for $r\geq 2$ are new. We remark that in [@OT] Oh and Thomann constructed almost sure global weak solutions to (\[nls\]) with respect to the Gibbs measure, such that at any time the law of these random solutions is again given by the Gibbs measure. The main point of the current paper is the almost sure *uniqueness* of solution with respect to the Gibbs measure.
Setup and the main theorem
--------------------------
In this section we setup the problem and state our main theorem. For a review of the background and previous works, see Section \[previous\].
### Wick ordering and Gibbs measure {#wick}
We will fix a probability space $(\Omega,\mathcal{B},\mathbb{P})$, and a set of independent complex Gaussian random variables $\{g_k\}_{k\in\mathbb{Z}^2}$ defined on $\Omega$ that are normalized, i.e. $\mathbb{E}g_k=0$ and $\mathbb{E}|g_k|^2=1$, and the law of $g_k$ is rotationally symmetric.
Let $\mathcal{V}=\mathcal{S}'(\mathbb{T}^2)$ be the space of distributions on $\mathbb{T}^2$. We define the $\mathcal{V}$-valued random variable $$\label{random}f=f(\omega):\omega\mapsto\sum_{k\in\mathbb{Z}^2}\frac{g_k(\omega)}{\langle k\rangle}e^{ik\cdot x},\quad \omega\in\Omega.$$ Let $\mathrm{d}\rho$ be the [Wiener]{} measure on $\mathcal{V}$, defined for Borel sets $E\subset\mathcal{V}$ by $$\rho(E)=\mathbb{P}(f^{-1}(E)),$$ so $\mathrm{d}\rho$ is the law of the random variable $f$. This measure $\mathrm{d}\rho$ is a [countably additive Gaussian measure supported in $\cap_{\varepsilon>0}H^{-\varepsilon}(\mathbb{T}^2)$, which we henceforth denote by $H^{0-}(\mathbb{T}^2)$ (similarly $H^{s-}(\mathbb{T}^2)=\cap_{\varepsilon>0}H^{s-\varepsilon}(\mathbb{T}^2)$), but not in $L^2(\mathbb{T}^2)$ (see e.g. [@Bogachev])]{}. Define the spectral truncation $\Pi_N$ by $$\label{truncN}\mathcal{F}_x\Pi_Nu(k)=\mathbf{1}_{\langle k\rangle\leq N}\cdot\mathcal{F}_xu(k),$$ where $\mathcal{F}_x$ is the space Fourier transform, $\langle k\rangle=\sqrt{|k|^2+1}$ and $\mathbf{1}_P$ denotes the indicator function of a set or property $P$, and define the expectation of truncated mass, $$\label{expect}
\sigma_N:=\frac{1}{(2\pi)^2}\mathbb{E}\|\Pi_Nf(\omega)\|_{L^2}^2=\sum_{\langle k\rangle\leq N}\frac{1}{\langle k\rangle^2}\sim \log N.$$
For each $N$ and each $p\geq 0$, define the Wick ordered powers, $$\label{wickpoly}
\begin{aligned}
W_N^{2p}(u)&=\sum_{j=0}^p(-1)^{p-j}{p\choose j}\frac{\sigma_N^{p-j}p!}{j!}|u|^{2j},\\
W_N^{2p+1}(u)&=\sum_{j=0}^p(-1)^{p-j}{{p+1}\choose {p-j}}\frac{\sigma_N^{p-j}p!}{j!}|u|^{2j}u,
\end{aligned}$$ and the canonical finite dimensional truncations for (\[nls\]), $$\label{truncnls}
\left\{
\begin{split}(i\partial_t+\Delta)u_N&=\Pi_NW_N^{2r+1}(u_N),\\
u_N(0)&=\Pi_Nu_{\mathrm{in}}.\end{split}
\right.$$ The following proposition ensures the convergence of the right hand side of (\[truncnls\]) as $N\to\infty$, and provides the definition of $W^{2r+1}(u)$ in (\[nls\]).
\[nonlin\] Let $n$ be a nonnegative integer. Then almost surely in $u$ with respect to the Wiener measure $\mathrm{d}\rho$, the limit $$\lim_{N\to\infty}W_N^n(\Pi_Nu)=\lim_{N\to\infty}\Pi_NW_N^n(\Pi_Nu)$$ exists in $H^{0-}(\mathbb{T}^2)$. We will denote this limit by $W^n(u)$.
For each $N$, we also define the truncated potential energy $$\label{potentialenergy}V_N[u]=\frac{1}{r+1}\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}W_N^{2r+2}(\Pi_Nu)\,\mathrm{d}x.$$ By Proposition \[nonlin\], the limit quantity $$\label{hamilton}V[u]=\lim_{N\to\infty}V_N[u]=\frac{1}{r+1}\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}W^{2r+2}(u)\,\mathrm{d}x$$ is defined $\mathrm{d}\rho$-almost surely in $u$. [One can verify that (\[truncnls\]) is a finite dimensional Hamiltonian system with Hamitonian $$\label{defham}\mathcal{H}_N[u]:=\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}|\nabla u|^2\,\mathrm{d}x+V_N[u].$$]{} [ This $\mathcal{H}_N[u]$, as well as the mass $\mathcal{M}[u]:=\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}|u|^2\,\mathrm{d}x,$ is conserved under the flow (\[truncnls\]).]{}
\[gibbsm\] Define the measure $\mathrm{d}\mu$ by $$\label{defgibbs}\mathrm{d}\mu= Z^{-1}e^{-V[u]}\,\mathrm{d}\rho,\quad Z=\int_{\mathcal{V}}e^{-V[u]}\,\mathrm{d}\rho(u)$$ Then it is mutually absolutely continuous with $\mathrm{d}\rho$, and the Radon-Nikodym derivative $Z^{-1}e^{-V[u]}$ belongs to $L^q(\mathrm{d}\rho)$ for any $1\leq q<\infty$. We call this $\mathrm{d}\mu$ the *Gibbs measure* for (\[nls\]).
[ Propositions \[nonlin\] and \[gibbsm\] stem from seminal works of Glimm and Jaffe [@GJ1], Simon [@Simon] and Nelson [@Nel1; @Nel2] in the context of quantum field theory. See also Da Prato-Tubaro [@DaT]. As stated, a proof of these propositions can be found in [@OT].]{}
### The main theorem
We can now state our main theorem.
\[main\] There exists a Borel set $\Sigma\subset\mathcal{V}$ with $\mu(\mathcal{V}\backslash\Sigma)=0$, such that $W^{2r+1}(u)\in H^{0-}(\mathbb{T}^2)$ is well-defined for $u\in\Sigma$. [ Furthermore:]{}
1. For each $u_{\mathrm{in}}\in\Sigma$ and each $t\in\mathbb{R}$, the solution $u_N(t)$ to (\[truncnls\]) converges to a unique limit $$\label{limittrunc}\lim_{N\to\infty}u_N(t)=u(t)$$ in $H^{0-}(\mathbb{T}^2)$, and $u(t)\in\Sigma$ for each $t\in\mathbb{R}$. This $u(t)$ solves (\[nls\]) in the distributional sense.\
2. [ The limit $u(t)$ in ]{} defines, for each $t\in\mathbb{R}$, a map from $\Sigma$ to itself: $u(t)=\Phi_tu_{\mathrm{in}}$. These maps then satisfy the usual group properties, and keep the Gibbs measure $\mathrm{d}\mu$ invariant, namely $$\mu(E)=\mu(\Phi_t(E))$$ for any Borel set $E\subset\Sigma$.
\[fixep\] [ In proving Theorem \[main\], we will replace $H^{0-}(\mathbb{T}^2)$ by $H^{-\varepsilon}(\mathbb{T}^2)$ where $0<\varepsilon\ll 1$ and throughout the proof we will fix this $\varepsilon$ (one then takes a countable intersection in $\varepsilon$).]{}
\[powerof2\] [(1) Since the Gibbs measure $\mathrm{d}\mu$ is mutually absolutely continuous with the Gaussian measure $\mathrm{d}\rho$, part (1) of Theorem \[main\] can also be viewed as an almost sure global well-posedness result with random initial data $u_{\mathrm{in}}=f(\omega)$ as in (\[random\]).]{}
\(2) The uniqueness in Theorem \[main\] is in the sense of unique limit of the solutions $u_N$ to the canonical truncations (\[truncnls\]). This is consistent with [the way similar results are stated]{} in the stochastic setting, see [@GIP; @Hairer0; @Hairer]. The trunaction frequency $N$ in (\[truncnls\]) can be any positive number. For simplicity, in the proof below we will assume that $N$ is a power of two. The general case follows from placing $N$ between two adjacent powers of two, say $\frac{N'}{2}$ and $N'$, and analyzing the difference $u_{N'}-u_N$ in the same way as $u_{N'}-u_{\frac{N'}{2}}$.
\(3) The advantage of the truncation (\[truncnls\]) is that it preserves its own (finite dimensional) Gibbs measure, and is thus suitable for global-in-time arguments. In [ establishing the]{} local theory, it is possible to replace (\[truncnls\]) with other canonical truncations, say by using a smooth Fourier multiplier instead of $\Pi_N$, or by truncating only the initial data but not the nonlinearity. The proof will essentially be the same, but needs slight adjustments in a few places. We will not pursue these matters here.
\(1) [ Theorem \[main\] is part of the program of constructing invariant Gibbs measures and their dynamics for the (renormalized) defocusing nonlinear Schrödinger equation (\[nls\]), see Section \[results1\] below. With Theorem \[main\] completing all cases[[^3]]{} with $d=2$, the remaining cases that are at least expected to be solvable[[^4]]{} are the invariance of Gibbs measure for $(d,r)=(3,1)$, and invariance of white noise for $(d,r)=(1,1)$. We expect both to be strictly harder than Theorem \[main\] as they are *critical in the probabilistic scaling* (though the $d=1$ case may be easier due to integrability), see Section \[background\].]{}
\(2) One may also try to construct invariant Gibbs measures for nonlinear wave equations with power nonlinearity. This is in general much easier than Schrödinger due to the derivative gain in Duhamel’s formula; for example the $d=2$ case was solved earlier by Oh-Thomann [@OT2]. Here the only remaining case is $(d,r)=(3,1)$, which is subcritical in the probabilistic scaling and likely can be done by applying the methods of this paper.
A review of previous works {#previous}
--------------------------
We start by reviewing previous results and methods on PDEs in the probabilistic setting. As the literature is now extensive, we will put emphasis on the works most relevant to the current paper.
### Invariant measures {#results1}
Since the pioneering works of Lebowitz-Rose-Speer [@LRS] and Bourgain [@Bourgain94; @Bourgain], there has been numerous results regarding invariant measures for nonlinear dispersive equations. Generally speaking, for any Hamiltonian dispersive equation one may construct the associated Gibbs measure $$\label{gibbsgen}\mathrm{d}\mu\sim e^{-\beta H}\prod_{x}\mathrm{d}x,$$ where $\beta>0$ and $H$ is the Hamiltonian. The definition (\[gibbsgen\]) is only formal; in some cases it can be justified [ by using the Gaussian measure as a reference measure and writing $\mathrm{d}\mu$ as a weighted Wiener measure.]{} For example the Hamiltonian for (\[nls\]) is $$H=\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}(|\nabla u|^2+\frac{1}{r+1}W^{2r+2}(u))\,\mathrm{d}x$$ and the Gibbs measure $$\mathrm{d}\mu\sim \underbrace{\exp\bigg[\frac{-1}{r+1}\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}W^{2r+2}(u)\,\mathrm{d}x\bigg]}_{\textrm{weight}}\cdot\underbrace{\exp\bigg[\frac{-1}{(2\pi)^2}\int_{\mathbb{T}^2}|\nabla u|^2\,\mathrm{d}x\bigg]\prod_{x\in\mathbb{T}^2}\,\mathrm{d}x}_{\textrm{{\color{black}Gaussian} measure}}$$ can be rigorously defined as a weighted Wiener measure, as in Proposition \[gibbsm\][^5]. Defining such Gibbs-type measures and studying their properties under various dynamics is a major problem in constructive quantum field theory.
The Gibbs measure $\mathrm{d}\mu$ for a given dispersive equation [ is *formally*]{} invariant due to a [‘formal Liouville’s Theorem’]{} and the conservation of Hamiltonian. It is of great interest to establish this invariance rigorously, as this would be the first step in studying the global dynamics from the statistical ensemble point of view. In [@Bourgain94; @Bourgain], Bourgain developed a systematic way of showing the invariance of $\mathrm{d}\mu$ from the invariance of finite dimensional Gibbs measures, provided one has local well-posedness or almost sure local well-posedness with respect to $\mathrm{d}\mu$.
Therefore, justifying the invariance of $\mathrm{d}\mu$ (and other similar formally invariant measures) basically reduces to proving almost sure local well-posedness on the support of $\mathrm{d}\mu$. As this support is very rough in high dimensions (namely $H^{1-\frac{d}{2}-}$ for (\[nls\]) in dimension $d$), most known results are limited to one dimension, or requires strong symmetry. For [ the]{} Schrödinger equation (\[nls\]) on the torus $\mathbb{T}^d$, Bourgain [@Bourgain94] solved the case $d=1$ and any $r$, and [@Bourgain] extended this to $d=2$ and $r=1$. These are the only results known to date. For wave equations slightly more is known; Oh-Thomann [@OT] solved the case $d=2$ and any $r$ ($d=1$ being much easier). In dimension [ $d = 3$]{} both problems remain open [ for the cubic equation]{} (see the scaling calculations in Section \[background\]).
Apart from the standard Schrödinger and wave models on tori, there are many results, again mostly in one dimension or on manifolds under radial symmetry, where invariance of Gibbs measure (or of associated weighted Wiener measures) is justified for various dispersive models on various background manifolds ([ see e.g. [@Tz0; @Oh; @Tz; @TTz; @NORS; @NRSS; @dS; @Deng; @Deng2; @DTV; @Giordi; @Thomann; @Sy] and references therein]{}). We also mention the recent developments of the compactness methods of Alveberio and Cruzeiro [@AC] where one explores the tightness of the sequence of finite dimensional measures and apply the theorems of Prokhorov and Skorokhod to obtain existence of weak solutions [ (see e.g. [@BTT; @NPST; @OT; @WangYue])]{}. These are less related to the current paper and we will not elaborate [ further]{}.
### Probabilistic well-posedness theory {#results2}
It has long been known that PDEs with *randomness* generally behave better in terms of local well-posedness (i.e. probabilistic well-posedness goes below the deterministic well-posedness threshold). Progress have been made in two parallel directions: random initial data problems and stochastically forced problems.
The first results along this line are due [ to the seminal works by]{} Bourgain [@Bourgain; @Bourgain2] [ in the random data setting]{} [ and later to]{} Da Prato-Debussche [@DD; @DD2] [ in the stochastic setting]{}. The idea in both works is to make a linear-nonlinear decomposition and observe the effect of *probabilistic smoothing*. For example, in [@Bourgain], the equation (\[nls\]) with $r=1$ on $\mathbb{T}^2$ is studied with random initial data in $H^{-\varepsilon}$ for some $0<\varepsilon\ll 1$, in which (\[nls\]) is ill-posed. However with randomness one may construct solution to (\[nls\]) that has the form $u=e^{it\Delta}u(0)+v$, where $u(0)\in H^{-\varepsilon}$ is the random initial data, and $v$ belongs to some *positive* Sobolev space in which (\[nls\]) is well-posed. In other words, this solution contains a rough random part $u_{\mathrm{lin}}:=e^{it\Delta}u(0)$ and a smooth remainder $v$. The point here is that, even though $u_{\mathrm{lin}}$ is rough, it has the explicit random structure which allows one to control the nonlinear interactions between $u_{\mathrm{lin}}$ and $u_{\mathrm{lin}}$, and between $u_{\mathrm{lin}}$ and $v$, in a more regular space.
Until recently the method of Bourgain, as well as its higher order variants which include some nonlinear interactions [ of $u_{\mathrm{lin}}$ with itself into the rough random part]{}[^6], has been the dominant strategy of exploiting randomness in local well-posedness theory for dispersive and wave equations with random data. After Bourgain’s pioneering work, there has been substantial success [ (for a sample of works, we refer the readers to [@Bourgain94; @Bourgain; @BTlocal; @CoOh; @Deng; @BB4; @NS; @HYue; @Bringmann; @KMV; @BOhP1; @DLM0; @KM] and references therein)]{}. [ However this method by itself]{} has its limitations and does not lead to optimal results in most cases.
A few years ago, a series of important works emerged, which greatly advanced the study of local well-posedness for stochastically forced PDEs, in fact reaching the optimal exponents in the parabolic case. These include the *theory of regularity structures* of Hairer [@Hairer0; @Hairer; @Hairer2; @Hairer3] and the *para-controlled calculus* of Gubinelli-Imkeller-Perkowski [@GIP; @GIP2]. A third method based on Wilsonian renormalization group analysis was independently proposed by Kupiainen in [@Kupia].
The theory of [ regularity structures]{} is based on the local-in-space properties of solutions at fine scales (so it is particularly suitable for parabolic equations); it builds a general theory of distributions which includes the profiles coming from the noise, and allows one to perform multiplications and thus analyze the nonlinearity. Since its success with the KPZ equation [@Hairer0] and the $\Phi^4_3$ model [@Hairer], this theory has been developed by Hairer and collaborators and is now powerful enough to solve [ a wide range of problems that are]{} subcritical according to a suitable [ parabolic]{} scaling. [ We will not [ get into]{} the details, but we refer the reader to [@FrH; @Hairer3; @Hairer4; @Hairer5; @HLab; @ChW; @MouWeXu; @MouWe3] and references therein for nice expositions of these ideas.]{}
The theory of para-controlled calculus, [ which is in spirit the point of departure of the present paper,]{} takes a different approach [and is based on the following idea.]{} [ In the approach of Bourgain and of Da Prato-Debussche mentioned above,]{} some nonlinear interactions between $u_{\mathrm{lin}}$ and $v$ may not have enough regularity despite [ $v$ being]{} more regular than $u_{\mathrm{lin}}$. [ A key observation however, is that]{} the only bad terms here are the high-low interactions where the high frequencies come from $u_{\mathrm{lin}}$ and [ the]{} low frequencies come from $v$, and such terms [ can be]{} *para-controlled* by the high-frequency inputs (which are nonlinear interactions of $u_{\mathrm{lin}}$ with itself). [ Here $f$ being]{} para-controlled by $g$ simply means that $f$ equals the high-low paraproduct of $g$ [ with some other function $h$, up to a smoother remainder]{}. With such structure one can show that these para-controlled terms have similar randomness structures as the nonlinear interactions of $u_{\mathrm{lin}}$ with itself and can [ then]{} be handled similarly as in Bourgain’s or Da Prato-Debussche’s approach, leaving an even smoother remainder. An example [@GIP] is the cubic heat equation with additive white noise on $\mathbb{T}^3$, $$(\partial_t-\Delta)u=u^3+\xi,$$ where $\xi$ is the spacetime white noise (the actual equation involves some subtle renormalization which we omit for simplicity). The solution one constructs has the form $$u=u_{\mathrm{lin}}+u_{\mathrm{cubic}}+X+Y,$$ where $$u_{\mathrm{lin}}=J\xi,\quad u_{\mathrm{cubic}}=J(u_{\mathrm{lin}}^3),\quad X=\sum_NJ(P_N(u_{\mathrm{lin}}^2)\cdot P_{\ll N}(u-u_{\mathrm{lin}})),$$ $J$ is the Duhamel operator associated with the heat equation, and $P_N$ etc. are standard Littlewood-Paley projections. The term $X$ para-controlled by the bilinear Gaussian $u_{\mathrm{lin}}^2$ will be constructed in some less regular space, which allows the remainder $Y$ to be constructed in a more regular space. The para-controlled calculus also has a higher order variant, see [@BaBe; @BaBeFr], which is believed to have [ comparable power to the theory of regularity structures. We refer the reader to [@CCh; @GP; @GP2; @GP3; @GP4; @MouWe3; @Perk; @BaBe; @BaBe2; @BaBeFr] and references therein for a nice exposition of these ideas and some other recent developments on this method. ]{}
Finally, we would like to mention two very recent results of Gubinelli-Koch-Oh [@GKO] and Bringmann [@Bringmann]. In [@GKO] the authors applied a version of para-controlled calculus to the [ stochastic]{} wave equation setting, and obtained almost sure local well-posedness for a quadratic wave equation with additive white noise on $\mathbb{T}^3$. In [@Bringmann] the author studied the [ nonlinear wave equation with quadratic derivative nonlinearity]{} on $\mathbb{R}^3$ and improved the [ known]{} well-posedness threshold with random initial data, again by analyzing high-low interactions. The interesting observation made in [@Bringmann] is that, in the para-controlled scheme above (or anything similar), [ one may in fact reduce matters to the high-frequency and low-frequency parts being independent]{}, which allows one to use more powerful probabilistic tools and obtain better estimates.
### Discussion {#discuss}
From the results in Section \[results2\] one notices that, for Schrödinger equations, the probabilistic improvement (defined as the difference between exponents of the deterministic $H^s$ well-posedness threshold and the obtained probabilistic $H^s$ well-posedness threshold) is much smaller compared to wave and heat equations[^7]. For example, the probabilistic improvement in [@Bourgain] for (\[nls\]) with $(d,r)=(2,1)$ is $\varepsilon\ll1$ and the improvement in [@NS] with $(d,r)=(3,2)$ is $\frac{1}{12}$, compared to the improvement $\approx1$ for wave obtained in [@OT] (and similarly for heat).
There are two reasons for this. First, heat equations are compatible with Hölder spaces $C^s$, which scales much higher than $H^s$, but a function with independent Gaussian Fourier coefficients that belongs to $H^s$ will automatically belong to $C^{s-}$ due to Khintchine’s inequality. This allows one to [ have a scaling at a higher regularity]{} [ and hence]{} be in a better situation when studying the heat equations. Such advantage is absent in Schrödinger and wave equations, since $C^s$ spaces are not compatible even with linear evolution, and cannot be used in any well-posedness theory. Second, the Duhamel evolution for heat equation gains two derivatives, and wave equation gains one. [ This allows for room to apply Sobolev embedding, and also reduces the task of controlling the nonlinearity to the task of making sense of products, which is still hard but at least more manageable.]{} In comparison, the Schrödinger Duhamel evolution has no smoothing effect, and it can be challenging to close the estimate even [ when the relevant products are well-defined.]{}
Probabilistic scaling and a general conjecture {#background}
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The proof of Theorem \[main\] consists of two parts: (a) proving almost sure local well-posedness for (\[nls\]) on the support of the Gibbs measure, and (b) applying invariance of truncated measures to extend local solutions to global ones. Apart from a few technical subtleties, part (b) is essentially an adaptation of the classical Bourgain’s proof [@Bourgain] and nothing is fundamentally new, so let us focus on the local theory.
The obvious difficulty here is that the Gibbs measure $\mathrm{d}\mu$ is supported in $H^{0-}$, while the (deterministic) scaling threshold, below which (\[nls\]) is ill-posed, is $s_c=1-\frac{1}{r}\to 1$ as $r\to\infty$. In the language of Section \[discuss\], one needs to obtain a probabilistic improvement $\approx 1$. Therefore, it is important to understand exactly how randomness allows us to beat scaling. Before describing our method in Section \[method\] below, we will first perform a heuristic analysis which, to the best of our knowledge, has not appeared in the literature before.
### The probabilistic scaling {#probsc}
Consider the Wick ordered nonlinear Schrodinger equation $$\label{nlsex}(i\partial_t+\Delta)u=W^{2r+1}(u),\quad u(0)=u_{\mathrm{in}}$$ on $\mathbb{T}^d$. For simplicity we will replace $W^{2r+1}(u)$ by the pure power $|u|^{2r}u$ below. The scaling critical threshold for (\[nlsex\]) is $$\label{detersc}
s_c=\frac{d}{2}-\frac{1}{r},$$ and (\[nlsex\]) is expected to be locally well-posed in $H^s$ only if $s\geq s_c$. This can be demonstrated in different ways, with the one most relevant to us as follows: suppose the initial data $u_{\mathrm{in}}$ has Fourier transform $\mathcal{F}_xu_{\mathrm{in}}(k)$ supported in $|k|\sim N$ with $|\mathcal{F}_xu_{\mathrm{in}}(k)|\sim N^{-\alpha}$ with $\alpha=s+\frac{d}{2}$, then $\|u_{\mathrm{in}}\|_{H^s}\sim 1$. If local well-posedness holds then one should expect that the second iteration (say at time $t=1$), $$u^{(2)}:=\int_0^1 e^{i(1-t')\Delta}(|e^{it'\Delta}u_{\mathrm{in}}|^{2r}e^{it'\Delta}u_{\mathrm{in}})\,\mathrm{d}t',$$ satisfies $\|u^{(2)}\|_{H^s}\lesssim 1$. [ By making Fourier expansions, we essentially get $$\label{naivedeter}\mathcal{F}_xu^{(2)}(k)\sim\sum_{\substack{k_1-\cdots+k_{2r+1}=k\\|k_j|\lesssim N}}\frac{1}{\langle\Sigma\rangle}\prod_{j=1}^{2r+1}\widehat{u_{\mathrm{in}}}(k_j),\quad \Sigma=|k|^2-|k_1|^2+\cdots-|k_{2r+1}|^2,$$ where complex conjugates are omitted. In the worst scenario this gives, up to log factors, $$|\mathcal{F}_xu^{(2)}(k)|\lesssim N^{-(2r+1)\alpha}\sup_{m\in\mathbb{Z}}\# S_m,$$ where $S_m=\{(k_1,\cdots,k_{2r+1}):k_1-\cdots+k_{2r+1}=k, \, |k_j|\lesssim N, \, \Sigma=m \}.$ By dimension counting one expects $\#S_m\lesssim N^{2rd-2}$, so in order for $\|u^{(2)}\|_{H^s}\lesssim 1$ we need $-(2r+1)\alpha+2rd-2\leq -\alpha,$ or equivalently $s\geq s_c.$]{}
Now what happens if one assumes the Fourier coefficients of initial data $\{\mathcal{F}_xu_{\mathrm{in}}(k)\}$ are *independent Gaussians* of size $N^{-\alpha}$? The sum (\[naivedeter\]) will then be a sum of products of independent Gaussian random variables, which is reminiscent of the classical Central Limit Theorem. Recall that in the latter we have a sum of $M$ independent random objects of unit size, and under certain general conditions, this sum scales only like $\sqrt{M}$ as opposed to $M$ if without randomness. In the same way, we would expect essentially a [‘square root gain’]{} here, that is, $$|\mathcal{F}_xu^{(2)}(k)|\lesssim N^{-(2r+1)\alpha}\sup_{m\in\mathbb{Z}}(\# S_m)^{\frac{1}{2}}\lesssim N^{-(2r+1)\alpha+rd-1},$$ so in order for $\|u^{(2)}\|_{H^s}\lesssim 1$ it suffices to have $-(2r+1)\alpha+rd-1\leq -\alpha$, or equivalently $$\label{probscale}s\geq s_p:=-\frac{1}{2r}.$$ Note [ that $s_p$ is independent of the dimension and that]{} we always have $s_p\leq s_c$. We will call this $s_p$ the critical threshold for *probabilistic scaling*[^8].
Of course the above argument is purely heuristic, and in particular ignores the important issue of high-low interactions (which will be a main difficulty in the current paper, see Section \[method\] below). Nevertheless we believe the following conjecture is natural:
\[generalconj\] Let $r$ and $d$ be positive integers and $s>s_p$. Then (\[nlsex\]) is almost surely locally well-posed (in the sense similar to Theorem \[main\]) with random initial data $$\label{randomdataex}u_{\mathrm{in}}=\sum_{k\in\mathbb{Z}^d}\frac{g_k}{\langle k\rangle^{\alpha}}e^{ik\cdot x},\quad\alpha=s+\frac{d}{2},$$ where $\{g_k\}$ are independent Gaussian random variables with $\mathbb{E}g_k=0$ and $\mathbb{E}|g_k|^2=1$. Note that almost surely the initial data $u_{\mathrm{in}}$ belongs to $H^{s-}(\mathbb{T}^d)$.
Since the Gibbs measure for (\[nls\]), defined in (\[defgibbs\]), is supported in $H^{0-}$ and $s_p=-\frac{1}{2r}$, (the local part of) Theorem \[main\] is a special case of Conjecture \[generalconj\]. Moreover when $r\to\infty$ we have $s_p\to 0$, so the result of Theorem \[main\] gets to be almost sharp (i.e. almost reaches the conjectured exponent) when $r\to\infty$.
This leads us to believe that Conjecture \[generalconj\] is reasonable, and may not be too far from reach; on the other hand, trying to prove it for fixed values of $r$ and $d$, even in the simplest case $(r,d)=(1,2)$, [requires new ideas.]{}
### More general settings
The heuristic discussions in the previous subsection can be extended to more general situations. These include, but are not limited to, the followings:
\(1) *Wave equations*. For wave equation (say with a power nonlinearity as in (\[nlsex\])) one can apply the same heuristics as in Section \[probsc\], where instead of (\[naivedeter\]) we have essentially $$\label{naivedeter2}\mathcal{F}_xu^{(2)}(k)\sim\frac{1}{\langle k\rangle}\sum_{\substack{k_1-\cdots+k_{2r+1}=k\\|k_j|\lesssim N}}\frac{1}{\langle\Sigma\rangle}\prod_{j=1}^{2r+1}\widehat{u_{\mathrm{in}}}(k_j),\quad \Sigma=|k|-|k_1|+\cdots-|k_{2r+1}|.$$
Assume now $|k|\sim N$, then compared to (\[naivedeter\]) one gains an extra factor $N^{-1}$ due to the antiderivative, while in the [ in the dimension counting argument one gains one less power of $N$]{} as $\Sigma$ is now linear instead of quadratic. In the deterministic setting this leads to the same scaling condition as [the Schrödinger equation]{}, but in the probabilistic setting this [ trade-off]{} leads to a *better* bound than [ in the Schrödinger case]{} as the one-dimension disadvantage gets ‘square-rooted’ [ by exploiting randomness as explained above.]{} This then gives $$|\mathcal{F}_xu^{(2)}(k)|\lesssim N^{-(2r+1)\alpha-1}N^{rd-\frac{1}{2}},$$ which leads to a *lower* probabilistic scaling threshold[, namely]{} $s_p^{\mathrm{wave}}=-\frac{3}{4r}$.
However, unlike Schrödinger, there is also a [ ‘high into low’]{} interaction, namely $|k|\sim 1$ in (\[naivedeter2\]), that needs to be addressed. A similar calculation using randomness and counting bounds yields heuristically that $$|\mathcal{F}_x{u^{(2)}}(k)|\lesssim N^{-(2r+1)\alpha}N^{rd-\frac{1}{2}},$$ which leads to the restriction $s\geq s_p':=\frac{-d-1}{2(2r+1)}$. Thus it is reasonable to conjecture that the wave equation is almost surely locally well-posed in $H^s\times H^{s-1}$ for $$s>\max(s_p^{\mathrm{wave}},s_p')=\max\big(\frac{-3}{4r},-\frac{d+1}{4r+2}\big);$$ in particular when $(r,d)=(1,3)$ the conjectured threshold is $H^{-\frac{2}{3}}$, which is [ below]{} $H^{-\frac{1}{2}-}$ where the Gibbs measure is supported.
\(2) *Other dispersion relations and/or nonlinearities*. For more general dispersion relations, say $\Lambda=\Lambda(k)$, the only thing above that changes is the counting bound for the set $$S_m=\{(k_1,\cdots,k_{2r+1}):k_1-\cdots+k_{2r+1}=k,\,\,\Sigma:=\Lambda(k)-\Lambda(k_1)+\cdots-\Lambda(k_{2r+1})\in[m,m+1]\}.$$ In contrast to parabolic equations (see [@Hairer]) where the exact form of [ the]{} elliptic part is irrelevant once the order is fixed, here the properties of $S_m$ [ depend]{} crucially on the choice of $\Lambda$, and [ have]{} to be analyzed on a case by case basis. For simple dispersive relations like Schrödinger, wave or gravity water wave (where $\Lambda(k)=\sqrt{|k|}$) this may be doable, but when $\Lambda$ gets more complicated (say a high degree polynomial), approaching the counterpart of Conjecture \[generalconj\] [ requires]{} getting sharp bounds for $\#S_m$, which [ in itself may be a hard problem in analytic number theory.]{}
For derivative nonlinearities, the scaling heuristics can still be carried out and the value of $s_p$ obtained in the same way as before (since such heuristics essentially [ take]{} into account only the high-high interactions). However the actual almost sure well-posedness threshold may be strictly higher than $s_p$ due to high-low interactions and derivative loss (in the same way that the deterministic theory for quasilinear equations does not quite reach scaling, see e.g. [@KRS; @ST]), which may be worth looking at first in some simple models. There is also the possibility of exponential nonlinearities but they are more of an ‘endpoint’ nature and will not be discussed here.
\(3) *Stochastic equations*. We may also consider wave and heat equations with additive noise (Schrödinger is also possible but has worse behavior), say of form $$\label{additive}(\partial_t^2-\Delta)u=|u|^{2r}u+\zeta,\quad\mathrm{or}\quad (\partial_t-\Delta)u=|u|^{2r}u+\zeta,$$ where $\zeta$ is the spacetime white noise which is essentially (after discretizing the time Fourier variable) $$\zeta=\sum_{k,\xi}g_{k,\xi}e^{i(k\cdot x+\xi t)},$$ where $g_{k,\xi}$ are independent normalized Gaussians.
The heat case of (\[additive\]) has been studied extensively, see the references in Section \[results2\]. [ In this case we can confirm that the scaling heuristics of Section \[probsc\] are consistent with that of [@Hairer]]{}. [ Indeed, note that]{} for (\[additive\]) the linear evolution $e^{it\Delta}u_{\mathrm{in}}$ in Section \[probsc\] is replaced by the linear noise term $$\psi(t)=\int_0^te^{(t-t')\Delta}\zeta(t')\,\mathrm{d}t'\sim\sum_{k,\xi}\frac{g_{k,\xi}}{|k|^2+|\xi|}e^{i(k\cdot x+\xi t)},$$ which belongs to $C_t^0H_x^{s-}$ for $s=-\frac{d}{2}+1$. The goal would then be to guarantee that the second iteration $$u^{(2)}(t)=\int_0^te^{(t-t')\Delta}(|\psi(s)|^{2r}\psi(t'))\,\mathrm{d}t'$$ belongs to the same space. By similar arguments, this time also taking into account the time Fourier variable, one can show that this leads to the restriction $r(d-2)<2$, which coincides with the subcriticality condition introduced in [@Hairer] in the case of (\[additive\]).
For the wave case of (\[additive\]), similar calculations lead to the subcriticality condition $r(d-2)<\frac{3}{2}$, [ which is]{} consistent with the [ results in]{} [@GKO; @GKO2]. In both cases, due to the particular choice of white noise, the high-to-low interactions studied in (1) [above]{} give the same condition on $(r,d)$.
[The random averaging operator method]{} {#method}
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We now turn to the proof of Theorem \[main\]. [ To bring about the main ideas we focus our attention here to the question]{} of almost sure local well-posedness of (\[nls\]) in the support of the Gibbs measure $\mathrm{d}\mu$, namely $H^{0-}(\mathbb{T}^2)$, and assume $u_{\mathrm{in}}=f(\omega)$ where $f$ is as in (\[random\]). We will also replace $W^{2r+1}(u)$ by the pure power $|u|^{2r}u$, like in Section \[probsc\].
### Main challenges
A naive attempt would be to follow [ Bourgain’s approach [@Bourgain]]{} and look for solutions to (\[nls\]) of form $u(t)=e^{it\Delta}u_{\mathrm{in}}+w$ where $w$ belongs to $C_t^0H_x^s$, or more precisely $X^{s,\frac{1}{2}+}$ (see Section \[funcspace\] for relevant definitions) for some positive $s$; this $w$ in particular will contain components of form $$\label{seconditer}
u^{(2)}(t)=I(|e^{it\Delta}f(\omega)|^{2r}e^{it\Delta}f(\omega)),\quad IF(t):=\int_0^t e^{i(t-t')\Delta}F(t')\,\mathrm{d}t'.$$However, it is shown in [@Bourgain] that even when $r=1$ (and obviously also for larger $r$), the $u^{(2)}$ defined in (\[seconditer\]) belongs to $X^{s,\frac{1}{2}+}$ only for $s<\frac{1}{2}$. As the space[^9] $X^{\frac{1}{2}-,\frac{1}{2}+}$ is still supercritical with respect to deterministic scaling [ for $d=2$]{} and $r\geq 2$, there will be no hope in solving (\[nls\]) using the above ansatz. One may [perform higher order Picard iterations,]{} but it turns out that regardless of the order, there is always some contribution in the remainder that has regularity $X^{\frac{1}{2}-,\frac{1}{2}+}$.
Here we make the first observation, namely that the [ poor]{} regularity of $u^{(2)}$ is only due to high-low interactions. In fact, by carrying out the analysis of Section \[probsc\], one can show that if the two highest frequencies in the input factors of (\[seconditer\]) are comparable, for example consider $$u_{(1)}^{(2)}(t)=\sum_NI(|e^{it\Delta}P_{N}f(\omega)|^2\cdot|e^{it\Delta}P_{\leq N}f(\omega)|^{2r-2}e^{it\Delta}P_{\leq N}f(\omega)),$$ where $P_N$ and $P_{\leq N}$ are standard Littlewood-Paley projections, then [ $u_{(1)}^{(2)}$]{} will belong to $X^{1-,\frac{1}{2}+}$ which is [(deterministically)]{} subcritical for any $r$. Therefore to solve (\[nls\]) it suffices to control the high-low interactions, say the ones of [ the]{} form $$\label{paracon}X=\sum_NI(e^{it\Delta}P_Nf(\omega)\cdot|P_{\ll N}u|^{2r}),$$ and to show that this term somehow has similar behavior as the linear evolution $e^{it\Delta}f(\omega)$.
The above idea is inspired by the paracontrolled analysis of Gubinelli-Imkeller-Perkowski [@GIP], see Section \[results2\], however due to the different nature of Schrödinger equation compared to heat and wave equations, their method does not work here. For example, suppose one defines a term $X$ by[^10] (\[paracon\]), which is paracontrolled by $e^{it\Delta}f(\omega)$, then one would need to have reasonable bounds on the low-frequency component $|P_{\ll N}u|^{2r}$. In the paracontrolled ansatz, one would like to construct $X$ in some less regular space (say $X^{\frac{1}{2}-,\frac{1}{2}+}$) and the remainder $Y=u-e^{it\Delta}f(\omega)-X$ in a more regular space (say $X^{1-,\frac{1}{2}+}$). Then the low frequency component can be expanded as $$|P_{\ll N}u|^{2r}=|P_{\ll N}e^{it\Delta}f(\omega)+P_{\ll N}X+P_{\ll N}Y|^{2r}.$$ Here, if all the factors are $P_{\ll N}e^{it\Delta}f(\omega)$, then the corresponding product can be bounded using estimates for multilinear Gaussians; however we also have the situation where all the factors are $P_{\ll N}X$. As $X$ is supposed to be constructed in the less regular space $X^{\frac{1}{2}-,\frac{1}{2}+}$, the only thing we know about $X$ is its regularity, and as this regularity is supercritical, this alone will not be able to guarantee any bound for $|P_{\ll N}X|^{2r}$ better than [ (say)]{} $|P_{\ll N}X|^{2r}\in X^{-10,\frac{1}{2}+}$ when $r$ is large, which is clearly not enough.
Another attempt would be to strengthen the definition of paracontrolling by requiring the low-frequency part to have frequency $\ll N^\alpha$ for some $\alpha\in(0,1)$, like in [@Bringmann]. In this case, however, this $\alpha$ cannot be too small since otherwise the high-high interaction (say) $$\label{newhh}u_{(\alpha)}^{(2)}(t)=\sum_NI(e^{it\Delta}P_{N}f(\omega)\overline{e^{it\Delta}P_{N^\alpha}f(\omega)}\cdot|e^{it\Delta}P_{\leq N^\alpha}f(\omega)|^{2r-2}e^{it\Delta}P_{\leq N^\alpha}f(\omega))$$ cannot be placed in a subcritical space. In fact, calculations show that $u_\alpha^{(2)}$ defined by (\[newhh\]) belongs to $X^{\frac{1+\alpha}{2}-,\frac{1}{2}+}$, so subcriticality implies $\alpha\geq 1-\frac{2}{r}$. When $r$ is large $\alpha$ has to be close to $1$ and the above issue persists.
[Note that when $r=2$, it might be possible to carry out the above with a small $\alpha$ by doing some refined analysis (which is by no means immediate in view of all the difficulties for the Schrödinger equation; see Section \[discuss\]). Our approach, which is described below, allows instead for a *unified* treatment for all values of $r$ by synthesizing the main underlying ideas and capturing the true randomness structure of the solution.]{}
### Random [averaging]{} operators
[ We propose a new method to resolve the issues above which goes beyond the para-controlled calculus [@GIP; @GKO2; @BaBe] and is based on the following two key observations.]{}
The first observation is that, in (\[paracon\]), we may replace the low-frequency part $|P_{\ll N}u|^{2r}$ by $|u_{\ll N}|^{2r}$, where $u_{\ll N}$ is the solution to (\[nls\]) with initial data $P_{\ll N}f(\omega)$, in the sense that the difference $|P_{\ll N}u|^{2r}-|u_{\ll N}|^{2r}$ contains at least one high-frequency factor and can be regarded as high-high interaction. The point here is that $|u_{\ll N}|^{2r}$, by definition, is a measurable function of $(g_k(\omega))_{\langle k\rangle\ll N}$, and is hence *independent* with the linear factor $e^{it\Delta}P_Nf(\omega)$. This independence will allow us to apply all the large deviation estimates, and will play an important role in the proof. [ Note that the same idea is also used in [@Bringmann].]{}
The second and more important observation is that, although the low-frequency factor $|P_{\ll N}X|^{2r}$ cannot be [controlled]{} with $X$ only assumed to be in $X^{\frac{1}{2}-,\frac{1}{2}+}$, in reality this $X$ is not an arbitrary function in $X^{\frac{1}{2}-,\frac{1}{2}+}$. This $X$ has its own structure - namely it is para-controlled [ in]{} itself. It is vital to be able to exploit this structure[^11] in order to complete the proof of Theorem \[main\]. [The idea of exploiting the randomness structure of low-frequency components was present in Bourgain [@Bourgain97], though in that case one has substantial smoothing and only the simplest structure is needed.]{}
In order to apply these two observations, especially the second [ one]{}, one possibility is to perform an iteration: in order [to control $|P_{\ll N}X|^{2r}$]{} we need to use the para-controlled structure of $X$, for example (where we identify $P_{\ll L}u$ and $u_{\ll L}$) $$\label{paracon2}P_{\ll N}X=\sum_{L\ll N}I(e^{it\Delta}P_Lf(\omega)\cdot|P_{\ll L}u|^{2r}),$$ [ which reduces the task to controlling $|P_{\ll L}u|^{2r}$ and $|P_{\ll L}X|^{2r}$]{}. We then continue this process until the frequency reaches $1$ and analyze the resulting multilinear expression. Unfortunately the structure of such multilinear expression is so complicated, that it is hard to extract from it anything useful.
[ In this paper we take a different approach]{}, in the spirit of our previous work [@DNY] in the deterministic setting, namely *we will extract all the randomness properties of $X$, as well as properties of the multilinearity $|\cdot|^{2r}$ when applied to these random objects, and turn them into two particular norm bounds for the operator $$\label{randomop}\mathcal{P}:y\mapsto \sum_NI(P_{N}y\cdot|P_{\ll N}X|^{2r}).$$* The norms we choose are the operator norm $\|\cdot\|_{\mathrm{OP}}$ and the Hilbert-Schmidt norm $\|\cdot\|_{\mathrm{HS}}$ with $\mathcal{P}$ viewed as a linear operator from the space $X^{s,\frac{1}{2}+}$ to itself (this does not depend on $s$ so we may in fact choose $s=0$); when the input $y$ in (\[randomop\]) is a linear Schrödinger flow, we get an operator from $H^s$ to $X^{s,\frac{1}{2}+}$, and will also measure the corresponding operator and Hilbert-Schmidt norms. Suppose the maximum frequency of $P_{\ll N}X$ in (\[randomop\]) is $L\ll N$, then roughly speaking, these norm bounds will look like $$\label{normintro}\|\mathcal{P}\|_{\mathrm{OP}}\lesssim L^{-\delta_0},\quad \|\mathcal{P}\|_{\mathrm{HS}}\lesssim N^{\frac{1}{2}+\delta_1}L^{-\frac{1}{2}},$$ where $\delta_1\ll\delta_0\ll 1$; note that these bounds are obviously false for arbitrary functions $X\in X^{\frac{1}{2}-,\frac{1}{2}+}$, so they really encode the randomness structure of $X$. See Proposition \[localmain\] for details.
This operator $\mathcal{P}$, which will be of central importance in our proof, depends on the quantity $X$ that has [an implicit]{} randomness structure. [ Moreover, in the Fourier variables, this operator can be viewed as a weighted average on smaller scales $L\ll N$. We are thus calling it a *random averaging operator*, which explains the name of our method.]{}
### The full ansatz {#secansatz}
We can now write down the full ansatz of the solution $u$ to (\[nls\]), namely that $$\label{fullansatz}u=e^{it\Delta}f(\omega)+\mathcal{P}(e^{it\Delta}f(\omega))+z,$$ where $z\in X^{1-,\frac{1}{2}+}$ is a smooth remainder, and the random [averaging]{} operator $\mathcal{P}$ is of form $$\mathcal{P}=\sum_{N}\sum_{L\ll N}\mathcal{P}_{NL},$$ where $\mathcal{P}_{NL}$ has the form (\[randomop\]) with the maximum frequency of $P_{\ll N}X$ in (\[randomop\]) being $L$. [ This $\mathcal{P}_{NL}$ is a Borel function of $(g_k(\omega))_{\langle k\rangle\leq L}$ - which is independent with $e^{it\Delta}P_Nf(\omega)$ - and satisfies (\[normintro\]).]{} See Section \[decompsol\] for the precise formulas.
With the ansatz (\[fullansatz\]), the proof of local well-posedness then goes by inducting on frequencies to show (\[normintro\]) and [ to]{} bound $z$ in $X^{1-,\frac{1}{2}+}$. In fact, suppose these are true for components of frequency $\ll N$, then the bounds (\[normintro\]) imply that the [part of $P_{\ll N}u$ (or $u_{\ll N}$) involving the random averaging operators]{} really behaves like a linear Schrödinger flow, so in $\mathcal{P}_{NL}$, see (\[randomop\]), we can effectively assume $X$ is a linear flow. [Hence]{} (\[normintro\]) follows from large deviation estimates for multilinear Gaussians and a $T^*T$ argument like the one of Bourgain [@Bourgain], and the estimate for $w$ follows from standard contraction mapping arguments. See Sections \[structuresol\] and \[multiest\] for details.
In Section \[notations\] we collect some of the notations and conventions that will be used in the proof. The rest of the paper is organized as follows. In Section \[prep0\] we introduce the gauge transform and reduce to a favorable nonlinearity, and define the norms that will be used in the proof below. In Section \[structuresol\] we identify the precise structure of the solution according to the ideas of Section \[method\], and reduce local well-posedness to some multilinear estimates, namely Proposition \[multi0\]. Section \[prep2\] then sets up the necessary tools (large deviation and counting estimates) needed in the proof of Proposition \[multi0\], and Section \[multiest\] contains the proof itself. Finally in Section \[global\] we apply an adapted version of Bourgain’s argument to extend local solutions to global ones and finish the proof of Theorem \[main\].
Notations and choice of parameters {#notations}
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Throughout the paper, the space and time Fourier transforms will be [ respectively]{} fixed as $$\label{fouriert}(\mathcal{F}_xu)(k)=\frac{1}{(2\pi)^2}\int_{\mathbb{T}^2}e^{-ik\cdot x}f(x)\,\mathrm{d}x,\quad (\mathcal{F}_tu)(\xi)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i\xi t}f(x)\,\mathrm{d}t.$$ We will be working in $(k,t)$ or $(k,\xi)$ variables, instead of the $x$ variable; so we will abbreviate $(\mathcal{F}_xu)(k)$ simply as $u_k$, [and will abuse notations and write $u=u_k(t)$.]{} The symbol $\widehat{u}$ will always represent time Fourier transform, so $(\mathcal{F}_{t,x}u)(k,\xi)=\widehat{u_k}(\xi)$.
Let the space [mean]{} $\mathcal{A}$ be defined by $\mathcal{A}u={\color{black} (\mathcal{F}_xu)(0)} = u_0$ (this may depend on time $t$ if $u$ does). Define the twisted spacetime Fourier transform $$\label{twistfourier}\widetilde{u}_k(\lambda)=\widetilde{u}(k,\lambda)=\widehat{u_k}(\lambda-|k|^2).$$ We also need to study functions $h_{kk^*}(t)$ of variables $k,k^*\in\mathbb{Z}^2$ and $t\in\mathbb{R}$, and $\mathfrak{h}_{kk'}(t,t')$ of variables $k,k'\in\mathbb{Z}^2$ and $t,t'\in\mathbb{R}$; for these we will define $$\label{twistfourier2}\widetilde{h}_{kk^*}(\lambda)=\widehat{h_{kk^*}}(\lambda-|k|^2),\quad\mathrm{and}\quad \widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')=2\pi \, \widehat{\mathfrak{h}_{kk'}}(\lambda-|k|^2,|k'|^2-\lambda'),$$where $\lambda$ and $\lambda'$ are Fourier variables corresponding to $t$ and $t'$ respectively.
Recall that $\langle k\rangle:=\sqrt{|k|^2+1}$, and $\mathbf{1}_P$ is the indicator function. [The cardinality of a finite set $E$ [ will be]{} denoted by $|E|$ or [ by]{} $\#E$.]{} We will be using smooth cutoff functions $\chi=\chi(z)$ which equal $1$ for $|z|\leq 1$ and equal $0$ for $|z|\geq 2$. For any Schwartz function $\varphi$ and any $0<\tau\ll 1$, we will define $\varphi_{\tau}(t)=\varphi(\tau^{-1}t)$.
For a complex number $z$ define $z^+=z$ and $z^- =\overline{z}$; we will also use the notation $z^{\iota}$ where $\iota$ will always be $\pm$.
In the proof we will encounter tuples $(k_1,\cdots,k_n)$, or maybe $(k_1^*,\cdots,k_n^*)$, with associated signs $\iota_1,\cdots,\iota_n\in\{\pm\}$; they are usually linked by some equation $\iota_1k_1+\cdots+\iota_nk_n=d$ or $\iota_1|k_1|^2+\cdots+\iota_n|k_n|^2=\alpha$, where $d$ and $\alpha$ are given, or by some expression $g_{k_1^*}^{\iota_1}\cdots g_{k_n^*}^{\iota_n}$.
\[dfpairing\] In the above context we say $(k_i,k_j)$ is a *pairing* in $\{k_1,\cdots,k_n\},$ if $k_i=k_j$ and $\iota_i=-\iota_j$. We say a pairing is *over-paired* if $k_i=k_j=k_\ell$ for some $\ell\not\in\{i,j\}$. Pairings and over-pairings in $\{k_1^*,\cdots,k_n^*\}$ are defined similarly.
For example, suppose $k=k_1-k_2+k_3+d$. If $k_1=k_2$, then $(k_1,k_2)$ is a pairing in $\{k_1,k_2,k_3\}$; if $k=k_1$ then $(k,k_1)$ is a pairing in $\{k,k_1,k_2,k_3\}$. If $k=k_1=k_2\neq k_3$, then $(k_1,k_2)$ is over-paired if considered as a pairing in $\{k,k_1,k_2,k_3\}$, but not if considered a pairing in $\{k_1,k_2,k_3\}$.
Recall Remark \[powerof2\] (2) that for the truncation $\Pi_N$ defined in (\[truncN\]), $N$ will be a power of two that is also $\gtrsim 1$. The same applies to other capital letters like $M$, $L$, $R$, etc.. Define also $\Pi_N^{\perp}=\mathrm{Id}-\Pi_N$ and $\Delta_N=\Pi_N-\Pi_{\frac{N}{2}}$, so that $$(\Delta_Nu)_k=\mathbf{1}_{N/2<\langle k\rangle\leq N}\cdot u_k.$$ Let $\mathcal{V}_N$ and $\mathcal{V}_N^\perp$ be the ranges of $\Pi_N$ and $\Pi_N^\perp$. For $N_1,\cdots,N_n$, we will define $\max^{(j)}(N_1,\cdots,N_n)$ to be the $j$-th maximal element among them, and denote it by $N^{(j)}$.
\[borel\] For any $N$ as above, we denote by $\mathcal{B}_{\leq N}$ the $\sigma$-algebra generated by the random variables $g_k$ for $\langle k\rangle\leq N$, and by $\mathcal{B}_{\leq N}^+$ the smallest $\sigma$-algebra containing both $\mathcal{B}_{\leq N}$ and the $\sigma$-algebra generated by the random variables $|g_k|^2$ for $k\in\mathbb{Z}^2$.
Recall that $\varepsilon$ is fixed by Remark \[fixep\]. Let $1\gg\delta_0\gg\delta$ be two fixed small positive constants depending on $r$ and $\varepsilon$ (think of $\delta_0=\delta^{1/50}$). [Define the parameters $$\label{defparam}
\gamma=\delta^{\frac{3}{4}},\quad \gamma_0=\delta^{\frac{5}{4}},\quad\kappa=\delta^{-4},\quad b=\frac{1}{2}+\delta^4,\quad b_1=b+\delta^4,\quad b_2=b-\delta^6,\quad a_0=2b-10\delta^6,$$ then we have the following hierarchy: $$\label{hierarchy}\varepsilon\gg\delta_0\gg
\gamma\gg\delta\gg\gamma_0\gg \delta\gamma_0\gg b-\frac{1}{2}=b_1-b=\kappa^{-1}\gg\delta^6.$$]{} Denote by $\theta$ any positive quantity that is small enough depending on $\delta$ (for example $\theta\ll\delta^{50}$). This $\theta$ may take different values at different places. [Let $C$ be any large absolute constant depending only on $r$, and $C_\theta$ be any large constant depending on $\theta$. Unless otherwise stated, The constants in $\lesssim$, $\ll$ and $O(\cdot)$ symbols will depend on $C_\theta$. Finally, if some statement $S$ about a random variable holds with probability $\mathbb{P}(S)\geq 1-{\color{black}C_\theta e^{-A^\theta}}$ for some $A>0$, we will say this $S$ is *$A$-certain*. ]{}
Acknowledgment
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The second author thanks Hendrik Weber for helpful comments regarding references on the $\phi^4$ model.
Equations, measures and norms {#prep0}
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Wick ordering and a gauge transform
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Consider a general polynomial $\mathcal{M}_n(u)$ of degree $n$, defined by [$$\label{orderm}[\mathcal{M}_n(u)]_k=\sum_{\iota_1k_1+\cdots+\iota_n k_n=k}a_{kk_1\cdots k_n}u_{k_1}^{\iota_1}\cdots u_{k_n}^{\iota_n},$$]{} [and similarly consider $$\label{orderm2}[\mathcal{H}_n(u)]_{kk'}=\sum_{\iota_1k_1+\cdots+\iota_n k_n+\iota k'=k}a_{kk'k_1\cdots k_n}u_{k_1}^{\iota_1}\cdots u_{k_n}^{\iota_n},$$ ]{} where $a_{kk_1\cdots k_n}$ and $a_{kk'k_1\cdots k_n}$ are constants. Recall the definition of pairings in Definition \[dfpairing\].
We say the polynomial in is *input-simple*, if $a_{kk_1\cdots k_n}=0$ unless each pairing in $\{k_1,\cdots,k_n\}$ is over-paired. Similarly we say it is *simple*, if $a_{kk_1\cdots k_n}=0$ unless each pairing in $\{k,k_1,\cdots,k_n\}$ is over-paired, and we say the polynomial in (\[orderm2\]) is simple, if $a_{kk'k_1\cdots k_n}=0$ unless each pairing in $\{k,k',k_1,\cdots,k_n\}$ is over-paired. These notions also apply to multilinear forms.
[ For $m: =\mathcal{A}|u|^2$, define]{} the following polynomials of degree $n\in\{2p,2p+1\}$ [(this $u$ may also be replaced by $v$)]{}: $$\label{pairfree}
\begin{aligned}:\mathrel{|u|^{2p}}:&=\sum_{j=0}^p(-1)^{p-j}{p\choose j}\frac{m^{p-j}p!}{j!}|u|^{2j},\\
:\mathrel{|u|^{2p}u}:&=\sum_{j=0}^p(-1)^{p-j}{{p+1}\choose {p-j}}\frac{m^{p-j}p!}{j!}|u|^{2j}u.
\end{aligned}$$ We will see in the proof of Proposition \[simpleterms\] that each of these is input-simple.
Define a gauge transform $v_N=\mathcal{G}_Nu_N$ associated with (\[truncnls\]) by $$\label{gauge}v_N(t)=u_N(t)\cdot\exp\bigg((r+1)i\int_0^t\mathcal{A}[W_N^{2r}(u_N)]\,\mathrm{d}t'\bigg).$$ Then $u_N$ solves (\[truncnls\]) if and only if $v_N$ solves the gauged equation $$\label{gauged}\left\{
\begin{split}(i\partial_t+\Delta)v_N&=\Pi_N\mathcal{Q}_N(v_N),\\
v_N(0)&=\Pi_Nu_{\mathrm{in}},
\end{split}
\right.$$ where $$\label{nonlinN}\mathcal{Q}_N(v)=W_N^{2r+1}(v)-(r+1)\mathcal{A}[W_N^{2r}(v)]v.$$ Since the gauge transform does not change the $t=0$ data, we will write $v_{\mathrm{in}}=u_{\mathrm{in}}$. [ The inverse of $\mathcal{G}_N$ is given by $$\label{invgauge}u_N(t)=v_N(t)\cdot\exp\bigg(-(r+1)i\int_0^t\mathcal{A}[W_N^{2r}(v_N)]\,\mathrm{d}t'\bigg),$$ since by (\[wickpoly\]), if $v=e^{i\alpha}u$ where $\alpha\in\mathbb{R}$, then $W^{n}(u)=W^{n}(v)$ for even $n$ and $W^n(u)=e^{i\alpha}W^n(v)$ for odd $n$.]{} Now assume $v_N$ is a solution to (\[gauged\]). Let $m_N$ be the truncated mass, which is conserved under (\[gauged\]), $$\label{truncmass}m_N=\mathcal{A}|v_N|^2=\sum_{\langle k\rangle\leq N}|(u_{\mathrm{in}})_k|^2,$$ and let [ $m_N^*:=m_N-\sigma_N$ where $\sigma_N$ is as in . Note that $m_N$ and $m_N^{\ast}$ are random terms if $u_{\mathrm{in}}=f(\omega)$ as in (\[random\]). We]{} have the following formula for $\mathcal{Q}_N$:
\[simpleterms\] We have $$\label{formulaqn}{\color{black}\mathcal{Q}_N(v_N)}=\sum_{l=0}^r{{r+1}\choose {r-l}}\frac{(m_N^*)^{r-l}r!}{l!}{\color{black}\mathcal{N}_{2l+1}(v_N)},$$ where $$\label{quintcubic}\mathcal{N}_{2l+1}(v)=:\mathrel{|v|^{2l}v}:-(l+1)(\mathcal{A}:\mathrel{|v|^{2l}}:)v.$$ Here $\mathcal{N}_{2l+1}$ is a simple polynomial of degree $2l+1$. By standard procedure, we can define a $(2l+1)$-multilinear form, which we still denote by $\mathcal{N}_{2l+1}$, such that it reduces to $\mathcal{N}_{2l+1}(v)$ when all inputs equal to $v$.
First we prove (\[formulaqn\]). By the definition of $\mathcal{Q}_N(v)$, see (\[nonlinN\]), it will suffice to obtain that $$\label{prop3.2:eq1}
{\color{black}W_N^{2r+1}(v_N)} =\sum_{l=0}^r{{r+1}\choose {r-l}}\frac{(m_N^*)^{r-l}r!}{l!} {\color{black}:\mathrel{|v_N|^{2l}v_N}:}$$ and $$\label{prop3.2:eq2}
{\color{black}(r+1)W_N^{2r}(v_N)} =\sum_{l=0}^r{{r+1}\choose {r-l}}\frac{(m_N^*)^{r-l}r!}{l!}(l+1){\color{black}:\mathrel{|v_N|^{2l}}:.}$$
By the definition of $:\mathrel{|v|^{2l}v}:$, see (\[pairfree\]), and combinatorial identities, we have $$\begin{aligned}
\text{RHS of ({\ref{prop3.2:eq1}})} & = \sum_{l=0}^r {r+1 \choose r-l} (m_N^*)^{r-l}r! \sum_{k=0}^l (-1)^{l-k} {l+1\choose l-k} \frac{m_N^{l-k}}{k!}{\color{black}\mathrel{|v_N|^{2k} v_N}}\\
&= \sum_{k=0}^r (-1)^{r-k} {r+1\choose r-k}\frac{r!}{k!}{\color{black}\mathrel{|v_N|^{2k} v_N}}
\sum_{l=k}^r {r-k\choose l-k}m_N^{l-k} (-m_N^*)^{r-l},\label{prop3.2:eq3}
\end{aligned}$$ which implies (\[prop3.2:eq1\]) due to binomial expansion.
Similarly we can calculate $$\begin{aligned}
\text{RHS of ({\ref{prop3.2:eq2}})} & = \sum_{l=0}^r {r+1 \choose r-l} (m_N^*)^{r-l}r! \, (l+1)\, \sum_{k=0}^l (-1)^{l-k} {l\choose k} \frac{m_N^{l-k}}{k!}{\color{black}\mathrel{|v_N|^{2k}}}\\
&= (r+1)\sum_{k=0}^r (-1)^{r-k} {r\choose k}\frac{r!}{k!}{\color{black}\mathrel{|v_N|^{2k}}}
\sum_{l=k}^r {r-k\choose l-k}m_N^{l-k} (-m_N^*)^{r-l},\label{prop3.2:eq4}
\end{aligned}$$ which implies (\[prop3.2:eq2\]).
Next we prove that $:\mathrel{|v|^{2p}}:$ and $:\mathrel{|v|^{2p}v}:$ are input-simple. Working in Fourier space, for any monomial $$\mathcal{X}:=(v_{k_1})^{a_1}(\overline{v_{k_1}})^{b_1}\cdots (v_{k_n})^{a_n}(\overline{v_{k_n}})^{b_n},$$ where the $k_j$’s are different, $a_j$ and $b_j$ are nonnegative integers, we will calculate the coefficient of $\mathcal{X}$ in the polynomial $:\mathrel{|v|^{2p}}:$ and $:\mathrel{|v|^{2p}v}:$, and will prove that this coefficient is zero provided $a_1=b_1=1$. Now clearly the coefficient of $\mathcal{X}$ in $|v|^{2p}$ and $|v|^{2p}v$, denoted by $[\mathcal{X}](|v|^{2p})$ and $[\mathcal{X}](|v|^{2p}v)$, are $$(|v|^{2p})=\frac{(p!)^2}{a_1!\cdots a_n!b_1!\cdots b_n!},\quad [\mathcal{X}](|v|^{2p}v)=\frac{p!(p+1)!}{a_1!\cdots a_n!b_1!\cdots b_n!},$$ under the assumptions $b_1+\cdots+b_n=p$ and $a_1+\cdots+a_n=p$ (or $a_1+\cdots+a_n=p+1$). [Recall that $m=\mathcal{A}|v|^2$,]{} we can calculate that $$\begin{split}\label{totalcoef}[\mathcal{X}](:\mathrel{|v|^{2p}}:)&=\sum_{l=0}^p(-1)^{p-l}\frac{p!}{l!}{p\choose l}(p-l)!(l!)^2\sum_{c_1+\cdots+c_n=p-l}\prod_{s=1}^n\frac{1}{c_s!(a_s-c_s)!(b_s-c_s)!}\\
&=(-1)^p(p!)^2{\color{black}\sum_{l=0}^p(-1)^{l}}\sum_{c_1+\cdots+c_n=p-l}\prod_{s=1}^n\frac{1}{c_s!(a_s-c_s)!(b_s-c_s)!}.
\end{split}$$ Now suppose $a_1=b_1=1$, then $c_1\in\{0,1\}$; clearly the terms [for $l$ and ]{}with $c_1=0$ exactly cancel the terms [for $l+1$ and ]{}with $c_1=1$, so $[\mathcal{X}](:\mathrel{|v|^{2p}}:)=0$. Similarly we can prove $[\mathcal{X}](:\mathrel{|v|^{2p}v}:)=0$.
Finally, we prove that $\mathcal{N}_{2p+1}(v)=:\mathrel{|v|^{2p}v}:-(p+1)(\mathcal{A}:\mathrel{|v|^{2p}}:)v$ is simple. By definition it suffices to prove that $$\mathcal{A}(\mathcal{N}_{2p+1}(v)\overline{v})=\mathcal{A}(\overline{v}:\mathrel{|v|^{2p}v}:)-(p+1)m\mathcal{A}:\mathrel{|v|^{2p}}:$$ is input-simple. We will actually show that this equals $\mathcal{A}:\mathrel{|v|}^{2p+2}:$ [whence]{} the result will follow. In fact, by (\[pairfree\]) we have $$\begin{split}\mathcal{A}:\mathrel{|v|^{2p+2}}:&=\sum_{l=0}^{p+1}(-1)^{p-l+1}{{p+1}\choose l}\frac{(p+1)!}{l!}m^{p-l+1}\mathcal{A}|v|^{2l},\\
\mathcal{A}(\overline{v}:\mathrel{|v|^{2p}v}:)&=\sum_{l=1}^{p+1}(-1)^{p-l+1}{{p+1}\choose l}\frac{p!}{(l-1)!}{\color{black}m^{p-l+1}}\mathcal{A}|v|^{2l},\\
-(p+1)m\mathcal{A}:\mathrel{|v|^{2p}}:&=\sum_{l=0}^{p}(-1)^{p-l+1}(p+1){{p}\choose l}\frac{p!}{l!}m^{p-l+1}\mathcal{A}|v|^{2l},
\end{split}$$ so the first line equals the sum of the second and third lines by direct calculation.
\[proper\] Later on we will consider general multilinear forms $\mathcal{N}_{n}$ which are simple, and can be written as $$\label{multiform}[{\color{black}\mathcal{N}_{n}(v^{(1)},\cdots,v^{(n)})]_{k}}=\sum_{\iota_1k_1+\cdots +\iota_nk_{n}=k}a_{kk_1\cdots k_{n}}(v_{k_1}^{(1)})^{\iota_1}\cdots (v_{k_{n}}^{(n)})^{\iota_n}.$$ We may assume the coefficient $a_{kk_1\cdots k_{n}}$ is symmetric in the $k_j$’s for which $\iota_j=+$, and also symmetric in the $k_j$’s for which $\iota_j=-$. Moreover, we assume that this coefficient only depends on the *set of pairings* among $\{k,k_1,\cdots,k_{n}\}$.
The multilinear form $\mathcal{N}_{2l+1}$ corresponding to (\[quintcubic\]) satisfies the above properties, and we will assume [ without any loss of generality]{} that $\iota_j=+$ (i.e. $\mathcal{N}_{2l+1}$ is linear in $v^{(j)}$) for $j$ odd, and $\iota_j=-$ (i.e. $\mathcal{N}_{2l+1}$ is conjugate linear in $v^{(j)}$) for $j$ even.
Finite and infinite dimensional measures
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Here we will summarize some properties of the infinite dimensional and finite dimensional (or truncated) Gaussian and Gibbs measures, that will be used later in the proof.
Recall that $\mathcal{V}_N$ and $\mathcal{V}_N^\perp$ are [ respectively the]{} ranges of the projections $\Pi_N$ and $\Pi_N^\perp$. We will identify $\mathcal{V}$ with $\mathcal{V}_N\times\mathcal{V}_N^\perp$. Let [$\mathrm{d}\rho_N$]{} and $\mathrm{d}\rho_N^\perp$ be the Gaussian measures defined on $\mathcal{V}_N$ and $\mathcal{V}_N^\perp$ respectively, such that $\mathrm{d}\rho=\mathrm{d}\rho_N\times\mathrm{d}\rho_N^\perp$. Define the measures $\mathrm{d}\mu_N^\circ$ on $\mathcal{V}_N$ and $\mathrm{d}\mu_N$ on $\mathcal{V}$ by $$\label{meastrunc}\mathrm{d}\mu_N^\circ=Z_N^{-1}e^{-V_N[u]}\,\mathrm{d}\rho_{N},\quad \mathrm{d}\mu_N=Z_N^{-1}e^{-V_N[u]}\,{\color{black}\mathrm{d}\rho};\quad Z_N=\int_{\mathcal{V}_N}e^{-V_N[u]}\,\mathrm{d}\rho_{N}(u),$$ then we have that $\mathrm{d}\mu_N=\mathrm{d}\mu_N^\circ\times\mathrm{d}\rho_N^\perp$. Recall also the measure $\mathrm{d}\mu$ defined in Proposition \[gibbsm\]; all these are probability measures.
\[measurefact\] When $N\to\infty$ we have $Z_N\to Z$, [with $0<Z<\infty$]{}. The sequence $Z_N^{-1}e^{-V_N[u]}$ converges to $Z^{-1}e^{-V[u]}$ almost surely, and also in $L^q(\mathrm{d}\rho)$ for any $1\leq q<\infty$. The measure $\mathrm{d}\mu_N$ converges to $\mathrm{d}\mu$ in the sense that the total variation of $\mu_N-\mu$ converges to $0$. Finally, the measure $\mathrm{d}\mu_N^\circ$ is invariant under the flows of (\[truncnls\]) and (\[gauged\]).
The convergence results are proved in [@OT]. The measure $\mathrm{d}\mu_N^\circ$ is invariant under (\[truncnls\]), because the latter is a finite dimensional Hamiltonian system, and [$$\mathrm{d}\mu_N^\circ(u_N)=\frac{1}{E_N}e^{-\mathcal{H}_N[u_N]-\mathcal{M}[u_N]}\,\mathrm{d}\mathcal{L}_N(u_N)$$]{} is its Gibbs measure (weighted by another conserved quantity), where $E_N$ is some constant, $\mathcal{H}_N$ and $\mathcal{M}$ are as in (\[defham\]), and $\mathrm{d}\mathcal{L}_N$ is the Lebesgue measure on the finite dimensional space $\mathcal{V}_N$.
To prove [ that]{} $\mathrm{d}\mu_N^\circ$ is invariant under (\[gauged\]), it suffices to show that it is preserved[^12] by the gauge transform $\mathcal{G}_N$. In fact, [by (\[wickpoly\]) and (\[defham\]) we know $\mathcal{H}_N[u_N]=\mathcal{H}_N[v_N]$ and $\mathcal{M}[u_N]=\mathcal{M}[v_N]$, so it suffices to prove that $\mathcal{G}_N$ preserves the Lebesgue measure $\mathrm{d}\mathcal{L}_N$.]{} Working in the coordinates $(r_k,\theta_k)_{\langle k\rangle\leq N}$ and $(r_k^*,\theta_k^*)_{\langle k\rangle\leq N}$, which are defined by $(u_N)_k=r_k e^{i\theta_k}$ and $(v_N)_k=r_k^* e^{i\theta_k^*}$, we can write the measure [$\mathrm{d}\mathcal{L}_N$]{} as $$\label{measureN}\mathrm{d}\mathcal{L}_N=\prod_{\langle k\rangle\leq N}r_k\mathrm{d}r_k\mathrm{d}\theta_k.$$ If [$v_N=\mathcal{G}_Nu_N$,]{} then we have $r_k^*=r_k$ and $\theta_k^*=\theta_k+F((r_j,\theta_j)_{\langle j\rangle\leq N})$, where $F$ may also depend on $t$, but does not depend on $k$. Moreover, by (\[wickpoly\]) and (\[gauge\]) we know that $F$ actually depends only on $r_j$ and [ on]{} the differences $\theta_j-\theta_\ell$, which [ are]{} invariant under the mapping $\theta_k\mapsto\theta_k^*$. It then follows that the transformation $(r_k,\theta_k)\mapsto (r_k^*,\theta_k^*)$ preserves the measure (\[measureN\]), [by a simple calculation of its Jacobian.]{}
Function spaces and linear estimates {#funcspace}
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From now on we will work with the equation (\[gauged\]) with the nonlinearity defined by (\[formulaqn\]) and (\[quintcubic\]), which has the form (\[multiform\]). Recall the well-known $X^{s,b}$ spaces (where $b$ may be replaced by $b_1$ or $b_2$) $$\label{xsb}\|u\|_{X^{s,b}}=\|\langle k\rangle^s\langle \lambda\rangle^b\,\widetilde{u}_k(\lambda)\|_{\ell_k^2L_\lambda^2}.$$ We will mostly consider $s=0$ and will denote $X^{0,b}=X^b$. In addition we introduce matrix norms which measure the functions $h=h_{kk^*}(t)$ and $\mathfrak{h}=\mathfrak{h}_{kk'}(t,s)$, namely $$\begin{aligned}
\label{matrixnorm1}\|h\|_{Y^{b}}&=\|\langle\lambda\rangle^{b}\, \widetilde{h}_{kk^*}(\lambda)\|_{\ell_{k^*}^2\to\ell_{k}^2L_\lambda^2},&{\color{black}\|\mathfrak{h}\|_{Y^{b,b}}}&=\|\langle\lambda\rangle^{b}\langle \lambda'\rangle^{-b}\, \widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')\|_{\ell_{k'}^2L_{\lambda'}^2\to\ell_{k}^2L_\lambda^2},\\
\label{matrixnorm2}
\|h\|_{Z^{b}}&=\|\langle\lambda\rangle^{b} \, \widetilde{h}_{kk^*}(\lambda)\|_{\ell_{k,k^*}^2L_\lambda^{2}},&{\color{black}\|\mathfrak{h}\|_{Z^{\widetilde{b},b}}}&=\|\langle\lambda\rangle^{\widetilde{b}}\langle \lambda'\rangle^{-b} \, \widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')\|_{\ell_{k,k'}^2L_{\lambda,\lambda'}^2},\end{aligned}$$ where [$\widetilde{b}\in\{b,b_1\}$]{}, $\|\cdot\|_{\ell_{k^*}^2\to\ell_{k}^2L_\lambda^2}$ and $\|\cdot\|_{\ell_{k'}^2L_\lambda'^2\to\ell_{k}^2L_\lambda^2}$ represent the operator norms of linear operators with the given kernels, for example $$\label{lptolp}{\color{black}\|\mathfrak{h}\|_{Y^{b,b}}}=\sup\bigg\{\bigg\|\sum_{k'}\int\mathrm{d}\mu\cdot\langle \lambda\rangle^{b}\langle\lambda'\rangle^{-b}\, \widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')y_{k'}(\lambda')\bigg\|_{\ell_k^2L_\lambda^2}:\|y_{k'}(\lambda')\|_{\ell_{k'}^2L_{\lambda'}^2}=1\bigg\}.$$ By definition one can verify that $$\label{matrixnorm0}{\color{black}\|\mathfrak{h}\|_{Y^{b,b}}}=\sup_{\|y\|_{X^b}=1}\bigg\|\sum_{k'}\int\mathrm{d}s\cdot \mathfrak{h}_{kk'}(t,s)y_{k'}(s)\bigg\|_{X^{b}}.$$For any of the above spaces, we can localize them in the standard way to a time interval $J$, $$\label{localize}\|u\|_{\mathcal{Z}(J)}=\inf\{\|v\|_{\mathcal{Z}}:v\equiv u\textrm{ on }J\}.$$ We will need the following simple estimates.
The norms $\|h\|_{Y^{b}}$ and $\|\mathfrak{h}\|_{Y^{b,b}}$ do not increase, when [$\widetilde{h}$ or $\widetilde{\mathfrak{h}}$ is multiplied by a function of $(k,\lambda)$, or a function of $k^*$ (or $(k',\lambda')$ for $\widetilde{\mathfrak{h}}$), which is at most $1$ in $l^\infty L^\infty$ or $l^\infty$ norms.]{}
Next, if [ $H$ is defined by]{} $$\label{operatorbd4}\widetilde{H}_{kk^*}(\lambda) =\sum_{k'}\int \mathrm{d}\lambda'\cdot{\color{black}\widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')}\, \widetilde{h}_{k'k^*}(\lambda'),$$ where ${\color{black}\widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')}$ is supported in $|k-k'|\lesssim L$, then for any $\alpha>0$ we have $$\label{operatorbd3}\bigg\|\bigg(1+\frac{|k-k^*|}{L}\bigg)^{\alpha}H\bigg\|_{Z^{b}}\lesssim \|\mathfrak{h}\|_{Y^{b,b}}\cdot\bigg\|\bigg(1+\frac{|k'-k^*|}{L}\bigg)^{\alpha}h\bigg\|_{Z^b}.$$
[The first statement follows directly from definition (\[lptolp\]).]{} Now let us prove (\[operatorbd3\]). We may fix $k^*$ and by translation invariance, we may assume $k^*=0$. [ Relabeling $\widetilde{H}_{k0}(\lambda) =:\widetilde{H}_k(\lambda)$ and $\widetilde{h}_{k'0}(\lambda) =:\widetilde{h}_{k'}(\lambda)$]{} we may decompose $$\widetilde{H}_k(\lambda)={\color{black}\sum_{M\geq L}}(\widetilde{H}^M)_k(\lambda);\quad {\color{black}(\widetilde{H}^M)_k(\lambda)=\left\{
\begin{aligned}&\mathbf{1}_{|k|\sim M}\widetilde{H}_k(\lambda),&M&> L,\\
&\mathbf{1}_{|k|\lesssim L}\widetilde{H}_k(\lambda),&M&= L,
\end{aligned}
\right.}$$ and similarly for $\widetilde{h}$, so that we have $$\label{dydsum} \bigg\|{\color{black}\langle\lambda\rangle^b}\bigg(1+\frac{|k|}{L}\bigg)^{\alpha}\widetilde{H}_{k}(\lambda)\bigg\|_{\ell_{k}^2L_\lambda^2}^2\sim{\color{black}\sum_{M\geq L}}L^{-2\alpha}M^{2\alpha}\|{\color{black}\langle\lambda\rangle^b}(\widetilde{H}^M)_k(\lambda)\|_{\ell_k^2L_\lambda^2}^2$$ and similarly for $h$. Since [$\widetilde{\mathfrak{h}}$ is supported in]{} $|k-k'|\lesssim L$, we have $$|(\widetilde{H}^M)_k(\lambda)|\leq\sum_{M'\sim M}\bigg|\sum_{k'}\int\mathrm{d}\lambda'\cdot\widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')(\widetilde{h}^{M'})_{k'}(\lambda')\bigg|,$$ therefore $$\|\langle \lambda\rangle^{b}(\widetilde{H}^M)_k(\lambda)\|_{\ell_k^2L_\lambda^2}^2\lesssim\|\langle \lambda\rangle^{b}\langle\lambda'\rangle^{-b}\widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')\|_{\ell_{k'}^2L_{\lambda'}^2\to\ell_k^2L_\lambda^2}^2\sum_{M'\sim M}\|\langle \lambda'\rangle^{b}(\widetilde{h}^{M'})_{k'}(\lambda')\|_{\ell_{k'}^2L_\lambda'^2}^2,$$ which, combined with (\[dydsum\]), implies (\[operatorbd3\]).
[Let $\chi$ be a smooth cutoff as in Section \[notations\], and define the time truncated Duhamel operator $$\label{duhameloper}\mathcal{I}F(t)=\chi(t)\int_0^te^{i(t-t')\Delta}\chi(t')F(t')\,\mathrm{d}t'.$$]{}
\[duhamelest\] [We have $2\mathcal{I}F(t)=\mathcal{J}F(t)-\chi(t)e^{it\Delta}\mathcal{J}F(0)$, where $\mathcal{J}$ is defined by $$\label{defjop}\mathcal{J}F(t)=\chi(t)\bigg(\int_{-\infty}^t-\int_t^{\infty}\bigg)e^{i(t-t')\Delta}\chi(t')F(t')\,\mathrm{d}t'.$$ Moreover we have the formula $$\label{trunckernel}\widetilde{\mathcal{J}F}(k,\lambda)=\int_{\mathbb{R}}\mathcal{J}(\lambda,\mu)\widetilde{F}(k,\mu)\,\mathrm{d}\mu,\quad |\partial_{\lambda,\mu}^\alpha\mathcal{J}(\lambda,\mu)|\lesssim_{\alpha,A}\frac{1}{\langle \lambda-\mu\rangle^{A}}\frac{1}{\langle \mu\rangle}.$$]{}
For the proof of Lemma \[duhamelest\], see the calculations in [@DNY], Lemma 3.1.
\[sttime\] Let $\varphi$ be any Schwartz function, recall that $\varphi_\tau(t)=\varphi(\tau^{-1}t)$ for any $0<\tau\ll1$. Then for any $u=u_k(t)$ and $\mathfrak{h}=\mathfrak{h}_{kk'}(t,t')$ we have [$$\label{sttime1}\|\varphi_{\tau}\cdot u\|_{X^{s,b}}\lesssim {\tau}^{b_1-b}\|u\|_{X^{s,b_1}},\quad \|\varphi_{\tau}(t)\cdot \mathfrak{h}\|_{Z^{b,b}}\lesssim {\tau}^{b_1-b}\|\mathfrak{h}\|_{Z^{b_1,b}},$$]{} provided that $u_k(0)=\mathfrak{h}_{kk'}(0,t')=0$.
Using definition of $Z^{\widetilde{b},b}$ norms and fixing the $(k',\lambda')$ variables, we can reduce the second inequality in (\[sttime1\]) to the first, and by fixing $k$ and conjugating by the linear Schrödinger flow, we can reduce the first to $$\|\langle\xi\rangle^{b}(\widehat{\varphi_{\tau}}*\widehat{v})(\xi)\|_{L^2}\lesssim {\tau}^{b_1-b} \, \|\langle \eta\rangle^{b_1}\widehat{v}(\eta)\|_{L^{2}}$$ for $v$ satisfying $v(0)=0$. Let $\widehat{v}=g_1+g_2$ where $$g_1(\xi)=\mathbf{1}_{|\xi|\geq {\tau}^{-1}}(\xi)\widehat{v}(\xi),\quad g_2(\sigma)=\mathbf{1}_{|\xi|< {\tau}^{-1}}(\xi)\widehat{v}(\xi).$$We will prove that $$\label{st21}\|\langle\xi\rangle^{b}(\widehat{\varphi_{\tau}}*g_1)(\xi)\|_{L^2}\lesssim {\tau}^{b_1-b} \, \|\langle \eta\rangle^{b_1}\widehat{v}(\eta)\|_{L^{2}},$$ $$\label{st22}\|\langle\xi\rangle^{b}(\widehat{\varphi_{\tau}}*g_2)(\xi)\|_{L^2}\lesssim {\tau}^{b_1-b} \,\|\langle \eta\rangle^{b_1}\widehat{v}(\eta)\|_{L^{2}}.$$ To prove (\[st21\]), we can reduce it to the $L^{2}\to L^2$ bound for the operator $$g(\eta)\mapsto\int_{\mathbb{R}}R(\xi,\eta)g(\eta)\,\mathrm{d}\eta,\quad R(\xi,\eta)=\mathbf{1}_{|\eta|\geq {\tau}^{-1}}\cdot \tau\widehat{\varphi}(\tau(\xi-\eta))\frac{\langle\xi\rangle^{b}}{\langle \eta\rangle^{b_1}}.$$ Since $$\mathbf{1}_{|\eta|\geq {\tau}^{-1}}\cdot\frac{\langle\xi\rangle^{b}}{\langle \eta\rangle^{b_1}}\lesssim {\tau}^{b_1-b}\frac{\langle T\xi\rangle^{b}}{\langle \tau\eta\rangle^{b_1}}\lesssim {\tau}^{b_1-b}\langle \tau(\xi-\eta)\rangle^{b},$$ it follows from Schur’s estimate that this $L^{2}\to L^2$ bound is at most$$\tau^{b_1-b}\|\tau\widehat{\varphi}(\tau\zeta)\langle \tau\zeta\rangle^{b}\|_{L_\zeta^1}\lesssim {\tau}^{b_1-b},$$ which proves (\[st21\]).
To prove (\[st22\]), note that by $v(0)=0$ we have $\int_{\mathbb{R}}\widehat{v}(\eta)\,\mathrm{d}\eta=0$, so $$\begin{split}|(\widehat{\varphi_{\tau}}*g_2)(\xi)|&=\bigg|-\tau\widehat{\varphi}(\tau\xi)\int_{|\eta|\geq \tau^{-1}}\widehat{v}(\eta)\,\mathrm{d}\eta-\int_{|\eta|<\tau^{-1}}\tau\widehat{v}(\eta)\big[\widehat{\varphi}(\tau\xi)-\widehat{\varphi}(\tau(\xi-\eta))\big]\,\mathrm{d}\eta\bigg|\\
&\lesssim \tau\langle \tau\xi\rangle^{-4}\int_{\mathbb{R}}\min(1,|\tau\eta|)|\widehat{v}(\eta)|\,\mathrm{d}\eta,
\end{split}$$ and by Hölder we have $$\int_{\mathbb{R}}\min(1,|\tau\eta|)|\widehat{v}(\eta)|\,\mathrm{d}\eta\lesssim \|\langle \eta\rangle^{b_1}\widehat{v}(\eta)\|_{L^{2}}\cdot\|\min(1,|\tau\eta|)\langle\eta\rangle^{-b_1}\|_{L^2}\lesssim {\tau}^{b_1- \frac{1}{2}}\|\langle \eta\rangle^{b_1}\widehat{v}(\eta)\|_{L^{2}}.$$ Using also the elementary bound $$\|\tau\langle \tau\xi\rangle^{-4}\langle\xi\rangle^{b}\|_{L^2}\lesssim {\tau}^{\frac{1}{2}-b},$$ we deduce (\[st22\]) and hence (\[sttime1\]).
Structure of solution: [random averaging operators]{} {#structuresol}
=====================================================
The decomposition {#decompsol}
-----------------
[We now fix a short time $0<\tau\ll 1$, and establish the local theory for (\[gauged\]), with initial data distributed according to the Gaussian measure $\mathrm{d}\rho_{N}$, on $J:=[-\tau,\tau]$.]{} By definition, this is equivalent to considering (\[gauged\]) with random initial data $u_{\mathrm{in}}=v_{\mathrm{in}}=f(\omega)$, which we will assume from now on, until the end of Section \[multiest\]. All functions that appear in the proof will be random (i.e. depends on $\omega$), whether or not we explicitly write $\omega$ in their expressions.
Recall [ that]{} the truncated mass $m_N$ defined in (\[truncmass\]) and the corresponding $m_N^*$ are random variables given by $$\label{truncmass2}m_N=\sum_{\langle k\rangle\leq N}\frac{|g_k|^2}{\langle k\rangle^2},\quad m_N^*=\sum_{\langle k\rangle\leq N}\frac{|g_k|^2-1}{\langle k\rangle^2}.$$ [ Note that they are Borel functions of $|g_k|^2$ for $\langle k\rangle\leq N$.]{} Let [$\nu_N:=m_N^*-m_{\frac{N}{2}}^*$.]{} By standard large deviation estimates we have[$$\label{massdiff}\mathbb{P}(|\nu_N|\geq AN^{-1})\leq Ce^{-C^{-1}A}$$]{} [ for any $A>0$, where $C$ is an absolute constant.]{} In particular, by removing a set of measure [$\leq C_\theta e^{-{\tau}^{-\theta}}$]{} (which will be done before proving any estimates) we may assume the following bounds, which [ are]{} used below without [ any further]{} mentioning: $$\label{simplebd}|g_k|\lesssim {\tau}^{-\theta}\langle k\rangle^\theta,\quad |m_N^*|\lesssim {\tau}^{-\theta},\quad {\color{black}|\nu_N|}\lesssim {\tau}^{-\theta} N^{-1+\theta}.$$
Our goal here is to obtain a quantitative estimate for the difference $y_N:=v_N-v_{\frac{N}{2}}$. By (\[gauged\]), this $y_N$ satisfies the equation [ $$\label{nlsdiff}
\left\{
\begin{split}(i\partial_t+\Delta)y_N&=\Pi_N\mathcal{Q}_N(y_N+v_{\frac{N}{2}})-\Pi_{\frac{N}{2}}\mathcal{Q}_{\frac{N}{2}}(v_{\frac{N}{2}}),\\
y_N(0)&=\Delta_Nf(\omega).
\end{split}
\right.$$]{} By (\[formulaqn\]) we can rewrite the above equation as $$\label{nlsdiff2}
\left\{
\begin{aligned}
(i\partial_t+\Delta)y_N&=\sum_{l=0}^rc_{rl} \, (m_N^*)^{r-l}\bigg\{\Pi_N\big[\mathcal{N}_{2l+1}(y_N+v_{\frac{N}{2}})-\mathcal{N}_{2l+1}(v_{\frac{N}{2}})\big]+\Delta_N\mathcal{N}_{2l+1}(v_{\frac{N}{2}})\bigg\}\\
&+\sum_{j=0}^rc_{rl}\, \big[(m_{\frac{N}{2}}^*+{\color{black}\nu_{N}})^{r-l}-(m_{\frac{N}{2}}^*)^{r-l}\big]\cdot\Pi_\frac{N}{2}\mathcal{N}_{2l+1}(v_{\frac{N}{2}}),\\
y_N(0)&=\Delta_Nf(\omega),
\end{aligned}
\right.$$ where $c_{rl}$ are constants that will not be important in the proof.
Let the set $$\label{setNL}\mathcal{K}:=\{(N,L)\in(2^{\mathbb{Z}})^2:2^{-1}\leq L< N^{1-\delta}\}.$$ For each $(N,L)\in\mathcal{K}$, we define the function $\psi_{N,L}$ as the solution to the (linear) equation $$\label{defpsi}
\left\{
\begin{aligned}
(i\partial_t+\Delta)\psi_{N,L}&=\sum_{l=0}^r(l+1) \, c_{rl}\, (m_N^*)^{r-l}\, \Pi_N\mathcal{N}_{2l+1}(\psi_{N,L},v_{L},\cdots,v_{L}),\\
\psi_{N,L}(0)&=\Delta_Nf(\omega).
\end{aligned}
\right.$$ [ It is important to place $\psi_{N, L}$ in the first position of $\mathcal{N}_{2l+1}$ in (\[defpsi\]), see Remark \[gamma2\].]{} By linearity we have, $$\label{linearity}(\psi_{N,L})_k=\sum_{k^*}\, H_{kk^*}^{N,L}\, \frac{g_{k^*}(\omega)}{\langle k^*\rangle},$$ where for $\frac{N}{2}<\langle k^*\rangle\leq N$ and $\langle k\rangle\leq N$, $H_{kk^*}^{N,L}=\varphi_k$ is the $k$-th mode of the solution $\varphi$ to the equation $$\label{defpsi2}
\left\{
\begin{aligned}
(i\partial_t+\Delta)\varphi&=\sum_{l=0}^r(l+1)c_{rl}(m_N^*)^{r-l}\Pi_N\mathcal{N}_{2l+1}(\varphi,v_{L},\cdots,v_{L}),\\
\varphi(0)&=e^{ik^*\cdot x},
\end{aligned}
\right.$$ and for other $(k,k^*)\in(\mathbb{Z}^2)^2$ define $H_{kk^*}^{N,L}=0$. By definition these $H_{kk^*}^{N,L}$, as well as [ the]{} $h_{kk^*}^{N,L}$ defined below, are $\mathcal{B}_{\leq N}$ measurable and [$\mathcal{B}_{\leq L}^+$]{} measurable [ in the sense of Definition \[borel\]]{}.
For any $N$, let $L_0$ be the largest $L$ satisfying $(N,L)\in\mathcal{K}$. We further define $$\label{matrices}\zeta_{N,L}: =\psi_{N,L} -\psi_{N,\frac{L}{2}},\qquad h^{N,L}: =H^{N,L}-H^{N,\frac{L}{2}}; \qquad z_N: =y_N-\psi_{N,L_0}.$$ Note that [$\psi_{N,\frac{1}{2}}=e^{it\Delta}(\Delta_Nf(\omega))$]{}, and [ that]{} $H_{kk^*}^{N,\frac{1}{2}}$ is [ $e^{-i|k|^2t}\mathbf{1}_{k=k^*}$ restricted to [ the frequency band]{} $\frac{N}{2}<\langle k\rangle\leq N$.]{} [Moreover $z_N$ is $\mathcal{B}_{\leq N}$ measurable, $z_N(0)=0$ and satisfies the equation]{} $$\begin{aligned}
{\color{black}(i\partial_t+\Delta)z_N}&=\sum_{l=0}^rc_{rl}(m_N^*)^{r-l}\cdot\Pi_N\big[\mathcal{N}_{2l+1}(z_N+\psi_{N,L_0}+v_{\frac{N}{2}})-\mathcal{N}_{2l+1}(v_{\frac{N}{2}})+\Delta_N\mathcal{N}_{2l+1}(v_{\frac{N}{2}})\big]\nonumber\\
&-\sum_{l=0}^rc_{rl}(m_N^*)^{r-l}\cdot\Pi_N\big[(l+1)\mathcal{N}_{2l+1}(\psi_{N,L_0},v_{L_0},\cdots,v_{L_0})\big]\nonumber\\
\label{equationz}&+\sum_{l=0}^rc_{rl}\big[(m_{\frac{N}{2}}^*+\nu_N)^{r-l}-(m_{\frac{N}{2}}^*)^{r-l}\big]\cdot\Pi_{\frac{N}{2}}\mathcal{N}_{2l+1}(v_{\frac{N}{2}}).\end{aligned}$$
[With the above construction, if we let $v=\lim_{N\to\infty}v_N$ be the gauged version of the solution $u$ to (\[nls\]), then we have $$\label{ansatz1}
v = e^{it\Delta}f(\omega)+\sum_{(N,L)\in\mathcal{K}}\zeta_{N,L}+z,\quad \text{where} \quad z=\sum_Nz_N.$$]{} [This is the ansatz (\[fullansatz\]) in Section \[secansatz\], where $\zeta_{N,L}$ can be viewed as a random [averaging]{} operator $\mathcal{P}_{NL}$, whose kernel is essentially given by $h^{N,L}$, applied to [the Gaussian free field]{} $e^{it\Delta}f(\omega)$. There are however two differences: (1) our $\mathcal{P}_{NL}$ is not exactly the one in (\[randomop\]), but an infinite iteration of the latter, because (\[randomop\]) has no smoothing effect; and (2) our $\mathcal{P}_{NL}$ is not exactly a Borel function of $(g_k)_{\langle k\rangle\leq L}$ as it also depends on $m_N^*$, but as it turns out this does not affect any probabilistic estimates, see Lemma \[largedev\].]{}
The a priori bounds {#apriori}
-------------------
We now state the local well-posedness result for (\[gauged\]). Its proof will occupy the rest of this section and Sections \[prep2\] and \[multiest\].
\[localmain\] Recall the relevant constants defined in (\[defparam\]), and that $\tau\ll 1$, $J=[-\tau,\tau]$. Then, ${\tau}^{-1}$-certainly, i.e. with probability $\geq 1-C_{\theta}e^{-{\tau}^{-\theta}}$, the following estimates hold for all $(N,L)\in\mathcal{K}$: $$\label{aprioriest}
\|h^{N,L}\|_{Y^{b}(J)}\leq L^{-\delta_0},\quad\|h^{N,L}\|_{Z^{b}(J)}\leq N^{\frac{1}{2}+\delta^\frac{5}{4}}L^{-\frac{1}{2}},\quad
\|z_N\|_{X^{b}(J)}\leq N^{-1+\gamma}.$$
### The extensions {#extension}
In proving Proposition \[localmain\] we will restrict $z_N$ and $h^{N,L}$ to $J$ and construct extensions of these restrictions [that are defined for all time]{}. This has to be done carefully so as to maintain the correct independence properties. We define these extensions inductively, as follows.
First, let $z_1^\dagger(t):=z_1(t)\chi_{\tau}(t)$ and $ {\color{black}\psi_{N,\frac{1}{2}}^\dagger(t)}: =\chi(t)e^{it\Delta}(\Delta_Nf(\omega))$, and define $H^{N,\frac{1}{2},\dagger}$ accordingly. Suppose $M\geq 1$ is a dyadic number and we have defined $z_N^\dagger$ for all $N\leq M$ and $h^{N,L,\dagger}$ for all [$(N,L)\in\mathcal{K}$]{} and $L<M$, then for $L\leq M$ we may define [ $$v_L^\dagger= \sum_{L'\leq L}y_{L'}^\dagger,\quad \text{ where } \quad y_L^\dagger=z_{L}^\dagger\, +\, \chi(t)e^{it\Delta}(\Delta_Lf(\omega))+\sum_{(L,R)\in\mathcal{K}}\zeta_{L,R}^{\dagger},$$ ]{} which is acceptable since for $(L,R)\in\mathcal{K}$ we must have $R<L^{1-\delta}\leq M$, so $h^{L,R,\dagger}$ and $H^{L,R,\dagger}$ are well-defined, hence $\zeta_{L,R}^\dagger$ and $\psi_{L,R}^\dagger$ can be defined by (\[linearity\]) and (\[matrices\]).
Next, for $(N,L)\in\mathcal{K}$ and $L=M$, we can define $H^{N,M,\dagger}$ such that for $\frac{N}{2}<\langle k^*\rangle\leq N$ and $\langle k\rangle\leq N$, $H_{kk^*}^{N,M,\dagger}=\varphi_k^\dagger$ is the $k$-th mode of the solution $\varphi^\dagger$ to the equation $$\label{daggered}\varphi^{\dagger}(t)=\chi(t)e^{it\Delta}(e^{ik^*\cdot x})-i\chi_{\tau}(t)\sum_{l=0}^r(l+1)\,c_{rl}\,(m_N^*)^{r-l}\cdot\mathcal{I}\Pi_N\mathcal{N}_{2l+1}\big(\varphi^\dagger,v_{M}^\dagger,\cdots,v_{M}^\dagger\big),$$ [provided this solution exists and is unique; otherwise simply define $H^{N,M,\dagger}=H^{N,M}\cdot\chi_\tau(t)$.]{} This defines $H^{N,M,\dagger}$ and hence also $h^{N,M,\dagger}$, $\psi_{N,M}^\dagger$ and $\zeta_{N,M}^\dagger$.
Finally we will define $z_{2M}$. As $\psi_{2M,L_0}^\dagger$ is already defined, where $L_0\leq M$ is the largest $L$ such that $(2M,L)\in\mathcal{K}$, we can define $z_{2M}^\dagger$ to be the unique fixed point of the mapping $$\label{fixedpoint}
\begin{aligned}z\mapsto &-i\chi_{\tau}(t)\sum_{l=0}^rc_{rl}\, (m_{2M}^*)^{r-l}\cdot\mathcal{I}\Pi_{2M}\bigg\{\mathcal{N}_{2l+1}\big(z+\psi_{2M,L_0}^\dagger+v_{M}^\dagger\big)-\mathcal{N}_{2l+1}\big(v_{M}^\dagger\big)\bigg\}\\
&+i\chi_{\tau}(t)\sum_{l=0}^r(l+1)c_{rl}\, (m_N^*)^{r-l}\cdot\mathcal{I}\Pi_{2M}\mathcal{N}_{2l+1}\big(\psi_{2M,L_0}^\dagger,v_{L_0}^{\dagger},\cdots,v_{L_0}^{\dagger}\big)
\\&-i\chi_{\tau}(t)\sum_{l=0}^rc_{rl}\, \big[(m_M^*+\nu_{2M})^{r-l}-(m_M^*)^{r-l}\big]\cdot\mathcal{I}\Pi_{M}\mathcal{N}_{2l+1}\big(v_M^{\dagger}\big)\\
&-i\chi_{\tau}(t)\sum_{l=0}^rc_{rl}\, (m_{2M}^*)^{r-l}\cdot\mathcal{I}\Delta_{2M}\mathcal{N}_{2j+1}\big(v_M^\dagger\big)
\end{aligned}$$ on the set $\mathcal{Z}=\{z:\|z\|_{X^b}\leq (2M)^{-1+\gamma}\}$, provided that this mapping is a contraction mapping from $\mathcal{Z}$ to itself. If it is not a contraction mapping, then simply define $z_{2M}^\dagger=z_{2M}\cdot\chi_{\tau}(t)$. This completes the inductive construction. [ One may then easily]{} verify that:
The $z_N^\dagger$ and $h^{N,L,\dagger}$ we constructed are indeed extensions of $z_N$ and $h^{N,L}$;
The $z_N^\dagger$ is supported in $\langle k\rangle\leq N$, and $h_{kk^*}^{N,L,\dagger}$ is supported in $\langle k\rangle\leq N$ and $\frac{N}{2}<\langle k^*\rangle\leq N$;
The random variable $h^{N,L,\dagger}$ is $\mathcal{B}_{\leq L}^+$ measurable, [and $z_N^\dagger$ and $h^{N,L,\dagger}$ are $\mathcal{B}_{\leq N}$ measurable;]{}
All the above are smooth and compactly supported in time $t\in[-2,2]$.
We will prove Proposition \[localmain\] by induction in $M$, but in the process we will need some auxiliary estimates. More precisely, we will prove the following result, which contains Proposition \[localmain\]:
\[localmain2\] [ Recall the relevant constants defined in (\[defparam\]), and that $\tau\ll 1$.]{} Consider the following statement which we call $\mathtt{Loc}(M)$ for $M\geq 1$: for any $(N,L)\in\mathcal{K}$ with $L<M$, we have $$\begin{aligned}
\label{induct1}
&\|h^{N,L,\dagger}\|_{Y^b}\leq L^{-\delta_0};\\
\label{induct2}
&\|h^{N,L,\dagger}\|_{Z^b}\leq N^{\frac{1}{2}+\gamma_0}L^{-\frac{1}{2}};\\
\label{induct3}&\bigg\|\bigg(1+\frac{|k-k^*|}{L}\bigg)^{\kappa}h_{kk^*}^{N,L,\dagger}\bigg\|_{Z^{b}}\leq N.\end{aligned}$$ Define the operators[^13] (where $0\leq l\leq r$) $$\begin{aligned}
\label{devop1}\mathcal{P}^+(w)&:=\chi_{\tau}(t)\cdot\mathcal{I}\Pi_N\big[\mathcal{N}_{2l+1}(w,v_L^\dagger,\cdots,v_L^\dagger)-\mathcal{N}_{2l+1}(w,v_\frac{L}{2}^\dagger,\cdots,v_{\frac{L}{2}}^\dagger)\big],\\
\label{defop2}\mathcal{P}^-(w)&:=\chi_{\tau}(t)\cdot\mathcal{I}\Pi_N\big[\mathcal{N}_{2l+1}(v_L^\dagger,w,v_L^\dagger,\cdots,v_L^\dagger)-\mathcal{N}_{2l+1}(v_\frac{L}{2}^\dagger,w,v_{\frac{L}{2}}^\dagger,\cdots,v_{\frac{L}{2}}^\dagger)\big],\end{aligned}$$ then for any $(N,L)\in\mathcal{K}$ [ as defined in ]{} with $L<M$ we have $$\label{induct4}\|\mathcal{P}^{\pm}\|_{X^{b}\to X^{b}}\leq \tau^{\theta}L^{-\delta_0^{\frac{1}{2}}}.$$ Let the kernel of $\mathcal{P}^+$ be $\mathfrak{h}_{kk'}^{N,L}(t,t')$, then for any $(N,L)\in\mathcal{K}$ with $L<M$ we have[ $$\label{induct5}\|\mathbf{1}_{|k|,|k'|\geq \frac{N}{4}}\cdot\mathfrak{h}_{kk'}^{N,L}(t,t')\|_{Z^{b,b}}\leq \tau^{\theta}N^{\frac{1}{2}+\gamma_0-\delta^3}L^{-\frac{1}{2}}.$$]{} Finally for any $N\leq M$ we have $$\label{induct6}\|z_N^\dagger\|_{X^b}\leq N^{-1+\gamma}.$$
Now suppose the statement $\mathtt{Loc}(M)$ is true for $\omega\in\Xi$, where $\Xi$ is a set, then the statement $\mathtt{Loc}(2M)$ is true for $\omega\in \Xi'$ where $\Xi'$ is another set such that $\mathbb{P}(\Xi\backslash\Xi')\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$. In particular, apart from a set of $\omega$ with probability $\leq C_{\theta}e^{-\tau^{-\theta}}$, the statement $\mathtt{Loc}(M)$ is true for all $M$.
[ The proof of Proposition \[localmain2\]:]{} reduction to multilinear estimates {#reductmulti}
--------------------------------------------------------------------------------
The heart of the proof of Proposition \[localmain2\] is a collection of (probabilistic) multilinear estimates for $\mathcal{N}_{2l+1}$. We will state them [ in Proposition \[multi0\] below]{} and show that they imply Proposition \[localmain2\]. [ We leave the proof of Proposition \[multi0\]]{} to Section \[multiest\].
\[multi0\] [ Recall the relevant constants defined in (\[defparam\]), and that $\tau\ll 1$.]{} Let the multilinear form $\mathcal{N}_{n}$ be as in (\[multiform\]), where $1\leq n\leq 2r+1$. We will also consider $\mathcal{N}_{n+1}$, in which we assume $\iota_1=+$. For each $1\leq j\leq n$, the input function $v^{(j)}$ satisfies one of the followings:
\(i) Type (G), where we define $L_j=1$, and $$\label{input1}{\color{black}(\widetilde{v^{(j)}})_{k_j}(\lambda_j)}=\mathbf{1}_{N_j/2< \langle k_j\rangle\leq N_j}\frac{g_{k_j}(\omega)}{\langle k_j\rangle}\widehat{\chi}(\lambda_j).$$
\(ii) Type (C), where $$\label{input2}{\color{black}(\widetilde{v^{(j)}})_{k_j}(\lambda_j)}=\sum_{N_j/2<\langle k_j^*\rangle\leq N_j}h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)\frac{g_{k_j^*}(\omega)}{\langle k_j^*\rangle},$$ with $h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)$ supported in the set $\big\{\langle k_j\rangle\leq N_j,\frac{N_j}{2}<\langle k_j^*\rangle\leq N_j\big\}$, $\mathcal{B}_{\leq N_j}$ mesurable and $\mathcal{B}_{\leq L_j}^+$ measurable for some $L_j\leq N_j^{1-\delta}$, and satisfying the bounds $$\label{input2+}
\begin{aligned}\|\langle \lambda_j\rangle^{b}h_{k_jk_j^*}^{(j)}(\lambda_j)\|_{\ell_{k_j^*}^2\to\ell_{k_j}^2L_{\lambda_j}^2}\lesssim L_j^{-\delta_0},\quad \| \langle \lambda_j\rangle^{b}h_{k_jk_j^*}^{(j)}(\lambda_j)\|_{\ell_{k_j,k_j^*}^2L_{\lambda_j}^2}&\lesssim N_j^{\frac{1}{2}+\gamma_0}L_j^{-\frac{1}{2}},\\\bigg\|\langle \lambda_j\rangle^{b}\bigg(1+\frac{|k_j-k_j^*|}{L_j}\bigg)^\kappa h_{k_jk_j^*}^{(j)}(\lambda_j)\bigg\|_{\ell_{k_j,k_j^*}^2L_{\lambda_j}^2}&\lesssim N_j.
\end{aligned}$$
\(iii) Type (D), where ${\color{black}(\widetilde{v^{(j)}})_{k_j}(\lambda_j)}$ is supported in $\{|k_j|\lesssim N_j\}$, and satisfies $$\label{input3}\|\langle\lambda_j\rangle^b{\color{black}(\widetilde{v^{(j)}})_{k_j}(\lambda_j)}\|_{\ell_{k_j}^2L_{\lambda_j}^2}\lesssim N_j^{-(1-\gamma)}.$$ [In each case, we will assume that [ derivatives in $\lambda_j$]{} of these functions satisfy the same bounds. This can always be guaranteed, since in practice everything will be compactly supported in time.]{}
Assume for $n_1\leq n$ that $v^{(j)}$ are of type (D) for $n_1+1\leq j\leq n$, and of type (G) or (C) for $1\leq j\leq n_1$. Let $\mathcal{G}$ and $\mathcal{C}$ be the sets of $j$ such that $v^{(j)}$ are of type (G) and (C) respectively, similarly denote by $\mathcal{D}:=\{n_1+1,\cdots,n\}$. Let $N^{(j)}=\max^{(j)}(N_1,\cdots,N_n)$ as before, and let [ $1\leq a\leq n$ be such that $N^{(1)}\sim N_a$.]{} Given $N_*\geq 1$, the followings hold $\tau^{-1}N_*$-certainly. [We emphasize that the exceptional set of $\omega$ removed does *not* depend on the choice of the functions $v_j(j\geq n_1+1)$.]{}
\(1) If $a\geq n_1+1$ (say $a=n$) and $N_*\gtrsim N^{(2)}$, then we have (recall $b_1=b+\delta^4$) $$\label{mainmult1}\|\mathcal{I}\mathcal{N}_{n}(v^{(1)},\cdots,v^{(n)})\|_{X^{b_1}}\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}} (N^{(1)})^{-1+\gamma}(N^{(2)})^{-\delta_0^{\frac{1}{3}}}.$$ Here the exceptional set does not depend on $N^{(1)}$.
\(2) If $a\leq n_1$ and $N_*\gtrsim N^{(1)}$, then we have $$\label{mainmult2}\|\mathcal{I}\mathcal{N}_{n}(v^{(1)},\cdots,v^{(n)})\|_{X^{b_1}}\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}} (N^{(1)}N^{(2)})^{-\frac{1}{2}(1-\gamma_0)}.$$ If moreover $\iota_a=-$, then we have the stronger bound $$\label{mainmult3}\|\mathcal{I}\mathcal{N}_{n}(v^{(1)},\cdots,v^{(n)})\|_{X^{b_1}}\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}} (N^{(1)})^{-(1-\gamma_0)}.$$ If moreover $\iota_a=+$ and $N^{(2)}\lesssim (N^{(1)})^{1-\delta}$, then we have stronger bound for the projected term $$\label{mainmult4}\|\mathcal{I}\Pi_{N^{(1)}}^\perp\mathcal{N}_{n}(v^{(1)},\cdots,v^{(n)})\|_{X^{b_1}}\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}} (N^{(1)})^{-(1-\frac{4\gamma}{5})}.$$
\(3) Now consider the operator $$\mathcal{Q}^+(w):=\mathcal{I}\Pi_{N_0}\mathcal{N}_{n+1}(\Pi_{N_0} w,v^{(1)},\cdots,v^{(n)}),$$ and [let its kernel be]{} [$\mathfrak{h}_{kk'}(t,t')$. If $N^{(1)}\lesssim N_0^{1-\delta}$]{} and $N_*\gtrsim N_0$, then we have [$$\label{mainmult5}\|\mathbf{1}_{|k|,|k'|\geq \frac{N_0}{4}}\cdot\mathfrak{h}_{kk'}(t,t')\|_{Z^{b_1,b}}\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}N_0^{\frac{1}{2}}(N^{(1)})^{-\frac{1}{2}+\gamma_0}.$$]{}
\[gamma2\] The improvement (\[mainmult4\]) is due to the exact projection $\Pi_{N^{(1)}}^\perp$. In fact this implies that in the expression (\[multiform\]) there exists some $1\leq a\leq n$ and some $\Gamma$, namely $\Gamma=(N^{(1)})^2-1$, such that $$\label{gammacon}|k|^2\geq \Gamma\geq |k_a|^2\quad\textrm{or}\quad |k|^2\leq \Gamma\leq |k_a|^2,\quad\textrm{and }N^{(1)}\sim N_a.$$ We call (\[gammacon\]) the *$\Gamma$-condition*. If we put some other projections in $\mathcal{N}_n$ that also guarantee (\[gammacon\]), for example $\Pi_M\mathcal{N}_{n}(\cdots,\Pi_M^\perp v^{(a)},\cdots)$ where $N^{(1)}\sim N_a$, then the same improvement (\[mainmult4\]) will remain true.
[In the proof below we will see that the $\Gamma$ condition provides the needed improvements in the case $N^{(1)}\sim N_a$ and $\iota_a=+$. This is the reason why we place $\psi_{N, L}$ in the first position of $\mathcal{N}_{2l+1}$ in (\[defpsi\]). On the other hand, the term where $\psi_{N, L}$ is placed in the second position can be handled using the improvement (\[mainmult3\]).]{}
To prove the statement $\mathtt{Loc}(2M)$ we start with (\[induct4\]) and (\[induct5\]), and may assume $L=M$. The proof for $\mathcal{P}^-$ in (\[induct4\]) will be similar, so let us consider $\mathcal{P}^+$. Since $v_M^\dagger=\sum_{L'\leq M}y_{L'}^\dagger$, by definition we can write $\mathcal{P}^+$ as a superposition of forms $$w\mapsto\chi_{\tau}(t)\cdot\mathcal{I}\Pi_N\mathcal{N}_{2l+1}(w,y_{N_2}^\dagger,\cdots,y_{N_{2l+1}}^\dagger),$$ where $\max(N_2,\cdots,N_{2l+1})=M$. As we have the decomposition $$y_L^\dagger\,=\,\chi(t)e^{it\Delta}(\Delta_Lf(\omega))\,+\, \sum _{(L,R)\in\mathcal{K}}\zeta_{L,R}^\dagger\,+\, z_L^\dagger$$by $\mathtt{Loc}(M)$ we know that each $y_{N_j}^{\dagger}$ can be decomposed into terms of type (G), type (C) (corresponding to some $L_j\lesssim N_j^{1-\delta}$), and type (D). The bound (\[induct4\]) is then a consequence of (\[sttime1\]) and (\[mainmult1\]), after removing a set of $\omega$ with measure $\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$ that is independent of $N$. Note that by (\[matrixnorm0\]), the $L=M$ case of (\[induct4\]) is equivalent to $$\label{inductmed}\|\mathfrak{h}^{N,M}\|_{Y^{b,b}}\leq \tau^{\theta}M^{-\delta_0^{\frac{1}{2}}}.$$Similarly (\[induct5\]) follows from (\[sttime1\]) and (\[mainmult5\]), because we have $$\tau^{b_1-b}\tau^{-\theta}N^{C\kappa^{-1}}N^{\frac{1}{2}}M^{-\frac{1}{2}+\gamma_0}\ll \tau^{\theta} N^{\frac{1}{2}+\gamma_0-2\delta^3}M^{-\frac{1}{2}}$$ using the fact that $M\lesssim N^{1-\delta}$. The set of $\omega$ removed here will depend on $N$, but it will have measure $\leq C_{\theta}e^{-(\tau^{-1}N)^\theta}$, so summing in $N\geq M$ we still get a set of measure $\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$.
[ Next we prove –, again assuming $L=M$. By (\[induct4\]) and $\mathtt{Loc}(M)$ we already know that the right hand side of (\[daggered\]) gives a contraction mapping in $X^b$, so (\[daggered\]) does have a unique solution. Subtracting]{} the equations (\[daggered\]) with $M$ and with $\frac{M}{2}$ instead of $M$, we deduce that $$\begin{gathered}
\label{matrixprod}h_{kk^*}^{N,M,\dagger}(t) = -i\sum_{l=0}^r(l+1)\,c_{rl}\, (m_N^*)^{r-l}\bigg\{\sum_{L\leq M}\sum_{k'}\int\mathrm{d}t'\cdot \mathfrak{h}_{kk'}^{N,L}(t,t')h_{k'k^*}^{N,M,\dagger}(t')\\+\sum_{L<M}\sum_{k'}\int\mathrm{d}t'\cdot \mathfrak{h}_{kk'}^{N,M}(t,t')h_{k'k^*}^{N,L,\dagger}(t'){\color{black}+\int\mathrm{d}t'\cdot \mathfrak{h}_{kk^*}^{N,M}(t,t')H_{k^*k^*}^{N,\frac{1}{2},\dagger}(t')}\bigg\},\end{gathered}$$ where $\mathfrak{h}_{kk'}^{N,L}(t,t')$ is the kernel corresponding to $\mathcal{P}^+$ in the $(k,k',t,t')$ variables. [Recall that we are already in a set where (\[simplebd\]) is true, which allows us to control $m_N^*$.]{} Now by the definition of $Y^b$ and $Y^{b,b}$ norms, the statement $\mathtt{Loc}(M)$ and (\[inductmed\]), we conclude that $$\begin{aligned}\|h^{N,M,\dagger}\|_{Y^b}&\lesssim\sum_{L\leq M}\|\mathfrak{h}^{N,L}\|_{Y^{b,b}}\cdot\|h^{N,M,\dagger}\|_{Y^b}+\|\mathfrak{h}^{N,M}\|_{Y^{b,b}}\bigg(\sum_{L<M}\|h^{N,L,\dagger}\|_{Y^b}+1\bigg)\\
&\lesssim \|h^{N,M,\dagger}\|_{Y^b}\cdot \sum_{L\leq M} \tau^{\theta}L^{-\delta_0^{\frac{1}{2}}}+\sum_{L<M}\tau^{\theta}M^{-\delta_0^{\frac{1}{2}}}L^{-\delta_0}\lesssim \tau^\theta \|h^{N,M,\dagger}\|_{Y^b}+\tau^\theta M^{-\delta_0^{\frac{1}{2}}},
\end{aligned}$$ which implies (\[induct1\]) [as desired]{}. In the same way we can prove (\[induct3\]) by using (\[operatorbd3\]), noting that [$\widetilde{\mathfrak{h}^{N,L}}$]{} is supported in $|k-k'|\lesssim L$. As for (\[induct2\]), recall that for $$\widetilde{H}_{kk^*}(\lambda)=\sum_{k'}\int\mathrm{d}\lambda'\cdot \widetilde{\mathfrak{h}}_{kk'}(\lambda,\lambda')h_{k'k^*}(\lambda')$$we have, [ by definition of the relevant norms, that]{} $$\|H\|_{l_{k,k^*}^2L_{\lambda}^2}\leq\min(\|\mathfrak{h}\|_{l_{k,k'}^2L_{\lambda,\lambda'}^2}\|h\|_{\ell_{k^*}^2\to \ell_{k'}^2L_{\lambda'}^2},\|\mathfrak{h}\|_{l_{k,k'}^2\to L_{\lambda,\lambda'}^2}\|h\|_{\ell_{k',k^*}^2L_{\lambda'}^2}).$$ Now in (\[matrixprod\]) we may assume $|k-k^*|\leq 2^{-10}N$ and $|k'-k^*|\leq 2^{-10}N$ (otherwise the bound follows trivially from (\[induct3\]) [ which we]{} just proved), so in particular $|k|,|k'|\geq \frac{N}{4}$ as $|k^*|\geq\frac{N}{2}$. Using the statement $\mathtt{Loc}(M)$ and (\[induct5\]) we get [ $$\begin{aligned}\|h^{N,M,\dagger}\|_{Z^b}&\lesssim\sum_{L\leq M}\|\mathfrak{h}^{N,L}\|_{Y^{b,b}}\cdot\|h^{N,M,\dagger}\|_{Z^b}+\|\mathbf{1}_{|k|,|k'|\geq \frac{N}{4}}\cdot\mathfrak{h}_{kk'}^{N,M}(t,t')\|_{Z^{b,b}}\bigg(\sum_{L<M}\|h^{N,L,\dagger}\|_{Y^b}+1\bigg)\\
&\lesssim \|h^{N,M,\dagger}\|_{Z^b}\cdot \sum_{L\leq M} \tau^{\theta}L^{-\delta_0^{\frac{1}{2}}}+\sum_{L<M}\tau^{\theta}N^{\frac{1}{2}+\gamma_0-\delta^3}M^{-\frac{1}{2}}L^{-\delta_0}\\&\lesssim \tau^\theta \|h^{N,M,\dagger}\|_{Y^b}+\tau^\theta N^{\frac{1}{2}+\gamma_0-\delta^3}M^{-\frac{1}{2}},
\end{aligned}$$]{} which proves (\[induct2\]).
Finally we prove (\[induct6\]) with $L=2M$, by showing that the mapping defined in (\[fixedpoint\]) is indeed a contraction mapping from the given set $\mathcal{Z}=\{z:\|z\|_{X^b}\leq (2M)^{-1+\gamma}\}$ to itself. Actually we will only prove that this mapping sends $\mathcal{Z}$ to $\mathcal{Z}$, as the difference estimate is done in the same way.
[ We will]{} separate the right hand side of (\[fixedpoint\]) into six groups, each of which has the form $$\chi_{\tau}(t)\cdot(m_{2M}^*)^{r-l}\mathcal{I}\Pi_{2M}\mathcal{N}_{2l+1}(v^{(1)},\cdots,v^{(2l+1)}),$$ where
1. At least two of the $v^{(j)}$ are equal to $z+\psi_{2M,L_0}^{\dagger}$, and others are either $z+\psi_{2M,L_0}^{\dagger}$ or $v_M^\dagger$;
2. We have $v^{(2)}=\psi_{2M,L_0}^{\dagger}$, and all others equal $v_M^\dagger$;
3. One of $v^{(1)}$ or $v^{(2)}$ equals $z$, and all others equal $v_M^\dagger$;
4. We have $v^{(1)}=\psi_{2M,L_0}^{\dagger}$, another $v^{(j)}$ equals $v_{M}^\dagger-v_{L_0}^\dagger$, and all others equal either $v_M^\dagger$ or $v_{L_0}^\dagger$;
5. The factor $(m_{2M}^*)^{r-l}$ [ is ]{}replaced by $(m_{M}^*+\nu_{2M})^{r-l}-(m_M^*)^{r-l}$ and all $v^{(j)}$ equal $v_M^\dagger$;
6. Same as (a), but with $\Delta_{2M}$ instead of $\Pi_{2M}$, and all $v^{(j)}$ equal $v_M^\dagger$.
By (\[sttime1\]), it will suffice to prove that each of these terms in [ (a) through (f), but]{} without the $\chi_{\tau}(t)$ factor, is bounded in $X^{b_1}$ by $\tau^{-\theta}(2M)^{-1+\gamma}$. Let [ one such term be denoted by]{} $\mathcal{M}$, and notice that we can decompose [ $$v_M^\dagger=\sum_{L\leq M}y_L^\dagger,\quad v_{L_0}^\dagger=\sum_{L\leq L_0}y_L^\dagger,\quad v_M-v_{L_0}=\sum_{L_0<L\leq M}y_L^\dagger,$$$$\psi_{2M,L_0}^\dagger=\chi(t)e^{it\Delta}(\Delta_{2M}f(\omega))+\sum_{L\leq L_0}\zeta_{2M,L}.$$]{}Moreover by what we have proved [ so far, we know that]{} $y_L^\dagger$ for $L\leq M$ can be decomposed into terms of types (G), (C) and (D), and that $\chi(t)e^{it\Delta}(\Delta_{2M}f(\omega))$ is of type (G), $\zeta_{2M,L}$ is of type (C), and $z$ is of type (D). By such decomposition we can reduce $\mathcal{M}$ to the terms studied in Proposition \[multi0\], with various choices of $N_j$ and $L_j$. [ We now proceed case by case]{}.
Case (a): Here we have at least two inputs $v^{(j)}$ with $N_j=2M$, so by either (\[mainmult1\]) or (\[mainmult2\]) we can bound $$\|\mathcal{M}\|_{X^{b_1}}\lesssim \tau^{-\theta}(2M)^{-1+\gamma_0+C\kappa^{-1}}$$ by removing a set of measure $\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$, which suffices.
Case (b): Here we have $v^{(2)}=N^{(1)}=2M$, while $\iota_2=-$. By (\[mainmult3\]) we have the same bound as above.
Case (c): This term, *with the $\chi_{\tau}(t)$ factor*, can be written as $$\sum_{L\leq M}\mathcal{P}_L^{\pm}(z),$$ where $\mathcal{P}_L^{\pm}$ are defined as in (\[devop1\]) and (\[defop2\]) (with subscript $L$ to indicate $L$ dependence). [If $L\leq L_0$ then]{} by (\[induct4\]) we can bound $$\|\chi_{\tau}(t)\cdot \mathcal{M}\|_{X^b}\leq(2M)^{-1+\gamma}\sum_{L\leq L_0}\tau^\theta L^{-\delta_0^{\frac{1}{2}}}\lesssim \tau^\theta(2M)^{-1+\gamma},$$ which suffices. Note that here no further set of $\omega$ needs to be removed; [if $L>L_0$ then this term can be bounded in the same way as in case (d) below, by removing a set of measure $\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$.]{}
Case (d): Here we have, due to the factor $v_M^\dagger-v_{L_0}^\dagger$, that $N^{(1)}=2M$ and $N^{(2)}\gtrsim M^{1-\delta}$; so by either (\[mainmult1\]) or (\[mainmult2\]) we can bound $$\|\mathcal{M}\|_{X^{b_1}}\lesssim \tau^{-\theta}M^{-1+\frac{\delta}{2}+C\gamma_0}$$ by removing a set of measure $\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$, which suffices.
Case (e): The bound for this term follows from the bound for $\nu_{2M}$ and the trivial bound (say (\[mainmult1\]) or (\[mainmult2\])) for the $\mathcal{N}_{2l+1}$ term.
Case (f): We may assume $N^{(1)}=N_a\sim M$. If either $v^{(a)}$ is of type (D) or $N^{(2)}\gtrsim M^{1-\delta}$ or $\iota_a=-$, we can reduce to one of the previous cases (namely (c) or (d) or (b)) and close as before; if $v^{(a)}$ is of type (G) or (C), $\iota_a=+$ and $N^{(2)}\ll M^{1-\delta}$, then the $\Delta_{2M}$ projection allows us to apply the improvement (\[mainmult4\]), which leads to $$\|\mathcal{M}\|_{X^{b_1}}\lesssim \tau^{-\theta}M^{-1+\frac{4}{5}\gamma+C\kappa^{-1}}$$ by removing a set of measure $\leq C_{\theta}e^{-(\tau^{-1}M)^\theta}$, which suffices. This completes the proof.
Large deviation and counting estimates {#prep2}
======================================
Proposition \[multi0\] will be proved in Section \[multiest\]. In this section we make some preparations for the proof, namely we introduce two large deviation estimates and some counting estimates for [ integer]{} lattice points.
Large deviation estimates
-------------------------
[We first prove the following large deviation estimate for multilinear Gaussians, which as far as we know is new.]{}
\[largedev\] Let $E\subset\mathbb{Z}^2$ [ be a finite subset]{}, and let [ $\mathcal{B}$ be]{} the $\sigma$-algebra generated by $\{g_k:k\in E\}$. Let $\mathcal{C}$ be a $\sigma$-algebra independent with $\mathcal{B}$, and let $\mathcal{C}^+$ be the smallest $\sigma$-algebra containing both $\mathcal{C}$ and the $\sigma$-algebra generated by $\{|g_k|^2:k\in E\}$. Consider the expression $$\label{indp}F(\omega)=\sum_{(k_1,\cdots,k_n)\in E^n}a_{k_1\cdots k_n}(\omega)\prod_{j=1}^ng_{k_j}(\omega)^{\iota_j},$$ where $n\leq 2r+1$, $\iota_j\in\{\pm\}$ and the coefficients $a_{k_1\cdots k_n}(\omega)$ are $\mathcal{C}^+$ measurable. Let $A\geq \#E$, then $A$-certainly we have [ $$\label{firstbound} |F(\omega)|\leq A^\theta M(\omega)^{\frac{1}{2}},$$]{} where $$\label{ortho}M(\omega)=\sum_{(X,Y)}\sum_{(k_{m}):m\not\in X\cup Y}\bigg(\sum_{\mathrm{pairing}\,(k_{i_s},k_{j_s}):1\leq s\leq p}|a_{k_1\cdots k_n}(\omega)|\bigg)^2.$$ In the summation (\[ortho\]) we require [ that]{} all $k_j\in E$, [ and that]{} $X:=\{i_1,\cdots,i_p\}$ and $Y:=\{j_1,\cdots,j_p\}$ are two disjoint subsets of $\{1,2,\cdots,n\}$. [ Recall also the definition of pairing in Definition \[dfpairing\]]{}.
Write in polar coordinates $g_k(\omega)=\rho_k(\omega)\eta_k(\omega)$ where $\rho_k=|g_k|$ and $\eta_k=\rho_k^{-1}g_k$, then all the $\rho_k$ and $\eta_k$ are independent, and each $\eta_k$ is uniformly distributed on the unit circle of $\mathbb{C}$. We may write $$\label{newexp1}F(\omega)=\sum_{(k_1,\cdots,k_n)\in E^n}b_{k_1\cdots k_n}(\omega)\prod_{j=1}^n\eta_{k_j}(\omega)^{\iota_j},\quad b_{k_1\cdots k_n}(\omega):=a_{k_1\cdots k_n}(\omega)\prod_{j=1}^n\rho_{k_j}(\omega).$$ Since $a_{k_1\cdots k_n}(\omega)$ are $\mathcal{C}^+$ measurable, we know that the collection $\{b_{k_1\cdots k_n}\}$ is independent with the collection $\{\eta_k:k\in E\}$. The goal is to prove that $$\label{largedevnew}\mathbb{P}(|F(\omega)|\geq {\color{black}BM_1(\omega)^{\frac{1}{2}}})\leq Ce^{-B^{1/n}},$$ where $C$ is an absolute constant, and $M_1(\omega)$ is the same as $M(\omega)$ but with [ the coefficients $a$]{} replaced by [ the coefficients $b$]{}. In fact, as $A\geq \#E$ we have $A$-certainly that $|b_{k_1\cdots k_n}(\omega)|\leq A^\theta|a_{k_1\cdots k_n}(\omega)|$, so (\[largedevnew\]) implies the desired bound.
We now prove (\[largedevnew\]). By independence, we may condition on the $\sigma$-algebra generated by $\{b_{k_1\cdots k_n}\}$ and prove (\[largedevnew\]) for the conditional probability, then take another expectation; therefore we may assume that $b_{k_1\cdots k_n}$ are constants (so $M_1(\omega)=M_1$ is a constant). Now let $\{h_k:k\in E\}$ be another set of i.i.d. normalized complex Gaussian random variables and define $$\label{newexp2}G=\sum_{(k_1,\cdots,k_n)\in E^n}|b_{k_1\cdots k_n}|\prod_{j=1}^nh_{k_j}^{\iota_j},$$ we want to compare $F$ and $G$ and show $\mathbb{E}|F|^{2d}\leq \mathbb{E}|G|^{2d}$ for any positive integer $d$. In fact, $$\begin{aligned}
\label{exp1}\mathbb{E}(|F|^{2d})&=\sum_{(k_j^i,\ell_j^i:1\leq i\leq d,1\leq j\leq n)}\prod_{i=1}^db_{k_1^i\cdots k_n^i}\overline{b_{\ell_1^i\cdots\ell_n^i}}\mathbb{E}\bigg(\prod_{i=1}^d\prod_{j=1}^n\eta_{k_j^i}^{\iota_j}\overline{\eta_{\ell_j^i}^{\iota_j}}\bigg),\\
\label{exp2}\mathbb{E}(|G|^{2d})&=\sum_{(k_j^i,\ell_j^i:1\leq i\leq d,1\leq j\leq n)}\prod_{i=1}^d|b_{k_1^i\cdots k_n^i}||b_{\ell_1^i\cdots\ell_n^i}|\mathbb{E}\bigg(\prod_{i=1}^d\prod_{j=1}^nh_{k_j^i}^{\iota_j}\overline{h_{\ell_j^i}^{\iota_j}}\bigg).\end{aligned}$$ The point is that we always have $$\bigg|\mathbb{E}\bigg(\prod_{i=1}^d\prod_{j=1}^n\eta_{k_j^i}^{\iota_j}\overline{\eta_{\ell_j^i}^{\iota_j}}\bigg)\bigg|\leq \mathrm{Re}\,\mathbb{E}\bigg(\prod_{i=1}^d\prod_{j=1}^nh_{k_j^i}^{\iota_j}\overline{h_{\ell_j^i}^{\iota_j}}\bigg).$$ In fact, by collecting all different factors we can write the expectations as $$\mathbb{E}\bigg(\prod_{\alpha}(\eta_{k^{(\alpha)}})^{x_\alpha}(\overline{\eta_{k^{(\alpha)}}})^{y_\alpha}\bigg)\quad\mathrm{and}\quad \mathbb{E}\bigg(\prod_{\alpha}(h_{k^{(\alpha)}})^{x_\alpha}(\overline{h_{k^{(\alpha)}}})^{y_\alpha}\bigg),$$ where the $k^{(\alpha)}$ are pairwise distinct. If $x_\alpha\neq y_\alpha$ for some $\alpha$, both expectations will be $0$; if $x_\alpha=y_\alpha$ for each $\alpha$, then the first expectation will be $1$ and the second expectation will be $\prod_{\alpha}x_\alpha!\geq 1$.
Now, since $G$ is an exact multilinear Gaussian expression, by the standard hypercontractivity estimate, [see [@OT],]{} we have $$\mathbb{E}|F|^{2d}\leq \mathbb{E}|G|^{2d}\leq(2d-1)^{nd}(\mathbb{E}|G|^2)^d,$$ so for any [ $D>0$]{} by using Chebyshev’s inequality and optimizing in $d$ we have $$\mathbb{P}(|F(\omega)|\geq D)\leq \min_d\bigg\{(2d-1)^{nd}\big(\frac{\mathbb{E}|G|^2}{D^2}\big)^d\bigg\}\leq C\exp\bigg\{\frac{-1}{2e}\big(\frac{\mathbb{E}|G|^2}{D^2}\big)^{\frac{1}{n}}\bigg\}$$ with some constant $C$ depending only on $n$. It then suffices to prove [$\mathbb{E}|G|^2\lesssim M_1$ with constants depending only on $n$.]{}
By dividing the sum (\[newexp2\]) into finitely many terms and rearranging the subscripts, we may assume $$k_{1}=\cdots=k_{j_1},\,\, k_{j_1+1}=\cdots=k_{j_2},\cdots,k_{j_{r-1}+1}=\cdots=k_{j_r},\,\,1\leq j_1<\cdots <j_r=n,$$ and the $k_{j_s}$ are different for $1\leq s\leq r$. Such a monomial that appears in (\[newexp2\]) has the form $$\prod_{s=1}^rh_{k_{j_s}}^{\beta_s}(\overline{h_{k_{j_s}}})^{\gamma_s},\quad \beta_s+\gamma_s=j_s-j_{s-1}\,\,(j_0=0),$$ where the factors for different $s$ are independent. We may also assume $\beta_s=\gamma_s$ for $1\leq s\leq q$ and $\beta_s\neq \gamma_s$ for $q+1\leq s\leq r$, and that $\iota_j$ has the same sign as $(-1)^j$ for $1\leq j\leq j_q$. Then we can further rewrite this monomial as a linear combination of $$\prod_{s=1}^p{\color{black}\beta_s!}\prod_{s=p+1}^q(|h_{k_{j_s}}|^{2\beta_s}-\beta_s!)\prod_{s=q+1}^rh_{k_{j_s}}^{\beta_s}(\overline{h_{k_{j_s}}})^{\gamma_s}$$ for $1\leq p\leq q$. Therefore, $G$ is a finite linear combination of expressions of [ the]{} form $$\sum_{k_{j_1},\cdots,k_{j_r}}|b_{k_{j_1},\cdots,k_{j_1},\cdots,k_{j_r},\cdots k_{j_r}}|\prod_{s=1}^p{\color{black}\beta_s!}\prod_{s=p+1}^q(|h_{k_{j_s}}|^{2\beta_s}-\gamma_s!)\prod_{s=q+1}^rh_{k_{j_s}}^{\beta_s}(\overline{h_{k_{j_s}}})^{\gamma_s}.$$ Due to independence and the fact that $\mathbb{E}(|h|^{2\beta}-\beta!)=\mathbb{E}(h^\beta(\overline{h})^\gamma)=0$ for a normalized Gaussian $h$ and $\beta\neq \gamma$, we conclude that $$\mathbb{E}|G|^2\lesssim\sum_{k_{j_{p+1}},\cdots k_{j_r}}\bigg(\sum_{k_{j_1},\cdots,k_{j_p}}|b_{k_{j_1},\cdots,k_{j_1},\cdots,k_{j_r},\cdots k_{j_r}}|\bigg)^2,$$ which is bounded by $M_1$ choosing $X=\{1,3,\cdots,j_p-1\}$ and $Y=\{2,4,\cdots,j_p\}$, [ since by]{} our assumptions $(k_{2i-1},k_{2i})$ is a pairing for $2i\leq j_p$. This completes the proof.
For the purpose of Section \[multiest\] we will also need the following lemma, which is [ a more general large deviation-type estimate]{} restricted to the no-pairing case:
\[lemma:5.1\] [ Let $\delta$ be as in Section \[notations\], $n\leq 2r+1$]{} and consider the following expression $$\label{multgauss}M(\omega)=\sum_{(k_1,\cdots,k_n)}\sum_{(k_1^*,\cdots,k_n^*)}\int \mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\cdot a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)\prod_{j=1}^ng_{k_j^*}(\omega)^{\iota_j}h^{(j)}_{k_jk_j^*}(\lambda_j,\omega)^{\pm},$$ where $a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)$ is a given function of $(k_1,\cdots,k_n,\lambda_1,\cdots,\lambda_n)$. Moreover in the summation we assume that there [ are]{} no pairings among $\{k_1^*,\cdots,k_n^*\}$, that $|k_j|\leq N_j, \frac{N_j}{2}<|k_j^*|\leq N_j$, and that $h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)$, as a random variable, is $\mathcal{B}_{\leq N_j^{1-\delta}}^+$ measurable. Let $N_*\geq\max( N_1,\cdots,N_n)$, then $N_*$-certainly (the exceptional set removed will depend on [ the coefficients $a$]{}), we have $$\label{multgauss1}|M(\omega)|\lesssim(N_*)^{\theta}\prod_{j=1}^n\|h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)\|_{\ell_{k_j^*}^2\to\ell_{k_j}^2L_{\lambda_j}^2}\cdot\|a_{k_1\cdots k_n}(\lambda_1,\cdots\lambda_n)\|_{\mathcal{L}},$$ where $\mathcal{L}$ is an auxiliary norm [ defined by]{} $$\label{auxnorm}\|a_{k_1\cdots k_n}(\lambda_1,\cdots\lambda_n)\|_{\mathcal{L}}^2:=\sum_{k_1,\cdots k_n}\int\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\cdot\big(\max_{1\leq j\leq n}\langle\lambda_j\rangle\big)^{\delta^6}(|a|^2+|\partial_{\lambda}a|^2).$$
Consider the big box $\{|\lambda_j|\leq (N_*)^{\delta^{-7}}\}$ and divide it into small boxes of size $(N_*)^{-\delta^{-1}}$. By exploiting the weight $(\max_{1\leq j\leq n}\langle\lambda_j\rangle)^{\delta^6}$ in (\[auxnorm\]) and using Poincaré’s inequality, we can find a function $b$ which is supported in the big box and is constant on each small box, such that $$\sup_{k_j,k_j^*}\|a-b\|_{L_{\lambda_1,\cdots,\lambda_n}^2}\lesssim (N_*)^{-\delta^{-1}}\|a\|_{\mathcal{L}}.$$ Exploiting this $(N_*)^{-\delta^{-1}}$ gain, summing over $|k_j|,|k_j^*|\lesssim N_*$ and using the simple bound $$\sup_{k_j,k_j^*}\bigg|\int \mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\cdot a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)\prod_{j=1}^n h_{k_jk_j^*}^{(j)}(\lambda_j)^\pm\bigg|\lesssim \sup_{k_j,k_j^*}\|a\|_{L_{\lambda_1,\cdots,\lambda_n}^2}\prod_{j=1}^n\sup_{k_j,k_j^*}\|h_{k_jk_j^*}^{(j)}(\lambda_j)\|_{L_{\lambda_j}^2}$$ suffices to bound the contribution with $a$ replaced by $a-b$; thus we may now replace $a$ by $b$ (or equivalently, assume $a$ is supported in the big box and is constant on each small box) and will prove (\[multgauss1\]) $N_*$-certainly, with the $\mathcal{L}$ norm replaced by the $l^2L^2$ norm, by induction.
By symmetry we may assume $N_1\geq\cdots\geq N_n$. Choose the smallest $q$ such that $N_q>2^{10}N_{q+1}$, then $N_1\sim N_q$ with constant depending only on $n$. Unless $N_1\leq C$, in which case (\[multgauss1\]) is trivial, we can conclude that $$h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)^\pm,\,1\leq j\leq n,\,\text{are $\mathcal{B}_{\leq N_1^{1-\delta}}^+$ measurable; $N_1^{1-\delta}\leq 2^{-10}N_q$,}$$ $$g_{k_j^*}(\omega)^{\pm},\,q+1\leq j\leq n,\,\text{are $\mathcal{B}_{\leq N_{q+1}}$ measurable; $N_{q+1}\leq 2^{-10}N_q$.}$$ Note that in this case, there is no pairing among $\{k_1^*,\cdots,k_n^*\}$ if and only if there is no pairing among $\{k_1^*,\cdots,k_q^*\}$ and no pairing among $\{k_{q+1}^*,\cdots,k_n^*\}$. We can then write $M(\omega)$ as $$M(\omega)=\sum_{k_1^*,\cdots,k_q^*}b_{k_1^*\cdots k_q^*}(\omega)\cdot\prod_{j=1}^q g_{k_j^*}(\omega)^{\iota_j},$$ where $$\begin{gathered}
\label{defofb}b_{k_1^*\cdots k_q^*}(\omega)=\sum_{k_1,\cdots,k_q}\int\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_q\prod_{j=1}^qh_{k_jk_j^*}^{(j)}(\lambda_j,\omega)^\pm\\\times\sum_{(k_{q+1},\cdots k_n)}\sum_{(k_{q+1}^*,\cdots,k_n^*)}\int \mathrm{d}\lambda_{q+1}\cdots\mathrm{d}\lambda_n\cdot a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)\prod_{j=q+1}^ng_{k_j^*}(\omega)^{\iota_j}h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)^\pm\end{gathered}$$ are $\mathcal{B}_{\leq 2^{-10}N_q}^+$ measurable. We then apply Lemma \[largedev\] and conclude that, after removing a set of $\omega$ with probability $\leq C_{\theta}e^{-(N_*)^{\theta}}$, we have $$\label{gaussprob}|M(\omega)|^2\lesssim (N_*)^\theta\sum_{k_1^*,\cdots k_q^*}|b_{k_1^*\cdots k_q^*}(\omega)|^2.$$ Now by (\[defofb\]) we have $$\begin{gathered}
\label{l2tol2}\sum_{k_1^*,\cdots k_q^*}|b_{k_1^*\cdots k_q^*}(\omega)|^2\lesssim\prod_{j=1}^q\|h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)\|_{\ell^2\to\ell^2L^2}^2\sum_{k_1,\cdots,k_q}\int\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_q\\\times\bigg|\sum_{(k_{q+1},\cdots k_n)}\sum_{(k_{q+1}^*,\cdots,k_n^*)}\int \mathrm{d}\lambda_{q+1}\cdots\mathrm{d}\lambda_n\cdot a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)\prod_{j=q+1}^ng_{k_j^*}(\omega)^{\iota_j}h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)^\pm\bigg|^2;\end{gathered}$$ by induction hypothesis, we get that $$\begin{gathered}
\label{fixedpt}\bigg|\sum_{(k_{q+1},\cdots k_n)}\sum_{(k_{q+1}^*,\cdots,k_n^*)}\int\mathrm{d}\lambda_{q+1}\cdots\mathrm{d}\lambda_n a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)\prod_{j=q+1}^ng_{k_j^*}(\omega)^{\iota_j}h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)^\pm\bigg|^2\\\lesssim (N_*)^{\theta}\prod_{j=q+1}^n\|h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)\|_{\ell_{k_j^*}^2\to\ell_{k_j}^2L_{\lambda_j}^2}^2\sum_{k_{q+1},\cdots k_n}\int \mathrm{d}\lambda_{q+1}\cdots\mathrm{d}\lambda_n\cdot|a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)|^2,\end{gathered}$$ up to a set of $\omega$ with probability $\leq C_{\theta}e^{-(N_*)^{\theta}}$, for any *fixed* $(k_j,\lambda_j)$ for $1\leq j\leq q$. By our assumption on [ the coefficients $a$]{}, the function $$(k_{q+1},\cdots, k_n,\lambda_{q+1},\cdots,\lambda_n)\mapsto a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)$$ which depends on the parameter $(k_j,\lambda_j)$ for $1\leq j\leq q$, has only $(N_*)^{C\delta^{-7}}$ different possibilities, so by removing a set of $\omega$ with probability $\leq C_{\theta}e^{-(N_*)^{\theta}}$, we may assume (\[fixedpt\]) holds for *all* $(k_j,\lambda_j)$, $1\leq j\leq q$. Thus we can sum (\[fixedpt\]) over $k_j$ and integrate over $\lambda_j$, and combine with (\[gaussprob\]) and (\[l2tol2\]) to get that $$|M(\omega)|^2\lesssim(N_*)^{\theta}\prod_{j=1}^n\|h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)\|_{\ell_{k_j^*}^2\to\ell_{k_j}^2L_{\lambda_j}^2}^2\sum_{k_1,\cdots,k_n}\int\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\cdot|a_{k_1\cdots k_n}(\lambda_1,\cdots,\lambda_n)|^2.$$ This completes the proof.
Counting estimates for lattice points
-------------------------------------
We start with a simple lemma, and then state the main [ integer]{} lattice point counting bounds that will be used in the proof below.
\[lem:counting\](1) Let $\mathcal{R}=\mathbb{Z}$ or $\mathbb{Z}[i]$. Then, given $0\neq m\in\mathcal{R}$, and $a_0,b_0\in\mathbb{C}$, the number of choices for $(a,b)\in\mathcal{R}^2$ that satisfy $$m=ab,\,\,|a-a_0|\leq M,\,\,|b-b_0|\leq N$$is $O(M^\theta N^\theta)$ with constant depending only on $\theta>0$.
\(2) Given dyadic numbers $N_1\gtrsim N_2\gtrsim N_3$, consider the set $$\begin{gathered}
\label{count}
S=\big\{(x,y,z)\in(\mathbb{Z}^2)^3:\iota_1 x+\iota_2y+\iota_3 z={\color{black}d},\,\,\iota_1|x|^2+\iota_2 |y|^2+\iota_3 |z|^2=\alpha,\\|x-a|\lesssim N_1,|y-b|\lesssim N_2,|z-c|\lesssim N_3\big\}.\end{gathered}$$ Assume also there is no pairing in $S$. Then, uniformly in $(a,b,c, {\color{black}d})\in(\mathbb{Z}^2)^4$ and $\alpha\in\mathbb{Z}$, we have $\#S\lesssim N_2^{1+\theta}N_3$. Moreover, if $\iota_1=\iota_2$, then we have the stronger bound $\#S\lesssim N_2^\theta N_3^2$.
\(1) This strengthened divisor estimate is essentially proved in [@DNY], Lemma 3.4. We know that $\mathcal{R}$ has unique factorization and satisfies the standard divisor estimate, namely the number of divisors of $0\neq m\in\mathcal{R}$ is $O(|m|^{\theta})$. Now suppose $\max(|a_0|,M)\geq\max(|b_0|,N)$, then $|m|\lesssim\max(|a_0|,M)^2$. We may assume $M_1\sim|a_0|\gg M^4$, and hence $|m|\lesssim M_1^2$.
We then claim that the number of divisors $a$ of $m$ that satisfies $|a-a_0|\leq M$ is at most two. In fact, suppose $a,b,c$ are different divisors of $m$ that belong to the ball $|x-a_0|\leq M$, then by unique factorization we have $\mathrm{lcm}(a,b,c)|m$, hence $$\frac{abc}{\gcd(a,b)\gcd(b,c)\gcd(c,a)}$$ divides $m$. As $|a|\sim M_1$ etc., and $|\gcd(a,b)|\leq |a-b|\lesssim M$ etc., we conclude that $$M_1^2\gtrsim|m|\geq\bigg|\frac{abc}{\gcd(a,b)\gcd(b,c)\gcd(c,a)}\bigg|\gtrsim M_1^3M^{-3},$$ contradicting the assumption $M_1\gg M^4$.
\(2) Let $x=(x_1,x_2)$, etc. If $\iota_1=\iota_2$, then with fixed $z$ (which has $O(N_3^2)$ choices), $x+y$ will be constant. Let $x-y=w$, then $$(w_1+iw_2)(w_1-iw_2)=|w|^2=2(|x|^2+|y|^2)-|x+y|^2$$ is constant. As $w$ belongs to a ball of radius $O(N_2)$ in $\mathbb{R}^2$, by (1) we know that the number of choices for $w$ is $O(N_2^\theta)$, hence $\#S=O(N_2^\theta N_3^2)$. Below we will assume that $\iota_1=+$ and $\iota_2=-$.
\(a) Suppose $\iota_3=+$, then we have that $$({\color{black}d_1}-z_1)(z_1-y_1)+({\color{black}d_2}-z_2)(z_2-y_2)=({\color{black}d}-z)\cdot(z-y)=\frac{|{\color{black}d}|^2-\alpha}{2}$$ is constant. If $({\color{black}d_1}-z_1)(z_1-y_1)\neq 0$ (or similarly if $({\color{black}d_2}-z_2)(z_2-y_2)\neq 0$), then with fixed $(y_2,z_2)$ (which has $O(N_2N_3)$ choices), $({\color{black}d_1}-z_1)(z_1-y_1)$ will be constant. As ${\color{black}d_1}-z_1$ belongs to an interval of size $O(N_3)$ in $\mathbb{R}$, and $z_1-y_1$ belongs to an interval of size $O(N_2)$ in $\mathbb{R}$, by (1) we know that the number of choices for $(y_1,z_1)$ is $O(N_2^\theta)$, so $\#S\lesssim N_2^\theta N_2N_3$.
If $({\color{black}d_1}-z_1)(z_1-y_1)=0$ and $({\color{black}d_2}-z_2)(z_2-y_2)=0$, as there is no pairing, we may assume that ${\color{black}d_1}=z_1$ and $z_2=y_2$ (or $z_1=y_1$ and ${\color{black}d_2}=z_2$, which is treated similarly), so $z_1={\color{black}d_1}$ and $x_2={\color{black}d_2}$ are fixed, $z_2$ has $O(N_3)$ choices and $x_1=y_1$ has $O(N_2)$ choices, which implies $\#S\lesssim N_2N_3$.
\(b) Suppose $\iota_3=-$, then similarly we have that $$({\color{black}d_1}+z_1)({\color{black}d_1}+y_1)+({\color{black}d_2}+z_2)({\color{black}d_2}+y_2)=({\color{black}d}+z)\cdot({\color{black}d}+y)=\frac{|{\color{black}d}|^2+\alpha}{2}$$ is constant. If $({\color{black}d_1}+z_1)({\color{black}d_1}+y_1)\neq 0$ (or similarly if $({\color{black}d_2}+z_2)({\color{black}d_2}+y_2)\neq 0$), then with fixed $(y_2,z_2)$ (which has $O(N_2N_3)$ choices), $({\color{black}d_1}+z_1)({\color{black}d_1}+y_1)$ will be constant. As ${\color{black}d_1}+z_1$ belongs to an interval of size $O(N_3)$ in $\mathbb{R}$, and ${\color{black}d_1}+y_1$ belongs to an interval of size $O(N_2)$ in $\mathbb{R}$, by (1) we know that the number of choices for $(y_1,z_1)$ is $O(N_2^\theta)$, so $\#S\lesssim N_2^\theta N_2N_3$.
If $({\color{black}d_1}+z_1)({\color{black}d_1}+y_1)=0$ and $({\color{black}d_2}+z_2)({\color{black}d_2}+y_2)=0$, as there is no pairing, we may assume that ${\color{black}d_1}+z_1=0$ and ${\color{black}d_2}+y_2=0$ (or ${\color{black}d_2}+z_2$ and ${\color{black}d_1}+y_1=0$, which is treated similarly), so $z_1=- {\color{black}d_1}$ and $y_2=- {\color{black}d_2}$ are fixed, $z_2$ has $O(N_3)$ choices and $y_1$ has $O(N_2)$ choices, which implies $\#S\lesssim N_2N_3$.
\[counting1\] [ Recall the relevant constants defined in (\[defparam\]).]{} The following bounds [are uniform in all parameters]{}. Given $d,d',k^0\in\mathbb{Z}^2$ and $k_j^0\in\mathbb{Z}^2(1\leq j\leq n)$, $\alpha,\Gamma\in\mathbb{R}$, $\iota,\iota_j\in\{\pm\}(1\leq j\leq n)$, and $2p\leq n$, as well as $M$, $N_j (0\leq j\leq n)$ and $R_i(1\leq i\leq p)$, such that for $1\leq i\leq p$ we have $N_{2i-1}\sim N_{2i}$, $\iota_{2i-1}=-\iota_{2i}$ and $R_{i}\lesssim N_{2i-1}^{1-\delta}$. Let $N^{(j)}=\max^{(j)}(N_1,\cdots,N_n)$, $N_{PR}=\max(N_1,\cdots,N_{2p})$ and let $N_*\gtrsim\max(N_0,N^{(1)})$. Also fix a subset $A$ of $\{1,\cdots,n\}$ that contains $\{1,\cdots,2p\}$, and recall the definition of the *$\Gamma$-condition* (\[gammacon\]). Consider the sets $$\begin{gathered}
\label{defset}\
S_1=\bigg\{(k,k_1,\cdots,k_n)\in(\mathbb{Z}^2)^{n+1}:\sum_{j=1}^n\iota_jk_j=k+d,\quad\sum_{j=1}^n\iota_j|k_j|^2=|k|^2+\alpha,\\|k_j-k_j^0|\lesssim N_j\,(1\leq j\leq n),\quad{\color{black}|k_{2i-1}-k_{2i}|\lesssim R_i(N_*)^{C\kappa^{-1}}\,(1\leq i\leq p)}\bigg\},\end{gathered}$$ $$\begin{gathered}
\label{defset2}
S_2=\bigg\{(k,k',k_1,\cdots,k_n)\in(\mathbb{Z}^2)^{n+2}:\iota k'+\sum_{j=1}^n\iota_jk_j=k+d,\quad\iota|k'|^2+\sum_{j=1}^n\iota_j|k_j|^2=|k|^2+\alpha,\\|k|,|k'|\lesssim N_0,\,\, |k_j-k_j^0|\lesssim N_j\,(1\leq j\leq n),\,\,{\color{black}|k_{2i-1}-k_{2i}|\lesssim R_i(N_*)^{C\kappa^{-1}}\,(1\leq i\leq p)}\bigg\},\end{gathered}$$ $$\label{extraeqn}{\color{black}S^{+}}=\bigg\{(k,k_1,\cdots,k_{n})\in(\mathbb{Z}^2)^{n+1}{\rm{\ and\ }}(k,k',k_1,\cdots,k_{n})\in(\mathbb{Z}^2)^{n+2}:\sum_{j\in A}\iota_jk_j=d'\bigg\},$$ $$\begin{gathered}
\label{defset4}
S_3=\bigg\{(k,k_1,\cdots,k_n)\in(\mathbb{Z}^2)^{n+1}:\sum_{j=1}^n\iota_jk_j=k+d,\quad\bigg||k|^2-\sum_{j=1}^n\iota_j|k_j|^2-\alpha\bigg|\lesssim M,\quad|k|\lesssim N_0,\\|k_j|\lesssim N_j \, (1\leq j\leq n),\quad{\color{black}|k_{2i-1}-k_{2i}|\lesssim R_i(N_*)^{C\kappa^{-1}}}\,{\color{black}(1\leq i\leq p)},\quad {\rm{and\ }}(\ref{gammacon}){\rm{\ holds}}\bigg\}.\end{gathered}$$
Assume that there is no pairing among the variables $k$, $k'$ and $k_j$ in the sets above. [Let $S_j^+=S_j\cap S^+$.]{} Then, for $S_1$ we have $$\label{bdset1}
(\#S_1)\cdot{\color{black}\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}}\lesssim (N_{PR})^{2\gamma_0}(N_*)^{C\kappa^{-1}}(N^{(1)}N^{(2)})^{-1}\prod_{j=1}^nN_j^2.$$ If $N^{(1)}\sim N_a$ and $\iota_a=-$, then we have $$\label{bdset2}
(\#S_1)\cdot{\color{black}\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}}\lesssim (N_{PR})^{2\gamma_0}(N_*)^{C\kappa^{-1}}(N^{(1)})^{-2}\prod_{j=1}^nN_j^2.$$ For $S_2$ and $S_3$ we have $$\label{bdset3}
(\#S_2)\cdot{\color{black}\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}}\lesssim (N_{PR})^{2\gamma_0}(N_*)^{C\kappa^{-1}}N_0(N^{(1)})^{-1}\prod_{j=1}^nN_j^2,$$$$\label{bdset5}
(\#S_3)\cdot{\color{black}\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}}\lesssim (N_{PR})^{2\gamma_0}(N_*)^{C\kappa^{-1}}M\frac{\max((N^{(2)})^2,|\alpha|)}{(N^{(2)})^2}(N^{(1)})^{-2}\prod_{j=1}^nN_j^2.$$
Finally, suppose we replace any of these $S_j$’s by the set [$S_j^+$.]{} Then (\[bdset1\])–(\[bdset3\]) hold with [ the]{} right hand side multiplied by an extra factor $$\label{exfactor}\big[\min\big(N^{(2)},\max_{j\in A,j\geq 2p+1}N_j\big)\big]^{-1}.$$ If $N^{(1)}\sim N_a$ and $a\in A$, then the stronger bound (\[bdset2\]) holds for $S_1^+$ with right hand side multiplied by an extra factor (\[exfactor\]), [regardless of whether $\iota_a=-$ or not.]{} [If $N^{(1)}\sim N_a$ and $2p+1\leq a\in A$, then (\[bdset3\]) holds for $S_2^+$ with right hand side multiplied by an extra factor $(N^{(1)})^{-1}$.]{} As for [$S_3^+$]{}, either it satisfies (\[bdset5\]) with the same extra factor (\[exfactor\]), or it satisfies $$\begin{gathered}
\label{bdset6}{\color{black}(\#S_3^+)}\cdot{\color{black}\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}}\lesssim (N_*)^{C\kappa^{-1}}M(N_{PR})^{2\gamma_0}\min\bigg(\frac{\max((N^{(2)})^2,|\alpha|)}{(N^{(1)}N^{(2)})^2}\\\times\big(\max^{(2)}\{N_j:2p+1\leq j\in A\}\big)^{-1},(N^{(1)})^{-1}(N^{(2)})^{-2}\bigg)\prod_{j=1}^nN_j^2.\end{gathered}$$
Let $a,b,c$ be such that $N^{(1)}\sim N_a$, $N^{(2)}\sim N_b$, and $\max(\{N_j:2p+1\leq j\in A\})\sim N_c$. In the proof below any factor that is $\lesssim (N_*)^{C\kappa^{-1}}$ will be negligible, so we will pretend they are $1$. For simplicity, let us first also ignore all $N_{2i-1}^{2\gamma_0}$ factors; at the end of the proof we will explain how to put them back.
\(1) We start with (\[bdset1\]). If $a,b\geq 2p+1$, we may fix all $k_j(j\not\in\{a,b\})$, and then apply Lemma \[lem:counting\] (2) to count the triple $(k,k_{a},k_{b})$. This gives $$(\#S_1)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim\bigg(\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{a,b\}}N_j^2\bigg)N_{a}N_{b},$$ which proves (\[bdset1\]) as $R_{j}\lesssim N_{2j-1}^{1-\delta}$. If $a\geq 2p+1$ and $b\leq 2p$ (say $b=1$), we may fix all $k_j(j\not\in\{1,a\})$, and then apply Lemma \[lem:counting\] (2) to count the triple $(k,k_{1},k_{a})$, noticing that $k_1$ belongs to a disc of radius $O(R_1)$ once $k_2$ is fixed. This gives $$(\#S_1)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim\bigg(N_1^2\prod_{i=2}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\neq a}N_j^2\bigg)N_{a}R_1\frac{N_1}{R_1},$$ which proves (\[bdset1\]). Finally, if $a\leq 2p$ (say $a=1$) then we may assume $b=2$. We may fix all $k_j(j\geq 3)$, and then apply Lemma \[lem:counting\] (2) to count the triple $(k,k_{1},k_{2})$, noticing that $k$ belongs to a disc of radius $O(R_1(N_*)^{C\kappa^{-1}})$ once all $k_j(j\geq 3)$ are fixed. This gives $$(\#S_1)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(\prod_{i=2}^pN_{2i-1}^3R_i\prod_{j\geq 2p+1}N_j^2\bigg)N_1R_1\frac{N_1}{R_1},$$ which proves (\[bdset1\]).
As for (\[bdset2\]) and (\[bdset3\]) we only need to consider $a$. If $a\geq 2p+1$ we may fix all $k_j(j\neq a)$, and then apply Lemma \[lem:counting\] (2) to count the pair $(k,k_{a})$ for (\[bdset2\]) (using the fact $\iota_{a}=-$) and the triple $(k,k',k_{a})$ for (\[bdset3\]), and get $$(\#S_1)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\neq a}N_j^2\bigg) ,$$ $$(\#S_2)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim\bigg(\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\neq a}N_j^2\bigg) N_{0}N_{a},$$ which proves (\[bdset2\]) and (\[bdset3\]). If $a\leq 2p$ (say $a=1$) we may assume $b=2$, in particular $N^{(1)}\sim N^{(2)}$ and (\[bdset2\]) follows from (\[bdset1\]); for (\[bdset3\]) we may fix all $k_j(j\geq 2)$ and then apply Lemma \[lem:counting\] (2) to count the triple $(k,k',k_{1})$, noticing that $k_1$ belongs to a disc of radius $O(R_1(N_*)^{C\kappa^{-1}})$ once $k_2$ is fixed, and get $$(\#S_2)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(N_1^2\prod_{i=2}^pN_{2i-1}^3R_i\prod_{j\geq 2p+1}N_j^2\bigg)N_0R_1\frac{N_1}{R_1},$$ which proves (\[bdset3\]).
\(2) Next we prove the improvements to (\[bdset1\])–(\[bdset3\]) for $S_j^{+}$. We start with (\[bdset1\]). If $a,b\not\in A$, we may fix all $k_j (c\neq j\in A)$ and apply (\[bdset1\]) to the rest variables and get $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim (N_aN_b)^{-1}\prod_{j\not\in A}N_j^2\prod_{i=1}^p N_{2i-1}^3R_i\prod_{j\in A,c\neq j\geq 2p+1}N_j^2,$$ which gains a factor $N_c^{-2}$ upon (\[bdset1\]). If $a\not\in A$ and $2p+1\leq b\in A$, we may fix all $k_j (b\neq j\in A)$ and apply (\[bdset1\]) to the rest variables and get $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim N_a^{-1}\prod_{j\not\in A}N_j^2\prod_{i=1}^p N_{2i-1}^3R_i\prod_{j\in A,b\neq j\geq 2p+1}N_j^2,$$ which gains a factor $N_b^{-1}$ upon (\[bdset1\]). If $a\not\in A$ and $b\leq 2p$ (say $b=1$), we may fix all $k_j(2\leq j\neq a)$, noticing that $k_c$ belongs to a ball of radius $\min(N_c,R_1(N_*)^{C\kappa^{-1}})$ once all $k_j(3\leq j\not\in \{a,c\})$ are fixed, and then apply Lemma \[lem:counting\] (2) to count the pair $(k,k_a)$ and get $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim\bigg(N_1^2\min(N_c^2,R_1^2)\prod_{i=2}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{a, c\}}N_j^2\bigg)N_a\frac{N_1}{R_1},$$which gains a factor $N_c^{-1}$ upon (\[bdset1\]).
Now if $2p+1\leq a\in A$ and either $b\not\in A$ or $2p+1\leq b\in A$, we may fix all $k_j(j\not\in\{a,b\})$ and apply Lemma \[lem:counting\] (2) to count the pair $(k,k_b)$ (if $b\not\in A$) or $(k_a,k_b)$ (if $2p+1\leq b\in A$), and get $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{a,b\}}N_j^2\bigg)N_b,$$ which gains a factor $N_b^{-1}$ upon the stronger bound (\[bdset2\]). If $2p+1\leq a\in A$ and $b\leq 2p$ (say $b=1$), we may fix all $k_j(j\not\in\{a,1,2\})$ and apply Lemma \[lem:counting\] (2) to count the [triple]{} $(k_a,k_1,k_2)$, noticing that $k_a$ belongs to a disc of radius $O(R_1(N_*)^{C\kappa^{-1}})$ once all $k_j(j\not\in\{a,1,2\})$ are fixed, and get $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(\prod_{i=2}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\neq a}N_j^2\bigg)N_1R_1\frac{N_1}{R_1},$$ which gains a factor $N_b^{-2}$ upon the stronger bound (\[bdset2\]). Finally, if $a\leq 2p$ (say $a=1$) then we may assume $b=2$. We may fix all $k_j(j\geq 3)$, noticing that $k_c$ belongs to a disc of radius $\min(N_c,R_1(N_*)^{C\kappa^{-1}})$ once all $k_j(3\leq j\neq c)$ are fixed, and then apply Lemma \[lem:counting\] (2) to count the pair $(k_{1},k_{2})$. This gives $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(\min(N_c,R_1)^2\prod_{i=2}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\neq c}N_j^2\bigg)N_1\frac{N_1}{R_1},$$ which gains a factor $N_c^{-1}$ upon the stronger bound (\[bdset2\]).
As for (\[bdset2\]) and (\[bdset3\]) we only need to consider $a$. If $a\not\in A$ we may fix all $k_j (c\neq j\in A)$ and apply (\[bdset2\]) or (\[bdset3\]) to the rest variables and get $${\color{black}(\#S_1^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim N_a^{-2}\prod_{j\not\in A}N_j^2\prod_{i=1}^p N_{2i-1}^3R_i\prod_{j\in A,c\neq j\geq 2p+1}N_j^2,$$ $${\color{black}(\#S_2^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim N_0N_a^{-1}\prod_{j\not\in A}N_j^2\prod_{i=1}^p N_{2i-1}^3R_i\prod_{j\in A,c\neq j\geq 2p+1}N_j^2,$$ which gains a factor $N_c^{-2}$ upon (\[bdset2\]) or (\[bdset3\]). If $a\in A$ then (\[bdset2\]) follows from the above proof for (\[bdset1\]); for (\[bdset3\]), if $2p+1\leq a\in A$ we may fix all $k_j (a\neq j\in A)$ and apply (\[bdset3\]) to the rest variables and get $${\color{black}(\#S_2^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim N_0\prod_{j\not\in A}N_j^2\prod_{i=1}^p N_{2i-1}^3R_i\prod_{j\in A,a\neq j\geq 2p+1}N_j^2,$$ which gains a factor $N_a^{-1}$ upon (\[bdset3\]); if $a\leq 2p$ (say $a=1$) we may fix all $k_j(2\leq j\in A)$, noticing that $k_c$ belongs to a ball of radius $\min(N_c,R_1(N_*)^{C\kappa^{-1}})$ once all $k_j(j\in A\backslash\{1,2,c\})$ are fixed, and apply (\[bdset3\]) to the rest variables and get$${\color{black}(\#S_2^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim\min(N_c,R_1)^2N_1^2\frac{N_1}{R_1}N_0\prod_{j\not\in A}N_j^2\prod_{i=2}^p N_{2i-1}^3R_i\prod_{j\in A,c\neq j\geq 2p+1}N_j^2,$$ which gains a factor $N_c^{-1}$ upon (\[bdset3\]).
\(3) Now we consider (\[bdset5\]) and its improvement. We may assume $\iota_a=+$ and $N^{(1)}\gg N^{(2)}$ (so $a\geq 2p+1$), since otherwise (\[bdset5\]) follows from (\[bdset2\]) or (\[bdset1\]) and similarly for the improvement. Now let $M_0=\max(|\alpha|,(N^{(2)})^2)$, if $M\gg M_0$ then we have $$\big||k|^2-|k_{a}|^2\big|\leq|\alpha|+\sum_{j\neq a}|k_j|^2+M\lesssim M;$$ combining with (\[gammacon\]) and Lemma \[lem:counting\] (1) we conclude that the number of choices for $|k_a|^2$, and thus $k_a$, is $O(M)$. We may fix $k_a$ and then count $k_j(j\neq a)$ to get $$(\#S_3)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim M\prod_{i=1}^p N_{2i-1}^3R_i\prod_{2p+1\leq j\neq a}N_j^2,$$ which proves (\[bdset5\]). As for $S_3^+$, [if $a\in A$ then the improvement of (\[bdset5\]) follows from the improvement of (\[bdset2\])]{}; if $a\not\in A$ we may fix $k_a$ and count $k_j(j\not\in\{a,c\})$ to get $${\color{black}(\#S_3^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim M\prod_{i=1}^p N_{2i-1}^3R_i\prod_{2p+1\leq j\not\in \{a,c\}}N_j^2,$$ which gains a factor $N_c^{-2}$ upon (\[bdset5\]).
Assume now $M\lesssim M_0$, then just like above we have $\big||k|^2-|k_a|^2\big|\lesssim M_0$, so $k$ has at most $O(M_0)$ choices, similarly $k_a$ has at most $O(M_0)$ choices. If $b\geq 2p+1$, we may assume $\iota_b=+$ (otherwise switch the roles of $k$ and $k_a$), then fix $k$ and $k_j(j\not\in\{a, b\})$ and apply Lemma \[lem:counting\] (2) to count the pair $(k_a,k_b)$ to get $$(\#S_3)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim M_0\bigg(\prod_{i=1}^p N_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{a,b\}}N_j^2\bigg)M,$$ which proves (\[bdset5\]); if $b\leq 2p$ (say $b=1$), we may fix $k$ and $k_j(j\not\in\{1,2,a\})$ and apply Lemma \[lem:counting\] (2) to count the [triple]{} $(k_1,k_2,k_a)$, noticing that $k_a$ belongs to a disc of radius $O(R_1(N_*)^{C\kappa^{-1}})$ once $k$ and $k_j(j\not\in\{1,2,a\})$ are fixed, and get $$(\#S_3)\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim \bigg(M_0\prod_{i=2}^p N_{2i-1}^3R_i\prod_{2p+1\leq j\neq a}N_j^2\bigg)N_1R_1M\frac{N_1}{R_1},$$ which proves (\[bdset5\]).
It remains to prove the improvement of (\[bdset5\]) for [$S_3^+$.]{} We may assume $a\not\in A$, since otherwise it follows from [the improvement of (\[bdset2\])]{}. Now if $b\not\in A$ we may fix all $k_j (c\neq j\in A)$ and apply (\[bdset5\]) to the rest variables and get $${\color{black}(\#S_3^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim MM_0(N_aN_b)^{-2}\prod_{j\not\in A}N_j^2\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\neq c}N_j^2,$$ which gains a factor $N_c^{-2}$ upon (\[bdset5\]). If $b\leq 2p$, say $b=1$, we may fix $k$ and $k_j(3\leq j\neq a)$, noticing that $k_c$ belongs to a disc of radius $\min(N_c,R_1(N_*)^{C\kappa^{-1}})$ once all $k_j(3\leq j\not\in\{ a,c\})$ are fixed, and then apply Lemma \[lem:counting\] (2) to count the pair $(k_1,k_2)$ and get $${\color{black}(\#S_3^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim MM_0\bigg(\min(N_c,R_1)^2\prod_{i=2}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{ a,c\}}N_j^2\bigg)N_1\frac{N_1}{R_1},$$ which gains a factor $N_c^{-1}$ upon (\[bdset5\]). Finally, assume $2p+1\leq b\in A$, then we will prove (\[bdset6\]). Let $\max^{(2)}\{N_j:2p+1\leq j\in A\}\sim N_d$, we may fix $k$ and $k_j(j\not\in \{a,b,d\})$, then apply Lemma \[lem:counting\] (2) to count the pair $(k_b,k_d)$ and get $$\label{lastone}{\color{black}(\#S_3^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim MM_0\bigg(\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{ a,b,d\}}N_j^2\bigg)N_d;$$alternatively we may choose to fix $k_j(j\not\in\{a,b\})$ then apply Lemma \[lem:counting\] (2) to count the pair $(k,k_a)$ and get $$\label{lasttwo}{\color{black}(\#S_3^+)}\prod_{i=1}^p\frac{N_{2i-1}}{R_i}\lesssim M\bigg(\prod_{i=1}^pN_{2i-1}^3R_i\prod_{2p+1\leq j\not\in\{ a,b\}}N_j^2\bigg)N_a,$$ and combining (\[lastone\]) and (\[lasttwo\]) yields (\[bdset6\]).
In the last part we will explain how to put back the powers $N_{2i-1}^{2\gamma_0}$. In fact, [in each estimate above we have the product $\prod_{i\geq 2}N_{2i-1}^3R_i$. ]{}As $R_i\lesssim N_{2i-1}^{1-\delta}$ and $N_{2i-1}\sim N_{2i}$ we have $$N_{2i-1}^3R_i\lesssim N_{2i-1}^2N_{2i}^2 \cdot N_{2i-1}^{-2\gamma_0},$$ which allows us to incorporate the extra factor $N_{2i-1}^{2\gamma_0}$ for $i\geq 2$. Thus we lose at most a factor $N_{1}^{2\gamma_0}$, which is acceptable as $N_1\lesssim N_{PR}$.
\[corcounting\] Recall $a_0>1$ defined in (\[defparam\]), and let all the parameters ($d$, $N_j$, $\iota_j$ etc.) be as in Proposition \[counting1\]. From the sets $S_j(1\leq j\leq 3)$ in Proposition \[counting1\] we may construct the quantities $\mathcal{E}_j$ as follows: each $\mathcal{E}_j$ is a sum over a set [${S}_j^{\mathrm{lin}}$.]{} This [${S}_j^{\mathrm{lin}}$]{} is formed from $S_j$ by removing [from its defining properties the one that involves the quadratic algebraic sum $\Sigma$]{} (this $\Sigma$ is $\iota_1|k_1|^2+\cdots +\iota_n|k_n|^2-|k|^2$ for $S_1$ and $S_3$, and $\iota_1|k_1|^2+\cdots +\iota_n|k_n|^2+\iota|k'|^2-|k|^2$ for $S_2$), and the summand is simply $\langle \Sigma-\alpha \rangle^{-a_0}$. Similarly define [$\mathcal{E}_j^+$]{} by [replacing $S_j$ with $S_j^+$.]{}
Then, the inequalities (\[bdset1\])–(\[bdset6\]), as well as their improvements, hold with $\# S_j$ replaced by $\mathcal{E}_j$ $(\#S_j^+$ replaced by $\mathcal{E}_j^{+}$), and with the factor $M$ on the right hand sides of (\[bdset5\]) and (\[bdset6\]) removed.
This is straightforward, by applying Proposition \[counting1\] for each value of $\Sigma$ and summing up using $a_0>1$ for (\[bdset1\])–(\[bdset3\]), and by dyadically decomposing $\langle\Sigma-\alpha\rangle$ and applying Proposition \[counting1\] for each dyadic piece for (\[bdset5\])–(\[bdset6\]).
Proof of the multilinear estimates {#multiest}
==================================
In this section we will prove Proposition \[multi0\] [ thus completing the local theory]{}. We will start with an estimate for general multilinear forms without pairing. [Given $d\in\mathbb{Z}^2$ and $\alpha\in\mathbb{R}$,]{} consider the following expressions: $$\label{mainexp1}
\mathcal{X}={\color{black}\sum_{\substack{(k,k_1,\cdots,k_n)\\\iota_1 k_1+\cdots+\iota_n k_n=k+d}}}\int\mathrm{d}\lambda\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\cdot\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)\overline{v_{k}(\lambda)}\prod_{j=1}^n[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j},$$ $$\begin{gathered}
\label{mainexp2}
\mathcal{Y}={\color{black}\sum_{\substack{(k,k',k_1,\cdots,k_n)\\\iota_1 k_1+\cdots+\iota_n k_n+\iota k'=k+d}}}\int \mathrm{d}\lambda\mathrm{d}\lambda'\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\\\times\eta\bigg(\lambda,\lambda-|k|^2-\iota(\lambda'-|k'|^2)-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)y_{kk'}(\lambda,\lambda')\prod_{j=1}^n[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j},\end{gathered}$$ where $d\in\mathbb{Z}^2$ and $\alpha\in\mathbb{R}$ are fixed, $\eta$ is a function that satisfies $$\label{kernelest2}|\eta(\lambda,\mu)|+|\partial_{\lambda,\mu}\eta(\lambda,\mu)|\lesssim\langle\mu\rangle^{-10}.$$ In the summation we always assume [ that]{} there is *no pairing*[^14] among the variables $k$, $k'$ and $k_j$.
We assume that the input functions $v^{(j)}$ are as in Proposition \[multi0\], where $v^{(j)}$ are of type (G) or (C) for $1\leq j\leq n_1$, and of type (D) for $n_1+1\leq j\leq n$. [Since we are working exclusively in the $\lambda_j$ spaces, we will abuse notations here and write $(v_{k_j}^{(j)})(\lambda_j)$ instead of $(\widetilde{v^{(j)}})_{k_j}(\lambda_j)$.]{} Let the parameters $N_j$, $L_j$, $N^{(j)}$ etc., and the sets $\mathcal{G}$ and $\mathcal{C}$ be as in that proposition. We further assume [ that]{} the functions $v_k(\lambda)$ and $y_{kk'}(\lambda,\lambda')$ satisfy $$\label{input0}\|\langle\lambda\rangle^bv_{k}(\lambda)\|_{\ell_{k}^2L_{\lambda}^2}\lesssim 1,\quad \|\langle\lambda\rangle^b\langle \lambda'\rangle^by_{kk'}(\lambda,\lambda')\|_{\ell_{k,k'}^2L_{\lambda,\lambda'}^2}\lesssim 1,$$ and that $v_{k}(\lambda)$ is supported in $\{|k|\lesssim N_0\}$ and $y_{kk'}(\lambda,\lambda')$ is supported in $\{|k|,|k'|\lesssim N_0\}$.
\[general\] Recall the relevant constants defined in (\[defparam\]), and that $\tau\ll 1$. Under all the above assumptions, there exist $p$ and $q$, and $N_{2p+l}\gtrsim R_{2p+l}\gtrsim L_{2p+l}(1\leq l\leq q)$ such that $2p+q\leq n_1$, that for $1\leq i\leq p$ we must have $N_{2i-1}\sim N_{2i}$ and $\iota_{2i-1}=-\iota_{2i}$, and that $2i-1$ and $2i$ do not both belong to $\mathcal{G}$. Define $R_i=\max(L_{2i-1},L_{2i})$ and let $N_*$ be fixed, then the following estimates hold $\tau^{-1}N_*$-certainly. Here, as in Proposition \[multi0\], the exceptional set of $\omega$ removed does *not* depend on the choice of the functions $v_j(j\geq n_1+1)$ or $v$ or $w$.
\(1) Assume $N_*\gtrsim\max(N_0,N^{(1)})$. Then we have $$\label{estimate1}|\mathcal{X}|^2\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}\mathcal{E}_1\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}\prod_{j=1}^nN_j^{-2}\prod_{j\geq n_1+1}N_j^{2\gamma}\prod_{2p+1\leq j\leq n_1}L_j^{-2\delta_0},$$ and similarly $$\label{estimate2}|\mathcal{Y}|^2\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}\mathcal{E}_2\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}\prod_{j=1}^nN_j^{-2}\prod_{j\geq n_1+1}N_j^{2\gamma}\prod_{2p+1\leq j\leq n_1}L_j^{-2\delta_0},$$ where $\mathcal{E}_1$ and $\mathcal{E}_2$ are the quantities defined in Corollary \[corcounting\], with $a_0=2b-10\delta^6$, and for *some* choice of the parameters in that corollary that [do not appear in the assumptions of the current proposition]{}. Moreover, if in the sum defining $\mathcal{X}$ we also assume the $\Gamma$-condition (\[gammacon\]), then (\[estimate1\]) holds with $\mathcal{E}_1$ replaced by $\mathcal{E}_3$ (see Corollary \[corcounting\] for the relevant definitions).
\(2) Assume $N_*\gtrsim\max(N_0,N^{(1)})$. Then we have $$\label{estimate3}|\mathcal{X}|^4\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}} \mathcal{E}_1{\color{black}\mathcal{E}_{1}^{+}}\bigg(\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}\bigg)^2\prod_{j=1}^nN_j^{-4}\prod_{j\geq n_1+1}N_j^{4\gamma}\prod_{2p+1\leq j\leq n_1}L_j^{40n^2},$$ and similarly $$\label{estimate4}|\mathcal{Y}|^4\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}\mathcal{E}_2{\color{black}\mathcal{E}_{2}^{+}}\bigg(\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}\bigg)^2\prod_{j=1}^nN_j^{-4}\prod_{j\geq n_1+1}N_j^{4\gamma}\prod_{2p+1\leq j\leq n_1}L_j^{40n^2},$$ where $\mathcal{E}_j$ and [$\mathcal{E}_j^{+}$]{} are the quantities defined in Corollary \[corcounting\], again for some choice of the parameters in that corollary that [do not appear in the assumptions of the current proposition]{}. In the set [$S^{+}$]{} in (\[extraeqn\]) the set $A$ will contain $\{1,2,\cdots,2p\}\cup \{n_1+1,\cdots,n\}$. Moreover, if in the sum defining $\mathcal{X}$ we also assume the $\Gamma$-condition (\[gammacon\]), then (\[estimate3\]) holds with [$\mathcal{E}_1\mathcal{E}_{1}^{+}$]{} replaced by [$\mathcal{E}_3\mathcal{E}_{3}^{+}$.]{}
\(3) Assume in addition that $N^{(1)}\sim N_n$ and $N_*\gtrsim N^{(2)}$. Then (\[estimate3\]) is true, with $N_n$ replaced by $N^{(2)}$ in both quantities $\mathcal{E}_1$ and [$\mathcal{E}_1^{+}$.]{} Moreover we have $$\label{estimate5}|\mathcal{X}|^4\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}(N^{(1)})^{-4(1-\gamma)}(N^{(2)})^{C\gamma}\widetilde{\mathcal{E}_1}{\color{black}\widetilde{\mathcal{E}_{1}^{+}}}\bigg(\prod_{i=1}^p\frac{N_{2i-1}^{1+2\gamma_0}}{R_i}\bigg)^2\prod_{j=1}^{n-1}N_j^{-4}\prod_{j=2p+1}^{2p+q}N_j^{2}R_j^{-2},$$where $\widetilde{\mathcal{E}_1}$ is the quantity defined in Corollary \[corcounting\], for some choice of the parameters in that corollary that [do not appear in the assumptions of the current proposition]{}, but with $N_n$ replaced by $N^{(2)}$. Similarly [$\widetilde{\mathcal{E}_{1}^{+}}$]{} is the quantity [$\mathcal{E}_1^+$]{}defined in Corollary \[corcounting\] with $A=\{1,\cdots,n\}$, but with $N_n$ replaced by $N^{(2)}$ and $N_{2p+l}$ [replaced by $R_{2p+l}(N_*)^{C\kappa^{-1}}$ for $1\leq l\leq q$.]{} Moreover, the exceptional set of $\omega$ removed is independent of $N^{(1)}$.
Proof of Proposition \[general\]
--------------------------------
We will prove Proposition \[general\] in the following three subsections. We will only prove the bounds for $\mathcal{X}$ without $\Gamma$-condition; with obvious modifications the proof also works for $\mathcal{Y}$ and for the version with $\Gamma$-condition. For simplicity we will omit the $\omega$ dependence, and may ignore any factors that are $\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}$.
Our proof will roughly follow an algorithm, indicated by the following steps. (1) Distinguish between the inputs $j\in\mathcal{D}$, where $v^{(j)}$ are bounded in $\ell^2L^2$, with $j\in\mathcal{G}\cup\mathcal{C}$. (2) Identify the pairings among $k_j(j\in\mathcal{G})$ and $k_j^*(j\in\mathcal{C})$ and reduce the sum of products of the $h^{(j)}$ functions over the paired variables to some functions $P^{(i)}$, see (\[contraction\]), that are also bounded in $\ell^2L^2$. (3) Estimate the sum in unpaired variables using Lemma \[lemma:5.1\] ([in Section \[case2\]]{} we will skip step (2) and estimate the whole sum including paired and unpaired variables using Lemma \[largedev\]). (4) Apply Cauchy-Schwartz to handle all the factors in $\ell^2L^2$, and
color[blue]{} then reduce to the $\mathcal{E}_j$ type quantities in Corollary \[corcounting\]. (5) [ When]{} necessary, apply a $\mathcal{T}^*\mathcal{T}$ argument and repeat the previous steps for the resulting kernel.
As the proof will be notation heavy, [ the reader may do a first reading making]{} the following simplifications without missing the core parts of the proof: (1) omit integration in any $\lambda_j$ and pretend $\lambda_j=0$ (so $v^{(j)}$ is a function of $k_j$ only and $h^{(j)}$ is a function of $k_j$ and $k_j^*$ only); (2) when identifying pairings, restrict to only simple pairings where (say) $k_i^*=k_j^*$ and does not equal any other $k_l^*$. These will make formulas like (\[contbd\]) simpler and the proofs more transparent.
[Throughout the proof we will fix the sets $U=\{1,2,\cdots,n\}$ and $V=\{1,2,\cdots,n-1\}$. We will (in this section only) introduce a shorthand notation for vectors: for a finite set $X$, define $k_{[X]}$ to be the vector $(k_j:j\in X)$; similarly define $\lambda_{[X]}$, $k_{[X]}^*$, etc., and define $\mathrm{d}\lambda_{[X]}=\prod_{j\in X}\mathrm{d}\lambda_j$.]{}
### A simple bound {#case1}
We first prove (\[estimate1\]). By definition we expand $$\begin{gathered}
\label{reduce1}
\mathcal{X}=\sum_{\substack{(k,k_{[U]})\\\iota_1 k_1+\cdots+\iota_n k_n=k+d}}\sum_{k_{[U]}^*}\int\mathrm{d}\lambda\mathrm{d}\lambda_{[U]}\\\times\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)\overline{v_{k}(\lambda)}\prod_{j=1}^{n_1}\, \frac{g_{k_j^*}^{\iota_j}}{\langle k_j^*\rangle} \, h_{k_jk_j^*}^{(j)}(\lambda_j)^\pm\prod_{j=n_1+1}^n[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j},\end{gathered}$$ recall that $k_{[U]}$ means $(k_1,\cdots,k_n)$, etc. The sum in $k_{[U]}^*$ is restricted to $\frac{N_j}{2}<\langle k_j^*\rangle\leq N_j$, and $h_{k_jk_j^*}^{(j)}(\lambda_j,\omega)$ is defined as in (\[input2\]) for $j\in\mathcal{C}$, and is defined to be $\mathbf{1}_{k_j=k_j^*}\widehat{\chi}(\lambda_j)$ for $j\in\mathcal{G}$.
Consider now the sum in $k_{[U]}^*$. By identifying all pairings among them (recall the definition of pairings in Definition \[dfpairing\]), we may assume there are $p$ sets $Y_i(1\leq i\leq p, 2p\leq n_1)$ and a set $Z$ that partitions $\{1,\cdots,n_1\}$, [ such that: (i) each $Y_i$ contains a pairing, (ii) the $k_j^*$ takes a single value for $j$ in each $Y_i$, (iii) this value is different for different $Y_i$ and is different from $k_j^*$ for $j\in Z$, and (iv) there is no pairing in $\{k_j^*:j\in Z\}$.]{} Then we manipulate this sum and rewrite it as a combination of two types of sums, namely (1) where we only require[^15]
that $k_j^*$ takes a single value for $j$ in each $Y_i$ and that there is no pairing in $\{k_j^*:j\in Z\}$, and (2) where there are more pairings in addition to [case (1)]{}, namely [ when]{} the value for $Y_i$ equals the value for some other $Y_{i'}$ or some $k_j^*$ for $j\in Z$. Since there are strictly more pairings in case (2) than in the sum we started with, we may repeat this process and eventually reduce to sums of type (1) only. The purpose of this manipulation is to ensure that the sum in $k_j^*\,(j\in Y_i)$ gives *exactly* [$$\label{contraction}P_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})=\sum_{k^*}\prod_{j\in Y_i}h_{k_jk^*}^{(j)}(\lambda_j)^{\pm}\langle k^*\rangle^{-q_3}(g_{k^*})^{q_1}(\overline{g_{k^*}})^{q_2},$$]{} where $q_1+q_2=q_3=|Y_i|$.
Note that $N_j$ for $j\in Y_i$ are all comparable. Without loss of generality we may assume $\{2i-1,2i\}\subset Y_i$ and $\iota_{2i-1}=-\iota_{2i}$. As $(k_{2i-1},k_{2i})$ is *not* a pairing, $2i-1$ and $2i$ cannot both belong to $\mathcal{G}$. Now we may assume $|k_{2i-1}-k^*|\lesssim L_{2i-1}(N_*)^{C\kappa^{-1}}$ and similarly for $k_{2i}$, since otherwise we gain a power $(N_*)^{-200n^2}$ due to the last bound in (\[input2+\]), which cancels any summation in any $(k_j,k_j^*)$ and the estimate will then [follow immediately]{}.
Let $R_i=\max(L_{2i-1},L_{2i})$, say $R_i=L_{2i-1}$, then we have $|k_{2i-1}-k_{2i}|\lesssim R_i(N_*)^{C\kappa^{-1}}$ for $1\leq i\leq p$. For (\[contraction\]) [ using the first two bounds in (\[input2+\]), we have that $$\label{contbd}
\begin{aligned}
&\quad\,\,\|P_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})\prod_{j\in Y_i}\langle \lambda_j\rangle^b\|_{\ell_{k_{[Y_i]}}^2L_{\lambda_{[Y_i]}}^2}^2\\&\lesssim\prod_{j\in Y_i}N_j^{-2}\cdot\|\langle \lambda_{2i}\rangle^bh_{k_{2i}k^*}^{(2i)}(\lambda_{2i})\|_{\ell_{k^*}^2\to\ell_{k_{2i}}^2L_{\lambda_{2i}}^2}^2\sum_{k^*}\prod_{2i\neq j\in Y_i}\|\langle \lambda_j\rangle^bh_{k_jk^*}^{(j)}(\lambda_j)\|_{\ell_{k_j}^2L_{\lambda_j}^2}^2\\&\lesssim \|\langle \lambda_{2i-1}\rangle^bh_{k_{2i-1}k^*}^{(2i-1)}(\lambda_{2i-1})\|_{\ell_{k_{2i-1}k^*}^2L_{\lambda_{2i-1}}^2}^2\prod_{j\in Y_i}N_j^{-2}\prod_{\substack{j\in Y_i\\j\neq 2i-1,2i}}L_{j}^{-2\delta_0}\\&\lesssim N_{2i-1}^{1+2\gamma_0}\, R_i^{-1}\, \prod_{j\in Y_i}N_j^{-2} \prod_{\substack{j\in Y_i\\j\neq 2i-1,2i}}L_{j}^{-2\delta_0}.
\end{aligned}$$]{} Now we have reduced [ the expression for $\mathcal{X}$]{} to $$\begin{gathered}
\label{reduce2}
\mathcal{X}=\sum_{\substack{(k,k_{[U]}):\\\iota_1 k_1+\cdots+\iota_n k_n=k+d}}\sum_{k_{[Z]}^*}\int\mathrm{d}\lambda\mathrm{d}\lambda_{[U]}\cdot\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)\\\times\overline{v_{k}(\lambda)}\prod_{i=1}^pP_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})\prod_{j\in Z}\frac{g_{k_j^*}^{\iota_j}}{\langle k_j^*\rangle}h_{k_jk_j^*}^{(j)}(\lambda_j)^\pm\prod_{j=n_1+1}^n[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j}.\end{gathered}$$ [Compared to (\[reduce1\]) it is important that there is no pairing in $k_{[Z]}^*$.]{} For simplicity of notations we will write $$\label{simpleexp}\mathcal{X}=\sum_{(1)}\int\mathfrak{F}\cdot\mathfrak{G},$$ where the symbol $\sum_{(1)}\int$ represents the sum in [ $k$ and $k_{[U\backslash Z]}$ and integration in $\lambda$ and $\lambda_{[U\backslash Z]}$, and the factor]{} $\mathfrak{F}$ is $$\mathfrak{F} :=\overline{v_{k}(\lambda)}\prod_{i=1}^pP_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})\prod_{j=n_1+1}^n[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j},$$ and the multilinear Gaussian $\mathfrak{G}$ given by $$\mathfrak{G} :=\sum_{k_{[Z]}}\sum_{k_{[Z]}^*}\int\mathrm{d}\lambda_{[Z]}\cdot\prod_{j\in Z}\frac{g_{k_j^*}^{\iota_j}}{\langle k_j^*\rangle}{\color{black}\langle\lambda_j\rangle^bh_{k_jk_j^*}^{(j)}(\lambda_j)^\pm}\cdot\mathfrak{A},$$ with coefficient $\mathfrak{A}$ of form $$\label{functiona}\mathfrak{A}:=\mathbf{1}_{\sum_{j\in Z}\iota_jk_j=d_0}\cdot\eta\bigg(\lambda,\alpha_0-\sum_{j\in Z}\iota_j(\lambda_j-|k_j|^2)\bigg)\prod_{j\in Z}\langle \lambda_j\rangle^{-b},$$ where $$\label{defnewd}d_0 :=k+d-\sum_{j\not\in Z}\iota_jk_j\in\mathbb{Z}^2,\quad \alpha_0:=\lambda-|k|^2-\alpha-\sum_{j\not\in Z}\iota_j(\lambda_j-|k_j|^2)\in\mathbb{R}.$$ The goal now is to estimate $\mathfrak{G}$. For fixed values of $(k,\lambda,k_{[U\backslash Z]},\lambda_{[U\backslash Z]})$, we may apply Lemma \[lemma:5.1\]; in order to make this uniform, we will apply the same reduction as in the proof of that lemma. Note that $(k,k_{[U\backslash Z]})$ has $\lesssim(N_*)^{C}$ choices; for $(\lambda,\lambda_{[U\backslash Z]})$, we may assume $|\lambda|\lesssim (N_*)^{\delta^{-7}}$ and $|\lambda_j|\lesssim (N_*)^{\delta^{-7}}$ since otherwise we may gain a power $\langle\lambda\rangle^{-\delta^6}$ or $\langle \lambda_j\rangle^{-\delta^6}$ from the weights in (\[input3\]), (\[functiona\]) or (\[input0\]), which is bounded by $(N_*)^{-\delta^{-1}}$ and use this gain to close. Once $\lambda$ and $\lambda_j$ are bounded, we can approximate them by integer multiples of $(N_*)^{-\delta^{-1}}$ and reduce to at most $(N_*)^{C\delta^{-7}}$ choices. In the end, after removing a set of probability $\leq C_{\theta}e^{-(\tau^{-1}N_*)^\theta}$ we can apply Lemma \[lemma:5.1\] for *all* choices of $(k,\lambda,k_{[U\backslash Z]},\lambda_{[U\backslash Z]})$, and use the first bound in (\[input2+\]) to get that $$\label{boundforg}|\mathfrak{G}|^2\lesssim\prod_{j\in Z}N_j^{-2}\prod_{j\in Z}L_j^{-2\delta_0}\cdot\|\mathfrak{A}\|_{\mathcal{L}}^2\\\lesssim\prod_{j\in Z}N_j^{-2}\prod_{j\in Z}L_j^{-2\delta_0}\sum_{k_{[Z]}:\sum_{j\in Z}\iota_jk_j=d_0}\bigg\langle\alpha_0+\sum_{j\in Z}\iota_j|k_j|^2\bigg\rangle^{-a_0}.$$ Finally applying Cauchy-Schwartz in the variables $(k,\lambda,k_{[U\backslash Z]},\lambda_{[U\backslash Z]})$, we deduce that $$|\mathcal{X}|^2\lesssim\bigg(\sum_{(1)}\int\langle\lambda\rangle^{2b}\prod_{j\in U\backslash Z}\langle\lambda_j\rangle^{2b}\cdot|\mathfrak{F}|^2\bigg)\bigg(\sum_{(1)}\int\langle\lambda\rangle^{-2b}\prod_{j\in U\backslash Z}\langle\lambda_j\rangle^{-2b}\cdot|\mathfrak{G}|^2 \bigg),$$ where the first parenthesis (together with some factors from the second parenthesis) gives [the product of all factors in (\[estimate1\]) except $\mathcal{E}_1$,]{} by using (\[input3\]), (\[input0\]) and (\[contbd\]); the second parenthesis, after applying (\[boundforg\]), integrating in $(\lambda,\lambda_{[U\backslash Z]})$ and pugging in (\[defnewd\]), reduces to $$\sum_{\substack{(k,k_{[U]}):\\\iota_1k_1+\cdots +\iota_nk_n=k+d}}\langle \Sigma-\alpha\rangle^{-a_0}\lesssim\mathcal{E}_1,$$ where $\Sigma=\iota_1|k_1|^2+\cdots+\iota_n|k_n|^2-|k|^2$ as in Corollary \[corcounting\]. This proves (\[estimate1\]).
### A general $\mathcal{T}^*\mathcal{T}$ argument {#case2}
Now we prove (\[estimate3\]), starting from (\[reduce2\]). Note that due to (\[input3\]) and (\[input0\]), the bound for $\mathcal{X}$ would follow from the $\ell_{k_n}^2L_{\lambda_n}^2\to \ell_{k}^2L_{\lambda}^2$ bound of the linear operator $\mathcal{T}$ with kernel $$\begin{gathered}
\label{kernel0}\mathcal{T}_{kk_n}(\lambda,\lambda_n)=\sum_{k_{[V]}:\iota_1 k_1+\cdots+\iota_n k_n=k+d}\,\sum_{k_{[Z]}^*}\int\mathrm{d}\lambda_{[V]}\cdot\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)\\\times\bigg(\prod_{i=1}^pP_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})\prod_{j\in Z}\frac{g_{k_j^*}^{\iota_j}}{\langle k_j^*\rangle}h_{k_jk_j^*}^{(j)}(\lambda_j)^\pm\prod_{j=n_1+1}^{n-1}[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j}\bigg)\langle\lambda\rangle^{-b}\langle\lambda_n\rangle^{-b}.\end{gathered}$$ We then calculate the kernel of [$\mathcal{O}=\mathcal{T}^*\mathcal{T}$,]{} which (similar to (\[simpleexp\])) can be written as[ $$\label{simpleexp2}\mathcal{O}_{k_nk_n'}(\lambda_n,\lambda_n')=\langle\lambda_n\rangle^{-b}\langle\lambda_n'\rangle^{-b}\sum_{(2)}\int\mathfrak{F}\cdot\mathfrak{G},$$]{} where the symbol $\sum_{(2)}\int$ represents the sum in $(k_{[V\backslash Z]},k_{[V\backslash Z]}')$ and integration in $(\lambda_{[V\backslash Z]},\lambda_{[V\backslash Z]}')$, the factor $\mathfrak{F}$ is independent of $(k_n,k_n',\lambda_n,\lambda_n')$, [ and is now defined as]{} $$\mathfrak{F}:=\prod_{i=1}^p\overline{P_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})}P_{k_{[Y_i]}'}^{(i)}(\lambda_{[Y_i]}')\prod_{j=n_1+1}^{n-1}\overline{[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j}}[v_{k_j'}^{(j)}(\lambda_j')]^{\iota_j},$$ and the multilinear Gaussian $\mathfrak{G}$ [ is now]{} given by [ $$\label{newdefg}\mathfrak{G}:=\sum_{(k_{[Z]}^*,k_{[Z]}'^*)}\prod_{j\in Z}g_{k_{j}^*}^{-\iota_j}g_{k_j'^*}^{\iota_j}\sum_{\substack{(k,k_{[Z]},k_{[Z]}'):\\\sum_{j\in Z}\iota_jk_j=k+d_0\\\sum_{j\in Z}\iota_jk_j'=k+d_0'}}\mathfrak{C},$$]{} with coefficient [$\mathfrak{C}$]{} of form [ $$\begin{gathered}
\mathfrak{C} :=\int\mathrm{d}\lambda\mathrm{d}\lambda_{[Z]}\mathrm{d}\lambda_{[Z]}'\cdot\langle\lambda\rangle^{-2b}\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j\in Z}\iota_j(\lambda_j-|k_j|^2)-\alpha_0\bigg)\\\times\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j\in Z}\iota_j(\lambda_j'-|k_j'|^2)-\alpha_0'\bigg)
\prod_{j\in Z}\overline{\frac{1}{\langle k_j^*\rangle}h_{k_jk_j^*}^{(j)}(\lambda_j)^\pm}\frac{1}{\langle k_j'^*\rangle}h_{k_j'k_j'^*}^{(j)}(\lambda_j')^\pm,\end{gathered}$$ ]{} where we now have $$\label{defnewd2}d_0:=d-\sum_{j\not\in Z}\iota_jk_j,\,\,d_0':=d-\sum_{j\not\in Z}\iota_jk_j';\quad \alpha_0:=\alpha+\sum_{j\not\in Z}\iota_j(\lambda_j-|k_j|^2),\,\,\alpha_0':=\alpha+\sum_{j\not\in Z}\iota_j(\lambda_j'-|k_j'|^2).$$ As in Section \[case1\] we may assume $|k_{2i-1}-k_{2i}|\lesssim R_i(N_*)^{C\kappa^{-1}}$ for $1\leq i\leq p$ and similarly for $k_{2i-1}'$ and $k_{2i}'$. The goal now is to estimate $\mathfrak{G}$ [ in ]{}. Let $L_+=\max\{L_j:j\in Z\}$, in view of the power $(L_+)^{40n^2}$ on the right hand side of (\[estimate3\]), we may assume $N_j\gg (L_+)^2$ for each $j\in Z$, otherwise we simply sum over $(k_j,k_j^*)$ and $(k_j',k_j'^*)$ and get rid of these variables. By the same arguments as in Section \[case1\], we may reduce to $\lesssim (N_*)^{C\delta^{-7}}$ choices for $(k_{[U\backslash Z]},k_{[U\backslash Z]}',\lambda_{[U\backslash Z]},\lambda_{[U\backslash Z]}')$; for each single choice, as [$\mathfrak{C}$]{} is $\mathcal{B}_{\leq L_+}^+$ measurable and there is no pairing in $k_{[Z]}^*$ or $k_{[Z]}'^*$, we may apply Lemma \[largedev\] and get [ $$\label{individual}|\mathfrak{G}|^2\lesssim\sum_{(k_{[Z\backslash W]}^*,k_{[Z\backslash W']}'^*)}\bigg(\sum_{k_{a_l}^*=k_{b_l}'^*(1\leq l\leq s)}\sum_{\substack{(k,k_{[Z]},k_{[Z]}'):\\\sum_{j\in Z}\iota_jk_j=k+d_0\\\sum_{j\in Z}\iota_jk_j'=k+d_0'}}|\mathfrak{C}|\bigg)^2,$$]{} where $W=\{a_1,\cdots,a_s\}$ and $W'=\{b_1,\cdots, b_s\}$ are subsets of $Z$, and we have $N_{a_l}\sim N_{b_l}$ and $\iota_{a_l}=\iota_{b_l}$ for $1\leq l\leq s$. As before we may assume $|k_j-k_j^*|\lesssim L_+(N_*)^{C\kappa^{-1}}$ and similarly for $k_j'-k_j'^*$, [ and]{} due to the $(L_+)^{40n^2}$ factor we may then fix the values of $k_j-k_j^*=e_j$ and $k_j'-k_j'^*=e_j'$. Therefore $k_{b_l}'-k_{a_l}=e_{b_l}'-e_{a_l}:=f_l$ is also fixed.
[Now the outer sum in (\[individual\]) can be viewed as a sum over $k_{[Z\backslash W]}$ and $k_{[Z\backslash W']}'$, and the inner sums can be viewed as a sum over $(k,k_{a_l},k_{b_l}':1\leq l\leq s)$ *that satisfies* $k_{b_l}'-k_{a_l}=f_l$.]{} When all these $k$-variables are fixed, we have $$\sup_{k_j,k_j^*}\|\langle\lambda_j\rangle^bh_{k_jk_j^*}^{(j)}(\lambda_j)\|_{L_{\lambda_j}^2}\lesssim 1,\quad \sup_{k_j',k_j'^*}\|\langle\lambda_j'\rangle^bh_{k_j'k_j'^*}^{(j)}(\lambda_j')\|_{L_{\lambda_j'}^2}\lesssim 1,$$ due to the first bound in (\[input2+\]). [ Using the algebra property of the norm $\|\langle\lambda\rangle^bh(\lambda)\|_{L^2}$ under convolution, we have]{} $$\begin{gathered}
\label{frakb} |\mathfrak{C}|\lesssim\prod_{j\in Z}N_j^{-2}\int\mathrm{d}\lambda\cdot\langle\lambda\rangle^{-2b}\bigg\langle\lambda-|k|^2+\sum_{j\in Z}\iota_j|k_j|^2-\alpha_0\bigg\rangle^{-b}\bigg\langle\lambda-|k|^2+\sum_{j\in Z}\iota_j|k_j'|^2-\alpha_0'\bigg\rangle^{-b}\\\lesssim \prod_{j\in Z}N_j^{-2}\cdot\bigg\langle|k|^2-\sum_{j\in Z}\iota_j|k_j|^2+\alpha_0\bigg\rangle^{-b}\bigg\langle|k|^2-\sum_{j\in Z}\iota_j|k_j'|^2+\alpha_0'\bigg\rangle^{-b}.\end{gathered}$$ [ With we can now bound]{} $\mathfrak{G}$ by[ $$\begin{gathered}
\label{boundsg}|\mathfrak{G}|^2\lesssim(L_+)^{40n^2}\prod_{j\in Z}N_j^{-4}\sum_{(k_{[Z\backslash W]},k_{[Z\backslash W']}')}\bigg(\sum_{(k,k_{a_l},k_{b_l}':1\leq l\leq s)}\bigg\langle|k|^2-\sum_{j\in Z\backslash W}\iota_j|k_j|^2-\sum_{l=1}^s\iota_{a_l}|k_{a_l}|^2+\alpha_0\bigg\rangle^{-b}\\\times\bigg\langle|k|^2-\sum_{j\in Z\backslash W'}\iota_j|k_j'|^2-\sum_{l=1}^s\iota_{b_l}|k_{b_l}'|^2+\alpha_0'\bigg\rangle^{-b}\bigg)^2,\end{gathered}$$]{} and multiplying out the square we get [ $$\label{2tolastsum}
|\mathfrak{G}|^2\lesssim(L_+)^{40n^2}\prod_{j\in Z}N_j^{-4}\sum_{(k_{[Z\backslash W]},k_{[Z\backslash W']}')}\sum_{\substack{(k,k_{a_l},k_{b_l}':1\leq l\leq s)\\(\accentset{\circ}{k},\accentset{\circ}{k_{a_l}},\accentset{\circ}{k_{b_l}'}:1\leq l\leq s)}}\langle \Upsilon\rangle^{-b}\cdot\langle\accentset{\circ}{\Upsilon}\rangle^{-b}\cdot\langle\Upsilon'\rangle^{-b}\cdot\langle\accentset{\circ}{\Upsilon'}\rangle^{-b},$$]{} where $$\Upsilon=|k|^2-\sum_{j\in Z\backslash W}\iota_j|k_j|^2-\sum_{l=1}^s\iota_{a_l}|k_{a_l}|^2+\alpha_0,\quad\accentset{\circ}{\Upsilon}=|\accentset{\circ}{k}|^2-\sum_{j\in Z\backslash W}\iota_j|k_j|^2-\sum_{l=1}^s\iota_{a_l}|\accentset{\circ}{k_{a_l}}|^2+\alpha_0,$$ $$\Upsilon'=|k|^2-\sum_{j\in Z\backslash W'}\iota_j|k_j'|^2-\sum_{l=1}^s\iota_{b_l}|k_{b_l}'|^2+\alpha_0',\quad \accentset{\circ}{\Upsilon'}=|\accentset{\circ}{k}|^2-\sum_{j\in Z\backslash W'}\iota_j|k_j'|^2-\sum_{l=1}^s\iota_{b_l}|\accentset{\circ}{k_{b_l}'}|^2+\alpha_0',$$with $\iota_{a_l}=\iota_{b_l}$, [ $\alpha_0$ and $\alpha'_0$ are as in ,]{} and the variables in the summation verify the following linear equations: $$\label{newequations}
\begin{aligned}\sum_{j\in Z\backslash W}\iota_jk_j+\sum_{l=1}^s\iota_{a_l}k_{a_l}-k&=\sum_{j\in Z\backslash W}\iota_jk_j+\sum_{l=1}^s\iota_{a_l}\accentset{\circ}{k_{a_l}}-\accentset{\circ}{k}=d_0,\\
\sum_{j\in Z\backslash W'}\iota_jk_j'+\sum_{l=1}^s\iota_{b_l}k_{b_l}'-k&=\sum_{j\in Z\backslash W'}\iota_jk_j'+\sum_{l=1}^s\iota_{b_l}\accentset{\circ}{k_{b_l}'}-\accentset{\circ}{k}=d_0',
\end{aligned}$$ [ with $d_0$ and $d'_0$ as in ,]{} as well as $k_{b_l}'-k_{a_l}=\accentset{\circ}{k_{b_l}'}-\accentset{\circ}{k_{a_l}}=f_l$.
By Cauchy-Schwartz, we may replace the summand on the right hand side of (\[2tolastsum\]) by $\langle\Upsilon\rangle^{-2b}\cdot\langle\accentset{\circ}{\Upsilon'}\rangle^{-2b}$ (or [ by]{} $\langle\accentset{\circ}{\Upsilon}\rangle^{-2b}\cdot\langle\Upsilon'\rangle^{-2b}$, which is treated [ similarly]{} by symmetry). Now going back to (\[simpleexp2\]) and applying Cauchy-Schwartz in the variables $(k_{[V\backslash Z]},k_{[V\backslash Z]}',\lambda_{[V\backslash Z]},\lambda_{[V\backslash Z]}')$, we get[ $$\begin{gathered}
|\mathcal{X}|^4\lesssim N_n^{-4(1-\gamma)}\sum_{k_n,k_n'}\int\mathrm{d}\lambda_n\mathrm{d}\lambda_n'|\mathcal{O}_{k_nk_n'}(\lambda_n,\lambda_n')|^2\lesssim N_n^{-4(1-\gamma)}\bigg(\sum_{(2)}\int\prod_{j\in V\backslash Z}\langle\lambda_j\rangle^{2b}\langle\lambda_j'\rangle^{2b}\cdot|\mathfrak{F}|^2\bigg)\\\times\bigg(\sum_{k_n,k_n'}\int\mathrm{d}\lambda_n\mathrm{d}\lambda_n'\cdot\langle\lambda_n\rangle^{-2b}\langle\lambda_n'\rangle^{-2b}\sum_{(2)}\int\prod_{j\in V\backslash Z}\langle\lambda_j\rangle^{-2b}\langle\lambda_j'\rangle^{-2b}\cdot|\mathfrak{G}|^2\bigg).\end{gathered}$$]{} The first parenthesis (together with some factors from the second parenthesis) give [the product of all factors in (\[estimate3\]) except $\mathcal{E}_1\mathcal{E}_1^{+}$, by using (\[input3\]) and (\[contbd\]).]{} The second parenthesis, after applying (\[2tolastsum\]) [with the summand $\langle\Upsilon\rangle^{-b}\cdot\langle\accentset{\circ}{\Upsilon}\rangle^{-b}\cdot\langle\Upsilon'\rangle^{-b}\cdot\langle\accentset{\circ}{\Upsilon'}\rangle^{-b}$ replaced by $\langle\Upsilon\rangle^{-2b}\langle\accentset{\circ}{\Upsilon'}\rangle^{-2b}$]{}, integrating in $\lambda_{[U\backslash Z]}$ and $\lambda_{[U\backslash Z]}'$, and plugging in (\[defnewd2\]), reduces to $$\label{reducetosum}\sum_{(k_{[U\backslash W]},k_{[U\backslash W']}')}\sum_{\substack{(k,k_{a_l},k_{b_l}':1\leq l\leq s)\\(\accentset{\circ}{k},\accentset{\circ}{k_{a_l}},\accentset{\circ}{k_{b_l}'}:1\leq l\leq s)}}\langle\Sigma-\alpha\rangle^{-2b}\langle\accentset{\circ}{\Sigma'}-\alpha\rangle^{-2b},$$ where $\Sigma$ and $\accentset{\circ}{\Sigma'}$ are respectively $$\Sigma=\sum_{j\not\in W}\iota_j|k_j|^2+\sum_{l=1}^s\iota_{a_l}|k_{a_l}|^2-|k|^2,\quad\accentset{\circ}{\Sigma'}=\sum_{j\not\in W'}\iota_j|k_j'|^2+\sum_{l=1}^s\iota_{b_l}|\accentset{\circ}{k_{b_l}'}|^2-|\accentset{\circ}{k}|^2,$$ and the variables in the summation satisfy $$\label{newequations2}
\begin{aligned}\sum_{j\not\in W}\iota_jk_j+\sum_{l=1}^s\iota_{a_l}k_{a_l}-k&=\sum_{j\not\in W}\iota_jk_j+\sum_{l=1}^s\iota_{a_l}\accentset{\circ}{k_{a_l}}-\accentset{\circ}{k}=d,\\
\sum_{j\not\in W'}\iota_jk_j'+\sum_{l=1}^s\iota_{b_l}k_{b_l}'-k&=\sum_{j\not\in W'}\iota_jk_j'+\sum_{l=1}^s\iota_{b_l}\accentset{\circ}{k_{b_l}'}-\accentset{\circ}{k}=d,
\end{aligned}$$ as well as $k_{b_l}'-k_{a_l}=\accentset{\circ}{k_{b_l}'}-\accentset{\circ}{k_{a_l}}=f_l$. [Now, when $k_{[U\backslash W]}$ and $(k,k_{a_l}:1\leq l\leq s)$ are fixed, the sum of $\langle\accentset{\circ}{\Sigma'}-\alpha\rangle^{-2b}$ over $k_{[U\backslash W']}'$ and $(\accentset{\circ}{k},\accentset{\circ}{k_{b_l}'}:1\leq l\leq s)$ can be bounded by $\mathcal{E}_1^{+}$ with $A=U\backslash W'$ in (\[extraeqn\]) due to the equations (\[newequations2\]); on the other hand the sum of $\langle\Sigma-\alpha\rangle^{-2b}$ over $k_{[U\backslash W]}$ and $(k,k_{a_l}:1\leq l\leq s)$ can be bounded by $\mathcal{E}_1$. This bounds the sum (\[reducetosum\]) by $\mathcal{E}_1\mathcal{E}_1^+$ and proves (\[estimate3\]).]{}
### A special $\mathcal{T}^*\mathcal{T}$ argument {#case3}
Assume now $N_n=N^{(1)}$ and $N_*\gtrsim N^{(2)}$. Again we only need to study the operator $\mathcal{T}$ given by the kernel (\[kernel0\]); note that $\mathcal{T}_{kk_n}(\lambda,\lambda_n)$ is supported in the set $\{(k,k_n):|k-\iota_nk_n+d|\lesssim N^{(2)}\}$, by the standard orthogonality argument it suffices to prove the same operator bound for $\widetilde{\mathcal{T}}$ which is $\mathcal{T}$ restricted to the set $\{k:|k-f|\leq N^{(2)}\}$, *uniformly* in $f\in\mathbb{Z}^2$. Below we will fix an $f$ and denote $\widetilde{\mathcal{T}}$ still by $\mathcal{T}$, so that in any summations below we may assume $|k-f|\lesssim N^{(2)}$ and $|k_n-\iota_n(f+d)|\lesssim N^{(2)}$ (same for $k_n'$). At this point the parameter $N^{(1)}$ or $N_n$ no longer explicitly appears in the estimate, so the set of $\omega$ we remove will be independent of it. Also we can prove (\[estimate3\]), with $N_n$ replaced by $N^{(2)}$ in both $\mathcal{E}_1$ and $\mathcal{E}_1^{\mathrm{ex}}$, by essentially repeating the proof in Section \[case2\] above [(and making the bound uniform in $f$ in exactly the same way as below)]{}; it remains to prove (\[estimate5\]).
We start with (\[simpleexp2\]) and now look for further pairings in $(k_{[Z]}^*,k_{[Z]}'^*)$ in the expression $\mathfrak{G}$ given by (\[newdefg\]). By repeating the same reduction step in Section \[case1\], we can find two partitions $(X_1,\cdots,X_q,W)$ and $(X_1',\cdots,X_q',W')$[^16] of the set $Z$, where $2p+q\leq n_1$, such that $N_j$ are all comparable for $j$ in each $X_l\cup X_l'$, and further reduce (\[simpleexp2\]) to a sum [ $$\label{simpleexp3}\mathcal{O}_{k_nk_n'}(\lambda_n,\lambda_n')=\langle\lambda_n\rangle^{-b}\langle\lambda_n'\rangle^{-b}\sum_{(3)}\int\mathfrak{F}^+\cdot\mathfrak{G}^+,$$]{} where the symbol $\sum_{(3)}\int$ represents the sum in [ $k_{[V\backslash W]}$ and $k_{[V\backslash W']}'$ and integration in $\lambda_{[V\backslash W]}$ and $\lambda_{[V\backslash W']}'$,]{} the factor $\mathfrak{F}^+$ is independent of $(k_n,k_n',\lambda_n,\lambda_n')$, $$\mathfrak{F}^+=\prod_{i=1}^p\overline{P_{k_{[Y_i]}}^{(i)}(\lambda_{[Y_i]})}P_{(k_{[Y_i]}')}^{(i)}(\lambda_{[Y_i]}')\prod_{j=n_1+1}^{n-1}\overline{[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j}}[v_{k_j'}^{(j)}(\lambda_j')]^{\iota_j}\prod_{l=1}^q{\color{black}Q_{k_{[X_l]},k_{[X_l']}'}^{(l)}(\lambda_{[X_l]},\lambda_{[X_l']}')},$$$$\label{defnewq}{\color{black}Q_{k_{[X_l]},k_{[X_l']}'}^{(l)}(\lambda_{[X_l]},\lambda_{[X_l']}')}:=\sum_{k^*}\prod_{j\in X_l}h_{k_jk^*}^{(j)}(\lambda_j)^{\pm}\prod_{j\in X_l'}h_{k_j'k^*}^{(j)}(\lambda_j')^{\pm}\langle k^*\rangle^{-q_3}(g_{k^*})^{q_1}(\overline{g_{k^*}})^{q_2},$$where $q_1+q_2=q_3=|X_l|+|X_l'|$, and the $P$ factors are defined in (\[contraction\]). We may also fix $a_l\in X_l$ and $b_l\in X_l'$ such that $\iota_{a_l}=\iota_{b_l}$; without loss of generality assume $b_l=2p+l$ for $1\leq l\leq q$. We can bound (\[defnewq\]) just like we bound (\[contraction\]) in (\[contbd\]), except that now it is possible to have $ X_l\cup X_l'\subset \mathcal{G}$. Let $R_{2p+l}=\max\{L_j:j\in X_l\cup X_l'\}\gtrsim L_{2p+l}$, then the same argument as in (\[contbd\]) gives $$\label{boundnewq}
{\color{black}\|Q_{k_{[X_l]},k_{[X_l']}'}^{(l)}(\lambda_{[X_l]},\lambda_{[X_l']}')\prod_{j\in X_l}\langle\lambda_j\rangle^b\prod_{j\in X_l'}\langle\lambda_j'\rangle^b\|_{\ell_{k_{[X_l]},k_{[X_l']}'}^2L_{\lambda_{[X_l]},\lambda_{[X_l']}'}^2}^2}\lesssim\prod_{j\in X_l}N_j^{-2}\prod_{j\in X_l'}N_j^{-2}\cdot N_{2p+l}^{2+2\gamma_0}R_{2p+l}^{-2}.$$ Finally, the multilinear Gaussian $\mathfrak{G}^+$ is given by $$\mathfrak{G}^+=\sum_{(k_{[W]},k_{[W']}')}\sum_{(k_{[W]}^*,k_{[W']}'^*)}\int\mathrm{d}\lambda_{[W]}\mathrm{d}\lambda_{[W']}'\cdot\prod_{j\in W}\overline{\frac{g_{k_j^*}^{\iota_j}}{\langle k_j^*\rangle}{\color{black}\langle\lambda_j\rangle^{b}}h_{k_jk_j^*}^{(j)}(\lambda_j)^\pm}\prod_{j\in W'}\frac{g_{k_j'^*}^{\iota_j}}{\langle k_j'^*\rangle}{\color{black}\langle\lambda_j'\rangle^{b}}h_{k_j'k_j'^*}^{(j)}(\lambda_j')^\pm\cdot\mathfrak{A^+},$$ where there is no pairing among $(k_{[W]}^*,k_{[W']}'^*)$, and coefficient $\mathfrak{A}^+$ of form $$\begin{gathered}
\mathfrak{A}^+=\sum_{\substack{k:\iota_1k_1+\cdots+\iota_nk_n=k+d\\\iota_1k_1'+\cdots+\iota_nk_n'=k+d}}\int\mathrm{d}\lambda\cdot\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)\\\times\langle\lambda\rangle^{-2b}\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j'-|k_j'|^2)-\alpha\bigg)\prod_{j\in W}\langle\lambda_j\rangle^{-b}\prod_{j\in W'}\langle\lambda_j'\rangle^{-b},\end{gathered}$$ [where the sum is over a single variable $k$.]{} As before we may also assume $|k_j-k^*|\lesssim R_{2p+l}(N_*)^{C\kappa^{-1}}$ for $j\in X_l$ in (\[defnewq\]) and similarly for $k_j'$ and $j\in X_l'$, so in particular $|k_{a_l}-k_{2p+l}'|\lesssim R_{2p+l}(N_*)^{C\kappa^{-1}}$.
The goal now is to estimate [$\mathfrak{G}^+$.]{} As before, we need to reduce to $(N_*)^{C\delta^{-7}}$ choices for $(k_{[U\backslash W]},\lambda_{[U\backslash W]})$ and $(k_{[U\backslash W]}',\lambda_{[U\backslash W]}')$, [and $f$.]{} By the same argument as in Sections \[case1\] and \[case2\], we may assume $|\lambda|\lesssim (N_*)^{\delta^{-7}}$ and $|\lambda_j|\lesssim (N_*)^{\delta^{-7}}$ [for $j\not\in W$ (similarly for $\lambda_j'$)]{} and get rid of these parameters; in the same way we may also fix $k_{[V\backslash W]}$ and [$k_{[V\backslash W']}'$,]{} as well as $f+d-\iota_nk_n$ and $f+d-\iota_nk_n'$. Letting $k=f+g$, we can rewrite $$\begin{gathered}
\mathfrak{A}^+=\sum_{|g|\leq N^{(2)}}{\color{black}\mathbf{1}_{\substack{\sum_{j\in W}\iota_jk_j=g+d_0\\\sum_{j\in W'}\iota_jk_j'=g+d_0'}}}\int\mathrm{d}\lambda\cdot\eta\bigg(\lambda,\lambda-2f\cdot g-|g|^2-\sum_{j\in V}\iota_j(\lambda_j-|k_j|^2)+\beta(f)+{\color{black}\gamma}\bigg)\\\times\langle\lambda\rangle^{-2b}\eta\bigg(\lambda,\lambda-2f\cdot g-|g|^2-\sum_{j\in V}\iota_j(\lambda_j'-|k_j'|^2)+\beta'(f)+{\color{black}\gamma'}\bigg){\color{black}\prod_{j\in W}\langle\lambda_j\rangle^{-b}\prod_{j\in W'}\langle\lambda_j'\rangle^{-b}},\end{gathered}$$ where $\gamma,\gamma'\in[0,1)$ are fixed, $d_0,d_0'\in \mathbb{Z}^2$, and $\beta(f),\beta'(f)$ are fixed integer-valued functions of $f$. One may assume $|d_0|,|d_0'|\lesssim N^{(2)}$ since otherwise $\mathfrak{A}^+\equiv 0$, then we may fix them and see that $f$ enters the whole expression only through the function $-2f\cdot g+\beta(f)$; moreover we may restrict $g$ to the set where $|-2f\cdot g+\beta(f)|\leq(N_*)^{\delta^{-7}}$ since otherwise either $\lambda$ or one $\lambda_j$ must be large and we close as before. The reduction to finitely many cases can then be done by invoking the following claim, [which will be proved at the end of this section:]{}
\[claim0\] Let the function $F_{f,\beta}(g)=-2f\cdot g+\beta$, with the particular domain $\mathrm{Dom}(F_{f,\beta})=\{|g|\leq N^{(2)}:|-2f\cdot g+\beta|\leq (N_*)^{\delta^{-7}}\}$. Then when $f\in\mathbb{Z}^2$ and $\beta\in\mathbb{Z}$ varies, the function $F_{f,\beta}$ (together with its domain) has finitely many, and in fact $\lesssim (N_*)^{C\delta^{-7}}$ possibilities.
From now on we may fix the value of $f$. By removing a set of probability $\leq C_\theta e^{-(\tau^{-1}N_*)^\theta}$, we can apply Lemma \[lemma:5.1\] and conclude that (recall that $a_0=2b-10\delta^6$) $$\begin{gathered}
|\mathfrak{G}^+|^2\lesssim\prod_{j\in W}N_j^{-2}\prod_{j\in W'}N_j^{-2}\sum_{(k_{[W]},k_{[W']}')}\int\mathrm{d}\lambda_{[W]}\mathrm{d}\lambda_{[W']}'\prod_{j\in W}\langle\lambda_j\rangle^{-a_0}\prod_{j\in W'}\langle\lambda_j'\rangle^{-a_0}\bigg[\sum_{\substack{k:\iota_1k_1+\cdots+\iota_nk_n=k+d\\\iota_1k_1'+\cdots+\iota_nk_n'=k+d}}\int \mathrm{d}\lambda\\\times\langle\lambda\rangle^{-2b}\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\alpha\bigg)\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j'-|k_j'|^2)-\alpha\bigg)\bigg]^2.\end{gathered}$$ The integral over $\lambda$ gives a factor $$\bigg\langle\Sigma-\sum_{j=1}^n\iota_j\lambda_j-\alpha\bigg\rangle^{-b}\bigg\langle\Sigma'-\sum_{j=1}^n\iota_j\lambda_j'-\alpha\bigg\rangle^{-b},$$ where $\Sigma$ and $\Sigma'$ are defined as $$\Sigma=\sum_{j=1}^n\iota_j|k_j|^2-|k|^2,\quad \Sigma'=\sum_{j=1}^n\iota_j|k_j'|^2-|k|^2.$$Since there is only one value of $k$ in the summation, we can reduce $$\label{boundfinalg}
\begin{aligned}
|{\color{black}\mathfrak{G}^+}|^2&\lesssim\sum_k\sum_{(k_{[W]},k_{[W']}')}\int\mathrm{d}\lambda_{[W]}\mathrm{d}\lambda_{[W']}'\prod_{j\in W}\langle\lambda_j\rangle^{-a_0}\prod_{j\in W'}\langle\lambda_j'\rangle^{-a_0}\\&\times\prod_{j\in W}N_j^{-2}\prod_{j\in W'}N_j^{-2}\bigg\langle\Sigma-\sum_{j=1}^n\iota_j\lambda_j-\alpha\bigg\rangle^{-2b}\bigg\langle\Sigma'-\sum_{j=1}^n\iota_j\lambda_j'-\alpha\bigg\rangle^{-2b}\\&\lesssim\prod_{j\in W}N_j^{-2}\prod_{j\in W'}N_j^{-2}\sum_k\sum_{(k_{[W]},k_{[W']}')}\bigg\langle\Sigma-\sum_{j\not\in W}\iota_j\lambda_j-\alpha\bigg\rangle^{-a_0}\bigg\langle\Sigma'-\sum_{j\not\in W'}\iota_j\lambda_j'-\alpha\bigg\rangle^{-a_0},
\end{aligned}$$[where in the summation over $k$ and $(k_{[W]},k_{[W']}')$ we assume that $\iota_1k_1+\cdots+\iota_nk_n=\iota_1k_1'+\cdots+\iota_nk_n'=k+d$.]{} Returning to (\[simpleexp3\]), by applying Cauchy-Schwartz in the variables $(k_{[V\backslash W]},\lambda_{[V\backslash W]})$ and $(k_{[V\backslash W']}',\lambda_{[V\backslash W']}')$ we conclude as before that $$\begin{gathered}
|\mathcal{X}|^4\lesssim (N^{(1)})^{-4(1-\gamma)}\bigg(\sum_{(3)}\int\prod_{j\in V\backslash W}\langle\lambda_j\rangle^{2b}\prod_{j\in V\backslash W'}\langle\lambda_j'\rangle^{2b}\cdot|\mathfrak{F}|^2\bigg)\\\times\bigg(\sum_{k_n,k_n'}\int\mathrm{d}\lambda_n\mathrm{d}\lambda_n'\cdot\langle\lambda_n\rangle^{-2b}\langle\lambda_n'\rangle^{-2b}\sum_{(3)}\int\prod_{j\in V\backslash W}\langle\lambda_j\rangle^{-2b}\prod_{j\in V\backslash W'}\langle\lambda_j'\rangle^{-2b}\cdot|\mathfrak{G}|^2\bigg).\end{gathered}$$ The first parenthesis (together with some factors from the second parenthesis) give [the product of all factors in (\[estimate5\]) except $\widetilde{\mathcal{E}_1}\widetilde{\mathcal{E}_1^{+}}$, by using (\[input3\]), (\[contbd\]) and (\[boundnewq\]).]{} The second parenthesis, after applying (\[boundfinalg\]) and integrating in $\lambda_{[U\backslash W]}$ and $\lambda_{[U\backslash W']}'$, reduces to $$\label{thirdsum}\sum_k \, \sum_{k_{[U]}:\iota_1k_1+\cdots+\iota_nk_n=k+d} \, \, \sum_{k_{[U]}':\iota_1k_1'+\cdots+\iota_nk_n'=k+d}\langle\Sigma-\alpha\rangle^{-a_0}\langle\Sigma'-\alpha\rangle^{-a_0}.$$ [Now when $(k,k_{[U]})$ are fixed, the sum of $\langle\Sigma'-\alpha\rangle^{-a_0}$ over $k_{[U]}'$ can be bounded by $\widetilde{\mathcal{E}_1^{+}}$ with $A=\{1,\cdots,n\}$ in (\[extraeqn\]), due to the linear equation $\iota_1k_1'+\cdots+\iota_nk_n'=k+d$, the fact that $k_{2p+l}'$ belongs to a disc of radius $O(R_{2p+l}(N_*)^{C\kappa^{-1}})$ once $k_{a_l}$ is fixed, and the fact that $|\iota_nk_n'-f-d|\lesssim N^{(2)}$. Moreover the sum of $\langle\Sigma-\alpha\rangle^{-a_0}$ over $(k,k_{[U]})$ can be bounded by $\widetilde{\mathcal{E}_1}$, due to the fact that $|\iota_nk_n-f-d|\lesssim N^{(2)}$. This then bounds (\[thirdsum\]) by $\widetilde{\mathcal{E}_1}\widetilde{\mathcal{E}_1^{+}}$ and proves (\[estimate5\]).]{}
Let $D=\mathrm{Dom}(F_{f,\beta})$. If $D$ contains three points $g_1,g_2,g_3$ that are not collinear, then we have $|f\cdot(g_1-g_2)|\lesssim (N_*)^{\delta^{-7}}$, $|f\cdot(g_1-g_3)|\lesssim (N_*)^{\delta^{-7}}$, and that $g_1-g_2$ and $g_1-g_3$ are linearly independent. This implies that $|f|\lesssim (N_*)^{2\delta^{-7}}$ and hence $|\beta|\lesssim (N_*)^{3\delta^{-7}}$ so the result is trivial. Now let us assume $D$ is contained in a line $\ell$; we may assume $\ell$ contains at least two points in the set $\{g:|g|\leq N^{(2)}\}$, otherwise $D$ is at most a singleton and the result is also trivial. Then the integer points in $\ell$ can be written as $p+q\sigma$, where $(p,q)\in(\mathbb{Z}^2)^2$ has at most $(N^{(2)})^{10}$ choices (so we may fix them), and hence $D=\{p+q\sigma:|p+q\sigma|\leq N^{(2)},|a\sigma+b|\leq (N_*)^{\delta^{-7}}\}$ where $a$ and $b$ are integers. Again as $|D|\geq 2$ we know that $|a|\lesssim (N_*)^{2\delta^{-7}}$ and $|b|\lesssim (N_*)^{3\delta^{-7}}$, so $F_{f,\beta}$ indeed has [$\lesssim (N_*)^{C\delta^{-7}}$]{} possibilities, as claimed.
\[extrarem\]For later use we will also consider the following variant of $\mathcal{X}$ (same for $\mathcal{Y}$): $$\begin{gathered}
\label{mainexp1+}
\mathcal{X}^+=\sum_{\substack{(k,k_1,\cdots,k_n)\\\iota_1 k_1+\cdots+\iota_n k_n=k+d}}\int\mathrm{d}\lambda\mathrm{d}\lambda_1\cdots\mathrm{d}\lambda_n\mathrm{d}\mu_1\cdots\mathrm{d}\mu_s\\\times\eta\bigg(\lambda,\lambda-|k|^2-\sum_{j=1}^n\iota_j(\lambda_j-|k_j|^2)-\sum_{j=1}^s\mu_j-\alpha\bigg)\overline{v_{k}(\lambda)}\prod_{j=1}^n[v_{k_j}^{(j)}(\lambda_j)]^{\iota_j}\prod_{j=1}^sw_j(\mu_j),\end{gathered}$$ where each $w_j$ satisfies $$\|\langle \mu_j\rangle^bw_j(\mu_j)\|_{L_{\mu_j}^2}\lesssim 1.$$ Then $\mathcal{X}^{+}$ will satisfy exactly the same estimates as $\mathcal{X}$ (same for $\mathcal{Y}$). In fact one can introduce a “virtual” variable $l_j$ which take a single value and view $w_j(\mu_j)$ as a function of $l_j$ and $\mu_j$ which has type (D), and repeat all the above proof with these new variables.
Proof of Proposition \[multi0\] {#conclude}
-------------------------------
With Corollary \[corcounting\] and Proposition \[general\], now we can prove Proposition \[multi0\]. Recall that we will abuse notation and write $(v_{k_j}^{(j)})(\lambda_j)$ instead of $(\widetilde{v^{(j)}})_{k_j}(\lambda_j)$. We will proceed in three steps; note that as before, in the proof below we will ignore any factor $\lesssim \tau^{-\theta}(N_*)^{C\kappa^{-1}}$.
*Step 1: reduction to estimating $\mathcal{X}$ and $\mathcal{Y}$*. [ First notice that, when the set of pairings among the variables involved in $\mathcal{N}_n$ is fixed, the coefficient in $\mathcal{N}_n$ will be a constant (see Remark \[proper\]). By Lemma \[duhamelest\], we may replace $\mathcal{I}$ by $\mathcal{J}$ in all estimates. Now by definition of the relevant norms, the kernel bound (\[trunckernel\]) and duality, we can reduce the desired estimates to the estimates of quantities of form $\mathcal{X}$ (for parts (1) and (2)) and $\mathcal{Y}$ (for part (3)) defined in (\[mainexp1\]) and (\[mainexp2\]), in fact with $d=\alpha=0$, except that the functions $v$ and $y$ introduced by duality only satisfy weaker bounds $$\label{weaker}\|\langle\lambda\rangle^{1-b_1}v_{k}(\lambda)\|_{\ell_{k}^2L_{\lambda}^2}\lesssim 1,\quad \|\langle\lambda\rangle^{1-b_1}\langle \lambda'\rangle^by_{kk'}(\lambda,\lambda')\|_{\ell_{k,k'}^2L_{\lambda,\lambda'}^2}\lesssim 1,$$ instead of (\[input0\]), and that there may be pairings in $\mathcal{X}$ and $\mathcal{Y}$ (but they will always be over-paired).]{} Now if $|\lambda|\leq (N_*)^{C_0}$ where $C_0$ is a large constant depending only on $n$, then since $b_1-b\sim 2b-1\sim\kappa^{-1}\sim\delta^4$, we can replace the power $\langle\lambda\rangle^{1-b_1}$ by $\langle\lambda\rangle^b$ in (\[weaker\]) to match (\[input0\]), at a price of losing a factor $(N_*)^{C\kappa^{-1}}$ which is acceptable. Now we will assume $|\lambda|\geq (N_*)^{C_0}$; below we will only consider part (1) of Proposition \[multi0\], since we have $N_*\gtrsim N^{(1)}$ in part (2) and $N_*\gtrsim N_0$ in part (3), and the proof will be similar and much easier.
Here the point is to use the weight $\langle\lambda\rangle^{1-b_1}$ in (\[weaker\]) to gain a power $\geq (N_*)^{-\frac{C_0}{3}}$, after which we still can assume $\|v_k(\lambda)\|_{\ell_k^2L_\lambda^2}\lesssim 1$. In view of this gain and the assumption $N_*\gtrsim N^{(2)}$, we may fix the values of $k_j$ and/or $k_j^*$ for each $1\leq j\leq n-1$. Moreover when $k_j$ and $k_j^*$ fixed the resulting function in $\lambda_j$ (let’s call them $w_j(\lambda_j)$) satisfies $\|\langle \lambda_j\rangle^bw_j(\lambda_j)\|_{L_{\lambda_j}^2}\lesssim 1$, which implies the corresponding $L_{\lambda_j}^1$ bound, so we may fix $\lambda_j(1\leq j\leq n-1)$ also. Finally as $$\int\langle\lambda_n\rangle^{2b}\|v_{k_n}^{(n)}(\lambda_n)\|_{\ell_{k_n}^2}^2\,\mathrm{d}\lambda_n=\|\langle\lambda_n\rangle^{b}v_{k_n}^{(n)}(\lambda_n)\|_{\ell_{k_n}^2L_{\lambda_n}^2}^2\lesssim (N^{(1)})^{-2(1-\gamma)},$$ we may also fix the value of $\lambda_n$, and reduce to $$\mathcal{X}=\sum_{k}\int v_{k}(\lambda)\mathrm{d}\lambda\cdot \eta(\lambda,\lambda-F(k))G_{k-d'},$$ where $\|G\|_{\ell^2}\sim (N^{(1)})^{-1+\gamma}$ and $F(k)$ is a function of $k$ which, as well as $d'$, depends on the choice of the other fixed variables. By first integrating in $\lambda$ using Cauchy-Schwartz and (\[kernelest2\]), then summing in $k$ using Cauchy-Schwartz again, we deduce that $$|\mathcal{X}|\lesssim \|v_k(\lambda)\|_{\ell_k^2L_\lambda^2}\cdot\|G\|_{\ell^2}\lesssim (N^{(1)})^{-1+\gamma},$$ which suffices in view of the gain $(N_*)^{-C_0/3}$.
*Step 2: the no-pairing case*. We have now reduced Proposition \[multi0\] to the estimates for the quantities $\mathcal{X}$ and $\mathcal{Y}$. If we assume there is no pairing, then we can apply Proposition \[general\], and then Corollary \[corcounting\]. Recall the new parameters such as $p$, $q$ and $R_j$ defined in Proposition \[general\]; denote $L_+=\max(L_{2p+1},\cdots L_{n_1})$ and $N_+=\max(N_{n_1+1},\cdots N_n)$. Also when we talk about an estimate in Proposition \[counting1\] we are actually talking about its counterpart in Corollary \[corcounting\].
In part (1), by combining (\[estimate3\]), (\[bdset1\]) and (\[bdset1\]) with the improvement factor (\[exfactor\]), with $N_n$ replaced by $N^{(2)}$ in both places, we obtain that $$|\mathcal{X}|\lesssim (N^{(1)})^{-1+\gamma}(N^{(2)})^{C\gamma}(L_+)^{40n^3}(N^{(2)})^{-\frac{1}{4}};$$ on the other hand by combining (\[estimate5\]), (\[bdset1\]) and (\[bdset1\]) with the improvement factor (\[exfactor\]), with the changes adapted to $\widetilde{\mathcal{E}_1}$ and $\widetilde{\mathcal{E}_1^{\mathrm{ex}}}$ indicated in Proposition \[general\], we obtain that $$|\mathcal{X}|\lesssim (N^{(1)})^{-1+\gamma}(N^{(2)})^{C\gamma}(L_+)^{-\frac{1}{4}},$$ noticing that $R_{2p+l}\gtrsim L_{2p+l}$ for $1\leq l\leq q$ and $N_j\gtrsim L_j$ for $2p+q+1\leq j\leq n_1$. Interpolating the above two bounds then gives (\[mainmult1\]).
In parts (2) and (3), we have $N^{(1)}\sim N_a$ and $a\in\mathcal{G}\cup\mathcal{C}$, in particular the extra factor (\[exfactor\]) is bounded by $(N_+)^{-1}$; [note that in case (3) one may have $a\in\mathcal{D}$ but in this case the extra factor will be replaced by $(N^{(1)})^{-1}$.]{} By combining (\[estimate1\]) and (\[bdset1\]) we obtain (noticing that $N_{PR}\lesssim N^{(2)}$) $$|\mathcal{X}|\lesssim (N^{(1)}N^{(2)})^{-\frac{1}{2}}(N^{(2)})^{\gamma_0}(N_+)^{C\gamma}(L_+)^{-\delta_0},$$ and by combining (\[estimate3\]) and (\[bdset1\]) together with the improvement factor (\[exfactor\]) we obtain that $$|\mathcal{X}|\lesssim (N^{(1)}N^{(2)})^{-\frac{1}{2}}(N^{(2)})^{\gamma_0}(N_+)^{C\gamma}(L_+)^{40n^3}(N_+)^{-\frac{1}{4}},$$ and interpolating the above two bounds gives (\[mainmult2\]); in the same way (\[mainmult3\]) follows from (\[estimate1\]), (\[estimate3\]), [(\[bdset2\]) and]{} (\[bdset2\]) with the improvement factor (\[exfactor\]), and (\[mainmult5\]) follows from (\[estimate2\]), (\[estimate4\]), [(\[bdset3\]) and (\[bdset3\]) with the suitable improvement factor.]{}
Finally consider (\[mainmult4\]); here we will define $N'=\max^{(2)}(N_{n_1+1},\cdots,N_n)$. Note that $\alpha=0$, so by combining (\[estimate1\]) and (\[bdset5\]) we get $$\label{finalbd1}|\mathcal{X}|\lesssim (N^{(1)})^{-1+\gamma_0}(N_+)^{\gamma}(N')^{C\gamma}(L_+)^{-\delta_0};$$ on the other hand, by combining (\[estimate3\]) and either (\[bdset5\]) with the improvement factor (\[exfactor\]) or (\[bdset6\]), we get that either $$\label{finalbd2}|\mathcal{X}|\lesssim (N^{(1)})^{-1+\gamma_0}(N_+)^{\gamma}(N')^{C\gamma}(L_+)^{40n^3}(N_+)^{-\frac{1}{4}},$$ or $$\label{finalbd3}|\mathcal{X}|\lesssim (N^{(1)})^{-1+\gamma_0}(N_+)^{\gamma}(N')^{C\gamma}(L_+)^{40n^3}{\color{black}\min\big((N')^{-\frac{1}{4}},(N^{(1)})^{\frac{1}{4}}(N_+)^{-\frac{1}{2}}\big)}.$$ Clearly interpolating (\[finalbd1\]) and (\[finalbd2\]) gives (\[mainmult4\]); suppose instead we have (\[finalbd1\]) and (\[finalbd3\]). Now if $N_+\geq (N^{(1)})^{\frac{2}{3}}$ and $L_+\geq (N_+)^{\frac{1}{(40n)^4}}$ then (\[finalbd1\]) implies (\[mainmult4\]); if $N_+\geq (N^{(1)})^{\frac{2}{3}}$ and $L_+\leq (N_+)^{\frac{1}{(40n)^4}}$ then (\[finalbd3\]) implies $|\mathcal{X}|\lesssim (N^{(1)})^{-1.01}$ which implies (\[mainmult4\]); if [$N_+\leq (N^{(1)})^{\frac{2}{3}}$]{} then interpolating (\[finalbd1\]) and (\[finalbd3\]) implies $$|\mathcal{X}|\lesssim (N^{(1)})^{-1+\gamma_0}(N_+)^{\gamma}\lesssim (N^{(1)})^{-1+\gamma_0+\frac{2\gamma}{3}}$$ which implies (\[mainmult4\]). Note also that for general $\alpha$, due to the factor $\frac{\max((N^{(2)})^2,|\alpha|)}{(N^{(2)})^2}$ on the right hand side of (\[bdset5\]), the above argument gives the bound $$\label{finalbd4}|\mathcal{X}|\lesssim (N^{(1)})^{-1+\frac{4\gamma}{5}}\max\bigg(1,\frac{|\alpha|^{\frac{1}{2}}}{N^{(2)}}\bigg).$$
*Step 3: the over-pairings*. We will only consider $\mathcal{X}$, $\mathcal{Y}$ is similar and easier since there cannot be any pairing between $\{k,k'\}$ and any $k_j$ due to $N^{(1)}\lesssim N_0^{1-\delta}$ and the restrictions $|k|,|k'|\geq \frac{N_0}{4}$ in (\[mainmult5\]). Now due to simplicity, any pairing in $\mathcal{X}$ must be an over-pairing; by collecting all these pairings we can find a partition $(A_1,\cdots,A_p,B)$ of $\{1,\cdots,n\}$ such that $|A_i|\geq 3$ and $k_j$ takes a single value for $j$ in each $A_i$, that this value is different for different $1\leq i\leq p$, and there is no over-pairing among $\{k_j:j\in B\}$. Then we can check that either there is no pairing among $\{k,k_j:j\in B\}$, or there is a unique over-pairing $k=k_{j_1}=k_{j_2}$ with $j_1,j_2\in B$ and $(\iota_{j_1},\iota_{j_2})\neq (-,-)$. In the latter case denote $\{j_1,j_2\}=A_0$ and replace $B$ by $B\backslash A_0$, so that there is no pairing among $\{k,k_j:j\in B\}$. Below we will focus on the first case, and leave to the end the necessary changes caused by $A_0$.
Now $\mathcal{X}$ is reduced to $$\label{newx}\mathcal{X}=\sum_{l_1,\cdots,l_p}\int\prod_{i=1}^p\prod_{j\in A_i}(v_{l_i}^{(j)}(\lambda_j))^{\pm}\mathrm{d}\lambda_j\cdot\mathcal{X}',$$where $l_i$ is the common value of $k_j$ for $j\in A_i$ (so that $|l_i|\lesssim N^{(2)}$), and $\mathcal{X}'$ is an expression of the same form as $\mathcal{X}$, but only involves the variables $(k,k_j)$ and $(\lambda,\lambda_j)$ for $j\in B$, with $d$ being a fixed linear combination of $l_i$, and $\alpha$ being a fixed linear combination of $|l_i|^2$. This gives $$|\mathcal{X}|\lesssim\sum_{l_1,\cdots,l_p}\prod_{i=1}^pM_{l_i}^{(i)}\cdot\sup_{l_1,\cdots,l_p}\bigg|\int\prod_{i=1}^p{\frac{1}{\color{black}M_{l_i}^{(i)}}}\prod_{j\in A_i}(v_{l_i}^{(j)}(\lambda_j))^{\pm}\mathrm{d}\lambda_j\cdot\mathcal{X}'\bigg|,\quad {\color{black}M_{l_i}^{(i)}:=\prod_{j\in A_i}\|\langle\lambda_j\rangle^{b_2}v_{l_i}^{(j)}(\lambda_j)\|_{L_{\lambda_j}^2},}$$ where recall that $b_2=b-\delta^6$. When each $l_i$ is fixed, by Remark \[extrarem\], the expression $$\int\prod_{i=1}^p{\frac{1}{\color{black}M_{l_i}^{(i)}}}\prod_{j\in A_i}(v_{l_i}^{(j)}(\lambda_j))^{\pm}\mathrm{d}\lambda_j\cdot\mathcal{X}'$$ can be estimated in the same way as $\mathcal{X}'$ [(replacing $b$ by $b_2$ will not change the proof),]{} which is done in *Step 2* above. We then only need to bound $$\sum_{l_1,\cdots,l_p}\prod_{i=1}^pM_{l_i}^{(i)}=\prod_{i=1}^p\sum_{l_i}M_{l_i}^{(i)},$$ which we establish in the following claim.
\[pairclaim\] Let $K_i=\max(N_j:j\in A_i)$ and $K_i'=\max^{(2)}(N_j:j\in A_i)$, then $\tau^{-1}N_*$-certainly we have that $$\label{pairbd}\sum_{l_i}M_{l_i}^{(i)}\lesssim \left\{
\begin{split}& K_i^{-1+\gamma}(K_i')^{-\frac{1}{3}}, & K_i&\sim N_j,j\in\mathcal{D};\\
& K_i^{-1+\theta},&K_i&\sim N_j,j\in\mathcal{G}\cup\mathcal{C}.
\end{split}
\right.$$
Let [$R_{l_i}^{(j)}=\|\langle\lambda_j\rangle^{b_2}v_{l_i}^{(j)}(\lambda_j)\|_{L_{\lambda_j}^2}$,]{} then we have $\|R^{(j)}\|_{\ell_{l_i}^\infty}\lesssim \|R^{(j)}\|_{\ell_{l_i}^2}\lesssim N_j^{-1+\gamma}$ if $j\in\mathcal{D}$, $\|R^{(j)}\|_{\ell_{l_i}^\infty}\lesssim N_j^{-1+\theta}$ and $\|R^{(j)}\|_{\ell_{l_i}^2}\lesssim N_j^{\theta}$ if $j\in\mathcal{G}$. [If $j\in\mathcal{C}$ we will apply Lemma \[largedev\], and again reduce to finitely many $\lambda_j$ by restricting the size of $\lambda_j$ and dividing into small intervals, and using the differentiability in $\lambda_j$ of $h_{k_jk_j^*}^{(j)}(\lambda_j)$, which is assumed in the statement of Proposition \[multi0\]. In the same way as before, by removing a set of probability $\leq C_\theta e^{-(\tau^{-1}N_*)^\theta}$ and omitting any $\tau^{-\theta}(N_*)^{C\kappa^{-1}}$ factors, we conclude that $$\|R^{(j)}\|_{\ell_{l_i}^\infty}\lesssim \|R^{(j)}\|_{\ell_{l_i}^2}\lesssim N_j^{-1}\|\langle\lambda_j\rangle^bh_{k_jk_j^*}^{(j)}(\lambda_j)\|_{\ell_{k_j,k_j^*}^2L_{\lambda_j}^2}\lesssim N_j^{-\frac{1}{2}+\gamma_0}L_{j}^{-\frac{1}{2}},$$ as well as $$\|R^{(j)}\|_{\ell_{l_i}^\infty}\lesssim N_j^{-1}L_j^2\sup_{k_j,k_j^*}\|\langle\lambda_j\rangle^bh_{k_jk_j^*}^{(j)}(\lambda_j)\|_{L_{\lambda_j}^2}\lesssim N_j^{-1}L_j^2,$$ which also implies $\|R^{(j)}\|_{\ell_{l_i}^\infty}\lesssim N_j^{-0.55}$.]{} Now let $K_i\sim N_j$ and [$K_i'\sim N_s$,]{} then if $j\in\mathcal{D}$, (\[pairbd\]) follows from applying Hölder and measuring $R^{(j)}$ and another factor other than $R^{(j)}$ or $R^{(s)}$ in $\ell_{l_i}^2$, and all other factors in $\ell_{l_i}^\infty$. If $j\in\mathcal{G}\cup\mathcal{C}$, then we may assume $|l_i|\sim K_i$ (otherwise (\[pairbd\]) follows trivially from the third inequality in (\[input2+\])), so the $N_j$ for $j\in A_i$ must all be comparable. We may then assume $j\in\mathcal{G}\cup\mathcal{C}$ for each $j\in A_i$, and (\[pairbd\]) follows from applying Hölder and measuring two factors in $\ell_{l_i}^2$ and the rest in $\ell_{l_i}^\infty$, such that at least one $R^{(j)}$ with $j\in\mathcal{G}$ is measured in $\ell_{l_i}^\infty$ if there is any. This completes the proof.
The general case of Proposition \[multi0\] then follows from the $\mathcal{X}'$ estimate, [namely the]{} no-pairing case in *Step 2*, combined with Claim \[pairclaim\]. More precisely, suppose $N^{(1)}=N_a$ with $a\in A_i$ for some $i$, then if $a\in\mathcal{D}$ the bound (\[pairbd\]) gives the power $(N^{(1)})^{-1+\gamma}$, while the power $(K_i')^{-\frac{1}{3}}$ in (\[pairbd\]), as well as the no-pairing case of the bounds (\[mainmult1\]) and (\[mainmult2\]), give the power $(N^{(2)})^{-\frac{1}{4}}$. If $a\in\mathcal{G}\cup\mathcal{C}$, then the power $(N^{(1)})^{-1+\theta}$ from (\[pairbd\]) is already enough. If $N^{(1)}=N_a$ with $a\in B$ then we simply apply the no-pairing case and use (\[pairbd\]) to [gain decay in $N^{(2)}$]{} when $N^{(2)}=N_j$ and $j\not\in B$. The only nontrivial case is (\[mainmult4\]), where there is no need to [gain decay in $N^{(2)}$]{}, but we have an extra factor $\lesssim\max(1,|\alpha|^{\frac{1}{2}})$ from (\[finalbd4\]), where $\alpha$ is a linear combination of $|l_i|^2$. By Claim \[pairclaim\] we have $$\langle\alpha\rangle^{\frac{1}{2}}\lesssim\max_{1\leq i\leq p}\min_{j\in A_i}N_j,\quad \prod_{i=1}^p\sum_{l_i}M_{l_i}^{(i)}\lesssim (N_*)^\theta\langle\alpha\rangle^{-\frac{1}{2}},$$ which cancels this extra factor and proves (\[mainmult4\]).
Finally we consider the case with $A_0$, say $k=k_{j_1}=k_{j_2}$ and $N_{j_1}\geq N_{j_2}$. Here we can check that $\mathcal{X}$ still has the form (\[newx\]), except that in $\mathcal{X}'$ the input $v_k(\lambda)$ is replaced by $$\widetilde{v}_k(\lambda)=\int_{\pm\lambda_0\pm\lambda_1\pm\lambda_2=\lambda} v_{k}(\lambda_0){\color{black}^{\pm}}\cdot v_{k}^{(j_1)}(\lambda_1){\color{black}^{\pm}}\cdot{\color{black}v_{k}^{(j_2)}}(\lambda_2){\color{black}^{\pm}}\,\mathrm{d}\lambda_1\mathrm{d}\lambda_2,$$ which satisfies, due to the same proof as in Claim \[pairclaim\], $$\label{pairbd2}
\begin{aligned}\|\langle\lambda\rangle^{b_2}\widetilde{v}_k(\lambda)\|_{\ell_k^2L_{\lambda}^2}&\lesssim \|\langle\lambda_0\rangle^{b_2}v_k(\lambda_0)\|_{\ell_k^2L_{\lambda_0}^2}\|\langle\lambda_1\rangle^{b_2}v_k^{(j_1)}(\lambda_1)\|_{\ell_k^\infty L_{\lambda_1}^2}\|\langle\lambda_2\rangle^{b_2}{\color{black}v_{k}^{(j_2)}}(\lambda_2)\|_{\ell_k^\infty L_{\lambda_2}^2}\\
&\lesssim\left\{
\begin{split}& N_{j_1}^{-1+\gamma}N_{j_2}^{-\frac{1}{3}},&j_1&\in\mathcal{D},\\
& N_{j_1}^{-1+\theta},&j_1&\in\mathcal{G}\cup\mathcal{C}.
\end{split}
\right.
\end{aligned}$$ The rest of proof now goes exactly as above using the additional bound (\[pairbd2\]), which has exactly the same gain as in Claim \[pairclaim\], and the set $A_0$ is treated together with the other sets $A_i$. This completes the proof of Proposition \[multi0\].
Stability and convergence {#stability}
-------------------------
[ Recall that $v_N$ is the solution of (\[gauged\]).]{} Proposition \[localmain\] already implies the convergence of $v_N$ on the short time interval $[-\tau,\tau]$. For the purpose of proving global well-posedness, we need some additional results, namely a stability and a commutator estimate. For the notations involved in the proof, see Sections \[decompsol\] and \[apriori\].
\[prop:stability\] Recall the relevant constants defined in (\[defparam\]), and that $\tau\ll 1$, $J=[-\tau, \tau]$. The following two statements hold $\tau^{-1}$-certainly.
\(1) For any $N\leq N'$, we have $$\label{commutator}\|v_N-\Pi_Nv_{N'}\|_{X^{b}(J)}\leq N^{-1+\gamma};$$
\(2) Let $w=\Pi_Nw$ be a solution to (\[gauged\]) on $J$, but with data $w(t_0)$ assigned at some $t_0\in J$ such that $\|w(t_0)-v_N(t_0)\|_{L^2}\leq AN^{-1+\gamma}(\log N)^\alpha$, where $\alpha\geq 0$ is an integer, then we have $$\label{perturbation2}
\|w-v_N\|_{X^{b}(J)}\leq BN^{-1+\gamma}(\log N)^{\alpha+1},$$ where $B$ depends only on $A$ and $\alpha$.
\(1) It suffices to prove $\|\Pi_Nv_{N'}^\dagger-v_N^\dagger\|_{X^{b}}\leq N^{-1+\gamma}$. We write $$\Pi_Nv_{N'}^\dagger-v_N^\dagger=\sum_{N<M\leq N'}\Pi_Ny_M^\dagger=\sum_{N<M\leq N'}(\Pi_N\psi_{M,L_0(M)}^\dagger+{\color{black}\Pi_Nz_M^\dagger}),$$ where $L_0(M)$ is the largest $L$ satisfying $(M,L)\in\mathcal{K}$. The bound for [$\Pi_Nz_M^\dagger$]{} follows from Proposition \[localmain2\], so it suffices to bound $\Pi_N\psi_{M,L_0}$, where $L_0=L_0(M)$. Let $\psi=\psi_{M,L_0}^\dagger$, then we have $$\label{expand0}\psi(t)={\color{black}\chi(t)} e^{it\Delta}(\Delta_Mf(\omega))-i\chi_{\tau}(t)\sum_{l=0}^r(l+1)c_{rl}(m_M^*)^{r-l}\cdot\mathcal{I}\Pi_M\mathcal{N}_{2l+1}\big(\psi,v_{L_0}^\dagger,\cdots,v_{L_0}^\dagger\big).$$ Since $N\leq \frac{M}{2}$, $\Pi_N\psi$ will solve the equation $$\begin{gathered}
\label{expand1}\Pi_N\psi(t)=-i\chi_{\tau}(t)\sum_{l=0}^r(l+1)c_{rl}(m_M^*)^{r-l}\cdot\mathcal{I}\Pi_N\mathcal{N}_{2l+1}\big(\Pi_N\psi,v_{L_0}^\dagger,\cdots,v_{L_0}^\dagger\big)\\-i\chi_{\tau}(t)\sum_{l=0}^r(l+1)c_{rl}(m_M^*)^{r-l}\cdot\mathcal{I}\Pi_N\mathcal{N}_{2l+1}\big(\Pi_N^\perp\psi,v_{L_0}^\dagger,\cdots,v_{L_0}^\dagger\big).\end{gathered}$$ Now $\tau^{-1}$-certainly we may assume (\[induct4\]) and the variant of (\[mainmult4\]) described in Remark \[gamma2\]. Note that (\[induct4\]) allows us to control the first line of (\[expand1\]); the second line of (\[expand1\]) is controlled by using (\[sttime1\]) and the variant of (\[mainmult4\]). In the end we get that $$\|{\color{black}\Pi_N\psi}\|_{X^{b}}\lesssim {\tau}^\theta\|\Pi_N\psi\|_{X^{b}}+{\tau}^\theta M^{-1+\frac{4\gamma}{5}},$$ which proves (\[commutator\]).
\(2) If $N\leq {\color{black} O_{A,\alpha}(1)}$ there is nothing to prove, so we may assume $N$ is large depending on [$(A,\alpha)$.]{} Let $\sigma=\mathcal{A}|w|^2-\mathcal{A}|v_N|^2$ (this is conserved), then we have [$$|\sigma|\lesssim(\|v_N(t_0)\|_{L^2}+\|w(t_0)\|_{L^2})\|w(t_0)-v_N(t_0)\|_{L^2}\lesssim\tau^{-\theta} AN^{-1+\gamma}(\log N)^{\alpha+1}.$$ Note the log loss due to the fact that $\|v_N\|_{L^2}^2\lesssim\tau^{-\theta}\log N$.]{} Recall that $v_N$ and $w$ satisfy the equations $$\label{equationsnew}
\left\{
\begin{aligned}(i\partial_t+\Delta)v_N&=\sum_{l=0}^rc_{rl}(m_N^*)^{r-l}\Pi_N\mathcal{N}_{2l+1}(v_N,\cdots,v_N),\\
(i\partial_t+\Delta)w&=\sum_{l=0}^rc_{rl}(m_N^*+\sigma)^{r-l}\Pi_N\mathcal{N}_{2l+1}(w,\cdots,w)
\end{aligned}
\right.$$ on $J$, so $z=w-v_N$ satisfies the equation $$\begin{gathered}
\label{equationsnew2}
(i\partial_t+\Delta)z=\sum_{l=0}^rc_{rl}[(m_N^*+\sigma)^{r-l}-(m_N^*)^{r-l}]\Pi_N\mathcal{N}_{2l+1}(v_N,\cdots,v_N)\\+\sum_{l=0}^rc_{rl}(m_N^*+\sigma)^{r-l}\Pi_N[\mathcal{N}_{2l+1}(z+v_N,\cdots,z+v_N)-\mathcal{N}_{2l+1}(v_N,\cdots,v_N)]\end{gathered}$$ on $J$, and $z_0=z(t_0)$ satisfies $\|z_0\|_{L^2}\leq AN^{-1+\gamma}(\log N)^\alpha$. In order to bound $\|z\|_{X^{b}(J)}$, it will suffice to prove that given $z_0$ and $\sigma$, the mapping $$\begin{gathered}
\label{equationsnew3}
z^\dagger\mapsto \chi(t-t_0)e^{i(t-t_0)\Delta}z_0-i\chi_{2\tau}(t-t_0)\sum_{l=0}^rc_{rl}[(m_N^*+\sigma)^{r-l}-(m_N^*)^{r-l}]\mathcal{I}_{t_0}\Pi_N\mathcal{N}_{2l+1}(v_N^\dagger,\cdots,v_N^\dagger)\\-i\chi_{2\tau}(t-t_0)\sum_{l=0}^rc_{rl}(m_N^*+\sigma)^{r-l}\mathcal{I}_{t_0}\Pi_N[\mathcal{N}_{2l+1}(z^\dagger+v_N^\dagger,\cdots,z^\dagger+v_N^\dagger)-\mathcal{N}_{2l+1}(v_N^\dagger,\cdots,v_N^\dagger)]\end{gathered}$$ is a contraction mapping from the set $\{z^\dagger:\|z^\dagger\|_{X^{b}}\leq AN^{-1+\gamma}(\log N)^{\alpha+1}\}$ to itself, where [ $$\mathcal{I}_{t_0}F(t)=\mathcal{I}F(t)-\chi(t)e^{i(t-t_0)\Delta}\mathcal{I}F(t_0);\quad \mathcal{I}_{t_0}F(t)=\chi(t)\int_{t_0}^t\chi(t')e^{i(t-t')\Delta}F(t')\,\mathrm{d}t'.$$]{}To this end we will decompose $v_N^\dagger=\sum_{N'\leq N}y_{N'}^\dagger$ and ($\tau^{-1}$-certainly) apply the estimates (\[induct4\]) and (\[mainmult1\]), in the same way as in the proof of (\[induct6\]). More precisely, we may use (\[induct4\]) to control the terms in (\[equationsnew3\]) that contain only one factor $z^\dagger$ (where we use the ${\tau}^\theta$ gain to ensure smallness), and use (\[mainmult1\]) to control the terms in (\[equationsnew3\]) that contain at least two factors $z^\dagger$ (where we use the gain of powers of $N$ to ensure smallness, noticing that $N$ is large enough compared to $B$). Note that in applying these estimates we need to replace $\mathcal{I}$ by $\mathcal{I}_{t_0}$ and $\chi_{\tau}(t)$ by $\chi_{2\tau}(t-t_0)$. This can be done because in Section \[reductmulti\] all estimates for $\chi_{\tau}(t)\cdot\mathcal{I}[\cdots]$ are deduced from (\[sttime1\]) and the corresponding estimates for $\mathcal{I}[\cdots]$; here by definition we have $\|\mathcal{I}_{t_0}F\|_{X^{\widetilde{b}}}\lesssim\|\mathcal{I}F\|_{X^{\widetilde{b}}}$ for $\widetilde{b}\in\{b,b_1\}$ which allows us to replace $\mathcal{I}$ by $\mathcal{I}_{t_0}$, and that $\mathcal{I}_{t_0}F(t_0)=0$ so (\[sttime1\]) is still applicable with $\mathcal{I}$ replaced by $\mathcal{I}_{t_0}$ and $\chi_{\tau}(t)$ replaced by $\chi_{2\tau}(t-t_0)$. The rest of the proof will be the same.
\[conv\] Recall the relevant constants defined in (\[defparam\]), the $\varepsilon$ fixed as in Remark \[fixep\], and that $\tau\ll 1$ and $J=[-\tau,\tau]$. Then the followings hold ${\tau}^{-1}$-certainly.
\(1) For any $N\leq N'$ we have $$\label{converge1}\|v_N-v_{N'}\|_{X^{-\theta,b_2}(J)}\leq {\tau}^{-\theta}N^{-\frac{\theta}{2}},$$ $$\label{converge2}\|(v_N-e^{it\Delta}v_N(0) {\color{black})}-(v_{N'}-e^{it\Delta}v_{N'}(0))\|_{X^{\frac{1}{2}-\gamma_0-\theta,b_2}(J)}\leq N^{-\frac{\theta}{2}}.$$ Note that the $X^{s,b}(J)$ bounds also imply the corresponding $C_t^0H_x^s(J)$ bounds.
\(2) Let $\mathcal{N}_n(v)$ be a polynomial, also viewed as a multilinear form $\mathcal{N}_n(v^{(1)},\cdots,v^{(n)})$, as in (\[multiform\]), but is only assumed to be input-simple (instead of simple). Then for any $N\leq N'$, the distance in $C_t^0H_x^{-\varepsilon}(J)$ of any two of the following expressions $$\label{converge}\mathcal{N}_n(v),\,\Pi_N\mathcal{N}_n(v),\,\Pi_{N'}\mathcal{N}_n(v):v\in\{v_N,v_{N'},\Pi_Nv_{N'}\}$$ is bounded by ${\tau}^{-\theta}N^{-\gamma}$. The same conclusion holds if $\mathcal{N}_n$ is replaced by $W_N^n$ or $W_{N'}^n$, or if $v$ is perturbed by any $w_N$ satisfying [$\|w_N\|_{X^{b}(J)}\leq AN^{-1+\gamma}(\log N)^\alpha$.]{} In the latter case the bound will be [$O_{A,\tau,\alpha}(1)N^{-\gamma}$.]{}
\(1) We only need to prove (\[converge2\]). By taking a summation we may assume $N'=2N$, and it suffices to prove that $\|y_{N'}-e^{it\Delta}(\Delta_{N'}f(\omega))\|_{X^{b_2}(J)}\leq (N')^{-\frac{1}{2}+\gamma_0+\frac{\theta}{2}}$. Now an extension of this function is given by $$y^\dagger=\sum_{L}\zeta_{N',L}^\dagger+z_{N'}^\dagger,$$ see Sections \[decompsol\] and \[apriori\], where $\zeta_{N',L}^\dagger$ is defined from $h^{N',L,\dagger}$ by (\[linearity\]) and (\[matrices\]), and $h^{N',L,\dagger}$ and $z_{N'}^\dagger$ satisfy (\[induct2\]) and (\[induct6\]). This controls the second term; to bound the first term, we use the $\mathcal{B}_{\leq L}^+$ measurability of $h^{N',L,\dagger}$ and Lemma \[largedev\], [and perform the same reduction step as in the proof of Claim \[pairclaim\] using differentiability in $\lambda$ of $\widetilde{h}_{kk^*}(\lambda)$ with $h=h^{N',L,\dagger}$,]{} to bound ${\tau}^{-1}N'$-certainly that $$\label{estfirst}\sum_L\|\zeta_{N,L}^\dagger\|_{X^{b_2}}\lesssim\sum_{L}(N')^{-1}\|h^{N',L,\dagger}\|_{Z^b}\lesssim (N')^{-\frac{1}{2}+\gamma_0+\frac{\theta}{2}}.$$ The right hand side of (\[estfirst\]) may involve a ${\tau}^{-\theta}$ factor, but this loss can always by recovered by adding a $\chi_{\tau}$ factor since $y^\dagger(0)=0$.
\(2) The bounds for $W^n$ follows from the bounds for $\mathcal{N}_n$ and the formulas (\[prop3.2:eq1\]) and (\[prop3.2:eq2\]), noticing that $:\mathrel{|v|^{2r}v}:$ and $:\mathrel{|v|^{2r}}:$ are input-simple. As for $\mathcal{N}_n$, by decomposing $$v_N^\dagger=\sum_{N'\leq N}y_{N'}^\dagger,\quad y_{N'}^\dagger=\chi(t)e^{it\Delta}(\Delta_{N'}f(\omega))+\sum_{L}\zeta_{N',L}^\dagger+z_{N'}^\dagger,$$ it suffices to ${\tau}^{-1}N^{(1)}$-certainly bound $\mathcal{N}_n(v^{(1)},\cdots,v^{(n)})$ in $C_t^0H_x^{-\varepsilon}(J)$ by ${\tau}^{-\theta}(N^{(1)})^{-\gamma}$ for $v^{(j)}$ as in the assumptions of Proposition \[multi0\]. The proof is a much easier variant of the arguments in Section \[case1\], so we will only sketch the most important points.
First, since the $\partial_t$ derivative of all the $v^{(j)}$’s are bounded by $(N^{(1)})^{C}$ (by restricting the size of $\lambda_j$ variables as we did in Section \[case1\]), by dividing $J$ into $(N^{(1)})^{\delta^{-1}}$ intervals we may reduce to $(N^{(1)})^{C\delta^{-1}}$ exceptional sets and thus fix a time $t\in J$. This gets rid of all the $\lambda_j$ variables (so we are considering $v_{k_j}^{(j)}$ and $h_{k_jk_j^*}^{(j)}$), and by a simple $H_t^{\frac{1}{2}+}\hookrightarrow C_t^0$ argument, the estimates (\[input2+\]) and (\[input3\]) remain true with the obvious changes. Now by repeating the arguments in Section \[case1\] (in a simplified situation without $\lambda_j$ integrations) and Section \[conclude\] (which deals with over-pairings) we get that $$\|\Delta_{N_0}\mathcal{N}_n(v^{(1)},\cdots,v^{(n)})\|_{L^2}^2\lesssim {\tau}^{-\theta}(N^{(1)})^{C\kappa^{-1}}(N^{(1)})^{C\gamma}\cdot(\#S)\prod_{j=1}^nN_j^{-2}\prod_{i=1}^p\frac{N_{2i-1}}{R_i},$$ where $N_{2i-1}\sim N_{2i}\gtrsim R_i$ and $$\begin{gathered}
S=\bigg\{(k,k_1,\cdots,k_n)\in(\mathbb{Z}^2)^{n+1}:\sum_{j=1}^n\iota_jk_j=k,\quad |k|\lesssim N_0,\\|k_j|\lesssim N_j\,(1\leq j\leq n),\quad|k_{2i-1}-k_{2i}|\lesssim R_i(N^{(1)})^{C\kappa^{-1}}\,(1\leq i\leq p)\bigg\},\end{gathered}$$ and a simple counting estimate yields $$\|\Delta_{N_0}\mathcal{N}_n(v^{(1)},\cdots,v^{(n)})\|_{L^2}^2\lesssim {\tau}^{-\theta}(N^{(1)})^{C\gamma}\min(1,(N^{(1)})^{-1}N_0)\lesssim {\tau}^{-\theta}N_0^{\varepsilon}(N^{(1)})^{-\gamma}$$ as by our choice $\gamma\ll\varepsilon$, which concludes the proof. With the $w_N$ perturbations the proof works the same way, except that the constants may depend also on $A$.
Before ending this section, we would like to shift the point of view from the probability space $(\Omega,\mathcal{B},\mathbb{P})$ to the spaces $\mathcal{V}$ and $\mathcal{V}_N$. Given [$0<\tau\ll 1$,]{} all the above proof has allowed us to identify a Borel set $E_\tau$ of $\mathcal{V}$ with $\rho(E_\tau)\geq 1-C_{\theta}e^{-\tau^{-\theta}}$, such that when $u_{\mathrm{in}}=v_{\mathrm{in}}\in E_\tau$, all the results in Sections \[structuresol\] and \[multiest\], including Propositions \[localmain\], \[localmain2\] and \[prop:stability\], are true.
In reality we will be using finite dimensional truncations of $E_\tau$, namely $E_{\tau}^{\overline{N}}=\Pi_{\overline{N}}E_\tau$. Clearly when $\Pi_{\overline{N}}u_{\mathrm{in}}\in E_{\tau}^{\overline{N}}$, all the quantitative estimates proved before will remain true if all the frequencies $N,N',L$, etc., are $\leq\overline{N}$. We moreover know that $\rho_{\overline{N}}(E_{\tau}^{\overline{N}})\geq \rho(E_\tau)\geq 1-C_{\theta}e^{-\tau^{-\theta}}$; since the Radon-Nikodym derivative $\frac{\mathrm{d}\mu_{\overline{N}}^\circ}{\mathrm{d}\rho_{\overline{N}}}$ is uniformly bounded in $L^2(\mathrm{d}\rho_{\overline{N}})$, we have that $$\label{measbound}\mu_{\overline{N}}^\circ(E_{\tau}^{\overline{N}})\geq 1-C\sqrt{\rho_{\overline{N}}(\mathcal{V}_{\overline{N}}\backslash E_{\tau}^{\overline{N}})}\geq 1-C_{\theta}e^{-\tau^{-\theta}}.$$ Finally, due to the gauge symmetry of (\[truncnls\]) and (\[gauged\]), we may assume that $E_\tau$ (and $E_\tau^{\overline{N}}$) is rotation invariant, i.e. $e^{i\alpha} E_\tau=E_\tau$ for $\alpha\in\mathbb{R}$.
Global well-posedness and measure invariance {#global}
============================================
In this section we will prove Theorem \[main\]. Recall the sets $E_\tau^{\overline{N}}$ defined at the end of Section \[stability\]. Denote the solution flow of (\[truncnls\]) by $\Phi_t^{N}$ and the solution flow of (\[gauged\]) by $\Psi_t^N$, which are mappings from $\mathcal{V}_N$ to itself. Define successively the sets $$\label{defsigma1}F_{T,K}^{\overline{N}}=\bigcap_{|j|\leq K}(\Psi_{\frac{jT}{K}}^{\overline{N}})^{-1}E_{\frac{T}{K}}^{\overline{N}},$$ $$\label{defsigma2}G_{T,K,A,D}^{\overline{N},\alpha}=\bigg\{v\in \mathcal{V}_{\overline{N}}:\exists t\in[-D,D]\textrm{ s.t. }\Psi_t^{\overline{N}}v=v'+v'',\,\,v'\in F_{T,K}^{\overline{N}},\,\,\|v''\|_{L^2}\leq A{\overline{N}}^{-1+\gamma}(\log\overline{N})^\alpha\bigg\},$$ $$\label{defsigma3}\Sigma=\bigcup_{D\geq 1}\bigcap_{T\geq 2^{10}D}\bigcup_{{\color{black}K\gg T};A,\alpha\geq 1}\limsup_{\overline{N}\to\infty}\Pi_{\overline{N}}^{-1}G_{T,K,A,D}^{\overline{N},\alpha}.$$ Here [$\Pi_{\overline{N}}^{-1}G=G\times\mathcal{V}_{\overline{N}}^{\perp}$ is the cylindrical set.]{} We understand that $T,K,A,D$ all belong to some given countable set (say powers of two), and $\alpha$ is an integer. All these sets are Borel, since in (\[defsigma2\]) one may replace the $\leq$ sign by the $<$ sign, and then restrict to rational $t$ by continuity. We will start by proving global well-posedness and then measure invariance.
\[globalexist\] The set $\Sigma$ satisfies $\mu(\mathcal{V}\backslash\Sigma)=0$, and $W^{2r+1}(u)\in H^{-\varepsilon}$ is well-defined for $u\in\Sigma$. For any $u_{\mathrm{in}}\in\Sigma$, the solutions $u_N(t)=\Phi_t^N\Pi_Nu_{\mathrm{in}}$ to (\[truncnls\]) converge to some $u(t)=\Phi_tu_{\mathrm{in}}$ in $C_t^0H_x^{-\varepsilon}([-T,T])$ for any $T>0$. This $u$ is a distributional solution to (\[nls\]), and $u(t)\in\Sigma$ for each $t$. The mappings $\Phi_t:\Sigma\to\Sigma$ satisfy $\Phi_0=\mathrm{Id}$ and $\Phi_{t+t'}=\Phi_t\Phi_{t'}$.
We first prove $\mu(\mathcal{V}\backslash\Sigma)=0$. By definition we have $$\label{setinclusion}\Sigma\supset\bigcap_{T\geq 2^{10}}\bigcup_{K\gg T}\limsup_{\overline{N}\to\infty}\Pi_{\overline{N}}^{-1}F_{T,K}^{\overline{N}},$$ so it suffices to prove for any fixed $T\geq 2^{10}$ that $$\sup_{K\gg T}\mu\bigg(\limsup_{\overline{N}\to\infty}\Pi_{\overline{N}}^{-1}F_{T,K}^{\overline{N}}\bigg)=1.$$ Now by Fatou’s lemma and the fact that the total variation of $\mu-\mu_{\overline{N}}$ converges to $0$, we have $$\mu\bigg(\limsup_{\overline{N}\to\infty}\Pi_{\overline{N}}^{-1}F_{T,K}^{\overline{N}}\bigg)\geq\limsup_{\overline{N}\to\infty}\mu_{\overline{N}}\big(\Pi_{\overline{N}}^{-1}F_{T,K}^{\overline{N}}\big)=\limsup_{\overline{N}\to\infty}\mu_{\overline{N}}^\circ\big(F_{T,K}^{\overline{N}}\big).$$ By invariance of $\mathrm{d}\mu_{\overline{N}}^\circ$ under the flow $\Psi_{t}^{\overline{N}}$ (Proposition \[measurefact\]) we know that $$\mu_{\overline{N}}^\circ\big(F_{T,K}^{\overline{N}}\big)\geq 1-(2K+1)\mu_{\overline{N}}^\circ\big(\mathcal{V}_{\overline{N}}\backslash E_{\frac{T}{K}}^{\overline{N}}\big)\geq 1-C_\theta Ke^{-(KT^{-1})^\theta}$$ uniformly in $\overline{N}$, and the right hand side converges to $1$ as $K\to\infty$, so $\mu(\mathcal{V}\backslash\Sigma)=0$.
Now suppose $u_{\mathrm{in}}\in\Sigma$. By definition we may choose some $D$, then for any $T\geq 2^{10}D$ we can find $(K,A,\alpha)$ such that $\Pi_{\overline{N}}u_{\mathrm{in}}\in G_{T,K,A,D}^{\overline{N},\alpha}$ for infinitely many $\overline{N}$. We may fix this $T$ (hence also $(K,A,\alpha)$) and this $\overline{N}$, so that $\|\Psi_{t_0}^{\overline{N}}\Pi_{\overline{N}}u_{\mathrm{in}}-v'\|_{L^2}\leq A\overline{N}^{-1+\gamma}(\log\overline{N})^\alpha$ for some $t_0\in[-D,D]$ and $v'\in F_{T,K}^{\overline{N}}$. We proceed in three steps.
*Step 1: analyzing $v'$*. We first prove that, for any $N\leq \overline{N}$ and $|j|\leq K$ there holds that $$\label{commute1}\|\Pi_N\Psi_{\frac{jT}{K}}^{\overline{N}}v'-\Psi_{\frac{jT}{K}}^N\Pi_Nv'\|_{L^2}\leq BN^{-1+\gamma}(\log N)^{|j|}$$ for some $B$ depending only on $(T,K)$. This is obviously true for $j=0$; suppose this is true for $j$, since $\Psi_{\frac{jT}{K}}^{\overline{N}}v'\in E_{\frac{T}{K}}^{\overline{N}}$, by Proposition \[prop:stability\] (1) we have $$\|\Pi_N\Psi_{\frac{(j\pm 1)T}{K}}^{\overline{N}}v'-\Psi_{\frac{\pm T}{K}}^N\Pi_N\Psi_{\frac{jT}{K}}^{\overline{N}}v'\|_{L^2}\leq N^{-1+\gamma}$$ [(note that as $K\gg T$ the local theory is applicable on intervals of length $\frac{T}{K}$),]{} and $$\|\Psi_{\frac{\pm T}{K}}^N\Pi_N\Psi_{\frac{jT}{K}}^{\overline{N}}v'-\Psi_{\frac{(j\pm 1)T}{K}}^N\Pi_Nv'\|_{L^2}\leq B'N^{-1+\gamma}(\log N)^{|j|+1}$$ by Proposition \[prop:stability\] (2) and (\[commute1\]), where $B'$ depends only on $B$ and $(T,K)$, so (\[commute1\]) holds also for $j\pm 1$ which concludes the inductive proof. By the same argument, we can show that (\[commute1\]) remains true with $\frac{jT}{K}$ replaced by any $t\in[-T,T]$ and $|j|$ replaced by $K$.
Similarly, since $\Psi_{\frac{jT}{K}}^{\overline{N}}v'\in E_{\frac{T}{K}}^{\overline{N}}$ for each $|j|\leq K$, by combining Propositions \[prop:stability\] and \[conv\], we conclude that for any $N\leq N'\leq\overline{N}$, $$\label{propv'}\sup_{t\in[-T,T]}\|\Psi_{t}^N\Pi_Nv'-\Psi_{t}^{N'}\Pi_{N'}v'\|_{H^{-\theta}}\leq O_{T,K}(1) N^{-\frac{\theta}{2}},$$ $$\label{propv'2}\sup_{t\in[-T,T]}\|W_N^n(\Psi_{t}^N\Pi_Nv')-W_{N'}^n(\Psi_{t}^{N'}\Pi_{N'}v')\|_{H^{-\varepsilon}}\leq O_{T,K}(1) N^{-\gamma},$$ and the same is true if $W_N^n$ and $W_{N'}^n$ in (\[propv’2\]) is replaced by $\Pi_NW_N^n$ and $\Pi_{N'}W_{N'}^n$.
*Step 2: linking $u_{\mathrm{in}}$ to $v'$*. Recall that $\|\Psi_{t_0}^{\overline{N}}\Pi_{\overline{N}}u_{\mathrm{in}}-v'\|_{L^2}\leq A\overline{N}^{-1+\gamma}(\log\overline{N})^\alpha$. Since $|t_0|\leq D\ll T$, by iterating Proposition \[prop:stability\] (2) we deduce that $\|\Pi_{\overline{N}}u_{\mathrm{in}}-\Psi_{-t_0}^{\overline{N}}v'\|_{L^2}\leq A'\overline{N}^{-1+\gamma}(\log\overline{N})^{\alpha+K}$ where $A'$ depends only on $(T,K,A,\alpha)$. Writing $-t_0=\frac{jT}{K}+t'$ with $|j|\leq 2^{-8} K$ and $|t'|\leq\frac{T}{K}$, we may apply Proposition \[prop:stability\] (2) again and combine this with (\[commute1\]) and similar estimates to deduce for any $N\leq\overline{N}$ that $$\label{link}\sup_{t\in[-\frac{T}{2},\frac{T}{2}]}\|\Psi_t^N\Pi_Nu_{\mathrm{in}}-\Psi_{t-t_0}^N\Pi_Nv'\|_{L^2}\leq BN^{-1+\gamma}(\log N)^{\alpha+2K}$$ with $B$ depending only on $(T,K,A,\alpha)$. By (\[propv’\]), (\[propv’2\]), (\[link\]) and Proposition \[conv\], we conclude for all $N\leq N'\leq \overline{N}$ that $$\label{propuin}\sup_{t\in[-\frac{T}{2},\frac{T}{2}]}\|\Psi_{t}^N\Pi_Nu_{\mathrm{in}}-\Psi_{t}^{N'}\Pi_{N'}u_{\mathrm{in}}\|_{H^{-\theta}}\leq O_{T,K,A,\alpha}(1) N^{-\frac{\theta}{2}},$$ $$\label{propuin2}\sup_{t\in[-\frac{T}{2},\frac{T}{2}]}\|W_N^n(\Psi_{t}^N\Pi_Nu_{\mathrm{in}})-W_{N'}^n(\Psi_{t}^{N'}\Pi_{N'}u_{\mathrm{in}})\|_{H^{-\varepsilon}}\leq O_{T,K,A,\alpha}(1) N^{-\gamma},$$ and the same is true for projections of $W_N^n$.
*Step 3: completing the proof*. Now, for fixed $(D,T,K,A,{\color{black}\alpha})$ we know that there exists infinitely many $\overline{N}$ such that (\[propuin\]) and (\[propuin2\]) are true for all $N\leq N'\leq \overline{N}$, so (\[propuin\]) and (\[propuin2\]) are simply true for all $N\leq N'$. This implies the convergence of $\Psi_{t}^N\Pi_Nu_{\mathrm{in}}$ in $C_t^0H_x^{-\varepsilon}([-\frac{T}{2},\frac{T}{2}])$, [and we will define $\Psi_t=\lim_{N\to\infty}\Psi_t^N\Pi_N$.]{} Since by the definition of gauge transform we have $$\label{linkphipsi}\Phi_t^N\Pi_Nu_{\mathrm{in}}=\Psi_t^N\Pi_Nu_{\mathrm{in}}\cdot e^{-iB_N(t)},\quad B_N(t)=(r+1)\int_0^t\mathcal{A}[W_N^{2r}(\Psi_t^N\Pi_Nu_{\mathrm{in}})]\,\mathrm{d}t',$$ (\[propuin\]) and (\[propuin2\]) also implies the convergence of $\Phi_{t}^N\Pi_Nu_{\mathrm{in}}$ in $C_t^0H_x^{-\varepsilon}([-\frac{T}{2},\frac{T}{2}])$, as well as the convergence of $\Pi_NW_N^{2r+1}(\Phi_t^N\Pi_Nu_{\mathrm{in}})$ in the same space. As $u_N=\Phi_{t}^N\Pi_Nu_{\mathrm{in}}$ solves the equation (\[truncnls\]) with right hand side being $\Pi_NW_N^{2r+1}(u_N)$, we know that the limit $u=\lim_{N\to\infty}u_N$ solves (\[nls\]) in the distributional sense. Let $u(t)=\Phi_tu$, the group properties of $\Phi_t$ follow from the group properties of $\Psi_t^N$ and limiting arguments similar to the above.
Finally we prove that $\Phi_{t_1}u_{\mathrm{in}}\in \Sigma$ for $u_{\mathrm{in}}\in\Sigma$ and any $t_1$. Let $D$ be associated with the assumption $u_{\mathrm{in}}\in\Sigma$, and fix $D_1\gg D+|t_1|$. For any $T\geq 2^{10}D_1$ there exists $(K,A,\alpha)$ such that $\Pi_{\overline{N}}u_{\mathrm{in}}\in G_{T,K,A,D}^{\overline{N},\alpha}$ for infinitely many $\overline{N}$. It suffices to show that for such $\overline{N}$ we must have $\Pi_{\overline{N}}\Phi_{t_1}u_{\mathrm{in}}\in G_{T,K,B,D_1}^{\overline{N},\alpha+3K}$ with $B$ depending only on $(T,K,A,\alpha)$. Since $\Pi_{\overline{N}}\Phi_{t_1}u_{\mathrm{in}}$ and $\Pi_{\overline{N}}\Psi_{t_1}u_{\mathrm{in}}$ only differs by a rotation and the sets we constructed are rotation invariant, we only need to prove the same thing for $\Pi_{\overline{N}}\Psi_{t_1}u_{\mathrm{in}}$.
Now, on the one hand we know for some $|t_0|\leq D$ that $\|\Pi_{\overline{N}}u_{\mathrm{in}}-\Psi_{-t_0}^{\overline{N}}v'\|_{L^2}\leq A'\overline{N}^{-1+\gamma}(\log\overline{N})^{\alpha+K}$ for some $A'$ depending only on $(T,K,A,\alpha)$ (see *Step 2*) and similarly $\|\Psi_{t_1}^{\overline{N}}\Pi_{\overline{N}}u_{\mathrm{in}}-\Psi_{t_1-t_0}^{\overline{N}}v'\|_{L^2}\leq A'\overline{N}^{-1+\gamma}(\log\overline{N})^{\alpha+2K}$. On the other hand by taking limits in (\[commute1\]) and (\[link\]) we also get $\|\Pi_{\overline{N}}\Psi_{t_1}u_{\mathrm{in}}-\Psi_{t_1}^{\overline{N}}\Pi_{\overline{N}}u_{\mathrm{in}}\|_{L^2}\leq A'\overline{N}^{-1+\gamma}(\log\overline{N})^{\alpha+2K}$, and hence $\|\Pi_{\overline{N}}\Psi_{t_1}u_{\mathrm{in}}-\Psi_{t_1-t_0}^{\overline{N}}v'\|_{L^2}\leq A'\overline{N}^{-1+\gamma}(\log\overline{N})^{\alpha+2K}$. Applying Proposition \[prop:stability\] (2) again we get that $\|\Psi_{t_0-t_1}^{\overline{N}}\Pi_{\overline{N}}\Psi_{t_1}u_{\mathrm{in}}-v'\|_{L^2}\leq B\overline{N}^{-1+\gamma}(\log\overline{N})^{\alpha+3K}$ with $|t_0-t_1|\leq D_1$ and $B\leq B(T,K,A, {\color{black}\alpha})$, thus by definition $\Pi_{\overline{N}}\Psi_{t_1}u_{\mathrm{in}}\in G_{T,K,B,D_1}^{\overline{N},\alpha+3K}$. This completes the proof.
\[last\] For any Borel subset $E\subset\Sigma$ and any $t_0\in\mathbb{R}$, we have $\mu(E)=\mu(\Phi_{t_0}E)$.
The map $\Phi_t$ is a limit of continuous mappings, so it is Borel measurable. By taking limits, we may assume the set $E$ is compact in $H^{-\varepsilon}$ topology. We may also assume that $|t_0|\leq 1$, and that for some fixed $(T,K,A,\alpha,D)$ with $K\gg T\geq 2^{10}D$ we have $E\subset\limsup_{\overline{N}\to\infty}G_{T,K,A,D}^{\overline{N},\alpha}$. By the proof of Proposition \[globalexist\] we can deduce that for [$u\in E$ and]{} $|t|\leq 2$, $$\label{commfin1}\|\Psi_t^{N}\Pi_Nu-\Pi_N\Psi_tu\|_{L^2}\lesssim N^{-1+\gamma}(\log N)^{\alpha+3K}$$ with constants depending on $(T,K,A,\alpha)$ (same below). Moreover, concerning the phase $B_N(t)$ involved in the gauge transform, namely $$B_N(t)=(r+1)\int_0^t\mathcal{A}[W_N^{2r}(\Psi_t^N\Pi_Nu)]\,\mathrm{d}t',$$ one can show that as $N\to\infty$, $B_N(t)$ converges to its limit $B(t)$ at a rate $\|B_N(t)-B(t)\|_{C_t^0([-2,2])}\lesssim N^{-1+\gamma}(\log N)^{\alpha+4K}$. In fact we may first reduce to short time intervals where local theory is applicable, then notice that $$\int_0^t\mathcal{A}[W_N^{2r}(\Psi_t^N\Pi_Nu)]\,\mathrm{d}t'=\mathcal{A}\mathcal{I}[W_N^{2r}(\Psi_t^N\Pi_Nu)]$$ for $|t|\ll 1$, and apply Proposition \[multi0\], more precisely (\[mainmult2\]), with the observation that the [mean]{} $\mathcal{A}$ restricts the two highest input frequencies in any multilinear expression $\mathcal{N}_n$ occurring in $W_N^{2r}$ to be comparable, i.e. $N^{(1)}\sim N^{(2)}$. We omit the details.
With the explicit convergence rate of $B_N(t)$, we see that (\[commfin1\]) holds with $\Psi_t^N$ and $\Psi_t$ replaced by $\Phi_t^N$ and $\Phi_t$, and with $3K$ replaced by $4K$. This gives, for $|t|\leq 1$, that $$\Pi_N\Phi_tE\subset\Phi_t^N\Pi_NE+\mathfrak{B}_{L^2}(A_1N^{-1+\gamma}(\log N)^{\alpha+4K})\subset\Phi_t^N(\Pi_NE+\mathfrak{B}_{L^2}(A_2N^{-1+\gamma}(\log N)^{\alpha+5K})),$$ where $A_{1,2}$ are constants depending only on $(T,K,A,\alpha)$, and $\mathfrak{B}_{L^2}(R)$ is the ball of radius $R$ in $L^2$ centered at the origin; note that the second subset relation follows from long-time stability, which is also a consequence of the proof of Proposition \[globalexist\]. By invariance of $\mathrm{d}\mu_N^\circ$ under $\Phi_t^N$ we have that $$\begin{gathered}
\mu_N(\Phi_{t_0}E)\leq\mu_N^\circ(\Pi_N\Phi_{t_0}E)\leq\mu_N^\circ\Phi_{t_0}^N(\Pi_NE+\mathfrak{B}_{L^2}(A_2N^{-1+\gamma}(\log N)^{\alpha+5K}))\\=\mu_N^\circ(\Pi_NE+\mathfrak{B}_{L^2}(A_2N^{-1+\gamma}(\log N)^{\alpha+5K})).\end{gathered}$$ It then suffices to prove that $$\label{compactness}\limsup_{N\to\infty}\Pi_N^{-1}(\Pi_NE+\mathfrak{B}_{L^2}(A_2N^{-1+\gamma}(\log N)^{\alpha+5K}))\subset E,$$ which would imply [$\mu(\Phi_{t_0}E)\leq \mu(E)$]{}, and conclude the proof by time reversibility[[^17]]{}. To prove (\[compactness\]), suppose $u$ is such that $\|\Pi_N(u-u_N)\|_{L^2}\leq A_2N^{-1+\gamma}(\log N)^{\alpha+5K}$ with $u_N\in E$ for infinitely many $N$, then by compactness we may assume $u_N\to v\in E$ in $H^{-\varepsilon}$, so $u_N\to u$ coordinate-wise and $u_N\to v$ coordinate-wise, hence $u=v\in E$ and the proof is complete.
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[^1]: $^1$Y. D. is funded in part by NSF-DMS-1900251.
[^2]: $^2$A.N. is funded in part by NSF DMS-1463714 and NSF-DMS 1800852.
[^3]: Brydges and Slade [@BrySl] proved that the natural construction of an invariant Gibbs measure is not possible for the focusing cubic NLS when $d=2$, unlike the case $d=1$ as proved by Lebowitz, Rose and Speer [@LRS]
[^4]: Gibbs measures in the cubic case of are not available for $d\geq 5$. This is intimately related to the nonexistence of a $\phi^4$ theory when $d \geq 5$ [@Aiz; @Fro]. We have learned that Copin and Aizenman recently also ruled out the existence of a $\phi^4$ theory in $d=4$.
[^5]: Strictly speaking the measure defined in Proposition \[gibbsm\] involves an additional weight which is an exponential of the $L^2$ mass. As the mass is also conserved, this does not affect any invariance properties.
[^6]: This usually results in a finite or infinite tree expansion.
[^7]: There are a few results [@Deng; @DLM0] which obtain a larger improvement for Schrödinger, but these are all in the non-compact setting where much better estimates ([such as]{} local smoothing) are available.
[^8]: This is associated with Gaussian random variables, but by the Central Limit Theorem, the scaling should be the same for more general types of random variables.
[^9]: This may be improved a little by switching to Fourier-Lebesgue spaces, but is still far from deterministic scaling when $r$ is large. On the other hand the Hölder spaces $C^s$, as pointed out in Section \[discuss\], are not suitable for Schrödinger equations.
[^10]: The description of para-controlled calculus in Section \[results2\] actually corresponds to $|P_{\ll N}u|^{2r}-|P_{\ll N}e^{it\Delta}f(\omega)|^{2r}$ instead of $|P_{\ll N}u|^{2r}$ in (\[paracon\]), but as will be clear below this does not matter. The choice of (\[paracon\]) is simpler for our purpose.
[^11]: We remark that the situation here is different from the high-order paracontrolled calculus described in for example [@BaBe]. In the latter one also has para-controlling formulas (with $\pi_<$ being the standard paraproduct) $X=\pi_<(v,Z)$, where $v$ has its own paracontrolled structure, but this structure is only used in bounding remainder terms involving things like $\pi_{\geq}(v,Z)$, instead of the para-controlled term $X$.
[^12]: For fixed time $t$, we can view $\mathcal{G}_N$ as a mapping from [$\mathcal{V}_N$]{} to itself, by requiring $u_N$ to solve (\[truncnls\]).
[^13]: [In fact we will prove stronger bounds where the low frequency inputs in (\[devop1\]) are replaced by $v_{L_1}^\dagger,\cdots v_{L_{2r}}^{\dagger}$ with $\max(L_j)=L$, and similarly for (\[defop2\]). But for simplicity we will just write (\[devop1\]) and (\[defop2\]).]{}
[^14]: [This requirement appears in the form of coefficients which are indicator functions of sets of form $\{k_j\neq k_l\}$. Such coefficients may lead to slightly different multilinear Gaussian expressions in the estimates below, but there will be at most $(N_*)^C$ possibilities where $N_*$ is a parameter to be defined below, and will not affect any estimates since our exceptional sets will always have measure at most $C_\theta e^{-(N_*)^\theta}$.]{}
[^15]: That is, we relax the requirement (iii) above, keeping only requirements (ii) and (iv).
[^16]: This $W$ and $W'$ are different from the $W$ and $W'$ of Section \[case2\].
[^17]: [By repeating the proof of Proposition \[globalexist\] we can show that $\Phi_t^N\Pi_Nu\to\Phi_tu$ in $H^{-\varepsilon}$ uniformly for $u\in E$ and $|t|\leq 2$, so $\Phi_{t_0}E$ is also compact in $H^{-\varepsilon}$ and satisfies similar properties as $E$.]{}
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---
abstract: 'Let $\mathcal{X}$ be a finite-dimensional complex vector space and let $k$ be a positive integer. An explicit formula for the $k$-reflexivity defect of the image of a generalized derivation on $L({\mathcal{X}})$, the space of all linear transformations on $\mathcal{X}$, is given. Using latter, we also study the $k$-reflexivity defect of the image of an elementary operator of the form $\Delta(T)=ATB-T$ ($T \in L(\mathcal{X})$).'
address: 'University of Ljubljana, IMFM, Jadranska ul. 19, 1000 Ljubljana, Slovenia'
author:
- Tina Rudolf
title: '$k$-reflexivity defect of the image of a generalized derivation'
---
Introduction
============
Let $\mathcal{X}$ be a finite-dimensional complex vector space and let $L(\mathcal{X})$ be the space of all linear transformations on $\mathcal{X}$. Let $k$ be a positive integer and denote by ${\mathcal{F}}_k$ the set of all elements in ${\mathbb{M}}_n$ of rank $k$ or less. The $k$-reflexive cover of a non-empty subset $\mathcal{S} \subseteq L(\mathcal{X})$ is defined by $${\rm Ref_{k}} \mathcal{S}=\{T \in L(\mathcal{X}): \forall \varepsilon >0, \, \forall x_1, \ldots,x_k \in \mathcal{X},\, \exists S \in \mathcal{S}: \, \|Tx_i-Sx_i\|<\varepsilon, \, i=1,\ldots,k\}.$$ It is easy to see that ${\rm Ref_{k}} \mathcal{S}$ is a linear subspace of $L(\mathcal{X})$. A linear subspace $\mathcal{S}$ is said to be $k$-reflexive if ${\rm Ref_{k}} \mathcal{S}=\mathcal{S}$. The $k$-reflexivity defect of a non-empty subset ${\mathcal{S}}$ is defined by ${\rm rd} _k(\mathcal{S})=\dim ({\rm Ref_{k}} \mathcal{S} / \mathcal{S})$. Since ${\mathcal{X}}$ is finite dimensional ${\rm rd} _k(\mathcal{S})=\dim ({\rm Ref_{k}} \mathcal{S})-\dim (\mathcal{S})$ holds. The annihilator of a non-empty subset ${\mathcal{S}}\subseteq {\mathbb{M}}_n$ is defined by ${\mathcal{S}}_\perp=\{C \in {\mathbb{M}}_n:\, {{\rm tr}}(CS)=0 \ \textup{for all} \ S \in {\mathcal{S}}\}$, where ${{\rm tr}}(\cdot)$ denotes the trace functional. It was shown in [@KL; @KL2] that $$\label{k-ref}
{{\rm Ref_{k}}}{\mathcal{S}}=\left({\mathcal{S}}_\perp \cap {\mathcal{F}}_k \right)_\perp$$ holds. The latter obviously implies that a $k$-reflexive space is also $j$-reflexive for all $j \geq k$.
Let $A,\,B \in L(\mathcal{X})$ be invertible linear transformations and let $\mathcal{S}$ be a linear subspace of $L(\mathcal{X})$. Let us denote $A\mathcal{S} B=\{ASB:\,S \in \mathcal{S}\}$ and $\mathcal{S}^\intercal=\{S^\intercal:\, S \in \mathcal{S}\}$. It is well known that transformations of the type $$\label{transf}
\mathcal{S} \mapsto A\mathcal{S} B = \{ASB:\,S \in \mathcal{S}\} \quad \textrm{and} \quad \mathcal{S} \mapsto \mathcal{S}^\intercal =\{S^\intercal:\, S \in \mathcal{S}\}$$ preserve the $k$-reflexivity defect. Hence, since $\mathcal{X}$ is a finite-dimensional complex vector space, one can assume that $\mathcal{X}=\mathbb{C}^n$ for some $n \in \mathbb{N}$ and $L(\mathcal{X})$ may be identified with $\mathbb{M}_n$, the algebra of all $n$-by-$n$ complex matrices. Throughout this paper we will be dealing with subspaces of $\mathbb{M}_n$ which have the decomposition of the form $$\mathcal{S}=\left(\begin{array}{ccc} \mathcal{S}_{11}&\ldots&\mathcal{S}_{1N}\\\vdots&&\vdots\\\mathcal{S}_{M1}& \ldots&\mathcal{S}_{MN}\end{array}\right),$$ where, for each pair of indices $(i,\,j)$, $\mathcal{S}_{ij}$ is a subspace of $\mathbb{M}_{m_i,n_j}$, the space of all $m_i$-by-$n_j$ complex matrices, and $\sum_{i=1}^Mm_i=\sum_{j=1}^Nn_j=n$. It is not hard to see that for spaces of this type one has $$\label{ref}
{\rm Ref_{k}} (\mathcal{S})=\left(\begin{array}{ccc} {\rm Ref_{k}} (\mathcal{S}_{11})&\ldots&{\rm Ref_{k}} (\mathcal{S}_{1N})\\\vdots&&\vdots\\{\rm Ref_{k}} (\mathcal{S}_{M1})&\ldots&{\rm Ref_{k}} (\mathcal{S}_{MN})\end{array}\right) \qquad \textrm{and} \qquad {\rm rd} _k\left(\mathcal{S}\right)=\sum_{i=1}^M \sum_{j=1}^N{\rm rd} _k \left(\mathcal{S}_{ij}\right).$$ In particular, $\mathcal{S}$ is $k$-reflexive if and only if $\mathcal{S}_{ij}$ is $k$-reflexive for every pair of indices $i\in \{1,\ldots,M\}$, $j \in \{1,\ldots,N\}$.
Elementary operators {#EO}
====================
Let $(A_1,\ldots,A_k)$ and $(B_1,\ldots,B_k)$ be arbitrary pairs of $n$-by-$n$ complex matrices. The elementary operator on $\mathbb{M}_n$ with coefficients $(A_1,\ldots,A_k)$ and $(B_1,\ldots,B_k)$ is defined by $$\label{delta}
\Delta (T)=A_1TB_1+\ldots+A_kTB_k, \qquad T \in \mathbb{M}_n.$$ If all $A_i$ are pairwise linearly independent and if the same holds for all $B_i$ ($1 \leq i \leq k$), then $\Delta$ is called elementary operator of length $k$. The simplest example of such operator is of course two-sided multiplication. Namely, let $A,\,B \in {\mathbb{M}}_n$ and let $\Delta$ be an elementary operator defined by $\Delta \left(T\right)=ATB$ for $T \in {\mathbb{M}}_n$. It is easy to see that the kernel and the image of $\Delta$ are reflexive spaces. In fact, if $\Delta$ is an elementary operator of length $k$ on ${\mathbb{M}}_n$, then by [@B1 Proposition 1.1] the space $\ker \Delta$ is $j$-reflexive for every $j \geq k$. It is reasonable to ask whether the same holds for ${{\rm im}}\Delta$ and we show that this is not generally the case.
\[anih\] Let $\Delta$ be an elementary operator on ${\mathbb{M}}_n$ with coefficients $\left(A_1,\, \ldots,\, A_k\right)$ and $\left(B_1,\, \ldots,\,B_k\right)$, defined by $\Delta \left(T\right)=A_1TB_1+A_2TB_2+\ldots+A_kTB_k$. Then there exists an elementary operator $\tilde{\Delta}$ such that $\left({{\rm im}}\Delta\right)_\perp=\ker \tilde{\Delta}$.
Define $\tilde{\Delta}\left(T\right)=B_1TA_1+B_2TA_2+\ldots+B_kTA_k$ for $T \in {\mathbb{M}}_n$. If $T$ is an arbitrary matrix, then ${{\rm tr}}(\Delta(T)C)={{\rm tr}}(T(B_1CA_1+\ldots+B_kCA_k))$ and therefore $C \in \left({{\rm im}}\Delta\right)_\perp$ if and only if $\tilde{\Delta}(C) \in ({\mathbb{M}}_n)_\perp=\{0\}$, that is, $C \in \ker \tilde{\Delta}$.
Next, we introduce some notation. For $k \in \mathbb{N}$ and $\alpha \in \mathbb{C}$, let $J_k(\alpha)$ denote the Jordan block of size $k$, i.e., $$J_k(\alpha)=\left(\begin{smallmatrix} \alpha&1&&&\\&&\ddots&\ddots&\\&&&\alpha&1\\&&&&\alpha\end{smallmatrix}\right)
\in \mathbb{M}_k.$$ In the following example we show that for any $n \geq 3$ there exists an inner derivation $\delta$ on ${\mathbb{M}}_n$ such that ${{\rm im}}\delta$ is not $(n-1)$-reflexive. Consequently, the image of such elementary operator of length $2$ is not $2$-reflexive.
Define $\delta\left(T\right)=J_n(0)T-TJ_n(0)$ for $T \in {\mathbb{M}}_n$. By , every subspace of ${\mathbb{M}}_n$ is $n$-reflexive, hence ${{\rm rd}}_n ({{\rm im}}\delta)=0$. It follows by Lemma \[anih\] that $({{\rm im}}\delta)_\perp$ is simply $\{J_n(0)\}'$, the commutant of the Jordan block $J_n(0)$. One can easily verify that $\{J_n(0)\}'$ is the algebra of all $n \times n$ upper triangular Toeplitz matrices which we will denote by $\mathfrak{T}_n$. Namely, $$\left({{\rm im}}\delta\right)_\perp=\left\{ \left( \begin{array}{cccccc} a_1&a_2&\ldots&\ldots&a_n\\ 0&a_1&a_2& &\vdots \\ \vdots&\ddots&\ddots&\ddots&\vdots \\ \vdots&&\ddots&\ddots&a_2 \\ 0&\ldots&\ldots&0&a_1\end{array}\right) :\, a_1,\,a_2, \ldots a_n \in {\mathbb{C}}\right\}.$$ By , ${{\rm im}}\delta$ is not $(n-1)$-reflexive space, since $\left({{\rm im}}\delta\right)_\perp \cap {\mathcal{F}}_{n-1} \subsetneq \left({{\rm im}}\delta\right)_\perp$. Note that also implies that ${{\rm rd}}_k \left({{\rm im}}\delta\right)=n-k$ for $1 \leq k \leq n-1$. Indeed, $\dim ({{\rm im}}\delta)=n^2-\dim (({{\rm im}}\delta)_\perp)=n^2-n$ and by we have $\dim ({{\rm Ref_{k}}}({{\rm im}}\delta))=n^2-\dim(({{\rm im}}\delta)_\perp \cap {\mathcal{F}}_k)=n^2-k$ for $1 \leq k \leq n-1$.
Generalized derivations {#GD}
=======================
Let $\Delta$ be an elementary operator of length $2$ on ${\mathbb{M}}_n$, i.e., a linear transformation of the form $\Delta(T)=A_1TB_1+A_2TB_2$ ($T \in \mathbb{M}_n$), where $A_1,\,A_2$ and $B_1,\,B_2$ are two pairs of linearly independent matrices. By [@B1 Proposition 1.1], one has ${{\rm rd}}_k(\ker \Delta)=0$ for all $k \geq 2$. In [@R] reflexivity of such elementary operator was studied and an explicit formula for the reflexivity defect of its kernel was given. This motivates the main subject of this paper, that is the $k$-reflexivity defect of the image of some special examples of elementary operators of length $2$.
Let $A,\,B \in \mathbb{M}_n$ be arbitrary matrices. Define the generalized derivation on $\mathbb{M}_n$ with coefficients $A$ and $B$ by $\Delta \,\left(T\right)=AT-TB$, $T \in \mathbb{M}_n$. Obviously, $\Delta$ is an example of an elementary operator of length $2$. Let $J_{p_1}(\lambda_1)\oplus \ldots \oplus J_{p_N}(\lambda_N)$ be the Jordan canonical form of $A$, where $\sum_{i=1}^Np_i=n$ and $\lambda_1, \ldots,\lambda_N$ are not necessarily distinct eigenvalues of $A$. Similarly, let $J_{r_1}(\mu_1)\oplus \ldots \oplus J_{r_M}(\mu_M)$ be the Jordan canonical form of $B$, where $\sum_{i=1}^Mr_i=n$ and $\mu_1, \ldots,\mu_M$ are not necessarily distinct eigenvalues of $B$. Let $R(i,\,j,\,k)$ be a non-negative integer defined by $$R(i,\,j,\,k):=\left\{\begin{array}{ccl}\min \{p_i,\,r_j\}-k & : & \textup{$\lambda_i=\mu_j$ and $k<\min\{p_i,\,r_j\}$,}\\ 0 & : & \textup{$\lambda_i \neq \mu_j$ or $k \geq \min\{p_i,\,r_j\}$}.\end{array}\right.$$
\[pp2\] With the above notation, the $k$-reflexivity defect of ${{\rm im}}\Delta$ can be expressed as $${{\rm rd}}_k({{\rm im}}\Delta)=\sum_{i=1}^N\sum_{j=1}^M R(i,\,j,\,k).$$ In particular, ${{\rm im}}\Delta$ is a $k$-reflexive space if and only if all roots of the greatest common divisor of $m_A$ and $m_B$ of $A$ and $B$, respectively, are of multiplicity at most $k$.
Let $\mathbf{0}_{p,r}$ denote the $p \times r$ zero matrix ($p,\,r \in {\mathbb{N}}$) and let $A$ and $B$ be as before the Proposition \[pp2\]. For $1 \leq i \leq N$ and $1 \leq j \leq M$ define the following elementary operators on ${\mathbb{M}}_{p_i,r_j}$ and ${\mathbb{M}}_{r_j,p_i}$, respectively, $$\begin{split}
\Delta_{p_i,r_j}(T) &= J_{p_i}(\lambda_i)T-TJ_{r_j}(\mu_j) \qquad (T \in {\mathbb{M}}_{p_i,r_j}), \\
\Delta_{r_j,p_i}(T) &= J_{r_j}(\mu_j)T-TJ_{p_i}(\lambda_i) \qquad (T \in {\mathbb{M}}_{r_j,p_i}).
\end{split}$$ Lemma \[anih\] yields $({{\rm im}}\Delta_{p_i,r_j})_\perp=\ker \Delta_{r_j,p_i}$. If $\lambda_i \neq \mu_j$, then $\Delta_{p_i,r_j}$ is bijective and ${{\rm im}}\Delta_{p_i,r_j}$ is a $k$-reflexive space for every $k \in {\mathbb{N}}$. Now assume that $\lambda_i=\mu_j$. It is not hard to see that $$\begin{split}
\ker \Delta_{r_j,p_i} &=\left\{\left(\begin{array}{cc}\mathbf{0}_{r_j,p_i-r_j} & T\end{array}\right): T \in \mathfrak{T}_{r_j} \right\} \qquad \textrm{if $r_j \leq p_i$,}\\
\ker \Delta_{r_j,p_i} &=\left\{ \left(\begin{array}{c} T \\ \mathbf{0}_{r_j-p_i,p_i}\end{array}\right): T \in \mathfrak{T}_{p_i}\right\} \qquad \textrm{if $r_j > p_i$.}
\end{split}$$ Let us denote $d=\min\{p_i,r_j\}$ and $D=\max\{p_i,r_j\}$. Since transformations of the type preserve $k$-reflexivity defect we can without any loss of generality assume that $\ker \Delta_{r_j,p_i}=\left\{\big(\begin{array}{cc} \mathbf{0}_{d,D-d} & T \end{array}\big): T \in \mathfrak{T}_d \right\}$ and therefore $\dim ({{\rm im}}\Delta)=d(D-1)$. The structure of the space $\ker \Delta_{r_j,p_i}$ yields that $\ker \Delta_{r_j,p_i} \cap {\mathcal{F}}_k$ is a linear space with the following property. If $k \geq d$, then $\ker \Delta_{r_j,p_i} \cap {\mathcal{F}}_k=\ker \Delta_{r_j,p_i}$. Otherwise, if $1 \leq k <d$, then $$\ker \Delta_{r_j,p_i} \cap {\mathcal{F}}_k=\left\{\left(\begin{array}{cc} \mathbf{0}_{k,d-k}& T \\ \mathbf{0}_{D-k,d-k}& \mathbf{0}_{D-k,k} \end{array}\right): T \in \mathfrak{T}_k\right\}.$$ Therefore, ${{\rm im}}\Delta_{p_i,r_j}$ is a $k$-reflexive space iff $k \geq d$ or $\lambda_i \neq \mu_j$. Otherwise, if $k<d$ and $\lambda_i=\mu_j$, one gets $\dim({{\rm Ref_{k}}}({{\rm im}}\Delta_{p_i,r_j}))=dD-k$. The result in general setting now follows by .
Let $A,\,B \in \mathbb{M}_n$ be as before the Proposition \[pp2\]. Let $\Delta: {\mathbb{M}}_n \rightarrow {\mathbb{M}}_n$ be an elementary operator defined by $\Delta (T)=ATB-T$. Let $R(i,\,j,\,k)$ be a non-negative integer defined by $$R(i,\,j,\,k):=\left\{\begin{array}{ccl}\min \{p_i,\,r_j\}-k & : & \textup{$\lambda_i,\,\mu_j \neq 0$, $\lambda_i=\frac{1}{\mu_j}$ and $k<\min\{p_i,\,r_j\}$,}\\ 0 & : & \textup{otherwise}.\end{array}\right.$$
With the above notation, the $k$-reflexivity defect of ${{\rm im}}\Delta$ can be expressed as $${{\rm rd}}_k({{\rm im}}\Delta)=\sum_{i=1}^N\sum_{j=1}^M R(i,\,j,\,k).$$
Define $\Delta_{p_i,r_j}(T)=J_{p_i}(\lambda_i)TJ_{r_j}(\mu_j)-T$ for $T \in {\mathbb{M}}_{p_i,r_j}$. By we get ${{\rm rd}}_k({{\rm im}}\Delta)=\sum_{i=1}^N\sum_{j=1}^M {{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})$, hence it suffices to determine ${{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})$. If $\lambda_i=\mu_j=0$, then it is not hard to see that for $T=(t_{uv}) \in {\mathbb{M}}_{p_i,r_j}$ we have $$\Delta_{p_i,r_j}(T)=-T+\left(\begin{array}{cccc}0&t_{21}&\ldots&t_{2,r_j-1}\\\vdots&\vdots&&\vdots\\0&t_{p_i,1}&\ldots&t_{p_i,r_j-1}\\0&0&\ldots&0 \end{array}\right),$$ therefore ${{\rm im}}\Delta_{p_i,r_j}={\mathbb{M}}_{p_i,r_j}$ and ${{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})=0$ for every positive integer $k$. If $\lambda_i=0$ and $\mu_j \neq 0$, then ${{\rm im}}\Delta_{p_i,r_j}=\{XJ_{r_j}(\mu_j):\, X \in {{\rm im}}\tilde{\Delta}_{p_i,r_j}\}$ where $\tilde{\Delta}_{p_i,r_j}: {\mathbb{M}}_{p_i,r_j} \rightarrow {\mathbb{M}}_{p_i,r_j}$ is a generalized derivation of the form $\tilde{\Delta}_{p_i,r_j}(T)=J_{p_i}(0)T-TJ_{r_j}(\mu_j)^{-1}$. Thus we have ${{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})={{\rm rd}}_k({{\rm im}}\tilde{\Delta}_{p_i,r_j})$. By [@HJ1 Example 6.2.13] one can easily see that inverting matrices preserves the sizes of Jordan blocks, hence Proposition \[pp2\] yields ${{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})=0$. Similarly, if $\lambda_i \neq 0$ and $\mu_j=0$ or if $\lambda_i \neq 0$, $\mu_j \neq 0$ and $\lambda_i \neq \frac{1}{\mu_j}$, then again Proposition \[pp2\] yields ${{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})=0$. Now assume that $\lambda_i,\,\mu_j \neq 0$ and that $\lambda_i = \frac{1}{\mu_j}$. As before, ${{\rm rd}}_k({{\rm im}}\Delta_{p_i,r_j})={{\rm rd}}_k({{\rm im}}\tilde{\Delta}_{p_i,r_j})$ where $\tilde{\Delta}_{p_i,r_j}: {\mathbb{M}}_{p_i,r_j} \rightarrow {\mathbb{M}}_{p_i,r_j}$ is a generalized derivation of the form $\tilde{\Delta}_{p_i,r_j}(T)=J_{p_i}(\lambda_i)T-TJ_{r_j}(\mu_j)^{-1}$. Now the Proposition \[pp2\] yields that ${{\rm rd}}_k({{\rm im}}\tilde{\Delta}_{p_i,r_j})=0$ if $k \geq \min\{p_i,r_j\}$ and ${{\rm rd}}_k({{\rm im}}\tilde{\Delta}_{p_i,r_j})=\min \{p_i,\,r_j\}-k$ if $k < \min\{p_i,r_j\}$. By one gets ${{\rm rd}}_k({{\rm im}}\Delta)=\sum_{i=1}^N\sum_{j=1}^M R(i,\,j,\,k)$.
Acknowledgements {#acknowledgements .unnumbered}
================
The author is thankful to the Slovenian Research Agency for their financial support.
[99]{} J. Bračič, [*Reflexivity of the commutant and local commutants of an algebraic operator*]{}, Linear Algebra Appl. 420 (2007) 20-28.
R. A. Horn, C. R. Johnson: [*Topics in matrix analysis*]{}, Corrected reprint of the 1991 original, Cambridge University Press, Cambridge, 1994.
J. Kraus, D. Larson: [*Some applications of a technique for constructing reflexive operator algebras*]{}, J. Operator Theory [**13**]{} (1985), 227-236.
J. Kraus, D. Larson: [*Reflexivity and distance formulae*]{}, Proc. London Math. Soc. [**53**]{} (1986), 340-356.
T. Rudolf, [*Reflexivity defect of kernels of the elementary operators of length $2$*]{}, Linear Algebra Appl. [**437**]{} (2012) 1366–1379.
|
---
author:
- 'A.T. Bajkova[^1]'
- 'V.V. Bobylev'
title: |
Re-determining the Galactic spiral density wave\
parameters from data on masers with\
trigonometric parallaxes
---
Introduction
============
Spectral analysis of residual velocities of different young galactic objects (HI clouds, OB stars, open star cluster younger than 50 Myr, masers) tracing the Galaxy spiral arms has been fulfilled, for example, by Clemens (1985), Bobylev, Bajkova & Stepanishchev (2008), Bobylev & Bajkova (2010). As a result there were determined the following parameters of the spiral density wave subject to the theory by Lin & Shu (1964): amplitude and wavelength of the perturbations, evoked by the spiral wave, pitch angle, phase of the Sun in the spiral wave. Nowadays galactic masers having high-precision trigonometric parallaxes, line-of-sight velocities and proper motions (Reid et al. 2009, Rygl et al. 2010) are of great interest.
The previous spectral analysis represents the simplest periodogram analysis of velocity perturbations based on conventional Fourier transform, what can be considered only as the first approximation of exact spectral analysis and can be applied adequately only in the case of small range of galactocentric distances (2-3 kpc). But analysis of modern data on galactic masers which are located in wide range of galactocentric distances ($3<$R$<14$ kpc) requires elaboration of more correct tools of spectral analysis accounting both logarithmic dependence from galactocentric distances and position angles of objects. For a detailed description of the new method of spectral analysis of velocity residuals see Bajkova & Bobylev (2012).
Our first study (Bobylev & Bajkova 2010) was based on an analysis of radial galactocentric velocities of only 28 Galactic masers. The second one (Bajkova & Bobylev 2012) dealt with 44 masers. Currently, high-precision VLBI measurements of parallaxes, line-of-site velocities, and proper motions are available for 58 Galactic masers, which is of great interest for our task. The aim of this present study is to re-determine the spiral density wave parameters by applying the recently proposed algorithm (Bajkova \$ Bobylev 2012) and new ones, described below, to more extensive data series.
Basic relations
===============
The velocity perturbations of Galactic objects produced by a spiral density wave (Lin & Shu 1964) are described by the relations $$\label{e-01}
V_R = - f_R \cos\chi,$$ $$\label{e-02}
\Delta V_{\theta} = f_{\theta} \sin\chi,$$ where $$\label{e-03}
\chi = m[\cot(i)\ln(R/R_{\circ})-\theta]+\chi_{\odot}$$ is the phase of the spiral density wave; $m$ is the number of spiral arms; $i$ is the pitch angle; $\chi_{\odot}$ is the phase of the Sun in the spiral density wave (Rohlfs 1977); $R_{\circ}$ is the galactocentric distance of the Sun; $\theta$ is the object’s position angle: $\tan\theta = y/(R_\circ-x)$, where $x,$ $y$ are the Galactic heliocentric rectangular coordinates of the object; $f_R$ and $f_\theta$ are the amplitudes of the radial and tangential perturbation components, respectively; $R$ is the distance of the object from the Galactic rotation axis, which is calculated using the heliocentric distance $r=1/\pi$: $$\label{e-003}
R^2=r^2\cos^2 b-2R_\circ r\cos b\cos l+R^2_\circ,$$ where $l$ and $b$ are the Galactic longitude and latitude of the object, respectively.
Equation (\[e-03\]) for the phase can be expressed in terms of the perturbation wavelength $\lambda$, which is equal to the distance between the neighboring spiral arms along the Galactic radius vector. The following relation is valid: $$\label{e-04}
\frac{2\pi R_{\circ}}{\lambda} = m\cot(i).$$ Equation (\[e-03\]) will then take the form $$\label{e-05}
\chi = \frac {2\pi R_{\circ}}{\lambda}
\ln(R/R_{\circ})-m\theta+\chi_{\odot}.$$
The question of determining the residual velocities is considered below. To goal of our spectral analysis of the series of measured velocities $V_{R_n}, \Delta V_{\theta_n}$ $n=1,2,\dots,N,$ where $N$ is the number of objects, is to extract the periodicity in accordance with model (\[e-01\])-(\[e-02\]) describing a spiral density wave with parameters $f_R,f_\theta,$ $\lambda,$ and $\chi_\odot$. If the wavelength $\lambda$ is known, then the pitch angle $i$ is easy to determine from Eq. (\[e-04\]) by specifying the number of arms $m$. Here, we adopt a two-armed model, i.e., $m=2$.
Methods
=======
Fourier transform-based analysis
--------------------------------
Let us represent series of velocity perturbations of galactic objects evoked by a spiral density wave (\[e-01\])-(\[e-02\]) in the most general, complex form: $$\label{e-06}
V_n=V_{R_n}+j\Delta V_{\theta_n},$$ where $j=\sqrt-1$, $n$ is a number of an object ($n=1,...,N$).
A periodogram analysis, which we consider here, requires calculation of power spectrum of series (\[e-06\]) expanded over orthogonal harmonic functions $$\exp[-j\frac{2\pi
R_{\circ}}{\lambda_k}\ln(R_n/R_{\circ})+jm\theta_n]$$ in accordance with expression (\[e-05\]) for the phase.
A complex spectrum of our series is: $$\begin{aligned}
\label{e-006}
\bar{V}_{\lambda_k}=\bar{V}_{\lambda_k}^{Re}+j\bar{V}_{\lambda_k}^{Im}
=\frac{1}{N}\sum_{n=1}^N (V_{R_n}+j\Delta
V_{\theta_n})\times\\
\times \exp[-j\frac{2\pi R_{\circ}}{\lambda_k}\ln(R_n/R_{\circ})+j
m\theta_n]\nonumber,\end{aligned}$$ where the upper indices $Re$ and $Im$ designate real and imaginary spectrum parts respectively.
Let us reduce the latter expression to a standard discrete Fourier transform in the following way: $$\begin{aligned}
\label{e-07}
\bar{V}_{\lambda_k}=\frac{1}{N}\sum_{n=1}^N
(V_{R_n}+j\Delta
V_{\theta_n})\times \exp(jm\theta_n)\times \\
\times \exp[-j\frac{2\pi
R_{\circ}}{\lambda_k}\ln(R_n/R_{\circ})]=\nonumber
\\
=\frac{1}{N}\sum_{n=1}^N V_n^{'}\exp[-j\frac{2\pi
R_{\circ}}{\lambda_k}\ln(R_n/R_{\circ})]\nonumber,\end{aligned}$$ where
$$\begin{aligned}
\label{e-08}
V_n^{'}=V_n^{Re}+V_n^{Im}=\\=[V_{R_n}\cos(m
\theta_n)-\Delta
V_{\theta_n}\sin(m\theta_n)]+\nonumber\\
+j[V_{R_n}\sin(m \theta_n)+\Delta V_{\theta_n}\cos(m
\theta_n)]\nonumber.\end{aligned}$$
And, finally, making the following change of variables $$\label{e-09}
R_n^{'}=R_{\circ}\ln(R_n/R_{\circ}),$$ we obtain standard Fourier transform of a new series $V_n^{'}$ (\[e-08\]), determined in point set $R_n^{'}$:
$$\label{e-10}
\bar{V}_{\lambda_k}=\frac{1}{N}\sum_{n=1}^N
V_n^{'}\exp[-j\frac{2\pi R_n^{'}}{\lambda_k}].$$
The periodogram $|\bar{V}_{\lambda_k}|^2$ is subject to further analysis. The peak of the periodogram determines the sought-for periodicity. The coordinate of the peak gives the wavelength $\lambda$ and, respectively, pith angle $i$ (see Eq.(\[e-04\])). Relation between a peak value of the periodogram $S_{peak}$ and perturbation amplitudes $f_R$ and $f_\theta$ is expressed as follows: $$\label{e-100}
f_R^2+f_{\theta}^2=2\times S_{peak}.$$
It is necessary to note that the spectral analysis of complex series (\[e-06\]) allows to determine $\lambda$ (or pitch angle $i$) and phase of the Sun $\chi_{\odot}$, but does not allow to estimate the amplitudes $f_R$ and $f_\theta$ separately. To determine them it is necessary to analyze radial $\{V_{R_n}\}$ and tangential $\{\Delta V_{\theta_n}\}$ velocity perturbations independently, as it has been shown by Bajkova & Bobylev (2012). Here we show how amplitudes of perturbations can be found if $\lambda$ and $\chi_{\odot}$ are known (for example, from previous complex analysis). We consider series $\{V_{R_n}\}$ (by analogy the same algorithm can be applied to series $\{\Delta
V_{\theta_n}\}$).
Let us represent Eq. (\[e-03\]) as $$\label{e-11}
\chi = \chi_1-m\theta,\label{e-11}$$ where $$\label{e-12}
\chi_1 = \frac {2\pi
R_{\circ}}{\lambda}\ln(R/R_{\circ})+\chi_{\odot}.$$ Substituting (\[e-11\]) into Eq. (\[e-01\]) for the perturbations at the $n$th point and performing standard trigonometric transformations, we will obtain [ $$\begin{aligned}
\label{e-13}
V_{R_n}=-f_R \cos(\chi_{1_n} - m\theta_n)=\\
=-f_R\cos\chi_{1_n}\cos m\theta_n - f_R\sin\chi_{1_n}\sin m\theta_n =\nonumber\\
=-f_R \cos\chi_{1_n}(\cos m\theta_n + \tan\chi_{1_n}\sin
m\theta_n)\nonumber.
\end{aligned}$$]{}
Let us designate $$\label{e-14}
V^{'}_{R}=-f_R\cos\chi_1,$$ Owing to the substitution (\[e-14\]), it then follows from (\[e-13\]) that $$\label{e-15}
V_{R_n}=V^{'}_{R_n}(\cos m\theta_n+\tan\chi_{1_n}
\sin m\theta_n).$$ Substituting known values of $\lambda$ and $\chi_{\odot}$ we can form from Eq. (\[e-15\]) a new data series $$\label{e-16}
V^{'}_{R_n}=V_{R_n}/(\cos m\theta_n+\tan\chi_{1_n}
\sin m\theta_n).$$ Again, using the substitution (\[e-09\]), we obtain a standard Fourier transform:
$$\label{e-17}
\bar{V}_{\lambda_k} = \frac{1} {N}\sum_{n=1}^{N} V^{'}_{R^{'}_n}
\exp\Bigl(-j\frac {2\pi R^{'}_n}{\lambda_k}\Bigr).$$
In this case relation between a peak value of the periodogram $S_{peak}$ and perturbation amplitudes $f_R$ is as follows:
$$\label{e-18} f_{R}^2=4\times S_{peak}.$$
Note, that a separate periodogram analysis based on operations (\[e-11\])-(\[e-17\]) can be realized as an iterative process of seeking for unknowns $\lambda$, $\chi_{\odot}$, and $f_{R}$ under optimization of some specific signal extraction quality criterium (Bajkova & Bobylev, 2012).
For numerical realization of Fourier transform (\[e-10\]) or (\[e-17\]) using fast Fourier transform (FFT) algorithms it is necessary to determine data $V_{n}^{'}(R^{'}) (n=1,\dots,N)$ on discrete grid $l=1,\dots,K=2^\alpha$, where $\alpha$ is integer, positive, $N\le K$; $\Delta_R$ is a discrete space. Coordinates of data are determined as follows: $l_n=[(R_n^{'}+|\min\{R_k^{'}\}|_{k=1,...,N})/\Delta_R]+1,
n=1,...,N$, where $[a]$ denotes an integer part of $a$. The sequence determined is considered as a periodical one with the period $D=K\times\Delta_R$. Obviously, the values of the $K$–point sequence are taken to be zero in the pixels into which no data fall.
The GMEM-based analysis
-----------------------
So far we have considered the simplest method of periodogram analysis based on linear Fourier transform. In the case where the data series are irregular, i.e., there are large gaps, the signal spectrum is distorted by large side lobes and it becomes difficult to distinguish the spectral component of the signal from spurious peaks. In this case, it may turn out to be useful to apply nonlinear methods of spectrum reconstruction from the available data. This problem is fundamentally resolvable if the sought for signal has a finite spectrum. Since our problem belongs to the class of problems on the extraction of polyharmonic functions from noise, we assume that this condition is met.
Here we propose a complex spectrum reconstruction algorithm based on well-known maximum entropy method (MEM). Since the spectrum is described by a complex-valued function, we apply a generalized form of MEM (GMEM) proposed and described in detail by Bajkova (1992) and Frieden & Bajkova (1994).
The spectrum $\bar{V}_k=\bar{V}_k^{Re}+j\bar{V}_k^{Im}$ and the data $V_{n}^{'}$ are related by the inverse Fourier transform: [ $$\begin{aligned}
\label{e-19}
\sum_{k=1}^{K} (\bar{V}_k^{Re}+j\bar{V}_k^{Im})
\exp(j\frac {2\pi (k-1)(l_n-1)}{K})=V_{n}^{'}.\end{aligned}$$]{}
Note that wavelength $\lambda_k$ and spatial frequency $(k-1)$ are related as follows: $$\label{e-190}
\lambda_k=\frac{D}{k-1}.$$
In our case, the reconstruction problem assumes finding the minimum of the following generalized entropy functional: [ $$\begin{aligned}
\label{e-20} E=\sum_{k=K_1}^{k=K_2} V_k^{Re+}\ln(aV_k^{Re+})
+V_k^{Re-}\ln(aV_k^{Re-})+\\
+V_k^{Im+}\ln(aV_k^{Im+})+V_k^{Im-}\ln(aV_k^{Im-})+\nonumber\\
+\sum_{n=1}^{n=N}\frac{(\eta_n^{Re})^2+(\eta_n^{Im})^2}{2\sigma_n^2},\nonumber\end{aligned}$$]{} where the sought–for variables $V_k^{Re}$ and $V_k^{Im}$ are represented as the difference of the positive and negative parts: $V_k^{Re}=V_k^{Re+}-V_k^{Re-}$ and $V_k^{Im}=V_k^{Im+}-Y_k^{Im-}$ respectively; in this case, $V_k^{Re+}, V_k^{Re-}, V_k^{Im+},
V_k^{Im-}\ge0$, $a\gg 1$ is the real–valued parameter responsible for the separation of the positive and negative parts of the sought-for variables with the required accuracy (in our case, we adopted $a=10^6)$, $K_1$ and $K_2$ are the a priori known lower and upper localization boundaries of the sought-for finite spectrum, $\eta_n^{Re}$ and $\eta_n^{Im}$ are a real and an imaginary parts respectively of the measurement error of the $n$th value of the series $V_n^{'}$ that obey a random law with a normal distribution with a zero mean and dispersion $\sigma_n$.
The constraints (\[e-19\]) on the unknowns, with accounting measurement errors, can be rewritten as
$$\begin{aligned}
\label{e-21}
\sum_{k=K1}^{k=K2}
((\bar{V}_k^{Re+}-\bar{V}_k^{Re-})+j(\bar{V}_k^{Im+}-\bar{V}_k^{Im-}))\times\\
\times\exp(j\frac {2\pi (k-1)(l_n-1)}{K})+\eta_n^{Re}+j\eta_n^{Im} = V_{n}^{'}\nonumber.\end{aligned}$$
We can see from (\[e-20\]), that the functional to be minimized consists of five terms, the first four ones are total entropy of the sought-for solution, the last one is $\chi^2$ measure of deviation between data and solution. Thus, the GMEM algorithm seeks for solutions not only for the spectrum unknowns, but also for measurement errors ($\eta_n^{Re}+j\eta_n^{Im}$). Therefore we can expect effective suppression of noise caused not only by non-uniformity of series but also by measurement errors. The optimization of functional (\[e-20\]) under conditions (\[e-21\]) can be done numerically using any gradient method. We used a steepest-descent method.
Data
====
Bobylev & Bajkova (2012) analyzed a sample of 44 masers with known high-precision trigonometric parallaxes, line-of-site velocities and proper motions. Since then the number of masers with such measurements has increased considerably. Now data on 58 masers are available in literature. Table 1 lists the input data on 58 masers associated with the youngest Galactic stellar objects (protostar objects of different mass, very massive supergiants, or T Tau stars). The references to majority of original data can be found in Bajkova & Bobylev (2012).
The input data, namely, trigonometric parallaxes and proper motions, were obtained by several groups using long time radio-interferometric observations carried out within the framework of different projects. One of them — the Japanese project VERA (VLBI Exploration of Radio Astrometry) (Honma et al. 2007) — is dedicated to observation of H$_2$O and SiO masers at 22 and 43 GHz, respectively. Note that higher observing frequency results in higher resolution and more accurate data. Methanol (CH$_3$OH) masers were observed at 12 GHz (VLBA, NRAO) and 8.4-GHz continuum radio-interferometric observations of radio stars were carried out with the same aim (Reid et al. 2009).
The line-of-site velocities $V_r(LSR)$ listed in Table 1 were determined with respect to the Local Standard of Rest by different authors from radio observations in CO emission lines. The parallaxes were determined, on average, with a relative error of $\sigma_\pi/\pi\approx5\%,$ and only in three regions the error exceeds the mean level. These are IRAS 16293-2422 ($\sigma_\pi/\pi=19\%$), G 23.43-0.20 ($\sigma_\pi/\pi=18\%$) and W 48 ($\sigma_\pi/\pi=14\%$).
![Coordinates of masers in the $XY$ Galactic plane (the Sun is located at the center of coordinate system).[]{data-label="f-0"}](fig00.eps){width="70mm"}
Fig. \[f-0\] shows the space distribution of masers projected onto the Galactic $XY$ plane. The objects can be seen to be widely scattered along the $x$, $y$ coordinates, implying a large scatter of position angles. Hence the spectrum analysis algorithm has to be constructed in order to correctly extract periodic signal from velocity perturbations traced by masers.
![Masers galactocentric radial velocities vs galactocentric distances $R$.[]{data-label="f-2"}](fig1.eps){width="70mm"}
Results
=======
First, we re-determined the parameters of the Galactic rotation curve using the data for 58 masers. The method employed is based on the well-known Bottlinger formulae (Ogorodnikov, 1965), where the angular velocity of Galactic rotation is expanded into a series up to 2-nd order terms in $r/R_0$ (Bobylev & Bajkova 2010). We adopt $R_0=8$ kpc and infer the following components of the peculiar solar velocity: $(U_\odot,V_\odot,W_\odot)=(7.4,16.6,8.53)\pm(1.0,0.8,0.5)$ km c$^{-1}$, and the following parameters of the Galactic rotation curve: $\Omega_0=-29.3 \pm0.6,$ km c$^{-1}$ kpc $^{-1}$, $\Omega'_0=+4.2\pm0.1$ km c$^{-1}$ kpc $^{-2}$, $\Omega''_0=-0.85\pm0.03,$ km c$^{-1}$ kpc $^{-3}$. The linear Galactic rotation velocity at $R=R_{\circ}$ then is equal to: $V_0=|R_0\Omega_0|=234\pm5$ km c$^{-1}$.
There is good agreement of our results with the results of analyzing masers by different authors. Based on a sample of 18 masers, McMillan & Binney (2010) showed that $\Omega_0$ lying within the range $29.9-31.6$ km s$^{-1}$ kpc$^{-1}$ at various $R_0$ was determined most reliably and obtained an estimate of $V_0=247\pm19$ km s$^{-1}$ for $R_0=7.8\pm0.4$ kpc. Based on a sample of 18 masers, Bovy, Hogg & Rix (2009) found $V_0=244\pm13$ km s$^{-1}$ at $R_0=8.2$ kpc. Using 44 masers, Bajkova & Bobylev (2012) found: $(U_\odot,V_\odot,W_\odot)=(7.6,17.8,8.3)\pm(1.5,1.4,1.2)$ km s$^{-1}$, $\Omega_0=-28.8 \pm0.8$ km s$^{-1}$ kpc$^{-1}$, $\Omega'_0=+4.18\pm0.15$ km s$^{-1}$ kpc$^{-2}$, $\Omega''_0=-0.87\pm0.06$ km s$^{-1}$ kpc$^{-3}$, $V_0=|R_0\Omega_0|=230\pm14$ km s$^{-1}$.
It is important that the rotation-curve parameters found are in good agreement with the results of analyzing young Galactic disk objects rotating most rapidly around the center: OB associations with $\Omega_0 =-31\pm1$ km s$^{-1}$ kpc$^{-1}$ (Mel’nik, Dambis, & Rastorguev 2001; Mel’nik & Dambis 2009), blue supergiants with $\Omega_0=-29.6\pm1.6$ km s$^{-1}$ kpc$^{-1}$ and $\Omega'_0= 4.76\pm0.32$ km s$^{-1}$ kpc$^{-2}$ (Zabolotskikh, Rastorguev & Dambis 2002), or OB3 stars with $\Omega_0 = -31.5\pm0.9$ km s$^{-1}$ kpc$^{-1}$, $\Omega^{'}_0 = +4.49\pm0.12$ km s$^{-1}$ kpc$^{-2}$ and $\Omega^{''}_0 = -1.05\pm0.38$ km s$^{-1}$ kpc$^{-3}$ (Bobylev & Bajkova 2011).
The galactocentric radial, $V_{R_n},$ and tangential, $V_{\theta_n}$ ($n=1,\dots,44,)$ velocities of the masers were determined from the relations $$V_{\theta_n}= U_n\sin \theta_n+(V_0+V_n)\cos \theta_n,\label{e-22}$$ $$V_{R_n}=-U_n\cos \theta_n+(V_0+V_n)\sin \theta_n,\label{e-23}$$ where $U_n, V_n$ are the heliocentric space velocities adjusted for solar peculiar motion.
The residual tangential velocities $\Delta V_{\theta_n}$ are obtained from the tangential velocities (\[e-22\]) minus the smooth rotation curve that is defined by the Galactic rotation parameters $\Omega_0,$ $\Omega'_0,$ and $\Omega''_0$ found. The radial velocities (\[e-23\]) depend only on one Galactic parameter $\Omega_0$ and do not depend on the rotation curve. As our experience showed, the data are so far insufficient to reliably extract the density wave from the tangential residual velocities of the masers. Therefore, here we determine the spiral density wave parameters only from galactocentric radial velocities.
Figure \[f-2\] shows the input radial velocity series $V_{R_n},
n=1,\dots,58$. Figure \[f-3\] shows transformed velocity series $V^{'}_{R^{'}_n}, n=1,\dots,58$ and main extracted harmonic. The periodogram obtained using a modified Fourier transform-based method is given in fig. \[f-4\]. An analysis of this periodogram allowed us to estimate the following spiral wave parameters: the amplitude of the radial perturbations $f_R=7.5 \pm 1.5$ km c$^{-1}$; the wavelength (interarm distance in the galactocentric direction) $\lambda=2.4 \pm 0.4$ kpc, and the phase of the Sun in the spiral density wave $\chi_\odot=-160 \pm 15^\circ$. The pitch angle of the spiral wave estimated from equation (\[e-04\]) for $m=2$ is $-5.5 \pm 1^\circ$. The significance level of the peak is $p=0.99$. Significance level was estimated using the simplest method based on Schuster theorem (Vityazev, 2001).The error bars are based on a Monte-Carlo simulation of 1000 random realizations of input data assuming that measurement errors obey the normal distribution. Note that the value for $f_R$ found by fitting the harmonic with $\lambda=2.4$ kpc to data is in good agreement with relation (\[e-18\]).
Power spectrum, obtained using the GMEM, is shown in fig.\[f-6\]. As we can see, this nonlinear reconstruction algorithm allowed us to get rid of the side lobes near the main peak almost completely and, thus, to increase considerably the significance of the extracted periodicity $(p=1.0)$ with $\lambda=2.4$ kpc.
For comparison, in our previous study (Bajkova & Bobylev 2012) we have obtained from data on 44 masers the following spiral density wave parameters: amplitude $f_R = 7.7\pm 1.7$ km s$^{-1}$, wavelength $\lambda=2.2^\pm 0.4$ kpc, pitch angle $i=-5\pm
0.9^\circ$, and the phase of the Sun $\chi_\odot= -147\pm
17^\circ$.
The parameters of Galactic spiral density wave obtained are in good agreement with those found by different authors by applying different methods to different Galactic tracer objects (Mel’nik et al. 2001; Zabolotskikh et al. 2002; Bobylev & Bajkova 2011 and many others).
![Transformed galactocentric radial velocities vs $R^{'}$ and extracted main harmonic fitted to the data (solid bold line).[]{data-label="f-3"}](fig2.eps){width="70mm"}
![Periodogram (spectrum power) of masers galactocentric radial velocities.[]{data-label="f-4"}](fig3.eps){width="70mm"}
Conclusions
===========
We used both Fourier transform-based spectral analysis technique and the generalized maximum entropy reconstruction method to extract a periodic signal from the galactocentric radial velocities of 58 masers with currently known high-precision trigonometric parallaxes, proper motions and line-of-site velocities. In accordance with Lin & Shu (1964) theory, the extracted periodic signal is associated with the Galactic spiral density wave. Masers span a wide range of galactocentric distances $3<$R$<14$ kpc and show a large scatter of position angles $\theta$ in the Galactic $XY$ plane, making it necessary an exact accounting for both the logarithmic nature of the argument and position angles. As a result, we re-determined the main parameters of the Galactic spiral density wave as follows: amplitude of radial velocities perturbations $f_R=7.5 \pm 1.5$ km s$^{-1}$, wavelength $\lambda=2.4 \pm 0.4$ kpc, pitch angle $-5.5 \pm
1^\circ$ and phase of the Sun in the density wave $\chi_\odot=-160
\pm 15^\circ$.
![The spectrum power reconstructed by the GMEM.[]{data-label="f-6"}](fig5.eps){width="70mm"}
This work was supported by the “Non - stationary processes in the Universe” Program of the Presidium of the Russian Academy of Science and the Program of State Support for Leading Scientific Schools of the Russian Federation (project. NSh–16245.2012.2, “Multi-wavelength Astrophysical Studies”).
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[^1]: Corresponding author:
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---
abstract: 'In the present work we investigate the main differences in the lead neutron skin thickness, binding energy, surface energy and density profiles obtained with two different density dependent hadron models. Our results are calculated within the Thomas-Fermi approximation with two different numerical prescriptions and compared with results obtained with a common parametrization of the non-linear Walecka model. The neutron skin thickness is a reflex of the equation of state properties. Hence, a direct correlation between the neutron skin thickness and the slope of the symmetry energy is found. We show that within the present approximations the asymmetry parameter for low momentum transfer polarized electron scattering is not sensitive to the model differences.'
author:
- 'S.S. Avancini'
- 'J.R. Marinelli'
- 'D.P. Menezes'
- 'M.M.W. Moraes'
- 'C. Providência'
title: Density dependent hadronic models and the relation between neutron stars and neutron skin thickness
---
PACS number(s): [21.65.+f,24.10.Jv,95.30.Tg,26.60.+c]{}
Introduction
============
The relation between neutron star properties which are obtained from adequate equations of state (EoS) and the neutron skin thickness has long been a topic of investigation in the literature. The details of this relation and the important quantities to be discussed have been well established in [@tb01], where it was shown that the difference between the neutron and the proton radii, the neutron skin thickness, is linearly correlated with the pressure of neutron matter at sub-nuclear densities. This is so because the properties of neutron stars are obtained from appropriate EoS whose symmetry energy depends on the density and also controls the size of the neutron skin thickness in heavy and asymmetric nuclei, as $^{208}$ Pb, for instance. It is important to remember that the EoS in neutron stars is also very isospin asymmetric due to the $\beta$- equilibrium constraint.
Hence, isospin asymmetry plays a major role in the understanding of the density dependence of the symmetry energy and the consequences it may arise [@steiner]. In [@hp01; @piek06] it was shown that the models that yield smaller neutron skins in heavy nuclei tend to yield smaller neutron star radii due to a softer EoS.
Neutron stars are believed to have a solid crust formed by nonuniform neutron rich matter in $\beta$-equilibrium above a liquid mantle. In the inner crust nuclei coexist with a gas of neutrons which have dripped out. The properties of this crust as, for instance, its thickness and pressure at the crust-core interface depend a lot on the density dependence of the EoS used to describe it [@haen00; @piek06]. On the other hand, it is well known [@chom04; @inst04] that the existence of phase transitions from liquid to gas phases in asymmetric nuclear matter (ANM) is intrinsically related with the instability regions which are limited by the spinodals. Instabilities in ANM described within relativistic mean field hadron models, both with constant and density dependent couplings at zero and finite temperatures have already been investigated [@inst04] and it was shown that the main differences occur at finite temperature and large isospin asymmetry close to the boundary of the instability regions. In neutral neutron-proton-electron (npe) matter the electrons are also included. In a thermodynamical calculation the instabilities almost completely disappear due to the high electron Fermi energy [@inst062].
However, in a dynamical calculation which includes the Coulomb interaction and allows for independent neutron, proton and electron fluctuations [@inst06; @coletivos], it is seen that the electron dynamics tends to restore the short wavelength instabilities although moderated by the high electron Fermi energy.
Moreover, it is also known that the liquid-gas phase transition in ANM can lead to an isospin distillation phenomenon, characterized by a larger proton fraction in the liquid phase than in the gas phase. This is due to the repulsive isovector channel of the nuclear interaction [@xu00; @ducoin06; @chomaz].
In a recent work the spinodal section and related quantities, as the neutron to proton density fluctuations responsible for the distillation effect, has been studied within different relativistic models [@inst062]. It was shown that the distillation effect within density dependent relativistic models decreases with density above a nuclear density of $\sim 0.02-0.03$ fm$^{-3}$, a result similar to the one obtained with the SLy230a parametrization of Skyrme interaction [@chabanat] and contrary to the results found with the more common relativistic parametrizations with no density dependent coupling parameters. In the last case the distillation effect becomes always larger as the density increases.
Also, the behavior of the symmetry energy obtained with density dependent models is closer to what one obtains with non-relativistic models than with other relativistic models with constant couplings [@inst04]. In an attempt to understand this behavior, a comparison between the non-relativistic Skyrme effective force and relativistic mean field models at subsaturation densities was performed [@comp]. It was shown that the relativistic models could also be reduced to an energy density functional similar to the one describing the Skyrme interaction.
There have already been some efforts in order to compare nuclear matter and finite nuclei properties obtained both with relativistic and non-relativistic models [@bao-li; @ring97] but there is no clear or obvious explanations for the differences. At very low densities both, the relativistic and the non-relativistic approaches predict a non-homogeneous phase commonly named [*pasta phase*]{}, formed by a competition between the long-range Coulomb repulsion and the short-range nuclear attraction [@pasta].
Based on the above arguments, it is very important that an accurate experimental measurement of the neutron skin thickness is achieved. This depends on a precise measurement of both the charge and the neutron radius. The charge radius is already known within a precision of one percent for most stable nuclei, using the well-known single-arm and non-polarized elastic electron scattering technique as well as the spectroscopy of muonic atoms [@vries] . For the neutron radius, our present knowledge has an uncertainty of about 0.2 fm [@horo]. However, using polarized electron beams it is possible to obtain the neutron distribution in nuclei in a fairly model independent way, as first discussed in [@Don] and, as a consequence, to obtain the desired neutron radius. In fact, the Parity Radius Experiment (PREX) at the Jefferson Laboratory [@prex] is currently running to measure the $^{208}$Pb neutron radius with an accuracy of less than 0.05 fm, using polarized electron scattering.
In the present work, we use two different hadronic models that incorporate density dependence in different ways. The first one, to which we refer next as the TW model is a density dependent hadronic model with the meson-to-nucleon couplings explicitly dependent of the density [@original; @tw]. In the following it is used to calculate the neutron skin thickness of $^{208}$Pb, which is a neutron-rich heavy nucleus. This model was chosen because it is based on a microscopic calculation, fits well many nuclei properties and, as stated above, has shown to provide results which are different from the usual NL3 [@nl3] and TM1 [@tm1] parametrizations for the non-linear Walecka model (NLWM), having a richer density dependence of the symmetry energy than most of the relativistic nuclear models. The original motivation for the development of this density dependent hadronic model [@flw; @lf] was to reproduce results obtained with the relativistic Dirac-Brueckner Hartree-Fock (DBHF) theory [@DB]. Later the DBHF calculations for nuclear matter were taken only as a guide for a suitable parametrization of the density dependence of the meson-nucleon coupling operators [@tw; @ring1]. Moreover, density dependent hadronic models can also be a useful tool in obtaining EoS for neutron stars even if hyperons are to be considered [@ddpeos], which is not the case if NL3 or TM1 are used. Both, NL3 and TM1, can only be used if the EoS is restricted to accommodate neutrons, protons and the leptons necessary to enforce $\beta$-stability. Once hyperons are included, the nucleons acquire a negative effective mass above $\sim 3-4 \rho_0$ densities [@compact; @alex], where $\rho_0$ is the nuclear saturation density.
The second model, that we refer to as NL$\omega\rho$ model, includes non-linear $\sigma-\rho$ and $\omega-\rho$ couplings [@hp01; @hp2001; @bunta; @bunta2] which allow to change the density dependence of the symmetry energy of the most common parametrizations of the NLWM that show essentially a linear behavior of the symmetry energy with density. However, the symmetry energy determines the behavior of isospin asymmetric matter and therefore is intrinsically related to the characteristics of the EoS that can describe neutron stars. Within this model the authors of [@hp01] have shown that the neutron skin thickness of $^{208}Pb$ was sensitive to the isovector channel of the nuclear interaction and there was a correlation between neutron skin thickness of nuclei and properties of neutron stars.
For the sake of completeness, the results of the present work, whenever possible are compared with the results obtained with the NL3 parametrization of the NLWM, known to describe finite nuclei properties well.
We perform two different numerical calculations to obtain the $^{208}$Pb properties: a Thomas-Fermi approximation based on the liquid-gas phase transition developed in [@gotas] and a Thomas-Fermi approximation based on a method proposed in [@ring], where a harmonic oscillator basis is used. We restrict ourselves to the Thomas-Fermi approximation because, as we show in the Results section at the end of the paper, for the purpose of obtaining correct surface energy and neutron-skin thickness, it is almost as good as the solution of the Dirac equation.
At this point it is worth mentioning that the scalar-isovector $\delta$ mesons, which play an important role in the isospin channel, could also be incorporated in our work as done in [@gaitanos; @inst04; @inst06] but in order to make the comparisons among different approximations as simple as possible, they will be included in a future work. Finally, as we are interested in nuclei ground state properties, all calculations are performed at zero temperature.
The TW density dependent hadronic model
=======================================
Next we describe the main quantities of the TW model, which has density dependent coupling parameters. The Lagrangian density reads: $${\cal L}=\bar \psi\left[\gamma_\mu\left(i\partial^{\mu}-\Gamma_v V^{\mu}-
\frac{\Gamma_{\rho}}{2} \boldsymbol{\tau} \cdot \mathbf {b}^\mu \right. \right.$$ $$\left. \left. -e \frac{(1+\tau_{i3})}{2} A^\mu \right)
-(M-\Gamma_s \phi)\right]\psi$$ $$+\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi
-m_s^2 \phi^2)
-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}$$ $$+\frac{1}{2}m_v^2 V_{\mu}V^{\mu}
-\frac{1}{4}\mathbf B_{\mu\nu}\cdot\mathbf B^{\mu\nu}+\frac{1}{2}
m_\rho^2 \mathbf b_{\mu}\cdot \mathbf b^{\mu}
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\label{lagtw}$$ where $\phi$, $V^\mu$, $\mathbf {b}^\mu$ and $A^{\mu}$ are the scalar-isoscalar, vector-isoscalar and vector-isovector meson fields and the photon field respectively, $\Omega_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ , $\mathbf B_{\mu\nu}=\partial_{\mu}\mathbf b_{\nu}-\partial_{\nu} \mathbf b_{\mu}
- \Gamma_\rho (\mathbf b_\mu \times \mathbf b_\nu)$, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and $\tau_{p3}=1$, and $\tau_{n3}=-1$. The parameters of the model are: the nucleon mass $M=939$ MeV, the masses of the mesons $m_s$, $m_v$, $m_\rho$, the electromagnetic coupling constant $e=\sqrt{4\pi/137}$ and the density dependent coupling constants $\Gamma_{s}$, $\Gamma_v$ and $\Gamma_{\rho}$, which are adjusted in order to reproduce some of the nuclear matter bulk properties shown in Table \[bulk\], using the following parametrization: $$\Gamma _{i}(\rho )=\Gamma _{i}(\rho _{sat})h_{i}(x),\quad x=\rho /\rho _{sat},
\label{paratw1}$$ with $$h_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}},\quad i=s,v$$ and $$h_{\rho }(x)=\exp [-a_{\rho }(x-1)], \label{paratw2}$$ with the values of the parameters $m_{i}$, $\Gamma _{i}(\rho_{sat})$, $a_{i}$, $b_{i}$, $c_{i}$ and $d_{i}$, $i=s,v,\rho $ given in [@tw]. This model does not include self-interaction terms for the meson fields (i.e. $\kappa =0$, $\lambda =0$ and $\xi=0$ ) as in NL3 or TM1 parametrizations for the NLWM.
The field equations of motion follow from the Euler-Lagrange equations. When they are obtained, some care has to be taken since the coupling operators depend on the baryon fields $\bar
\psi$ and $\psi$ through the density. When the partial derivatives of $\cal{L}$ are performed relatively to the fields $\bar \psi$ and $\psi$, they yield extra terms due to the functional dependence of the coupling operators. The new terms are absent in the usual Quantum Hadrodynamic (QHD, NLWM) models [@sw; @nl3; @tm1]. The equations of motion for the fields read:
$$\begin{aligned}
(\partial_\mu\partial^{\mu} + m_{\phi}^2)\phi &=&
\Gamma_s \bar \psi \psi , \label{PHI} \\
\partial_{\nu} \Omega^{\mu\nu} + m_v^2 V^{\mu} &=&
\Gamma_v \bar \psi \gamma^{\mu} \psi, \label{OME}\\
\partial_{\nu} {\mathbf {B}}^{~\mu\nu} + m_\rho^2
{\mathbf b}^\mu &=&
\frac{\Gamma_\rho} {2} \bar \psi \boldsymbol {\tau} \gamma^{\mu}
\psi, \label{RHO}\\
\partial_{\nu} F^{\mu\nu} &=& \frac{e}{2}\bar \psi
(1+\tau_3 )\gamma^{\mu} \psi, \label{EM}\\
\left[ \gamma_{\mu}(i\partial^\mu -\Sigma^{\mu}) -M^{\ast}
\right] \psi&=&0 ~, \label{DIRAC}\end{aligned}$$
where $M^{\ast}=M-\Gamma_s \phi$. Notice that in the equation of motion for the baryon field $\psi$ the vector self-energy consists of two terms, $\Sigma_{\mu}$ = $\Sigma^{(0)}_\mu$ + $\Sigma^{R}_{\mu}$, where: $$\Sigma^{(0)}_\mu = \Gamma_{\omega} V_\mu
+\frac{\Gamma_{\rho}}{2} \boldsymbol {\tau}\cdot {\mathbf b}_{\mu}
+ \frac{e}{2} (1+\tau_3 ) A_{\mu},$$ $$\Sigma^{R}_{\mu}=\left( \frac{\partial \Gamma_v}{\partial
{\rho}} V^\nu j_\nu + \frac{1}{2}
\frac{\partial \Gamma_\rho}{\partial \rho} {\mathbf b}_\nu \cdot \mathbf j_3^\nu - \frac{\partial
\Gamma_\phi}{\partial {\rho}} \bar \psi \psi \phi\right)
u_\mu ~, \label{REAR}$$ where $\Sigma^{(0)}_\mu$ is the usual vector self-energy, $\hat\rho u_\mu=j_\mu$ with $u^2=1$ $j_\nu=\bar \psi \gamma_\nu \psi$, $\mathbf j_3^\nu= \bar \psi {\boldsymbol \tau} \gamma^\nu \psi$ and, as a result of the derivative of the Lagrangian with respect to $\rho$ a new term appears, $\Sigma^{R}_{\mu}$, which is called rearrangement self-energy and has been shown to play an essential rôle in the applications of the theory. This term guarantees the thermodynamical consistency and the energy-momentum conservation. For more detailed calculations, at zero and finite temperatures, please refer to [@previous].
In the static case there are no currents in the nucleus and the spatial vector components are zero. Therefore, the mesonic equations of motion become: $$\nabla^2 \phi = m_s^2\phi- \Gamma_s \rho_s,
\label{elphi}$$ $$\nabla^2 V_0 = m_v^2 V_0 - \Gamma_v \rho,
\label{elV0}$$ $$\nabla^2 b_0 =m_\rho^2 b_0 -\frac{\Gamma_\rho}{2} \rho_3,
\label{elb0}$$ $$\nabla^2 A_0 =-e \rho_p,\label{elA0}$$ where $\rho_s=<\bar \psi \psi>$ is the scalar density, $\rho=\rho_p+\rho_n$, $\rho_3=\rho_p-\rho_n$ and $\rho_p$ and $\rho_n$ are the proton and neutron densities.
Thomas-Fermi approximation
--------------------------
We first define the functional $$\Omega= E - \mu_p B_p - \mu_n B_n ,
\label{Omega}$$ where $E$ is the energy, $\mu_p$ ($\mu_n$) is the proton (neutron) chemical potential and $B_p$ ($B_n$) is the proton (neutron) number. Within the semi-classical Thomas-Fermi approximation, the energy of the nuclear system with particles described by the one-body phase-space distribution function $ f({\mathbf r},{\mathbf p},t)$ at position $\mathbf r$, instant $t$ with momentum $\mathbf p$ is given by $$E= \sum_i \gamma \int \d^3r \frac{\d^3p}{(2\pi)^3}\,
f_i({\mathbf r},{\mathbf p},t)
\left(\sqrt{{\mathbf p}^2 + {M^*}^2}
+ {\cal V}_{i}\right)$$ $$+ \frac{1}{2} \int \d^3r \left[
(\nabla \phi)^2 + m_s^2 \phi^2 -(\nabla V_0)^2 - m_v^2 V_0^2 \right.$$ $$\left. - (\nabla b_0)^2 - m_\rho^2 b_0^2 - (\nabla A_0)^2 \right]$$ where $${\cal V}_{p}= \Gamma_v V_0 + \frac{\Gamma_\rho}{2} b_0 + e A_0\;,
\quad {\cal V}_{n}= \Gamma_v V_0 - \frac{\Gamma_\rho}{2} b_0 \;,$$
$\gamma=2$ refers to the spin multiplicity and the distribution functions for protons and neutrons are $$f_i=\theta(k_{Fi}^2(r)-p^2), ~~~~~~~i=p,n~.$$ In this approach, the scalar, proton and neutron densities become: $$\rho_s(r)= \frac{\gamma}{2\pi^2} \sum_{i=p,n} \int_0^{k_{Fi}(r)}
p^2 \d p \frac{M^*}{\epsilon}$$ with $\epsilon=\sqrt{p^2+{M^*}^2}$ and $$B_i= \int \d^3r \rho_i, \quad
\rho_i(r)= \frac{\gamma}{6\pi^2}k_{Fi}^3(r).$$ From the above expressions we get for (\[Omega\]) $$\Omega= \int \d^3r \left( \frac{1}{2} \left [
(\nabla \phi)^2 -(\nabla V_0)^2 - (\nabla b_0)^2 - (\nabla A_0)^2
\right] + V_{ef}\right)$$ with $$V_{ef}= \frac{1}{2} \left[
m_s^2 \phi^2 -m_v^2 V_0^2 -m_\rho^2 b_0^2 \right] - \mu_p \rho_p-\mu_n \rho_n$$ $$+ \frac{\gamma}{2 \pi^2} \sum_{i=p,n} \int_0^{k_{Fi}} \d p p^2 \epsilon +
\Gamma_v V_0 \rho + \Gamma_{\rho} \frac{b_0}{2} \rho_3
+ e A_0 \rho_p
\label{vef}$$
Minimization of $\Omega$ with respect to $ k_{Fi}(r),\, i=p,n$, gives rise to the following conditions $$k_{Fp}^2\left(\mu_p-\sqrt{k_{Fp}^2+{M^*}^2}-
\Gamma_v V_0- \frac{\Gamma_\rho}{2} b_0 -e A_0 - \Sigma_0^R \right)=0$$ and $$k_{Fn}^2\left(\mu_n-\sqrt{k_{Fn}^2+{M^*}^2}-
\Gamma_v V_0+ \frac{\Gamma_\rho}{2} b_0 - \Sigma_0^R \right)=0,$$ where the rearrangement term is $$\Sigma_0^R=\frac{\partial\, \Gamma_v}{\partial \rho}\, \rho\, V_0+\frac{\partial\, \Gamma_\rho}{\partial \rho}\, \rho_3\, \frac{b_0}{2}-
\frac{\partial\, \Gamma_s}{\partial \rho}\, \rho_s\, \phi .$$ From the above equations we obtain $k_{Fp}=0$ and $k_{Fn}=0$ or, for $k_{Fp}$ or $k_{Fn}$ different from zero, $$\mu_p=\sqrt{k_{Fp}^2+{M^*}^2}+ \Gamma_v V_0 + \frac{\Gamma_\rho}{2} b_0 +
e A_0 + \Sigma_0^R, \label{mup}$$ $$\mu_n=\sqrt{k_{Fn}^2+{M^*}^2}+ \Gamma_v V_0 - \frac{\Gamma_\rho}{2} b_0
+\Sigma_0^R.
\label{mun}$$ The values of $k_{Fp}$ and $k_{Fn}$ are obtained inverting these two last equations.
Such density dependences in the coupling parameters do not affect the energy functional but of course affect its derivative such as the pressure density and the chemical potentials. As already discussed in the literature [@inst04; @inst06; @inst062; @ddpeos], the rearrangement term is crucial in obtaining different behaviors in physical properties related to the chemical potentials or to their derivatives with respect to the density, such as spinodal regions, as compared with the more common NL3 or TM1 parametrizations.
NL${\omega\rho}$ model
======================
The Lagrangian density that incorporates the extra non-linear $\sigma-\rho$ and $\omega-\rho$ couplings [@hp01; @hp2001; @bunta; @bunta2] reads
$${\cal L}=\bar \psi\left[\gamma_\mu\left(i\partial^{\mu}-g_v V^{\mu}-
\frac{g_{\rho}}{2} {\boldsymbol {\tau}} \cdot \mathbf {b}^\mu \right. \right.$$ $$\left. \left. -e \frac{(1+\tau_{i3})}{2} A^\mu \right)
-(M-g_s \phi)\right]\psi$$ $$+\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi
-m_s^2 \phi^2)
-\frac{1}{3!}\kappa \phi ^{3}
-\frac{1}{4!}\lambda \phi ^{4}
-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}$$ $$+\frac{1}{2}m_v^2 V_{\mu}V^{\mu}
-\frac{1}{4}\mathbf B_{\mu\nu}\cdot\mathbf B^{\mu\nu}+\frac{1}{2}
m_\rho^2 \mathbf b_{\mu}\cdot \mathbf b^{\mu}
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ $$+g_\rho^2\mathbf b_{\mu}\cdot \mathbf b^{\mu}[\Lambda_s
g_s^2\phi^2+\Lambda_v g_v^2 V_\mu V^\mu],
\label{lagacoplada}$$ where $\Omega_{\mu\nu}$, $\mathbf B_{\mu\nu}$ and $F_{\mu\nu}$ are defined after eq.(\[lagtw\]). The parameters of the model are again the masses and the couplings, which are now constants, i.e., $g_s$ replaces $\Gamma_s$, $g_v$ replaces $\Gamma_v$ and $g_\rho$ replaces $\Gamma_\rho$. Non-linear $\sigma$ terms are also included. We have followed the prescription of [@hp01], where the starting point was the NL3 parametrization and the $g_\rho$ coupling was adjusted for each value of the coupling $\Lambda_i$ studied in such a way that for $k_F=1.15$ fm$^{-1}$ (not the saturation point) the symmetry energy is 25.68 MeV. In the present work we set $\Lambda_s=0$ as in [@bunta2]. Notice that other possibilities for this model with $\sigma-\rho$ and $\omega-\rho$ couplings have already been discussed in the literature as in [@piek06], for instance.
The mesonic equations of motion in the Thomas-Fermi approximation become
$$\nabla^2 \phi = m_s^2\phi- g_s \rho_s + \frac{\kappa}{2} \phi^2 +
\frac{\lambda}{6} \phi^3
\label{elphi2}$$
$$\nabla^2 V_0 = m_v^2 V_0 - g_v \rho + 2 \Lambda_v g_v^2\, V_0\, g_{\rho}^2
b_0^2,
\label{elV02}$$
$$\nabla^2 b_0 =m_\rho^2 b_0 -\frac{g_\rho}{2} \rho_3 + 2 \Lambda_v
g_{\rho}^2 b_0 g_v^2 V_0^2,
\label{elb02}$$
$$\nabla^2 A_0 =-e \rho_p,\label{elA02}$$
and the expression for the energy reads
$$E= \sum_i \int \d^3r \left(
\gamma \int_0^{k_{Fi}(r)} \frac{\d^3p}{(2\pi)^3}\,
\sqrt{{\mathbf p}^2 + {M^*}^2} \right.$$ $$+\frac{1}{2} \left[ (\nabla \phi)^2 + m_s^2 \phi^2
-(\nabla V_0)^2 - m_v^2 V_0^2 \right.$$ $$- (\nabla b_0)^2 - m_\rho^2 b_0^2 - (\nabla A_0)^2$$ $$\left. + g_v V_0 \rho + \frac{g_\rho}{2} \rho_3 b_0 + e A_0 \rho_p \right]$$ $$\left. +\frac{\kappa}{6} \phi^3 + \frac{\lambda}{24} \phi^4
-\Lambda_v g_v^2 V_0^2 g_\rho^2 b_0^2 \right).$$
All other expressions are very similar to the ones obtained from the TW model and can be read off from them bearing in mind that the density dependent couplings have to be replaced by the constant couplings. In particular the chemical potentials do not contain the rearrangement term $\Sigma_0^R$.
Numerical result via a nucleation process
=========================================
At this point, eqs. (\[elphi\]-\[elA0\]) for the TW model and eqs. (\[elphi2\]-\[elA02\]) for the NL$\omega\rho$ model have to be solved numerically in a self-consistent way and hence, initial and boundary conditions for each equation are necessary. One of the methods we use here is based on a prescription given in [@gotas], where these conditions are obtained from a situation of phase coexistence in a mean field approximation with classical meson fields and no electromagnetic interaction. The method is well explained in [@gotas] and, as we are using different models here, just the main equations are written next.
For the TW model, the equilibrium equations for homogeneous matter for the fields are: $$\begin{aligned}
m_{s}^2\phi
- \Gamma_{s} ~\rho_{s } &=&0, \label{phitw} \\
m_{v}^2 V_0 - \Gamma_{v}~ \rho&=&0, \label{V0tw}\\
m_\rho^2 b_0 - \frac{\Gamma_{\rho}}{2}~ \rho_3&=&0,
\label{b0tw}\end{aligned}$$ and for the energy and pressure density: $${\cal E}=\frac{1}{\pi^2} \sum_i \int_0^{k_{Fi}} p^2 dp~
\sqrt{{\mathbf p}^2 + {M^*}^2}$$ $$+\frac{m_{s}^2}{2} \phi^2 + \frac{m_{v}^2}{2} V_0^2
+ \frac{m_{\rho}^2}{2} b_0^2,
\label{enermfa}$$ $$P=\frac{1}{3 \pi^2} \sum_i \int_0^{k_{Fi}} \frac{p^4 dp}{\epsilon}
- \frac{m_{s}^2}{2} \phi^2 + \frac{m_{v}^2}{2} V_0^2
+ \frac{m_{\rho}^2}{2} b_0^2$$ + . \[pressmfa\]
For the NL$\omega\rho$ model, the equilibrium equations for homogenous matter, energy density and pressure become: $$m_s^2\phi- g_s \rho_s + \frac{\kappa}{2} \phi^2 +
\frac{\lambda}{6} \phi^3 =0,
\label{phi2}$$ $$m_v^2 V_0 - g_v \rho + 2 \Lambda_v g_v^2\, V_0\, g_{\rho}^2
b_0^2 =0,
\label{V02}$$ $$m_\rho^2 b_0 -\frac{g_\rho}{2} \rho_3 + 2 \Lambda_v
g_{\rho}^2 b_0 g_v^2 V_0^2 = 0,
\label{b02}$$
$${\cal E}= \frac{1}{\pi^2} \sum_i \int_0^{k_{Fi}}
p^2 dp \sqrt{{\mathbf p}^2 + {M^*}^2}$$ $$+\frac{1}{2} \left[m_s^2 \phi^2 - m_v^2 V_0^2 - m_\rho^2 b_0^2 \right]
+ g_v V_0 \rho + \frac{g_\rho}{2} \rho_3 b_0$$ $$+\frac{\kappa}{6} \phi^3 + \frac{\lambda}{24} \phi^4
-\Lambda_v g_v^2 V_0^2 g_\rho^2 b_0^2.$$ and $$P=\frac{1}{3 \pi^2} \sum_i \int_0^{k_{Fi}} \frac{p^4 dp}{\epsilon}
- \frac{m_{s}^2}{2} \phi^2 + \frac{m_{v}^2}{2} V_0^2
+ \frac{m_{\rho}^2}{2} b_0^2$$ - \^3 - \^4 + \_v g\_v\^2 V\_0\^2 g\_\^2 b\_0\^2.
Based on the geometrical construction and Gibbs conditions for phase coexistence, i.e., the pressure and both chemical potentials are equal in both phases, we build the binodal section given in Fig. \[binodais\]. Notice that we have defined the proton fraction of the system as $$y_p = \frac{\rho_p}{\rho}.$$ The binodal section yields the boundary conditions which we need. For the same pressure, two points, with different proton fractions are found. For each of these points, the meson fields and the densities are well defined and used as the initial and boundary conditions in eqs. (\[elphi\]-\[elA0\]), which are then solved. Once the meson fields are obtained, all the quantities that depend on them, as the energy, pressure densities, chemical potentials, baryonic densities, etc are also computed. The solution is a droplet with a certain proton fraction surrounded by a gas of neutrons. If stable nuclei are calculated, the gas vanishes because the energy of the system lies below the neutron drip line and the finite nuclei properties are easily calculated. This is the general method, but the results depend strongly on the model used because of the reasons discussed in Section VI.
--------------------------------------------------------------------------------------------------------------------------------- --
![Binodal section for the NL3, TW and NL$\omega\rho$ parametrizations.[]{data-label="binodais"}](fig1.eps "fig:"){width="8.cm"}
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Numerical result within a harmonic oscillator basis
===================================================
Here a different prescription for solving the equations of motion and the thermodynamical quantities within the Thomas-Fermi approximation is used. According to [@ring], meson field equations of motion of the Klein-Gordon type with sources can be carried out by an expansion in a complete set of basis states. The harmonic oscillator functions with orbital angular momentum equal to zero are then chosen. The oscillator length is given by $$b_B=\frac{b_0}{\sqrt 2}, \quad b_0=\sqrt{\frac{\hbar}{M \omega_0}},$$ where $M$ is the nucleon mass and $\omega_0$ is the oscillator frequency. The meson fields and their corresponding inhomogeneous part can be expanded as $$\Lambda(r)=\sum_{n=1}^{n_B} \Lambda_n R_{n0}(r), \quad
S_\Lambda(r)=\sum_{n=1}^{n_B} S_n^\Lambda R_{n0}(r),
\label{ans}$$ where $\Lambda(r)=\phi(r),V_0(r),b_0(r)$ and $$R_{n l}(r)=\frac{N_{nl}}{b_0^{3/2}} x^l L^{l+1/2}_{n-1}(x^2)exp(-x^2/2),$$ where $x=r/b_0$ is the radius measured in units of the oscillator length, $$N_{nl}=\sqrt{2(n-1)!/(l+n-1/2)!}
\label{norm}$$ is the normalization constant and $L^m_n(x^2)$ are the associated Laguerre polynomials. For the calculation of the meson fields $l=0$ in the expressions given below. Once the ansatz given by eqs.(\[ans\]) are substituted into eqs.(\[elphi\]- \[elb0\]), a set of inhomogeneous equations is obtained: $$\sum_{n'=1}^{n_B} {\cal H}_{n n'} \Lambda_{n'}=S_n^\Lambda$$ where $${\cal H}_{n n'}= \delta_{n n'}\left( b_B^{-2} (2(n-1)+3/2) + m^2_\Lambda
\right)$$ $$+ \delta_{n n'+1} b_B^{-2} \sqrt{n(n+1/2)}+ \delta_{n+1 n'} b_B^{-2}
\sqrt{n'(n'+1/2)}.$$ Only the massive fields can be calculated with this method because the convergence of the Coulomb field, which has a long range, is very slow. The Green’s function method is then chosen to describe the electromagnetic interaction:
$$A_{0}(r)=e~\int~r'^{2}dr'\rho_{p}(r')G_c(r,r'), \label{green}$$
with $$G_c(r,r')= \biggl\{ \begin{array}{c}
1/r {~~~~\rm for} ~~r>r' \\
1/r' {~~~~\rm for} ~r'>r. \end{array}$$
Results
=======
Parity Violating Electron Scattering and the Neutron Radius
-----------------------------------------------------------
We start this section by defining the asymmetry for polarized electron scattering of a hadronic target as $$\mathcal{A}=\frac{d\sigma_{+}/d\Omega-d\sigma_{-}/d\Omega}{d\sigma_{+}/d\Omega+d\sigma_{-}/d\Omega},$$ where $d\sigma_{\pm}/d\Omega$ is the differential cross section for initially polarized electrons with positive($+$) and negative ($-$) helicities. As the electromagnetic interaction is not sensitive to the above difference, the asymmetry becomes dependent of the weak interaction between the electron and the target. Moreover, we know from the Standard Model that the neutral Z-boson couples more strongly to the neutron than to the proton. Those reasonings were then used in [@Don] to first propose a clean way to determine the neutron distribution in nuclei. If we consider elastic scattering on an even-even target nucleus, the asymmetry can be written in the form:
$$\mathcal{A}=\frac{Gq^2}{2\pi\alpha\sqrt{2}}a[\beta_{V}^{p}+\beta_{V}^{n}\frac{\rho_{n}(q)}{\rho_{p}(q)}].\label{ass}$$
In the above expression, G, $\alpha$, $a$ and $\beta_{V}^{p,n}$ are Standard Model coupling constants as defined in [@Don], $q$ is the transferred momentum by the electron to the nucleus and,
$$\rho_{n(p)}(q)=\int~d^{3}r~j_{0}(qr)\rho_{n(p)}(\mathbf {r}),$$
$\rho_{n(p)}(\mathbf {r})$ being the neutron (proton) distribution in configuration space and $j_{0}$ the spherical Bessel function of order zero. It is then clear that a small $q$ measurement of the asymmetry gives the neutron radius of the distribution once the proton radius is well known. The proton and neutron mean-square radius are defined as
$$R_{i}^{2}=\frac{\int~d^{3}r r^2 \rho_{i}(\mathbf {r})}{\int~d^{3}r
\rho_{i}(\mathbf {r})}, \quad i=p,n.$$
The neutron skin thickness is defined as $$\theta=R_n-R_p. \label{skin}$$
In the PREX experiment mentioned in the Introduction, the asymmetry is expected to be measured at $q\approx0.4~fm^{-1}$ [@prex]. Also, because the target is a heavy nucleus ($^{208}$Pb), the above results for the asymmetry should be reconsidered for a detailed comparison with the experiment, since they were obtained using a Plane Wave Born Approximation for the electron [@horo98]. For our present purposes, eq. (\[ass\]) is sufficient to illustrate the sensitivity to the different model parametrizations and is used next in the presentation of our numerical results.
The surface energy per unit area of the droplets in the small surface thickness approximation, excluding the electromagnetic field, reads [@gotas] $$\sigma=\int_0^\infty \d r \left[ \left(\frac{\d \phi}{\d r}\right)^2-
\left(\frac{\d V_0}{\d r}\right)^2 -
\left(\frac{\d b_0}{\d r}\right)^2 \right].
\label{sig}$$ However, as the electromagnetic interaction does not contribute to surface properties directly, we have kept the same definition for the surface energy.
In Table II we show the neutron and proton radius, the neutron skin thickness, the binding energy and the surface energy obtained within the Thomas-Fermi approximation and the two different numerical prescriptions described in the previous sections. All the results are sensitive to the numerical calculation although the analytical approximation is the same. When the nucleation method is performed, the neutron radius is systematically larger, what results in a thicker neutron skin. This is correlated with the fact that the surface energy is lower within the nucleation calculation than within the harmonic oscillator method. Within the same numerical prescription, the neutron skin thickness is smaller with the TW model than with the NL3. As the coupling strength $\Lambda_v$ increases in the NL$\omega\rho$ model, the results move from the original NL3 to the TW results for all quantities, except the proton radius, which oscillates a little. We have also included the results obtained with the HS parametrization [@hs] because we have used this parametrization in order to compare the TF and the Dirac results for the cross sections, as discussed in the following. As this parametrization is known not to give as good results as the other parametrizations of the NLWM for finite nuclei, we do not comment on the results it provides. Notice that the experimental radius for the protons is obtained from the charge radius $R_c$ and it is given by $R_p=\sqrt{R_c^2 -0.64}$ in fm [@ring]. Our results can be compared with experimental and other theoretical results found in the literature. The proton radius, which is known to better than 0.001 fm is better described within the TW model. This quantity is practically independent of the $\omega-\rho$ interaction strength in the NL$\omega\rho$ model as far as the HO numerical prescription is used. The neutron radius, on the other had, is strongly model dependent with drastic consequences in the neutron skin thickness calculation. The experimental values for $\theta$ are still very uncertain and all our results fall inside the experimental confidence interval. We shall comment on possible restrictions to the neutron skin thickness in the next section. NL3 provides the best results for the binding energy.
In [@nl3], the results shown for the proton and neutron radius are respectively 5.52 and 5.85 fm, yielding a skin of 0.33 fm, larger than ours. Notice, however, that in [@nl3] the Dirac equation was explicity solved. In [@piek06], the authors obtained a value of 0.21 fm for the neutron skin thickness and a binding energy of -7.89 MeV within a different parametrization of the NL$\omega\rho$ model. Again in this case the Dirac equation was solved.
In Fig. \[ronp1\] we show the difference between neutron and proton densities at the $Pb$ surface for the models discussed in the present work with the Thomas-Fermi approximation solved in a harmonic oscillator basis. While the curves deviate a little in between 6.0 and 8.0 fm, at the very surface they are similar, but a small discrepancy, reflecting the differences in the neutron skin can be seen.
In Fig. \[ronpcomp1\] we display again the difference between neutron and proton densities within both numerical calculations of the TW and NL3 models. These two Thomas-Fermi calculations should have given more similar results. However the nucleation method predicts a very small surface energy for the NL3 parametrization, and therefore, a large radius. This may be related to the choice of the boundary conditions and a deeper comparison between the two methods will be pursued.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Difference between neutron and proton densities obtained with the Thomas-Fermi approach solved in a harmonic oscillator basis for the models discussed in the present work.[]{data-label="ronp1"}](fig2.eps "fig:"){width="8.cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Difference between neutron and proton densities obtained with the Thomas-Fermi approach solved with both numerical prescriptions for the TW model.[]{data-label="ronpcomp1"}](fig3.eps "fig:"){width="8.cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
Next we present our results for the asymmetry given by eq. (\[ass\]) as a function of the transfered momentum. We begin with Fig.\[asym0\] which displays the results for the HS parametrization of the Walecka model. The curve labeled [*no structure*]{} means the case where $Z\rho_{n}(r)=N\rho_{p}(r)$ and the other two curves are obtained within the TF approximation and the full solution of the Dirac equation in the Hartree approximation. At the momentum transfer values of recent experimental interest (around $\simeq 0.4$ fm$^{-1}$), the curves are almost identical. A careful analysis of the same results in a different scale shows us that the asymmetry changes $12$ and $11$ percent respectively within the Dirac and TF approximations in comparison with the [*no structure*]{} case. Since it is the measurement of the asymmetry in this low momentum transfer region that will provide the accurate result for the neutron skin thickness, we have restricted our calculations to the TF approximation, as stated in the Introduction.
------------------------------------------------------------------------------------------------------------------------- --
![Parametrization HS, comparison Thomas-Fermi-HO versus Dirac-HO[]{data-label="asym0"}](fig4a.eps "fig:"){width="8.cm"}
![Parametrization HS, comparison Thomas-Fermi-HO versus Dirac-HO[]{data-label="asym0"}](fig4b.eps "fig:"){width="8.cm"}
------------------------------------------------------------------------------------------------------------------------- --
In Fig. \[asym2\]a we show the asymmetry obtained with the NL3 model for both numerical calculations in the TF approximation, i.e, nucleation and HO expansion methods. In this case, the agreement is very satisfactory even for larger $q$-values, although the small numerical discrepancies is reflected in a $\sim 10$ percent difference in the predicted neutron skin thickness, as can be seen from Table II. Finally, in Fig. \[asym2\]b our results for the NL$\omega\rho$ (using two different values for the $\omega-\rho$ coupling constant) and the TW models within the HO numerical prescription are shown. Again, at low momentum transfers, all curves coincide. However, it should be noticed that even for two different model parametrizations which lead us to identical neutron skin thicknesses, a measurement of the asymmetry in a higher $q$-region with a modest experimental precision, can distinguish between them. Also, we should expect that the asymmetry presents more structure in this high momentum transfer region if we solve the Dirac equation instead of using the TF approach, once the high $q$ value region is much more sensitive to the central part of the neutron distribution, which is known to be [*flat*]{} in the TF approximation. These differences can be seen in Fig.\[asym0\].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
![Asymmetry obtained with a) NL3 with both numerical prescriptions and b)parametrizations NL$\omega\rho$ and TW[]{data-label="asym2"}](fig5a.eps "fig:"){width="8.cm"}
![Asymmetry obtained with a) NL3 with both numerical prescriptions and b)parametrizations NL$\omega\rho$ and TW[]{data-label="asym2"}](fig5b.eps "fig:"){width="8.cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
Different EoS, different neutron skins
======================================
For the sake of completeness, at this point, we discuss some of the differences between the TW, the NL$\omega\rho$ models and the NL3 parametrization of the NLWM.
From Fig. \[binodais\] one can see that the largest possible pressure for a phase coexistence in the TW model is much lower, and appears at a lower proton fraction than the NL3 model. This gives rise to a thinner crust within the TW model, which may imply that the more exotic [*pasta*]{} shapes will not form [@haen00]. The NL$\omega\rho$ model goes on a different direction, i.e., the pressure becomes higher than the one obtained with the NL3 as the $\Lambda_v$ coupling is turned on.
Although the nuclear matter properties fitted to parametrize the models are quite similar (see Table \[bulk\]), the way the EoS behaves when extrapolated to higher or lower densities can vary a lot from a density dependent hadron model to one of the parametrizations of the NLWM. Moreover, as seen from Table \[bulk\], although the effective mass at saturation density is lower with the TW than with the NL3, it can accommodate hyperons if an EoS for stellar matter is necessary, contrary to the usual NL3 parametrization [@ddpeos; @alex; @compact].
------------------------- -------- ------------------ ------------------ ------------------- ------- -- -- -- -- --
NL3 NL$\omega\rho$ TW
[@nl3] [@bunta] [@tw]
$\Lambda_v=0.01$ $\Lambda_v=0.02$ $\Lambda_v=0.025$
$B/A$ (MeV) 16.3 16.3 16.3 16.3 16.3
$\rho_0$ (fm$^{-3}$) 0.148 0.148 0.148 0.148 0.153
$K$ (MeV) 271 271 271 271 240
${\cal E}_{sym.}$ (MeV) 37.4 34.9 33.1 32.3 32.0
$M^*/M$ 0.60 0.60 0.60 0.60 0.56
$L$ (MeV) 118 88 68 61 55
$K_{sym}$ (MeV) 100 -46 -53 -34 -124
------------------------- -------- ------------------ ------------------ ------------------- ------- -- -- -- -- --
: Nuclear matter properties.[]{data-label="prop"}
\[bulk\]
Another quantity of interest in asymmetric nuclear matter is the nuclear bulk symmetry energy, shown in Table \[bulk\] for the saturation point. The differences in the symmetry energy at densities larger than the nuclear saturation density is still not well established, but has already been extensively discussed in the literature even for the TW model [@inst04; @inst062; @ddpeos; @bao-li]. Again, for the sake of completeness we reproduce these results here because the neutron skin thickness and the neutron star EoS are related by this quantity [@tb01; @steiner; @hp01; @piek06], which is usually defined as $ {\cal E}_{sym} =\left. \frac{1}{2} \frac{\partial^2 {\cal E}/\rho}
{\partial \delta^2} \right|_{\delta=0}$, with $\delta=-\rho_3/\rho=1-2y_p$. The symmetry energy can be analytically rewritten as $${\cal E}_{sym}= \frac{k_F^2}{6 \epsilon_F}+ \frac{\Gamma_\rho^2}
{8 m_\rho^2} \rho, \label{esym}$$ for the TW model and as $${\cal E}_{sym}= \frac{k_F^2}{6 \epsilon_F}+ \frac{g_\rho^2}{8 {m^*_\rho}^2} \rho
\label{esymd}$$ with the effective $\rho$-meson mass defined as [@hp01] $${m^*_\rho}^2=m_\rho^2+ 2 g_v^2 g_\rho^2 \Lambda_v V_0^2$$ for the NL$\omega\rho$ model. In both cases $$k_{Fp}=k_F(1+\delta)^{1/3},\qquad k_{Fn}=k_F(1-\delta)^{1/3},$$ with $k_F=(1.5 \pi^2\rho)^{1/3}$ and $\epsilon_F=\sqrt{k_F^2+{M^*}^2}$. In equations (\[esym\]) and (\[esymd\]) the second term dominates at large densities. It is seen that the non-linear $\rho-\omega$ terms introduce a non-linear density behavior in the symmetry energy of the NLWM parametrizations such as NL3 and TM1. In TW the non-linear density behavior enters through the density dependent coupling parameters. These non-linear density behavior is important because the linear behavior of NL3 and TM1 parametrizations predicts too high symmetry energy at densities of importance for neutron star matter which has direct influence on the proton fraction dependence with density.
---------------------------------------------------------------------------------------------------------------------- --
![Symmetry energy for the NL3, TW and NL$\omega\rho$ models.[]{data-label="esymfig"}](fig6.eps "fig:"){width="8.cm"}
---------------------------------------------------------------------------------------------------------------------- --
From Fig. \[esymfig\], it is easily seen that the symmetry energy obtained with the TW model behaves in a very different way, as compared with NL3. In [@piek06] a relation between the symmetry energy and the nuclear binding energy is discussed : the harder the EoS, the more the symmetry energy rises with density. The density dependence discussed in [@piek06] is of the type introduced in [@hp01; @bunta] through the inclusion of a $\sigma-\rho$ and/or $\omega-\rho$ couplings and then, similar with the NL$\omega\rho$ model discussed here. One can observe that as the strength of the coupling increases, the symmetry energy gets closer to the TW curve. In fact, in [@inst062] it was shown that once this kind of coupling is introduced with a reasonable strength, the symmetry energy at low densities tends to behave as the TW model.
The symmetry energy can be expanded around the nuclear saturation density and reads $${\cal E}_{sym}(\rho)={\cal E}_{sym}(\rho_0) +
\frac{L}{3} \left(\frac{\rho-\rho_0}{\rho_0}\right) +
\frac{K_{sym}}{18} {\left(\frac{\rho-\rho_0}{\rho_0}\right)}^2,$$ where $L$ and $K_{sym}$ are respectively the slope and the curvature of the nuclear symmetry energy at $\rho_0$ and they are calculated from $$L=3 \rho_0 \frac{\partial {\cal E}_{sym}(\rho)}{\partial \rho}|_{\rho=\rho_0}
\quad
K_{sym}=9 \rho_0^2 \frac{\partial^2 {\cal E}_{sym}(\rho)}{\partial^2 \rho}|_
{\rho=\rho_0}.$$ These two quantities can provide important information on the symmetry energy at both high and low densities because they characterize the density dependence of the energy symmetry. In a recent work [@bao-an], the authors found a correlation between the slope of the symmetry energy and the neutron skin thickness. In their work 21 sets of the non-relativistic Skyrme potential were investigated and only 4 of them were shown to have $L$ values consistent with the values extracted from experimental isospin diffusion data from heavy ion collisions. In fact, the extracted value was $L=88 \pm 25$ MeV [@tsang], which gives a very strong constraint on the density dependence of the nuclear symmetry energy and consequently on the EoS as well. A detailed analysis of Table I shows that, if this constraint is to be taken seriously, neither the NL3 nor the TW model satisfy it. Nevertheless, the NL$\omega\rho$ slope interpolates beautifully between the NL3 and TW slope values. Once again it is seen that the increase in $\Lambda_v$ approximates the NL3 model values for the slope and energy symmetry to the TW values. Moreover, we have also tried to find a correlation between the $\theta$ values shown in Table II and $L$ values displayed in Table I. We found that, as far as some numerical imprecision are considered, larger values of $L$ correspond to larger values of the neutron skin, as seen in Fig. \[correlation\].
------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Correlation between the neutron skin $\theta$ and the slope of the symmetry energy $L$.[]{data-label="correlation"}](fig7.eps "fig:"){width="8.cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------- --
Let’s now go back to the problem of solving the differential equations within the nucleation numerical prescription. As we need boundary conditions arising from the liquid-gas phase coexistence in order to solve eqs. (\[elphi\]-\[elA0\]) for the TW model and eqs. (\[elphi2\]-\[elA02\]) for the NL$\omega\rho$ model, the binodal sections are essential and the spinodal sections, which separate the regions of stable to unstable matter are also of interest. If we had displayed the binodals in a $\rho_p$ versus $\rho_n$ plot, as it is done with the spinodals in Fig \[spinodais\], we could see that the spinodals surfaces lie inside the binodal sections and share the critical point corresponding to the highest pressure.
In Fig. \[spinodais\] the spinodals for the three different models discussed in this work are shown. Once again, some of these results can also be found in the recent literature [@inst04; @inst062], but we include them here to make a direct link with the binodals. The instability of the ANM system is essentially determined by density fluctuations in the isoscalar channel. Although the spinodals are, by themselves, not relevant in calculations performed at the thermodynamical equilibrium, the isospin channel is very sensitive to the instabilities occurring below the nuclear saturation density. The spinodal is determined by the values of pressure, proton fraction and density for which the determinant of $${\cal F}_{ij}=\left(\frac{\partial^2{\cal F}}{\partial \rho_i\partial\rho_j}
\right)_T,
\label{stability}$$ where $\cal F$ is the free energy density, goes to zero. A detailed analysis of this quantity can be found in [@mc03; @inst062].
-------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Spinodal section in terms of $\rho_p$ versus $\rho_n$ for the NL3, TW and NL$\omega\rho$ models.[]{data-label="spinodais"}](fig8.eps "fig:"){width="8.cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------- --
[From Fig. \[spinodais\], it is seen that the instability region in the $\rho_p/\rho_n$ plane, defined by the inner section of the spinodal curve is larger for the TW than for the NL3 model. The size of the instability region depends on the derivative of the chemical potentials with respect to the neutron and proton densities. At low densities different models exhibit different behaviors.]{}
The presence of the rearrangement term in the TW model also plays a decisive role. Even though a relatively large compensation exists between scalar and vector mesons in the isoscalar channels within the rearrangement term at low densities, the spinodal region is defined by the derivative of the chemical potential and therefore of the rearrangement term.
Next we examine the spinodals obtained with different coupling strengths for the NL$\omega\rho$ model. As seen in Fig. \[spinodais\], there is almost no difference between the different curves. They all fall around the original NL3 curve but once again, they tend to the TW curve as the coupling strength increases. However, contrary to the TW model, it was shown in [@inst06] that the direction of the instability in NL$_{\omega\rho}$ increases distillation as the density increases, and the larger the coupling $\Lambda_v$ the larger the effect.
Finally, to end this section, let’s make our points clear: we have used a simple mean field theory approach to obtain the boundary conditions for the equations of motion of the meson fields in the nucleation prescription. These boundary conditions depend on the model used and are intrinsically related with the liquid-gas phase transition which, in turn, can be well understood by studying the coexistence surfaces of the corresponding models. On the other hand, the neutron skin thickness shows a linear correlation with the slope of the symmetry energy, as already pointed out in [@bao-an] for non-relativistic models. Based on the different behaviors found with density dependent hadronic models and the NLWM, an obvious consequence is the fact that the neutron skin thickness depends on the choice of the model.
Conclusions
===========
We have calculated the $^{208}Pb$ neutron skin thickness with two different density dependent hadronic models, the TW and the NL$\omega\rho$ model, and one of the most used parametrizations of the NLWM, the NL3. The calculations were done within the Thomas-Fermi approximation, which gives quite accurate results for the asymmetry in the momentum transfer range of interest for the calculation of neutron skins. In implementing the numerical results two different prescriptions were used: the first one based on the nucleation process and the second one based on the harmonic oscillator basis method. We have seen that when the nucleation method is performed, the neutron radius is systematically larger, what results in a thicker neutron skin. This is a consequence of the fact that the surface energy is lower within the nucleation calculation than within the harmonic oscillator method. Within the same numerical prescription, the neutron skin thickness is smaller with the TW model than with the NL3. As the coupling strength $\Lambda_v$ increases in the NL$\omega\rho$ model, the neutron skin thickness moves from the original NL3 towards the TW results. We have also found that although the neutron skin thickness is model dependent, the asymmetry at low momentum transfers (below 0.5 fm$^{-1}$) is very similar for all models and all numerical prescriptions. As $q$ increases, the asymmetry also becomes model dependent. The density profiles obtained from the solution of the Dirac equation exhibits oscillations near the center of the nucleus, behavior which is not reproduced within the Thomas-Fermi approximation. This fact shows up in the asymmetry at large momentum transfers and therefore all the calculations should be reproduced by solving the Dirac equation. This calculation is already under investigation.
It is worth mentioning that the neutron skin thickness has shown to give hints on the equations of state that are suitable to describe neutron stars. Moreover, in [@bao-an] a correlation between the slope of the symmetry energy and the neutron skin thickness was found for Skyrme-type models. We have observed that this correlation was also present in the density dependent models we have studied in the present work.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was partially supported by CNPq(Brazil), CAPES(Brazil)/GRICES (Portugal) under project 100/03 and FEDER/FCT (Portugal) under the projects POCTI/FP/63419/2005 and POCTI/FP/63918/2005.
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----------------------------------- --------------- ------- ------- ---------------- ------- ------------ -- -- -- --
model approximation $R_n$ $R_p$ $\theta$ $B/A$ $\sigma$
(fm) (fm) (fm) MeV Mev/fm$^2$
NL3 TF+nucleation 5.88 5.65 0.24 -7.77 0.76
NL3 TF+HO 5.79 5.57 0.22 -7.79 0.96
NL$\omega\rho$, $\Lambda_v=0.01$ TF+HO 5.77 5.57 0.20 -7.73 0.98
NL$\omega\rho$, $\Lambda_v=0.02$ TF+HO 5.75 5.57 0.17 -7.65 0.99
NL$\omega\rho$, $\Lambda_v=0.025$ TF+HO 5.74 5.58 0.16 -7.63 1.00
TW TF+nucleation 5.71 5.50 0.22 -6.42 1.08
TW TF+HO 5.68 5.52 0.16 -7.46 1.10
HS TF+HO 5.70 5.47 0.24 -6.10 1.37
exp.[@fricke] 5.44
exp. [@audi] -7.87
exp. [@kraszna] $0.12\pm 0.07$
exp. [@hintz] $0.20\pm 0.04$
----------------------------------- --------------- ------- ------- ---------------- ------- ------------ -- -- -- --
: $^{208}$ Pb properties[]{data-label="lead"}
|
---
abstract: 'Recognition and identification of unknown targets is a crucial task in surveillance and security systems. Electronic Support Measures (ESM) are one of the most effective sensors for identification, especially for maritime and air–to–ground applications. In typical surveillance systems multiple ESM sensors are usually deployed along with kinematic sensors like radar. Different ESM sensors may produce different types of reports ready to be sent to the fusion center. The focus of this paper is to develop a new architecture for target recognition and identification when non–homogeneous ESM and possibly kinematic reports are received at the fusion center. The new fusion architecture is evaluated using simulations to show the benefit of utilizing different ESM reports such as attributes and signal level ESM data.'
author:
-
-
bibliography:
- 'references.bib'
title: Object Recognition and Identification Using ESM Data
---
|
---
abstract: 'The South Pole, which hosts the IceCube Neutrino Observatory, has a complete and around-the-clock exposure to the Galactic Center. Hence, it is an ideal location to search for gamma rays of PeV energy coming from the Galactic Center. However, it is hard to detect air showers initiated by these gamma rays using cosmic-ray particle detectors due to the low elevation of the Galactic Center. The use of antennas to measure the radio footprint of these air showers will help in this case, and would allow for a 24/7 operation time. So far, only air showers with energies well above $10^{16}$ eV have been detected with the radio technique. Thus, the energy threshold has to be lowered for the detection of gamma-ray showers of PeV energy. This can be achieved by optimizing the frequency band in order to obtain a higher level of signal-to-noise ratio. With such an approach, PeV gamma-ray showers with high inclination can be measured at the South Pole.'
author:
- 'A. Balagopal V.$^{*}$, A. Haungs, T. Huege, F. G. Schr[ö]{}der'
bibliography:
- 'MainDoc.bib'
date: 'Received: date / Accepted: date'
title: 'Search for PeVatrons at the Galactic Center using a radio air-shower array at the South Pole'
---
Introduction {#intro}
============
The study of air showers using radio detection techniques, to date, has been mainly applied in the case of charged cosmic-ray measurements and neutrino searches [@Huege:2016veh][@Schroder:2016hrv]. Such showers have been detected with energy thresholds of at least a few tens of PeV. We show that this technique can also be used for PeV gamma ray astronomy, by lowering the energy threshold. This can be done by extending the frequency band of measurement to higher frequencies than those used by current radio air shower arrays.
The Galactic Center has been identified as a source of gamma rays of TeV energy by H.E.S.S. [@Abramowski:2016mir]. The source of this excess of TeV gamma rays has been traced to a PeVatron near the Galactic Center, in particular, close to the black hole Sgr A\*. The H.E.S.S. data prefer a power law spectrum of E$^{-2.3}$ with no cut-off. The spectrum of TeV gamma rays can be extrapolated to PeV energies. Detection of PeV gamma rays approaching from the Galactic Center, would strengthen the evidence of the existence of such a PeVatron.
Current efforts to look for PeVatrons from the Galactic plane with the IceCube Observatory involve the measurement of the neutrino and muon fluxes from possible sources in the northern sky and the southern sky, respectively [@GonzalezGarcia:2009jc][@Halzen:2009us]. Such searches then aim at explaining the contribution of PeVatrons to the knee of the cosmic-ray spectrum. Observing such PeVatrons using down-going muons will restrict the visible sky to that within the nearly vertical zenith angle range, due to limitations in the detector volume. Hence, Galactic Center observations with the help of down-going muons would be restricted.
Gamma rays of PeV energy, upon entering the Earth’s atmosphere will produce air showers, similar to those produced by cosmic rays. These air showers can be detected on the ground using particle detectors and radio antennas. The Galactic Center is always visible at the South Pole, at an angle of $29^\circ$ above the horizon (zenith angle of $61^\circ$). Hence the IceCube Observatory at the South Pole is an ideal location to search for gamma rays from the Galactic Center. The number of gamma rays arriving at the IceCube Observatory from the Galactic Center with energies above 0.8 PeV is estimated to be around 11.5 events per year, from a simple extrapolation of the spectrum measured by H.E.S.S. (See appendix \[sec:flux\]).
The IceCube Neutrino Observatory [@Achterberg:2006md], the 1 km$^3$ array for the detection of astrophysical neutrinos, has a surface component of ice-Cherenkov particle detectors (IceTop) used for the detection of cosmic-rays [@IceCube:2012nn]. It is planned to upgrade IceTop using scintillators [@HuberSam]. It is also foreseen to have a large surface array of scintillation detectors and air-Cherenkov telescopes as a part of IceCube-Gen2 [@JvanSanten] [@2015ICRC...34.1070E][@Seckel:2015aex][@IceAct2017]. A surface array of radio antennas in addition to this could potentially increase the accuracy for the detection of air showers (especially inclined air showers) and for the determination of mass composition. RASTA, a previous study that was made using test antennas at the South Pole explored the possibility of improving veto capabilities and cosmic-ray studies at IceCube [@Boser:2010sw]. Apart from this, in-ice radio measurements at the South Pole is also being made using ARA, that aims at measuring the radio signals produced by high energy neutrinos inside ice [@Allison:2015eky].
[.49]{} {width="8cm"}
[.49]{} {width="8cm"}
A surface array of radio antennas at the South Pole can be used to search for air showers produced by PeV gamma rays arriving from the Galactic Center. Inclined air showers of PeV energy will be hard to detect and reconstruct effectively using particle detectors, since a major part of the shower dies out by the time it reaches the detector array. This is especially the case for showers induced by gamma rays. Gamma-ray showers have lesser muonic content when compared to hadronic showers. In particular, the showers induced by protons have a significantly larger fraction of muons than gamma-ray showers. Muons, unlike electrons and positrons, are the most prominent component of inclined air showers of PeV energy, that will reach the ground. The low muonic content of gamma-ray showers results in fewer pulses in the IceTop tanks and the future scintillation detectors. In contrast, the radio signal from a shower with the same primary energy survives and can be detected on the ground by an array of radio antennas. Thus by comparing the radio emission to the number of muons detected on the ground one can distinguish between showers initiated by gamma rays and those by other nuclei.
Radio emission of air showers develops mainly due to the deflection of the electrons and positrons of the shower in the Earth’s magnetic field (Geomagnetic effect). This results in a time-varying current that produces radio pulses [@Kahn206][@Scholten:2007ky]. Another contribution to radio emission comes from the Askaryan effect, which is due to the charge excess at the shower front that forms as the shower propagates through the atmosphere [@Askaryan_1961][@Askaryan_1965]. This effect, which has a smaller influence in air showers than the geomagnetic effect, causes a small asymmetry in the total radio emission. At higher frequencies, a Cherenkov ring is visible in the radio footprint, due to the time compression of radio pulses caused by the refractive index of air [@deVries:2011pa]. For example, at frequencies such as that covered by the band 50-350 MHz, which is used by the lower frequency component of the Square Kilometer Array (SKA-LOW) [@7928622], the Cherenkov ring is visible. Such a Cherenkov ring is only marginal in the frequency band 30-80 MHz, which is the frequency range used by most of the existing radio air shower experiments, e.g. AERA [@1748-0221-7-10-P10011], Tunka-Rex [@Bezyazeekov:2015rpa] and LOFAR [@Schellart:2013bba].
An inclined shower produced by a gamma ray from the Galactic Center will leave a large radio footprint on the ground, whose diameter ranges from several 100 m to km depending on the angle of inclination. Recent studies of inclined air showers by The Auger Engineering Radio Array (AERA) have experimentally proven this [@Kambeitz:2016rqu]. The footprint detected on the ground is elliptical in shape because of pure geometrical reasons [@Huege:2015lga]. The inclined air showers detected by AERA have energies higher than $10^{18}$ eV. Similar characteristics will be seen by showers of PeV energy arriving at the IceCube location. Figure \[fig:ldf\] shows the different simulated amplitudes delivered to 81 antennas, on an area of 1 km$^2$, at the location of the IceTop stations [@IceCube:2012nn]; that is with one antenna placed at the center of the two Cherenkov tanks that form an IceTop station (see also Figure \[fig:SNRmap\]). The lateral distribution of the amplitudes in the figure are those from gamma-ray induced showers and proton induced showers at these antenna locations. Here, the zenith angle is fixed to $61^\circ$. These showers have a geomagnetic angle (angle between the Earth’s magnetic field and the shower axis) of $\alpha = 79^\circ$. For illustration, the frequencies are split into bands with a width of 70 MHz each, and range from 30 MHz all the way up to 380 MHz. The proton showers have lower amplitudes than gamma-ray showers since they have lower electromagnetic content. The plot shows the mean amplitude along with the spread about the mean with 30 simulated showers for gamma-ray and proton primaries. We can see that the the lateral distribution for these showers change with the frequency of observation. At frequencies above 100 MHz, we start to see the Cherenkov ring in such a distribution. At very high frequencies like those above 300 MHz, the emission becomes extremely localized, giving non-zero values of the amplitude only on the Cherenkov ring. That is, the radio signal dies out gradually at every other location at such high frequencies. So far, it has been considered that air showers from cosmic rays with an energy range greater than $10^{16}$ eV can be measured using the radio detection technique. At energies lower than this, the background overwhelms the radio signal from an air shower [@Huege:2016veh]. This is especially the case for the frequency range of 30-80 MHz. This makes it hard to measure air showers at low energies, unless interferometric methods are used. Thus in order to measure air showers produced by PeV gamma rays, we explore a different method to lower the energy threshold. This paper focuses on this aspect, especially on the optimization of the observing frequency bands in order to lower the threshold energy for the detection of gamma-ray air showers.
Simulation of air showers
=========================
A thorough study of the air showers that are produced by the incoming gamma rays is needed for predicting the radio signal that will be detected on the ground. For this purpose, air shower simulations were performed using CoREAS [@Huege:2013vt], which is the radio extension of CORSIKA [@Heck:1998vt]. We use CORSIKA-7.4005 with hadronic interaction models FLUKA-2011.2c.2 and SIBYLL-2.1 [@PhysRevD.80.094003]. Later simulations used CORSIKA-7.5700 with SIBYLL-2.3. This was not seen to change the received radio signals from air showers significantly (since the main difference between the versions is in the muonic content of the hadronic interactions). A total of 1579 simulations have been done for this study. The simulations used the atmosphere of the South Pole (South pole atmosphere for Oct. 01, 1997 provided in CORSIKA) with an observation level of 2838 m above the sea level. All the showers have been simulated using the thinning option (with a thinning level of $2.7 \times 10^{-7}$).
The showers simulated are those of gamma-ray primaries with energies ranging from 1-10 PeV. The azimuth angle for preliminary studies were fixed so that the shower axis is oriented anti-parallel to the Magnetic North ($\phi$ = 0), thereby giving a geomagnetic angle of $79^\circ$ for a shower with a zenith angle of $61^\circ$. At the South Pole, the magnetic field is inclined at an angle of $18^\circ$ with respect to the vertical, with an intensity of 55.2 $\mu$T. The zenith angle is fixed to $61^\circ$ for a major portion of the simulations since this is the inclination of the Galactic Center at the South Pole. The core position was set at the center of the IceTop array, i.e. at (0,0). For comparison, proton showers were also simulated, with the same parameters. The simulations included 81 antennas, each at the center of an IceTop station. This resulted in an array where the average antenna spacing is around 125 m. An inner infill array of antennas with much denser spacing (approximately 90 m) is also present, since such a structure is present for the IceTop stations also. The entire array covers an area of around 1 km$^2$.
The output from CoREAS simulations gives the signal strength at each of these antenna stations in units of $\mu$V/m. This has to be folded through the response of an antenna, in order to estimate the measurable signal. For this purpose, a simple half wave dipole antenna with resonance at 150 MHz was simulated using NEC2++ [@necpp]. Antennas in the east-west and in the north-south direction (with respect to the magnetic field) were used at the location of each station, to extract the complete signal from the air shower. Here, the z-component is neglected. This can be safely done because of the small angle between the magnetic field and the vertical, thereby resulting in a smaller z component of electric field as compared to the x and y components, even for the inclined air showers.
The simulations that are performed here are simplified, since the main focus is to understand the required experimental setup for lowering the energy threshold in order to detect PeV gamma rays from the Galactic Center. Specific effects like the impact of an optimized type of antenna or other details that can be important for a particular experimental setup are ignored in this context and can be included in the case of a more detailed study.
[.99]{} ![(a) Brightness of the Galactic radio background radiation given by the Cane function [@Cane]. (b) Total noise temperature as a function of frequency. \[fig:Cane\]](brightness_cane.eps "fig:"){height=".69\columnwidth"} \[fig:Cane(a)\]
[.99]{} ![(a) Brightness of the Galactic radio background radiation given by the Cane function [@Cane]. (b) Total noise temperature as a function of frequency. \[fig:Cane\]](noisevstempCane.eps "fig:"){height=".75\columnwidth"} \[fig:Cane(b)\]
[.49]{} {width="8cm"}
[.49]{} {width="8cm"}
Inclusion of a noise model
==========================
One of the major challenges for the detection of radio signals from showers of PeV energy is the lower level of signal, when compared to the background noise as discussed in section \[intro\]. There can be external as well as internal (thermal) sources of noise for radio air shower experiments. The external sources of noise range from Galactic noise through man-made noise to noise contributed by atmospheric events. At the South Pole, the external contribution mainly comes from Galactic noise as the contributions from other elements are expected to be much lower in comparison. Existing air shower experiments point out that measurement of signals from inclined air showers of PeV energy range using the band of 30-80 MHz is hard to achieve. Within this frequency band, the signal will be dominated by noise, especially for showers in the PeV energy range.
In this study, a simplified and average model of diffuse Galactic noise developed by Cane [@Cane] is used. It has already been shown by measurements from RASTA and ARA that the Cane model describes the Galactic noise measured at the South Pole with a reasonable accuracy [@MikeRATSA][@Allison:2011wk]. The noise is given in units of brightness (Galactic Brightness background) in this model, as can be seen in Figure \[fig:Cane\]. The corresponding brightness temperature, obtained from the relation $T = \frac{1}{2k_{\mathrm{B}}}\frac{c^{2}}{\nu^{2}} B(\nu)$, is used for determining the Galactic contribution to the total noise. In addition to the Galactic noise, there is also a contribution from the thermal component, which arises due to the electronic boards and other equipments related to the experimental setup (internal noise). A thermal noise of 300 K is used here. With a very simple hardware, the thermal noise contribution could even be more than 300 K. Much lower noise levels of a few 10 K can be achieved by dedicated hardware optimization.
The noise temperature can be related to the power received in the antenna by $P = k_{\mathrm{B}} T \delta\nu$, where $\delta\nu$ is the frequency interval within which the power is extracted and $k_{\mathrm{B}}$ is the Boltzmann constant. From Figure \[fig:Cane\], we see that the Galactic noise diminishes as the frequency increases. At frequencies above $\approx$ 150 MHz, we become mainly limited by the thermal noise.
The expected noise for a given frequency band can be expressed as time traces from the predicted noise temperature within this band (See appendix \[sec:noisetrace\]). Noise traces extracted like this can be compared to signals from air showers as shown in Figure \[fig:traces\]. Here, the signals considered are those for a gamma-ray shower of 10 PeV energy and inclined at an angle of $61^\circ$.
It is clearly seen that the signal-to-noise ratio increases as we move to higher frequencies, as is expected from the behavior of the Galactic noise. Here, the signal-to-noise-ratio is determined as $\mathrm{SNR} = S^2/N^2$ where $S$ is the maximum of the Hilbert envelope over the signal and $N$ is the RMS noise in the specified frequency band. The suppression of the Galactic noise beyond 150 MHz is visible in the time traces of the noise. It becomes obvious that moving on to higher frequencies will enable us to have a higher level of signal-to-noise ratio (SNR), provided the antenna falls within the footprint of the shower.
Optimizing the observing frequency band
=======================================
Although it is clear from Figure \[fig:traces\] that using frequency bands that are higher than the standard band (30-80 MHz) will help us in enhancing the signal-to-noise ratio, the exact band that should be used for maximizing the chances of observation is still unclear. It is of course, possible to measure in wide band frequencies, and then to digitally filter into the required frequency range. But this will increase the cost of the experiment considerably, since the usage of higher frequencies require a greater sampling rate and hence better communication facility, memory, ADC, etc. Thus, a detailed study is made to estimate the frequency range that will give a maximum signal-to-noise ratio (and thereby maximize the detection probability), and can hence be used for the experiment, which is the focus in the following section.
A close inspection of the shower footprint at higher frequencies reveals that there are three regions of interest: on the Cherenkov ring, inside the Cherenkov ring, and outside the Cherenkov ring. It is desirable to have a high value of SNR in all of these regions for maximizing the probability of detection in the entire antenna array.
[.99]{}
![SNR seen in a typical antenna inside, on and outside the Cherenkov ring respectively, at various frequency bands, for one typical shower induced by a 10 PeV gamma-ray primary with zenith angle $= 61^\circ$ and $\alpha = 79^\circ$. \[fig:BWscan\]](stn45_resonance150MHz_insidecher_pwrscale.eps){width="8.1cm"}
[.99]{} ![SNR seen in a typical antenna inside, on and outside the Cherenkov ring respectively, at various frequency bands, for one typical shower induced by a 10 PeV gamma-ray primary with zenith angle $= 61^\circ$ and $\alpha = 79^\circ$. \[fig:BWscan\]](stn54_resonance150MHz_atcher_pwrscale.eps "fig:"){width="8.1cm"}
[.99]{} ![SNR seen in a typical antenna inside, on and outside the Cherenkov ring respectively, at various frequency bands, for one typical shower induced by a 10 PeV gamma-ray primary with zenith angle $= 61^\circ$ and $\alpha = 79^\circ$. \[fig:BWscan\]](stn33_resonance150MHz_outsidecher_pwrscale.eps "fig:"){width="8.1cm"}
![SNR map of a 10 PeV gamma-ray shower ($\theta = 61^{\circ}$) at 100-190 MHz. The black dots represent the 81 antenna positions. The antenna on the Cherenkov ring used for the frequency band scan is shown by the star shape. The square shape represents the antenna outside the Cherenkov ring and the pentagon that inside the Cherenkov ring.[]{data-label="fig:SNRmap"}](dense_array_SNR_interp_100-190MHznew_markedants.eps){width="1.05\columnwidth"}
A scan of the possible frequency bands that can be used for the measurement of air showers of energy 10 PeV is made. That is, we can construct a heat map of the SNR in different frequency bands. The frequencies for the heat map range from 30 MHz to 150 MHz for the lower edge of the frequency band and from 80 MHz to 350 MHz for the upper edge of the band. Such a scan is made for antenna stations at each region mentioned above. This is shown in Figure \[fig:BWscan\] for a typical gamma-ray shower with a zenith angle of $61^\circ$ and an energy of 10 PeV.
It is obvious that the typical frequency band of 30-80 MHz (lower left bins in Figure \[fig:BWscan\]) is not ideal for obtaining an optimal level of SNR. In the figure, the brightest zone for each region on the shower footprint shows the ideal frequency band, where a maximum range of SNR is obtained. Taking measurements at frequencies like 100-190 MHz gives a higher SNR. All bands where a value of SNR less than 10 is obtained are set to the color white, since this is the typical threshold for detection in an individual antenna station [@Aab:2015vta]. The bands with high SNR become especially crucial, when the energy threshold is attempted to be lowered. A map of the SNR that is measurable by the antennas is shown in Figure \[fig:SNRmap\]. The black dots represent the 81 antennas considered in the simulations. The antennas considered for the frequency band scan in Figure \[fig:BWscan\] are also marked here. The SNR map shown in the figure is obtained for the frequency band 100-190 MHz[^1].
Showers of other zenith angles and other primaries also show a similar behavior in the frequency band scan. As the zenith angle and the primary type changes, there is a variation in the scaling of SNR. This results from the change in the total electromagnetic content (for different primary type) and the different spread of the signal strength on the ground (for different zenith angle). There is a direct relation between the spread in the diameter of the Cherenkov ring and the inclination of the shower. Thus, the frequency bands with a higher value of SNR are the same for showers of other primaries and other zenith angles as that for a shower of zenith angle $61^\circ$ (shown in Figure \[fig:BWscan\]).
The observed signal-to-noise ratio in the antennas will depend on the energy of the shower, the zenith angle, and the azimuth angle (resulting in varying values of the Geomagnetic angle). The study of SNR in these parameter spaces is described in the following sections. The variation of the SNR with respect to the changing position of the shower maximum is not taken into account over here.
[.99]{}
![Zenith angle dependence of the SNR for showers produced by 10 PeV primary gamma ray with $\phi$ = 0. Each bin contains a typical shower for the respective zenith angle. At $\theta$ = $70^\circ$ the shower illuminates almost the entire array (b,c). \[fig:zenith\]](gamma30-80MHz_samedist.eps){width="1\linewidth"}
[.99]{} ![Zenith angle dependence of the SNR for showers produced by 10 PeV primary gamma ray with $\phi$ = 0. Each bin contains a typical shower for the respective zenith angle. At $\theta$ = $70^\circ$ the shower illuminates almost the entire array (b,c). \[fig:zenith\]](gamma50-350MHz_samedist.eps "fig:"){width="1\linewidth"}
[.99]{} ![Zenith angle dependence of the SNR for showers produced by 10 PeV primary gamma ray with $\phi$ = 0. Each bin contains a typical shower for the respective zenith angle. At $\theta$ = $70^\circ$ the shower illuminates almost the entire array (b,c). \[fig:zenith\]](gamma100-190MHz_samedist.eps "fig:"){width="1\linewidth"}
{width="1\linewidth"}
Dependence on the zenith angle {#sec:zenith}
------------------------------
The evolution of the SNR with the zenith angle can be looked at for different frequency bands. This evolution is looked at for antenna stations at various perpendicular distances to the shower axis (which is equivalent to the radial distance of the antennas to the shower axis in the shower plane). Such an evolution is shown for zenith angles ranging from $0^\circ$ to $70^\circ$ in Figure \[fig:zenith\], for the bands 30-80 MHz, 100-190 MHz and 50-350 MHz.
For the standard band of 30-80 MHz, the signal-to-noise ratio is significantly lower than that for the bands 50-350 MHz and 100-190 MHz. Among all the three bands, the highest level of signal-to-noise ratio is obtained for 100-190 MHz for all zenith angles, as expected. In particular, for showers of greater inclination, a higher signal-to-noise ratio is achieved in most of the antennas if we use the higher frequency bands. The areas where the Cherenkov ring falls on the antennas are visible for the higher frequencies. These are the really bright regions seen for each zenith angle and appears only for the more inclined showers.
At lower zenith angles, a major part of the shower is lost because of clipping effects. The high observation level at the South Pole is the reason for the showers getting clipped off. The distance to the shower maximum at these zenith angles is about a few kilometers, while that for showers of $70^\circ$ inclination is in the order of tens of kilometers. The clipping of the shower at lower zenith angles causes the radio emission to be underdeveloped for detection. This is also the reason for the appearance of the Cherenkov ring only for zenith angles $\gtrsim 30^\circ$.
In Figure \[fig:zenith\], the distances of the antennas from the shower axis fall within the range of 50 m to approximately 520 m, but only the antennas with a SNR $>10$ can detect these showers. For vertical showers, these are the antennas with distances of $\approx$ 100 meters and for inclined showers, these are the antennas that are even as far away as 500 m. This range corresponds to the required minimum spacing to detect these showers. That is, for vertical showers the antennas could at most have a spacing of 100 m and for inclined showers with $\theta \gtrsim 60^\circ$ a spacing of 300 m is sufficient to achieve a threshold of 10 PeV. It is a known feature that the farther the shower maximum is from the observation level, the greater is the radius of the Cherenkov ring. This is purely due to geometric effects of shower propagation. The propagation of the Cherenkov ring signature in the figure as the zenith angle increases is a manifestation of this. For an observation level of 2838 m above sea level, the average distance at which the Cherenkov ring falls is $\mathrm{d_{Ch}} \approx 250$ m for a shower of zenith angle $70^\circ$ and is $\mathrm{d_{Ch}} \approx 150$ m for a shower of zenith angle $60^\circ$.
The total energy fluence of the radio signal at the ground increases up to the zenith angle where clipping effects are no longer observed. On an average, it was seen that for 10 PeV gamma-ray showers, the total radiated energy does not get clipped-off for zenith angles greater than $50^\circ$. For zenith angles greater than this, the total energy in the radio footprint remains nearly the same, but the area increases. This results in a lower power per unit area on the ground, causing a decrease in the SNR. The relatively lower signal-to-noise ratio for the $70^{\circ}$ shower in Figure \[fig:zenith\] as compared to the $60^{\circ}$ shower is an effect of this.
[.45]{} {width="1\linewidth"}
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[.45]{} {width="1\linewidth"}
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Dependence on the azimuth angle {#sec:azimuth}
-------------------------------
Another parameter that the signal-to-noise ratio depends on is the azimuth angle of the shower. Variations in the azimuth angle result in changes in the geomagnetic angle. As a shower of zenith angle $61^{\circ}$ covers a range of azimuth angles from -180 to 180 degrees, the geomagnetic angle (at the South Pole, where the magnetic field is inclined to the vertical direction by $18^{\circ}$) varies from $43^{\circ}$ to $79^{\circ}$. This leads to an amplitude variation by a factor of $\frac{\mathrm{sin(43^{\circ})}}{\mathrm{sin(79^{\circ})}} = 0.7$. We find that for gamma-ray showers with these range of orientations and with an energy of 10 PeV, the maximum value of the SNR varied with a standard deviation of $\mathrm{\sigma_{SNR}} = 264$ with a mean value 1518. That is, with changing the azimuth angle there is a variation in the maximum value of the SNR by 17.4$\%$ about the mean. Apart from this, there is also a variation of the amplitude at a fixed azimuth angle due to shower-to-shower fluctuations which comes to 3.7$\%$ on an average. This will be further discussed in section \[sec:energy\].
We can thus infer that for inclined air showers at the South Pole, there is not a strong variation of the signal-to-noise ratio as the azimuth angle varies. This is shown in Figure \[fig:azimuth\]. Here, gamma-ray showers each with an energy of 10 PeV and an inclination of $61^{\circ}$ and with varying azimuth angles are shown. Thus it is justified to study the other effects only at one particular azimuth angle.
Dependence on the primary energy {#sec:energy}
--------------------------------
![The evolution of maximum SNR and maximum amplitude for gamma-ray showers with $\alpha$ = $79^{\circ}$ and $\theta$ = $61^{\circ}$ in the frequency band 100-190 MHz. The points represent the mean value and the standard deviation arising due to shower-to-shower fluctuations. The best fit to both set of simulated data points are shown, which has an $E^2$ nature for the maximum SNR and $E^1$ nature for the maximum amplitude. []{data-label="fig:evolution"}](SNRandAmpevolutionlabel.eps){width="1\linewidth"}
The signal that is observed by the antennas will obviously depend on the energy of the primary particle. The SNR becomes weaker as the energy of the primary particle decreases. The signal-to-noise ratio of showers with gamma-ray and proton primaries with energies ranging from 1 PeV to 9 PeV, are shown in Figure \[fig:energy\]. These are showers with zenith angles of $61^{\circ}$, $40^{\circ}$ and $70^{\circ}$, and are filtered to the band 100-190 MHz.
If we use the optimal frequency band like 100-190 MHz, we will be able to lower the threshold of detection down to 1 PeV for gamma-ray showers which have a zenith angle of $61^{\circ}$. For detection, it is required that a minimum of three antennas have a SNR above 10. For $61^{\circ}$ showers, we can achieve this, provided we have at least three antennas within a distance of $\sim 50-180$ m from the shower axis. This is mainly the area where the Cherenkov ring falls on the antenna array that gives a higher level of SNR. For proton showers of $61^{\circ}$ inclination, it is possible to lower the energy threshold to the level of 2 PeV in the band 100-190 MHz.
In a similar manner, the energy threshold can be lowered for showers with zenith angles $40^{\circ}$ and $70^{\circ}$ as shown in Figure \[fig:energy\]. For showers with $\theta = 40^{\circ}$, we need at least three antennas within a distance of $\sim 80$ m from the shower axis. This means that a much denser array is needed in this case. In the case of the $70^{\circ}$ showers, the minimum energy that can be detected is 2 PeV and is nearly independent of how dense the array spacing is.
The showers shown in Figure \[fig:energy\] are sample showers in these energy ranges. They will also have shower-to-shower fluctuations, because of which the amplitude detected in each antenna station will differ. Taking such fluctuations into account, gamma-ray induced air showers with zenith angles of $61^{\circ}$ and azimuth angles of $0^{\circ}$ were simulated with 11 simulations at each energy. Figure \[fig:evolution\] shows the fluctuations in the maximum SNR and the maximum amplitude for these gamma-ray showers with energies ranging from 1-9 PeV. This is shown in the figure for a frequency range of 100-190 MHz. These showers were seen to have an average relative standard deviation of the maximum SNR of 7.6$\%$ for all energies. Similarly, a 3.7$\%$ variation in the maximum amplitude is obtained.
It is seen that there is a clear correlation between the maximum SNR (or maximum amplitude) obtained and the energy of the primary particle. The maximum SNR was seen to be proportional to $E^2$ and the maximum amplitude $\propto E$. A fit of $\mathrm{SNR_{max}}$ = (17.04 $\pm 0.43$) $\times$ $E^{2.03 \pm 0.02}$) was obtained. Similarly, the maximum amplitude was seen to be related to the energy as $\mathrm{Amp_{max}}$ = (8.04 $\pm 0.10$) $\times$ $E^{1.01 \pm 0.01}$)
Detection of air showers using the radio technique in the PeV energy range is something that has not been achieved so far. This study shows that such a detection is possible if the measurement is taken in the optimum frequency range, e.g. 100-190 MHz. This means that by using this frequency range, for radio air shower detectors, it is possible to search for gamma rays of PeV energy arriving at the South Pole, from the Galactic Center.
Such a method can indeed be used at other locations on the Earth, for increasing the probability of detection of lower energy air shower events. However, the Galactic Center may not be visible at all times. The exact threshold for detection may vary depending on the observation level, the magnetic field at these locations and the dimensions of the antenna array. The noise conditions of these areas will also affect the measurement. The use of the optimum frequencies will nevertheless increase the detection rate of inclined air showers and will lower the energy threshold. In addition, by using interferometric methods, the very conservative condition of SNR $>$ 10 in 3 antennas can certainly be achieved.
![Efficiency of detection at different energies. An SNR $>$ 10 in at least 3 antennas is applied as the condition for detection. This is tested for 100 $\gamma$-ray showers with $\theta = 61^\circ$ in each energy bin in the frequency band 100-190 MHz. \[fig:efficiency\]](efficiency_diffen.eps){width="1\linewidth"}
Efficiency of detection
=======================
The results quoted in section \[sec:energy\] will depend on fluctuations between different showers of PeV energy, that arrive at the antenna array with different azimuth angles and different core positions. These factors, along with shower-to-shower fluctuations, will affect the rate of detection of air showers produced by PeV gamma rays.
To have an estimate of this, 170 simulations of gamma-ray induced showers with an energy of 1 PeV and zenith angle of $61^{\circ}$ were performed. These simulations had random azimuth angles and random core positions. Out of these showers, those with their core positions lying within a radius $\approx$ 564 m (corresponding to an area of 1 km$^2$) from the center of the array were chosen. This reduced the sample size to 140 events. If more than three antennas in the array have SNR $> 10$, the shower is detected. Upon conducting this test it was seen that these showers were detected with an efficiency of 47$\%$.
To have a better estimate of the energy where an efficiency of 100$\%$ is reached and the energy where the efficiency goes down to 0$\%$, simulations were done from 0.6 PeV to 1.8 PeV with 100 simulations in each energy bin, and with a bin width of 0.1 PeV. It was seen that an efficiency of 100$\%$ is reached for an energy of 1.4 PeV and the efficiency goes down to 0$\%$ below 0.7 PeV. The efficiency curve for the simulated showers is shown in Figure \[fig:efficiency\].
Discussion
==========
As discussed in section \[sec:energy\] it is possible to lower the energy threshold for radio detection by using an optimum frequency band. In the case of gamma-ray showers with a zenith angle of $61^{\circ}$, it is possible to lower the threshold down to 1 PeV for the frequency band of 100-190 MHz, if we have an antenna array with an average spacing of 125 m at the South Pole. At other experimental locations, the threshold may vary depending on the specific environment of the region. This method can be used not only for the specific purpose of PeVatron detection, but also for improving our current understanding of air showers, e.g. the study of mass composition at energies starting from the PeV range. The results presented here may vary depending on the exact noise that is present at the site of the experimental setup. On comparing with other available sky maps, the noise model by Cane predicts a level of noise that is slightly lower. For example, at a frequency of 110 MHz, the Cane model is seen to show around 15 $\%$ less amplitude in the level of noise than that of other noise models like LFmap. This will introduce second order fluctuations in the SNR and has been neglected here. A more detailed study should also take these fluctuations into account. There will also be fluctuations depending on the local sidereal time. A thermal noise level of 300 K is considered in this study. Today, antennas with much lower system noise are available; e.g. the SKA-LOW prototype antenna, SKALA, has a system noise of about 40 K only [@2015ExA....39..567D]. The uncertainty arising from CoREAS can be estimated from the experimental tests made on CoREAS so far. Different air shower experiments determined CoREAS to be accurate on an absolute scale to better than 20$\%$ at frequencies up to 80 MHz [@LOPES:2015eya][@Apel:2016gws]. This means that the uncertainty in the threshold due to the use of CoREAS is likely smaller than 20$\%$.
The detection potential of such an antenna array will also depend on the triggering capability. Triggers provided to the antennas by the IceTop array will not be fully efficient for PeV gamma-ray showers that are inclined with a zenith angle of $61^{\circ}$. Triggering is possible only if a particle from the air shower hits one of the IceTop tanks. The triggering capabilities of the future scintillator array is not yet studied in detail. Alternatively, if self-triggering of the antenna array is used, the energy threshold will rise depending on the broad-band radio interferences at the experimental site. The ARIANNA experiment has demonstrated that the conditions at Antarctica can be excellent for self-triggering [@Barwick:2016mxm].
A major challenge for the detection of these gamma rays is the background cosmic-ray flux which will be much larger than the gamma-ray flux. At energies above 0.8 PeV, a maximum of 8 gamma-ray events can be expected from the Galactic Center in one year, for a radio array with an area of 1 km$^2$. Out of these, 5 events will be above 1.4 PeV, where we have a full efficiency of detection, assuming that events considered to be detectable will also be triggered. In order to distinguish these gamma-ray events, in a point-source scenario, from the background cosmic-ray events an angular resolution of 0.1$^\circ$ or better, and a minimum gamma-hadron separation factor of 10 is required for a detection within a 5$\sigma$ confidence level in 3 years.
Conclusions
===========
We have performed a simulation study for the detection of air showers produced by primary gamma rays of PeV energy. The focus is on showers of zenith angle $61^{\circ}$, since this is the direction from which PeV gamma rays will approach the IceCube Observatory from the Galactic Center. In order to find the best measurement parameters, CoREAS simulations have been done, assuming an antenna array at the positions of the IceTop stations.
The signal-to-noise ratio received at these antennas has mainly been focused on in this analysis. A scan of the possible frequency bands within which the experiment can operate shows that there is a range of frequencies within which the SNR is at the optimum level. One of these frequency bands, namely 100-190 MHz, has been used here for studying other shower dependencies. This is the first study that shows that moving to this frequency range will help in the detection of inclined air showers. It will even help in lowering the energy threshold for gamma-ray showers with a zenith angle of $61^{\circ}$ down to $\approx$ 1 PeV at the South Pole.
For a hybrid array of 1 km$^2$ area and an average antenna spacing of 125 m, with an operating frequency band of 100-190 MHz, $61^{\circ}$ gamma-ray showers of 1 PeV can be detected with an efficiency of 47$\%$. This number was determined for showers whose cores fall within a circular region around the array center covering an area of 1 km$^2$. An antenna array with an average spacing of 125 m has been used in this case. We can reach a full efficiency above 1.4 PeV, and have a non-zero rate of detection above 0.8 PeV. Due to the simplifications in the simulation studies, the experimentally achievable thresholds may vary by a few tens of percents. They could even be lowered further with the usage of sophisticated hardware or by using interferometric detection techniques. Using even the simple radio setup considered in this paper at the IceCube location will give us a chance for detecting PeV gamma rays from the Galactic Center.
We would like to acknowledge the support of the IceCube collaboration and the members of the IceCube-Gen2 group at KIT and DESY-Zeuthen. We would also like to thank Anna Nelles and Larissa Paul for their very useful inputs.
Estimation of the number of gamma-ray events with PeV energies {#sec:flux}
==============================================================
It is essential to have an estimate of the number of gamma-ray events of PeV energy expected to approach the detector from the direction of the Galactic Center. For this, we have to extrapolate the flux of TeV gammas that has been observed by H.E.S.S.. A simple extrapolation to PeV energies, without any cut-off (which is preferred by the H.E.S.S. data points), is shown in Figure \[fig:extrapolation\]. A spectrum with cut-off at energies like 1 PeV, 10 PeV or 100 PeV is also possible. Here we mainly consider the best-case scenario, that is a spectrum without any cut-off. A spectrum of $\frac{dN}{dE}\propto E^{-2.32}$ is used for the extrapolation, which is the best fit to the H.E.S.S. data points.
![Flux as seen by H.E.S.S. [@Abramowski:2016mir], with a simple $E^{-2.32}$ extrapolation. The gamma-ray flux will get attenuated due to interactions with the CMB. The detectable flux takes the detection efficiency of the radio array into account. \[fig:extrapolation\]](flux_atten_eff_PeVscale.eps){width="1\linewidth"}
.
The extrapolated flux will also get attenuated due to the CMB [@Vernetto:2016alq]. This leads to a survival probability of the gamma rays from the extrapolated flux given by $\approx \frac{1}{e^{L_{\mathrm{dis}}/L_{\mathrm{atten}}}}$, where $L_{\mathrm{dis}}$ is the distance traveled by the gamma rays (here it is the distance between the Earth and the Galactic Center $\approx$ 8.5 kpc) and $L_{\mathrm{atten}}$ is the attenuation lengths of the gamma rays at different energies. The resulting spectrum after attenuation is also shown in Figure \[fig:extrapolation\]. Finally, the efficiency of detection of the radio array at various energies is also taken into consideration. The resulting flux that will be seen by the antenna array is shown by the black curve in the figure. From this method of extrapolation, an estimate of the expected number of events above PeV energies, detected in a year in a array of area 1 km$^2$, has been derived. Since the Galactic Center lies at an inclination of $61^\circ$ at the South Pole, the area of coverage of the array has to be weighted by a geometry factor of cosine($61^\circ$).
The expected number of events above 0.8 PeV, where we have a non-zero efficiency of detection, and above 1.4 PeV, where we have a full efficiency, are shown in Table \[table:flux\]. Here, we have made the assumption that the events that can be detected by the radio array will also be triggered. The number of events before and after folding through the detection efficiency are shown in the table. These numbers are evaluated using the attenuated flux of gamma rays.
[c|c|c]{} &
----------------------------------
$N\mathrm{_{events}}$($>$ $E_0$)
(1 yr)
----------------------------------
: Estimated number of events per year obtained from the extrapolation of the attenuated gamma-ray flux, with and without detector efficiency limits. \[table:flux\]
&
----------------------------------
$N\mathrm{_{events}}$($>$ $E_0$)
$\times$ efficiency
----------------------------------
: Estimated number of events per year obtained from the extrapolation of the attenuated gamma-ray flux, with and without detector efficiency limits. \[table:flux\]
\
--------------------
$E_0$ = 0.8 PeV
(efficiency $> 0$)
--------------------
: Estimated number of events per year obtained from the extrapolation of the attenuated gamma-ray flux, with and without detector efficiency limits. \[table:flux\]
& 11.5 & 7.9\
-------------------
$E_0$ = 1.4 PeV
(full efficiency)
-------------------
: Estimated number of events per year obtained from the extrapolation of the attenuated gamma-ray flux, with and without detector efficiency limits. \[table:flux\]
& 5.1 & 5.1
To obtain an estimate of the required gamma-hadron separation factor, we compared the expected number of gamma rays in 3 years with the number of cosmic rays that IceTop can see in 3 years within a region of the sky with a diameter of 0.1$^\circ$. Above 0.8 PeV, the expected number of cosmic rays is $\approx$ 289.3 and gamma rays is $\approx$ 23.8. This means that for a detection within a confidence level of 5$\sigma$, we will need a separation factor of $\approx$ 12.7 On the other hand, if we consider the number of events above 1.4 PeV (gamma rays $\approx$ 15.4 and cosmic rays $\approx$ 99.5), we require a separation factor of $\approx$ 10.5. The gamma-hadron separation can be done by using the information of the shower maximum or by using the different muon content of showers from these primaries. The optimization of this requires a separate, deeper study.
Generating a noise trace {#sec:noisetrace}
========================
A model for the Galactic noise developed by Cane [@Cane] is used for the following discussion. The Galactic noise is provided in units of brightness ($ B(\nu)$) and is expressed as a function of frequency. Assuming the source of Galactic noise to be a blackbody and hence using the Rayleigh-Jeans law, we can relate the brightness to its brightness temperature.
$$B(\nu) = 2k_{\mathrm{B}}T\frac{\nu^{2}}{c^{2}} \hspace{7mm} [ \text{W}\text{m}^{-2} \text{sr}^{-1} \text{Hz}^{-1}]$$
where $\mathrm{k_B}$ is the Boltzmann’s constant and T is the brightness temperature.\
We can add thermal noise due to the electronics of the receiving system to the brightness temperature in order to obtain the total noise temperature. $$T_{ \mathrm{tot}} = T_{ \mathrm{brightness}} + T_{ \mathrm{thermal}}$$
The electromagnetic power of the noise obtained in the frequency band $\delta \nu$ from solid angle d$\Omega$ by an antenna of effective area of $A_{ \mathrm{eff}}$ is, $$P_{\nu}(\theta,\phi) = \frac{1}{2} B(\nu) \mathrm{d}\Omega \;A_{ \mathrm{eff}}(\theta,\phi)\; \delta \nu$$ Here, a factor of 1/2 has to be added in order to account for the fact that the antenna can extract power only from one of the polarizations of the incoming electromagnetic wave.\
The Poynting flux per unit frequency is obtained by integrating the brightness over the solid angle. $$\begin{aligned}
S =& \int B(\nu) \mathrm{d}\Omega \hspace{7mm} [ \text{W}\text{m}^{-2} \text{Hz}^{-1}] \\
=& \;\frac{2k_\mathrm{B}\nu^2}{c^2} \int T(\theta, \phi) \mathrm{d}\Omega
\end{aligned}$$ The Poynting flux within the frequency interval of $\delta \nu$ is then, $$\begin{aligned}
S_{\nu} =& \;S \delta \nu \hspace{7mm} [ \text{W}\text{m}^{-2} ]\\
=& \;\frac{2k_\mathrm{B} \nu^2 \delta \nu}{c^2} \int T(\theta, \phi) \mathrm{d}\Omega
\end{aligned}$$ Again, the Poynting flux extracted at the antenna is $S_{\mathrm{rec}}=\frac{S_{\nu}}{2}$ for reasons of polarization matching.\
We can relate the Poynting flux to the electric field delivered to the antenna as, $$|\overrightarrow{S}| = \frac{1}{2nZ_0}|\overrightarrow{E}|^2$$ where $Z_0 = 376.7303$ Ohm is the vacuum impedance.\
Taking the refractive index of air to be 1, the amplitude of the electric field at the antenna because of the Galactic noise can then be obtained from the Poynting flux as, $$\begin{aligned}
|\overrightarrow{E}|=& \sqrt{S_{\mathrm{rec}}2Z_0} \hspace{7mm} [ \text{V/m}]\\
=& \sqrt{\frac{1}{2}2Z_0\frac{2k_\mathrm{B}\nu^2 \delta \nu}{c^2} \int T(\theta, \phi) \; \mathrm{d}\Omega}
\end{aligned}$$ Thus, the voltage developed at the antenna is $$V(\nu) = \overrightarrow{E}\cdotp \overrightarrow{l_{\mathrm{eff}}} \hspace{7mm} [ \text{V} ]$$
Since we have already taken into account that the polarization should match, we can multiply the modulus of the field and the modulus of the antenna height to obtain the voltage. Of course, this simplification cannot be done for a noise model with directional dependence. The received voltage is now given by
$$ V(\nu) = \sqrt{2Z_0\frac{\mathrm{k_{B}} \nu^2 \delta \nu}{c^2} \int T(\theta,\phi)\; |\overrightarrow{l_{\mathrm{eff}}}(\theta, \phi)|^2 \; \mathrm{d}\Omega}
$$
Since the model used has the temperature to be independent of $\theta$ and $\phi$, $T(\theta,\phi)=T$ can be taken out of the integral.\
The amplitude extracted from the model has no phase information of the incoming noise. We can add random phases to the amplitude since noise indeed behaves randomly. $$\begin{aligned}
V(\nu) = V(\nu) * \mathrm{exp}(-i\varphi)
\end{aligned}$$ $\varphi$ is a random number that is generated between 0 and 2$\pi$.
Finally, we can convert the amplitude to the time domain using Inverse Fourier Transform:
$$V(\nu)\longrightarrow V(t)$$
Of course, this is only an average behavior of the noise. One can also assume variations in the extracted amplitude about this average noise.
[^1]: This was produced by running a CoREAS simulation in parallel mode with 3750 antennas using the hadronic interaction model UrQMD instead of FLUKA
|
---
abstract: 'Compressed sensing provided a data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sampling rate. In many real-world applications, a signal of interest is typically sparse not in the canonical basis but in a certain transform domain, such as wavelets or the finite difference. The theory of compressed sensing was extended to the analysis sparsity model but known extensions are limited to specific choices of sensing matrix and sparsifying transform. In this paper, we propose a unified theory for robust recovery of sparse signals in a general transform domain by convex programming. In particular, our results apply to general acquisition and sparsity models and show how the number of measurements for recovery depends on properties of measurement and sparsifying transforms. Moreover, we also provide extensions of our results to the scenarios where the atoms in the transform has varying incoherence parameters and the unknown signal exhibits a structured sparsity pattern. In particular, for the partial Fourier recovery of sparse signals over a circulant transform, our main results suggest a uniformly random sampling. Numerical results demonstrate that the variable density random sampling by our main results provides superior recovery performance over known sampling strategies.'
author:
- |
Kiryung Lee, Yanjun Li , Kyong Hwan Jin,\
and Jong Chul Ye, [^1]
title: |
Unified Theory for Recovery of Sparse Signals\
in a General Transform Domain
---
Compressed sensing, analysis sparsity model, sparsifying transform, total variation, incoherence, variable density sampling.
Introduction {#sec:intro}
============
The theory of compressed sensing (CS) [@donoho2006compressed; @candes2006robust] provided a new data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms are guaranteed to reconstruct the unknown sparse signal from the linear measurements taken at a provably near optimal rate. Reconstruction algorithms with performance guarantees include modern optimization algorithms for $\ell_1$-norm-based convex optimization formulations (e.g., [@beck2009fast; @boyd2011distributed]) and iterative greedy algorithms (e.g., [@needell2009cosamp; @dai2009subspace; @blumensath2009iterative; @foucart2011hard]).
The canonical sparsity model in CS assumes that the unknown signal $f \in \cz^d$ is $s$-sparse in the standard coordinate basis. In other words, $\norm{f}_0 \leq s$, where $\norm{\cdot}_0$ counts the number of nonzero elements. The acquisition process in CS is linear and represented by a sensing matrix $A \in \cz^{m \times d}$ so that the $m$ linear measurements in $b \in \cz^m$ is given by $$b = A f + w,$$ where $w \in \cz^m$ denotes additive noise to the measurements and satisfies $\norm{w}_2 \leq \epsilon$. For certain random sensing matrices, it was shown that an estimate $\hat{f}$ given by $$\label{eq:bpdn}
\hat{f} = \argmin_{\tilde{f} \in \cz^d} \norm{\tilde{f}}_1 \quad \mathrm{subject~to} \quad \norm{b - A \tilde{f}}_2 \leq \epsilon$$ satisfies $\norm{\hat{f} - f}_2 \leq c_1 \epsilon$ with high probability, provided that $m \geq C s \log^\alpha d$ for some $\alpha \in \mathbb{N}$ and numerical constants $C$ and $c_1$. In particular, in the noiseless case ($\epsilon = 0$), the estimate $\hat{f}$ coincides with the ground truth signal $f$. For example, Candes and Tao [@candes2005decoding] showed that the above guarantees hold for a Gaussian sensing matrix $A$ whose entries are i.i.d. following $\mathcal{N}(0,1/m)$ via the restricted isometry property (RIP). Recent results with a sharper sample complexity of $m \geq 2 s \log(d/s)$ were derived using the Gaussian width of a tangent cone [@chandrasekaran2012convex; @amelunxen2014living].
In fact, the original idea of compressed sensing [@bresler1999image] was motivated by a need to accelerate various imaging modalities. The sensing matrix $A$ in these applications takes observations in a measurement transform domain, i.e. $$\label{eq:structred_sensing_mtx}
A = \sqrt{\frac{n}{m}} S_\Omega \Psi,$$ where $\Psi \in \cz^{n \times d}$ is the matrix representation of the measurement transform and the sampling operator $S_\Omega: \cz^n \to \cz^m$ takes the $m$ elements indexed by $\Omega = \{\omega_1,\omega_2,\ldots,\omega_m\}$. For example, when $\Psi$ is a discrete Fourier transform (DFT) matrix, $A$ is a partial Fourier matrix. The aforementioned near optimal performance guarantees were shown for a partial Fourier sensing matrix $A$ obtained using a random set $\Omega$ [@candes2006robust; @candes2006near; @rudelson2008sparse; @candes2011probabilistic] and generalized for the case where the rows of $\Psi \in \cz^{n \times d}$ correspond to an incoherent tight frame for $\cz^d$ [@rauhut2010compressive].
However, in numerous imaging applications, a signal of interest is not sparse in the standard coordinate basis. The theory of compressed sensing was accordingly extended to the so-called synthesis and analysis sparsity models [@candes2011compressed; @nam2013cosparse; @giryes2014greedy]. The synthesis sparsity model assumes that $f \in \cz^d$ is represented as a linear combination of few atoms in a dictionary $D \in \cz^{d \times N}$. Equivalently, $f$ is represented as $f = D u$ with an $s$-sparse coefficient vector $u \in \cz^N$. Compressed sensing with the synthesis sparsity model can be interpreted as conventional compressed sensing of $u$ using a sensing matrix $A D$ where $u$ is $s$-sparse in the standard basis. In particular when $A$ is a Gaussian matrix and $D$ is an orthogonal matrix, conventional performance guarantees carry over to the synthesis sparsity model. On the other hand, the analysis sparsity model, which is motivated from sparse representation in harmonic analysis [@mallat2008wavelet], assumes that the transform $\Phi f \in \cz^N$ of $f$ via $\Phi \in \cz^{N \times d}$ is $s$-sparse. In fact, a signal of interest in practical applications often follows the analysis sparsity model via various transforms including finite difference and wavelet [@mallat2008wavelet], contourlet [@do2005contourlet], curvelet [@starck2002curvelet], and Gabor transforms. Thus, the analysis sparsity model has been used as an effective regularizer for classical inverse problems in signal processing (e.g., denoising and deconvolution) and compressed sensing imaging (e.g., [@lustig2008compressed]).
Unlike the previous results with the canonical sparsity model, the theory of compressed sensing with the analysis model has been relatively less explored and known results are limited to specific cases. Candes et al. [@candes2011compressed] considered the recovery of $f \in \cz^d$ such that $\Phi f$ is $s$-sparse for a transform $\Phi \in \cz^{N \times d}$ satisfying $\Phi^* \Phi = I_d$, i.e. the columns of $\Phi^*$ correspond to a tight frame for $\cz^d$. They showed that $$\label{eq:bpdn_tfm}
\hat{f} = \argmin_{\tilde{f} \in \cz^d} \norm{\Phi \tilde{f}}_1 \quad \mathrm{subject~to} \quad \norm{A \tilde{f} - b}_2 \leq \epsilon$$ has the error bound given by $\norm{\hat{f} - f}_2 \leq c_1 \epsilon$, provided that the sensing matrix $A \Phi^*$ satisfies the RIP. In particular for a Gaussian sensing matrix $A$, their performance guarantee holds with high probability for $m = O(s \log(d/s))$ for any $\Phi$ satisfying $\Phi^* \Phi = I_d$. Indeed the performance guarantee by Candes et al. [@candes2011compressed] applies beyond the case of a Gaussian sensing matrix $A$. Krahmer and Ward [@krahmer2011new] showed that if $A \in \cz^{m \times d}$ and $D \in \cz^{d \times N}$ satisfy the RIP and $\varepsilon \in \mathbb{R}^d$ is a Rademacher sequence with random $\pm 1$ entries, then $A \diag(\varepsilon) D$ satisfies the RIP, where $\diag(\varepsilon) \in \cz^{d \times d}$ denotes the diagonal matrix whose diagonal entries are $\varepsilon$. However, applying a random sign before the acquisition might not be feasible in certain applications. In another line of research, it was shown [@nam2013cosparse; @giryes2014greedy] that greedy algorithms for the analysis sparsity model provide performance guarantees if the sensing matrix $A$ is a near isometry when acting on all transform-sparse $f$ such that $\Phi f$ is sparse. Again for $\Phi$ satisfying $\Phi^* \Phi = I_d$, this condition on $A$ is less demanding than the RIP of $A \Phi^*$ since the latter implies the former. However, it has not been studied how such a relaxation translates into less-demanding sample complexity.
A special analysis sparsity model associated with the finite difference transform $\Phi$ has been of particular interest in signal processing and imaging applications. The corresponding convex surrogate $\norm{\Phi f}_1$, known as the total variation (TV), has been popularly used as an effective regularizer for solving inverse problems. Needell and Ward [@needell2013stable] provided performance guarantees for TV minimization with a partial Fourier sensing matrix $A$ in terms of the RIP of $A D$ with a Haar wavelet dictionary $D$. Their result was further refined by Krahmer and Ward [@krahmer2014stable] with a clever idea of variable density sampling adopting the local incoherence parameters. Remarkably, Krahmer and Ward [@krahmer2014stable] provided performance guarantees at the sample complexity of $m = O(s \log^3 s \log^2 d)$. However, these results on TV minimization rely on a special embedding theorem that relates the Haar transform to the finite difference transform, which holds only for signals of dimension two or higher. Therefore, the results by Needell and Ward [@needell2013stable] and by Krahmer and Ward [@krahmer2014stable] do not apply to the 1D case and more importantly do not generalize to other sparsifying transforms.
Recently, inspired by analogous results for the canonical sparsity model [@chandrasekaran2012convex; @amelunxen2014living], Kabanava et al. [@kabanava2015robust] derived a performance guarantee of TV minimization with a Gaussian sensing matrix $A$ at the sample complexity of $m > d \,[1 - \{ 1-(s+1)/d \}^2/\pi]$. Cai and Xu [@cai2015guarantees] showed a similar result with $m \geq C \sqrt{s d} \log d$. Compared to the previous result by Krahmer and Ward [@krahmer2014stable], the above results [@kabanava2015robust; @cai2015guarantees] apply to the 1D case but at a significantly suboptimal sample complexity. More importantly, the use of a Gaussian sensing matrix might not be relevant to practical applications. Along a similar analysis strategy, Kabanava and Rauhut [@kabanava2015analysis] showed performance guarantees for with a Gaussian sensing matrix $A \in \mathbb{R}^{m \times d}$ and a frame analysis operator $\Phi \in \mathbb{R}^{N \times d}$ roughly at the sample complexity of $m \geq 2 \kappa s \log (2N/s)$, where $\kappa$ denotes the ratio of upper and lower frame bounds, i.e. $\kappa$ is the condition number of the frame operator $\Phi^* \Phi$. The sample complexity of this result for a tight frame is near optimal. However, their analysis is restricted to a Gaussian sensing matrix $A$ and does not generalize to other sensing matrices.
Contributions
-------------
Our main contribution is to derive performance guarantees for compressed sensing of analysis-sparse vectors by for general classes of sensing matrix $A$ given in the form of with measurement transform $\Psi$ and (redundant) analysis transform $\Phi$. Unlike the previous works, performance guarantees in this paper apply without being restricted to a particular choice of $\Phi$ and $\Psi$[^2]. The number of measurements implying these guarantees depends on certain properties of $(\Phi,\Psi)$ and this result identifies a class of measurement and sparsity models allowing recovery at a near optimal sampling rate. Moreover, when $\Psi$ and $\Phi$ have a few strongly correlated atoms, adopting the idea by Krahmer and Ward [@krahmer2014stable], we propose to acquire linear measurements using random sampling with respect to a variable density designed with the correlations between $\Psi$ and $\Phi$. This modified acquisition enables recovery at a lower sampling rate. We also extend the results to group sparsity models. This extension applies to various popular regularized recovery methods including the isotropic-total-variation minimization. In special cases where $\Psi$ is Fourier and $\Phi$ is a circulant matrix, our theory suggests a uniformly random sampling or its variation. For example, for the total variation minimization, unlike the common belief in practice, a sampling strategy that combines the acquisition of the lowest frequency and uniformly random sampling on the other frequencies provides better reconstruction than known variable density random sampling strategies.
Our main idea is inspired by a previous work on structured matrix completion by Chen and Chi [@chen2014robust]. They showed that a structured low-rank matrix (e.g., a low-rank Hankel matrix) is successfully recovered from partially observed entries by minimizing the nuclear norm. Indeed, their structured low-rank matrix completion can be interpreted as follows. The unknown structured matrix $M$ is given as the image $T x$ of the generator $x$ via a linear map $T$. Then the completion of the structured matrix $M$ with the low-rankness prior is equivalent to the completion of the generator $x$ with low-rankness in the transform domain via $T$. Similarly, compressed sensing with the analysis sparsity model is equivalent to the recovery of $x \in \cz^n$ with the prior that $T x \in \cz^N$ is $s$-sparse where the transform $T$ is given as $T = \Phi \Psi^\dagger$. Therefore the two problems are analogous to each other in the sense that their priors correspond to atomic sparsity [@chandrasekaran2012convex] in their respective transform domains. Moreover, in both the structured low-rank matrix model and the transform-domain sparsity model, $T$ is not necessarily surjective, which implies that $T T^*$ may be rank-deficient. This violates an important technical condition known as the *isotropy* property and many crucial steps in the proofs of existing performance guarantees for CS break down. To overcome this difficulty, we adopt the clever idea by Chen and Chi [@chen2014robust] through the aforementioned analogy and derive near optimal performance guarantees for the recovery of sparse signals in a transform domain without resorting to the isotropy property. However, besides this similarity, our results are significantly different from the analogous results [@chen2014robust] in the following sense. Chen and Chi [@chen2014robust] assumed that $T$ is restricted to a set of special linear operators that generate structured matrices and their analysis indeed relies critically on strong properties satisfied by such linear operators (e.g., $T$ needs to satisfy $T^* T = I_n$). Contrarily, we only assume a mild condition that $T$ is injective and the resulting performance guarantees apply to more general cases. For example, in compressed sensing with the analysis sparsity model, $T = \Phi \Psi^\dagger$ is non-unitary if the measurement transform $\Psi$ is non-unitary (e.g., the Radon transform) or the sparsifying transform $\Phi$ is non-unitary (e.g., biorthogonal wavelet and data-adaptive transforms).
We illustrate our theory through extensive Monte Carlo simulations. The variable density sampling in this paper provides an improved recovery performance over the previously suggested sampling strategies. For example, when $\Phi$ and $\Psi$ have atoms showing high correlation (e.g. Fourier and wavelet), our theory suggests to sample more densely in the lower frequencies, which enables successful recovery from fewer observations. Rather surprisingly, when $\Phi$ and $\Psi$ respectively correspond to the finite difference and the Fourier transforms, our theory suggests a special sampling density that always takes the lowest frequency and chooses the other frequency components randomly using the uniform density. This is contrary to the common belief in compressed sensing but the proposed sampling strategy turns out be more successful than previous works not only in theory but also empirically. These numerical results strongly support out new theory.
Organization
------------
The rest of this paper is organized as follows. The main results are presented in Sections \[sec:main\_result\] and \[sec:main\_result\_vd\], where the proofs are deferred to Sections \[sec:pf\_main\_result\_noiseless\] and \[sec:pf\_main\_result\_noisy\]. We extend the theory to the group sparsity models in Section \[sec:main\_result\_gs\] and study a special case of circulant transforms in Section \[sec:circulant\]. After demonstrating empirical observations in Section \[sec:numres\], which supports our main results, we conclude the paper with some final remarks in Section \[sec:concl\].
Notations
---------
For a positive integer $N$, we will use a shorthand notation $[N]$ for the set $\{1,\ldots,N\}$. For a vector $z \in \cz^N$, let $z[k]$ denote the $k$th element of $z$ for $k \in [N]$, i.e. $z = [z[1], z[2], \ldots, z[N]]^\transpose$. The Hadamard product of two vectors $x,y \in \cz^n$ is denoted by $x \odot y$. The circular convolution of two vectors $x,y \in \cz^n$ is denoted by $x \circledast y$. The Kronecker product of two matrices $A$ and $B$ is denoted by $A\otimes B$. The operator norm from $\ell_p^n$ to $\ell_q^N$ will be denoted by $\norm{\cdot}_{p \to q}$. For brevity, the spectral norm $\norm{\cdot}_{2 \to 2}$ will be written without subscript as $\norm{\cdot}$. For a matrix $A$, its Hermitian transpose and its Moore-Penrose pseudo inverse are respectively written as $A^*$ and $A^\dagger$.
For $J \subset [N]$, the coordinate projection with respect to $J$, denoted by $\Pi_J$, is defined as $$(\Pi_J z)[k] :=
\begin{dcases}
z[k] & k \in J, \\
0 & \mathrm{otherwise}.
\end{dcases}$$ The complex signum function, denoted by $\mbox{sgn}(\cdot): \cz^N \to \cz^N$, is defined as $$\label{eq:sgn}
(\mbox{sgn}(z))[k] :=
\begin{dcases}
\frac{z[k]}{|z[k]|} & z[k] \neq 0, \\
0 & \mathrm{otherwise}.
\end{dcases}$$ Let $e_1,\ldots,e_n$ denote the standard basis vectors for $\cz^n$. In other words, $e_k$ is the $k$th column of the $n$-by-$n$ identity matrix.
Recovery of Sparse Signals in a General Transform Domain {#sec:main_result}
========================================================
Let $T: \cz^n \to \cz^N$ be a linear map that we call a “transform” in this paper. Let $\Omega = \{\omega_1,\omega_2,\ldots,\omega_m\}$ denote the multi-set of $m$ sampling indices out of $[n] := \{1,\ldots,n\}$ with possible repetition of elements. Given $\Omega$, the sampling operator $S_\Omega: \cz^n \to \cz^m$ is defined so that the $j$th element of $S_\Omega x \in \cz^m$ is the $\omega_j$th element of $x \in \cz^n$ for $j = 1,\ldots,m$. We are interested in recovering an unknown signal $x \in \cz^n$ from its partial entries at $\Omega$ when the transform $T x$ is known $s$-sparse a priori, i.e. $\norm{T x}_0 \leq s$. Compressed sensing with the analysis sparsity model is an instance of this problem formulation as shown in the following. Recall that $\Psi$ is of full column rank. Let $T: \cz^n \to \cz^N$ be defined by $T = \Phi \Psi^\dagger$ where $\Psi^\dagger$ denotes the Moore-Penrose pseudo inverse of $\Psi$. Let $x \in \cz^n$ denote the vector containing fully sampled measurements, i.e. $x = \Psi f$. Since $f = \Psi^\dagger x$, we have $\Phi f = \Phi \Psi^\dagger x = T x$. Thus the recovery of $f$ from $b = \sqrt{n/m} S_\Omega \Psi f$ with the prior that $\Phi f$ is $s$-sparse is equivalent to the recovery of $x$ from $b = \sqrt{n/m} S_\Omega x$ with the prior that $T x$ is $s$-sparse.
In the noise-free scenario where partial entries of $x$ are observed exactly, we propose to estimate $x$ as the minimizer to the following optimization problem: $$\label{eq:ell1min}
\minimize_{g \in \cz^n} \, \norm{T g}_1 \quad \mathrm{subject~to} \quad S_\Omega g = S_\Omega x.$$ We provide a sufficient condition for recovering $x$ exactly by in the following theorem.
\[thm:uniqueness\] Suppose $T: \cz^n \to \cz^N$ is injective. Let $\gamma$ be defined by $$\label{eq:defgamma}
\gamma := \argmin_{\tilde{\gamma} > 0} \norm{\tilde{\gamma} T^* T - I_n},$$ where $\widetilde{T} = (T^\dagger)^*$. Let $\mu$ be given by $$\label{eq:incoherence}
\mu = \max_{k \in [n]} \max\left\{ n \norm{\gamma^{1/2} T e_k}_\infty^2, n \norm{\gamma^{-1/2} \widetilde{T} e_k}_\infty^2 \right\}.$$
Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Suppose $T x$ is $s$-sparse. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$m \geq \frac{C(1+\beta) \mu s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right].$$
See Section \[sec:pf\_main\_result\_noiseless\].
All the results in this section including Theorem \[thm:uniqueness\] as well as other similar results [@candes2011probabilistic; @gross2011recovering; @chen2014robust], all derived using the golfing scheme, provide an *instance* recovery guarantee that applies to a single arbitrary instance of $x$, which is a weaker result than the *uniform* recovery guarantee that applies to the set of all transform-sparse signals. In compressed sensing with the canonical sparsity model, an instance guarantee is given from a fewer measurements than the uniform guarantee [@candes2011probabilistic] by a poly-log factor. For matrix completion problems, the RIP do not hold and known results [@gross2011recovering; @chen2014robust] only provide an instance guarantee. We suspect that this is the case with the recovery problem in this paper.
Note that $\norm{T}_{1 \to 2}$ denotes the largest $\ell_2$-norm among all columns of $T$. Similarly, $\norm{T^\dagger}_{2 \to \infty} = \norm{\widetilde{T}}_{1 \to 2}$ denotes the largest $\ell_2$-norm among all columns of $\widetilde{T}$. In a special case when $T^* T = I_n$, we have $T^\dagger = T^*$. Thus $\norm{T}_{1 \to 2} = \norm{T^\dagger}_{2 \to \infty} = 1$. In general, if $\norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} = O(N^{\alpha_1})$ for some $\alpha_1 \in \mathbb{N}$ and $$\frac{1}{1 - \norm{\gamma T^* T - I_n}} = O(\log^{\alpha_2} N)$$ for some $\alpha_2 \in \mathbb{N}$, then we get a performance guarantee at a near optimal scaling of sample complexity of $m = O(\mu s \log^\alpha N)$, where $\alpha = \alpha_1 + \alpha_2$. These are mild conditions and easily satisfied by transforms that arise in practical applications.
In the noisy scenario where partial entries of $x$ are observed with additive noise, we propose to estimate $x$ by solving the following optimization problem: $$\label{eq:ell1min_noisy}
\minimize_{g \in \cz^n} \, \norm{T g}_1
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} g - S_{\Omega'} x^\sharp}_2 \leq \epsilon,$$ where $x^\sharp$ denotes a noisy version of $x$, $\Omega'$ denotes the set of all unique elements in $\Omega$, and $S_{\Omega'}^*$ is the adjoint of the sampling operator $S_{\Omega'}$ that fills missing entries at outside $\Omega'$ with 0.
\[thm:stability1\] Suppose the hypotheses of Theorem \[thm:uniqueness\] hold. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying $$\label{eq:xharp_fid}
\norm{S_{\Omega'} (x - x^\sharp)}_2 \leq \epsilon.$$ Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon,$$
See Section \[subsec:pf:thm:stability1\].
We can tighten the upper bound on the estimation error in Theorem \[thm:stability1\] when $T^* T$ is well conditioned. To this end, we will use the following theorem, which is obtained by modifying [@lee2013oblique Theorem 3.1].
\[thm:rboplike\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameters $\mu$ and $\gamma$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Then with probability $1-\xi$, we have $$\label{eq:rboplike}
\max_{|\widetilde{J}| \leq s} \left\| \Pi_{\widetilde{J}} \left(\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger - T T^\dagger \right) \Pi_{\widetilde{J}} \right\| \leq \delta,
%\max_{|\widetilde{J}| \leq s} \left\| \Pi_{\widetilde{J}} \left(\frac{n}{m} B_\Omega - B\right) \Pi_{\widetilde{J}} \right\| \leq \delta,$$ provided $$\begin{aligned}
\label{eq:rboplike:cond1}
m {} & \geq \frac{C_1 \delta^{-2} \mu s \log^2 s \log N \log m}{1 - \norm{\gamma T^* T - I_n}},
\intertext{and}
\label{eq:rboplike:cond2}
m {} & \geq C_2 \delta^{-2} \mu s \log (\xi^{-1}).\end{aligned}$$
See Appendix \[sec:pf:thm:rboplike\].
Using Theorem \[thm:rboplike\], we provide another sufficient condition for stable recovery of sparse signals in a transform domain, which has a smaller noise amplification factor.
\[thm:stability2\] Suppose $T: \cz^n \to \cz^N$ is injective. Suppose $T$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameters $\mu$ and $\gamma$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying . Then there exist numerical constants $C, c > 0$ for which the following holds. With probability $1 - N^{-4}$, we have $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left[ 14\sqrt{N} + \frac{n}{m} \left\{ \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) +1 \right\} \right] \epsilon,$$ provided $$\label{eq:noisy_samp_comp}
m \geq \frac{C \mu s \log^4 N}{1 - \norm{\gamma T^* T - I_n}}.$$ Furthermore, if $T^* T = I_n$, then with probability $1 - N^{-4}$, $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left( 14\sqrt{N} + \frac{n}{m} R \right) \epsilon,$$ where $R$ is the count of the most repeated elements in $\Omega$.
See Section \[subsec:pf:thm:stability2\].
By the definition of $\Omega'$, we have $|\Omega|-|\Omega'| \leq m-1$. Suppose that $\norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} = O(1)$. The resulting noise amplification factor by Theorem \[thm:stability2\] is $O(\sqrt{N} + n)$, which is already smaller than $O(\sqrt{N} n)$ by Theorem \[thm:stability1\]. In fact, the distribution of $|\Omega'|$ is explicitly given by $$\mathbb{P}\left( |\Omega'| = \ell \right) = \frac{\stirling{m}{\ell} n!}{n^m (n-\ell)!},$$ where $\stirling{m}{\ell}$ denotes the Stirling number of the second kind defined by $$\stirling{m}{\ell} := \frac{1}{\ell!} \sum_{j=0}^\ell(-1)^{\ell-j} {\ell \choose j} j^m.$$ It would be possible to compute a more tight probabilistic upper bound on $|\Omega|-|\Omega'|$ with its distribution. However, because of the other factor $\sqrt{N}$, regardless of $|\Omega|-|\Omega'|$, the noise amplification factor by Theorem \[thm:stability2\] cannot be improved to $O(1)$ as shown for compressed sensing with the canonical sparsity model [@candes2011probabilistic] or with a special analysis model [@krahmer2014stable]. We admit that this suboptimality in noise amplification is a limitation of our analysis. It will be interesting to see whether one can obtain near optimal noise amplification for a general transform $T$.
Incoherence-Dependent Variable Density Sampling {#sec:main_result_vd}
===============================================
In the results of the previous section, the number of measurements is proportional to the incoherence parameter $\mu$ in , which is the worst case $\ell_\infty$-norm among $\{T e_k\}_{k=1}^n$ and $\{\widetilde{T} e_k\}_{k=1}^n$. In certain scenarios, these $\ell_\infty$-norms are unevenly distributed. For example, in compressed sensing with the analysis sparsity model, $T$ is given by $T = \Phi \Psi^\dagger$ with the sensing transform $\Psi \in \cz^{n \times d}$ and the sparsifying transform $\Phi \in \cz^{N \times d}$. If $\Psi$ and $\Phi$ correspond to the DFT and DWT (discrete wavelet transform), respectively, low-frequency atoms have larger correlations. Thus there are a few $T e_k$s that dominate the others with large $\ell_\infty$-norms. Krahmer and Ward [@krahmer2014stable] proposed a clever idea of sampling measurements with respect to a variable density adapted to the local incoherence parameters, which are $\{\norm{T e_k}_\infty\}_{k=1}^n$ and $\{\norm{\widetilde{T} e_k}_\infty\}_{k=1}^n$.[^3] Then the sample complexity depends on not the worst case incoherence parameter but the average of the local incoherence parameters. In this section, adopting the idea by Krahmer and Ward [@krahmer2014stable], we extend the results in Section \[sec:main\_result\] to the case where the local incoherence parameters are unevenly distributed. The following theorem is analogous to Theorem \[thm:uniqueness\] and provides a sufficient condition for recovery of sparse signals in a transform domain when measurements are sampled according to a variable density.
\[thm:uniqueness\_vd\] Suppose $T: \cz^n \to \cz^N$ is injective. Let $\mu_k$ and $\tilde{\mu}_k$ be defined by $$\label{eq:loc_incoherenceT}
\mu_k = n \norm{\gamma^{1/2} T e_k}_\infty^2
\quad \mathrm{and} \quad
\tilde{\mu}_k = n \norm{\gamma^{-1/2} \widetilde{T} e_k}_\infty^2$$ for all $k \in [n]$, where $\gamma$ is defined in from $T$ and $\widetilde{T} = (T^\dagger)^*$. Let $\Omega = \{\omega_1, \ldots, \omega_m\}$ be a multi-set of random indices where $\omega_k$s are independent copies of a random variable $\omega$ with the following distribution: $$\label{eq:variable_density}
\mathbb{P}(\omega = k) = \frac{\sqrt{\mu_k \tilde{\mu}_k}}{\sum_{j=1}^n {\sqrt{\mu_j \tilde{\mu}_j}}}, \quad \forall k \in [n].$$ Suppose $T x$ is $s$-sparse. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$m \geq \frac{C(1+\beta) \bar{\mu} s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right],$$ where $\bar{\mu}$ is defined as $$\label{eq:defbarmu}
\bar{\mu} := \frac{1}{n} \sum_{k=1}^n \sqrt{\mu_k \tilde{\mu}_k}
%\bar{\mu} := \left( \frac{1}{n} \sum_{k=1}^n \mu_k \tilde{\mu}_k \right)^{1/2}.$$
Compared to Theorem \[thm:uniqueness\], Theorem \[thm:uniqueness\_vd\] provides a performance guarantee at a smaller sample complexity, where the worst-case incoherence parameter is replaced by the average incoherence parameter $\bar{\mu}$. Note that $\bar{\mu}$ is always no greater than the worst-case incoherence parameter $(\max_k \mu_k)^{1/2} (\max_k \tilde{\mu}_k)^{1/2}$. In particular when there exist dominant $\mu_k$s or $\tilde{\mu}_k$s compared to other incoherence parameters, the sample complexity of Theorem \[thm:uniqueness\_vd\] is much smaller than that of Theorem \[thm:uniqueness\].
Define $$\label{eq:defnu}
\nu := [\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}]^\transpose
\quad \mathrm{and} \quad
\tilde{\nu} := [\sqrt{\tilde{\mu}_1}, \ldots, \sqrt{\tilde{\mu}_n}]^\transpose.$$ Without loss of generality, we may assume that $\mu_k$s and $\tilde{\mu}_k$s are strictly positive. Then all entries of $\nu$ and $\tilde{\nu}$ are nonzero.
Using $\nu$ and $\tilde{\nu}$, we construct a pair of weighted transforms $W$ and $\widetilde{W}$ as $$\label{eq:defW}
W = \sqrt{\bar{\mu}} T [\mbox{diag}(\nu)]^{-1}
\quad \mathrm{and} \quad
\widetilde{W} = \sqrt{\bar{\mu}} \widetilde{T} [\mbox{diag}(\tilde{\nu})]^{-1}.$$ Then, $W$ and $\widetilde{W}$ satisfy $$\label{eq:incoW}
\max_{k \in [n]} \norm{\gamma^{1/2} W e_k}_\infty \leq \sqrt{\frac{\bar{\mu}}{n}}
\quad \mathrm{and} \quad
\max_{k \in [n]} \norm{\gamma^{-1/2} \widetilde{W} e_k}_\infty \leq \sqrt{\frac{\bar{\mu}}{n}}.$$ Furthermore, we have $$\label{eq:isoW}
\begin{aligned}
\mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m W S_\Omega^* S_\Omega \widetilde{W}^* \right)
{} & = \mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m W e_{\omega_j} e_{\omega_j}^* \widetilde{W}^* \right) \\
{} & = \mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m \bar{\mu} T [\mbox{diag}(\nu)]^{-1} e_{\omega_j} e_{\omega_j}^* [\mbox{diag}(\tilde{\nu})]^{-1} (\widetilde{T})^* \right) \\
{} & = \mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m \frac{\bar{\mu}}{\sqrt{\mu_{\omega_j} \tilde{\mu}_{\omega_j}}} T e_{\omega_j} e_{\omega_j}^* T^\dagger \right) \\
{} & = \frac{1}{m} \sum_{j=1}^m T \widetilde{T}^\dagger = T T^\dagger.
\end{aligned}$$ Since $\mbox{diag}(\tilde{\nu})$ is an invertible matrix, $T$ and $W$ span the same subspace and $T T^\dagger$ is an orthogonal projection onto the span of $W$. Furthermore, $$\langle W e_{k'}, \widetilde{W} e_k \rangle = 0, \quad \forall k \neq k'.$$
Let $g = \sqrt{\bar{\mu}} [\mbox{diag}(\nu)]^{-1} g'$. Then $T g = W g'$. Thus is equivalent to $$\label{eq:ell1min_vd}
\minimize_{g' \in \cz^n} \, \norm{W g'}_1
\quad \mathrm{subject~to} \quad S_\Omega g' = \bar{\mu}^{-1/2} S_\Omega [\mbox{diag}(\nu)] x.$$
Applying Theorem \[thm:uniqueness\] to with incoherence parameter $\bar{\mu}$ completes the proof.
In the case when we are given sampled measurements corrupted with additive noise, Theorems \[thm:stability1\] and \[thm:stability2\] are modified according to the change of the distribution for choosing random sample indices.
\[thm:stability1\_vd\] Suppose the hypotheses of Theorem \[thm:uniqueness\_vd\] hold. Let $x^\sharp$ be a noisy version of $x$ that satisfies . Let $\hat{x}$ be the minimizer to $$\label{eq:ell1min_noisy_vd}
\minimize_{g \in \cz^n} \, \norm{T g}_1
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} [\rho \odot (g - x^\sharp)]}_2 \leq \epsilon,$$ where $\rho := \bar{\mu}^{-1/2} [\sqrt{\mu_1}, \dots, \sqrt{\mu_n}]^\transpose$. Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}}{\min_{k \in [n]} \tilde{\mu}_k}
\cdot
\left[ 2 + 28 \sqrt{N} \left( \frac{\max_{k \in [n]} \mu_k}{\min_{k \in [n]} \tilde{\mu}_k} \cdot 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right] \epsilon.$$
Like Theorem \[thm:uniqueness\_vd\], Theorem \[thm:stability1\_vd\] provides a performance guarantee at a lower sample complexity (in order) when compared to Theorem \[thm:stability1\]. On the other hand, the noise amplification factor of Theorem \[thm:stability1\_vd\] is larger than that of Theorem \[thm:stability1\] by a factor that depends on the distribution of the local incoherence parameters $\{(\mu_k,\tilde{\mu}_k)\}_{k=1}^n$.
Let $W$ and $\widetilde{W}$ be defined in . Let $\nu$ and $\tilde{\nu}$ be defined in . Define $$\label{eq:defLambda}
\Lambda := \sqrt{\bar{\mu}} [\mbox{diag}(\nu)]^{-1}
\quad \mathrm{and} \quad
\widetilde{\Lambda} = \sqrt{\bar{\mu}} [\mbox{diag}(\tilde{\nu})]^{-1}.$$ Then $W = T \Lambda$ and is equivalent to $$\label{eq:ell1min_noisy_vd2}
\minimize_{g' \in \cz^n} \, \norm{W g'}_1
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} g' - S_{\Omega'} \Lambda^{-1} x^\sharp}_2 \leq \epsilon.$$
Let $\breve{x}$ denote the minimizer to . Then, the minimizer $\hat{x}$ to is represented as $\hat{x} = \Lambda \breve{x}$, i.e. $\breve{x} = \Lambda^{-1} \hat{x}$. By applying Theorem \[thm:stability1\] to , we obtain $$\label{eq:pf_thm_stability1_vd:bnd}
\begin{aligned}
\norm{T \hat{x} - T x}_2
{} & = \norm{W \breve{x} - W \Lambda^{-1} x}_2 \\
{} & \leq \left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T \Lambda}_{1 \to 2} \norm{\widetilde{\Lambda}^{-1} T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon \norm{W}.
\end{aligned}$$
The proof completes by applying the following inequalities to : $$\norm{\widetilde{\Lambda}^{-1} T^\dagger}_{2 \to \infty} \leq \norm{\widetilde{\Lambda}^{-1}} \norm{T^\dagger}_{2 \to \infty}
= \frac{\bar{\mu} \norm{T^\dagger}_{2 \to \infty}}{\min_{k \in [n]} \tilde{\mu}_k}$$ and $$\norm{T \Lambda}_{1 \to 2} \leq \norm{T}_{1 \to 2} \norm{\Lambda}
= \frac{\max_{k \in [n]} \mu_k \norm{T^\dagger}_{2 \to \infty}}{\bar{\mu}}.$$
\[thm:stability2\_vd\] Suppose $T: \cz^n \to \cz^N$ is injective. Suppose that $T$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameters $\{\mu_k\}_{k=1}^n$, $\{\tilde{\mu}_k\}_{k=1}^n$, and $\gamma$. Let $\bar{\mu}$ be defined in . Let $\Omega = \{\omega_1, \ldots, \omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. copies of a random variable $\omega$ with the distribution in . Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying . Then there exist numerical constants $C, c > 0$ for which the following holds. With probability $1 - N^{-4}$, we have $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}}{\min_{k \in [n]} \tilde{\mu}_k}
\cdot
\left[ 14\sqrt{N} + \frac{n}{m} \left\{ \frac{\max_{k \in [n]} \mu_k}{\min_{k \in [n]} \tilde{\mu}_k} \cdot \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) +1 \right\} \right] \epsilon,$$ provided $$m \geq \frac{C \bar{\mu} s \log^4 N}{1 - \norm{\gamma T^* T - I_n}}.$$ Furthermore, if $T^* T = I_n$, then with probability $1 - N^{-4}$, $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}}{\min_{k \in [n]} \tilde{\mu}_k}
\cdot
\left( 14\sqrt{N} + \frac{n}{m} \cdot \frac{\max_{k \in [n]} \mu_k}{\min_{k \in [n]} \tilde{\mu}_k} \cdot R \right) \epsilon,$$ where $R$ is the count of the most repeated elements in $\Omega$.
Let $W$ and $\widetilde{W}$ be defined in . Let $\nu$ and $\tilde{\nu}$ be defined in . Let $\Lambda$ and $\widetilde{\Lambda}$ be defined in . In the proof of Theorem \[thm:stability1\_vd\], we have shown that is equivalent to . In the proof of Theorem \[thm:uniqueness\_vd\], we have shown that $W$ and $\widetilde{W}$ satisfy \[eq:incoW,eq:isoW\]. Therefore, applying Theorem \[thm:stability2\] to with incoherence parameter $\bar{\mu}$ completes the proof.
Extension to Group Sparsity Models {#sec:main_result_gs}
==================================
In Sections \[sec:main\_result\] and \[sec:main\_result\_vd\], we considered the sparsity model in a transform domain. In applications, the transform $T x$ exhibits additional structures – group sparsity. For example, in compressed sensing of 2D signals, the sparsifying transform $\Phi = [\Phi_1^\top, \Phi_2^\top]^\top$ can be the concatenation of the horizontal and vertical finite difference operators $\Phi_1$ and $\Phi_2$. Anisotropic total variation encourages the sparsity of $\Phi f$ [@krahmer2014stable]. On the other hand, one can choose isotropic total variation, assuming that $\Phi_1 f$ and $\Phi_2 f$ are jointly sparse (see the experiments section). For another example, in compressed sensing of color images or hyperspectral images, the sparse codes acquired from applying the analysis operator to different channels are usually assumed to be jointly sparse. To exploit such structures, we extend the results in the previous sections to group sparsity models in a transform domain. More specifically, we assume that the transform $T x$ of unknown signal $x$ via $T: \cz^n \to \cz^L$ is $(s,t)$ strongly group sparse in the following sense.
Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$, i.e. $\bigcup_{j \in [N]} \calG_j = [L]$ and $\calG_j \bigcap \calG_{j'} = \emptyset$ for $j \neq j'$. A vector $z \in \cz^L$ is $(s,t)$ strongly group sparse with respect to $\calG$ if there exists $J \subset [N]$ such that $$\supp{z} \subset \calG_J,
\quad
|\calG_J| \leq t,
\quad \mathrm{and} \quad
|J| \leq s,$$ where $\calG_J$ is defined by $$\calG_J := \bigcup_{j \in J} \calG_j.$$
An atomic norm for this group sparsity model is given by $$\tnorm{z}_{\calG,1} := \sum_{j \in [N]} \norm{\Pi_{\calG_j} z}_2.$$ The dual norm of $\tnorm{\cdot}_{\calG,1}$ is defined by $$\tnorm{z}_{\calG,\infty} := \sup_{\zeta \in \cz^L: \tnorm{\zeta}_{\calG,1} \leq 1} |\langle \zeta, z \rangle|,$$ which is equivalently rewritten as $$\tnorm{z}_{\calG,\infty} = \max_{j \in [N]} \norm{\Pi_{\calG_j} z}_2.$$
A subgradient $\tilde{z}$ of $\tnorm{\cdot}_{\calG,1}$ at $z$ is given by $$\label{eq:subgrad_mixed}
\Pi_{\calG_j} \tilde{z}
=
\begin{dcases}
\frac{\Pi_{\calG_j} z}{\norm{\Pi_{\calG_j} z}_2}, & \Pi_{\calG_j} z \neq 0, \\
0, & \mathrm{otherwise},
\end{dcases}
\qquad \forall j \in [N].$$
In a special case where $\calG_j = \{j\}$ for all $j \in [N]$, the strong group sparsity level reduces to the conventional sparsity model. The analogy between the two models is summarized in Table \[tab:replace\_notation\]. All the results in the previous section generalize to the strong group sparsity model according to this analogy. In the below, we state the extended results as Theorems, the proofs of which are obtained in a straightforward way by modifying the proofs of analogous theorems and lemmas for the usual sparsity model according to Table \[tab:replace\_notation\]. Thus, we do not repeat the proofs in this section. (The only exception is Lemma \[lemma:E3\] and we provide an analogous Lemma \[lemma:E3’\] in Section \[subsec:riplike\_lemmas\].)
[>p[20ex]{}|C|C]{} & Sparsity Model & Group Sparsity Model\
$\mbox{dim}(R(T))$ & $N$ & $L = \sum_{j=1}^N |\calG_j|$\
& & $\gsupp{z} = \{ j :~ \Pi_{\calG_j} z \neq 0 \}$\
& & $\supp{z} = \bigcup_{j \in \gsupp{z}} \calG_j$\
group sparsity level & $\norm{z}_0 = |\supp{z}|$ & $\tnorm{z}_{\calG,0} = |\gsupp{z}|$\
total sparsity level & $\norm{z}_0 = |\supp{z}|$ & $\norm{z}_0 = |\supp{z}|$\
atomic norm & $\norm{z}_1 = \sum_{j=1}^N |\Pi_{\{j\}} z|$ & $\tnorm{\cdot}_{\calG,1} = \sum_{j=1}^N \norm{\Pi_{\calG_j} z}_2$\
dual norm & $\norm{\cdot}_\infty = \max_{j \in [N]} |\Pi_{\{j\}} z|$ & $\tnorm{\cdot}_{\calG,\infty} = \max_{j \in [N]} \norm{\Pi_{\calG_j} z}_2$\
subgradient & &\
of atomic norm & &\
incoherence & &\
parameter & &\
local incoherence & &\
parameters & &\
When $T x$ is strongly group sparse, we propose to estimate $T x$ by solving the following optimization problem, which generalizes : $$\label{eq:mixedmin}
\minimize_{g \in \cz^n} \, \tnorm{T g}_{\calG,1}
\quad \mathrm{subject~to} \quad S_\Omega g = S_\Omega x.$$ Then the following theorem, which is analogous to Theorem \[thm:uniqueness\], provides a performance guarantee for .
\[thm:uniqueness\_gs\] Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$. Suppose $T: \cz^n \to \cz^L$ is injective. Let $\mu_\calG$ be given by $$\label{eq:incoherence_gs}
\mu_\calG = \max_{k \in [n]} \max_{j \in [N]}
\max\left\{
n \norm{\gamma^{1/2} \Pi_{\calG_j} T e_k}_2^2,
n \norm{\gamma^{-1/2} \Pi_{\calG_j} \widetilde{T} e_k}_2^2
\right\},$$ where $\gamma$ is defined in from $T$ and $\widetilde{T} = (T^\dagger)^*$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Suppose $T x$ is $(s,t)$ strongly group sparse with respect $\calG$. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$\label{eq:samprate_gs}
m \geq \frac{C(1+\beta) \mu_\calG s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right].$$
Suppose that $\mu$ and $\mu_\calG$ are the smallest constants satisfying corresponding incoherence conditions. If $t = \ell s$ and $|\calG_j| = \ell$ for all $j \in [N]$, then $$\norm{\Pi_{\calG_j} T e_k}_2 \leq \sqrt{|\calG_j|} \norm{T e_k}_\infty, \quad \forall j \in [N], ~ \forall k \in [n].$$ Thus, we have $\mu_\calG \leq \ell \mu$ and a sufficient condition for is given by $$m \geq \frac{C(1+\beta) \mu t}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right].$$ In other words, the sample complexity is proportional to the total sparsity level $t$ and there is no gain from the group structure. This inequality is tight if each $\Pi_{\calG_j} T e_k$ have nonzero elements of the same magnitude. Contrarily, if nonzero elements of each $\Pi_{\calG_j} T e_k$ vary a lot in their magnitudes, $\mu_\calG$ is smaller than $\ell \mu$ and there is gain from the group sparsity structure.
In the presence of noise to measurements, we generalize the optimization formulation for recovery in as follows: $$\label{eq:mixedmin_noisy}
\minimize_{g \in \cz^n} \, \tnorm{T g}_{\calG,1}
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} g - S_{\Omega'} x^\sharp}_2 \leq \epsilon.$$
The following theorem, analogous to Theorem \[thm:stability1\], provides a performance guarantee for .
\[thm:stability1\_gs\] Suppose the hypotheses of Theorem \[thm:uniqueness\_gs\] hold. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying $$\norm{S_{\Omega'} (x - x^\sharp)}_2 \leq \epsilon.$$ Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon.$$
The results for recovery using a variable density sampling designed from local incoherence parameters generalize in a similar way. We state the results in the following theorems.
\[thm:uniqueness\_vd\_gs\] Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$. Suppose $T: \cz^n \to \cz^N$ is injective. Let $\mu_{\calG,k}$ and $\tilde{\mu}_{\calG,k}$ be given by $$\label{eq:loc_incoherenceT_gs}
\mu_{\calG,k} = \max_{j \in [N]} n \norm{\gamma^{1/2} \Pi_{\calG_j} T e_k}_2^2
\quad \mathrm{and} \quad
\tilde{\mu}_{\calG,k} = \max_{j \in [N]} n \norm{\gamma^{-1/2} \Pi_{\calG_j} \widetilde{T} e_k}_2^2,$$ where $\gamma$ is defined in from $T$ and $\widetilde{T} = (T^\dagger)^*$. Let $\Omega = \{\omega_1, \ldots, \omega_m\}$ be a multi-set of random indices where $\omega_k$s are independent copies of a random variable $\omega$ with the following distribution: $$\label{eq:variable_density_gs}
\mathbb{P}(\omega = k) = \frac{\sqrt{\mu_{\calG,k} \tilde{\mu}_{\calG,k}}}{\sum_{j=1}^n \sqrt{\mu_{\calG,j} \tilde{\mu}_{\calG,j}}}, \quad \forall k \in [n].$$ Suppose $T x$ is $(s,t)$ strongly group sparse with respect $\calG$. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$m \geq \frac{C(1+\beta) \bar{\mu}_\calG s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right],$$ where $\bar{\mu}$ is defined as $$\bar{\mu}_\calG := \frac{1}{n} \sum_{k=1}^n \sqrt{\mu_{\calG,k} \tilde{\mu}_{\calG,k}}.$$
\[thm:stability1\_vd\_gs\] Suppose the hypotheses of Theorem \[thm:uniqueness\_vd\_gs\] hold. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying $$\label{eq:xharp_fid_gs}
\norm{S_{\Omega'} x - S_{\Omega'} x^\sharp}_2 \leq \epsilon.$$ Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}_\calG}{\min_{k \in [n]} \tilde{\mu}_{\calG,k}}
\cdot
\left[ 2 + 28 \sqrt{N} \left( \frac{\max_{k \in [n]} \mu_{\calG,k}}{\min_{k \in [n]} \tilde{\mu}_{\calG,k}} \cdot 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right] \epsilon.$$
Circulant Transforms {#sec:circulant}
====================
Many sparsifying transforms fall into the category of circulant transforms, including the identity transform. A block transform (e.g., block DCT), when applied to all overlapping patches of a signal (sliding window with stride $1$, including the wrap-around patches at the edges), is a union of circulant transforms applied to the signal [@pfister2015learning]. In this section, we consider only the case when the measurement matrix $\Psi\in\cz^{n\times n}$ is the DFT matrix, and compute the variable density sampling distribution for sparsity with respect to a circulant transform, and distribution for joint sparsity with respect to a union of circulant transforms. We show that, if the circulant transforms are injective, the distributions and correspond to the uniform distribution on $[n]$. On the other hand, some circulant transforms are not injective (e.g., the finite difference operator for 1D total variation or 2D isotropic total variation). Even in this case, we show that the “variable” density sampling distributions are a variation of the uniform distribution.
Injective Circulant Transforms
------------------------------
We say $\Phi\in \cz^{n\times n}$ is a circulant transform (circulant matrix), if $\Phi f = \phi \circledast f$ is the circular convolution of $f$ with some vector $\phi \in \cz^n$, i.e. the matrix representation of $\Phi$ is given by $$\label{eq:circulant}
\Phi = \begin{bmatrix}
\phi[1] & \phi[n] & \phi[n-1] & \cdots & \phi[2] \\
\phi[2] & \phi[1] & \phi[n] & \cdots & \phi[3] \\
\phi[3] & \phi[2] & \phi[1] & \cdots & \phi[4] \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\phi[n] & \phi[n-1] & \phi[n-2] & \cdots & \phi[1] \\
\end{bmatrix}$$ The identity transform is a circulant transform with $\phi=e_1$.
By linearity and shift invariance of circular convolution, a circulant transform $\Phi$ can always be diagonalized by the DFT matrix $\Psi$: $$\label{eq:diagonalize}
\Phi = \Psi^* \diag(\lambda) \Psi.$$ where $\lambda = \sqrt{n}\Psi \phi$ is the (unnormalized) DFT of $\phi$. The same argument also applies to a 2D circulant transform (circulant block circulant matrix) and the 2D DFT matrix. To avoid verbosity, we use circulant transform $\Phi$ and DFT matrix $\Psi$ to denote both 1D and 2D transforms.
\[thm:circulant1\] For the DFT matrix $\Psi\in\cz^{n\times n}$ and an invertible circulant transform $\Phi\in\cz^{n\times n}$, the sampling density distribution is the uniform distribution on $[n]$.
By the diagonalization in , we have $$\begin{aligned}
& T = \Phi \Psi^\dagger = \Psi^* \diag(\lambda),\\
& \widetilde{T} = (T^\dagger)^* = \Psi^* \diag(\tilde{\lambda}),\end{aligned}$$ where $\tilde{\lambda}$ is the complex conjugate of the element-wise inverse of $\lambda$, i.e. $\tilde{\lambda}[k]=(\lambda[k]^*)^{-1}$. Then, the following choice of parameters $\mu_k$ and $\tilde{\mu}_k$ satisfy : $$\begin{aligned}
& \mu_k = n\gamma \norm{Te_k}_\infty^2 = n\gamma |\lambda[k]|^2 \norm{\Psi^*e_k}_\infty^2 = \gamma |\lambda[k]|^2,\\
& \tilde{\mu}_k = n\gamma^{-1} \norm{\widetilde{T}e_k}_\infty^2 = n\gamma^{-1} |\tilde{\lambda}[k]|^2 \norm{\Psi^*e_k}_\infty^2 = \gamma^{-1} |\lambda[k]|^{-2},\end{aligned}$$ where $\Psi^*e_k$ is the $k$th column of the discrete Fourier basis and has infinity norm $1/\sqrt{n}$. Therefore, $\mu_k\tilde{\mu}_k = 1$ for all $k\in [n]$, and the distribution in is $\mathbb{P}(\omega = k) = 1/n$, i.e. the uniform distribution on $[n]$.
Applying a sparsifying circulant transform to a signal is equivalent to passing the signal through a sparsifying filter. One may also pass the signal through a bank of sparsifying filters. The filter bank is equivalent to a union of circulant transforms, concatenated as $$\label{eq:concatenation}
\Phi = [\Phi_1^\transpose, \Phi_2^\transpose,\dots, \Phi_\ell^\transpose]^\transpose \in\cz^{L\times n},$$ where $L=\ell n$. For example, the patch transform sparsity model [@pfister2015learning] corresponds to this case. The 2D isotropic total variation model (see Section \[sec:TV\]), as another example, has an additional joint sparsity structure. For the latter example, let us consider a particular partition $\calG = \{\calG_1, \ldots, \calG_n\}$ given by $$\label{eq:partition}
\calG_k = \{(j-1)n+k\}_{j=1}^\ell, \qquad \forall k\in [n].$$ For group sparsity on this union of circulant transforms, we have a result similar to Theorem \[thm:circulant1\].
\[thm:circulant2\] For the DFT matrix $\Psi\in\cz^{n\times n}$, an injective transform $\Phi\in\cz^{L\times n}$ defined in , and the partition defined in , the sampling density distribution is the uniform distribution on $[n]$.
Similar to , we have the following factorization for the concatenated transform: $$\label{eq:diagonalize2}
\Phi = (I_\ell \otimes \Psi^*) [\diag(\lambda_1),\diag(\lambda_2),\dots, \diag(\lambda_\ell)]^\top \Psi,$$ where $\lambda_j$ is the (unnormalized) DFT of $\phi_j$, the convolution kernel of the $j$th cirulant transform $\Phi_j$. Hence $T$ and $\widetilde{T}$ are $$\begin{aligned}
& T = \Phi \Psi^\dagger = (I_\ell \otimes \Psi^*)[\diag(\lambda_1),\diag(\lambda_2),\dots, \diag(\lambda_\ell)]^\top,\\
& \widetilde{T} = (T^\dagger)^* = T(T^*T)^{-1} = (I_\ell \otimes \Psi^*) [\diag(\tilde{\lambda}_1),\diag(\tilde{\lambda}_2),\dots, \diag(\tilde{\lambda}_\ell)]^\top,\end{aligned}$$ where $\tilde{\lambda}_j[k] = \lambda_j[k]/\bigl(\sum_{j'=1}^{\ell} |\lambda_{j'}[k]|^2\bigr)$. Then, $\mu_{\calG,k}$ and $\tilde{\mu}_{\calG,k}$ in are given respectively by $$\begin{aligned}
\mu_{\calG,k} & = n\gamma \max_{k'\in[n]}\norm{\Pi_{\calG_{k'}}Te_k}_2^2 \\
& = n\gamma \max_{k'\in[n]} \norm{\Pi_{\calG_{k'}}\bigl\{\bigl[\lambda_1[k],\lambda_2[k],\dots,\lambda_\ell[k]\bigr]^\transpose \otimes (\Psi^*e_k) \bigl\} }_2^2 \\
& = n\gamma \max_{k'\in[n]} |e_{k'}^\transpose \Psi^*e_k| \norm{\bigl[\lambda_1[k],\lambda_2[k],\dots,\lambda_\ell[k]\bigr]^\transpose}_2^2
= \gamma \bigl(\sum_{j=1}^{\ell} |\lambda_{j}[k]|^2\bigr)\end{aligned}$$ and $$\begin{aligned}
\tilde{\mu}_{\calG,k} & = n\gamma^{-1} \max_{k'\in[n]}\norm{\Pi_{\calG_{k'}}\widetilde{T}e_k}_2^2 \\
& = n\gamma^{-1} \max_{k'\in[n]} \norm{\Pi_{\calG_{k'}}\bigl\{\bigl[\tilde{\lambda}_1[k],\tilde{\lambda}_2[k],\dots,\tilde{\lambda}_\ell[k]\bigr]^\transpose \otimes (\Psi^*e_k) \bigl\} }_2^2 \\
& = \gamma^{-1} \bigl(\sum_{j=1}^{\ell} |\tilde{\lambda}_{j}[k]|^2\bigr)
= \gamma^{-1} \bigl(\sum_{j=1}^{\ell} |\lambda_{j}[k]|^2\bigr)^{-1}.\end{aligned}$$ Therefore, $\mu_{\calG,k}\tilde{\mu}_{\calG,k} = 1$ for all $k\in [n]$, and the distribution in is $\mathbb{P}(\omega = k) = 1/n$, i.e. the uniform distribution on $[n]$.
Non-injective Circulant Transforms {#sec:TV}
----------------------------------
In this section, we consider non-injective circulant transforms, such as the finite difference operator for 1D total variation, or the union of vertical and horizontal finite difference operators for 2D isotropic total variation. Since the spectral responses of these transforms are zero at certain frequencies, they are invariant to changes in the corresponding frequency components. Therefore, unless these frequency components are sampled in the measurement, they cannot be recovered from $\ell_1$-norm minimization or mixed norm minimization . Assuming without loss of generality that the null frequencies are the first $n_0$ columns in $\Psi^*$ ($n_0<\min\{n,m\}$), we adopt the following two-step sampling scheme:
1. Always sample indices $\omega_k = k$ for $k\in [n_0]$.
2. Generate a multi-set of $m-n_0$ indices $\{\omega_{n_0+1},\omega_{n_0+2},\dots,\omega_m\}$, which are i.i.d. following a distribution on $\{n_0+1,n_0+2,\dots,n\}$.
We can compute the sampling density on $\{n_0+1,n_0+2,\dots,n\}$ based on and (or and ) by removing the zero columns in $T=\Phi\Psi^*$ (replacing $T$ with $T[e_{n_0+1},e_{n_0+2},\dots,e_{n}]$). Next, we state variations of Theorems \[thm:circulant1\] and \[thm:circulant2\] in these cases.
\[cor:circulant1\] Suppose circulant transform $\Phi$ in satisfies $\lambda[k] = 0$ for $k\in[n_0]$. Then the sampling density in Step 2), for $\ell_1$-norm minimization based on , is the uniform distribution on $\{n_0+1,n_0+2,\dots,n\}$.
\[cor:circulant2\] Suppose concatenated transform $\Phi$ in satisfies $\lambda_j[k] = 0$ for all $j\in[\ell]$ and $k\in[n_0]$. Then the sampling density in Step 2), for mixed norm minimization based on , is the uniform distribution on $\{n_0+1,n_0+2,\dots,n\}$.
These results are direct consequences of Theorems \[thm:circulant1\] and \[thm:circulant2\], whose proofs translate with no changes other than removing the zero columns in $T$ (the columns indexed by $[n_0]$).
Next, we specialize these results to total variation minimization. We define the 1D finite difference operator $\Phi_{\mathrm{TV},n}$ by , where $$\phi[k] = \begin{cases}
1 & k = 1\\
-1 & k = 2\\
0 & k\in\{3,4,\dots, n\}.
\end{cases}$$ Then $\norm{f}_{\mathrm{TV}} = \norm{\Phi_{\mathrm{TV},n} f}_1$ is the 1D total variation of $f$. Clearly, the circulant transform $\Phi_{\mathrm{TV},n}$ is not injective, since its null space contains the direct current (DC) component – $\Psi^*e_1$. By Corollary \[cor:circulant1\], we adopt the following sampling scheme for total variation minimization:
1. Always sample index $\omega_1 = 1$.
2. Generate a multi-set of $m-1$ indices $\{\omega_2,\omega_3,\dots,\omega_m\}$, which are i.i.d. following the uniform distribution on $\{2,3,\dots,n\}$.
Total variation is more commonly used for 2D signals (e.g., images). For a 2D signal $f$ of size $n_1\times n_2$, the finite difference operator is a concatenation of the vertical and horizontal finite difference operators: $$\Phi_{\mathrm{TV},n_1,n_2} = \begin{bmatrix}
I_{n_2} \otimes \Phi_{\mathrm{TV},n_1} \\
\Phi_{\mathrm{TV},n_2} \otimes I_{n_1}
\end{bmatrix}.$$ The anisotropic and isotropic total variations of $f$ are defined by $$\begin{aligned}
& \norm{f}_{\mathrm{TV, aniso}} = \norm{\Phi_{\mathrm{TV},n_1,n_2} f}_1,\\
& \norm{f}_{\mathrm{TV, iso}} = \tnorm{\Phi_{\mathrm{TV},n_1,n_2} f}_{\calG,1},\end{aligned}$$ where the partion $\calG=\{\calG_1,\dots,\calG_n\}$ is defined by for $\ell=2$ and $n=n_1n_2$. Let the measurement operator $\Psi$ be the 2D DFT on signals of size $n_1\times n_2$. Similar to the 1D case, the DC component $\Psi^* e_1$ belongs to the null space of $\Phi_{\mathrm{TV},n_1,n_2}$. By Corollary \[cor:circulant2\], we use the same two-step sampling scheme as in the 1D case.
Proof of Theorem \[thm:uniqueness\] {#sec:pf_main_result_noiseless}
===================================
In this section, we prove Theorem \[thm:uniqueness\], which provides a sufficient condition for exact recovery of sparse signals in a transform domain from noiseless observations. The proof of Theorem \[thm:uniqueness\] is based on the golfing scheme [@gross2011recovering], which was originally proposed for matrix completion [@gross2011recovering] and later adopted to compressed sensing [@candes2011probabilistic] and to structured matrix completion [@chen2014robust].
The golfing scheme constructs an inexact dual certificate. The notion of a dual certificate was originally proposed for compressed sensing (cf. [@candes2006near]). To reconstruct an $s$-sparse $f \in \cz^d$ from $b = A f$, it was proposed to estimate $f$ as the solution to $$\minimize_{\tilde{f} \in \cz^d} \, \norm{\tilde{f}}_1 \quad \mathrm{subject~to} \quad b = A \tilde{f}.$$ A dual certificate is a subgradient $v \in \cz^d$ of $\norm{\cdot}_1$ at $f$ such that $f$ is orthogonal to all null vectors of the sensing matrix $A$. In matrix completion, the objective function is replaced from the $\ell_1$-norm to the nuclear norm and the sensing matrix is replaced by a pointwise sampling operator. Gross [@gross2011recovering] proposed the clever golfing scheme that constructs an inexact dual certificate, which is close to the exact dual certificate, and showed a low-rank matrix is exactly reconstructed from partial entries sampled at a near optimal rate. Candes and Plan [@candes2011probabilistic] adopted the golfing scheme back to compressed sensing and showed that exact recovery is guaranteed from $m = O(s \log d)$ incoherent measurements, which improves on the previous performance guarantee with $m = O(s \log^4 d)$.
Chen and Chi [@chen2014robust] adopted the golfing scheme to structured matrix completion. Let $T: \cz^n \to \cz^{n_1 \times n_1}$ be a linear operator that maps a vector $x \in \cz^n$ to a structured matrix $T x \in \cz^{n_1 \times n_2}$ (e.g., a Hankel matrix). Chen and Chi [@chen2014robust] proposed to estimate $x$ by $$\minimize_{g \in \cz^n} \, \norm{T g}_* \quad \mathrm{subject~to} \quad S_\Omega g = S_\Omega x.$$ This can be interpreted as recovery of low-rank matrices in a special transform domain, where $T$ maps the standard basis vectors $\{e_1,\ldots,e_n\}$ to unit-norm matrices with disjoint supports. Unlike compressed sensing or matrix completion, in structured matrix completion, the dimension of the vector space where the unknown signal is rearranged as a structured low-rank matrix is larger than the dimension of the vector space where measurements are sampled. In other words, $T$ is a tall matrix and is not surjective. With this nontrivial difference, the conventional approaches [@gross2011recovering; @candes2011probabilistic] do not apply directly to structured matrix completion. Chen and Chi [@chen2014robust] cleverly modified the definition of an inexact dual certificate and the golfing scheme accordingly and provided performance guarantee at a near optimal sample complexity. We adopt their approach to recovery of sparse signals in a transform domain, where the nuclear norm is replaced by the $\ell_1$-norm[^4]. Notably, we extend the theory to the case where $T$ is not necessarily a unitary transform (e.g., $T^* T = I_n$), which is the case in various practical applications.
We first present the following lemma that extends the notion of an inexact dual certificate for recovery of sparse signals in a transform domain where the transform $T$ is not necessarily unitary.
\[lemma:uniqueness\] Suppose that $T: \cz^n \to \cz^N$ is injective. Let $J \subset [N]$ denote the support of $T x$, i.e. the elements of $J$ correspond to the locations of the nonzero elements in $T x$. Let $\Omega$ be a multi-set that consists of elements in $[n]$ with possible repetitions. Let $\Omega'$ denote the set of all distinct elements in $\Omega$. Suppose that $$\label{eq:local_isometry_rank_deficient}
\left\|
\frac{n}{m} \Pi_J T S_\Omega^* S_\Omega T^\dagger \Pi_J
- \Pi_J T T^\dagger \Pi_J
\right\|
\leq \frac{1}{2}.$$ If there exists a vector $v \in \cz^N$ satisfying $$\label{eq:dualcert_vanish}
(T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^* v = 0,$$ $$\label{eq:dualcert_sgn}
\norm{\Pi_J (v - \mbox{\upshape sgn}(T x))}_2 \leq \frac{1}{7 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}},$$ and $$\label{eq:dualcert_bnd}
\norm{(I_N - \Pi_J) v}_\infty \leq \frac{1}{2},$$ then $x$ is the unique minimizer to (\[eq:ell1min\]).
See Section \[subsec:pf:lemma:uniqueness\].
The next lemma shows that such an inexact dual certificate exists with hight probability under the hypothesis of Theorem \[thm:uniqueness\].
\[lemma:existence\] Suppose the hypotheses of Theorem \[thm:uniqueness\] hold. Then, with probability $1 - e^{-\beta} - 1/n$, there exists a vector $v \in \cz^N$ satisfying \[eq:dualcert\_vanish,eq:dualcert\_sgn,eq:dualcert\_bnd\].
See Section \[subsec:pf:lemma:existence\].
Lemma \[lemma:existence\] together with Lemma \[lemma:uniqueness\] implies Theorem \[thm:uniqueness\]. Indeed, we only need to verify . By Lemma \[lemma:E1\], holds with probability at least $1 - 2/n$ provided that $m \geq 32 \mu s \max\{1/(1-\norm{T^* T - I_n}), 1/6\} \log n$. This completes the proof of Theorem \[thm:uniqueness\].
In the next section, we will introduce fundamental estimates that will be used in the proofs of Lemmas \[lemma:uniqueness\] and \[lemma:existence\].
Lemmas on fundamental estimates {#subsec:riplike_lemmas}
-------------------------------
The following lemmas provides estimates on various functions of the random matrix $T S_\Omega^* S_\Omega T^\dagger$, which are analogous to the corresponding estimates for RIPless compressed sensing [@candes2011probabilistic].
Similarly to RIPless compressed sensing, we also employ the notion of incoherence. In fact, our incoherence assumption in is analogous to a generalized version for anisotropic compressed sensing [@rudelson2013reconstruction; @lee2013oblique; @kueng2014ripless].
One important distinction from compressed sensing is that the *isotropy* property is not satisfied. Indeed, since the random indices in $\Omega$ are i.i.d. following the uniform distribution on $[n]$, the random matrix $T S_\Omega^* S_\Omega T^\dagger$ satisfies $$\frac{n}{m} \mathbb{E} T S_\Omega^* S_\Omega T^\dagger = T T^\dagger.$$ While $T T^\dagger$ is an orthogonal projection (idempotent and self-adjoint), it is not necessarily an identity operator. The rank-deficiency of $T T^\dagger$ requires new analysis in Lemmas \[lemma:uniqueness\] and \[lemma:existence\] compared to the previous results for compressed sensing with the canonical sparsity model [@candes2011probabilistic]. However, the fundamental estimates measure the deviations of functions of $T S_\Omega^* S_\Omega T^\dagger$ from their expectations and do not require the isotropy property ($T^* T = I_n$). We present the following lemmas for the fundamental estimates, whose proofs are deferred to the appendix.
\[lemma:E1\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for $\delta > 0$, $$\mathbb{P}\left(\left\| \Pi_J \left(\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger - T T^\dagger \right) \Pi_J \right\| \geq \delta\right)
\leq 2s \exp\left( -\frac{m}{s\mu} \cdot \frac{\delta^2/2}{1/(1-\norm{\gamma T^* T - I_n})+\delta/3} \right).$$
See Appendix \[sec:pf:lemma:E1\].
\[lemma:E2\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $q \in \cz^N$ be a fixed vector. Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for each $t \leq 1/2$, $$\mathbb{P}\left(\left\|\Pi_J \left(\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger - T T^\dagger \right) \Pi_J q\right\|_2 \geq t \norm{\Pi_J q}_2\right)
\leq \exp\left\{ -\frac{1}{4} \left( t \sqrt{\frac{m(1-\norm{\gamma T^* T - I_n})}{s\mu}} - 1 \right)^2 \right\}.$$
See Appendix \[sec:pf:lemma:E2\].
\[lemma:E3\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $q \in \cz^N$ be a fixed vector. Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for each $t > 0$, $$\begin{aligned}
{} & \mathbb{P}\left(\left\|\Pi_{[N] \setminus J} \left( \frac{n}{m} \widetilde{T} S_\Omega^* S_\Omega T^* - \widetilde{T} T^* \right) \Pi_J q\right\|_\infty \geq t \norm{\Pi_J q}_2\right) \\
{} & \quad \leq 2N \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).\end{aligned}$$
See Appendix \[sec:pf:lemma:E3\].
\[lemma:E3’\] Suppose $T: \cz^n \to \cz^L$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $q \in \cz^L$ be a fixed vector. Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$. Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\calG_J = \bigcup_{j \in J} \calG_j$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for each $t > 0$, $$\begin{aligned}
{} & \mathbb{P}\left( \left\|\Pi_{[L] \setminus \calG_J} \left( \frac{n}{m} \widetilde{T} S_\Omega^* S_\Omega T^* - \widetilde{T} T^* \right) \Pi_{\calG_J} q\right\|_{\calG,\infty} \geq t \norm{\Pi_{\calG_J} q}_2\right) \\
{} & \quad \leq 2 \max_{j \in [N]} |\calG_j| N \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).\end{aligned}$$
See Appendix \[sec:pf:lemma:E3’\].
Proof of Lemma \[lemma:uniqueness\] {#subsec:pf:lemma:uniqueness}
-----------------------------------
Our proof essentially adapts the arguments of Chen and Chi [@chen2014robust Appendix B] for the structured low-rank matrix completion problem. There are two key differences in the two proofs. First, the $\ell_1$-norm replaces the nuclear norm. Second, $T$ is a general injective transform, which is not necessarily unitary. These differences require nontrivial modifications of crucial steps in the proof. Furthermore, the upper bound on the deviation of $v$ from $\mbox{sgn}(T x)$ in is sharpened by optimizing parameters. This improvement also applies to the previous work [@chen2014robust].
Let $\hat{x} = x + h$ be the minimizer to (\[eq:ell1min\]). We show that $T h = 0$ in two complementary cases. Then by the injectivity of $T$, $h = 0$, or equivalently, $\hat{x} = x$.
**Case 1:** We first consider the case when $h$ satisfies $$\label{eq:case1}
\norm{\Pi_J T h}_2 \leq 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T h}_2.$$
Since $\mbox{sgn}(\Pi_{[N] \setminus J} T h)$ and $\mbox{sgn}(T x)$ have disjoint supports, it follows that $\mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h)$ and $\mbox{sgn}(T x)$ coincide on $J$. Furthermore, $\norm{\mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h)}_\infty \leq 1$. Therefore, $\mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h)$ is a valid sub-gradient of the $\ell_1$-norm at $T x$. Then it follows that $$\label{eq:pf_lemma_uniqueness:ineq1}
\begin{aligned}
\norm{T x + T h}_1
{} & \geq \norm{T x}_1 + \langle \mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ Th \rangle \\
{} & = \norm{T x}_1
+ \langle v, T h \rangle
+ \langle \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ Th \rangle
- \langle v - \mbox{sgn}(T x), Th \rangle.
\end{aligned}$$
In fact, $\langle v, T h \rangle = 0$ as shown below. The inner product of $T h$ and $v$ is decomposed as $$\label{eq:pf_lemma_uniqueness:ip}
\langle v, Th \rangle
= \langle v, (I_N - T T^\dagger) Th) \rangle
+ \langle v, (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger) Th \rangle
+ \langle v, T S_{\Omega'}^* S_{\Omega'} T^\dagger Th \rangle.$$ Indeed, all three terms in the right-hand side of are 0. Since $T T^\dagger$ is the orthogonal projection onto the range space of $T$, the first term is 0. The second term is 0 by the assumption on $v$ in . Since $\hat{x}$ is feasible for , $S_\Omega \hat{x} = S_\Omega x$. Thus $S_\Omega h = S_\Omega (\hat{x} - x) = 0$, i.e. $e_\omega^* h = 0$ for all $\omega \in \Omega$, which also implies $S_{\Omega'} h = 0$. Then it follows that $S_{\Omega'} T^\dagger T h = S_{\Omega'} h = 0$. Thus the third term of the right-hand side of is 0.
Since the $\mbox{sgn}(\cdot)$ operator commutes with $\Pi_{[N] \setminus J}$ and $\Pi_{[N] \setminus J}$ is idempotent, we get $$\begin{aligned}
\langle \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ T h \rangle
{} & = \langle \Pi_{[N] \setminus J} \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ T h \rangle \\
{} & = \langle \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ \Pi_{[N] \setminus J} T h \rangle \\
{} & = \norm{\Pi_{[N] \setminus J} T h}_1.\end{aligned}$$
Then implies $$\label{eq:pf_lemma_uniqueness:ineq2}
\norm{T x + T h}_1 \geq \norm{T x}_1 + \norm{\Pi_{[N] \setminus J} T h}_1 - \langle v - \mbox{sgn}(T x), Th \rangle.$$
We derive an upper bound on the magnitude of the last term in the right-hand side of given by
\[eq:pf\_lemma\_uniqueness:ineq3\] $$\begin{aligned}
| \langle v - \mbox{sgn}(T x), Th \rangle |
{} & = | \langle \Pi_J (v - \mbox{sgn}(T x)), Th \rangle + \langle \Pi_{[N] \setminus J} (v - \mbox{sgn}(T x)), Th \rangle | \nonumber \\
{} & \leq | \langle \Pi_J (v - \mbox{sgn}(T x)), Th \rangle | + | \langle \Pi_{[N] \setminus J} v, Th \rangle | \label{eq:pf_lemma_uniqueness:ineq3a} \\
{} & \leq \norm{\Pi_J (v - \mbox{sgn}(T x))}_2 \norm{\Pi_J Th}_2
+ \norm{\Pi_{[N] \setminus J} v}_\infty \norm{\Pi_{[N] \setminus J} T h}_1 \label{eq:pf_lemma_uniqueness:ineq3b} \\
{} & \leq \frac{1}{7 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}} \norm{\Pi_J Th}_2 + \frac{1}{2} \norm{\Pi_{[N] \setminus J} T h}_1, \label{eq:pf_lemma_uniqueness:ineq3c}\end{aligned}$$
where holds by the triangle inequality and the fact that $T x$ is supported on $J$; by Hölder’s inequality; by the assumptions on $v$ in and .
We continue by applying to and get $$\begin{aligned}
\norm{T x + T h}_1
{} & \geq \norm{T x}_1 - \frac{1}{7 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}} \norm{\Pi_J Th}_2 + \frac{1}{2} \norm{\Pi_{[N] \setminus J} T h}_1 \\
{} & \geq \norm{T x}_1 - \frac{3}{7} \norm{\Pi_{[N] \setminus J} T h}_2 + \frac{1}{2} \norm{\Pi_{[N] \setminus J} T h}_2 \\
{} & = \norm{T x}_1 + \frac{1}{14} \norm{\Pi_{[N] \setminus J} T h}_2,\end{aligned}$$ where the second step follows from .
Then, $\norm{T \hat{x}}_1 \geq \norm{T x}_1 \geq \norm{T \hat{x}}_1$, which implies $\Pi_{[N] \setminus J} T h = 0$. By (\[eq:case1\]), we also have $\Pi_J T h = 0$. Therefore, it follows that $T h = 0$.
**Case 2:** Next, we consider the complementary case when $h$ satisfies $$\label{eq:case2}
\norm{\Pi_J T h}_2 > 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T h}_2.$$
In the previous case, we have shown that $S_\Omega h = 0$. Thus $S_\Omega T^\dagger T h = 0$. Then together with $(I_N - T T^\dagger) T = 0$, we get $$\begin{aligned}
\left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) T h = 0,\end{aligned}$$ which implies $$\label{eq:pf_lemma_uniqueness:ineq4}
\begin{aligned}
0 {} & \geq \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) T h \right\rangle \\
{} & = \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_J T h \right\rangle \\
{} & \quad + \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_{[N] \setminus J} T h \right\rangle.
\end{aligned}$$
The magnitude of the first term in the right-hand side of is lower-bounded by $$\label{eq:pf_lemma_uniqueness:lb}
\begin{aligned}
{} & \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_J T h \right\rangle \right| \\
{} & = \left| \langle \Pi_J T h, \Pi_J T h \rangle \right|
- \left| \left\langle \Pi_J T h, \left( T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger \right) \Pi_J T h \right\rangle \right| \\
{} & \geq \norm{\Pi_J T h}_2^2 - \left\|\Pi_J T T^\dagger \Pi_J - \frac{n}{m} \Pi_J T S_\Omega^* S_\Omega T^\dagger \Pi_J\right\| \norm{\Pi_J T h}_2^2 \\
{} & \geq \frac{1}{2} \norm{\Pi_J T h}_2^2,
\end{aligned}$$ where the last step follows from the assumption in .
Next, we derive an upper bound on the second term in the right-hand side of . To this end, we first computes the operator norm of $T e_k e_k^* T^\dagger$ for $k \in [n]$. In fact, $\norm{T e_k e_k^* T^\dagger} = \norm{T e_k}_2 \norm{\widetilde{T} e_k}_2$, where $\widetilde{T}$ is the adjoint of $T^\dagger$. Therefore, we only need to compute $\norm{T e_k}_2$ and $\norm{\widetilde{T} e_k}_2$. First, $\norm{T e_k}_2$ is upper-bounded by $$\begin{aligned}
\max_{k \in [n]} \norm{T e_k}_2 = \norm{T}_{1 \to 2}.\end{aligned}$$ On the other hand, $\norm{\widetilde{T} e_k}_2$ is upper-bounded by $$\begin{aligned}
\max_{k \in [n]} \norm{\widetilde{T} e_k}_2
= \norm{\widetilde{T}}_{1 \to 2} = \norm{T^\dagger}_{2 \to \infty},\end{aligned}$$ where the last step holds since $\widetilde{T}$ is the adjoint operator of $T^\dagger$ and $\ell_\infty^n$ is the dual space of $\ell_1^n$. By the above upper bounds on $\norm{T e_k}_2$ and $\norm{\widetilde{T} e_k}_2$, we get $$\begin{aligned}
\norm{T e_k e_k^* T^\dagger} \leq \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}.\end{aligned}$$ Then the operator norm of $ \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger $ is upper-bounded by $$\label{eq:pf_lemma_uniqueness:ineq5}
\begin{aligned}
\left\| \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right\|
{} & \leq \frac{n}{m} \left( \norm{T e_{\omega_1} e_{\omega_1}^* T^\dagger + I_N - T T^\dagger} + \sum_{j=2}^m \norm{T e_{\omega_j} e_{\omega_j}^* T^\dagger}_2 \right) \\
{} & \leq \frac{n}{m} \left( \max(\norm{T e_{\omega_1} e_{\omega_1}^* T^\dagger}, \norm{I_N - T T^\dagger}) + \sum_{j=2}^m \norm{T e_{\omega_j} e_{\omega_j}^* T^\dagger}_2 \right) \\
{} & \leq n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty},
\end{aligned}$$ where the second step follows since $(T e_{\omega_1} e_{\omega_1}^* T^\dagger)^*(I_N - T T^\dagger) = 0$ and $(I_N - T T^\dagger) (T e_{\omega_1} e_{\omega_1}^* T^\dagger)^* = 0$. The second term in the right-hand side of is then upper-bounded by $$\label{eq:pf_lemma_uniqueness:ub}
\begin{aligned}
{} & \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_{[N] \setminus J} T h \right\rangle \right| \\
{} & \leq \left\| \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right\| \norm{\Pi_J T h}_2 \norm{\Pi_{[N] \setminus J} T h}_2 \\
{} & \leq n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_J T h}_2 \norm{\Pi_{[N] \setminus J} T h}_2,
\end{aligned}$$ where the last step follows from .
Applying and to provides $$\begin{aligned}
0 {} & \geq \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_J T h \right\rangle \right| \\
{} & \quad - \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_{[N] \setminus J} T h \right\rangle \right| \\
{} & \geq \frac{1}{2} \norm{\Pi_J T h}_2^2 - n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_J T h}_2 \norm{\Pi_{[N] \setminus J} T h}_2 \\
{} & \geq \frac{1}{2} \norm{\Pi_J T h}_2^2 - \frac{1}{3} \norm{\Pi_J T h}_2^2 \\
{} & = \frac{1}{6} \norm{\Pi_J T h}_2^2 \geq 0,\end{aligned}$$ where the second inequality follows from . Then it is implied that $\Pi_J T h = 0$. By (\[eq:case2\]), we also have $\Pi_{[N] \setminus J} T h = 0$. Therefore, $T h = 0$, which completes the proof.
Proof of Lemma \[lemma:existence\] {#subsec:pf:lemma:existence}
----------------------------------
We construct a dual certificate $v$ using a golfing scheme. Since the isotropy is not satisfied, the original golfing scheme needs to be modified accordingly. We adopt the version for structured matrix completion [@chen2014robust].
Recall that the elements of $\Omega = \{\omega_1, \ldots, \omega_m\}$ are i.i.d. following the uniform distribution on $[n]$. We partition the multi-set $\Omega$ into $\ell$ multi-sets so that $\Omega_1$ consists of the first $m_1$ elements of $\Omega$, $\Omega_2$ consists of the next $m_2$ elements of $\Omega$, and so on, where $\sum_{i=1}^\ell m_i = m$. Then, $\Omega_i$s are mutually independent and each $\Omega_i$ consists of i.i.d. random indices.
The version of the golfing scheme in this paper generates a dual certificate $v \in \cz^N$ from intermediate vectors $q_i \in \cz^N$ for $i=0,\ldots,\ell-1$ by $$v = \sum_{i=1}^\ell \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - T T^\dagger \right) q_{i-1},$$ where $q_i$s are generated as follows: first, initialize $q_0 = \mbox{sgn}(T x)$; next, generate $q_i$s recursively by $$q_i = \Pi_J \left( \widetilde{T} T^* - \frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* \right) q_{i-1}, \quad i = 1,\ldots,\ell-1.$$ Here, $\widetilde{T}$ denotes the adjoint of $T^\dagger$.
Note that $$\begin{aligned}
{} & (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^*
\left( \frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) \\
{} & = \frac{n}{m_i} (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^* \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^*
+ (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^* (I_N - \widetilde{T} T^*) \\
{} & = \frac{n}{m_i} (\widetilde{T} T^* - \widetilde{T} S_{\Omega'}^* S_{\Omega'} T^*) \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^*
+ (\widetilde{T} T^* - \widetilde{T} S_{\Omega'}^* S_{\Omega'} T^*) (I_N - \widetilde{T} T^*) = 0.\end{aligned}$$ Thus it follows that $v$ satisfies .
The rest of the proof is devoted to show that $v$ satisfies and , which follows similarly to the proof of [@candes2011probabilistic Lemma 3.3]. For completeness, we verify that the arguments in [@candes2011probabilistic] are valid in our setting (with neither isotropy nor self-adjointness).
We show that $q_i$ satisfies the following two properties with high probability for each $i \in [\ell]$: first, $$\label{eq:decay_qi}
\norm{q_i}_2 \leq c_i \norm{q_{i-1}}_2$$ and, second, $$\label{eq:bnd_qi_infty}
\left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \right\|_\infty \leq t_i \norm{q_{i-1}}_2.$$
Let $p_1(i)$ (resp. $p_2(i)$) denote the probability that the inequality in (resp. ) does not hold. Since $q_{i-1}$ is independent of $\Omega_i$, by Lemma \[lemma:E2\], $p_1(i)$ is upper-bounded by $$p_1(i) \leq \exp\left( - \frac{1}{4}(c_i \sqrt{m_i(1-\norm{\gamma T^* T - I_n})/(s\mu)}-1)^2 \right).$$ Therefore, $p_1(i) \leq \frac{1}{\alpha} e^{-\beta}$ if $$\label{eq:bndmi_p1}
m_i \geq \frac{2+8(\beta + \log \alpha)}{c_i^2} \cdot \frac{\mu s}{1-\norm{\gamma T^* T - I_n}}.$$ On the other hand, note that $q_{i-1} = \Pi_J q_{i-1}$. Then it follows that $$\begin{aligned}
\left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \right\|_\infty
{} & = \left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) \Pi_J q_{i-1} \right\|_\infty \\
{} & = \left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* - \widetilde{T} T^* \right) \Pi_J q_{i-1} \right\|_\infty.\end{aligned}$$ Again, we use the fact that $q_{i-1}$ is independent of $\Omega_i$. Then, by Lemma \[lemma:E3\], $p_2(i)$ is upper-bounded by $$p_2(i) \leq 2N \exp\left( - \frac{3t_i^2 m_i}{6\mu/(1-\norm{\gamma T^* T - I_n}) + 2 \mu \sqrt{s} t_i} \right).$$ Therefore, $p_2(i) \leq \frac{1}{\alpha} e^{-\beta}$ if $$\label{eq:bndmi_p2}
m_i \geq \left( \frac{2}{t_i^2 s (1-\norm{\gamma T^* T - I_n})} + \frac{2}{3t_i\sqrt{s}} \right) (\beta + \log(2\alpha) + \log N) s\mu.$$
We set the parameters similarly to the proof of [@candes2011probabilistic Lemma 3.3] as follows: $$\label{eq:param}
\begin{aligned}
\ell {} & = \left\lceil \frac{\log_2 s}{2} + \log_2 n
+ \log_2 \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right\rceil + 3, \\
c_i {} & =
\begin{cases}
1/\lceil 2 \sqrt{\log N} ~ \rceil & i=1,2,3, \\
1/2 & 3 \leq i \leq \ell,
\end{cases} \\
t_i {} & =
\begin{cases}
1/\lceil 4 \sqrt{s} ~\rceil & i=1,2,3, \\
\log N / \lceil 4 \sqrt{s} ~\rceil & 3 \leq i \leq \ell,
\end{cases}\\
m_i {} & = \lceil 10(1+\log 6+\beta) \mu s c_i^{-2} \rceil, \quad \forall i. \\
\end{aligned}$$ By the construction of $v$, we have $$\begin{aligned}
\Pi_J v {} & = \sum_{i=1}^\ell \Pi_J \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \\
{} & = \sum_{i=1}^\ell \left[ \Pi_J q_{i-1} - \Pi_J \left(\widetilde{T} T^* - \frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* \right) q_{i-1} \right] \\
{} & = \sum_{i=1}^\ell \left( q_{i-1} - q_i \right)
= q_0 - q_\ell \\
{} & = \mbox{sgn}(T x) - q_\ell = \Pi_J \mbox{sgn}(T x) - q_\ell.\end{aligned}$$ Therefore, implies $$\norm{\Pi_J (v - \mbox{sgn}(T x))}_2
= \norm{q_\ell}_2
\leq \prod_{i=1}^\ell c_i \norm{\mbox{sgn}(T x)}_2
\leq \frac{\sqrt{s}}{2^\ell \log N}.$$
Next, by and , we have $$\label{eq:dualcert_bnd_proof}
\begin{aligned}
\norm{\Pi_{[N] \setminus J} v}_\infty
{} & \leq \sum_{i=1}^\ell \left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \right\|_\infty \\
{} & \leq \sum_{i=1}^\ell t_i \norm{q_{i-1}}_2 \\
{} & \leq \sqrt{s} \left(t_1 + \sum_{i=2}^\ell t_i \prod_{j=1}^{i-1} c_j\right).
\end{aligned}$$ By setting parameters as in , the right-hand side in is further upper-bounded by $$\frac{1}{4} \left( 1 + \frac{1}{2\sqrt{\log N}} + \frac{\log N}{4 \log N} + \cdots \right) < \frac{1}{2}.$$ Then, we have shown that $v$ satisfies .
It remains to show that and hold with the desired probability. From and , it follows that $$p_j(i) \leq \frac{1}{6} e^{-\beta}, \quad \forall i \in [\ell], ~ \forall j=1,2.$$ In particular, we have $$\sum_{j=1}^2 \sum_{i=1}^3 p_j(i) \leq e^{-\beta}.$$ This implies that the first three $\Omega_i$s satisfy and except with probability $e^{-\beta}$.
On the other hand, we also have $$p_1(i) + p_2(i) < \frac{1}{3}, \quad \forall i = 4,\ldots,\ell.$$ In other words, the probability that $\Omega_i$ satisfies and is at least $2/3$. The union bound doesn’t show that $\Omega_i$ satisfies and for all $i \geq 4$ with the desired probability.
As in the proof of [@candes2011probabilistic Lemma 3.3], we adopt the oversampling and refinement strategy by Gross [@gross2011recovering]. Recall that each random index set $\Omega_i$ consists of i.i.d. random indices following the uniform distribution on $[n]$. Thus $\Omega_i$s are mutually independent. In particular, we set $\Omega_i$s are of the same cardinality in . Therefore, $\Omega_i$s are i.i.d. random variables. We generate a few extra copies of $\Omega_i$ for $i = \ell+1,\ldots,\ell'+3$ where $\ell' = 3(\ell-3)$. Then, by Hoeffding’s inequality, there exist at least $\ell-3$ $\Omega_i$s for $i \geq 4$ that satisfy and with probability $1 - 1/n$. (We refer more technical details for this step to [@candes2011probabilistic Section III.B].) Therefore, there are $\ell$ good $\Omega_i$s satisfying and with probability $1 - e^{-\beta} - 1/n$, and the dual certificate $v$ is constructed from these good $\Omega_i$s. The total number of samples for this construction requires $$m \geq \frac{40(1+\log 6 +\beta)\mu s (3\log N + 3\ell)}{1 - \norm{\gamma T^* T - I_n}},$$ which can be simplified as $$m \geq \frac{C(1+\beta) \mu s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right]$$ for a numerical constant $C$.
Proofs for Theorems \[thm:stability1\] and \[thm:stability2\] {#sec:pf_main_result_noisy}
=============================================================
In this section, we prove Theorems \[thm:stability1\] and \[thm:stability2\], which provide sufficient conditions for stable recovery of sparse signals in a transform domain from noisy data.
Proof of Theorem \[thm:stability1\] {#subsec:pf:thm:stability1}
-----------------------------------
Let $h := \hat{x} - x$. Since $\hat{x}$ is the minimizer to , it follows that $$\label{eq:pf_thm_stability:bnd1}
\begin{aligned}
\norm{T x}_1
{} & \geq \norm{T \hat{x}}_1 = \norm{T x + T h}_1 \\
{} & \geq \norm{T x + T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_1 - \norm{T S_{\Omega'}^* S_{\Omega'} h}_1.
\end{aligned}$$
Since $x$ and $\hat{x}$ are feasible for , it follows that $$\label{eq:noisy_feas}
\begin{aligned}
{} & \max\left( \norm{T S_{\Omega'}^* S_{\Omega'} (\hat{x} - x^\sharp)}_2, ~ \norm{T S_{\Omega'}^* S_{\Omega'} (x - x^\sharp)}_2 \right) \\
{} & \leq \norm{T} \max\left( \norm{S_{\Omega'} (\hat{x} - x^\sharp)}_2, ~ \norm{S_{\Omega'} (x - x^\sharp)}_2 \right) \leq \epsilon \norm{T}.
\end{aligned}$$ Therefore, by the triangle inequality, we have $$\label{eq:tube_cstr}
\norm{T S_{\Omega'}^* S_{\Omega'} h}_2 = \norm{T S_{\Omega'}^* S_{\Omega'} (\hat{x} - x)}_2
\leq \norm{T S_{\Omega'}^* S_{\Omega'} (\hat{x} - x^\sharp)}_2 + \norm{T S_{\Omega'}^* S_{\Omega'} (x - x^\sharp)}_2 \leq 2 \epsilon \norm{T}.$$
The rest of the proof will compute upper bounds on $\norm{T h}_2$ in two complementary cases similarly to the proof of Lemma \[lemma:uniqueness\]. Unlike the noiseless case ($x^\sharp = x$ and $\epsilon = 0$) in Lemma \[eq:bnd\_qi\_infty\], the condition in does not necessarily imply $S_\Omega h = 0$. In fact, the proof of Lemma \[lemma:uniqueness\] critically depends on the condition $S_\Omega h = 0$. Essentially, we replace $h$ by $(I_n - S_{\Omega'}^* S_{\Omega'}) h$. Then it follows that $$\label{eq:noisy_vanish}
%B_{\Omega'} (B - B_{\Omega'}) T h = 0.
S_{\Omega'} (I_n - S_{\Omega'}^* S_{\Omega'}) h = 0.$$
**Case 1:** We first consider the case when $(I_n - S_{\Omega'}^* S_{\Omega'}) h$ satisfies $$\label{eq:case1noisy}
\norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
\leq 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2,$$ where $J$ denotes the support of $T x$.
Under , similarly to the proof of Lemma \[lemma:uniqueness\], we have $$\label{eq:pf_thm_stability:bnd2}
\norm{T x + T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_1
\geq \norm{T x}_1 + \frac{1}{14} \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2.$$ Combining and provides $$\label{eq:pf_thm_stability:bnd3}
\norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
\leq 14 \norm{T S_{\Omega'}^* S_{\Omega'} h}_1
\leq 14 \sqrt{N} \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \leq 28\sqrt{N} \epsilon \norm{T}.$$ On the other hand, implies $$\label{eq:pf_thm_stability:bnd4}
\begin{aligned}
\norm{T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
{} & \leq \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2 + \norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2 \\
{} & \leq (1 + 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}) \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2.
\end{aligned}$$ Therefore, combining \[eq:tube\_cstr,eq:pf\_thm\_stability:bnd3,eq:pf\_thm\_stability:bnd4\] provides $$\label{eq:pf_thm_stability:bnd5}
\norm{T h}_2 \leq \left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon \norm{T}.$$
**Case 2:** Next, we consider the complementary case when $(I_n - S_{\Omega'}^* S_{\Omega'}) h$ satisfies $$\label{eq:case2noisy}
\norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
> 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2.$$
Again, similarly to the proof of Lemma \[lemma:uniqueness\], we get $(I_n - S_{\Omega'}^* S_{\Omega'}) h = 0$. Therefore, $$\norm{T h}_2 \leq \norm{T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2 + \norm{T S_{\Omega'}^* S_{\Omega'} T h}_2 \leq 2 \epsilon \norm{T},$$ which is smaller than the upper bound on $\norm{T h}_2$ in the previous case.
By applying $\norm{h}_2 \leq \norm{T h}_2 / \sigma_{\min}(T)$ to , we obtain the desired upper bound on $\norm{h}_2$. This completes the proof.
Proof of Theorem \[thm:stability2\] {#subsec:pf:thm:stability2}
-----------------------------------
By Theorem \[thm:rboplike\], the condition in implies that $\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger$ satisfies the condition in with constant $\delta = 1/3$. Note that the estimates in Lemmas \[lemma:E1,lemma:E2,lemma:E3\] are implied by . Therefore, the rest of the proof will be identical to that of Theorem \[thm:stability2\] except that we compute a tighter upper bound on $\norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2$ as follows.
First, we decompose $\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h$ as $$\label{eq:pf_thm_stability2:decomp}
\begin{aligned}
\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h
{} & = \Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) T h \\
{} & \quad + \Pi_J \frac{n}{m} T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h \\
{} & \quad + \Pi_J \left( \frac{n}{m} - 1 \right) T S_{\Omega'}^* S_{\Omega'} h.
\end{aligned}$$
Let $J_1$ correspond to the indices of the $s$-largest coefficients of $\Pi_{[N] \setminus J} T h$; $J_2$ to the indices of the next $s$-largest coefficients of $\Pi_{[N] \setminus J} T h$, and so on. Then the $\ell_2$-norm first term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd1}
\begin{aligned}
{} & \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) T h\right\|_2 \\
{} & \leq \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J \cup J_1} T h\right\|_2 \\
{} & \quad + \sum_{i \geq 2} \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J_i} T h\right\|_2.
\end{aligned}$$ By , the first term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd2}
\left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J \cup J_1} T h\right\|_2
\leq \frac{1}{3} \norm{\Pi_{J \cup J_1} T h}_2
\leq \frac{1}{3} \norm{T h}_2.$$
By , the second term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd3}
\begin{aligned}
{} & \sum_{i \geq 2} \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J_i} T h\right\|_2 \\
{} & \leq \frac{1}{3} \sum_{i \geq 2} \norm{\Pi_{J_i} T h}_2
\leq \frac{1}{3\sqrt{s}} \sum_{i \geq 1} \norm{\Pi_{J_i} T h}_1
= \frac{1}{3\sqrt{s}} \norm{\Pi_{[N] \setminus J} T h}_1.
\end{aligned}$$ Since $\hat{x}$ is the minimizer to , we have the so called “cone” constraint: $$\norm{T x}_1 \geq \norm{T \hat{x}}_1 \geq \norm{Tx + T h}_1 \geq \norm{\Pi_J T x}_1 - \norm{\Pi_J T h}_1 + \norm{\Pi_{[N] \setminus J} T h}_1 - \norm{\Pi_{[N] \setminus J} T x}_1,$$ which implies $$\norm{\Pi_{[N] \setminus J} T h}_1 \leq 2 \norm{\Pi_{[N] \setminus J} T x}_1 + \norm{\Pi_J T h}_1 = \norm{\Pi_J T h}_1,$$ where the last step follows since $T x$ is supported on $J$. Therefore, implies $$\label{eq:pf_thm_stability2:bnd4}
\sum_{i \geq 2} \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J_i} T h\right\|_2
\leq \frac{1}{3\sqrt{s}} \norm{\Pi_{[N] \setminus J} T h}_1
\leq \frac{1}{3\sqrt{s}} \norm{\Pi_J T h}_1
\leq \frac{1}{3} \norm{\Pi_J T h}_2.$$
Plugging and to provides $$\label{eq:pf_thm_stability2:bnd5}
\left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) T h\right\|_2
\leq \frac{2}{3} \norm{T h}_2.$$
Next, we derive an upper bound on the $\ell_2$-norm of the second term in the right-hand side of . Since $\Omega'$ consists of distinct elements in $\Omega$, it follows that $$S_{\Omega'}^* S_{\Omega'} S_\Omega^* S_\Omega
= \sum_{k \in \Omega} S_{\Omega'}^* S_{\Omega'} e_k e_k^*
= \sum_{k \in \Omega} e_k e_k^*
= S_\Omega^* S_\Omega.$$ Furthermore, $S_{\Omega'}^* S_{\Omega'}$ is idempotent. Therefore, we obtain the following identity: $$\label{eq:pf_thm_stability2:id1}
S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'} = (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) S_{\Omega'}^* S_{\Omega'}.$$ From , we get $$\label{eq:pf_thm_stability2:bnd6}
\begin{aligned}
\norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h}_2
{} & = \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & = \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) T^\dagger T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) T^\dagger} \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq \max_k \norm{T e_k e_k^* T^\dagger} (|\Omega|-|\Omega'|) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2.
\end{aligned}$$ Then the $\ell_2$-norm of the second term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd7}
\begin{aligned}
\left\|\Pi_J \frac{n}{m} T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h\right\|_2
{} & \leq \frac{n}{m} \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h}_2 \\
{} & \leq \frac{n}{m} \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2.
\end{aligned}$$
By applying and to , then combining the result with and , we get $$\begin{aligned}
\norm{T h}_2
\leq 2 \epsilon \norm{T} + 28 \sqrt{N} \epsilon \norm{T} + \frac{2}{3} \norm{T h}_2
+ 2 \frac{n}{m} \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) \epsilon \norm{T}
+ 2 \left( \frac{n}{m} - 1 \right) \epsilon \norm{T},\end{aligned}$$ which implies $$\norm{T h}_2 \leq 6 \left[ 14\sqrt{N} + \frac{n}{m} \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) + 1 \right) \right] \epsilon \norm{T}.$$
In the case when $T^* T = I_n$, $T e_k e_k^* T^\dagger$s correspond to orthogonal projections onto mutually orthogonal one-dimensional subspaces. Recall that the summands in $T S_\Omega^* S_\Omega T^\dagger$ repeat at most $R$ times. Then the summands in $T S_\Omega^* S_\Omega T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger$ repeat at most $R - 1$ times. Therefore, we get a sharper estimate than that in given by $$\label{eq:pf_thm_stability2:bnd8}
\begin{aligned}
\norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h}_2
{} & \leq \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) T^\dagger} \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq (R - 1) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2.
\end{aligned}$$ This completes the proof for the second claim.
Numerical Results {#sec:numres}
=================
In this section, we conduct numerical experiments of solving and , with partial Fourier measurements and several different sparsifying transforms. We compare different sampling schemes in Monte-Carlo experiments, and observe that the variable density sampling schemes proposed in and yield superior recovery results in terms of success rate.
The optimization problems and are solved using Alternating Direction Method of Multipliers (ADMM) [@boyd2011distributed]. For example, for $T=\Phi\Psi^*\in\cz^{n\times n}$, is rewritten in the following form with two linear constraints, and solved by ADMM with two additional terms in the augmented Lagrangian. $$\begin{array}{ll}
\displaystyle \minimize_{y \in \cz^n} & \norm{y}_1 \\
\mathrm{subject~to} & T g = y,\\
& S_\Omega g = S_\Omega x.
\end{array}$$ In all experiments, the ADMM algorithm runs for 1,000 iterations.
1D Signals
----------
For 1D signals, we use the DFT $\Psi\in\cz^{n\times n}$, and two sparsifying transforms $\Psi_{\mathrm{TV}, n}, \Psi_{\mathrm{W}, n, \ell}\in \cz^{n\times n}$, denoting the finite difference operator and the discrete Haar wavelet [@mallat2008wavelet] at level $\ell$, respectively. For the finite difference operator $\Psi_{\mathrm{TV}, n}$, we use the two-step scheme in Section \[sec:TV\]. In Step 2), we either use the uniform density according to Corollary \[cor:circulant1\] (see Fig. \[fig:1d\_a\]), or the distribution restricted to $\{2,3,\dots, n\}$ computed from the other transform $\Psi_{\mathrm{W}, n, \ell}$ (see Fig. \[fig:1d\_b\]). For the discrete Haar wavelet $\Psi_{\mathrm{W}, n, \ell}$, we test two distributions on $[n]$ – the uniform distribution (see Fig. \[fig:1d\_a\]) and the variable density distribution (see Fig. \[fig:1d\_b\]).
In the numerical experiments, we choose $n = 512$, and synthesize signals $f$ at sparsity levels $s = 16, 32, 48, \dots, 128$ with respect to the above two sparsifying transforms. We use the Haar wavelet at level $\ell = 6$. To make the experiments more realistic, we synthesize the sparse signals by thresholding a real-world signal (see Fig. \[fig:1d\_sig\]) in the transform domains. For the discrete Haar wavelet, we analyze the signal using $\Psi_{\mathrm{W}, n, \ell}$, zero out small coefficients to achieve a certain sparsity level, and synthesize the signal using $\Psi_{\mathrm{W}, n, \ell}^*$. Due to the non-injectivity of the finite difference operator, we replace the last row of $\Psi_{\mathrm{TV}, n}$ by $[1,1,\dots, 1]^\top \in\cz^n$ in the signal analysis and synthesis steps.
For every sampling scheme, we repeat the experiment for $m = 16, 32, 48, \dots, 512$. We run 50 Monte-Carlo experiments for each setting. An instance is declared a success if the following reconstruction signal-to-noise ratio (RSNR) of the solution $\hat{g}$ to exceeds $60$dB. $$\mathrm{RSNR} = -20\log_{10}\left(\frac{\norm{\hat{g}-x}_2}{\norm{x}_2}\right).$$ The success rate is computed for every pair $(s, m)$ and shown in Fig. \[fig:1d\_c\] – \[fig:1d\_f\].
For signal recovery using the Haar wavelet, the variable density sampling scheme yields higher success rate than the uniform density sampling scheme (see Fig. \[fig:1d\_c\] and \[fig:1d\_d\]). For 1D TV, the density computed using 1D TV, which is uniform, yields higher success rate than that computed from the Haar wavelet (see Fig. \[fig:1d\_e\] and \[fig:1d\_f\]). These experiments show that one can recover sparse signals more successfully using the sampling density computed from the specific sparsifying transform used in the recovery, which is adaptive to local incoherence between the transform and the measurement. It is commonly believed among practitioners that sampling more densely in the low frequency region, where the energy of a natural signal is concentrated, always provides a better reconstruction. However, that turns out not be the case when one cares about perfect recovery of exactly sparse signals in a certain transform domain from noise-free measurements.
\
\
\
2D Signals
----------
We also run numerical experiments on a 2D signal, the modified Shepp-Logan phantom (see Fig. \[fig:2d\_sig\]). We minimize the anisotropic and isotropic 2D total variations, which correspond to solving and for the 2D finite difference operator $\Phi_{\mathrm{TV},n_1,n_2}$.
For the two types of total variations, we use the two-step scheme in Section \[sec:TV\]. In Step 2), we use the densities in and for anisotropic and isotropic total variations, respectively. As a comparison, we use the variable density proposed by Krahmer and Ward [@krahmer2014stable], which is computed with the 2D separable Haar wavelet $\Psi_{\mathrm{W}, n_2, \log_2n_2}\otimes \Psi_{\mathrm{W}, n_1, \log_2n_1}$.
We use a phantom of size $n = n_1\times n_2 = 256\times 256$, and repeat the experiments for $m= n/32, n/16,3n/32,\dots, n/4$. We run 50 Monte-Carlo experiments for each setting, and compute the success rate for every chioce of $m$, as shown in Fig. \[fig:2d\_d\] and \[fig:2d\_e\].
For both anisotropic TV minimization and isotropic TV minimization , the success rates using sampling density computed from TV are higher than those using sampling density computed from separable Haar wavelet. Although the performances of two sampling schemes are relatively close for anisotropic TV minimization (see Fig. \[fig:2d\_d\]), the advantage of the density computed from TV over that computed from wavelet is more pronounced for isotropic TV minimization (see Fig. \[fig:2d\_e\]). Contrary to common belief, sampling low frequencies more densely does not always lead to superior recovery. We suggest using sampling densities and tailored to the specific sparsifying transform and the measurement operator.
Conclusion {#sec:concl}
==========
In this paper, we established a unified theory for recovery of sparse signals in a transform domain. Our theory guaranteed robust recovery from noisy measurements by convex programming and apply without relying on a particular choice of measurement and sparsifying transforms. We quantified the sufficient sampling rate using functions of the two transforms, and this result identifies a class of measurement and sparsity models enabling recovery at a near optimal sampling rate. We also proposed a variable sampling density designed with incoherence parameters of the two transforms, which provided recovery guarantee at a lower sampling rate than previous works in various scenarios. Furthermore, we extended the result to the group-sparsity models so that it also applies to the popular isotropic total variation minimization. In particular, for the partial Fourier recovery of sparse signals over a circulant transform, our theory suggests a uniformly random sampling or its variation. Our numerical results showed that our variable density random sampling strategy outperforms other known sampling strategies in various scenarios. This suggests that our new theory is indeed universally useful.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank referees for their valuable comments and suggestions.
Bernstein inequalities
======================
\[thm:mtx\_bernstein\_ineq\] Let $\{X_j\} \in \cz^{d \times d}$ be a finite sequence of independent random matrices. Suppose that $\mathbb{E} X_j = 0$ and $\norm{X_j} \leq B$ almost surely for all $j$ and $$\max\left( \left\| \sum_j \mathbb{E} X_j X_j^* \right\|,~ \left\| \sum_j \mathbb{E} X_j^* X_j \right\| \right) \leq \sigma^2.$$ Then for all $t \geq 0$, $$\mathbb{P} \left( \left\| \sum_j X_j \right\| \geq t \right)
\leq 2d \exp \left( \frac{-t^2/2}{\sigma^2 + Bt/3} \right).$$
\[thm:vec\_bernstein\_ineq\] Let $\{v_j\} \in \cz^d$ be a finite sequence of independent random vectors. Suppose that $\mathbb{E} v_j = 0$ and $\norm{v_j}_2 \leq B$ almost surely for all $j$ and $\mathbb{E} \sum_j \norm{v_j}_2^2 \leq \sigma^2$. Then for all $0 \leq t \leq \sigma^2/B$, $$\mathbb{P} \left( \left\| \sum_j v_j \right\|_2 \geq t \right)
\leq \exp \left( - \frac{t^2}{8 \sigma^2} + \frac{1}{4} \right).$$
Proof of Lemma \[lemma:E1\] {#sec:pf:lemma:E1}
===========================
Define $$X_j := \Pi_J (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_J, \quad \forall j \in [m].$$ Then, $X_j$ satisfies $\mathbb{E} X_j = 0$ and $\norm{X_j} \leq \mu s$ for all $j$. Since $$\begin{aligned}
X_j^* X_j
{} & = n^2 \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & \quad - n \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T T^\dagger \Pi_J \\
{} & \quad - n \Pi_J T T^\dagger \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & \quad + \Pi_J T T^\dagger \Pi_J T T^\dagger \Pi_J,\end{aligned}$$ it follows that $$\begin{aligned}
\mathbb{E} X_j^* X_j
{} & = \mathbb{E} n^2 \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J - \Pi_J T T^\dagger \Pi_J T T^\dagger \Pi_J \\
{} & \leq \mathbb{E} n^2 e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & = \mathbb{E} n^2 \norm{\gamma^{1/2} \Pi_J T e_{\omega_j}}_2^2 \gamma^{-1} \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & \leq \mathbb{E} n \mu s \gamma^{-1} \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & = \mu s \gamma^{-1} \Pi_J \widetilde{T} T^\dagger \Pi_J.\end{aligned}$$ By symmetry, we also have $$\mathbb{E} X_j X_j^* \leq \mu s \gamma \Pi_J T T^* \Pi_J.$$ Therefore, $$\max\left( \left\| \sum_{j=1}^m \mathbb{E} X_j X_j^* \right\|,~ \left\| \sum_{j=1}^m \mathbb{E} X_j^* X_j \right\| \right)
\leq m \mu s \max( \norm{\gamma^{-1} \widetilde{T} T^\dagger}, \norm{\gamma T T^*} )
\leq \frac{m \mu s}{1 - \norm{\gamma T^* T - I_n}}.$$ Applying the above results to Theorem \[thm:mtx\_bernstein\_ineq\] with $t = m \delta$ completes the proof.
Proof of Lemma \[lemma:E2\] {#sec:pf:lemma:E2}
===========================
Define $$v_j := \Pi_J (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_J q, \quad \forall j \in [m].$$ Then, $v_j$ satisfies $\mathbb{E} v_j = 0$ and $$\begin{aligned}
\norm{v_j}_2
{} & \leq \norm{\Pi_J n T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q}_2 + \norm{\Pi_J T T^\dagger \Pi_J q}_2 \\
{} & \leq (\norm{\gamma^{1/2} \Pi_J \sqrt{n} T e_{\omega_j}}_2 \norm{\gamma^{-1/2} \Pi_J \sqrt{n} \widetilde{T} e_{\omega_j}}_2 + 1) \norm{\Pi_J q}_2 \\
{} & \leq (s\mu+1) \norm{\Pi_J q}_2.\end{aligned}$$ Furthermore, $$\begin{aligned}
\mathbb{E} \norm{v_j}_2^2
{} & = \mathbb{E} n^2 q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q
- q^* \Pi_J T T^\dagger \Pi_J q \\
{} & \leq n \norm{\gamma^{1/2} \Pi_J T e_{\omega_j}}_2^2 \mathbb{E} n \gamma^{-1} q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q \\
{} & \leq \mu s \gamma^{-1} \mathbb{E} n q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q \\
{} & = \mu s \gamma^{-1} q^* \Pi_J \widetilde{T} T^\dagger \Pi_J q \\
{} & \leq \frac{\mu s \norm{\Pi_J q}_2^2}{1 - \norm{\gamma T^* T - I_n}},\end{aligned}$$ where the second inequality follows by the incoherence property. Applying the above results to Theorem \[thm:vec\_bernstein\_ineq\] completes the proof.
Proof of Lemma \[lemma:E3\] {#sec:pf:lemma:E3}
===========================
Let $i \in [n] \setminus J$ be arbitrarily fixed. Define $$w_j := \langle e_i, (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_J q \rangle, \quad \forall j \in [m].$$ Then, $w_j$ satisfies $\mathbb{E} w_j = 0$ and $$\begin{aligned}
|w_j|
{} & \leq |e_j^* n T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q| + \norm{T T^\dagger \Pi_J q}_\infty \\
{} & \leq (\norm{\gamma^{1/2} \sqrt{n} T e_{\omega_j}}_\infty \norm{\gamma^{-1/2} \sqrt{n} \Pi_J \widetilde{T} e_{\omega_j}}_2 + 1) \norm{\Pi_J q}_2 \\
{} & \leq (\sqrt{s} \mu+1) \norm{\Pi_J q}_2.\end{aligned}$$ Furthermore, $$\begin{aligned}
\mathbb{E} |w_j|^2
{} & = \mathbb{E} n^2 q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* e_i e_i^* T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q
- q^* \Pi_J T T^\dagger e_i e_i^* T T^\dagger \Pi_J q \\
{} & \leq n \norm{\gamma^{1/2} T e_{\omega_j}}_\infty^2 \gamma^{-1} \mathbb{E} n q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q \\
{} & \leq \mu \gamma^{-1} q^* \Pi_J \widetilde{T} T^\dagger \Pi_J q \\
{} & \leq \frac{\mu \norm{\Pi_J q}_2^2}{1 - \norm{\gamma T^* T - I_n}}.\end{aligned}$$ Applying the above results to Theorem \[thm:mtx\_bernstein\_ineq\] gives $$\mathbb{P}\left(\left| \left\langle e_i, \left(\frac{n}{m} B_\Omega^* - B\right) \Pi_J q \right\rangle \right| \geq t \norm{\Pi_J q}_2\right)
\leq \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).$$ Combine this for $i \in [N]$ with the union bound completes the proof.
Proof of Lemma \[lemma:E3’\] {#sec:pf:lemma:E3'}
============================
Let $i \in [n] \setminus J$ be arbitrarily fixed. Define $$v_j := \Pi_{\calG_i} (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_{\calG_J} q, \quad \forall j \in [m].$$ Then, $v_j$ satisfies $\mathbb{E} v_j = 0$ and $$\begin{aligned}
\norm{v_j}_2
{} & \leq \norm{\Pi_{\calG_i} n T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_{\calG_J} q}_2 + \norm{\Pi_{\calG_i} T T^\dagger \Pi_{\calG_J} q}_2 \\
{} & \leq (\norm{\gamma^{1/2} \sqrt{n} \Pi_{\calG_i} T e_{\omega_j}}_2 \norm{\gamma^{-1/2} \sqrt{n} \Pi_{\calG_J} \widetilde{T} e_{\omega_j}}_2 + 1) \norm{\Pi_{\calG_J} q}_2 \\
{} & \leq (\sqrt{s} \mu_\calG + 1) \norm{\Pi_{\calG_J} q}_2.\end{aligned}$$ Furthermore, $$\begin{aligned}
\mathbb{E} v_j^* v_j
{} & = \mathbb{E} n^2 q^* \Pi_{\calG_J} \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_{\calG_i} T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_{\calG_J} q
- q^* \Pi_{\calG_J} T T^\dagger \Pi_{\calG_i} T T^\dagger \Pi_{\calG_J} q \\
{} & \leq n \norm{\gamma^{1/2} \Pi_{\calG_i} T e_{\omega_j}}_2^2 \gamma^{-1} \mathbb{E} n q^* \Pi_{\calG_J} \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_{\calG_J} q \\
{} & \leq \mu_\calG \gamma^{-1} q^* \Pi_{\calG_J} \widetilde{T} T^\dagger \Pi_{\calG_J} q \\
{} & \leq \frac{\mu_\calG \norm{\Pi_{\calG_J} q}_2^2}{1 - \norm{\gamma T^* T - I_n}}.\end{aligned}$$ Note that $$v_j v_j^* \leq v_j^* v_j I_L, \quad \forall j \in [m].$$ Applying the above results to Theorem \[thm:vec\_bernstein\_ineq\] gives $$\mathbb{P}\left(\left\| \Pi_{\calG_i} \left(\frac{n}{m} B_\Omega^* - B\right) \Pi_{\calG_J} q \right\|_2 \geq t \norm{\Pi_{\calG_J} q}_2\right)
\leq \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).$$ Combine this for $i \in [N]$ with the union bound completes the proof.
Proof of Theorem \[thm:rboplike\] {#sec:pf:thm:rboplike}
=================================
Theorem \[thm:rboplike\] is analogous to [@lee2013oblique Theorem 3.1]. It has been shown that if $T$ is of full row rank, then the deviation of $\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger$ from $T T^\dagger = I_N$ is small with high probability [@lee2013oblique Theorem 3.1]. On the contrary, Theorem \[thm:rboplike\] assumes that $T$ is of full column rank and shows that the deviation of $\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger$ from $T T^\dagger$, which is not necessarily $I_N$, is small with high probability.
The proof of Theorem \[thm:rboplike\] is obtained from that of [@lee2013oblique Theorem 3.1] by replacing $I_N$ by $T T^\dagger$. For example, the isotropy condition $$\frac{n}{m} \mathbb{E} T S_\Omega^* S_\Omega T^\dagger = I_N$$ is replaced by $$\frac{n}{m} \mathbb{E} T S_\Omega^* S_\Omega T^\dagger = T T^\dagger.$$ For a matrix $M \in \cz^{N \times N}$, the term $\theta_s(M)$, previously defined by in [@lee2013oblique] $$\theta_s(M) := \max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}} (M - I_N) \Pi_{\widetilde{J}}}$$ is replaced by $$\theta_s(M) := \max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}} (M - T T^\dagger) \Pi_{\widetilde{J}}}$$ Clearly, $T T^\dagger = I_N$ if $T$ has full row rank. However, in the hypothesis of Theorem \[thm:rboplike\], $T$ has full column rank and $T T^\dagger$ may be rank deficient. Since the modifications are rather straightforward, we omit the details of the proof and refer them to [@lee2013oblique Appendix E].
By modifying [@lee2013oblique Theorem 3.1] and its proof as shown above, is implied by and $$m \geq C_1 \delta^{-2} K_T \mu s \log^2 s \log N \log m,$$ where the factor $K_T$ is given by $$\begin{aligned}
K_T
{} & = \left\{
\left(2 + \max_{|\widetilde{J}| \leq s} \left\|\Pi_{\widetilde{J}} (\gamma T T^* - T T^\dagger) \Pi_{\widetilde{J}}\right\|\right)^{1/2}
+
\left(2 + \max_{|\widetilde{J}| \leq s} \left\|\Pi_{\widetilde{J}} (\gamma^{-1} \widetilde{T} T^\dagger - T T^\dagger) \Pi_{\widetilde{J}}\right\|\right)^{1/2}
\right\}^2 \\
{} & \leq 4 + 2 \max\left(
\max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}}(\gamma T T^* - T T^\dagger)\Pi_{\widetilde{J}}}
,~
\max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}}(\gamma^{-1} \widetilde{T} T^\dagger - T T^\dagger)\Pi_{\widetilde{J}}}
\right) \\
{} & \leq 4 + 2 \max\left(
\norm{\gamma T T^* - T T^\dagger}
,~
\norm{\gamma^{-1} \widetilde{T} T^\dagger - T T^\dagger}
\right) \\
{} & = 4 + 2 \max\left(
\norm{\gamma T^* T - I_n}
,~
\norm{\gamma^{-1} T^\dagger \widetilde{T} - I_n}
\right).\end{aligned}$$ Finally, we verify that $$\norm{\gamma^{-1} T^\dagger \widetilde{T} - I_n} \leq \frac{1}{\norm{\gamma T^* T - I_n}},$$ where the upper bound dominates $\norm{\gamma T^* T - I_n}$. This completes the proof.
[^1]: This work was supported in part by the National Science Foundation under grants IIS 14-47879 and Korea Science and Engineering Foundation under grants NRF-2013M3A9B2076548 and NRF-2016R1A2B3008104. K. Lee is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: kiryung@ece.gatech.edu). Y. Li is with the Coordinated Science Laboratory and the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (e-mail: yli145@illinois.edu). K.H. Jin is with the Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (e-mail: kyong.jin@epfl.ch). J.C. Ye is with the Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejon 305-701, Korea (e-mail: jong.ye@kaist.ac.kr).
[^2]: A Gaussian sensing matrix is also obtained in the form of by choosing $\Psi$ as a Gaussian matrix.
[^3]: Krahmer and Ward [@krahmer2014stable] considered orthonormal $\Phi$ and $\Psi$ respectively corresponding to the DFT and the Haar DWT. In this case, $T^\dagger = T^*$. Thus, $\widetilde{T} = T$.
[^4]: A subset of the authors [@ye2016compressive] sharpened the original analysis of the completion of structured low-rank matrices by Chen and Chi [@chen2014robust] particularly on the noise propagation in the recovery. In this paper, we generalize the improved version [@ye2016compressive].
|
---
abstract: 'Recent investigations of excitonic absorption spectra in cuprous oxide ($\mathrm{Cu_{2}O}$) have shown that it is indispensable to account for the complex valence band structure in the theory of excitons. In $\mathrm{Cu_{2}O}$ parity is a good quantum number and thus the exciton spectrum falls into two parts: The dipole-active exciton states of negative parity and odd angular momentum, which can be observed in one-photon absorption ($\Gamma_4^-$ symmetry) and the exciton states of positive parity and even angular momentum, which can be observed in two-photon absorption ($\Gamma_5^+$ symmetry). The unexpected observation of $D$ excitons in two-photon absorption has given first evidence that the dispersion properties of the $\Gamma_5^+$ orbital valence band is giving rise to a coupling of the yellow and green exciton series. However, a first theoretical treatment by Ch. Uihlein *et al.* \[Phys. Rev. B **23**, 2731 (1981)\] was based on a simplified spherical model. The observation of $F$ excitons in one-photon absorption is a further proof of a coupling between yellow and green exciton states. Detailed investigations on the fine structure splitting of the $F$ exciton by F. Schweiner *et al.* \[Phys. Rev. B **93**, 195203 (2016)\] have proved the importance of a more realistic theoretical treatment including terms with cubic symmetry. In this paper we show that the even and odd parity exciton system can be consistently described within the same theoretical approach. However, the Hamiltonian of the even parity system needs, in comparison to the odd exciton case, modifications to account for the very small radius of the yellow and green $1S$ exciton. In the presented treatment we take special care of the central-cell corrections, which comprise a reduced screening of the Coulomb potential at distances comparable to the polaron radius, the exchange interaction being responsible for the exciton splitting into ortho and para states, and the inclusion of terms in the fourth power of $p$ in the kinetic energy being consistent with $O_{\mathrm{h}}$ symmetry. Since the yellow $1S$ exciton state is coupled to all other states of positive parity, we show how the central-cell corrections affect the whole even exciton series. The close resonance of the $1S$ green exciton with states of the yellow exciton series has a strong impact on the energies and oscillator strengths of all implied states. The consistency between theory and experiment with respect to energies and oscillator strengths for the even and odd exciton system in $\mathrm{Cu_{2}O}$ is a convincing proof for the validity of the applied theory.'
author:
- Frank Schweiner
- Jörg Main
- Günter Wunner
- Christoph Uihlein
title: 'The even exciton series in Cu$_2$O'
---
Introduction \[sec:Introduction\]
=================================
Excitons are the quanta of fundamental optical excitations in both insulators and semiconductors in the visible and ultraviolet spectrum of light. The Coulomb interaction between electron and hole leads to a hydrogen-like series of excitonic states [@TOE]. Cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ is a prime example where one can even identify four different excitonic series (yellow, green, blue, and violet) being related to the two topmost valence bands and the two lowest conduction bands [@GRE]. Recently, the yellow series could be followed up to a spectacular high principal quantum number of $n=25$ [@GRE]. This outstanding experiment has launched the new field of research of giant Rydberg excitons and led to a variety of new theoretical and experimental investigations on the topic of excitons in $\mathrm{Cu_{2}O}$ [@28; @QC; @175; @75; @76; @50; @80; @100; @125; @78; @79; @150; @74; @77; @225].
$\mathrm{Cu_{2}O}$ has octahedral symmetry $O_{\mathrm{h}}$ so that the symmetry of the bands can be assigned by the irreducible representations $\Gamma_{i}^{\pm}$ of $O_{\mathrm{h}}$. The yellow and green exciton series share the same threefold degenerate $\Gamma_5^+$ orbital valence band state. This state splits due to spin-orbit interaction into an upper twofold degenerate $\Gamma_7^+$ valence band (yellow series) and a lower fourfold degenerate $\Gamma_8^+$ valence band (green series). The band structure of both bands is essentially determined by the anisotropic dispersion properties of the orbital state. The threefold degeneracy of the orbital state is lifted as soon as a non-zero $k$ vector gets involved, with new eigenvectors depending on the orientation of $\boldsymbol{k}$. A consequence of the splitting of the orbital state is a partial quenching of the spin-orbit interaction. This $\boldsymbol{k}$ dependent quenching is not only responsible for a remarkable non-parabolicity of the two top valence bands but leads likewise to a $\boldsymbol{k}$ dependent mixing of the $\Gamma_7^+$ and $\Gamma_8^+$ Bloch states and can thus cause a mixing of the yellow and green exciton series. A mixing of both series is favored by the large Rydberg energy of approximately $100\,\mathrm{meV}$, a corresponding large exciton extension in $k$ space and the small spin-orbit splitting of only $130\,\mathrm{meV}$.
A Hamiltonian that is able to cope with a coupled system of yellow and green excitons must take explicit care of the dispersion properties of the orbital valence band state and has to include the spin-orbit interaction. Such a kind of Hamiltonian was first introduced by Uihlein *et al.* [@7] for explaining the unexpected fine structure splitting observed in the two-photon absorption spectrum of $\mathrm{Cu_{2}O}$. They used a simplified spherical dispersion model for the $\Gamma_5^+$ orbital valence band with an identical splitting into longitudinal and transverse states independent of the orientation of $\boldsymbol{k}$. This simplification had the appealing advantage that the total angular momentum remains a good quantum number so that the exciton problem could be reduced to calculate the eigenvalues of a system of coupled radial wave functions. A problem in their paper is the incorrect notation of the $1S$ green and $2S$ yellow excitons states. Both notations need to be exchanged to be consistent with their calculations. Although the spherical model can explain many details of the experimental findings, one has to be aware of its limitations. A more realistic Hamiltonian being compliant with the real band structure by including terms of cubic symmetry has already proved its validity by explaining the puzzling fine structure of the odd parity states in $\mathrm{Cu_{2}O}$ [@100]. The intention of this paper is to show that the same kind of Hamiltonian can likewise describe the fine splitting of the even parity excitons.
However, when comparing the even parity and odd parity exciton systems, it is obvious that the even exciton system is a much more challenging problem. One reason for this is the close resonance of the green $1S$ exciton with the even parity states of the yellow series with principal quantum number $n\geq 2$. This requires a very careful calculation of the binding energy of the green $1S$ exciton. Furthermore, the binding energy of the yellow $1S$ exciton is much larger than expected from a simple hydrogen like series, inter alia, due to a less effective screening of the Coulomb potential at distances comparable to the polaron radius. Moreover, a breakdown of the electronic screening is expected at even much shorter distances, but a proper treatment is exceeding the limits of the continuum approximation. Hence, we introduce in this paper a $\delta$-function like central cell correction term that should account for all kinds of short range perturbations affecting the immediate neighborhood of the central cell. The magnitude of this term is treated as a free parameter that can be adjusted to the experimental findings. It is important to note that a change of this parameter leads to a significant shift of the green $1S$ exciton with respect to the higher order states of the yellow series and has therefore a high impact on the energies and the compositions of the resulting coupled exciton states. Taking this in mind it is fundamental that one can likewise achieve a match to the relative oscillator strengths of the involved states.
Dealing with the even parity system of $\mathrm{Cu_{2}O}$ is also confronting us with the problem of a proper treatment of the $1S$ exciton with respect to its very small radius since a small exciton radius means a large extension of the exciton in $k$ space. The challenge is therefore to meet the band structure of the valence band in a much larger vicinity of the $\Gamma$ point. For coping with this situation, we include in the kinetic energy of the hole all terms in the fourth power of $p$ being compliant with the octahedral symmetry of $\mathrm{Cu_{2}O}$. The parameters of these terms are carefully adjusted to get a best fit to the band structure in the part of the $k$ space being relevant for the $1S$ exciton.
Despite of all these modifications, it is important to note that the Hamiltonian is essentially the same as the one being applied to the odd exciton system [@100]. The fundamental modifications presented in this paper are irrelevant for the odd parity system because of their $\delta$ function like nature or their specific form affecting only exciton states with a small radius. Hence, we present a consistent theoretical model for the complete exciton spectrum of $\mathrm{Cu_{2}O}$.
Comparing our results to experimental data, we can prove very good agreement as regards not only the energies but also the oscillator strengths since our method of solving the Schrödinger equation allows us also to calculate relative oscillator strengths for one-photon and two-photon absorption. This agreement between theory and experiment is important not only for the investigation of exciton spectra in electric or combined electric and magnetic fields. A correct theoretical description of excitons is indispensable if Rydberg excitons will be used in the future in quantum information technology, or used to attain a deeper understanding of quasi-particle interactions in semiconductors [@77; @GRE]. Furthermore, this agreement is a prerequisite for a future search for exceptional points in the exciton spectrum [@50].
The paper is organized as follows: Having presented the Hamiltonian of excitons in $\mathrm{Cu_{2}O}$ when considering the complete valence band structure Sec. \[sec:Hamiltonian\], we discuss all corrections to this Hamiltonian due to the small radius of the $1S$ exciton in Sec. \[sec:ccc\]. In Sec. \[sec:eosc\] we show how to solve the Schrödinger equation using a complete basis and how to calculate relative oscillator strengths for one-photon and two-photon absorption. In Sec. \[sec:results\] we discuss the complete yellow and green exciton spectrum of $\mathrm{Cu_{2}O}$ paying attention to the exciton states with a small principal quantum number and especially to the green $1S$ exciton state. Finally, we give a short summary and outlook in Sec. \[sec:Summary-and-outlook\].
Hamiltonian \[sec:Hamiltonian\]
===============================
In this section we present the Hamiltonian of excitons in $\mathrm{Cu_{2}O}$, which accounts for the complete valence band structure of this semiconductor. This Hamiltonian describes the exciton states of odd parity with a principal quantum number $n\geq 3$ very well [@100; @125]. However, for the exciton states of even parity and for the $2P$ exciton corrections to this Hamiltonian are needed, which will be described in Sec. \[sec:ccc\].
The lowest $\Gamma_6^+$ conduction band in $\mathrm{Cu_{2}O}$ is almost parabolic in the vicinity of the $\Gamma$ point and the kinetic energy can be described by the simple expression $$H_{\mathrm{e}}\!\left(\boldsymbol{p}_{\mathrm{e}}\right)=\frac{\boldsymbol{p}_{\mathrm{e}}^{2}}{2m_{\mathrm{e}}},\label{eq:He}$$ with the effective electron mass $m_{\mathrm{e}}$. Since $\mathrm{Cu_{2}O}$ has cubic symmetry, we use the irreducible representations $\Gamma_{i}^{\pm}$ of the cubic group $O_{\mathrm{h}}$ to assign the symmetry of the bands.
Due to interband interactions and nonparabolicities of the three uppermost valence bands in $\mathrm{Cu_{2}O}$, the kinetic energy of the hole is given by the more complex expression [@80; @100; @125], $$\begin{aligned}
H_{\mathrm{h}}\!\left(\boldsymbol{p}_{\mathrm{h}}\right) & = & H_{\mathrm{so}}+\left(1/2\hbar^{2}m_{0}\right)\left\{ \hbar^{2}\left(\gamma_{1}+4\gamma_{2}\right)\boldsymbol{p}_{\mathrm{h}}^{2}\right.\phantom{\frac{1}{1}}\nonumber \\
& + & 2\left(\eta_{1}+2\eta_{2}\right)\boldsymbol{p}_{\mathrm{h}}^{2}\left(\boldsymbol{I}\cdot\boldsymbol{S}_{\mathrm{h}}\right)\phantom{\frac{1}{1}}\nonumber \\
& - & 6\gamma_{2}\left(p_{\mathrm{h}1}^{2}\boldsymbol{I}_{1}^{2}+\mathrm{c.p.}\right)-12\eta_{2}\left(p_{\mathrm{h}1}^{2}\boldsymbol{I}_{1}\boldsymbol{S}_{\mathrm{h}1}+\mathrm{c.p.}\right)\phantom{\frac{1}{1}}\nonumber \\
& - & 12\gamma_{3}\left(\left\{ p_{\mathrm{h}1},p_{\mathrm{h}2}\right\} \left\{ \boldsymbol{I}_{1},\boldsymbol{I}_{2}\right\} +\mathrm{c.p.}\right)\phantom{\frac{1}{1}}\nonumber \\
& - & \left.12\eta_{3}\left(\left\{ p_{\mathrm{h}1},p_{\mathrm{h}2}\right\} \left(\boldsymbol{I}_{1}\boldsymbol{S}_{\mathrm{h}2}+\boldsymbol{I}_{2}\boldsymbol{S}_{\mathrm{h}1}\right)+\mathrm{c.p.}\right)\right\} \phantom{\frac{1}{1}}\label{eq:Hh}\end{aligned}$$ with $\left\{ a,b\right\} =\frac{1}{2}\left(ab+ba\right)$, the free electron mass $m_{0}$, and c.p. denoting cyclic permutation. The quasi-spin $I=1$ describes the threefold degenerate valence band and is a convenient abstraction to denote the three orbital Bloch functions $xy$, $yz$, and $zx$ [@25]. The parameters $\gamma_{i}$, which are called the first three Luttinger parameters, and the parameters $\eta_{i}$ describe the behavior and the anisotropic effective mass of the hole in the vicinity of the $\Gamma$ point. The spin-orbit coupling, which enters Eq. (\[eq:Hh\]), is given by $$H_{\mathrm{so}}=\frac{2}{3}\Delta\left(1+\frac{1}{\hbar^{2}}\boldsymbol{I}\cdot\boldsymbol{S}_{\mathrm{h}}\right).\label{eq:soc}$$
In a first approximation, the interaction between the electron and the hole is described by a screened Coulomb potential $$V\!\left(\boldsymbol{r}_{e}-\boldsymbol{r}_{h}\right)=-\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon_{\mathrm{s}1}}\frac{1}{\left|\boldsymbol{r}_{e}-\boldsymbol{r}_{h}\right|}$$ with the dielectric constant $\varepsilon_{\mathrm{s}1}=7.5$. For small relative distances $r=\left|\boldsymbol{r}\right|=\left|\boldsymbol{r}_{e}-\boldsymbol{r}_{h}\right|$ corrections to this potential and to the kinetic energies $H_{\mathrm{e}}\left(\boldsymbol{p}_{\mathrm{e}}\right)$ and $H_{\mathrm{h}}\left(\boldsymbol{p}_{\mathrm{h}}\right)$ are needed, which will be described in Sec. \[sec:ccc\] and which will be denoted here by $V^{\mathrm{CCC}}$.
After introducing relative and center of mass coordinates [@90] and setting the position and momentum of the center of mass to zero, the complete Hamiltonian of the relative motion finally reads [@17_17; @7] $$\begin{aligned}
H & = & E_{\mathrm{g}}+V\!\left(\boldsymbol{r}\right)+H_{\mathrm{e}}\!\left(\boldsymbol{p}\right)+H_{\mathrm{h}}\!\left(\boldsymbol{p}\right)+V_{\mathrm{CCC}}\label{eq:H}\end{aligned}$$ with the energy $E_{\mathrm{g}}$ of the band gap.
Note that by setting the total momentum to zero, we neglect polariton effects, even though in experiments the polaritonic part is always present. However, when considering the experimental results of Refs. [@9_1; @9; @8], the polariton effect on the $1S$ exciton is on the order of tens of $\mu\mathrm{eV}$ and, hence, much smaller than the energy shifts considered here. Furthermore, in Ref. [@83] criteria for the experimental observability of polariton effects are given. Inserting the material parameters of $\mathrm{Cu_{2}O}$ and the experimental linewidths of the exciton states observed in Refs. [@GRE; @28], it can be shown that polariton effects are not observable for the exciton states of $n\geq 2$. We will discuss this in greater detail in a future work.
Central-cell corrections \[sec:ccc\]
====================================
Due to its small radius, the $1S$ exciton in $\mathrm{Cu_{2}O}$ is an exciton intermediate between a Frenkel and a Wannier exciton [@TOE]. Hence, appropriate corrections are needed to describe this exciton state correctly. The corrections, which allow for the best possible description of the exciton problem within the continuum approximation of the solid, are called central-cell corrections and have first been treated by Uihlein *et al* [@6; @7] and Kavoulakis *et al* [@1] for $\mathrm{Cu_{2}O}$. While Uihlein *et al* [@7] accounted for these corrections only in a simplified way by using a semi-empirical contact potential $V=-V_0\delta\!\left(\boldsymbol{r}\right)$, the treatment of Kavoulakis *et al* [@1] did non account for the band structure and the effect of the central-cell corrections was discussed only on the $1S$ state and only using perturbation theory. By considering the complete valence band structure of $\mathrm{Cu_{2}O}$ in combination with a non-perturbative treatment of the central-cell corrections, we present a more accurate treatment of the whole yellow exciton series in $\mathrm{Cu_{2}O}$. Corrections beyond the frame of the continuum approximation will not be treated here. However, these corrections may describe remaining small deviations between experimental and theoretical results.
The central-cell corrections as discussed in Ref. [@1] comprise three effects, which are (i) the appearance of terms of higher-order in the momentum $\boldsymbol{p}$ in the kinetic energies of electron and hole, (ii) the momentum- and frequency-dependence of the dielectric function $\varepsilon$, and (iii) the appearance of an exchange interaction, which depends on the momentum of the center of mass.
Band structure of Cu$_2$O \[sec:Bandstruc\]
-------------------------------------------
Since the radius of the yellow $1S$ exciton is small, the extension of its wave function in momentum space is accordingly large. Hence, we have to consider terms of the fourth power of $\boldsymbol{p}$ in the kinetic energy of the electron and the hole. The inclusion of $p^4$ terms in Eqs. (\[eq:He\]) and (\[eq:Hh\]) leads to an extended and modified Hamiltonian in the sense of Altarelli, Baldereschi and Lipari [@17_17_18; @17_17_26; @7_11; @17_17; @17_15] or Suzuki and Hensel [@44_12].
The extended Hamiltonian must be compatible with the symmetry $O_{\mathrm{h}}$ of the crystal and transform according to the irreducible representation $\Gamma_1^+$. All the terms of the fourth power of $\boldsymbol{p}$ span a fifteen-dimensional space with the basis functions $$p_i^4,\quad p_i^3 p_j,\quad p_i^2 p_j^2,\quad p_i p_j p_k^2$$ with $i,j,k\in\{1,2,3\}$ and $i\neq j\neq k\neq i$. Including the quasi spin $I$ and using group theory, one can find six linear combinations of $p^4$ terms, which transform according to $\Gamma_1^+$ [@G3] (see Appendix \[sec:p4\]). Using the results of Appendix \[sec:p4\], we can write the kinetic energy of the electron and the hole as
$$\begin{aligned}
H_{\mathrm{e}}\!\left(\boldsymbol{p}_{\mathrm{e}}\right) & = & \frac{1}{2\hbar^2 m_{\mathrm{e}}}\left\{\left(\hbar^2+\lambda_{1}a^2\boldsymbol{p}_{\mathrm{e}}^{2}\right)\boldsymbol{p}_{\mathrm{e}}^{2}+\lambda_{2}a^2\left[p_{\mathrm{e}1}^2 p_{\mathrm{e}2}^2+\mathrm{c.p.}\right]\right\}\label{eq:Hecorr}\end{aligned}$$
and $$\begin{aligned}
H_{\mathrm{h}}\!\left(\boldsymbol{p}_{\mathrm{h}}\right) & = & H_{\mathrm{so}}+\frac{1}{2\hbar^{4}m_{0}}\left\{ \left(\gamma_{1}+4\gamma_{2}\right)\hbar^{2}\left(\hbar^{2}+\xi_{1}a^{2}\boldsymbol{p}_{\mathrm{h}}^{2}\right)\boldsymbol{p}_{\mathrm{h}}^{2}+\xi_{2}a^{2}\hbar^{2}\left[p_{\mathrm{h}1}^{2}p_{\mathrm{h}2}^{2}+\mathrm{c.p.}\right]\right.\nonumber \\
\nonumber \\
& - & 6\gamma_{2}\left(\hbar^{2}+\xi_{3}a^{2}\boldsymbol{p}_{\mathrm{h}}^{2}\right)\left[p_{\mathrm{h}1}^{2}\boldsymbol{I}_{1}^{2}+\mathrm{c.p.}\right]-12\gamma_{3}\left(\hbar^{2}+\xi_{4}a^{2}\boldsymbol{p}_{\mathrm{h}}^{2}\right)\left[p_{\mathrm{h}1}p_{\mathrm{h}2}\left\{ \boldsymbol{I}_{1},\boldsymbol{I}_{2}\right\} +\mathrm{c.p.}\right]\nonumber \\
\nonumber \\
& + & 2\left(\eta_{1}+2\eta_{2}\right)\hbar^{2}\left[\boldsymbol{p}_{\mathrm{h}}^{2}\;\boldsymbol{I}\cdot\boldsymbol{S}_{\mathrm{h}}\right]-12\eta_{2}\hbar^{2}\left[p_{\mathrm{h}1}^{2}\boldsymbol{I}_{1}\boldsymbol{S}_{\mathrm{h}1}+\mathrm{c.p.}\right]-12\eta_{3}\hbar^{2}\left[p_{\mathrm{h}1}p_{\mathrm{h}2}\left(\boldsymbol{I}_{1}\boldsymbol{S}_{\mathrm{h}2}+\boldsymbol{I}_{2}\boldsymbol{S}_{\mathrm{h}1}\right)+\mathrm{c.p.}\right]\nonumber \\
\nonumber \\
& - & \left. 6\xi_{5}a^{2}\left[\left(p_{\mathrm{h}1}^{4}+6p_{\mathrm{h}2}^{2}p_{\mathrm{h}3}^{2}\right)\boldsymbol{I}_{1}^{2}+\mathrm{c.p.}\right]-12\xi_{6}a^{2}\left[\left(p_{\mathrm{h}1}^{2}+p_{\mathrm{h}2}^{2}-6p_{\mathrm{h}3}^{2}\right)p_{\mathrm{h}1}p_{\mathrm{h}2}\left\{ \boldsymbol{I}_{1},\,\boldsymbol{I}_{2}\right\} +\mathrm{c.p.}\right] \right\}\label{eq:Hhcorr}\end{aligned}$$
with the lattice constant $a$ and the unknown parameters $\lambda_i$ and $\xi_i$. Note that the values of parameters $\eta_i$ are smaller than the Luttinger parameters $\gamma_i$ (see Table \[tab:1\]). Hence, we expect the terms of the form $p^4 IS_{\mathrm{h}}$ to be negligibly small.
![Fits to the band structure obtained via spin density functional theory calculations [@20] (black linespoints) for (a) conduction band and (b) valence bands of $\mathrm{Cu_{2}O}$ for the \[100\] direction using the expressions (\[eq:Hecorr\]) and (\[eq:Hhcorr\]) (red lines). The green solid line shows the function $\left|\Phi_{1S}\left(\boldsymbol{k}\right)\right|^2$ for $a_{\mathrm{exc}}^{\left(1S\right)}=a$ in units of $a^3$. One can see that the differences between the fit using quartic terms and the fit of Ref. [@80] (blue dashed lines) neglecting these terms are small in the range of extension of $\left|\Phi_{1S}\left(\boldsymbol{k}\right)\right|^2$. Note that $\left|\Phi_{1S}\!\left(\boldsymbol{k}\right)\right|^2$ is not shown in the lower panel for reasons of clarity.\[fig:FigGX\]](Fig1.pdf){width="1.0\columnwidth"}
![Same as Fig. \[fig:FigGX\] for the \[110\] direction.\[fig:FigGM\]](Fig2.pdf){width="1.0\columnwidth"}
After replacing $H_{\mathrm{e}}\!\left(\boldsymbol{p}_{\mathrm{e}}\right)\rightarrow H_{\mathrm{e}}\!\left(\hbar\boldsymbol{k}\right)$ and $H_{\mathrm{h}}\!\left(\boldsymbol{p}_{\mathrm{h}}\right)\rightarrow -H_{\mathrm{h}}\!\left(\hbar\boldsymbol{k}\right)$, we can determine the eigenvalues of these Hamiltonians and fit them as in Ref. [@80] for $\left|\boldsymbol{k}\right|<\pi/a$ to the band structure of $\mathrm{Cu_{2}O}$ obtained via spin density functional theory calculations [@20].
To obtain a reliable result, we perform a least-squares fit with a weighting function. Even though the exciton ground state will show deviations from a pure hydrogen-like $1S$ state, we expect that the radial probability density can be described qualitatively by that function. Hence, we use the modulus squared of the Fourier transform $\Phi_{1S}\!\left(\boldsymbol{k}\right)=\mathcal{F}\left(\Psi_{1S}\right)\left(\boldsymbol{k}\right)$ of the hydrogen-like function $$\Psi_{1S}\!\left(\boldsymbol{r}\right)=\frac{1}{\sqrt{\pi \left(a_{\mathrm{exc}}^{\left(1S\right)}\right)^3}}e^{-r/a_{\mathrm{exc}}^{\left(1S\right)}}$$ as the weighting function for the fit. It reads [@IPEP] $$\begin{aligned}
\left|\Phi_{1S}\!\left(\boldsymbol{k}\right)\right|^2 & \sim & \left|\frac{1}{\sqrt{(2\pi)^3}}\int\mathrm{d}\boldsymbol{r}\;\Psi_{1S}\!\left(\boldsymbol{r}\right)e^{-i\boldsymbol{k}\boldsymbol{r}}\right|^2\nonumber \\
\nonumber \\
& = & \frac{8 \left(a_{\mathrm{exc}}^{\left(1S\right)}\right)^3}{\pi^2 \left(1+k^2 \left(a_{\mathrm{exc}}^{\left(1S\right)}\right)^2\right)^4}\end{aligned}$$ with the radius $a_{\mathrm{exc}}^{\left(1S\right)}$ of the $1S$ exciton state. Although we do not *a priori* know the true value of $a_{\mathrm{exc}}^{\left(1S\right)}$, the experimental value of the binding energy of the $1S$ state [@TOE; @7] as well as the calculations of Ref. [@1] indicate that it is on the order of one or two times the lattice constant $a=0.427\,\mathrm{nm}$ [@JPCS27; @P6; @DB_45]. For the fit to the band structure we assume a small value of $a_{\mathrm{exc}}^{\left(1S\right)}=a$ as a lower limit in the sense of a safe estimate since then the extension of the exciton wave function in Fourier space is larger. In doing so, we will now show that even if the radius of the $1S$ exciton were smaller or equal to the lattice constant $a$, there would not be contributions of the $p^4$ terms of the band structure.
The results of the fit are depicted as red solid lines in Figs. \[fig:FigGX\], \[fig:FigGM\], and \[fig:FigGR\]. For a comparison, we also show the fit neglecting the quartic terms in the momenta (blue dashed lines) [@80]. The values of the fit parameters are $$\begin{aligned}
{4}
\lambda_1 = & -1.109\times 10^{-2} ,\quad && \lambda_2 = && -2.052\times 10^{-2}, \nonumber \\
\xi_1 = & -1.389\times 10^{-1} ,\quad && \xi_4 = && -1.518\times 10^{-1}, \nonumber \\
\xi_2 = & \quad\: 2.353\times 10^{-3} ,\quad && \xi_5 = && \quad\: 9.692\times 10^{-4}, \nonumber \\
\xi_3 = & -1.523\times 10^{-1} ,\quad && \xi_6 = && -8.385\times 10^{-4}. \end{aligned}$$ As can be seen, e.g., from Fig. \[fig:FigGM\], the fit including the quartic terms is only slightly better than the fit with the quadratic terms for small $k$. A clear difference between the fits can be seen only for large values of $k$ as regards the valence bands: Since some of the pre-factors of the quartic terms are positive, the energy of the valence bands in the fitted model increases for larger values of $k$.
Considering the minor differences between the fits for small $k$ and the small extension of the $1S$ exciton function in $k$ space even for $a_{\mathrm{exc}}^{\left(1S\right)}=a$ (see, e.g., Fig. \[fig:FigGX\]), the quartic terms will hardly affect this exciton state and can be neglected. These arguments still hold if, e.g., $a_{\mathrm{exc}}^{\left(1S\right)}=0.2a$ is assumed.
In the work of Ref. [@1] the introduction of $p^4$ terms seemed necessary to explain the experimentally observed large mass of the $1S$ exciton. However, the experimental observations are already well described by quadratic terms in $p$ when considering the complete valence band structure [@100]. As we already stated in Ref. [@150], a simple restriction to the $\Gamma_7^+$ band neglecting the $\Gamma_8^+$ band and considering the nonparabolicity of the $\Gamma_7^+$ band via $p^4$ terms as has been done in Ref. [@1] does not treat the problem correctly.
![Same as Fig. \[fig:FigGX\] for the \[111\] direction.\[fig:FigGR\]](Fig3.pdf){width="1.0\columnwidth"}
Dielectric constant \[sec:eps\]
-------------------------------
In the case of the $1S$ exciton in $\mathrm{Cu_{2}O}$ the relative motion of the electron and the hole is sufficiently fast that phonons cannot follow it and corrections on the dielectric constant need to be considered.
In general, the electron and the hole are coupled to longitudinal optical phonons via the Fröhlich interaction [@AP3; @2] and to longitudinal acoustic phonons via the deformation potential coupling [@M1_7; @2]. While in the case of optical phonons the ions of the solid are displaced in anti-phase and thus create a dipole moment in the unit cell of a polar crystal, the ions are displaced in phase in the case of acoustic phonons and no dipole moment is created. Hence, one expects that the interaction between electron or hole and optical phonons is much larger than the interaction with acoustic phonons in polar crystals [@HP0; @SO].
If the frequency of the relative motion of electron and hole is high enough so that the ions of the solid cannot follow it, the Coulomb interaction between electron and hole is screened by the high-frequency or background dielectric constant $\varepsilon_{\mathrm{b}}$ [@TOE; @SOK1_82L1]. This dielectric constant describes the electronic polarization, which can follow the motion of electron and hole very quickly [@PAE].
For lower frequencies of the relative motion the contribution of the phonons to the screening becomes important and the dielectric function $\varepsilon$ becomes frequency dependent. In many semiconductors the frequency of the relative motion in exciton states with a principal quantum number of $n\geq 2$ is so small that the low-frequency or static dielectric constant $\varepsilon_{\mathrm{s}}$ can be used [@SO], which involves the electronic polarization *and* the displacement of the ions [@PAE]. Note that we use the notation $\varepsilon_{\mathrm{b}}$, $\varepsilon_{\mathrm{s}}$ instead of $\varepsilon_{\infty}$, $\varepsilon_{0}$ to avoid the risk of confusion with the electric permittivity $\varepsilon_{0}$ [@PAE; @SO].
The transition from $-e^2/4\pi\varepsilon_{0}\varepsilon_{\mathrm{s}}r$ to $-e^2/4\pi\varepsilon_{0}\varepsilon_{\mathrm{b}}r$, which takes place when the frequency of the electron or the hole is of the same size as the frequency of the phonon [@PAE], had been investigated in detail by Haken in Refs. [@HP4_21; @HP8; @HP9; @HP0; @HP1; @PAE]. He considered at first the interaction between electron or hole and the phonons and then constructed the exciton from the resulting particles with polarisation clouds, i.e., the polarons. The change of the Coulomb interaction between both particles was then explained in terms of an exchange of phonons, i.e., of virtual quanta of the polarization field [@PAE].
The final result for the interaction in the transition region between $-e^2/4\pi\varepsilon\varepsilon_{\mathrm{s}}r$ and $-e^2/4\pi\varepsilon\varepsilon_{\mathrm{b}}r$ was the so-called Haken potential [@HP8; @HP9; @HP1; @TOE; @SO; @IPEP], $$\begin{aligned}
V\!\left(r\right) & = & -\frac{e^2}{4\pi\varepsilon_0 r}\left[\frac{1}{\varepsilon_{\mathrm{s}}}+\frac{1}{2\varepsilon^*}\left(e^{-r/\rho_{\mathrm{h}}}+e^{-r/\rho_{\mathrm{e}}}\right)\right].\label{eq:Haken}\end{aligned}$$ Here $\rho_{\mathrm{e}}$ and $\rho_{\mathrm{h}}$ denote the polaron radii $$\rho_{\mathrm{e/h}}=\sqrt{\frac{\hbar}{2m_{\mathrm{e/h}}^*\omega_{\mathrm{LO}}}}$$ with the frequency $\omega_{\mathrm{LO}}$ of the optical phonon and $\varepsilon^*$ is given by $$\frac{1}{\varepsilon^*}=\frac{1}{\varepsilon_{\mathrm{b}}}-\frac{1}{\varepsilon_{\mathrm{s}}}.$$ Note that in the result of Haken [@HP0; @HP4] the polaron masses $m_i^*$ instead of the bare electron and hole masses have to be used in the polaron radii and the kinetic energies. Furthermore, the lattice relaxation due to the interaction of excitons and phonons decreases the band gap energy for electrons and holes. However, since the value of $E_{\mathrm{g}}$ for $\mathrm{Cu_{2}O}$ has been determined in Ref. [@GRE] from the experimental exciton spectrum, the polaron effect is already accounted for in the band gap energy [@SO].
Note that the above results were derived in the simple band model and by assuming only one optical phonon branch contributing to the Fröhlich interaction. To the best of our knowledge there is no model accounting for more than one optical phonon branch [@HP4; @1; @PAE], which complicates the correct treatment of $\mathrm{Cu_{2}O}$, where two LO phonons contribute to the Fröhlich interaction. Even though there are theories for polarons in the degenerate band case [@22; @E3; @HP7], we will use only the leading, spherically symmetric terms, in which only the isotropic effective mass of the hole or only the Luttinger parameter $\gamma_1$ enters. Of course, there are further terms of cubic symmetry, which also depend on the other Luttinger parameters. However, since already $\gamma_1$ is at least by a factor of $2$ larger than the other Luttinger parameters, we expect the further terms in the Haken potentials to be smaller than the leading term used here. Since the effect of the Haken potential on the exciton spectrum is not crucial, as will be seen from Fig. 6, the neglection of further terms in the polaron potentials will then be *a posteriori* justified.
Furthermore, the Haken potential (\[eq:Haken\]) cannot describe the non-Coulombic electron-hole interaction for very small values of $r$, which is due to the finite size of electron and hole [@TOE]. The conditions of validity of the potential (\[eq:Haken\]) have, e.g., been discussed by Haken in Ref. [@PAE].
When treating the Haken potential numerically for different polar crystals, the experimental and theoretical binding energies of the exciton states sometimes do not agree, for which reason corrections, sometimes phenomenologically, to the Haken potential have been introduced [@HP5; @HP2; @HP3; @HP6] leading to clearly better results. One of these refined formulas is the potential proposed by Pollmann and Büttner [@HP6; @HP4] $$\begin{aligned}
V\!\left(r\right) & = & -\frac{e^2}{4\pi\varepsilon_0 r}\nonumber \\
\nonumber \\
& \times & \left[\frac{1}{\varepsilon_{\mathrm{s}}}+\frac{1}{\varepsilon^*}\left(\frac{m_{\mathrm{h}}}{\Delta m}e^{-r/\rho_{\mathrm{h}}}-\frac{m_{\mathrm{e}}}{\Delta m}e^{-r/\rho_{\mathrm{e}}}\right)\right],\label{eq:PB}\end{aligned}$$ in which the bare electron and hole masses have to be used and where $\Delta m$ is given by $\Delta m=m_{\mathrm{h}}-m_{\mathrm{e}}$. Hence, we take the statements given above as a reason to propose the following phenomenological potentials for $\mathrm{Cu_{2}O}$, which are motivated by the formula of Haken and by the formula of Pollmann and Büttner:
$$\begin{aligned}
V^{\mathrm{H}}\!\left(r\right)=-\frac{e^{2}}{4\pi\varepsilon_{0}r}\left[\frac{1}{\varepsilon_{\mathrm{s}1}}+\frac{1}{2\varepsilon_{1}^{*}}\left(e^{-r/\rho_{\mathrm{h}1}}+e^{-r/\rho_{\mathrm{e}1}}\right)+\frac{1}{2\varepsilon_{2}^{*}}\left(e^{-r/\rho_{\mathrm{h}2}}+e^{-r/\rho_{\mathrm{e}2}}\right)\right]\label{eq:Haken2}\end{aligned}$$
and $$\begin{aligned}
V^{\mathrm{PB}}\!\left(r\right) & = & -\frac{e^{2}}{4\pi\varepsilon_{0}r}\left[\frac{1}{\varepsilon_{\mathrm{s}1}}+\frac{1}{\varepsilon_{1}^{*}}\left(\frac{m_{0}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{h}1}}-\frac{m_{\mathrm{e}}\gamma_{1}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{e}1}}\right)\right.\nonumber \\
& & \;\,\quad\qquad\qquad+\left.\frac{1}{\varepsilon_{2}^{*}}\left(\frac{m_{0}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{h}2}}-\frac{m_{\mathrm{e}}\gamma_{1}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{e}2}}\right)\right].\label{eq:PB2}\end{aligned}$$
Here we use $$\frac{1}{\varepsilon_{i}^{*}}=\frac{1}{\varepsilon_{\mathrm{b}i}}-\frac{1}{\varepsilon_{\mathrm{s}i}}$$ and $$\rho_{\mathrm{e}i}=\sqrt{\frac{\hbar}{2m_{\mathrm{e}}\omega_{\mathrm{LOi}}}},\qquad\rho_{\mathrm{h}i}=\sqrt{\frac{\hbar\gamma_1}{2m_0\omega_{\mathrm{LOi}}}},$$ where the energies of the phonons and the values of the dielectric constants are given by [@1] $$\hbar\omega_{\mathrm{LO1}}=18.7\,\mathrm{meV},\qquad\hbar\omega_{\mathrm{LO\,2}}=87\,\mathrm{meV}$$ and $${\varepsilon_{\mathrm{s}1}}=7.5,\qquad{\varepsilon_{\mathrm{b}1}}={\varepsilon_{\mathrm{s}2}}=7.11,\qquad{\varepsilon_{\mathrm{b}2}}=6.46.$$ As has been done in Ref. [@HP4] for perovskite $\mathrm{CH_{3}NH_{3}PbI_{3}}$, we use $V_{\mathrm{H}}$ or $V_{\mathrm{PB}}$ in the Schrödinger equation without an additional fit parameter and find out which of these potentials describes the exciton spectrum of $\mathrm{Cu_{2}O}$ best. Since for the polaron radii $\rho_{\mathrm{e}}$ and $\rho_{\mathrm{h}}$ $1.6a\leq \rho \leq 4.4a$ holds, we expect the Haken or the Pollmann-Büttner potential to have a significant influence on the exciton states with $n\leq 2$.
As the Fröhlich coupling constant is small in $\mathrm{Cu_{2}O}$, i.e., it is $\alpha^{\mathrm{F}}\lesssim 0.2$ for the two optical phonons and both the electron and the hole [@20], the bare electron and hole masses differ from the polaron masses by at most 3%. Hence, we can calculate with the bare masses when using $V_{\mathrm{H}}$.
Besides the frequency dependence of the dielectric function also its momentum dependence becomes important if the exciton radius is on the order of the lattice constant. This momentum dependence of the dielectric function arises from the electronic polarization [@1_23; @1].
When treating the excitons of $\mathrm{Cu_{2}O}$ in momentum space, the wave functions of the $n\geq 2$ states are localized about $k=0$ so that for these states the $k$ dependence of $\varepsilon$ is not important. However, for the $1S$ state $a_{1S}\approx a$ holds and thus this state is screened by $\varepsilon$ at higher momenta $k$ [@1]. Considering the Coulomb interaction for the $1S$ exciton in $k$ space, $$V(k,\,\omega)=-\frac{1}{\sqrt{(2\pi)^3}}\,\frac{e^2}{\varepsilon_0\varepsilon(k,\,\omega) k^2},\label{eq:CoulF}$$ Kavoulakis *et al* [@1] derived a correction term by assuming $$\begin{aligned}
\frac{1}{\varepsilon(k,\omega)} & \approx & \frac{1}{\varepsilon_{\mathrm{b}}-d(ka)^2}\approx \frac{1}{\varepsilon_{\mathrm{b}}}+\frac{d(ka)^2}{\varepsilon_{\mathrm{b}}^2}\label{eq:epsexp}\end{aligned}$$ valid for $\,E_{\mathrm{g}}/\hbar\gg \omega\gg \omega_{\mathrm{LO}}$ with a small unknown constant $d$. Inserting Eq. (\[eq:epsexp\]) in Eq. (\[eq:CoulF\]) and Fourier transforming the second expression, one obtains the following correction term to the Coulomb interaction: $$V_{d}=-da^2 \frac{e^2}{\varepsilon_0\varepsilon_{\mathrm{b}}^2}V_{\mathrm{uc}}\,\delta\!\left(\boldsymbol{r}\right)=-V_0 V_{\mathrm{uc}}\delta\!\left(\boldsymbol{r}\right).\label{eq:deltaV0}$$ Following the calculation of Ref. [@1_23] on the dielectric function and using the lowest $\Gamma_8^-$ conduction band and the highest $\Gamma_7^+$ valence band, Kavoulakis *et al* [@1] estimated the value of $d$ to $d\approx 0.18$ [@1].
Note that in general a Kronecker delta would appear in Eq. (\[eq:deltaV0\]) [@TOE]. However, as we treat the exciton problem in the continuum approximation, this Kronecker delta is replaced by the delta function times the volume $V_{\mathrm{uc}}=a^3$ of one unit cell. Thus, the parameter $V_0$ has the unit of an energy.
We have already stated above that the Haken potential cannot describe the electron-hole interaction correctly for very small $r$. Therefore, we now assume that the potential (\[eq:deltaV0\]) is not only due to the momentum dependence of the dielectric function but that it also accounts for deviations from the Haken potential at small $r$. Hence, we will treat $V_0$ as an unknown fit parameter in the following.
Exchange interaction \[sec:exchange\]
-------------------------------------
In the Wannier equation or Hamiltonian of excitons the exchange interaction is generally not included but regarded as a correction to the hydrogen-like solution [@TOE]. Recently, we have presented a comprehensive discussion of the exchange interaction in $\mathrm{Cu_{2}O}$ [@150]. We could show, in accordance with Ref. [@1], that corrections to the exchange interaction due to a finite momentum $\hbar\boldsymbol{K}$ of the center of mass of the exciton are negligibly small. Hence, only the $K$ independent part of the exchange interaction [@7; @E2; @E3; @150] $$H_{\mathrm{exch}}=J_{0}\left(\frac{1}{4}-\frac{1}{\hbar^{2}}\boldsymbol{S}_{\mathrm{e}}\cdot\boldsymbol{S}_{\mathrm{h}}\right) V_{\mathrm{uc}}\delta\!\left(\boldsymbol{r}\right)\label{eq:Hexch}$$ needs to be considered. Within the simple hydrogen-like model the exchange interaction would only affect the $nS$ exciton states as these states have a nonvanishing probability density at $r=0$. However, when considering the complete valence band structure, the exciton states with even or with odd values of $L$ are coupled, and thus the exchange interaction will affect the whole even exciton series.
It is well known from experiments that the splitting between the yellow $1S$ ortho and the yellow $1S$ para exciton amounts to about $12\,\mathrm{meV}$ [@1_6a; @E2_16; @1_6c]. Hence, we have to choose the value of $\tilde{J}_{0}$ such that this splitting is reflected in the theoretical spectrum.
Summary \[sec:sumccc\]
----------------------
Following the explanations given in Secs. \[sec:eps\] and \[sec:exchange\], the term $V_{\mathrm{CCC}}$ in the Hamiltonian of Eq. (\[eq:H\]) takes one of the following forms:
$$\begin{aligned}
V_{\mathrm{CCC}}^{\mathrm{H}}\!\left(\boldsymbol{r}\right) & = & -\frac{e^{2}}{4\pi\varepsilon_{0}r}\left[\frac{1}{2\varepsilon_{1}^{*}}\left(e^{-r/\rho_{\mathrm{h}1}}+e^{-r/\rho_{\mathrm{e}1}}\right)+\frac{1}{2\varepsilon_{2}^{*}}\left(e^{-r/\rho_{\mathrm{h}2}}+e^{-r/\rho_{\mathrm{e}2}}\right)\right]\nonumber\\
& & +\left[-V_0 +J_{0}\left(\frac{1}{4}-\frac{1}{\hbar^{2}}\boldsymbol{S}_{\mathrm{e}}\cdot\boldsymbol{S}_{\mathrm{h}}\right)\right]V_{\mathrm{uc}}\delta\!\left(\boldsymbol{r}\right),\label{eq:HCCC_H}\\
\nonumber\\
\nonumber\\
V_{\mathrm{CCC}}^{\mathrm{PB}}\!\left(\boldsymbol{r}\right) & = & -\frac{e^{2}}{4\pi\varepsilon_{0}r}\left[\frac{1}{\varepsilon_{1}^{*}}\left(\frac{m_{0}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{h}1}}-\frac{m_{\mathrm{e}}\gamma_{1}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{e}1}}\right)\right.\nonumber \\
& & \;\,\quad\qquad\qquad+\left.\frac{1}{\varepsilon_{2}^{*}}\left(\frac{m_{0}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{h}2}}-\frac{m_{\mathrm{e}}\gamma_{1}}{m_{0}-m_{\mathrm{e}}\gamma_{1}}e^{-r/\rho_{\mathrm{e}2}}\right)\right]\nonumber\\
& & +\left[-V_0 +J_{0}\left(\frac{1}{4}-\frac{1}{\hbar^{2}}\boldsymbol{S}_{\mathrm{e}}\cdot\boldsymbol{S}_{\mathrm{h}}\right)\right]V_{\mathrm{uc}}\delta\!\left(\boldsymbol{r}\right),\label{eq:HCCC_PB}\end{aligned}$$
\[eq:HCCC\]
\[cf. Eqs. (\[eq:Haken2\]), (\[eq:PB2\]), (\[eq:deltaV0\]), and (\[eq:Hexch\])\]. Note that while the operators with $\delta\left(\boldsymbol{r}\right)$ affect only the exciton series with even values of $L$, the Haken or Pollmann and Büttner potential affect all exciton states [@PAE]. A comparison of our results with the experimental values of Refs. [@GRE; @7; @28; @DB_49; @HO] will allow us, in Sec. \[sec:results\], to determine the size of the unknown parameters $V_0$ and $J_0$.
Eigenvalues and oscillator strengths \[sec:eosc\]
=================================================
In this section we describe how the Schrödinger equation corresponding to the Hamiltonian (\[eq:H\]) is solved in a complete basis. Furthermore, we discuss how to calculate oscillator strengths for two-photon absorption. An appropriate basis to solve the Schrödinger equation has been presented in detail in Ref. [@100]. Hence, we recapitulate only the most important points.
As regards the angular momentum part of the basis, we have to consider that the different operators in the Hamiltonian couple the quasi spin $I$, the hole spin $S_{\mathrm{h}}$, and the angular momentum $L$ of the exciton. Hence, we introduce the effective hole spin $J=I+S_{\mathrm{h}}$, the angular momentum $F=L+J$, and the total angular momentum $F_{t}=F+S_{\mathrm{e}}$. For the radial part of the exciton wave function we use the Coulomb-Sturmian functions [@S1] $$U_{NL}\!\left(r\right)=N_{NL}^{(\alpha)}\left(2\rho\right)^{L}e^{-\rho}L_{N}^{2L+1}\left(2\rho\right)\label{eq:U}$$ with $\rho=r/\alpha$, an arbitrary convergence or scaling parameter $\alpha$, the associated Laguerre polynomials $L_{n}^{m}\left(x\right)$, and a normalization factor $$N_{NL}^{(\alpha)}=\frac{2}{\sqrt{\alpha^3}}\left[\frac{N!}{\left(N+L+1\right)\left(N+2L+1\right)!}\right]^{\frac{1}{2}}.\label{eq:normaliz}$$ The radial quantum number $N$ is related to the principal quantum number $n$ via $n=N+L+1$. Finally, we use the following ansatz for the exciton wave function
$$\begin{aligned}
\left|\Psi\right\rangle & = & \sum_{NLJFF_{t}M_{F_{t}}}c_{NLJFF_{t}M_{F_{t}}}\left|\Pi\right\rangle,\\
\nonumber \\
\left|\Pi\right\rangle & = & \left|N,\, L;\,\left(I,\, S_{\mathrm{h}}\right)\, J;\, F,\, S_{\mathrm{e}};\, F_{t},\, M_{F_{t}}\right\rangle\label{eq:basis}\end{aligned}$$
\[eq:ansatz\]
with real coefficients $c$. The parenthesis and semicolons in Eq. (\[eq:basis\]) are meant to illustrate the coupling scheme of the spins and the angular momenta. Since the $z$ axis is a fourfold axis, it is sufficient to use only $M_{F_t}$ quantum numbers which differ by $\pm 4$ in Eq. (\[eq:ansatz\]).
We now express the Hamiltonian (\[eq:H\]) in terms of irreducible tensors [@ED; @7_11; @44]. Inserting the ansatz (\[eq:ansatz\]) in the Schrödinger equation $H\Psi=E\Psi$ and multiplying from the left with another basis state $\left\langle \Pi'\right|$, we obtain a matrix representation of the Schrödinger equation of the form $$\boldsymbol{D}\boldsymbol{c}=E\boldsymbol{M}\boldsymbol{c}.\label{eq:gev}$$ The vector $\boldsymbol{c}$ contains the coefficients of the ansatz (\[eq:ansatz\]). All matrix elements, which enter the symmetric matrices $\boldsymbol{D}$ and $\boldsymbol{M}$ and which have not been treated in Ref. [@100], are given in Appendix \[sub:Matrix-elements\]. The generalized eigenvalue problem (\[eq:gev\]) is finally solved using an appropriate LAPACK routine [@Lapack]. The material parameters used in our calculation are listed in Table \[tab:1\].
Since the basis cannot be infinitely large, the values of the quantum numbers are chosen in the following way: For each value of $n=N+L+1$ we use $$\begin{aligned}
L & = & 0,\,\ldots,\, n-1,\nonumber \\
J & = & 1/2,\,3/2,\nonumber \\
F & = & \left|L-J\right|,\,\ldots,\,\min\left(L+J,\, F_{\mathrm{max}}\right),\\
F_{t} & = & F-1/2,\, F+1/2,\nonumber \\
M_{F_{t}} & = & -F_{t},\,\ldots,\, F_{t}.\nonumber \end{aligned}$$ The value $F_{\mathrm{max}}$ and the maximum value of $n$ are chosen appropriately large so that the eigenvalues converge. Additionally, we can use the scaling parameter $\alpha$ to enhance convergence. However, it should be noted that the value of $\alpha$ does not influence the theoretical results for the exciton energies in any way, i.e., the converged results do not depend on the value of $\alpha$.
--------------------------------------- ------------------------------------------------------------ --------------
band gap energy $E_{\mathrm{g}}=2.17202\,\mathrm{eV}$
electron mass $m_{\mathrm{e}}=0.99\, m_{0}$ [@M2]
spin-orbit coupling $\Delta=0.131\,\mathrm{eV}$ [@80]
valence band parameters $\gamma_{1}=1.76$ [@80; @100]
$\gamma_2=0.7532$ [@80; @100]
$\gamma_3=-0.3668$ [@80; @100]
$\eta_1=-0.020$ [@80; @100]
$\eta_2=-0.0037$ [@80; @100]
$\eta_3=-0.0337$ [@80; @100]
lattice constant $a=0.42696\,\mathrm{nm}$ [@20_24]
dielectric constants $\varepsilon_{\mathrm{s}1}=7.5$ [@SOK1_82L1]
$\varepsilon_{\mathrm{b}1}=\varepsilon_{\mathrm{s}2}=7.11$ [@SOK1_82L1]
$\varepsilon_{\mathrm{b}2}=6.46$ [@SOK1_82L1]
energy of $\Gamma_{4}^{-}$-LO phonons $\hbar\omega_{\mathrm{LO1}}=18.7\,\mathrm{meV}$ [@1]
$\hbar\omega_{\mathrm{LO2}}=87\,\mathrm{meV}$ [@1]
--------------------------------------- ------------------------------------------------------------ --------------
: Material parameters used in the calculations. Instead of the band gap energy $E_{\mathrm{g}}=2.17208\,\mathrm{eV}$ of Ref. [@GRE] a slightly smaller value is used to obtain a better agreement with experimental values in Sec. \[sec:results\].\[tab:1\]
Note that the presence of the delta functions in Eq. (\[eq:HCCC\]) makes the whole problem more complicated than in Ref. [@100] since not only the eigenvalues but also the wave functions at $r=0$ have to converge. However, for a specific value of $\alpha$ it is not possible to obtain convergence for all exciton states of interest. Therefore, we solve the Schrödinger equation initially without the $\delta\!\left(\boldsymbol{r}\right)$ dependent terms. We then select the converged eigenvectors and with these we set up a second generalized eigenvalue problem now including the $\delta\!\left(\boldsymbol{r}\right)$ dependent terms. This problem is again solved using an appropriate LAPACK routine [@Lapack] and provides the correct converged eigenvalues of the complete Hamiltonian (\[eq:H\]).
Having solved the eigenvalue problem, we can use the eigenvectors to determine relative oscillator strengths. The determination of relative oscillator strengths in one-photon absorption has been presented in detail in Refs. [@100; @125]. While in one photon absorption excitons of symmetry $\Gamma_4^-$ are dipole-allowed [@100], the selection rules for two-photon absorption [@6_15a; @6_15b; @6_15c] are different and excitons of symmetry $\Gamma_5^+$ can be optically excited.
When considering one-photon absorption one generally treats the operator $\boldsymbol{A}\boldsymbol{p}$ with the vector potential $\boldsymbol{A}$ of the radiation field in first order perturbation theory. The dipole operator then transforms according to the irreducible representation $D^1$ of the full rotation group. In two-photon absorption one needs the operator $\boldsymbol{A}\boldsymbol{p}$ twice and thus the product $D^1\otimes D^1=D^0 \oplus D^1 \oplus D^2$ has to be considered [@G3]. In $\mathrm{Cu_{2}O}$ the reduction of these irreducible representations by the cubic group $O_{\mathrm{h}}$ has to be considered and one obtains $$\Gamma_4^-\otimes\Gamma_4^-=\Gamma_1^+\oplus\Gamma_4^+\oplus\left(\Gamma_3^+\oplus\Gamma_5^+\right).$$ In two-photon absorption the spin $S=S_{\mathrm{e}}+S_{\mathrm{h}}=0$ remains unchanged and the exciton state must have an $L=0$ component. Hence, the correct expression for the relative oscillator strength is given by $$f_{\mathrm{rel}}\sim\left|\lim_{r\rightarrow0}\,_{T}\left\langle 1,\,M'_{F_t}\middle|\Psi\left(\boldsymbol{r}\right)\right\rangle\right|^2,\label{eq:frelT}$$ with the wave function $\left|\Psi\right\rangle$ of Eq. (\[eq:ansatz\]) and the state $\left|F'_t,\,M'_{F_t}\right\rangle_T$, which is a short notation for $$\begin{aligned}
& & \left|\left(S_{\mathrm{e}},\,S_{\mathrm{h}}\right)\,S,\,I;\,I+S,\,L;\,F'_t,\,M'_{F_t}\right\rangle\nonumber\\
& = & \left|\left(1/2,\,1/2\right)\,0,\,1;\,1,\,0;\,F'_t,\,M'_{F_t}\right\rangle.\label{eq:stateT}\end{aligned}$$ Note that the coupling scheme of the spins and angular momenta in Eq. (\[eq:stateT\]) given by $$S_{\mathrm{e}}+S_{\mathrm{h}}=S\quad\rightarrow\quad(I+S)+L=F'_t$$ is different from the one of Eq. (\[eq:basis\]) due to the requirement that $S$ must be a good quantum number.
It can be shown that the state $\left|1,\,M'_{F_t}\right\rangle_T$ transforms according to the irreducible representation $\Gamma_5^+$ of $O_{\mathrm{h}}$ [@G3], for which reason only exciton states of this symmetry can be excited in two-photon absorption. By choosing particular directions of the polarization of the light, e.g., by choosing one photon being polarized in $x$ direction and one photon being polarized in $y$ direction, only one component of the $\Gamma_5^+$ exciton states, the $xy$ component, can be excited optically. We consider this case in the following and hence use $M'_{F_t}=0$ in Eq. (\[eq:frelT\]). Finally, we wish to note that the exciton states of symmetry $\Gamma_5^+$ can weakly be observed in one-photon absorption in quadrupole approximation [@7].
Results and discussion \[sec:results\]
======================================
In this section we determine the values of the parameters $J_0$ and $V_0$ and discuss the complete exciton spectrum of $\mathrm{Cu_{2}O}$.
The parameter $J_0$ describes the strength of the exchange interaction. It is well known that the exchange interaction mainly affects the $1S$ exciton and that the splitting between the ortho and the para exciton state amounts to $11.8\,\mathrm{meV}$ [@1_6a; @E2_16; @1_6c; @7; @DB_49]. By choosing $$J_0=0.792\pm 0.068\,\mathrm{eV}\label{eq:J0}$$ we obtain the correct value of this splitting irrespective of whether using the Haken or the Pollman-Büttner potential \[cf. Eq. (\[eq:HCCC\])\].
The figures \[fig:Fig4\] and \[fig:Fig5\] show the effect of the correction with the coefficient $V_0$ on the spectrum for the Haken and the Pollman-Büttner potential, respectivley. As can be seen from these figures, the exchange splitting of the $1S$ state hardly changes when varying the value $V_0$. Hence, we can determine $V_0$ almost independently of $J_0$.
To find the optimum value of $V_0$, we compare our results to the energies of the even parity exciton states given in Refs. [@7; @80; @78; @HO; @DB_49; @7_22]. However, we can see from Figs. \[fig:Fig4\] and \[fig:Fig5\] that there is no value of $V_0$ for which all theoretical results take the values of the experimentally determined energies. This is not unexpected since the central-cell corrections are only an attempt to account for the specific properties of the $1S$ exciton within the continuum limit of Wannier excitons and are not an exact description of this exciton state. Hence, we do not expect a perfect agreement between theory and experiment.
Small deviations from the experimental values could also be explained by small uncertainties in the Luttinger parameters $\gamma_i$, $\eta_i$ [@80; @100] or the band gap energy [@GRE] as well as by a finite temperature or small strains in the crystal. On the other hand, it is also possible that the experimental values are affected by uncertainties. This can be seen, e.g., when comparing the slightly different experimental results of Refs. [@80] and [@78; @HO].
Note that the almost perfect agreement between theoretical and experimental results in Refs. [@6; @7] could only be obtained by taking also $\gamma_1'$, $\mu'$ and $\Delta$ as fit parameters to the experiment. However, these parameters are connected to the band structure in $\mathrm{Cu_{2}O}$ [@20] and cannot be chosen arbitrarily [@28; @100].
![Behavior of the even exciton states as functions of $V_0$ when using $V_{\mathrm{CCC}}^{\mathrm{H}}$ \[see Eq. (\[eq:HCCC\_H\])\]. The color bar shows the relative oscillator strengths for two-photon absorption. The blue straight lines denote the position of the dipole-allowed $\Gamma_5^+$ $S$ and $D$ exciton states observed in the experiment. We also show the positions of the $1S$ para excitons $(1S_{\mathrm{y/g}}^{\mathrm{p}})$. The gray area indicates the optimum range of $V_0=0.539\pm\,0.027\,\mathrm{eV}$, where the ratio of the relative oscillator strengths of the yellow $2S$ and the green $1S$ state amounts to $\sim 16$. The effect of the central-cell corrections on the whole even exciton spectrum is evident. For further information see text.\[fig:Fig4\]](Fig4.pdf){width="1.0\columnwidth"}
![Same calculation as in Fig. \[fig:Fig4\] but with $V_{\mathrm{CCC}}^{\mathrm{PB}}$ \[see Eq. (\[eq:HCCC\_PB\])\]. One can see only slight differences for the $n=1$ and $n=2$ exciton states when comparing the results to Fig. \[fig:Fig4\]. The gray area indicates the optimum range of $V_0=0.694\pm\,0.027\,\mathrm{eV}$. \[fig:Fig5\]](Fig5.pdf){width="1.0\columnwidth"}
-- -- -- -- ----- ------------------------------------------------------------------------------
$0$ $\Gamma_{2}^{+}$
$1$ $\Gamma_{5}^{+}$
$0$ $\Gamma_{2}^{-}$
$1$ $\Gamma_{5}^{-}$
$1$ $\Gamma_{4}^{-}$
$2$ $\Gamma_{3}^{-}\oplus\Gamma_{5}^{-}$
$1$ $\Gamma_{5}^{+}$
$2$ $\Gamma_{3}^{+}\oplus\Gamma_{4}^{+}$
$2$ $\Gamma_{3}^{+}\oplus\Gamma_{4}^{+}$
$3$ $\Gamma_{1}^{+}\oplus\Gamma_{4}^{+}\oplus\Gamma_{5}^{+}$
$2$ $\Gamma_{3}^{-}\oplus\Gamma_{4}^{-}$
$3$ $\Gamma_{1}^{-}\oplus\Gamma_{4}^{-}\oplus\Gamma_{5}^{-}$
$3$ $\Gamma_{1}^{-}\oplus\Gamma_{4}^{-}\oplus\Gamma_{5}^{-}$
$4$ $\Gamma_{2}^{-}\oplus\Gamma_{3}^{-}\oplus\Gamma_{4}^{-}\oplus\Gamma_{5}^{-}$
$1$ $\Gamma_{5}^{+}$
$2$ $\Gamma_{3}^{+}\oplus\Gamma_{4}^{+}$
-- -- -- -- ----- ------------------------------------------------------------------------------
: Decomposition of the irreducible representations of the rotation group or the angular momentum states by the cubic group $O_{\mathrm{h}}$. Note that the quasi-spin $I$ already enters the momentum $F$ via $J$. The irreducible representations denote the symmetry of the envelope function $\left(L\right)$, the combined symmetry of envelope and hole $\left(F\right)$ or the complete symmetry of the exciton $\left(F_t\right)$.\[tab:2\]
{width="2.0\columnwidth"}
$E_{\mathrm{exp}}$ [\[]{}eV[\]]{} $E_{\mathrm{theor}}$ [\[]{}eV[\]]{} $f_{\mathrm{rel}}$ gp [\[]{}%[\]]{} $E_{\mathrm{exp}}$ [\[]{}eV[\]]{} $E_{\mathrm{theor}}$ [\[]{}eV[\]]{} $f_{\mathrm{rel}}$ gp [\[]{}%[\]]{}
-- -- ----------------------------------- ------------------------------------- -------------------- ------------------ -- -- -- -- ----------------------------------- ------------------------------------- -------------------- ------------------ -- --
$E_{\mathrm{exp}}$ [\[]{}eV[\]]{} $E_{\mathrm{theor}}$ [\[]{}eV[\]]{} $f_{\mathrm{rel}}$ gp [\[]{}%[\]]{} $E_{\mathrm{exp}}$ [\[]{}eV[\]]{} $E_{\mathrm{theor}}$ [\[]{}eV[\]]{} $f_{\mathrm{rel}}$ gp [\[]{}%[\]]{}
-- -- ----------------------------------- ------------------------------------- -------------------- ------------------ -- -- -- -- ----------------------------------- ------------------------------------- -------------------- ------------------ -- --
It can be seen from Figs. \[fig:Fig4\] and \[fig:Fig5\] that the oscillator strength of the exciton state at $E\approx 2.143\,\mathrm{eV}$ changes rapidly with increasing $V_0$. From the experimental results of Refs. [@6; @7] we know that the two exciton states at $E=2.1378\,\mathrm{eV}$ and $E=2.1544\,\mathrm{eV}$ are well separated from the other exciton states and that the phonon background is small. Hence, the ratio of the relative two-photon oscillator strengths can be calculated quite accurately to $\sim\!16$.
We now choose the value of $V_0$ such that the ratio of the calculated two-photon oscillator strengths reaches the same value and obtain $$V_0=0.539\pm\,0.027\,\mathrm{eV}\label{eq:V0H}$$ when using the Haken potential \[cf. Eq. (\[eq:HCCC\_H\])\] or $$V_0=0.694\pm\,0.027\,\mathrm{eV}\label{eq:V0PB}$$ when using the Pollmann-Büttner potential \[cf. Eq. (\[eq:HCCC\_PB\])\]. Note that the error bars for $V_0$ are chosen such that the ratio of the oscillator strengths lies between $14$ and $18$.
Having determined the most suitable values of $V_0$ and $J_0$, we can now turn our attention to the exciton Bohr radius $a_{\mathrm{exc}}^{\left(1S\right)}$ of the $1S$ ortho exciton and to the correct assignment of the $n=2$ exciton states.
To determine the radius $a_{\mathrm{exc}}^{\left(1S\right)}$, we evaluate $$\begin{aligned}
\left\langle\Psi\middle| r\middle|\Psi\right\rangle & = & \sum_{N'}\sum_{NLJFF_t M_{F_t}}c_{N'LJFF_t M_{F_t}}c_{NLJFF_t M_{F_t}}\nonumber\\
\nonumber\\
& \times & \sum_{j=-2}^{2}\frac{\alpha\left(R_2 \right)^{j}_{NL}}{N+L+j+1}\delta_{N',\,N+j}\label{eq:PsirPsi}\end{aligned}$$ with the wave function $\Psi$ of Eq. (\[eq:ansatz\]) and compare the result with the formula [@GRE_1] $$\left\langle r\right\rangle=\frac{1}{2}a_{\mathrm{exc}}\left[3n^2-L\left(L+1\right)\right]\label{eq:aexcH}$$ known from the hydrogen atom, where we set $n=1$ and $L=0$. Note that the function $\left(R_2 \right)^{j}_{NL}$ in Eq. (\[eq:PsirPsi\]) is taken from the recursion relations of the Coulomb-Sturmian functions in the Appendix of Ref. [@100]. We obtain $$a_{\mathrm{exc}}^{\left(1S\right)}\approx 0.793\,\mathrm{nm}\approx 1.86\,a$$ when using the Haken potential or $$a_{\mathrm{exc}}^{\left(1S\right)}\approx 0.810\,\mathrm{nm}\approx 1.90\,a$$ when using the Pollmann-Büttner potential. In both cases the radius of the $1S$ ortho exciton is large enough that the corrections to the kinetic energy discussed in Sec. \[sec:Bandstruc\] can certainly be neglected.
Let us now proceed to the correct assignment of the $n=2$ exciton states. Since in the investigation of Uihlein *et al* [@6; @7] the wrong values for the Luttinger parameters were used (cf. Ref. [@100]), it is not clear whether the state at $E=2.1544\,\mathrm{eV}$ can still be assigned as the yellow $2S$ ortho exciton state and the state at $E=2.1378\,\mathrm{eV}$ as the green $1S$ ortho exciton state when using the correct Luttinger parameters.
To demonstrate from which hydrogen-like states the experimentally observed exciton states originate, we find it instructive to start from the hydrogen-like spectrum with almost all material parameters set to zero and then increase these material parameters successively to their true values. This is shown in Fig. \[fig:Fig6\].
At first all material parameters except for $\gamma_1'$ are set to zero, so that a true hydrogen-like spectrum is obtained, where the yellow (y) and green (g) exciton states are degenerate. This spectrum is shown in the panel (a) of Fig. \[fig:Fig6\]. When increasing the spin-orbit coupling constant $\Delta$ in Fig. \[fig:Fig6\](a), the degeneracy between the green and the yellow exciton series is lifted. The increase of the Luttinger parameters $\mu'$ and $\delta'$ in the panels (b) and (c) furthermore lifts the degeneracy between the exciton states of different angular momentum $L$. The Haken potential does not change degeneracies but slightly lowers the energy of the exciton states in Fig. \[fig:Fig6\](d). The exchange energy described by the constant $J_0$ lifts the degenercy between ortho and para exciton states in Fig. \[fig:Fig6\](e). As the operator $\delta\!\left(\boldsymbol{r}\right)$ affects only the states of even parity (blue lines), the energy of the odd exciton states (red lines) remains unchanged in Fig. \[fig:Fig6\](f). Note that we increase $\Delta$ in two steps to its true value of $\Delta=0.131\,\mathrm{eV}$ for reasons of clarity. Hence, at the bottom of Fig. \[fig:Fig6\](g) all material values have been increased to their true values. For a comparison, we show in panel (h) the position of the experimentally observed states. Following the exciton states from panel (a) to (g), it is possible to assign them with the notation $nL^{\mathrm{p/o}}_{\mathrm{y/g}}$, where the upper index denotes a para or an ortho exciton state and the lower index a yellow or a green state.
The results presented in Fig. \[fig:Fig6\] suggest to assign the exciton state at $E=2.1378\,\mathrm{eV}$ to the green $1S$ ortho exciton state. However, one can observe an anticrossing between the green $1S$ state and the yellow $2S$ state, which is indicated by a green arrow in Fig. \[fig:Fig6\](g). Hence, the assignment has to be changed. As a proof, we can calculate the percentage of the $J=3/2$ component of these states, i.e., their green part, by evaluating $$\mathrm{gp}=\left\langle\Psi\middle| P\middle|\Psi\right\rangle$$ with the projection operator $$P=\sum_{M_J=-3/2}^{3/2}\left|\frac{3}{2},\,M_J\right\rangle\left\langle\frac{3}{2},\,M_J\right|$$ and the exciton wave function $\left|\Psi\right\rangle$ (see also Appendix \[sub:green-part\]).
The green part gp of the state at $E\approx 2.1544\,\mathrm{eV}$ is distinctly higher ($\mathrm{gp}\approx 40\%$) than the green part of the exciton state at $E\approx 2.1378\,\mathrm{eV}$ ($\mathrm{gp}\approx 11\%$). However, since also $\mathrm{gp}\approx 40\%$ is significantly smaller than one, we see that the assignment of this exciton state as the ground state of the green series is questionable and shows the significant deviations from the hydrogen-like model. The green $1S$ exciton state is distributed over the yellow states. Note that in Ref. [@7] also the state of higher energy had a larger green part than the state of lower energy. However, in Fig. 2 of Ref. [@7] the assignment is reversed since the limit of $\mu'\rightarrow 0$ was used to designate the states. It seems obvious that a similar anticrossing between the green $1S$ state and the yellow $2S$ state was disregarded. A considerable effect of the interaction between the green and yellow series is the change in the oscillator strength of the states. The oscillator strength of the $2S_y$ state is much smaller than expected when assuming two independent, i.e., green and yellow, series [@6; @7] (cf. also Tables \[tab:3\] and \[tab:4\]).
For reasons of completeness, we give the size of the green $1S$ and the yellow $2S$ state by evaluating Eq. (\[eq:PsirPsi\]). Since these states are strongly mixed and a correct assignment with a principal quantum number $n$ is not possible, we do not use the formula (\[eq:aexcH\]). We obtain
$$\begin{aligned}
\left\langle r\right\rangle\left(2S_y\right) & \approx & 4.32\,\mathrm{nm}\approx 10.1\,a,\\
\left\langle r\right\rangle\left(1S_g\right) & \approx & 5.32\,\mathrm{nm}\approx 12.5\,a,\end{aligned}$$
when using the Haken potential or
$$\begin{aligned}
\left\langle r\right\rangle\left(2S_y\right) & \approx & 4.39\,\mathrm{nm}\approx 10.3\,a,\\
\left\langle r\right\rangle\left(1S_g\right) & \approx & 4.09\,\mathrm{nm}\approx 9.58\,a,\end{aligned}$$
when using the Pollmann-Büttner potential. We see that in both cases the values of $\left\langle r\right\rangle$ for the green $1S$ and the yellow $2S$ state are of the same size. This is expected due to the strong mixing of both states.
The resonance of the green $1S$ state with the yellow exciton series and the mixing of all even exciton states via the cubic band structure leads to an admixture of $D$ and $G$ states to the green $1S$ state. Hence, the three $\Gamma_5^+$ states which we assigned with $1S_{\mathrm{g}}$ are elliptically deformed and invariant only under the subgroup $D_{4\mathrm{h}}$ of $O_{\mathrm{h}}$ [@100; @G3]. The lower symmetry of the envelope function allows for a smaller mean distance between electron and hole in a specific direction, which leads to a gain of energy due to the Coulomb interaction [@100]. As regards the $xy$-component, the symmetry axis of the according subgroup $D_{4\mathrm{h}}$ is the $z$-axis of the crystal. Since for this state the expectation values $\left\langle\Psi\middle| x^2\middle|\Psi\right\rangle$ and $\left\langle\Psi\middle| y^2\middle|\Psi\right\rangle$ are identical, we can calculate the semi-principal axes of the elliptically deformed state by evaluating $$\begin{aligned}
\left\langle\Psi\middle| x^2\middle|\Psi\right\rangle & = & \left\langle\Psi\middle|\frac{1}{2}\left(r^2-z^2\right)\middle|\Psi\right\rangle\nonumber\\
& = & \sum_{N'L'J'F'F_t' M_{F_t}'}\sum_{NLJFF_t M_{F_t}}\nonumber\\
\nonumber\\
& & c_{N'L'J'F'F_t' M_{F_t}'}c_{NLJFF_t M_{F_t}}\nonumber\\
\nonumber\\
& \times & \alpha^2\left\langle\Pi'\middle|\frac{1}{3}r^2-\frac{1}{3\sqrt{6}}X^{(2)}_0\middle|\Pi\right\rangle
\label{eq:Psix2Psi}\end{aligned}$$ and $$\begin{aligned}
\left\langle\Psi\middle| z^2\middle|\Psi\right\rangle & = & \sum_{N'L'J'F'F_t' M_{F_t}'}\sum_{NLJFF_t M_{F_t}}\nonumber\\
\nonumber\\
& & c_{N'L'J'F'F_t' M_{F_t}'}c_{NLJFF_t M_{F_t}}\nonumber\\
\nonumber\\
& \times & \alpha^2\left\langle\Pi'\middle|\frac{1}{3}\sqrt{\frac{2}{3}}X^{(2)}_0+\frac{1}{3}r^2\middle|\Pi\right\rangle
\label{eq:Psix2Psi}\end{aligned}$$ with the wave function $\Psi$ of Eq. (\[eq:ansatz\]) and the matrix elements $\langle\Pi'|X^{(2)}_0|\Pi\rangle$ and $\langle\Pi'|r^2|\Pi\rangle$ listed in the Appendix of Ref. [@125]. We obtain $$\begin{aligned}
\left\langle x^2\right\rangle & \approx & 116.4\,a^2,\nonumber\\
\left\langle z^2\right\rangle & \approx & 29.9\,a^2,\end{aligned}$$ when using the Haken potential or $$\begin{aligned}
\left\langle x^2\right\rangle & \approx & 68.6\,a^2,\nonumber\\
\left\langle z^2\right\rangle & \approx & 25.1\,a^2,\end{aligned}$$ when using the Pollmann-Büttner potential. The significant differences in $\left\langle x^2\right\rangle$ and $\left\langle z^2\right\rangle$ show again the strong resonance of the green $1S$ state with the yellow series as well as the strong admixture of states with $L\geq 2$. We finally want to note that, due to the coupling of the yellow and green series, the green $1S$ has to be regarded as an excited state in the complete exciton spectrum and not as the ground state of the green series. In particular, the green $1S$ state is orthogonal to the true ground state of the complete spectrum, i.e., to the yellow $1S$ state.
Let us now discuss the other exciton states. To determine the number of para and ortho exciton states as well as their degeneracies for the different values of $L$, one can use group theoretical considerations. In the spherical approximation, in which the cubic part of the Hamiltonian is neglected $\left(\delta'=0\right)$, the momentum $F=J+L$ is a good quantum number for the states of negative parity since the exchange interaction does not act on these states. The states of positive parity can be classified by the total momentum $F_t=F+S_{\mathrm{e}}$ in the spherical approximation.
If the complete cubic Hamiltonian is treated, the reduction of the irreducible representations $D^{F}$ or $D^{F_t}$ of the rotation group by the cubic group $O_{\mathrm{h}}$ has to be considered [@G1]. This is shown in Table \[tab:2\]. As has already been stated in Ref. [@100], a normal spin one transforms according to the irreducible representation $\Gamma_{4}^{+}$ of the cubic group whereas the quasi-spin $I$ transforms according to $\Gamma_{5}^{+}=\Gamma_{4}^{+}\otimes\Gamma_{2}^{+}$. Therefore, one has to include the additional factor $\Gamma_{2}^{+}$ when determining the symmetry of an exciton state [@7; @28; @100]. This symmetry is given by the symmetry of the envelope function, the valence band, and the conduction band: $$\Gamma_{\mathrm{exc}}=\Gamma_{\mathrm{env}}\otimes\Gamma_{\mathrm{v}}\otimes\Gamma_{\mathrm{c}}.$$ Only states of symmetry $\Gamma_4^-$ are dipole allowed in one-photon absorption and only states of symmetry $\Gamma_5^+$ are dipole allowed in two-photon absorption. Hence, we see from Table \[tab:2\] that there are at the most one $P$ state and four $F$ states or one $S$ and two $D$ states for each principal quantum number $n$, which can be observed in experiments.
Since the exchange interaction does not act on the exciton states with negative parity, one can use the irreducible representations of the second column of Table \[tab:2\] to classify these exciton states [@28]. For the exciton states of positive parity the irreducible representations of the third column are needed. Note that the cubic part of the Hamiltonian mixes the $S$ and $D$ exciton states of symmetry $\Gamma_5^+$. Hence, the exchange interaction acts only on the $D$ excitons of symmetry $\Gamma_5^+$ via their $S$ component. The degeneracies between the $D$ states of symmetry $\Gamma_3^+$ and $\Gamma_4^+$ or $\Gamma_1^+$ and $\Gamma_4^+$ is not lifted, respectively (cf. the third column of Table \[tab:2\]).
Since neither $J$ nor $F$ are good quantum numbers due to the cubic symmetry of our Hamiltonian, we do not use the nomenclature $n^{2J+1}L_{F}$ of Refs. [@6; @7]. Although $L$ is likewise no good quantum number, the assignment of the exciton states by using $S$, $P$, $D$, $F$ and $G$ to denote the angular momentum is still common (see, e.g., Refs. [@80; @28]). Hence, we feel obliged to classify the states by introducing the notation $nL_{\mathrm{y}/\mathrm{g}}$ for comparison with other works but also stress that this is generally not instructive due to the large deviations from the hydrogen-like model (cf. also Ref. [@100]). By the index y or g we denote the yellow or the green exciton series, respectively. To be more correct, we will also give the symmetry of the exciton states in terms of the irreducible representations of Table \[tab:2\]. These symmetries can be determined by regarding the eigenvectors of the generalized eigenvalue problem (\[eq:gev\]) [@100].
In the Tables \[tab:3\] and \[tab:4\] we now give a direct comparison between experimental and theoretical exciton energies for all states with $n\leq 5$. One can see that the results with the Haken potential listed in Tab. \[tab:3\] show a better agreement with the experimental values than the results with the Pollmann-Büttner potential listed in Tab. \[tab:4\]. Hence, we have chosen the central-cell corrections with the Haken potential for the calculation of Fig. \[fig:Fig6\].
The Haken potential or the Pollmann-Büttner potential also slightly affects the odd exciton series and especially the $2P$ exciton state. These potentials shift the energy of the $\Gamma_4^-$ (resp. $\Gamma_8^-$) $2P$ exciton state by an amount of $210\,\mathrm{\upmu eV}$ (Haken) or $880\,\mathrm{\upmu eV}$ (Pollmann-Büttner) towards lower energies.
Summary and outlook \[sec:Summary-and-outlook\]
===============================================
We have treated the exciton spectrum of $\mathrm{Cu_{2}O}$ considering the complete valence band structure, the exchange interaction, and the central-cell corrections. A thorough discussion of the central-cell corrections revealed that only the frequency and momentum dependence of the dielectric function $\varepsilon\left(k,\,\omega\right)$ have to be accounted for. Due to the estimated size of the $1S$ exciton Bohr radius, corrections to the kinetic energy can be neglected. Hence, only the two parameters $V_{0}$ and $J_{0}$ are decisive for the relative position of the exciton states. While $J_0$ describes the splitting of the exciton states into ortho and para components, $V_0$ changes the relative energy of the states but leaves this splitting between ortho and para component of the same exciton state almost unchanged. Hence, these parameters could be determined almost independently. This means that our results are not very sensitive to the choice of the parameters used. Instead, there is only one combination of both parameters $J_0$ and $V_0$ given in Eqs. (\[eq:J0\])-(\[eq:V0PB\]), for which our results are in good agreement with the experiment.
We have shown that the central-cell corrections considerably affect the complete even exciton series since the valence band structure couples the $1S$ state to higher exciton states. The frequency dependence of the dielectric function also slightly affects the odd exciton series and lowers, in particular, the energy of the $2P$ exciton state. Furthermore, we have demonstrated that due to the coupling of the yellow and the green exciton series the green $1S$ exciton state is distributed over all yellow states.
In contrast to earlier works [@78], we have presented a closed theory of the complete exciton series in $\mathrm{Cu_{2}O}$, where we explicitly give the correction potentials (\[eq:HCCC\_H\]) or (\[eq:HCCC\_PB\]). Hence, the introduction of quantum defects or the introduction of different exchange parameters for different exciton states, which take the effect of the central cell corrections into account only phenomenologically, is redundant [@78; @80].
The results of our theory show a very good agreement with experimental values (see Table. \[tab:3\]). Therefore, we are confident that an according extension of our theory will allow for the calculation of exciton spectra in $\mathrm{Cu_{2}O}$ in electric or in combined electric and magnetic fields.
We thank J. Heckötter, M. Bayer, D. Fröhlich, M. A[ß]{}mann, and D. Dizdarevic for helpful discussions.
$p^4$-Terms \[sec:p4\]
======================
As has already been stated in Sec. \[sec:Bandstruc\], the terms of the fourth power of $\boldsymbol{p}$ span a fifteen dimensional space with the basis functions $$p_i^4,\quad p_i^3 p_j,\quad p_i^2 p_j^2,\quad p_i p_j p_k^2$$ with $i,j,k\in\{1,2,3\}$ and $i\neq j\neq k\neq i$. The six linear combinations of $p^4$ terms (including the quasi spin $I$), which transform according to $\Gamma_1^+$ [@G3] read in terms of irreducible tensors
$$\begin{aligned}
\mathrm{(I):}\quad & p^4, \\
\displaybreak[2]
\mathrm{(II):}\quad & P^{(4)}\left(\Gamma_1^+\right), \\
\displaybreak[2]
\mathrm{(III):}\quad & p^2\left(P^{(2)}\cdot I^{(2)}\right), \\
\displaybreak[2]
\mathrm{(IV):}\quad & p^2\left[P^{(2)}\times I^{(2)}\right]^{(4)}\left(\Gamma_1^+\right), \\
\displaybreak[2]
\mathrm{(V):}\quad & \left[P^{(4)}\times I^{(2)}\right]^{(4)}\left(\Gamma_1^+\right), \\
\displaybreak[2]
\mathrm{(VI):}\quad & \left[P^{(4)}\times I^{(2)}\right]^{(6)}\left(\Gamma_1^+\right),\end{aligned}$$
with
$$T^{(4)}\left(\Gamma_1^+\right)=\sqrt{\frac{5}{24}}\sum_{k=\pm 4}T^{(4)}_{k}+\sqrt{\frac{7}{12}}T^{(4)}_{0},$$
and $$T^{(6)}\left(\Gamma_1^+\right)=-\frac{\sqrt{7}}{4}\sum_{k=\pm 4}T^{(6)}_{k}+\frac{1}{\sqrt{8}}T^{(6)}_{0}.$$
One can choose appropriate linear combinations of the states (I)-(VI):
$$\begin{aligned}
\frac{1}{5}\mathrm{(I)}-\frac{1}{3\sqrt{30}}\mathrm{(II)} = &\: \left[p_1^2 p_2^2 +\mathrm{c.p.}\right]
\\
\displaybreak[1]
\nonumber \\
\frac{2}{3}\hbar^2 \mathrm{(I)}+\frac{2}{45}\mathrm{(III)}+\frac{1}{18}\sqrt{\frac{24}{5}}\mathrm{(IV)} = &\: \boldsymbol{p}^2\left[p_1^2 \boldsymbol{I}_1^2+\mathrm{c.p.}\right]
\\
\displaybreak[1]
\nonumber \\
\frac{1}{30}\mathrm{(III)}-\frac{1}{36}\sqrt{\frac{24}{5}}\mathrm{(IV)} = &\: \boldsymbol{p}^2\left[p_1 p_2\left\{\boldsymbol{I}_1,\,\boldsymbol{I}_2\right\}+\mathrm{c.p.}\right]
\\
\displaybreak[1]
\nonumber \\
\frac{6}{5}\hbar^2 \mathrm{(I)}-\frac{8}{9\sqrt{30}}\hbar^2 \mathrm{(II)}-\frac{4}{27}\sqrt{\frac{7}{11}}\mathrm{(V)}+\frac{1}{9}\sqrt{\frac{14}{33}}\mathrm{(VI)} = &\: \left[\left(p_1^4+6p_2^2 p_3^2\right) \boldsymbol{I}_1^2+\mathrm{c.p.}\right]
\\
\displaybreak[1]
\nonumber \\
-\frac{1}{18}\sqrt{\frac{7}{11}}\mathrm{(V)}-\frac{1}{9}\sqrt{\frac{14}{33}}\mathrm{(IV)} = &\: \left[\left(p_1^2+ p_2^2 -6p_3^2\right)p_1 p_2 \left\{\boldsymbol{I}_1,\,\boldsymbol{I}_2\right\}+\mathrm{c.p.}\right]\end{aligned}$$
with $\left\{ a,b\right\} =\frac{1}{2}\left(ab+ba\right)$ and c.p. denoting cyclic permutation. These linear combinations enter the generalized expressions of the kinetic energy of the hole and the electron in Sec. \[sec:Bandstruc\].
Oscillator strengths \[sub:Oscillator-strengths\]
=================================================
We now give the formula for the expression $$\lim_{r\rightarrow0}\,_{T}\left\langle 1,\,M'_{F_t}\middle|\Psi\left(\boldsymbol{r}\right)\right\rangle,$$ which is needed for the evaluation of the relative oscillator strength $f_{\mathrm{rel}}$ (\[eq:frelT\]) in two photon absorption experiments. Using the wave function of Eq. (\[eq:ansatz\]), we find $$\begin{aligned}
&\: \lim_{r\rightarrow0}\,_{T}\left\langle 1,\,M'_{F_t}\middle|\Psi\left(\boldsymbol{r}\right)\right\rangle\nonumber \\
= &\: \sum_{NFF_{t}}\sum_{M_{S_{\mathrm{e}}}}c_{N0FFF_{t}M'_{F_t}}\;\sqrt{\frac{2}{\alpha^3}}\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(-1\right)^{F-2M_{S_{\mathrm{e}}}+\frac{1}{2}}\left[(2F+1)(2F_{t}+1)\right]^{\frac{1}{2}}\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(\begin{array}{ccc}
F & \frac{1}{2} & F_{t}\\
M'_{F_t}-M_{S_{\mathrm{e}}} & M_{S_{\mathrm{e}}} & -M'_{F_t}
\end{array}\right)\nonumber \\\
\displaybreak[2]
\nonumber \\
\times &\: \left(\begin{array}{ccc}
1 & \frac{1}{2} & F\\
M'_{F_t} & -M_{S_{\mathrm{e}}} & M_{S_{\mathrm{e}}}-M'_{F_t}
\end{array}\right).\end{aligned}$$
Green part of $\Psi$ \[sub:green-part\]
=======================================
Here we give the formula for the scalar product which is needed to calculate the green part of the wave function $\Psi$ as $$\mathrm{gp}=\sum_{M_J=-3/2}^{3/2}\left\langle\Psi\middle|\frac{3}{2},\,M_J\right\rangle\left\langle\frac{3}{2},\,M_J\middle|\Psi\right\rangle.$$ We find $$\begin{aligned}
&\: \left\langle\Psi\middle|\frac{3}{2},\,M_J\right\rangle\left\langle\frac{3}{2},\,M_J\middle|\Psi\right\rangle\nonumber \\
\displaybreak[2]
\nonumber \\
= &\: \sum_{j=-1}^{1}\sum_{NLFF_{t}M_{F_t}}\sum_{F'F_{t}'M_{F_t}'}\sum_{M_{S_{\mathrm{e}}}M_L}\frac{\left(R_{1}\right)_{NL}^{j}}{N+L+j+1}\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: c_{(N+j)LJF'F_{t}'M_{F_t}'}c_{NLJFF_{t}M_{F_t}}\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(-1\right)^{F+F'+M_{F_t}+M_{F_t}'-2J+2M_J-1}\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left[(2F+1)(2F_{t}+1)(2F'+1)(2F_{t}'+1)\right]^{\frac{1}{2}}\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(\begin{array}{ccc}
F & \frac{1}{2} & F_{t}\\
M_L+M_J & M_{S_{\mathrm{e}}} & -M_{F_t}
\end{array}\right)\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(\begin{array}{ccc}
L & J & F\\
M_L & M_J & -M_L-M_J
\end{array}\right)\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(\begin{array}{ccc}
F' & \frac{1}{2} & F_{t}'\\
M_L+M_J & M_{S_{\mathrm{e}}} & -M_{F_t}'
\end{array}\right)\nonumber \\
\displaybreak[2]
\nonumber \\
\times &\: \left(\begin{array}{ccc}
L & J & F'\\
M_L & M_J & -M_L-M_J
\end{array}\right).\end{aligned}$$
The function $\left(R_{1}\right)_{NL}^{j}$ is taken from the recursion relations of the Coulomb-Sturmian functions in the Appendix of Ref. [@100].
Matrix elements \[sub:Matrix-elements\]
=======================================
In this section we give the matrix elements of the terms in Eq. (\[eq:HCCC\]) in the basis of Eq. (\[eq:basis\]) in Hartree units. The normalization factor $N_{NL}^{\left(\alpha\right)}$ is given in Eq. (\[eq:normaliz\]). All other matrix elements, which enter the symmetric matrices $\boldsymbol{D}$ and $\boldsymbol{M}$ in Eq. (\[eq:gev\]) and which are not given here, are listed in the Appendix of Ref. [@100].
$$\begin{aligned}
\left\langle \Pi'\left|\delta\left(\boldsymbol{r}\right)\right|\Pi\right\rangle = &\: \delta_{L'0}\delta_{L0}\delta_{JJ'}\delta_{F_{t}F'_{t}}\delta_{M_{F_{t}}M'_{F_{t}}}\:\frac{1}{\pi}\left(-1\right)^{F_{t}+F'+F+J+\frac{1}{2}}\nonumber \\
\nonumber \\
\times &\: \left[\left(2F_{t}+1\right)\left(2F+1\right)\left(2F'+1\right)\right]^{\frac{1}{2}}\left\{ \begin{array}{ccc}
F' & F_{t} & \frac{1}{2}\\
F_{t} & F & 0
\end{array}\right\} \left\{ \begin{array}{ccc}
0 & F' & J\\
F & 0 & 0
\end{array}\right\}, \label{eq:matdelta}
\\
\displaybreak[1]
\nonumber \\
\left\langle \Pi'\left|\boldsymbol{S}_{\mathrm{e}}\cdot\boldsymbol{S}_{\mathrm{h}}\,\delta\left(\boldsymbol{r}\right)\right|\Pi\right\rangle = &\: \delta_{L'0}\delta_{L0}\delta_{F_{t}F'_{t}}\delta_{M_{F_{t}}M'_{F_{t}}}\:\frac{3}{2\pi}\left(-1\right)^{F_t+F'+F+J+J'}\nonumber \\
\nonumber \\
\times &\: \left[\left(2F+1\right)\left(2F'+1\right)\left(2J+1\right)\left(2J'+1\right)\right]^{\frac{1}{2}}\nonumber \\
\nonumber \\
\times &\: \left\{ \begin{array}{ccc}
F' & F & 1\\
\frac{1}{2} & \frac{1}{2} & F_t
\end{array}\right\} \left\{ \begin{array}{ccc}
F & F' & 1\\
J' & J & 0
\end{array}\right\} \left\{ \begin{array}{ccc}
\frac{1}{2} & J' & 1\\
J & \frac{1}{2} & 1
\end{array}\right\},
\\
\displaybreak[1]
\nonumber \\
\left\langle \Pi'\left|\frac{1}{r}e^{-r/\rho}\right|\Pi\right\rangle = &\: \delta_{LL'}\delta_{JJ'}\delta_{F_{t}F_{t}'}\delta_{M_{F_{t}}M_{F_{t}}'}\:\frac{1}{4}N_{N'L}^{\left(1\right)}N_{NL}^{\left(1\right)}\:\sum_{k=0}^{N'}\:\sum_{j=0}^{N}\left(-1\right)^{k+j}\nonumber \\
\nonumber \\
\times &\:
\binom{N'+2L+1}{N'-k}\binom{N+2L+1}{N-j}\frac{\left(2L+k+j+1\right)!}{k!\,j!}\left[\frac{2\rho}{1+2\rho}\right]^{\left(2+2L+k+j\right)}.\end{aligned}$$
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abstract: 'Weak value amplification (WVA) is a concept that has been extensively used in a myriad of applications with the aim of rendering measurable tiny changes of a variable of interest. In spite of this, there is still an on-going debate about its [*true*]{} nature and whether is really needed for achieving high sensitivity. Here we aim at solving the puzzle, using some basic concepts from quantum estimation theory, highlighting what the use of the WVA concept can offer and what it can not. While WVA cannot be used to go beyond some fundamental sensitivity limits that arise from considering the full nature of the quantum states, WVA can notwithstanding enhance the sensitivity of [*real*]{} detection schemes that are limited by many other things apart from the quantum nature of the states involved, i.e. [*technical noise*]{}. Importantly, it can do that in a straightforward and easily accessible manner.'
author:
- 'Juan P. Torres'
- 'Luis José Salazar-Serrano'
title: 'Weak value amplification: a view from quantum estimation theory that highlights what it is and what isn’t'
---
Introduction {#introduction .unnumbered}
============
Weak value amplification (WVA) [@aharonov1988] is a concept that has been used under a great variety of experimental conditions [@hosten2008; @zhou2012; @ben_dixon2009; @pfeifer2011; @howell2010_freq; @egan2012; @xu_guo2013] to reveal tiny changes of a variable of interest. In all those cases, a priori sensitivity limits were not due to the quantum nature of the light used ([*photon statistics*]{}), but instead to the insufficient resolution of the detection system, what might be termed generally as [*technical noise*]{}. WVA was a feasible choice to go beyond this limitation. In spite of this extensive evidence, “its interpretation has historically been a subject of confusion" [@dressel2014]. For instance, while some authors [@jordan2014] show that “weak-value-amplification techniques (which only use a small fraction of the photons) compare favorably with standard techniques (which use all of them)", others [@knee2014] claim that WVA “does not offer any fundamental metrological advantage" , or that WVA [@ferrie2014] “does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection”. However, these conclusions are criticized by others based on the idea that “the assumptions in their statistical analysis are irrelevant for realistic experimental situations” [@vaidman2014]. The problem might reside in
Here we make use of some simple, but fundamental, results from quantum estimation theory [@helstrom1976] to show that there are two sides to consider when analyzing in which sense WVA can be useful. On the one hand, the technique generally makes use of linear-optics unitary operations. Therefore, it cannot modify the statistics of photons involved. Basic quantum estimation theory states that the post-selection of an appropriate output state, the basic element in WVA, cannot be better than the use of the input state [@nielsen2000]. Moreover, WVA uses some selected, appropriate but partial, information about the quantum state that cannot be better that considering the full state. Indeed, due to the unitarian nature of the operations involved, it should be equally good any transformation of the input state than performing no transformation at all. In other words, when considering only the quantum nature of the light used, WVA cannot enhance the precision of measurements [@lijian2015].
On the other hand, a more general analysis that goes beyond only considering the quantum nature of the light, shows that WVA can be useful when certain technical limitations are considered. In this sense, it might increase the ultimate resolution of the detection system by effectively lowering the value of the smallest quantity that can detected. In most scenarios, although not always [@torres2012], the signal detected is severely depleted, due to the quasi-orthogonality of the input and output states selected. However, in many applications, limitations are not related to the low intensity of the signal [@hosten2008], but to the smallest change that the detector can measure irrespectively of the intensity level of the signal.
A potential advantage of our approach is that we make use of the concept of trace distance, a clear and direct measure of the degree of distinguishability of two quantum states. Indeed, the trace distance gives us the minimum probability of error of distinguishing two quantum states that can be achieved under the best detection system one can imagine [@helstrom1976]. Measuring tiny quantities is essentially equivalent to distinguishing between nearly parallel quantum states. Therefore we offer a very basic and physical understanding of how WVA works, based on the idea of how WVA transforms very close quantum states, which can be useful to the general physics reader.
Here were we use an approach slightly different from what other analysis of WVA do, where most of the times the tool used to estimate its usefulness is the Fisher information. Contrary to how we use the trace distance here, to set a sensitivity bound only considering how the quantum state changes for different values of the variable of interest, the Fisher information requires to know the probability distribution of possible experimental outcomes for a given value of the variable of interest. Therefore, it can look for sensitivity bounds for measurements by including [*technical characteristics*]{} of specific detection schemes [@jordan2014]. A brief comparison between both approaches will be done towards the end of this paper.
One word of caution will be useful here. The concept of weak value amplification is presented for the most part in the framework of Quantum Mechanics theory, where it was born. It can be readily understood in terms of constructive and destructive interference between probability amplitudes [@duck1989]. Interference is a fundamental concept in any theory based on waves, such as classical electromagnetism. Therefore, the concept of weak value amplification can also be described in many scenarios in terms of interference of classical waves [@howell2010]. Indeed, most of the experimental implementations of the concept, since its first demonstration in 1991 [@ritchie1991], belong to this type and can be understood without resorting to a quantum theory formalism.
An example of the application of the weak value amplification concept: measuring small temporal delays with large bandwidth pulses. {#an-example-of-the-application-of-the-weak-value-amplification-concept-measuring-small-temporal-delays-with-large-bandwidth-pulses. .unnumbered}
-----------------------------------------------------------------------------------------------------------------------------------
For the sake of example, we consider a specific weak amplification scheme [@brunner2010], depicted in Fig. 1, which has been recently demonstrated experimentally [@xu_guo2013; @salazar2014]. It aims at measuring very small temporal delays $\tau$, or correspondingly tiny phase changes [@strubi2013], with the help of optical pulses of much larger duration. We consider this specific case because it contains the main ingredients of a typical WVA scheme, explained below, and it allows to derive analytical expressions of all quantities involved, which facilitates the analysis of main results. Moreover, the scheme makes use of linear optics elements only and also works with large-bandwidth partially-coherent light [@li_guo2011].
In general, a WVA scheme requires three main ingredients: a) the consideration of two subsystems (here two degrees of freedom: the polarisation and the spectrum of an optical pulse) that are weakly coupled (here we make use of a polarisation-dependent temporal delay that is introduced with the help of a Michelson interferometer); b) the [*pre-selection*]{} of the input state of both subsystems; and c) the [*post-selection*]{} of the state in one of the subsystems (the state of polarisation) and the measurement of the state of the remaining subsystem (the spectrum of the pulse). With appropriate [*pre-*]{} and [*post-selection*]{} of the polarisation of the output light, tiny changes of the temporal delay $\tau$ can cause anomalously large changes of its spectrum, rendering in principle detectable very small temporal delays.
![Weak value amplification scheme aimed at detecting extremely small temporal delays. The input pulse polarisation state is selected to be left-circular by using a polariser, a quarter-wave plate (QWP) and a half-wave plate (HWP). A first polarising beam splitter (PBS$_1$) splits the input into two orthogonal linear polarisations that propagate along different arms of the interferometer. An additional QWP is introduced in each arm to rotate the beam polarisation by $90^{\circ}$ to allow the recombination of both beams, delayed by a temporal delay $\tau$, in a single beam by the same PBS. After PBS$_1$, the output polarisation state is selected with a liquid crystal variable retarder (LCVR) followed by a second polarising beam splitter (PBS$_2$). The variable retarder is used to set the parameter $\theta$ experimentally. Finally, the spectrum of each output beam is measured using an optical spectrum analyzer (OSA). ($\hat{x}$,$\hat{y}$) and ($\hat{u}$,$\hat{v}$) correspond to two sets of orthogonal polarisations. Figure drawn by one of the authors (Luis-Jose Salazar Serrano).[]{data-label="figure_scheme"}](figure1.eps){width="70.00000%"}
Let us be more specific about how all these ingredients are realized in the scheme depicted in Fig. 1. An input coherent laser beam ($N$ photons) shows circular polarisation, ${\bf e}_{\mathrm{in}}=1/\sqrt{2}\,\left(
\hat{x}-i\hat{y} \right)$, and a Gaussian shape with temporal width $T_0$ (Full-width-half maximum, $\tau \ll T_0$). The normalized temporal and spectral shapes of the pulse read $$\begin{aligned}
& & \Psi(t)=\left( \frac{4 \ln 2}{\pi T_0^2}\right)^{1/4} \exp
\left(
-\frac{2 \ln 2 t^2}{T_0^2} \right) \nonumber \\
& & \Psi(f)=\left(\frac{\pi T_0^2}{\ln 2}\right)^{1/4} \exp \left(
-\frac{\pi^2 T_0^2 f^2}{2 \ln 2} \right).\end{aligned}$$ The input beam is divided into the two arms of a Michelson interferometer with the help of a polarising beam splitter (PBS$_1$). Light beams with orthogonal polarisations traversing each arm of the interferometer are delayed $\tau_0$ and $\tau_0+\tau$, respectively, which constitute the weak coupling between the two degrees of freedom. After recombination of the two orthogonal signals in the same PBS$_1$, the combination of a liquid-crystal variable retarder (LCVR) and a second polarising beam splitter (PBS$_2$) performs the post-selection of the polarisation of the output state, projecting the incoming signal into the polarisation states $\hat{u}=1/\sqrt{2} \left[
\hat{x}+\hat{y}\exp(i\theta) \right]$ and $\hat{v}=1/\sqrt{2}
\left[ \hat{x}-\hat{y}\exp(i\theta) \right]$. The amplitudes of the signals in the two output ports write (not normalized) $$\begin{aligned}
& & \Phi_u(\tau)=\frac{\Psi(\Omega)}{2} \exp
\left[ i(\omega_0 +\Omega) \tau_0 \right] \left\{1+
\exp \left[ i (\omega_0+\Omega) \tau-i\Gamma \right] \right\} \label{projections1} \\
& & \Phi_v(\tau)=\frac{\Psi(\Omega)}{2} \exp \left[ i \left(
\omega_0+ \Omega \right)\tau_0 \right] \left\{ 1-\exp \left[ i
(\omega_0+\Omega) \tau -i\Gamma \right] \right\},
\label{projections2}\end{aligned}$$ where $\Gamma=\pi/2+\theta$.
![Spectrum measured at the output. (a) and (b): Spectral shape of the mode functions for $\tau=0$ (solid blue line) and $\tau=100$ as (dashed green line). In (a) the post-selection angle $\theta$ is $97.2^{\circ}$, so as to fulfil the condition $\omega_0 \tau-\Gamma=\pi$. In (b) the angle $\theta$ is $96.7^{\circ}$. (c) Shift of the centroid of the spectrum of the output pulse after projection into the polarisation state $\hat{u}$ in PBS$_2$, as a function of the post-selection angle $\theta$. Green solid line: $\tau=10$ as; Dotted red line: $\tau=50$ as, and dashed blue line: $\tau=100$ as. Label [**I**]{} corresponds to $\theta=96.7^{\circ}$ \[mode for $\tau=100$ as shown in (b)\]. Label [**II**]{} corresponds to $\theta=97.2^{\circ}$, where the condition $\omega_0 \tau-\Gamma=\pi$ is fulfiled \[mode for $\tau=100$ shown in (a)\]. It yields the minimum mode overlap between states with $\tau=0$ and $\tau \neq 0$. Data: $\lambda_0=1.5\, \mu$m and $T_0=100$ fs.[]{data-label="figure_modes"}](figure2.eps){width="90.00000%"}
After the signal projection performed after PBS$_2$, the WVA scheme distinguishes different states, corresponding to different values of the temporal delay $\tau$, by measuring the spectrum of the outgoing signal in the selected output port. The different spectra obtained for delays $\tau=0$ and $\tau=100$ as, for two different polarisation projections, are shown in Figures 2 (a) and 2 (b). To characterize different modes one can measure, for instance, the centroid of the spectrum. Fig. 2 (c) shows the centroid shift of the output signal for $\tau \ne 0$, which reads $$\Delta
f=-\frac{\tau\,\ln 2 }{\pi T_0^2} \frac{\gamma \sin
\left(\omega_0 \tau - \Gamma\right)}{1+\gamma \cos \left(\omega_0
\tau-\Gamma\right)}, \label{centroid_shift}$$ The differential power between both signals (with $\tau=0$ and $\tau \ne 0$) reads $$\frac{P_{\mathrm{out}}(\tau)-P_{\mathrm{out}}(\tau=0)}{P_{\mathrm{in}}}=\frac{1}{2}\,
\left[ \cos \Gamma-\cos \left(\omega_0 \tau -\Gamma \right)
\right]$$ When there is no polarisation-dependent time delay ($\tau=0$), the centroid of the spectrum of the output signal is the same than the centroid of the input laser beam, i.e., there is no shift of the centroid ($\Delta f=0$). However, the presence of a small $\tau$ can produce a large and measurable shift of the centroid of the spectrum of the signal.
Results {#results .unnumbered}
=======
View of weak value amplification from quantum estimation theory {#view-of-weak-value-amplification-from-quantum-estimation-theory .unnumbered}
---------------------------------------------------------------
Detecting the presence ($\tau \neq 0$) or absence ($\tau=0$) of a temporal delay between the two coherent orthogonally-polarised beams after recombination in PBS$_1$, but before traversing PBS$_2$, is equivalent to detecting which of the two quantum states, $$|\Phi_0 \rangle=|\Phi (\tau_0) \rangle_x |\Phi (\tau_0) \rangle_y$$ or $$|\Phi_1 \rangle=|\Phi(\tau_0) \rangle_x |\Phi (\tau_0+\tau)
\rangle_y \label{state1}$$ is the output quantum state which describes the coherent pulse leaving PBS$_1$. $(x,y)$ designates the corresponding polarisations. The spectral shape (mode function) $\Phi$ writes $$\Phi(\tau_0+\tau) = \Psi(\Omega) \exp \left[ i (\omega_0+\Omega)
(\tau_0+\tau) \right], \label{modes_input}$$ where $\omega_0$ is the central frequency of the laser pulse, $\Omega=2\pi f$ is the angular frequency deviation from the the center frequency and $\Psi(\Omega)$ is the spectral shape of the input coherent laser signal.
The minimum probability of error that can be made when distinguishing between two quantum states is related to the trace distance between the states [@fuchs1999]. For two pure state, $\Phi_0$ and $\Phi_1$, the (minimum) probability of error is [@helstrom1976; @ou1996; @englert1996] $$\label{average_error}
P_{\mathrm{error}}=\frac{1}{2}\left(1-\sqrt{1-|\langle
\Phi_0|\Phi_1\rangle|^2} \right).$$ For $\Phi_0=\Phi_1$, $P_{\mathrm{error}}=0.5$. On the contrary, to be successful in distinguishing two quantum states with low probability of error ($P_{\mathrm{error}} \sim 0$) requires $|\langle \Phi_0|\Phi_1\rangle| \sim 0$, i.e., the two states should be close to orthogonal.
The coherent broadband states considered here can be generally described as single-mode quantum states where the mode is the corresponding spectral shape of the light pulse. Let us consider two single-mode coherent beams $$\begin{aligned}
& & |\alpha \rangle=\exp \left( -\frac{|\alpha|^2}{2} \right)
\sum_{n=0}^{\infty} \frac{\alpha^n \left(
A^{\dagger}\right)^n}{n!}|0\rangle \nonumber \\
& & |\beta \rangle=\exp \left( -\frac{|\beta|^2}{2} \right)
\sum_{n=0}^{\infty} \frac{\beta^n \left(
B^{\dagger}\right)^n}{n!}|0\rangle,\end{aligned}$$ where $A$ and $B$ are the two modes $$\begin{aligned}
& & A^{\dagger}=\int d\Omega F(\Omega) a^{\dagger}(\Omega) \nonumber \\
& & B^{\dagger}=\int d\Omega G(\Omega) a^{\dagger}(\Omega),\end{aligned}$$ and $|\alpha|^2$ and $|\beta|^2$ are the mean number of photons in modes $A$ and $B$, respectively. The mode functions $F$ and $G$ are assumed to be normalized, i.e., $\int d\Omega
|F(\Omega)|^2=\int d\Omega |G(\Omega)|^2=1$. The overlap between the quantum states, $|\langle \beta|\alpha \rangle|^2$, reads $$\label{overlap1}|\langle \beta|\alpha \rangle|^2=\exp \left(
-|\alpha|^2-|\beta|^2 + \rho \alpha \beta^{*}+\rho^{*}\alpha^{*}
\beta\right),$$ where we introduce the mode overlap $\rho$ that reads $$\label{overlap2} \rho=\int d\Omega F(\Omega) \left[ G(\Omega)
\right]^{*}.$$ In order to obtain Eq. (\[overlap1\]) we have made use of $\langle 0|B^n \left[ A^{\dagger}\right]^m |0 \rangle=n! \rho^n
\delta_{nm}$. For $\rho=1$ (coherent beams in the same mode but with possibly different mean photon numbers) we recover the well-known formula for single-mode coherent beams [@glauber1963]: $|\langle \beta|\alpha
\rangle|^2=\exp\left(-|\alpha-\beta|^2 \right)$.
Making use of Eqs. (\[modes\_input\]), (\[overlap1\]) and (\[overlap2\]) we obtain $$\begin{aligned}
& & |\langle \Phi_0|\Phi_1\rangle|^2 \nonumber \\
& & =|\langle \Phi(\tau_0)|\Phi(\tau_0)\rangle_x|^2 |\langle
\Phi(\tau_0)|\Phi(\tau_0+\tau)\rangle_y|^2 \nonumber \\
& & = \exp \left[-N \left( 1-\gamma \cos \omega_0 \tau\right)
\right], \label{input_result}\end{aligned}$$ where $$\gamma= \exp \left( -\ln 2 \,\frac{\tau^2}{T_0^2} \right).$$
![Mode overlap and insertion loss as a function of the post-selection angle. Mode overlap $\rho$ of the mode functions corresponding to the quantum states with $\tau=0$ and $\tau=100$ as, as a function of the post-selection angle $\theta$ (solid blue line). The insertion loss, given by $10\log_{10}\,P_{\mathrm{out}}/P_{\mathrm{in}}$ is indicated by the dotted green line. The minimum mode overlap, and maximum insertion loss, corresponds to the post-selection angle $\theta$ that fulfils the condition $\omega_0 \tau-\Gamma=\pi$, which corresponds to $\theta=97.2^{\circ}$. Data: $\lambda_0=1.5 \,\mu$m, $T_0=100$ fs.[]{data-label="figure_overlap_loss"}](figure3.eps){width="50.00000%"}
In the WVA scheme considered here, the signal after PBS$_2$ is projected into the orthogonal polarisation states $\hat{u}$ and $\hat{v}$, and as a result the signals in both output ports are given by Eqs. (\[projections1\]) and (\[projections2\]). Making use of Eqs. (\[projections1\]), (\[projections2\]) and (\[overlap2\]) one obtains that the mode overlap (for $\Phi_u$) reads $$\rho =\frac{1+\cos \Gamma+\gamma \cos
\omega_0 \tau+\gamma \cos (\omega_0 \tau-\Gamma) - i \left[ \sin
\Gamma+\gamma \sin \omega_0 \tau+\gamma \sin (\omega_0
\tau-\Gamma) \right]}{2\left[1+\cos \Gamma \right]^{1/2}
\left[1+\gamma \cos (\omega_0 \tau-\Gamma) \right]^{1/2}}.$$ For $\tau=0$, and therefore $\gamma=1$, we obtain $\rho=1$. Fig. 3 shows the mode overlap of the signal in the corresponding output port for a delay of $\tau=100$ as. The mode overlap has a minimum for $\omega_0 \tau-\Gamma=\pi$, where the two mode functions becomes easily distinguishable, as shown in Fig. 2 (a). The effect of the polarisation projection, a key ingredient of the WVA scheme, can be understood as a change of the mode overlap ([*mode distinguishability*]{}) between states with different delay $\tau$.
However, an enhanced mode distinguishability in this output port is accompanied by a corresponding increase of the insertion loss, as it can be seen in Fig. 3. The insertion loss, $P_{\mathrm{out}}(\tau)/P_{\mathrm{in}}=1/2\, \left[
1+\gamma \cos (\omega_0 \tau-\Gamma)\right]$, is the largest when the modes are close to orthogonal ($\rho \sim 0$). Both effects indeed compensate, as it should be, since WVA implements unitary transformations, and the trace distance between quantum states is preserved under unitary transformations. The quantum overlap between the states reads $$\begin{aligned}
& & |\langle \Phi_u(\tau_0)|\Phi_u(\tau_0+\tau)\rangle|^2=|\langle
\Phi_v(\tau_0)|\Phi_v(\tau_0+\tau)\rangle|^2 \nonumber \\
& & =\exp \left[-\frac{N}{2} \left( 1-\gamma \cos \omega_0
\tau\right) \right],\end{aligned}$$ so $$\begin{aligned}
& & |\langle \Phi_0|\Phi_1 \rangle|^2 \nonumber \\
& & =|\langle \Phi_u(\tau_0)|\Phi_u(\tau_0+\tau)\rangle_u|^2
|\langle \Phi_v(\tau_0)|\Phi_v(\tau_0+\tau)\rangle_v|^2 \nonumber \\
& & =\exp \left[-N \left( 1-\gamma \cos \omega_0 \tau\right)
\right] \label{output_result},\end{aligned}$$ which is the same result \[see Eq. (\[input\_result\])\] obtained for the signal after PBS$_1$, but before PBS$_2$.
We can also see the previous results from a slightly different perspective making use of the Cramér-Rao inequality [@helstrom1976]. The WVA scheme considered throughout can be thought as a way of estimating the value of the single parameter $\tau$ with the help of a light pulse in a coherent state $|\alpha
\rangle$. Since the quantum state is pure, the minimum variance that can show any unbiased estimation of the parameter $\tau$, the Cramér-Rao inequality, reads $$\mathrm{Var} \left( \hat{\tau} \right) \ge \frac{1}{4}\left[ \langle
\frac{\partial \alpha}{\partial \tau}| \frac{\partial
\alpha}{\partial \tau} \rangle-\left| \langle \alpha |
\frac{\partial \alpha}{\partial \tau}\rangle\right|^2
\right]^{-1}, \label{cramer1}$$ Making use of Eq. (\[state1\]), one obtains that here the Cramér-Rao inequality reads [@derivative] $$\mathrm{Var} \left( \hat{\tau} \right) \ge \frac{1}{2N \left(
\omega_0^2+B^2\right)} \label{cramer2}$$ where $B=\sqrt{2 \ln 2}/T_0$ is the rms bandwidth in angular frequency of the pulse. In all cases of interest $B \ll
\omega_0$. The Cramér-Rao inequality is a fundamental limit that set a bound to the minimum variance that any measurement can achieve. It is unchanged by unitary transformations and only depends on the quantum state considered.
Inspection of Eqs. (\[input\_result\]) and (\[output\_result\]) seems to indicate that a measurement after projection in any basis, the core element of the weak amplification scheme, provides no fundamental metrological advantage. Notice that this result implies that the only relevant factor limiting the sensitivity of detection is the quantum nature of the light used (a [*coherent state*]{} in our case). To obtain this result, we are implicitly assuming that a) we have full access to all relevant characteristics of the output signals; and b) detectors are ideal, and can detect any change, as small as it might be, if enough signal power is used. If this is the case, weak value amplification provides no enhancement of the sensitivity.
However, this can be far from truth in many realistic experimental situations. In the laboratory, the quantum nature of light is an important factor, but not the only one, limiting the capacity to measure tiny changes of variables of interest. On the one hand, most of the times we detect only certain characteristic of the output signals, probably the most relevant, but this is still partial information about the quantum state. On the other hand, detectors are not ideal and noteworthy limitations to its performance can appear. To name a few, they might no longer work properly above a certain photon number input, electronics and signal processing of data can limit the resolution beyond what is allowed by the specific quantum nature of light, conditions in the laboratory can change randomly effectively reducing the sensitivity achievable in the experiment. Surely, all of these are [*technical*]{} rather than [*fundamental*]{} limitations, but in many situations the ultimate limit might be [*technical*]{} rather than [*fundamental*]{}. In this scenario, we show below that weak value amplification can be a [*valuable*]{} and an [*easy*]{} option to overcome all of these technical limitations, as it has been demonstrated in numerous experiments.
![Reduction of the probability of error using a weak value amplification scheme. (a) Minimum probability of error as a function of the photon number $N$ that leaves the interferometer. The two points highlighted corresponds to $N=10^6$, which yields $P_{\mathrm{error}}=1.3 \times 10^{-1}$, and $N=10^7$, which yields $P_{\mathrm{error}}=9.3 \times 10^{-5}$. (b) Number of photons ($N_{\mathrm{out}}$) after projection in the polarisation state $\hat{u}=1/\sqrt{2} \left[ \hat{x}+\hat{y}\exp(i\theta)
\right]$, as a function of the angle $\theta$. The input number of photons is $N=10^7$. The dot corresponds to the point $N_{\mathrm{out}}=10^6$ and $\theta=53.2^{\circ}$. Pulse width: $T_0$=1 ps; temporal delay: $\tau$= 1 as.[]{data-label="figure_sat"}](figure4.eps){width="90.00000%"}
Discussion {#discussion .unnumbered}
==========
Advantages of using weak value amplification (I): when the detector cannot work above a certain photon number. {#advantages-of-using-weak-value-amplification-i-when-the-detector-cannot-work-above-a-certain-photon-number. .unnumbered}
--------------------------------------------------------------------------------------------------------------
Let us suppose that we have at hand light detectors that cannot be used with more than $N_0$ photons. Any limitation on the detection time or the signal power would produce such limitation. The technical advantages of using WVA in this scenario has been previously pointed out [@jordan2014]. Here we make this apparent from a quantum estimation point of view, and quantify this advantage.
Fig. 4(a) shows the minimum probability of error as a function of the number of photons ($N$) entering (and leaving) the interferometer. For $N_0=10^6$, inspection of the figure shows that the probability of error is $P_\mathrm{error}=1.3 \times
10^{-1}$. This is the best we can do with this experimental scheme and these particular detectors without resorting to weak value amplification. However, if we project the output signal from the interferometer into a specific polarisation state, and increase the flux of photons, we can decrease the probability of error, without necessarily going to a regime of high depletion of the signal [@torres2012]. For instance, with $\theta=53.2^{\circ}$, and a flux of photons of $N=10^7$, so that after projection $N_{\mathrm{out}}=10^6$ photons reach the detector, the probability of error is decreased to $P_{\mathrm{error}}=9.3 \times 10^{-5}$, effectively enhancing the sensitivity of the experimental scheme (see Fig. 4(b)). The probability of error can be further decreased, also for other projections, at the expense of further increasing the input signal $N$.
In general, the minimum quantum overlap achievable between the states without any projection is $$|\langle \Phi_0|\Phi_1\rangle|^2=\exp \left[-N_0 \left( 1-\gamma
\cos \omega_0 \tau\right) \right],$$ while making use of projection in a weak value amplification scheme is $$|\langle \Phi_0|\Phi_1\rangle|^2=\exp
\left[-\frac{2N_0 \left(1-\gamma \cos \omega_0 \tau
\right)}{1+\gamma \cos \left(\omega_0 \tau-\Gamma-\pi/2 \right)}
\right]. \label{enhancement}$$ Eq. (\[enhancement\]) shows that when the number of photons that the detection scheme can handle is limited ($N_0$), projection into a particular polarisation state, at the expense of increasing the signal level, is advantageous. From a quantum estimation point of view, WVA increases the minimum probability of error reachable, since the projection makes possible to use the maximum number of photons available ($N_0$) with a corresponding enhanced mode overlap. Notice that the effect of using different polarisation projections can be beautifully understood as reshaping of the balance between signal level and mode overlap.
Advantages of using weak value amplification (II): when the detector cannot differentiate between two signals {#advantages-of-using-weak-value-amplification-ii-when-the-detector-cannot-differentiate-between-two-signals .unnumbered}
-------------------------------------------------------------------------------------------------------------
As second example, let us consider that specific experimental conditions makes hard, even impossible, to detect very similar modes, i.e., with mode overlap $\rho \sim 1$. We can represent this by assuming that there is an [*effective*]{} mode overlap ($\rho_{\mathrm{eff}}$) which takes into account all relevant experimental limitations of a specific set-up, given by $$\rho \Longrightarrow \rho_{\mathrm{eff}}=1-(1-\rho)\exp \left[
-\left(\frac{\rho}{a}\right)^n \right].$$ Fig. 5 shows an example where we assume that detected signals corresponding to $\rho > 0.9$ cannot be safely distinguished due to technical restrictions of the detection system. For $\rho > 0.9$, $\rho_{\mathrm{eff}}=1$, so the detection system cannot distinguish the states of interest even by increasing the level of the signal. On the contrary, for smaller values of $\rho$, accessible making use of a weak amplification scheme, this limitation does not exist since the detection system can resolve this modes when enough signal is present.
![Effective mode overlap. For $\rho>0.9$ the detection system cannot distinguish the states of interest. Data: $a=0.9$ and $n=100$.[]{data-label="figure_effective_rho"}](figure5.eps){width="50.00000%"}
Advantages of using weak value amplification (III): enhancement of the Fisher information {#advantages-of-using-weak-value-amplification-iii-enhancement-of-the-fisher-information .unnumbered}
-----------------------------------------------------------------------------------------
Up to now, we have used the concept of trace distance to look for the minimum probability of error achievable in [*any*]{} measurement when using a given quantum state. In doing that, we only considered how the quantum state changes for different values of the variable to be measured, without any consideration of how this quantum state is going to be detected. If we would like to include in the analysis additional characteristics of the detection scheme, one can use the concept of Fisher information, that requires to consider the probability distribution of possible experimental outcomes for a given value of the variable of interest. In this approach, one chooses different probability distributions to describe formally [*characteristics*]{} of specific detection scheme [@jordan2014].
Let us assume that to estimate the value of the delay $\tau$, we measure the shift of the centroid ($\Delta f$) of the spectrum $\Phi_u(\tau)$, given by Eq. (\[projections2\]). A particular detection scheme will obtain a set of results $\left\{ (\Delta
f)_i \right\}$, $i=1..N$ for a given delay $\tau$. $N$ is the number of photons detected. The Fisher information $I(\tau)$ provides a bound of $\mathrm{Var}\left( \hat{\tau} \right)$ for any unbiased estimator when the probability distribution $p(\left\{ (\Delta f)_i \right\}|\tau)$ of obtaining the set $\left\{ (\Delta f)_i \right\}$, for a given $\tau$, is known. If we assume that the probability distribution $p(\left\{(\Delta f)_i
\right\} |\tau)$ is Gaussian, with mean value $\Delta f$ given by Eq. (\[centroid\_shift\]) and variance $\sigma^2$, determined by the errors inherent to the detection process, the Fisher information reads [@fisher1] $$I(\tau)=\frac{N}{\sigma^2} \left[ \frac{\partial \Delta
f}{\partial \tau}\right]^2$$ where $$\frac{\partial \Delta f}{\partial \tau}=\frac{\gamma B^2 \left[
B^2 \tau^2 \sin \phi-\omega_0 \tau \left( \gamma+\cos
\phi\right)-\sin \phi \left( 1+\gamma \cos \phi\right)
\right]}{2\pi\, \left( 1+\gamma \cos \phi\right)^2}$$ and $\phi=\omega_0 \tau-\Gamma$.
For $\phi=0$, i.e., the angle of post-selection is $\theta=-\pi/2+\omega_0 \tau$, the Fisher information is $$I_0=\frac{N_0}{2} \left( 1+\gamma\right) \times \frac{\gamma^2 B^4
(\omega_0 \tau)^2}{2\pi\sigma^2 (1+\gamma)^2}= \frac{\gamma^2 B^4
(\omega_0 \tau)^2}{4\pi\sigma^2 (1+\gamma)} \label{fisher_bound1}$$
Notice that $\theta=-\pi/2$ corresponds to considering equal input and output polarization state, i.e., no weak value amplification scheme. For $\phi=\pi$, where the angle of post-selection is $\theta=\pi/2+\omega_0 \tau$, we have $$I_{\pi}=\frac{N_0}{2} \left( 1-\gamma\right) \times \frac{\gamma^2
B^4 (\omega_0 \tau)^2}{2\pi\sigma^2 (1-\gamma)^2}= \frac{\gamma^2
B^4 (\omega_0 \tau)^2}{4\pi\sigma^2 (1-\gamma)}
\label{fisher_bound2}$$ $\theta=\pi/2$ corresponds to considering an output polarisation state orthogonal to the input polarisation state i.e., when the effect of weak value amplification is most dramatic, as it can be easily observed in Fig. 2(a). The Fisher bound for $\Phi=\pi$ is a factor $I_{\pi}/I_0=(1+\gamma)/(1-\gamma)$ larger than the bound for $\Phi=0$, so WVA achieves enhancement of the Fisher information. This Fisher information enhancement effect, which does not happen always, it has been observed for certain WVA schemes [@viza2013; @jordan2014].
There is no contradiction between the facts that the minimum probability of error, obtained by making use of the concept of trace distance, is not changed by WVA, while at the same time there can be enhancement of the Fisher information. By selecting a particular probability distribution to evaluate the Fisher information, we include information about the detection scheme. In our case, we estimate the value of $\tau$ by measuring the $\tau$-dependent shift of the centroid of the spectrum of the signal in one output port after PBS$_2$, which is only part of all the information available, given by the full signal in Eqs. (\[projections1\]) and (\[projections2\]). We also assumed a Gaussian probability distribution with a constant variance $\sigma^2$ independent of $\tau$. The Cramér-Rao bound we have derived here depends on the full information available (the quantum state) before any particular detection. An unitary transformation, as WVA is, does not modify the bound. On the contrary, the Fisher information, by using a particular probability distribution to describe the possible outcomes in an particular experiment, selects certain aspects of the quantum state to be measured ([*partial information*]{}), and this bound can change in a WVA scheme, although the bound should be always above the Cramér-Rao bound. In this restrictive scenario, the use of certain polarization projections can be preferable.
The existence and nature of these different bounds might possibly explain certain confusion about the capabilities of WVA, whether WVA is considered to provide any metrological advantage or not. On the one hand, if we consider the trace distance, or the quantum Cramér-Rao inequality, without any consideration about how the quantum states are detected, post-selection inherent in WVA does not lower the minimum probability of error achievable, so from this point of view WVA offers no metrological advantage. On the other hand, in certain scenarios, the Fisher information, when it takes into account [*information about the detection scheme*]{}, can be enhanced due to post-selection. In this sense, one can think of WVA as an advantageous way to optimize a particular detection scheme.
Conclusions {#conclusions .unnumbered}
===========
WVA schemes makes use of linear optics unitary transformations. Therefore, if the only limitations in a measurement are due to the quantum nature ([*intrinsic statistics*]{}) of the light, for instance, the presence of Shot noise in the case of coherent beams, WVA does not offer any advantage regarding any decrease of the minimum probability of error achievable. This is shown by making use of the trace distance between quantum states or the Cramér-Rao inequality, which set sensitivity bounds that are independent of any particular post-selection. However, notice that this implicitly assume that full information about the quantum states used can be made available, and detectors are ideal, so they can detect any change of the variable of interest, as small as it might be, provided there is enough signal power.
Nevertheless, these assumptions are in many situations of interest far from true. These limitations, sometimes refereed as [*technical noise*]{}, even though not fundamental (one can always imagine using a better detector or a different detection scheme) are nonetheless important, since they limit the accuracy of specific detection systems at hand. In these scenarios, the importance of weak value amplification is that by decreasing the mode overlap associated with the states to be measured and possibly increasing the intensity of the signal, the weak value amplification scheme allows, in principle, to distinguish them with lower probability of error.
We have explored some of these scenarios from an quantum estimation theory point of view. For instance, we have seen that when the number of photons usable in the measurement is limited, the minimum probability of error achievable can be effectively decreased with weak value amplification. We have also analyzed how weak value amplification can differentiate between [*in practice*]{}-indistinguishable states by decreasing the mode overlap between its corresponding mode functions.
Finally we have discussed how the confusion about the usefulness of weak value amplification can possibly derive from considering different bounds related to how much sensitivity can, in principle, be achieved when estimating a certain variable of interest. One might possibly say that the advantages of WVA [*have nothing to do with fundamental limits and should not be viewed as addressing fundamental questions of quantum mechanics*]{} [@caves2014]. However, [*from a practical rather than fundamental point of view*]{}, the use of WVA can be advantageous in experiments where sensitivity is limited by experimental (technical), rather than fundamental, uncertainties. In any case, if a certain measurement is [*optimum*]{} depends on its capability to effectively reach any bound that might exist.
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Glauber, R. J. Coherent and incoherent states of the radiation field. *Phys. Rev.* **131**, 2766–2788 (1966).
Let us define $Q=\langle\frac{\partial \alpha}{\partial \tau}| \frac{\partial
\alpha}{\partial \tau} \rangle-\left| \langle {\small \alpha} | \frac{\partial \alpha}{\partial \tau} \rangle\right|^2$. For a coherent product state of the form $|\alpha\rangle=\prod_i |\alpha_i \rangle$, where the index $i$ refers to different frequency modes, one obtains that $Q=\sum_i Q_i$, where $Q_i=\langle \frac{\partial \alpha_i}{\partial \tau}| \frac{\partial \alpha_i}{\partial \tau} \rangle-\left| \langle
{\small \alpha_i} | \frac{\partial \alpha_i}{\partial \tau} \rangle\right|^2$. If $\alpha_i=\beta_i \exp(i\varphi_i)$, where $|\beta_i|^2$ is the mean number of photons in frequency mode $i$ and only $\varphi_i$ depends on the parameter $\tau$ as $\varphi_i=(\omega_0+\Omega_i)\tau$, one obtains that $|\partial
\alpha_i /\partial \tau \rangle=i \left(\partial \varphi_i/\partial \tau\right) \alpha_i a_i^{\dagger}|\alpha_i
\rangle$, where $a_i^{\dagger}$ is the creation operator of the corresponding frequency mode.
The unknown parameter of interest has value $\tau$. After repeated measurements to estimate its value, we obtain a distribution of outcomes $\left\{ x \right\}$ which can be characterized by a probability distribution $p(x|\tau)$ that depends on the value of $\tau$. The Fisher information reads $I(\tau)=-\int dx p(x|\tau)\partial_{\tau}^2 p(x|\tau)$. The variance of any unbiased estimator that makes use of the ensemble $\left\{ x \right\}$ is bounded from below by $\mathrm{Var}(\hat{\tau})
\ge 1/I(\tau)$. When the Fisher function can be written as $I[\eta(\tau)]$, where $\eta$ is the variable that we measure, the Fisher information can be written as $I(\tau) = I(\eta)\left(\frac{\partial\eta}{\partial\tau}\right)^2$.
Viza, G. I. et al. Weak-values technique for velocity measurements. *Opt. Lett.* **38**, 2949-2952 (2013).
Combes, J. & Ferrie, C. & Zhang, J., and Carlton M. Caves, C. M. Quantum limits on postselected, probabilistic quantum metrology, *Phys. Rev. A* **89**, 052117 (2014).
Figure Captions {#figure-captions .unnumbered}
===============
Figure1 {#figure1 .unnumbered}
-------
Weak value amplification scheme aimed at detecting extremely small temporal delays. The input pulse polarisation state is selected to be left-circular by using a polariser, a quarter-wave plate (QWP) and a half-wave plate (HWP). A first polarising beam splitter (PBS$_1$) splits the input into two orthogonal linear polarisations that propagate along different arms of the interferometer. An additional QWP is introduced in each arm to rotate the beam polarisation by $90^{\circ}$ to allow the recombination of both beams, delayed by a temporal delay $\tau$, in a single beam by the same PBS. After PBS$_1$, the output polarisation state is selected with a liquid crystal variable retarder (LCVR) followed by a second polarising beam splitter (PBS$_2$). The variable retarder is used to set the parameter $\theta$ experimentally. Finally, the spectrum of each output beam is measured using an optical spectrum analyzer (OSA). ($\hat{x}$,$\hat{y}$) and ($\hat{u}$,$\hat{v}$) correspond to two sets of orthogonal polarisations.
Figure2 {#figure2 .unnumbered}
-------
Spectrum measured at the output. (a) and (b): Spectral shape of the mode functions for $\tau=0$ (solid blue line) and $\tau=100$ as (dashed green line). In (a) the post-selection angle $\theta$ is $97.2^{\circ}$, so as to fulfil the condition $\omega_0 \tau-\Gamma=\pi$. In (b) the angle $\theta$ is $96.7^{\circ}$. (c) Shift of the centroid of the spectrum of the output pulse after projection into the polarisation state $\hat{u}$ in PBS$_2$, as a function of the post-selection angle $\theta$. Green solid line: $\tau=10$ as; Dotted red line: $\tau=50$ as, and dashed blue line: $\tau=100$ as. Label [**I**]{} corresponds to $\theta=96.7^{\circ}$ \[mode for $\tau=100$ as shown in (b)\]. Label [**II**]{} corresponds to $\theta=97.2^{\circ}$, where the condition $\omega_0 \tau-\Gamma=\pi$ is fulfiled \[mode for $\tau=100$ shown in (a)\]. It yields the minimum mode overlap between states with $\tau=0$ and $\tau \neq 0$. Data: $\lambda_0=1.5\, \mu$m and $T_0=100$ fs.
Figure3 {#figure3 .unnumbered}
-------
Mode overlap and insertion loss as a function of the post-selection angle. Mode overlap $\rho$ of the mode functions corresponding to the quantum states with $\tau=0$ and $\tau=100$ as, as a function of the post-selection angle $\theta$ (solid blue line). The insertion loss, given by $10\log_{10}\,P_{\mathrm{out}}/P_{\mathrm{in}}$ is indicated by the dotted green line. The minimum mode overlap, and maximum insertion loss, corresponds to the post-selection angle $\theta$ that fulfils the condition $\omega_0 \tau-\Gamma=\pi$, which corresponds to $\theta=97.2^{\circ}$. Data: $\lambda_0=1.5 \,\mu$m, $T_0=100$ fs.
Figure4 {#figure4 .unnumbered}
-------
Reduction of the probability of error using a weak value amplification scheme. (a) Minimum probability of error as a function of the photon number $N$ that leaves the interferometer. The two points highlighted corresponds to $N=10^6$, which yields $P_{\mathrm{error}}=1.3 \times 10^{-1}$, and $N=10^7$, which yields $P_{\mathrm{error}}=9.3 \times 10^{-5}$. (b) Number of photons ($N_{\mathrm{out}}$) after projection in the polarisation state $\hat{u}=1/\sqrt{2} \left[ \hat{x}+\hat{y}\exp(i\theta)
\right]$, as a function of the angle $\theta$. The input number of photons is $N=10^7$. The dot corresponds to the point $N_{\mathrm{out}}=10^6$ and $\theta=53.2^{\circ}$. Pulse width: $T_0$=1 ps; temporal delay: $\tau$= 1 as.
Figure5 {#figure5 .unnumbered}
-------
Effective mode overlap. For $\rho>0.9$ the detection system cannot distinguish the states of interest. Data: $a=0.9$ and $n=100$.
|
---
abstract: 'Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.'
address:
- 'Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria'
- 'Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria'
author:
- Ansgar Jüngel
- Stefan Schuchnigg
title: ' Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations'
---
[^1]
Introduction {#sec.intro}
============
Evolution equations often contain some structural information reflecting inherent physical properties such as positivity of solutions, conservation laws, and entropy dissipation. Numerical schemes should be designed in such a way that these structural features are preserved on the discrete level in order to obtain accurate and stable algorithms. In the last decades, concepts of structure-preserving schemes, geometric integration, and compatible discretization have been developed [@CMO11], but much less is known about the preservation of entropy dissipation and large-time asymptotics.
Entropy-stable schemes were derived by Tadmor already in the 1980s [@Tad87] in the context of conservation laws, thus without (physical) diffusion. Later, entropy-dissipative schemes were developed for (finite-volume) discretizations of diffusion equations in [@Bes12; @Fil08; @GlGa09]. In [@CaGu15], a finite-volume scheme which preserves the gradient-flow structure and hence the entropy structure is proposed. All these schemes are based on the implicit Euler method and are of first order (in time) only. Higher-order time schemes with entropy-dissipating properties are investigated in very few papers. A second-order predictor-corrector approximation was suggested in [@LiYu14], while higher-order semi-implicit Runge-Kutta (DIRK) methods, together with a spatial fourth-order central finite-difference discretization, were investigated in [@BFR15]. In [@BEJ14; @JuMi15], multistep time approximations were employed, but they can be at most of second order and they dissipate only one entropy and not all functionals dissipated by the continuous equation. In this paper, we remove these restrictions by investigating time-discrete Runge-Kutta schemes of order $p\ge 1$ for general diffusion equations.
We stress the fact that we are interested in the analysis of entropy-dissipating schemes by “translating” properties for the continuous equation to the semi-discrete level, i.e., we study the stability of the schemes. However, we will not investigate convergence, stiffness, or computational issues here (see e.g. [@BFR15]).
More precisely, we consider time discretizations of the abstract Cauchy problem $$\label{1.eq}
\pa_t u(t) + A[u(t)] = 0, \quad t>0, \quad u(0)=u^0,$$ where $A:D(A)\to X'$ is a (differential) operator defined on $D(A)\subset X$ and $X$ is a Banach space with dual $X'$. In this paper, we restrict ourselves to diffusion operators $A[u]$ defined on some Sobolev space with solutions $u:\Omega\times(0,\infty)\to{{\mathbb R}}^n$, which may be vector-valued. A typical example is $A[u]={\operatorname{div}}(a(u)\na u)$ defined on $X=L^2(\Omega)$ with domain $D(A)=H^2(\Omega)$, where $a:{{\mathbb R}}\to{{\mathbb R}}$ is a smooth function (see section \[sec.de\]). Equation often possesses a Lyapunov functional $H[u]=\int_\Omega h(u)dx$ (here called [*entropy*]{}), where $h:{{\mathbb R}}^n\to{{\mathbb R}}$, such that $$\frac{dH}{dt}[u] = \int_\Omega h'(u)\pa_t u dx
= -\int_\Omega h'(u)A[u] dx \le 0,$$ at least when the [*entropy production*]{} $\int_\Omega h'(u)A[u] dx$ is nonnegative, Here, $h'$ is the derivative of $h$ and $h'(u)A[u]$ is interpreted as the inner product of $h'(u)$ and $A[u]$ in ${{\mathbb R}}^n$. Furthermore, if $h$ is convex, the convex Sobolev inequality $\int_\Omega h'(u)A[u]dx
\ge \kappa H[u]$ for some $\kappa>0$ may hold [@CJS15], which implies that $dH/dt\le-\kappa H$ and hence exponential convergence of $H[u]$ to zero with rate $\kappa$. The aim is to design a higher-order time-discrete scheme which preserves this entropy-dissipation property.
To this end, we propose the following semi-discrete Runge-Kutta approximation of : Given $u^{k-1}\in X$, define $$\label{1.rk}
u^{k} = u^{k-1} + \tau\sum_{i=1}^s b_iK_i, \quad
K_i = -A\bigg[u^{k-1} + \tau\sum_{j=1}^s a_{ij}K_j\bigg], \quad i=1,\ldots,s,$$ where $t^{k}$ are the time steps, $\tau=t^{k}-t^{k-1}>0$ is the uniform time step size, $u^k$ approximates $u(t^k)$, and $s\ge 1$ denotes the number of Runge-Kutta stages. Since the Cauchy problem is autonomous, the knots $c_1,\ldots,c_s$ are not needed here. In concrete examples (see below), $u^k$ are functions from $\Omega$ to ${{\mathbb R}}^n$. If $a_{ij}=0$ for $j\ge i$, the Runge-Kutta scheme is explicit, otherwise it is implicit and a nonlinear system of size $s$ has to be solved to compute $K_i$. We assume that scheme is solvable for $u^k:\Omega\to{{\mathbb R}}^n$.
Given $h:{{\mathbb R}}^n\to{{\mathbb R}}$, we wish to determine conditions under which the functional $$\label{1.H}
H[u^k] = \int_\Omega h(u^k(x))dx$$ is dissipated by the numerical scheme , $$\label{1.ed}
H[u^k] + \tau\int_\Omega A[u^k]h'(u^k)dx \le H[u^{k-1}], \quad k\in{{\mathbb N}}.$$ In many examples (see below), $\int_\Omega A[u^k]h'(u^k)dx\ge 0$ and thus, the function $k\mapsto H[u^k]$ is decreasing. Such a property is the first step in proving the preservation of the large-time asymptotics of the numerical scheme (see Remark \[rem.exp\]).
Our main results, stated on an informal level, are as follows:
(i) We determine an abstract condition under which the discrete entropy-dissipation inequality holds for sufficiently small $\tau^k>0$. This condition is made explicit for special choices of $A$ and $h$, yielding entropy-dissipative implicit or explicit Runge-Kutta schemes of any order.
(ii) Numerical experiments for the porous-medium equation indicate that $\tau^k$ may be chosen independent of the time step $k$, thus yielding discrete entropy dissipation for all discrete times.
(iii) We show that for Runge-Kutta schemes of order $p\ge 2$, the abstract condition in (i) is exactly the criterion of Liero and Mielke [@LiMi13] to conclude geodesic 0-convexity of the entropy. In particular, it is related to the Bakry-Emery condition [@BaEm85].
Let us describe the main results in more detail. We recall that the Runge-Kutta scheme is consistent if $\sum_{j=1}^s a_{ij}=c_i$ and $\sum_{i=1}^s b_i=1$. Furthermore, if $\sum_{i=1}^s b_ic_i=\frac12$, it is at least of order two [@HNW93 Chap. II]. We introduce the number $$\label{1.CRK}
C_{\rm RK} = 2\sum_{i=1}^sb_i(1-c_i),$$ which takes only three values: $$\begin{aligned}
C_{\rm RK} &= 0\quad\mbox{for the implicit Euler scheme}, \\
C_{\rm RK} &= 1\quad\mbox{for any Runge-Kutta scheme of order }p\ge 2, \\
C_{\rm RK} &= 2\quad\mbox{for the explicit Euler scheme}.\end{aligned}$$
The [*first main result*]{} is an abstract entropy-dissipation property of scheme for entropies of type .
\[thm.ed\] Let $h\in C^2({{\mathbb R}}^n)$, let $A:D(A)\to X'$ be Fréchet differentiable with Fréchet derivative $DA[u]:X\to X'$ at $u\in D(A)$, and let $(u^k)$ be the Runge-Kutta solution to . Suppose that $$\label{1.I0}
I_0^k :=
\int_\Omega \big(C_{\rm RK}h'(u^k)DA[u^k](A[u^k])+h''(u^k)(A[u^k])^2\big)dx > 0.$$ Then there exists $\tau^k>0$ such that for all $0<\tau\le\tau^k$, $$\label{1.edi}
H[u^k] + \tau\int_\Omega A[u^k]h'(u^k)dx \le H[u^{k-1}].$$
We assume that the solutions to are sufficiently regular such that the integral can be defined. In the vector-valued case, $h''(u^k)$ is the Hessian matrix and we interpret $h''(u^k)(A[u^k])^2$ as the product $A[u^k]^\top h''(u^k)A[u^k]$. For Runge-Kutta schemes of order $p\ge 2$ (for which $C_{\rm RK}=1$), the integral corresponds exactly to the second-order time derivative of $H[u(t)]$ for solutions $u(t)$ to the [*continuous*]{} equation . Observe that the entropy-dissipation estimate is only [*local*]{}, since the time step restriction depends on the time step $k$. For implicit Euler schemes (and convex entropies $h$), it is known that $\tau^k$ can be chosen independent of $k$. For general Runge-Kutta methods, we cannot prove rigorously that $\tau^k$ stays bounded from below as $k\to\infty$. However, our numerical experiments in section \[sec.num\] indicate that inequality holds for sufficiently small $\tau>0$ uniformly in $k$.
\[rem.exp\] If the convex Sobolev inequality $\int_\Omega A[u^k]h'(u^k)dx\ge
\kappa H[u^k]$ holds for some constant $\kappa>0$ and if there exists $\tau^*>0$ such that $\tau^k\ge\tau^*>0$ for all $k\in{{\mathbb N}}$, we infer from that for $\tau:=\tau^*$, $$H[u^k] \le (1+\kappa\tau)^{-k}H[u^0] = \exp(-\eta\kappa t^k)H[u^0],
\quad\mbox{where }\eta = \frac{\log(1+\kappa\tau)}{\kappa\tau}
< 1,$$ which implies exponential decay of the discrete entropy with rate $\eta\kappa$. This rate converges to the continuous rate $\kappa$ as $\tau\to 0$ and therefore, it is asymptotically sharp.
Theorem \[thm.ed\] can be generalized to a larger class of entropies, namely to so-called [*first-order entropies*]{} $$\label{1.F}
F[u^k] = \int_\Omega |\na f(u^k)|^2 dx,$$ where, for simplicity, we consider only the scalar case with $f:{{\mathbb R}}\to{{\mathbb R}}$. An important example is the Fisher information with $f(u)=\sqrt{u}$.
\[thm.ed2\] Let $f\in C^2({{\mathbb R}})$, let $A:D(A)\to X'$ be Fréchet differentiable, and let $(u^k)$ be the Runge-Kutta solution to . Assume that the boundary condition $\na f(u^k)\cdot\nu=0$ on $\pa\Omega$ is satisfied. Furthermore, suppose that $$\label{1.I1}
\begin{aligned}
I_1^k :=
\int_\Omega \Big( & |\na(f'(u^k)A[u^k]|^2
- C_{\rm RK}\Delta f(u^k)f'(u^k)DA[u^k](A[u^k]) \\
&{}- \Delta f(u^k)f''(u^k)(A[u^k])^2\Big)dx > 0.
\end{aligned}$$ Then there exists $\tau^k>0$ such that for all $0<\tau\le\tau^k$, $$F[u^k] + \tau\int_\Omega A[u^k]f'(u^k)dx \le F[u^{k-1}].$$
The key idea of the proof of Theorem \[thm.ed\] (and similarly for Theorem \[thm.ed2\]) is a concavity property of the difference of the entropies at two consecutive time steps with respect to the time step size $\tau$. To explain this idea, let $u:=u^k$ be fixed and introduce $v(\tau):=u^{k-1}$. Clearly, $v(0)=u$. A formal Taylor expansion of $G(\tau):=H[u]-H[v(\tau)]$ yields $$H[u^k]-H[u^{k-1}] = G(\tau) = G(0) + \tau G'(0) + \frac{\tau^2}{2}G''(\xi^k),$$ where $0<\xi^k<\tau$. A computation, made explicit in section \[sec.meth\], shows that $G'(0)=\int_\Omega A[u^k]h'(u^k)dx$ and $G''(0)=-I_0^k$. Now, if $G''(0)<0$, there exists $\tau^k>0$ such that $G''(\tau)\le 0$ for $\tau\in [0,\tau^k]$ and in particular $G''(\xi^k)\le 0$. Consequently, $G(\tau)\le \tau G'(0)$, which equals . The definition of $v(\tau)$ assumes implicitly that is [*backward*]{} solvable. We prove in Proposition \[prop.back\] below that this property holds if the operator $A$ is a smooth self-mapping on $X$.
\[rem.tauk\]Since $(u^k)$ is expected to converge to the stationary solution, $\lim_{k\to\infty}I_0^k=0$. Thus, in principle, for larger values of $k$, we expect that $\tau^k$ becomes smaller and smaller, thus restricting the choice of time step sizes $\tau$. However, practically, the situation is better. For instance, for the implicit Euler scheme, if $h$ is convex, we obtain $$H[u^k] - H[u^{k-1}] \le \int_\Omega h'(u^k)(u^k-u^{k-1})dx
= -\tau\int_\Omega h'(u^k)A[u^k]dx$$ for [*any*]{} value of $\tau>0$. Moreover, for other (higher-order) Runge-Kutta schemes, the numerical experiments in section \[sec.num\] indicate that there exists $\tau^*>0$ such that $G''(\tau)\le 0$ holds for all $\tau\in[0,\tau^*]$ uniformly in $k\in{{\mathbb N}}$. In this situation, inequality holds for all $0<\tau\le\tau^*$, and thus our estimate is global. In fact, the function $G''$ is numerically even nonincreasing in some interval $[0,\tau^*]$ but we are not able to prove this analytically.
The [*second main result*]{} is the specification of the abstract conditions and for a number of examples: a quasilinear diffusion equation, porous-medium or fast-diffusion equations, a linear diffusion system, and the fourth-order Derrida-Lebowitz-Speer-Spohn equation (see sections \[sec.de\]-\[sec.dlss\] for details). For instance, for the porous-medium equation $$\pa_t u = \Delta(u^\beta) \mbox{ in }\Omega,\ t>0, \quad
\na u^\beta\cdot\nu = 0 \mbox{ on }\pa\Omega, \quad u(0)=u^0,$$ we show that the Runge-Kutta scheme scheme satisfies $$H[u^k] + \tau\beta\int_\Omega(u^k)^{\alpha+\beta-2}|\na u^k|^2 dx
\le H[u^{k-1}], \quad\mbox{where }
H[u]=\frac{1}{\alpha(\alpha+1)}\int_\Omega u^{\alpha+1} dx,$$ for $0<\tau\le\tau^k$ and all $(\alpha,\beta)$ belonging to some region in $[0,\infty)^2$ (see Figure \[fig.0th\] below). For $\alpha=0$, we write $H[u]=\int_\Omega u(\log u-1)dx$. In one space dimension and for Runge-Kutta schemes of order $p\ge 2$, this region becomes $-2<\alpha-\beta<1$, which is the same condition as for the continuous equation (except the boundary values). Furthermore, the first-order entropy is dissipated for Runge-Kutta schemes of order $p\ge 2$, in one space dimension, $$F[u^k] + \tau C_{\alpha,\beta}\int_\Omega (u^k)^{\alpha+\beta-2}(u^k)_{xx}^2 dx
\le F[u^{k-1}], \quad\mbox{where }
F[u]=\int_\Omega (u^{\alpha/2})_x^2 dx,$$ for $0<\tau\le\tau^k$ and all $(\alpha,\beta)$ belonging to the region shown in Figure \[fig.1st\] below, and $C_{\alpha,\beta}>0$ is some constant. This region is smaller than the region of admissible values $(\alpha,\beta)$ for the continuous entropy. The borders of that region are indicated in the figure by dashed lines.
The proof of the above results, and namely of $G''(0)<0$, is based on systematic integration by parts [@JuMa06]. The idea of the method is to formulate integration by parts as manipulations with polynomials and to conclude the inequality $G''(0)<0$ from a polynomial decision problem. This problem can be solved directly or by using computer algebra software.
Our [*third main result*]{} is the relation to geodesic 0-convexity of the entropy and the Bakry-Emery approach when $C_{\rm RK}=1$ (Runge-Kutta scheme of order $p\ge 2$). Liero and Mielke formulate in [@LiMi13] the abstract Cauchy problem as the gradient flow $$\pa_t u = -K[u]DH[u], \quad t>0, \quad u(0)=u^0,$$ where the Onsager operator $K[u]$ describes the sum of diffusion and reaction terms. For instance, if $A[u]={\operatorname{div}}(a(u)\na u)$, we can write $A[u]={\operatorname{div}}(a(u)h''(u)^{-1}\na h'(u))$ and thus, identifying $h'(u)$ and $DH[u]$, we have $K[u]\xi={\operatorname{div}}(a(u)h''(u)^{-1}\na\xi)$. It is shown in [@LiMi13] that the entropy $H$ is geodesic $\lambda$-convex if the inequality $$\label{1.M}
M(u,\xi) := \langle \xi,DA[u]K[u]\xi\rangle - \frac12\langle\xi,DK[u]A[u]\xi\rangle
\ge \lambda\langle\xi,K[u]\xi\rangle$$ holds for all suitable $u$ and $\xi$. We will prove in section \[sec.meth\] that $$G''(0) = 2M(u^k,h'(u^k)).$$ Hence, if $G''(0)\le 0$ then with $\lambda=0$ is satisfied for $u=u^k$ and $\xi=h'(u^k)$, yielding geodesic $0$-convexity for the semi-discrete entropy. Moreover, if $G''(0)\le -\lambda G'(0)$ then we obtain geodesic $\lambda$-convexity. Since $G'(0)=-dH[u]/dt$ and $G''(0)=-d^2H[u]/dt^2$ in the continuous setting, the inequality $G''(0)\le -\lambda G'(0)$ can be written as $$\frac{d^2H}{dt^2}[u] \ge -\lambda \frac{dH}{dt}[u],$$ which corresponds to a variant of the Bakry-Emery condition [@BaEm85], yielding exponential convergence of $H[u]$ (if $\tau^k\ge\tau^*>0$ for all $k$). Thus, our results constitute a first step towards a [*discrete Bakry-Emery approach*]{}.
The paper is organized as follows. The abstract method, i.e. the proof of backward solvability and of Theorems \[thm.ed\] and \[thm.ed2\], is presented in section \[sec.meth\]. The method is applied in the subsequent sections to a scalar diffusion equation (section \[sec.de\]), the porous-medium equation (section \[sec.pme\]), a linear diffusion system (section \[sec.sys\]), and the fourth-order Derrida-Lebowitz-Speer-Spohn equation (section \[sec.dlss\]). Finally, section \[sec.num\] is devoted to some numerical experiments showing that $G''$ is negative in some interval $[0,\tau^*]$.
The abstract method {#sec.meth}
===================
In this section, we show that the Runge-Kutta scheme is backward solvable if $A$ is a self-mapping and we prove Theorems \[thm.ed\] and \[thm.ed2\].
\[prop.back\] Let $(\tau,u^{k})\in[0,\infty)\times X$, where $X$ is some Banach space, and let $A\in C^2(X,X)$ be a self-mapping. Then there exists $\tau_0>0$, a neighborhood $V\subset X$ of $u^{k}$, and a function $v\in C^2([0,\tau_0);X)$ such that holds for $u^{k-1}:=v(\tau)$. Moreover, $$\label{v0}
v(0)=0, \quad v'(0)=A[u], \quad\mbox{and}\quad v''(0)=C_{\rm RK}DA[u](A[u]).$$
The self-mapping assumption is strong for differential operators $A$ but it is somehow natural in the context of Runge-Kutta methods and valid for smooth solutions.
The idea of the proof is to apply the implicit function theorem in Banach spaces (see [@Dei85 Corollary 15.1]). To this end, we set $u:=u^k$ and define the mapping $J=(J_0,\ldots,J_{s}):{{\mathbb R}}\times X^{s+1}\to X^{s+1}$ by $$\begin{aligned}
J_0(\tau,y) &= v - u + \tau\sum_{i=1}^s b_i k_i, \quad\mbox{where }
y=(k_1,\ldots,k_s,v), \\
J_i(\tau,y) &= k_i + A\bigg[v + \tau\sum_{j=1}^s a_{ij}k_j\bigg],
\quad i=1,\ldots,s.\end{aligned}$$ The Fréchet derivative of $J$ in the direction of $(\tau_h,y_h)$, where $y_h=(k_{h1},\ldots,k_{hs},v_h)$, reads as $$\begin{aligned}
DJ_0(\tau,y)(\tau_h,y_h) &= v_h + \tau_h\sum_{i=1}^s b_i k_{i}
+ \tau\sum_{i=1}^s b_i k_{hi}, \\
DJ_i(\tau,y)(\tau_h,y_h) &= k_{hi}
+ DA\bigg[v + \tau\sum_{j=1}^s a_{ij}k_j\bigg]
\bigg(v_h + \tau_h\sum_{j=1}^s a_{ij}k_{j} + \tau\sum_{j=1}^s a_{ij}k_{hj}\bigg), \end{aligned}$$ where $i=1,\ldots,s$. Let $\tau_0=0$ and $y_0=(-A[u],\ldots,-A[u],u)$. Then $J(\tau_0,y_0)=0$ and $$DJ_0(\tau_0,y_0)(0,y_h) = v_h, \quad
DJ_i(\tau_0,y_0)(0,y_h) = k_{ih} + DA[u](v_h), \quad i=1,\ldots,s.$$ The mapping $y_h\mapsto DJ(\tau_0,y_0)(0,y_h)$ is clearly an isomorphism from $X^{s+1}$ onto $X^{s+1}$. By the implicit function theorem, there exist an interval $U\subset [0,\tau_0)$, a neighborhood $V\subset X^{s+1}$ of $y_0$, and a function $(k,v)\in
C^2([0,\tau_0);V)$ such that $(k,v)(0)=(-A[u],\ldots,-A[u],u)$ and $J(\tau,k(\tau),v(\tau))=0$ for all $\tau\in[0,\tau_0)$.
Implicit differentiation of $J(\tau,k(\tau),v(\tau))=0$ yields $$\begin{aligned}
0 &= v'(\tau) + \sum_{i=1}^s b_i k_i(\tau) + \tau\sum_{i=1}^s b_i k_i'(\tau), \\
0 &= k_i'(\tau) + DA\bigg[v + \tau\sum_{j=1}^s a_{ij}k_j(\tau)\bigg]
\bigg(v'(\tau) + \sum_{j=1}^s a_{ij} k_j(\tau) + \tau\sum_{j=1}^s a_{ij}
k_j'(\tau)\bigg),\end{aligned}$$ where $i=1,\ldots,s$ and $\tau\in[0,\tau_0)$. Using $\sum_{i=1}^s b_i=1$ and $\sum_{j=1}^s a_{ij}=c_i$, we infer that $$\begin{aligned}
v'(0) &= -\sum_{i=1}^s b_i k_i(0) = \sum_{i=1}^s b_iA[u] = A[u], \nonumber \\
k_i'(0) &= -DA[u]\bigg(A[u] - \sum_{j=1}^s a_{ij}A[u]\bigg)
= -(1-c_i)DA[u](A[u]). \label{aux1}\end{aligned}$$ Differentiating $J_0(\tau,k(\tau),v(\tau))=0$ twice leads to $$0 = v''(\tau) + 2\sum_{i=1}^s b_i k_i'(\tau) + \tau\sum_{i=1}^s b_ik_i''(\tau).$$ Because of , this reads at $\tau=0$ as $$v''(0) = -2\sum_{i=1}^s b_ik_i'(0) = 2\sum_{i=1}^s b_i(1-c_i)DA[u](A[u])
= C_{\rm RK}DA[u](A[u]).$$ This finishes the proof.
We prove now Theorems \[thm.ed\] and \[thm.ed2\].
We set $u:=u^k$. By Proposition \[prop.back\], there exists a backward solution $v\in C^2([0,\tau_0))$ such that $v(0)=u$, $v'(0)=A[u]$, and $v''(0)=C_{\rm RK}DA[u](A[u])$. Furthermore, the function $G(\tau)=\int_\Omega(h(u)-h(v(\tau)))dx$ satisfies $G(0)=0$, $$\begin{aligned}
G'(0) &= -\int_\Omega h'(v(0))v'(0)dx = -\int_\Omega h'(u)A[u]dx, \\
G''(0) &= -\int_\Omega\big(h'(v(0))v''(0)
+ h''(v(0))v'(0)^2\big)dx \\
&= -\int_\Omega\big(C_{\rm RK}h'(u)DA[u](A[u]) + h''(u)(A[u])^2\big)dx
= -I_0^k < 0,\end{aligned}$$ using the assumption. By continuity, there exists $0<\tau^k<\tau_0$ such that $G''(\xi)\le 0$ for $0\le\xi\le\tau^k$. Then the Taylor expansion $G(\tau)=G(0)+G'(0)\tau+\frac12G''(\xi)\tau^2
\le G'(0)\tau$ concludes the proof.
Following the lines of the previous proof, it is sufficient to compute $G'(0)$ and $G''(0)$, where now $G(\tau)=\int_\Omega(|\na f(u)|^2-|\na f(v(\tau))|^2)dx$. Using integration by parts and the boundary condition $\na f(v)\cdot\nu=0$ on $\pa\Omega$, we compute $$G'(0) = -\int_\Omega\na f(v(0))\cdot\na\big( f'(v(0))v'(0)\big)dx
= \int_\Omega \Delta f(u)f'(v(\tau))A[u]dx,$$ since $v(0)=u$ and $v'(0)=A[u]$. Furthermore, again integrating by parts, $$\begin{aligned}
G''(\tau) &= -\int_\Omega\Big(\big|\na\big(f'(v(\tau))v'(\tau)\big)\big|^2
+ \na f(v(\tau))\cdot\na\big(f''(v(\tau))(v'(\tau))^2\big) \\
&\phantom{xx}{}+ \na f(v(\tau))\cdot\na\big(f'(v(\tau))v''(\tau)\big)\Big)dx \\
&= -\int_\Omega\Big(\big|\na\big(f'(v(\tau))v'(\tau)\big)\big|^2
- \Delta f(v(\tau))f''(v(\tau))(v'(\tau))^2 \\
&\phantom{xx}{}- \Delta f(v(\tau))f'(v(\tau))v''(\tau)\Big)dx.\end{aligned}$$ Since $v''(0)=C_{\rm RK}DA[u](A[u])$, this reduces at $\tau=0$ to $$\begin{aligned}
G''(0) &= -\int_\Omega\Big(|\na(f'(u)A[u])|^2 - \Delta f(u)f''(u)(A[u])^2
- C_{\rm RK}\Delta f(u)f'(u)DA[u](A[u])\Big)dx.\end{aligned}$$ This expression equals $-I_1^k$, and the result follows.
Finally, we show that $G''(0)$ for entropies is related to the geodesic convexity condition of [@LiMi13].
Let $A[u]=K(u)DH[u]$ for some symmetric operator $K:D(A)\to X$ and Fréchet derivative $DH[u]$, let $G$ be defined as in the proof of Theorem \[thm.ed\] for a solution $u^k$ to the Runge-Kutta scheme of order $p\ge 2$, and let $M(u,\xi)$ be given by . Then $$G''(0) = -2M(u^k,DH[u^k]).$$
The proof is just a (formal) calculation. Recall that for Runge-Kutta schemes of order $p\ge 2$, we have $C_{\rm RK}=1$. Set $u:=u^k$ and identify $DH[u]$ with $\xi=h'(u)$. Inserting the expression $DA[u](v)=DK[u](v)h'(u)+ K[u]h''(u)v$ into the definition of $G''(0)$, we find that $$\begin{aligned}
-G''(0) &= \langle\xi,DA[u](A[u])\rangle + \langle A[u],h''(u)A[u]\rangle \\
&= \big\langle\xi,DK[u](A[u])\xi + K[u]h''(u)A[u]\big\rangle
+ \langle A[u],h''(u)A[u]\rangle \\
&= \langle\xi,DK[u](K[u]\xi)\xi\rangle + \langle\xi,K[u]h''(u)K[u]\xi\rangle
+ \langle K[u]\xi,h''(u)K[u]\xi\rangle \\
&= \langle\xi,DK[u](K[u]\xi)\xi\rangle + 2\langle\xi,K[u]h''(u)K[u]\xi\rangle,\end{aligned}$$ since $K[u]$ is assumed to be symmetric. Rearranging the terms, we obtain $$\begin{aligned}
-G''(0) &= 2\langle\xi,DK[u](K[u]\xi)\xi\rangle
+ 2\langle\xi,K[u]h''(u)K[u]\xi\rangle
- \langle\xi,DK[u](K[u]\xi)\rangle \\
&= 2\langle\xi,DA[u](K[u]\xi)\xi\rangle - \langle\xi,DK[u](A[u])\rangle
= 2M(u,\xi),\end{aligned}$$ which proves the claim.
Scalar diffusion equation {#sec.de}
=========================
In this section, we analyze time-discrete Runge-Kutta schemes of the diffusion equation $$\label{de.eq}
\pa_t u = {\operatorname{div}}(a(u)\na u), \quad t>0, \quad u(0)=u^0,$$ with periodic or homogeneous Neumann boundary conditions. This equation, also including a drift term, was analyzed in [@LiMi13] in the context of geodesic convexity. Our results are similar to those in [@LiMi13] but we consider the time-discrete and not the continuous equation and we employ systematic integration by parts [@JuMa06].
Setting $\mu(u)=a(u)/h''(u)$, we can write the diffusion equation as a formal gradient flow: $$\pa_t u = -A[u] := {\operatorname{div}}(\mu(u)\na h'(u)), \quad t>0.$$ We prove that the Runge-Kutta scheme dissipates all convex entropies subject to some conditions on the functions $\mu$ and $h$.
\[thm.de\] Let $\Omega\subset{{\mathbb R}}^d$ be convex with smooth boundary. Let $(u^k)$ be a sequence of (smooth) solutions to the Runge-Kutta scheme of the diffusion equation . Let $k\in{{\mathbb N}}$ be fixed and $u^k$ be not equal to the constant steady state of . We suppose that for all admissible $u$, it holds that $a(u)\ge 0$, $h''(u)\ge 0$, $$\begin{aligned}
\label{de.cond1}
& b(u) := \frac23(C_{\rm RK}+1)\int_{u_0}^u \mu(v)\mu'(v)h''(v)dv \ge 0, \\
& \frac{d-1}{d}b(u) \le (C_{\rm RK}+1)h''(u)\mu(u)^2, \label{de.cond2} \\
& (C_{\rm RK}+2)\mu(u)\mu''(u) + (C_{\rm RK}-1)\mu'(u)^2 < 0. \label{de.cond3}\end{aligned}$$ Then there exists $\tau^k>0$ such that for all $0<\tau<\tau^k$, $$H[u^k] + \tau\int_\Omega h''(u^k)a(u^k)|\na u^k|^2 dx \le H[u^{k-1}].$$
Conditions - correspond to (4.12) in [@LiMi13]. Condition is satisfied for concave functions $\mu$, except for the explicit Euler scheme ($C_{\rm RK}=2$) for which we need additionally $4\mu\mu''+(\mu')^2<0$. For the implicit Euler scheme, we may allow even for nonconcave mobilities $\mu$, e.g. $\mu(u)=u^\gamma$ for $1<\gamma<2$.
According to Theorem \[thm.ed\], we only need to show that $I_0^k=-G''(0)>0$. To simplify, we set $u:=u^k$. First, we observe that the boundary condition $\na u\cdot\nu=0$ on $\Omega$ implies that $0=\pa_t\na u\cdot\nu=\na\pa_t u\cdot\nu=-\na A[u]\cdot\nu$ on $\pa\Omega$. Using $DA[u](A[u]) = {\operatorname{div}}(a'(u)A[u]\na u+a(u)\na A[u])=\Delta(a(u)A[u])$, the abbreviation $\xi=h'(u)$, and integration by parts, we compute $$\begin{aligned}
G''(0) &= -\int_\Omega\Big(C_{\rm RK}h'(u)\Delta(a(u)A[u])
+ h''(u)\big({\operatorname{div}}(\mu(u)\na h'(u))\big)^2\Big)dx \\
&= \int_\Omega\Big(C_{\rm RK}\na h'(u)\cdot\na(a(u)A[u])
- h''(u)\big(\mu'(u)\na u\cdot\na h'(u)+\mu(u)\Delta h'(u)\big)^2\Big)dx \\
&= -\int_\Omega\bigg(C_{\rm RK}\Delta\xi a(u)A[u]
+ h''(u)\left(\frac{\mu'(u)}{h''(u)}|\na\xi|^2 + \mu(u)\Delta\xi\right)^2
\bigg)dx.\end{aligned}$$ The boundary integrals vanish since $\na u\cdot\nu=\na A[u]\cdot\nu=0$ on $\pa\Omega$. Replacing $A[u]$ by ${\operatorname{div}}(\mu(u)\na\xi)
=\mu(u)\Delta\xi+\mu'(u)|\na\xi|^2/h''(u)$ and expanding the square, we arrive at $$\begin{aligned}
G''(0) &= -\int_\Omega\bigg(\big(C_{\rm RK}a(u)\mu(u)+h''(u)\mu(u)^2\big)
(\Delta\xi)^2 \nonumber \\
&\phantom{xx}{}+ \left(C_{\rm RK}a(u)\frac{\mu'(u)}{h''(u)}+2\mu(u)\mu'(u)\right)
\Delta\xi|\na\xi|^2 + \frac{\mu'(u)^2}{h''(u)}|\na\xi|^4\bigg)dx
\label{diffeq.G2} \\
&= -\int_\Omega\big((C_{\rm RK}+1)h''(u)\mu(u)^2 \xi_L^2
+ (C_{\rm RK}+2)\mu(u)\mu'(u)\xi_L\xi_G^2 + \mu'(u)^2 h''(u)^{-1}\xi_G^4\big)dx,
\nonumber\end{aligned}$$ where we have employed the identity $a(u)=\mu(u)h''(u)$ and the abbreviations $\xi_G=|\na\xi|$ and $\xi_L=\Delta\xi$.
We apply now the method of systematic integration by parts [@JuMa06]. The idea is to identify useful integration-by-parts formulas and to add them to $G''(0)$ without changing the sign of $G''(0)$. The first formula is given by $$\label{de.ibp1}
\int_\Omega{\operatorname{div}}\big(\Gamma_1(u)(\na^2\xi-\Delta\xi{\mathbb I})\cdot\na\xi\big)dx
= \int_{\pa\Omega}\Gamma_1(u)\na\xi^\top(\na^2\xi-\Delta\xi{\mathbb I})\nu ds,$$ where $\Gamma_1(u)\le 0$ is an arbitrary (smooth) scalar function which still needs to be chosen, and $\mathbb{I}$ is the unit matrix in ${{\mathbb R}}^{d\times d}$. The left-hand side can be expanded as $$\begin{aligned}
\int_\Omega & \left(\frac{\Gamma_1'(u)}{h''(u)}\na\xi^\top
(\na^2\xi-\Delta\xi{\mathbb I})
\na\xi + \Gamma_1(u)\na^2\xi:(\na^2\xi-\Delta\xi{\mathbb I})\right)dx \\
&= \int_\Omega\left(\frac{\Gamma_1(u)}{h''(u)}\xi_{GHG}
- \frac{\Gamma_1'(u)}{h''(u)}\xi_L\xi_G^2
+ \Gamma_1(u)\xi_H^2 - \Gamma_1(u)\xi_L^2\right)dx,\end{aligned}$$ where we have set $\xi_{GHG}=\na\xi^\top\na^2\xi\na\xi$ and $\xi_H=|\na^2\xi|$. The boundary integral in becomes $$\int_{\pa\Omega}\Gamma_1(u)\left(\frac12\na(|\na\xi|^2)
-\Delta\xi\na\xi\right)\cdot\nu ds
= \frac12\int_{\pa\Omega}\Gamma_1(u)\na(|\na\xi|^2)\cdot\nu ds \ge 0,$$ since $\Gamma_1(u)\le 0$, $\na\xi\cdot\nu=0$ on $\pa\Omega$, and it holds that $\na(|\na\xi|^2)\cdot\nu\le 0$ on $\pa\Omega$ for all smooth functions satisfying $\na\xi\cdot\nu=0$ on $\pa\Omega$ [@LiMi13 Prop. 4.2]. Here we need the convexity of $\Omega$. Thus, the first integration-by-parts formula becomes $$\label{de.J1}
J_1 := \int_\Omega\left(\frac{\Gamma_1'(u)}{h''(u)}\xi_{GHG}
- \frac{\Gamma_1'(u)}{h''(u)}\xi_L\xi_G^2 + \Gamma_1(u)\xi_H^2
- \Gamma_1(u)\xi_L^2\right)dx \ge 0.$$ The second formula reads as $$\begin{aligned}
\label{de.J2}
0 &= \int_\Omega{\operatorname{div}}\big(\Gamma_2(u)|\na\xi|^2\na\xi)dx \\
&= \int_\Omega\left(\frac{\Gamma_2'(u)}{h''(u)}\xi_G^4 + 2\Gamma_2(u)\xi_{GHG}
+ \Gamma_2(u)\xi_L\xi_G^2\right)dx =: J_2, \nonumber\end{aligned}$$ where $\Gamma_2$ is an arbitrary scalar function. The goal is to find functions $\Gamma_1(u)\le 0$ and $\Gamma_2(u)$ such that $G''(0)\le G''(0)+J_1+J_2 < 0$.
According to [@JuMa08], the computations simplify if we introduce the variables $\xi_R$ and $\xi_S$ satisfying $$(d-1)\xi_G^2\xi_S = \xi_{GHG} - \frac{1}{d}\xi_L\xi_G^2, \quad
\xi_H^2 = \frac{1}{d}\xi_L^2 + d(d-1)\xi_S^2 + \xi_R^2.$$ The existence of $\xi_R$ follows from the inequality $$\xi_H^2 = |\na^2\xi|^2 \ge \frac{1}{d}(\Delta\xi)^2 + \frac{d}{d-1}
\left(\frac{\na\xi^\top\na^2\xi\na\xi}{\na\xi^2} - \frac{\Delta\xi}{d}\right)^2
= \frac{1}{d}\xi_L^2 + d(d-1)\xi_S^2,$$ which is proven in [@JuMa08 Lemma 2.1]. Then $$\label{de.poly}
G''(0) \le G''(0) + J_1 + J_2
= -\int_\Omega\big(a_1\xi_L^2 + a_2 \xi_L\xi_G^2 + a_3\xi_G^4
+ a_4\xi_S\xi_G^2 + a_5\xi_R^2 + a_6\xi_S^2\big)dx,$$ where $$\label{de.ai}
\begin{aligned}
a_1 &= (C_{\rm RK}+1)h''(u)\mu(u)^2 + \left(1-\frac{1}{d}\right)\Gamma_1(u), \\
a_2 &= (C_{\rm RK}+2)\mu(u)\mu'(u)
+ \left(1-\frac{1}{d}\right)\frac{\Gamma_1'(u)}{h''(u)}
- \left(\frac{2}{d}+1\right)\Gamma_2(u), \\
a_3 &= \frac{\mu'(u)^2-\Gamma_2'(u)}{h''(u)}, \quad
a_4 = -(d-1)\left(\frac{\Gamma_1'(u)}{h''(u)} + 2\Gamma_2(u)\right), \\
a_5 &= -\Gamma_1(u), \quad a_6 = -d(d-1)\Gamma_1(u).
\end{aligned}$$ The aim now is to determine conditions on $a_1,\ldots,a_6$ such that the polynomial $P(\xi)=a_1\xi_L^2 + a_2 \xi_L\xi_G^2 + a_3\xi_G^4 + a_4\xi_S\xi_G^2
+ a_5\xi_R^2 + a_6\xi_S^2$ is nonnegative as this implies that $G''(0)\le 0$. In the general case, this leads to nonlinear ordinary differential equations for $\Gamma_1$ and $\Gamma_2$ which cannot be easily solved. A possible solution is to require that the coefficients of the mixed terms vanish, i.e. $a_2=a_4=0$, and that the remaining coefficients are nonnegative. The case $d=1$ being simpler than the general case (since $J_1$ is not necessary), we assume that $d>1$. Then $a_4=0$ implies that $\Gamma_1'(u)/h''(u)=-2\Gamma_2(u)$. Replacing $\Gamma_1'(u)/h''(u)$ by $-2\Gamma_2(u)$ in $a_2=0$ gives $$\Gamma_2(u) = \frac{C_{\rm RK}+2}{3}\mu(u)\mu'(u).$$ On the other hand, replacing $\Gamma_2(u)$ by $-\Gamma_1'(u)/(2h''(u))$ in $a_2=0$, we find that $$\Gamma_1'(u) = -\frac23(C_{\rm RK}+2)\mu(u)\mu'(u)h''(u)$$ or, after integration, $$\Gamma_1(u) = -\frac23(C_{\rm RK}+2)\int_{u_0}^u\mu(v)\mu'(v)h''(v)dv.$$ These functions have to satisfy the conditions $$\begin{aligned}
a_1 &\ge 0\quad\mbox{or}\quad \frac{d-1}{d}\Gamma_1(u) \ge
-(C_{\rm RK}+1)h''(u)\mu(u)^2, \\
a_3 &\ge 0\quad\mbox{or}\quad
(C_{\rm RK}+2)\mu(u)\mu''(u) + (C_{\rm RK}-1)\mu'(u)^2 \le 0, \\
a_5 &\ge 0\quad\mbox{or}\quad \Gamma_1(u)\le 0\quad\mbox{for all }u,\end{aligned}$$ Note that $a_1\ge 0$ and $a_5\ge 0$ correspond to and , respectively. This shows that $P(\xi)\ge 0$ for all $\xi\in{{\mathbb R}}^4$ and $G''(0)\le 0$.
If $G''(0)=0$, the nonnegative polynomial $P$, which depends on $x\in\Omega$ via $\xi$, has to vanish. In particular, $a_3\xi_G^4=a_3|\na u|^4=0$ in $\Omega$. As $a_3>0$ by assumption, $u(x)=\mbox{const.}$ for $x\in\Omega$. This contradicts the hypothesis that $u$ is not a steady state. Consequently, $G''(0)<0$, and we finish the proof by setting $b(u)=-\Gamma_1(u)$.
Porous-medium equation {#sec.pme}
======================
The results of the previous section can be applied in principle to the Runge-Kutta scheme for the porous-medium or fast-diffusion equation $$\label{pm.eq}
\pa_t u = \Delta(u^\beta)\quad\mbox{in }\Omega,\ t>0, \quad
\na u^\beta\cdot\nu=0\quad\mbox{on }\pa\Omega, \quad u(0)=u^0,$$ where $\beta>0$. It can be seen that conditions - are not optimal for particular entropies. This is not surprising since we have neglected the mixed terms in the polynomial in (i.e. $a_2=a_4=0$) which is not optimal. In this section, we make a different approach by making an ansatz for the functions $\Gamma_1$ and $\Gamma_2$, considering both zeroth-order and first-order entropies.
Zeroth-order entropies
----------------------
We prove the following result.
\[thm.pm\] Let $\Omega\subset{{\mathbb R}}^d$ be convex with smooth boundary. Let $(u^k)$ be a sequence of (smooth) solutions to the Runge-Kutta scheme for . Let the entropy be given by $H[u]=\alpha^{-1}(\alpha+1)^{-1}\int_\Omega u^{\alpha+1}dx$ with $\alpha>0$, let $k\in{{\mathbb N}}$, and let $u^k$ be not the constant steady state of . There exists a nonempty region $R_0(d)\subset(0,\infty)^2$ and $\tau^k>0$ such that for all $(\alpha,\beta)\in R_0(d)$ and $0<\tau\le\tau^k$, $$H[u^k] + \tau\beta\int_\Omega (u^k)^{\alpha+\beta-2}|\na u^k|^2 dx
\le H[u^{k-1}], \quad k\in{{\mathbb N}}.$$ In one space dimension, we have $$\begin{aligned}
&\mbox{implicit Euler:} & R_0(1) &= (0,\infty)^2, \\
&\mbox{Runge-Kutta of order }p\ge 2: &
R_0(1) &= \big\{(\alpha,\beta)\in(0,\infty)^2: -2<\alpha-\beta<1\}, \\
&\mbox{explicit Euler:} &
R_0(1) &= \big\{(\alpha,\beta)\in(0,\infty)^2: -1<\alpha-\beta<1\}.\end{aligned}$$
For the implicit Euler scheme, the theorem shows that any positive values for $(\alpha,\beta)$ is admissible which corresponds to the continuous situation. For the Runge-Kutta case with $C_{\rm RK}=1$, our condition is more restrictive. As expected, the explicit Euler scheme requires the most restrictive condition. The set $R_0(d)$ is illustrated in Figure \[fig.0th\] for $d=2$ and $d=10$.
![Set $R_0(d)$ of all $(\alpha,\beta)$ for which the zeroth-order entropy is dissipating. Left column: $d=2$, right column: $d=10$. Top row: explicit Euler scheme with $C_{\rm RK}=2$, middle row: implicit Euler scheme with $C_{\rm RK}=1$, bottom row: Runge-Kutta scheme of order $p\ge 2$ with $C_{\rm RK}=0$.[]{data-label="fig.0th"}](pme_zeroth_d2_c2.eps "fig:"){width="65mm" height="55mm"} ![Set $R_0(d)$ of all $(\alpha,\beta)$ for which the zeroth-order entropy is dissipating. Left column: $d=2$, right column: $d=10$. Top row: explicit Euler scheme with $C_{\rm RK}=2$, middle row: implicit Euler scheme with $C_{\rm RK}=1$, bottom row: Runge-Kutta scheme of order $p\ge 2$ with $C_{\rm RK}=0$.[]{data-label="fig.0th"}](pme_zeroth_d10_c2.eps "fig:"){width="65mm" height="55mm"} ![Set $R_0(d)$ of all $(\alpha,\beta)$ for which the zeroth-order entropy is dissipating. Left column: $d=2$, right column: $d=10$. Top row: explicit Euler scheme with $C_{\rm RK}=2$, middle row: implicit Euler scheme with $C_{\rm RK}=1$, bottom row: Runge-Kutta scheme of order $p\ge 2$ with $C_{\rm RK}=0$.[]{data-label="fig.0th"}](pme_zeroth_d2_c1.eps "fig:"){width="65mm" height="55mm"} ![Set $R_0(d)$ of all $(\alpha,\beta)$ for which the zeroth-order entropy is dissipating. Left column: $d=2$, right column: $d=10$. Top row: explicit Euler scheme with $C_{\rm RK}=2$, middle row: implicit Euler scheme with $C_{\rm RK}=1$, bottom row: Runge-Kutta scheme of order $p\ge 2$ with $C_{\rm RK}=0$.[]{data-label="fig.0th"}](pme_zeroth_d10_c1.eps "fig:"){width="65mm" height="55mm"} ![Set $R_0(d)$ of all $(\alpha,\beta)$ for which the zeroth-order entropy is dissipating. Left column: $d=2$, right column: $d=10$. Top row: explicit Euler scheme with $C_{\rm RK}=2$, middle row: implicit Euler scheme with $C_{\rm RK}=1$, bottom row: Runge-Kutta scheme of order $p\ge 2$ with $C_{\rm RK}=0$.[]{data-label="fig.0th"}](pme_zeroth_d2_c0.eps "fig:"){width="65mm" height="55mm"} ![Set $R_0(d)$ of all $(\alpha,\beta)$ for which the zeroth-order entropy is dissipating. Left column: $d=2$, right column: $d=10$. Top row: explicit Euler scheme with $C_{\rm RK}=2$, middle row: implicit Euler scheme with $C_{\rm RK}=1$, bottom row: Runge-Kutta scheme of order $p\ge 2$ with $C_{\rm RK}=0$.[]{data-label="fig.0th"}](pme_zeroth_d10_c0.eps "fig:"){width="65mm" height="55mm"}
Since $k\in{{\mathbb N}}$ is fixed, we set $u:=u^k$. We choose the functions $$\Gamma_1(u) = c_1\beta^2 u^{2\beta-\alpha-1}, \quad
\Gamma_2(u) = c_2\beta^2 u^{2\beta-2\alpha-1}.$$ It holds $h''(u)=u^{\alpha-1}$ and $\mu(u)=\beta u^{\beta-\alpha}$. Then the coefficients in are as follows: $$\begin{aligned}
a_1 &= \beta^2\big((C_{\rm RK}+1) + (1-\tfrac{1}{d})c_1\big)u^{2\beta-\alpha-1}, \\
a_2 &= \beta^2\big((C_{\rm RK}+2)(\beta-\alpha)
+ (1-\tfrac{1}{d})(2\beta-\alpha-1)c_1
- (\tfrac{2}{d}+1)c_2\big)u^{2\beta-2\alpha-1}, \\
a_3 &= \beta^2\big((\beta-\alpha)^2 - (2\beta-2\alpha-1)c_2\big)
u^{2\beta-3\alpha-2}, \\
a_4 &= -\beta^2(d-1)\big((2\beta-\alpha-1)c_1+2c_2\big)u^{2\beta-2\alpha-1}, \\
a_5 &= -\beta^2 c_1 u^{2\beta-\alpha-1}, \quad
a_6 = -\beta^2 d(d-1) c_1 u^{2\beta-\alpha-1}.\end{aligned}$$ Introducing the variables $\eta_j = \xi_j/u^{\alpha}$ for $j\in\{G,L,R,S\}$, we can write as $$\begin{aligned}
& G''(0)\le G''(0)+J_1+J_2 = -\beta^2\int_\Omega u^{2\beta+\alpha-1}Q(\eta)dx, \\
&\mbox{where }
Q(\eta) = b_1\eta_L^2 + b_2\eta_L\eta_G^2 + b_3\eta_G^4
+ b_4\eta_S\eta_G^2 + b_5\eta_R^2 + b_6\eta_S^2\end{aligned}$$ with coefficients $$\begin{aligned}
b_1 &= (C_{\rm RK}+1)+(1-\tfrac{1}{d})c_1, \\
b_2 &= (C_{\rm RK}+2)(\beta-\alpha) + (1-\tfrac{1}{d})(2\beta-\alpha-1)c_1
- (\tfrac{2}{d}+1)c_2, \\
b_3 &= (\beta-\alpha)^2 - (2\beta-2\alpha-1)c_2, \\
b_4 &= -(d-1)\big((2\beta-\alpha-1)c_1+2c_2\big), \\
b_5 &= -c_1, \quad b_6 = -d(d-1) c_1.\end{aligned}$$
We need to determine all $(\alpha,\beta)$ such that there exist $c_1\le 0$, $c_2\in{{\mathbb R}}$ such that $Q(\eta)\ge 0$ for all $\eta=(\eta_G,\eta_L,\eta_R,\eta_S)$. Without loss of generality, we exclude the cases $b_1=b_2=0$ and $b_4=b_6=0$ since they lead to parameters $(\alpha,\beta)$ included in the region calculated below. Thus, let $b_1>0$ and $b_6>0$. These inequalities give the bound $-(C_{\rm RK}+1)/(1-1/d)<c_1<0$. Thus, we may introduce the parameter $\lambda\in(0,1)$ by setting $c_1=-\lambda(C_{\rm RK}+1)/(1-1/d)$. The polynomial $Q(\eta)$ can be rewritten as $$\begin{aligned}
Q(\eta) &= b_1\left(\eta_L+\frac{b_2}{2b_1}\eta_G^2\right)^2
+ b_6\left(\eta_S+\frac{b_4}{2b_6}\eta_G^2\right)^2
+ b_5\eta_R^2 + \eta_G^4\left(b_3-\frac{b_2^2}{4b_1}-\frac{b_4^2}{4b_6}\right) \\
&\ge\eta_G^4\left(b_3-\frac{b_4^2}{4b_6}-\frac{b_2^2}{4b_1}\right)
=: \frac{\eta_G^4 (C_{\rm RK}+1)}{4b_1b_6}R(c_2;\lambda,\alpha,\beta),\end{aligned}$$ where $R(c_2;\lambda,\alpha,\beta)$ is a quadratic polynomial in $c_2$ with the nonpositive leading term $-d^2(4-3\lambda) + 4(2-3\lambda)d - 4$. The polynomial $R(c_2;\lambda,\alpha,\beta)$ is nonnegative for some $c_2$ if and only if its discriminant $4d^2\lambda(1-\lambda)S(\lambda;\alpha,\beta)$ is nonnegative. Here, $S(\lambda;\alpha,\beta)$ is a quadratic polynomial in $\lambda$. In order to derive the conditions on $(\alpha,\beta)$ such that $S(\lambda;\alpha,\beta)\ge 0$ for some $\lambda\in(0,1)$, we employ the computer-algebra system [Mathematica]{}. The result of the command
> Resolve[Exists[LAMBDA, S[LAMBDA] >= 0 && LAMBDA > 0
> && LAMBDA < 1], Reals]
gives all $(\alpha,\beta)\in{{\mathbb R}}^2$ such that there exist $c_1\le 0$, $c_2\in{{\mathbb R}}$ such that $Q(\eta)\ge 0$. The interior of this region equals the set $R_0(d)$, defined in the statement of the theorem. This shows that $G''(0)\le 0$ for all $(\alpha,\beta)\in R_0(d)$.
If $G''(0)=0$, the nonnegative polynomial $Q$ has to vanish. In particular, $b_1\eta_L^2=0$. If $\eta_L=0$ in $\Omega$, the boundary conditions imply that $u$ is constant, which contradicts our assumption that $u$ is not the steady state. Thus $b_1=0$. Similarly, $b_2=b_3=b_4=0$. This gives a system of four inhomogeneous linear equations for $(c_1,c_2)$ which is unsolvable. Consequently, $G''(0)<0$.
The set $R_0(d)$ is nonempty since, e.g., $(1,1)\in R_0(d)$. Indeed, choosing $c_1=-1$ and $c_2=0$, we find that $Q(\eta)=(C_{\rm RK}+\frac{1}{d})\eta_L^2
+ \eta_R^2 + d(d-1)\eta_S^2\ge 0$.
In one space dimension, the situation simplifies since the Laplacian coincides with the Hessian and thus, the integration-by-parts formula is not needed. Then (see ) $$G''(0) = G''(0)+J_1
= -\beta^2\int_\Omega u^{2\beta+\alpha-1}\big(a_1\xi_L^2 + a_2\xi_L\xi_G^2
+ a_3\xi_G^4\big)dx,$$ where $$a_1 = C_{\rm RK}+1, \quad
a_2 = (C_{\rm RK}+2)(\beta-\alpha) - 3c_2, \quad
a_3 = (\beta-\alpha)^2 - (2\beta-2\alpha-1)c_2.$$ The polynomial $P(\xi)=\xi_G^4(a_1 y^2 + a_2 y + a_3)$ with $y=\xi_L/\xi_G^2$ is nonnegative if and only if $a_1\ge 0$ and $4a_1a_3-a_2^2\ge 0$, which is equivalent to $$\label{pm.aux}
-9c_2^2 + 2\big((C_{\rm RK}-2)(\alpha-\beta)+2(C_{\rm RK}+1)\big)c_2
- C_{\rm RK}^2(\alpha-\beta)^2 \ge 0.$$ This inequality has a solution $c_2\in{{\mathbb R}}$ if and only if the quadratic polynomial has real roots, i.e. if its discriminant is nonnegative, $$\begin{aligned}
0 &\le \big((C_{\rm RK}-2)(\alpha-\beta)+2(C_{\rm RK}+1)\big)^2
- 9C_{\rm RK}^2(\alpha-\beta)^2 \\
&= 4(C_{\rm RK}+1)\left(-(2C_{\rm RK}-1)(\alpha-\beta)^2
+ (C_{\rm RK}-2)(\alpha-\beta) + (C_{\rm RK}+1)\right). \end{aligned}$$ The polynomial $-(2C_{\rm RK}-1)z^2 + (C_{\rm RK}-2)z + (C_{\rm RK}+1)$ with $z=\alpha-\beta$ is always nonnegative if $C_{\rm RK}=0$ (implicit Euler). For $C_{\rm RK}=1$ and $C_{\rm RK}=2$, this property holds if and only if $-(C_{\rm RK}+1)/(2C_{\rm RK}-1)\le\alpha-\beta\le 1$. This concludes the proof.
First-order entropies
---------------------
We consider the one-dimensional case and first-order entropies with $f(u)=u^{\alpha/2}$, $\alpha>0$.
\[thm.pm1\] Let $\Omega\subset{{\mathbb R}}$ be a bounded interval. Let $(u^k)$ be a sequence of (smooth) solutions to the Runge-Kutta scheme of order $p\ge 2$ for in one space dimension. Let the entropy be given by $F[u]=\int_\Omega (u^{\alpha/2})_x^2 dx$ with $\alpha>0$, let $k\in{{\mathbb N}}$ be fixed, and let $u^k$ be not the constant steady state of . There exists a nonempty region $R_1\in[0,\infty)^2$ and $\tau^k>0$ such that for all $(\alpha,\beta)\in R_1$, there is a constant $C_{\alpha,\beta}>0$ such that for all $0<\tau\le\tau^k$, $$F[u^k] + \tau C_{\alpha,\beta}\int_\Omega (u^k)^{\alpha+\beta-3}(u^k_{xx})^2 dx
\le F[u^{k-1}], \quad k\in{{\mathbb N}}.$$
Figure \[fig.1st\] illustrates the set $R_1$. The set of admissible values $(\alpha,\beta)$ for the continuous equation is given by $\{-2\le\alpha-2\beta<1\}$ (the borders of this set are depicted in the figure by the dashed lines).
![Set of all $(\alpha,\beta)$ for which the discrete first-order entropy for solutions to the one-dimensional porous-medium equation is dissipating. The continuous first-order entropy is dissipated for $-2\le\alpha-2\beta<1$. The borders of this set is indicated in the figure by dashed lines.[]{data-label="fig.1st"}](pme_first_d1.eps){width="85mm"}
First, we compute $G'(0)$ according to Theorem \[thm.ed2\]: $$G'(0) = -\alpha\int_\Omega u^{\alpha/2-1}(u^{\alpha/2})_{xx}
(u^\beta)_{xx} dx.$$ We show that $G'(0)$ is nonpositive in a certain range of values $(\alpha,\beta)$. We formulate $G'(0)$ as $$G'(0) = -\frac{\alpha^2\beta}{4}\int_\Omega u^{\alpha+\beta-1}
\big((\alpha-2)(\beta-1)\xi_1^4 + (\alpha+2\beta-4)\xi_1^2\xi_2 + 2\xi_2^2\big)dx,$$ where $\xi_1=u_x/u$, $\xi_2=u_{xx}/u$. We employ the integration-by-parts formula $$0 = \int_\Omega(u^{\alpha+\beta-4}u_x^3)_x dx
= \int_\Omega u^{\alpha+\beta-1}\big((\alpha+\beta-4)\xi_1^4 + 3\xi_1^2\xi_2\big)dx
=: J.$$ Therefore, $$G'(0) = G'(0)-\frac{\alpha^2\beta}{4}cJ
= -\frac{\alpha^2\beta}{4}\int_\Omega u^{\alpha+\beta-1}P(\xi)dx,$$ where $$P(\xi) = \big((\alpha-2)(\beta-1)+(\alpha+\beta-4)c\big)\xi_1^4
+ \big(\alpha+2\beta-4+3c\big)\xi_1^2\xi_2 + 2\xi_2^2.$$ This polynomial is nonnegative if and only if $$8\big((\alpha-2)(\beta-1)+(\alpha+\beta-4)c\big)
- (\alpha+2\beta-4+3c)^2\ge 0,$$ which is equivalent to $$g(c) := -9c^2 + 2(\alpha-2\beta-4)c - (\alpha-2\beta)^2 \ge 0.$$ The maximizing value $c^*=(\alpha-2\beta-4)/9$, obtained from $g'(c)=0$, yields $$g(c^*) = -\frac89(\alpha-2\beta-1)(\alpha-2\beta+2) \ge 0$$ and consequently $G'(0)\le 0$ if $-2\le\alpha-2\beta\le 1$. This condition is the same as in [@CJS15 Theorem 13] for the continuous equation.
Next, we turn to the proof of $G''(0)<0$. The proof of Theorem \[thm.ed2\] shows that $$\begin{aligned}
G''(0) &= -\frac{\alpha}{2}\int_\Omega\bigg(\frac{\alpha}{2}
\big(u^{\alpha/2-1}(u^\beta)_{xx}\big)_x^2 - \bigg(\frac{\alpha}{2}-1\bigg)
u^{\alpha/2-2}(u^{\alpha/2})_{xx}(u^\beta)_{xx}^2 \\
&\phantom{xx}{}- \beta C_{\rm RK}u^{\alpha/2-1}(u^{\alpha/2})_{xx}
\big(u^{\beta-1}(u^\beta)_{xx}\big)_{xx}\bigg)dx.\end{aligned}$$ We integrate by parts in the last term and use $(\beta u^{\beta-1}(u^\beta)_{xx})_x=0$ on $\pa\Omega$: $$\begin{aligned}
G''(0) &= -\frac18\alpha^2\beta^2\int_\Omega u^{\alpha+2\beta-2} \\
&\phantom{xx}{}\times
\big(a_1\xi_1^6 + a_2\xi_1^4\xi_2 + a_3\xi_1^3\xi_3 + a_4\xi_1^2\xi_2^2
+ a_5\xi_1\xi_2\xi_3 + a_6\xi_2^3 + a_7\xi_3^2\big)dx,\end{aligned}$$ where $\xi_1=u_x/u$, $\xi_2=u_{xx}/u$, $\xi_3=u_{xxx}/u$, and $$\begin{aligned}
a_1 &= (\beta-1)\big(2C_{\rm RK}\alpha^2\beta - 3C_{\rm RK}\alpha^2
+ 2\alpha\beta^2-2(5C_{\rm RK}+3)\alpha\beta + (15C_{\rm RK}+4)\alpha \\
&\phantom{xx}{}+ 2\beta^3 - 14\beta^2 + 4(3C_{\rm RK}+7)\beta
- 2(9C_{\rm RK} + 8)\big), \\
a_2 &= (\beta-1)\big(4C_{\rm RK}\alpha^2 + (8C_{\rm RK}+7)\alpha\beta
- (32C_{\rm RK}+9)\alpha + 12\beta^2 - 2(8C_{\rm RK}+25)\beta \\
&\phantom{xx}{}+ 6(8C_{\rm RK} + 7)\big), \\
a_3 &= C_{\rm RK}\alpha^2 + 2\alpha\beta - (5C_{\rm RK}+2)\alpha
+ 4(C_{\rm RK}+1)\beta^2 - 2(5C_{\rm RK}+8)\beta + 12(C_{\rm RK} + 1), \\
a_4 &= 2(\beta-1)\big(2(4C_{\rm RK}+1)\alpha + 9\beta - (16C_{\rm RK} + 13)\big), \\
a_5 &= 2(2C_{\rm RK}+1)\alpha + 4(2C_{\rm RK}+3)\beta - 16(C_{\rm RK}+1), \\
a_6 &= 2-\alpha, \quad a_7 = 2(C_{\rm RK}+1).\end{aligned}$$ We employ three integration-by-parts formulas: $$\begin{aligned}
0 &= \int_\Omega\big(u^{\alpha+2\beta-5}u_{xx}^2 u_x\big)_x dx
= \int_\Omega u^{\alpha+2\beta-2}\big((\alpha+2\beta-5)\xi_1^2\xi_2^2
+ 2\xi_1\xi_2\xi_3 + \xi_2^3\big)dx =: J_1, \\
0 &= \int_\Omega\big(u^{\alpha+2\beta-6}u_{xx}u_x^3\big)_x dx
= \int_\Omega u^{\alpha+2\beta-2}\big((\alpha+2\beta-6)\xi_1^4\xi_2
+ \xi_1^3\xi_3 + 3\xi_1^2\xi_2^2\big)dx =: J_2, \\
0 &= \int_\Omega\big(u^{\alpha+2\beta-7}u_x^5\big)_x dx
= \int_\Omega u^{\alpha+2\beta-2}\big((\alpha+2\beta-7)\xi_1^6
+ 5\xi_1^4\xi_2\big)dx =: J_3.\end{aligned}$$ Then $$\begin{aligned}
& G''(0) = G''(0)-\frac18\alpha^2\beta^2(c_1J_1+c_2J_2+c_3J_3)
= -\frac18\alpha^2\beta^2\int_\Omega u^{\alpha+2\beta-2}P(\xi)dx, \\
&\mbox{where }P(\xi) = b_1\xi_1^6 + b_2\xi_1^4\xi_2 + b_3\xi_1^3\xi_3
+ b_4\xi_1^2\xi_2^2 + b_5\xi_1\xi_2\xi_3 + b_6\xi_2^3 + b_7\xi_3^2,\end{aligned}$$ and the coefficients are given by $$\begin{aligned}
b_1 &= a_1 + (\alpha+2\beta-7)c_3, &
b_2 &= a_2 + (\alpha+2\beta-6)c_2 + 5c_3, \\
b_3 &= a_3 + c_2, & b_4 &= a_4 + (\alpha+2\beta-5)c_1 + 3c_2, \\
b_5 &= a_5 + 2c_1, & b_6 &= a_6 + c_1, \\
b_7 &= a_7. \end{aligned}$$ Choosing $c_1=-a_6$, we eliminate the cubic term $\xi_2^3$. Furthermore, setting, $x=\xi_2/\xi_1^2$ and $y=\xi_3/\xi_1^3$, we can write the polynomial $P$ as a quadratic polynomial in $(x,y)$: $$Q(x,y)=\xi_1^6 P(\xi)
= b_1 + b_2 x + b_3 y + b_4 x^2 + b_5 xy + b_7 y^2.$$
The following lemma is a consequence of the proof of Lemma 2.2 in [@JuMi09].
The polynomial $p(x,y) = A + B x + C y + D x^2 + E xy + F y^2$ with $F>0$ is nonnegative for all $(x,y)\in{{\mathbb R}}^2$ if and only if $$\begin{aligned}
&\text{\rm (i)}\phantom{.}\quad 4DF-E^2>0 \quad\mbox{and}\quad
A(4DF-E^2) - B^2F - C^2D + BCE \ge 0, \quad\mbox{or} \\
&\text{\rm (ii)}\quad 4DF-E^2=0\quad\mbox{and}\quad 2BF-CE=0\quad\mbox{and}\quad
4AF-C^2\ge 0.\end{aligned}$$
Note that in case $4DF-E^2=0$ and $E\neq 0$, we may replace $2BF-CE=0$ by the condition $2BEF = CE^2 = 4CDF$ or (since $F>0$) $BE=2CD$.
The first inequality in case (i), $$\begin{aligned}
0 &< 4b_4b_7 - b_5^2
= -(C_{\rm RK}+1)(2C_{\rm RK}+1)\alpha^2
+ (2C_{\rm RK}+2)(4C_{\rm RK}-3)\alpha\beta
+ (9C_{\rm RK}+9)\alpha \\
&\phantom{xx}{}- 2C_{\rm RK}(4C_{\rm RK}+3)\beta^2
+ (8C_{\rm RK}+12)\beta + (3C_{\rm RK}+3)c_2 - (12C_{\rm RK} + 14),\end{aligned}$$ is linear in $c_2$ and provides a lower bound for $c_2$: $$\begin{aligned}
c_2 &> \frac{1}{3(C_{\rm RK}+1)}\Big((C_{\rm RK}+1)(2C_{\rm RK}+1)\alpha^2
- (2C_{\rm RK}+2)(4C_{\rm RK}-3)\alpha\beta
- (9C_{\rm RK}+9)\alpha \\
&\phantom{xx}{}+ 2C_{\rm RK}(4C_{\rm RK}+3)\beta^2
- (8C_{\rm RK}+12)\beta + (12C_{\rm RK} + 14)\Big) =: c_2^*.\end{aligned}$$ The second inequality in case (i) becomes $$0\le b_1(4b_4b_7-b_5^2) - b_2^2b_7 - b_3^2b_4 + b_2b_3b_5
= -50(C_{\rm RK}+1)c_3^2 + p_1(\alpha,\beta,c_2)c_3 + p_2(\alpha,\beta,c_2),$$ where $p_1$ and $p_2$ are some polynomials in $\alpha$, $\beta$, and $c_2$. This quadratic expression in $c_3$ is nonnegative if and only if its discriminant is nonnegative, $$\begin{aligned}
0 &\le -200(C_{\rm RK}+1)p_2(\alpha,\beta,c_2) - p_1(\alpha,\beta,c_2)^2 \\
&= -8\big(4b_4b_7 - b_5^2\big)
\big(25c_2^2 + p_3(\alpha,\beta)c_2 + p_4(\alpha,\beta)\big),\end{aligned}$$ where $p_3(\alpha,\beta)$ and $p_4(\alpha,\beta)$ are some polynomials in $\alpha$ and $\beta$. The factor $4b_4b_7 - b_5^2$ is positive, so we have to ensure that $R_{\alpha,\beta}(c_2)=25c_2^2 + p_3(\alpha,\beta)c_2 + p_4(\alpha,\beta)\le 0$ for some $c_2>c_2^*$. Therefore we must ensure that the rightmost root of $R_{\alpha,\beta}(c_2)$ is larger or equal than the lower bound for $c_2$, i.e., $-p_3(\alpha,\beta)+\sqrt{p_3^2(\alpha,\beta)-100p_4(\alpha,\beta)}\ge 50c^*_2$. For $C_{\rm RK}=1$, the values $(\alpha,\beta)$ for which there exists $c_2>c_2^*$ such that $R_{\alpha,\beta}(c_2)\le 0$ is depicted in Figure \[fig.1st\]. In case (ii), we may immediately calculate $c_2$ and $c_3$ but this results in a region which is already contained in the first one. This shows that $G''(0)\le 0$.
If $G''(0)=0$, the polynomial $Q$ vanishes. Thus, either $u_x/u=\xi_1=0$ or $P(\xi)=0$ in $\Omega$. The first case is impossible since $u$ is not constant in $\Omega$. As $b_7=a_7=2(C_{\rm RK}+1)>0$, the second case $P(\xi)=0$ implies that $\xi_3=0$. Hence, $u$ is a quadratic polynomial. In view of the boundary conditions, $u$ must be constant, but this contradicts our assumption. Hence, $G''(0)<0$.
Linear diffusion system {#sec.sys}
=======================
We consider the following linear diffusion system: $$\label{sys.eq}
\pa_t u_1 - \rho_1\Delta u_1 = \mu(u_2-u_1), \quad
\pa_t u_2 - \rho_2\Delta u_2 = \mu(u_1-u_2),$$ with initial and homogeneous Neumann boundary conditions, $\rho_1$, $\rho_2$, $\mu>0$, and the entropy $$\label{sys.H}
H[u] = \int_\Omega h(u)dx
= \int_\Omega\sum_{i=1}^2 u_i(\log u_i-1)dx,$$ where $u=(u_1,u_2)$. If the initial data is nonnegative, the maximum principle shows that the solutions to are nonnegative too.
\[thm.sys\] Let $(u^k)$ be a sequence of (smooth) nonnegative solutions to the Runge-Kutta scheme for with $C_{\rm RK}=1$ and $\rho:=\rho_1=\rho_2$. Let the entropy $H$ be given by . Let $k\in{{\mathbb N}}$ be fixed and let $u^k$ be not the steady state of . Then there exists $\tau^k>0$ such that for all $0<\tau<\tau^k$, $$H[u^k] + \tau\int_\Omega\bigg(\rho\sum_{i=1}^2\frac{|\na u_i^k|^2}{u_i^k}
+ \mu(\log u_1^k-\log u_2^k)(u_1^k-u_2^k)\bigg)dx \le H[u^{k-1}].$$
Note that we need equal diffusivities $\rho_1=\rho_2$ and higher-order schemes ($C_{\rm RK}=1$). These conditions are in accordance of [@LiMi13], where the continuous equation was studied. In order to highlight the step where these conditions are needed, the following proof is slightly more general than actually needed.
We fix $k\in{{\mathbb N}}$ and set $u:=u^k$. Let $A[u]=(A_1[u],A_2[u])
=(\rho_1\Delta u_1+\mu(u_2-u_1),\rho_2\Delta u_2+\mu(u_1-u_2))$. Since $A$ is linear, $DA[u](h)=A[h]$. Thus, $$G''(0) = -\int_\Omega\big(C_{\rm RK}
h'(u)^\top A[A[u]] + A[u]^\top h''(u)A[u]\big)dx
= -G_{1} - G_{2}.$$ In the following, we set $\pa_i h=\pa h/\pa u_i$ for $i=1,2$. We integrate by parts twice, using the boundary conditions $\na u_i\cdot\nu=0$ and $\na A_i[u]\cdot\nu=0$ on $\pa\Omega$, and collect the terms: $$\begin{aligned}
G_{1} &= C_{\rm RK}\int_\Omega\Big(
\pa_1 h(u)\big(\rho_1\Delta A_1[u] + \mu(A_2[u]-A_1[u])\big) \\
&\phantom{xx}{}
+ \pa_2 h(u)\big(\rho_2\Delta A_2[u] + \mu(A_1[u]-A_2[u])\big)\Big)dx \\
&= C_{\rm RK}\int_\Omega\Big(\rho_1\Delta\pa_1 h(u)A_1[u]
+ \rho_2\Delta\pa_2 h(u)A_2[u] \\
&\phantom{xx}{}+ \mu(\pa_1 h(u)-\pa_2 h(u))(A_2[u]-A_1[u])\Big)dx \\
&= C_{\rm RK}\int_\Omega\Big(\rho_1\big(\pa_1^2 h(u)\Delta u_1
+\pa_1^3 h(u)|\na u_1|^2\big)\big(\rho_1 \Delta u_1+\mu(u_2-u_1)\big) \\
&\phantom{xx}{}+ \rho_2\big(\pa_2^2 h(u)\Delta u_2
+ \pa_2^3 h(u)|\na u_2|^2\big)\big(\rho_2 \Delta u_2+\mu(u_1-u_2)\big) \\
&\phantom{xx}{}+ \mu(\pa_2 h(u)-\pa_1 h(u))\big(\rho_1\Delta u_1-\rho_2\Delta u_2
+ 2\mu(u_2-u_1)\big)\Big)dx \\
&= C_{\rm RK}\int_\Omega\Big(\rho_1^2\pa_1^2h(u)(\Delta u_1)^2
+ \rho_2^2\pa_2^2h(u)(\Delta u_2)^2
+ \rho_1^2\pa_1^3h(u)\Delta u_1|\na u_1|^2 \\
&\phantom{xx}{}+ \rho_2^2\pa_2^3h(u)\Delta u_2|\na u_2|^2
+ \rho_1 \mu\big(\pa_1^2h(u)(u_2-u_1) + \pa_2h(u)-\pa_1h(u)\big)\Delta u_1 \\
&\phantom{xx}{}
+ \rho_2 \mu\big(\pa_2^2h(u)(u_1-u_2) + \pa_1h(u)-\pa_2h(u)\big)\Delta u_2
+ \rho_1 \mu \pa_1^3h(u)(u_2-u_1)|\na u_1|^2 \\
&\phantom{xx}{}+ \rho_2 \mu \pa_2^3h(u)(u_1-u_2)|\na u_2|^2
+ 2\mu^2(\pa_2h(u)-\pa_1h(u))(u_2-u_1)\Big)dx.\end{aligned}$$ Furthermore, $$\begin{aligned}
G_{2} &= \int_\Omega\Big(\pa_1^2h(u)\big(\rho_1\Delta u_1+\mu(u_2-u_1)\big)^2
+ \pa_2^2h(u)\big(\rho_2\Delta u_2+\mu(u_1-u_2)\big)^2\Big)dx \\
&= \int_\Omega\Big(\rho_1^2\pa_1^2h(u)(\Delta u_1)^2
+ \rho_2^2\pa_2^2h(u)(\Delta u_2)^2 + 2\rho_1 \mu\pa_1^2h(u)(u_2-u_1)\Delta u_1 \\
&\phantom{xx}{}+ 2\rho_2 \mu\pa_2^2h(u)(u_1-u_2)\Delta u_2
+ \mu^2(\pa_1^2h(u)+\pa_2^2h(u))(u_1-u_2)^2\Big)dx.\end{aligned}$$ Adding $G_1$ and $G_2$, we arrive at $$\begin{aligned}
G''(0) &= -\sum_{i=1}^2\int_\Omega\Big(\rho_i^2(C_{\rm RK}+1)\pa_i^2h(u)
(\Delta u_i)^2 + \rho_i^2C_{\rm RK}\pa_i^3h(u)\Delta u_i|\na u_i|^2\Big)dx \\
&\phantom{xx}{}- \int_\Omega\Big(\rho_1\mu\big((C_{\rm RK}+2)\pa_1^2h(u)(u_2-u_1)
+ C_{\rm RK}(\pa_2h(u)-\pa_1h(u))\big)\Delta u_1 \\
&\phantom{xx}{}+ \rho_2\mu\big((C_{\rm RK}+2)\pa_2^2h(u)(u_1-u_2)
+ C_{\rm RK}(\pa_1h(u)-\pa_2h(u))\big)\Delta u_2 \\
&\phantom{xx}{} + \rho_1 \mu C_{\rm RK}\pa_1^3h(u)(u_2-u_1)|\na u_1|^2
+ \rho_2 \mu C_{\rm RK}\pa_2^3h(u)(u_1-u_2)|\na u_2|^2\Big)dx \\
&\phantom{xx}{}
- \int_\Omega \mu^2\Big(2(\pa_1h(u)-\pa_2h(u))
+(\pa_1^2h(u)+\pa_2^2h(u))(u_1-u_2)\Big)(u_1-u_2)dx \\
&= -I_2 - I_1 - I_0.\end{aligned}$$
The idea of [@LiMi13] is to show that each integral $I_i$, involving only derivatives of order $i$, is nonnegative. In contrast to [@LiMi13], we employ systematic integration by parts, which allows for a simpler and more general proof in our context. For the term $I_2$, we use the following integration-by-parts formula: $$0 = \int_\Omega{\operatorname{div}}\big(u_i^{-2}|\na u_i|^3\big)dx
= \int_\Omega\big(-2u_i^{-3}|\na u_i|^4 + 3u_i^{-2}\Delta u_i|\na u_i|^2\big)dx
=: J_i.$$ Then, for $\eps>0$, $$\begin{aligned}
&I_2 - c\sum_{i=1}^2\rho_i^2 J_i - \eps\sum_{i=1}^2 u_i^{-3}|\na u_i|^4dx \\
&= \sum_{i=1}^2\rho^2_i\int_\Omega\Big((C_{\rm RK}+1)u_i^{-1}(\Delta u_i)^2
- (3c + C_{\rm RK})u_i^{-2}\Delta u_i|\na u_i|^2
+ (2c-\eps) u_i^{-3}|\na u_i|^4\Big)dx.\end{aligned}$$ The integrand defines a quadratic polynomial in the variables $\Delta u_i$ and $|\na u_i|^2$ and is nonnegative if its discriminant satisfies $4(2c-\eps)(C_{\rm RK}+1)-(3c+C_{\rm RK})^2\ge 0$. It turns out that this inequality holds true for $C_{\rm RK}\in\{0,1\}$ if we choose $c=2/3$ and $\eps>0$ sufficiently small. When $C_{\rm RK}=2$, we can show only that $I_2\ge 0$ which is not sufficient to prove that $G''(0)<0$ (see below). We conclude that $$\label{sys.I2}
I_2\ge \eps\sum_{i=1}^2\int_\Omega u_i^{-3}|\na u_i|^4 dx.$$
Integrating by parts in $I_1$ in order to obtain only first-order derivatives, we find after some rearrangements that $$\begin{aligned}
I_1 &= \mu\int_\Omega\big(a_1|\na\log u_1|^2 + a_2\na\log u_1\cdot\na\log u_2
+ a_3|\na\log u_2|^2\big)dx, \quad\mbox{where} \\
a_1 &= 2\rho_1(C_{\rm RK} u_1+u_2), \quad
a_3 = 2\rho_2(C_{\rm RK}u_2+u_1), \\
a_2 &= -(C_{\rm RK}(\rho_1+\rho_2)+2\rho_2)u_1
- (C_{\rm RK}(\rho_1+\rho_2)+2\rho_1)u_2.\end{aligned}$$ The integrand is nonnegative if and only if $4a_1a_3 - a_2^2\ge 0$ for all $(u_1,u_2)$. We compute: $$\begin{aligned}
C_{\rm RK}=0: &\quad 4a_1a_3-a_2^2 = -4(\rho_1u_2-\rho_2u_1)^2, \\
C_{\rm RK}=1: &\quad 4a_1a_3-a_2^2 = (\rho_1-\rho_2)\big(\rho_1(u_1^2+6u_1u_2+9u_2^2)
- \rho_2(9u_1^2+6u_1u_2+u_2^2)\big), \\
C_{\rm RK}=2: &\quad 4a_1a_3-a_2^2 = -4\big(\rho_1(u_1+2u_2) - \rho_2(2u_1+u_2)\big).\end{aligned}$$ Thus, $4a_1a_3 - a_2^2\ge 0$ is possible only if $\rho_1=\rho_2$ and $C_{\rm RK}=1$.
Finally, we see immediately that the remaining term $$I_0 = \mu^2\int_\Omega\bigg(2(\log u_1-\log u_2)(u_1-u_2)
+ \bigg(\frac{1}{u_1}+\frac{1}{u_2}\bigg)(u_1-u_2)^2\bigg)dx$$ is nonnegative. This shows that $G''(0)\le 0$. If $G''(0)=0$, we infer from that $u_i=\mbox{const.}$, but this contradicts our hypothesis that $u_i$ is not a steady state.
The Derrida-Lebowith-Speer-Spohn equation {#sec.dlss}
=========================================
Consider the one-dimensional fourth-order equation $$\label{dlss.eq}
\pa_t u = -(u(\log u)_{xx})_{xx} \quad\mbox{in }\Omega, \ t>o, \quad
u(0)=u^0$$ with periodic boundary conditions. This equation appears as a scaling limit of the so-called (time-discrete) Toom model, which describes interface fluctuations in a two-dimensional spin system [@DLSS91]. The variable $u$ is the limit of a random variable related to the deviation of the spin interface from a straight line. The multi-dimensional version of models the eectron density $u$ in a quantum semiconductor, und the equation is the zero-temperature, zero-field approximation of the quantum drift-diffusion model [@Jue09]. For existence results for , we refer to [@JuMa08] and references therein.
To simplify our calculations, we analyze only the logarithmic entropy $H[u]=\int_\Omega u(\log u-1)dx$. It is possible to verify condition also for entropies of the form $\int_\Omega u^\alpha dx$, but it turns out that only sufficiently small $\alpha>0$ are admissible (about $0<\alpha<0.15\ldots$) and the computations are very tedious. Therefore, we restrict ourselves to the case $\alpha=0$.
Let $(u^k)$ be a sequence of (smooth) solutions to the Runge-Kutta scheme with $C_{\rm RK}=1$ for . Let the entropy be given by $H[u]=\int_\Omega u(\log u-1)dx$, let $k\in{{\mathbb N}}$ be fixed, and let $u^k$ be not a steady state. Then there exists $\tau^k>0$ such that for all $0<\tau<\tau^k$, $$H[u^k] + \tau q\int_\Omega u(\log u)_x^8 dx
+ \tau\int_\Omega u(\log u)_{xx}^2 dx \le H[u^{k-1}], \quad
q\approx 0.0045.$$
First, we observe that $G'(0)=-\int_\Omega (u(\log u)_{xx})_{xx}\log udx
=-\int_\Omega u(\log u)_{xx}^2 dx$. With $A[u]=(u(\log u)_{xx})_{xx}$ and $DA[u](h) = \big(h_{xx} - 2(\log u)_xh_x + (\log u)_x^2 h\big)_{xx}$, we can write $G''(0)=-I_0^k$ according to as $$\begin{aligned}
G''(0) &= -\int_\Omega\Big(\log u\big(A[u]_{xx}
- 2(\log u)_xA[u]_x + (\log u)_x^2A[u]\big)_{xx} + \frac{1}{u}A[u]^2\Big)dx \\
&= -\int_\Omega\Big((\log u)_{xx}\big(A[u]_{xx}
- 2(\log u)_xA[u]_x + (\log u)_x^2A[u]\big) + \frac{1}{u}A[u]^2\Big)dx \\
&= -\int_\Omega\Big(\big(v_{xxxx} + 2(v_x v_{xx})_x
+ v_x^2 v_{xx}\big)
A[u] + \frac{1}{u}A[u]^2\Big)dx,\end{aligned}$$ where we have integrated by parts several times and have set $v=\log u$. Then $A[u]=u(v_x^2 v_{xx} + 2v_x v_{xxx} + v_{xx}^2
+ v_{xxxx})$ and, with the abbreviations $\xi_1=v_x,\ldots,\xi_4=v_{xxxx}$, $$\begin{aligned}
G''(0) &= -\int_\Omega u\Big(2\xi_1^4\xi_2^2
+ 8\xi_1^3\xi_2\xi_3
+ 5\xi_1^2\xi_2^3 + 4\xi_1^2\xi_2\xi_4
+ 8\xi_1^2\xi_3^2 + 10\xi_1\xi_2^2\xi_3 \\
&\phantom{xx}{}+ 8\xi_1\xi_3\xi_4 + 3\xi_2^4
+ 5\xi_2^2\xi_4 + 2\xi_4^2\Big)dx.\end{aligned}$$ We employ the following integration-by-parts formulas: $$\begin{aligned}
0 &= \int_\Omega(uv_x^7)_x dx = \int_\Omega u(\xi_1^8 + 7\xi_1^6\xi_2)dx =: J_1, \\
0 &= \int_\Omega(uv_{xx}v_x^5)_x dx
= \int_\Omega u(\xi_1^6\xi_2 + \xi_1^5\xi_3 + 5\xi_1^4\xi_2^2)dx =: J_2, \\
0 &= \int_\Omega(uv_{xxx}v_x^4)_x dx
= \int_\Omega u(\xi_1^5\xi_3 + \xi_1^4\xi_4 + 4\xi_1^3\xi_2\xi_3)dx =: J_3, \\
0 &= \int_\Omega(uv_{xx}^2v_x^3)_x dx
= \int_\Omega u(\xi_1^4\xi_2^2 + 2\xi_1^3\xi_2\xi_3 + 3\xi_1^2\xi_2^3)dx =: J_4, \\
0 &= \int_\Omega(uv_{xx}v_{xxx}v_x^2)_x dx
= \int_\Omega u(\xi_1^3\xi_2\xi_3 + \xi_1^2\xi_2\xi_4 + \xi_1^2\xi_3^2
+ 2\xi_1\xi_2^2\xi_3)dx =: J_5, \\
0 &= \int_\Omega(uv_{xxx}^2v_x)_x dx
= \int_\Omega u(\xi_1^2\xi_3^2 + 2\xi_1\xi_3\xi_4 + \xi_2\xi_3^2)dx =: J_6, \\
0 &= \int_\Omega(uv_{xx}^3v_x)_x dx
= \int_\Omega u(\xi_1^2\xi_2^3 + 3\xi_1\xi_2^2\xi_3 + \xi_2^4)dx =: J_7, \\
0 &= \int_\Omega(uv_{xxx}v_{xx}^2)_x dx
= \int_\Omega u(\xi_1\xi_2^2\xi_3 + 2\xi_2\xi_3^2 + \xi_2^2\xi_4)dx =: J_8.\end{aligned}$$ Then $$\begin{aligned}
G''(0) &=
G''(0) - 4\sum_{i=1}^8 c_iJ_i = -\int_\Omega u\Big(a_1\xi_1^8 + a_2\xi_1^6\xi_2
+ a_3\xi_1^5\xi_3 + a_4\xi_1^4\xi_2^2
+ a_5\xi_1^4\xi_4 \\
&\phantom{xx}{} + a_6\xi_1^3\xi_2\xi_3
+ a_7\xi_1^2\xi_2^3 + a_8\xi_1^2\xi_2\xi_4 + a_9\xi_1^2\xi_3^2
+ a_{10}\xi_1\xi_2^2\xi_3 + a_{11}\xi_1\xi_3\xi_4 + a_{12}\xi_2^4 \\
&\phantom{xx}{}
+ a_{13}\xi_2^2\xi_4 + a_{14}\xi_2\xi_3^2 + a_{15}\xi_4^2\Big)dx,\end{aligned}$$ where $$\begin{aligned}
a_1 &= 4c_1, & a_2 &= 28c_1 + 4c_2, & a_3 &= 4c_2 + 4c_3, \\
a_4 &= 2 + 20c_2 + 4c_4, & a_5 &= 4c_3, & a_6 &= 8 + 16c_3 + 8c_4 + 4c_5, \\
a_7 &= 5 + 12c_4 + 4c_7, & a_8 &= 4 + 4c_5, & a_9 &= 8 + 4c_5 + 4c_6, \\
a_{10} &= 10 + 8c_5 + 12c_7 + 4c_8, & a_{11} &= 8 + 8c_6, & a_{12} &= 3 + 4c_7, \\
a_{13} &= 5 + 4c_8, & a_{14} &= 4c_6 + 8c_8, & a_{15} &= 2. & & \end{aligned}$$ Next, we eliminate all terms involving $\xi_4$ by formulating the following square: $$\begin{aligned}
& G''(0) = -\int_\Omega u\bigg[a_{15}
\bigg(\xi_4 + \frac{a_5}{2a_{15}}\xi_1^4 + \frac{a_8}{2a_{15}}\xi_1^2\xi_2
+ \frac{a_{11}}{2a_{15}}\xi_1\xi_3 + \frac{a_{13}}{2a_{15}}\xi_2^2\bigg)^2 \\
&\phantom{xx}{}+ \bigg(a_1-\frac{a_5^2}{4a_{15}}\bigg)\xi_1^8
+ \bigg(a_2-\frac{a_5a_8}{2a_{15}}\bigg)\xi_1^6\xi_2
+ \bigg(a_3-\frac{a_5a_{11}}{2a_{15}}\bigg)\xi_1^5\xi_3 \\
&\phantom{xx}{}+ \bigg(a_4-\frac{a_8^2}{4a_{15}}-\frac{a_5a_{13}}{2a_{15}}\bigg)
\xi_1^4\xi_2^2 + \bigg(a_6-\frac{a_8a_{11}}{2a_{15}}\bigg)\xi_1^3\xi_2\xi_3
+ \bigg(a_7-\frac{a_8a_{13}}{2a_{15}}\bigg)\xi_1^2\xi_2^3 \\
&\phantom{xx}{}
+ \bigg(a_9-\frac{a_{11}^2}{4a_{15}}\bigg)\xi_1^2\xi_3^2
+ \bigg(a_{10}-\frac{a_{11}a_{13}}{2a_{15}}\bigg)\xi_1\xi_2^2\xi_3
+ \bigg(a_{12}-\frac{a_{13}^2}{4a_{15}}\bigg)\xi_2^4
+ a_{14}\xi_2\xi_3^2\bigg]dx.\end{aligned}$$ We eliminate all terms involving $\xi_3$ and set the corresponding coefficients to zero. From $a_{14}=0$ we conclude that $c_6=-2c_8$. Furthermore, $$\begin{aligned}
a_9-\frac{a_{11}^2}{4a_{15}} &= 0\ &\mbox{gives}\qquad
c_5 &= 8c_8^2-6c_8, \\
a_{10}-\frac{a_{11}a_{13}}{2a_{15}} &= 0\ &\mbox{gives}\qquad
c_7 &= -\frac{20}{3}c_8^2 + \frac83 c_8, \\
a_6-\frac{a_8a_{11}}{2a_{15}} &= 0\ &\mbox{gives}\qquad
c_4 &= -2c_3 - 16c_8^3 + 16c_8^2 - 5c_8, \\
a_3-\frac{a_5a_{11}}{2a_{15}} &= 0\ &\mbox{gives}\qquad
c_2 &= c_3 - 4c_3c_8.\end{aligned}$$ By these choices, we obtain $$b_{12} :=a_{12}-\frac{a_{11}^2}{4a_{15}}
= -\frac{86}{3}c_8^2 + \frac{17}{3}c_8 - \frac18.$$ This quadratic polynomial in $c_8$ admits its maximal value at $c_8^*=17/172$ with value $b_{12}=20/129$. The integral can now be written as $$G''(0) \le -\int_\Omega u\big(b_1\xi_1^8 + b_2\xi_1^6\xi_2
+ b_4\xi_1^4\xi_2^2 + b_7\xi_1^2\xi_2^3 + b_{12}\xi_2^4\big)dx,$$ where $$\begin{aligned}
& b_1 = a_1 - \frac{a_5^2}{4a_{15}}
= 4c_1 - 2c_3^2, \\
& b_2 = a_2 - \frac{a_5a_8}{2a_{15}}
= 28c_1 - 32c_3c_8^2 + 8c_3c_8, \\
& b_4 = a_4 - \frac{a_8^2}{4a_{15}} - \frac{a_5a_{13}}{2a_{15}}
= 7c_3 - 84c_3c_8 - 128c_8^4 + 128c_8^3 - 40c_8^2 + 4c_8, \\
& b_7 = a_7 - \frac{a_8a_{13}}{2a_{15}}
= -24c_3 - 244c_8^3 + \frac{448}{3}c_8^2 - \frac{70}{3}c_8.\end{aligned}$$ If $b_4=2b_2b_{12}/b_7 + b_7^2/(4b_{12})$, we can write the integal as the sum of two squares, noting that $b_{12}$ is positive, $$G''(0) \le -\int_\Omega u\bigg(b_{12}
\bigg(\xi_2^2 + \frac{b_7}{2b_{12}}\xi_1^2\xi_2 + \frac{b_2}{b_7}\xi_1^4\bigg)^2
+ \bigg(b_1 - \frac{b_2^2b_{12}}{b_7^2}\bigg)\xi_1^8\bigg)dx.$$ The expression $b_4b_7-2b_2b_{12} - b_7^3/(4b_{12})=0$ defines a polynomial in $(c_1,c_3)$ which is linear in $c_1$. Solving it for $c_1$ gives $$c_1 = \frac{449307}{175}c_3^3 + \frac{741681}{2150}c_3^2
+ \frac{35780649411}{2393160700}c_3 + \frac{34135130165539}{163091166664200}.$$ It remains to show that $p(c_3):=b_1 - b_2^2b_{12}/b_7^2$, which is a polynomial of fourth order in $c_3$, is positive. Choosing $c_3^*=-0.029$, we find that $p(c_3^*)\approx 0.0045>0$. This shows that $$G''(0) \le -q(c_3^*)\int_\Omega u\xi_1^8 dx
= -q(c_3^*)\int_\Omega u(\log u)_x^8 dx \le 0.$$ Finally, if $G''(0)=0$, we infer that $u$ is constant which is excluded. Therefore, $G''(0)<0$, which ends the proof.
Numerical examples {#sec.num}
==================
The aim of this section is to explore the numerical behavior of the second-order derivative of the function $G(\tau)$, defined in the introduction, for the porous-medium equation in one space dimension. The equation is discretized by standard finite differences, and we employ periodic boundary conditions. The discrete solution $u_i^k$ approximates the solution $u(x_i,t^k)$ to with $x_i=i\triangle x$, $t^k=k\tau$, and $\triangle x$, $\tau$ are the space and time step sizes, respectively. We choose the Barenblatt profile $$\label{num.ic}
u^0(x) = t_0^{-1/(\beta+1)}\max\bigg(0,C-\frac{\beta-1}{2\beta(\beta+1)}
\frac{(x-1/2)^2}{t_0^{2/(\beta+1)}}\bigg)^{1/(\beta-1)}, \quad 0\le x\le 1,$$ where $$t_0=0.01, \quad
C = \frac{\beta-1}{2\beta(\beta+1)}\frac{(x_R-1/2)^2}{t_0^{2/(\beta+1)}}, \quad
x_R = \frac14,$$ as the initial datum. Its support is contained in $[\frac12-x_R,\frac12+x_R]$; see Figure \[fig.evol\] (left). We choose the exponent $\beta=2$. The semi-logarithmic plot of the discrete entropy $H_d[u^k]=\sum_{i=0}^N (u_i^k)^\alpha \triangle x$ with $\alpha=5$ versus time is illustrated in Figure \[fig.evol\] (right), using the implicit Euler scheme with parameters $\tau=10^{-4}$ and the number of grid points $N=1/\triangle x=64$. The decay is exponential for “large” times. The nonlinear discrete system is solved by Newton’s method with the tolerance ${\tt tol}=10^{-15}$. We have highlighted four time steps $t_i$ at which we will compute numerically the function $G(\tau)$ for the following Runge-Kutta schemes: $$\begin{aligned}
&\mbox{explicit Euler scheme:} & u^k-u^{k-1} &= -\tau A[u^{k-1}], \\
&\mbox{implicit Euler scheme:} & u^k-u^{k-1} &= -\tau A[u^{k}], \\
&\mbox{second-order trapezoidal rule:} & u^k-u^{k-1}
&= -\frac{\tau}{2}(A[u^k]+A[u^{k-1}]), \\
&\mbox{third-order Simpson rule:} & u^k-u^{k-1}
&= -\frac{\tau}{6}(A[u^k]+4A[(u^k+u^{k-1})/2]+A[u^{k-1}]).\end{aligned}$$
![Left: Evolution of the intial datum for $\beta=2$ at various time steps $t_i$, $i=0,1,2,3$. Right: Semi-logarithmic plot of the discrete entropy $H_d[u^k]$ versus time.[]{data-label="fig.evol"}](GSS_usnap_PME_3.eps "fig:"){width="75mm"} ![Left: Evolution of the intial datum for $\beta=2$ at various time steps $t_i$, $i=0,1,2,3$. Right: Semi-logarithmic plot of the discrete entropy $H_d[u^k]$ versus time.[]{data-label="fig.evol"}](GSS_entropy_PME_3.eps "fig:"){width="75mm"}
We set as before $u:=u^k$, $v(\tau):=u^{k-1}$ and compute $G(\tau)=H_d[u]-H_d[v(\tau)]$ and the discrete second-order derivative $\pa^2 G$ of $G$ (using central differences). The result is presented in Figure \[fig.G\]. As expected, the discrete derivative $\pa^2G$ is negative on a (small) interval for all times $t_i$, $i=1,2,3$. We observe that $\pa^2 G$ is even slightly decreasing, but we expect that it becomes positive for sufficiently large values of $\tau$. Clearly, the values for $\pa^2G$ tend to zero as we approach the steady state (see Remark \[rem.tauk\]). This experiment indicates that $\tau^k$ from Theorem \[thm.ed\] is bounded from below by $\tau^*=3\cdot 10^{-4}$, for instance.
![Numerical evaluation of the discrete version of $G''(\tau)$ for various Runge-Kutta schemes at the time steps $t_i$. Top left: explicit Euler scheme; top right: implicit Euler scheme; bottom left: implicit trapezoidal rule; bottom right: Simpson rule.[]{data-label="fig.G"}](GSS_PME_3.eps){width="140mm"}
In order to understand the behavior of $G(\tau)$ in a better way, it is convenient to study the discrete version of the quotient $$\label{num.Q}
Q(\tau) := \frac{G''(\tau)}{\|u^{\alpha+2\beta-2}u_x^4\|_{L^1}}.$$ Indeed, the analysis in Section \[sec.pme\] gives an estimate of the type $G''(0)\le -C\int_\Omega u^{2\beta+\alpha-5}u_x^4 dx$ for some constant $C>0$. Thus, we expect that for sufficiently small $\tau>0$, $Q(\tau)$ is bounded from above by some negative constant. This expectation is confirmed in Figure \[fig.quot\]. In the examples, $Q(\tau)$ is a decreasing function of $\tau$, and $Q(0)$ is decreasing with increasing time.
![Numerical evaluation of the discrete version of $Q(\tau)$, defined in , for various Runge-Kutta schemes at the time steps $t_i$. Top left: explicit Euler scheme; top right: implicit Euler scheme; bottom left: implicit trapezoidal rule; bottom right: Simpson rule.[]{data-label="fig.quot"}](GSS_rel_PME_3.eps){width="140mm"}
All these results indicate that the threshold parameter $\tau^k$ in Theorem \[thm.ed\] can be chosen independently of the time step $k$.
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[^1]: The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P24304, P27352, and W1245
|
---
author:
- 'B. Ruijl'
- 'T. Ueda'
- 'J.A.M. Vermaseren'
bibliography:
- 'myref.bib'
title: '[<span style="font-variant:small-caps;">Form</span>]{} version 4.2'
---
Introduction {#sec:introduction}
============
We introduce a new version of the symbolic manipulation toolkit [<span style="font-variant:small-caps;">Form</span>]{} [@Vermaseren:2000nd; @Kuipers:2012rf], named [<span style="font-variant:small-caps;">Form</span>]{} 4.2. [<span style="font-variant:small-caps;">Form</span>]{} aims to be a high-performance symbol manipulator, even in cases where there are billions of terms that take up several terabytes of disk space.
This release is an update from [<span style="font-variant:small-caps;">Form</span>]{} 4.1, which was released in 2013. In the latest release we have fixed more than 50 bugs and have introduced more than 20 new features. For an overview of all changes made from [<span style="font-variant:small-caps;">Form</span>]{} 4.1 to [<span style="font-variant:small-caps;">Form</span>]{} 4.2, please see the Changelog [@Changelog]. The driving force of the development of new features and improvements in [<span style="font-variant:small-caps;">Form</span>]{} is the use of the program in actual research projects. Many of the new features in [<span style="font-variant:small-caps;">Form</span>]{} 4.2 were inspired by the use in <span style="font-variant:small-caps;">Forcer</span> [@FORCER] but should be useful in other environments as well. The correctness, efficiency and limitations of the new features have been extensively tested through its physics applications.
The latest version of [<span style="font-variant:small-caps;">Form</span>]{} can be obtained from GitHub [@Repository], where installation instructions can also be found. The installation of [<span style="font-variant:small-caps;">Form</span>]{}, including the threaded version [<span style="font-variant:small-caps;">TForm</span>]{} [@Tentyukov:2007mu], on `x86-64` machines with normal Linux distributions should be rather straightforward.
Whenever new features are introduced, users will resort to creative applications that are beyond the imagination of the designers. This can mean that certain restrictions, that were originally thought to never cause problems, can become an obstacle to such innovative use. Hence, if the user encounters issues, please report them to the GitHub Issue Tracker [@Issues]. One can also search for known issues in the Issue Tracker.
The layout of this work is as follows. In section \[sec:features\] we show some new features and in section \[sec:summary\] we give a summary.
Notable new features {#sec:features}
====================
In this section we present some notable new features of [<span style="font-variant:small-caps;">Form</span>]{} 4.2. For more details of all features available in [<span style="font-variant:small-caps;">Form</span>]{} 4.2, please see the reference manual [@Manual].
Generating all matches
----------------------
The [‘\$=12‘\#=12‘\_=12]{}[Identify]{} statement, often abbreviated as [‘\$=12‘\#=12‘\_=12]{}[id]{}, is one of core statements of [<span style="font-variant:small-caps;">Form</span>]{}. This statement tries to match a given pattern and performs algebraic replacements. [<span style="font-variant:small-caps;">Form</span>]{} version 4.2 introduces a new option [‘\$=12‘\#=12‘\_=12]{}[all]{}, which tries to generate *all* possible matches instead of just the first. For example,
CF v,f,s;
L F = v(1,2,3,4);
id all v(?a,?b) = f(?a)*s(?b);
Print +s;
.end
gives the following result:
F =
+ f*s(1,2,3,4)
+ f(1)*s(2,3,4)
+ f(1,2)*s(3,4)
+ f(1,2,3)*s(4)
+ f(1,2,3,4)*s
;
The [‘\$=12‘\#=12‘\_=12]{}[id all]{} statement can be used in many different scenarios. One is to find all automorphisms (symmetries) of a graph, described by the vertex structure [‘\$=12‘\#=12‘\_=12]{}[vx]{}, which have (undirected) momenta assigned to all edges connected to the vertex as function arguments [@jvLL2016].
[(90,90)(-40,-50)]{}
(0,0)[45]{}[Black]{}[White]{} (-75,0)(-45,0) (75,0)(45,0) (31.819,31.819)(-31.819,-31.819) (0,0)[5]{}[White]{}[White]{} (-31.819,31.819)(31.819,-31.819) (-60,10) [$Q_1$]{} (60,10) [$Q_2$]{} (-50,25) [$p_1$]{} (50,25) [$p_3$]{} (0,55) [$p_2$]{} (0,-55) [$p_5$]{} (-10,25) [$p_7$]{} (10,25) [$p_8$]{} (-50,-20)[$p_6$]{} (50,-20)[$p_4$]{}
In figure \[fig:no\] we show a graph for which we find all eight automorphisms using the following code:
V Q,Q1,Q2,p1,...,p8;
CF map,vx(s); * note that the vertex structure vx is symmetric
L F = vx(Q1,p1,p6)*vx(p1,p2,p7)*vx(p2,p3,p8)*
vx(p3,p4,Q2)*vx(p4,p5,p7)*vx(p5,p6,p8);
id all vx(Q1?,p1?,p6?)*vx(p1?,p2?,p7?)*vx(p2?,p3?,p8?)*
vx(p3?,p4?,Q2?)*vx(p4?,p5?,p7?)*vx(p5?,p6?,p8?) =
map(Q1,Q2,p1,p2,p3,p4,p5,p6,p7,p8);
Print +s;
.end
which yields
F =
+ map(Q1,Q2,p1,p2,p3,p4,p5,p6,p7,p8)
+ map(Q1,Q2,p1,p7,p4,p3,p8,p6,p2,p5)
+ map(Q1,Q2,p6,p5,p4,p3,p2,p1,p8,p7)
+ map(Q1,Q2,p6,p8,p3,p4,p7,p1,p5,p2)
+ map(Q2,Q1,p3,p2,p1,p6,p5,p4,p8,p7)
+ map(Q2,Q1,p3,p8,p6,p1,p7,p4,p2,p5)
+ map(Q2,Q1,p4,p5,p6,p1,p2,p3,p7,p8)
+ map(Q2,Q1,p4,p7,p1,p6,p8,p3,p5,p2)
;
The `map` function provides a mapping from the old edge labeling to a new one, which leaves the graph invariant.
Output optimization
-------------------
Output optimization of polynomials was introduced in [<span style="font-variant:small-caps;">Form</span>]{} 4.1 [@Kuipers:2013pba]. It relies on the idea that the number of arithmetic operations required to evaluate a polynomial can be reduced by pulling variables outside of brackets (Horner’s rule) and by removing common subexpressions. Finding the best order in which to extract variables (a Horner scheme) is difficult and [<span style="font-variant:small-caps;">Form</span>]{} has different algorithms for it, namely [‘\$=12‘\#=12‘\_=12]{}[Format O1]{} to [‘\$=12‘\#=12‘\_=12]{}[Format O4]{}.
In [<span style="font-variant:small-caps;">Form</span>]{} 4.2 the performance of the output optimization has been increased, by improving the common subexpression detection. The mode that gives the best results in [<span style="font-variant:small-caps;">Form</span>]{} 4.1, [‘\$=12‘\#=12‘\_=12]{}[Format O3]{}, has been improved to be less dependent on a user-given exploration-exploitation constant, based on the work of [@Ruijl:2013epa] and [@Ruijl:2014hha].
Furthermore, a new algorithm has been added, based on local search methods [@Ruijl:2014spa]. This option is called [‘\$=12‘\#=12‘\_=12]{}[Format O4]{} and it generally produces better results and is faster than the Monte Carlo Tree Search method used in [‘\$=12‘\#=12‘\_=12]{}[Format O3]{}. An example is presented below:
S a,b,c,d,e,f,g,h,i,j,k,l,m,n;
L G = (4*a^4+b+c+d + i^4 + g*n^3)^10 +
(a*h + e + f*i*j + g + h)^8 + (i + j + k + l + m + n)^12;
Format O4,saIter=300; * use 300 iterations for optimization
.sort
#optimize G
#write "Optimized with Horner scheme: `optimscheme_'"
#write "Number of operations in output: `optimvalue_'"
#clearoptimize
.end
When running with [‘\$=12‘\#=12‘\_=12]{}[tform]{}, every thread runs its own search and the best value is selected. This gives better results than using fewer cores with a higher number of optimization iterations, [‘\$=12‘\#=12‘\_=12]{}[saIter]{}.
Automatic series expansion of rational functions
------------------------------------------------
The [‘\$=12‘\#=12‘\_=12]{}[Polyratfun]{}, which was introduced in [<span style="font-variant:small-caps;">Form</span>]{} 4.0 to treat multivariate rational functions as coefficients of terms, now supports automatic expansion in one variable. In the example below the rational function represented by the function [‘\$=12‘\#=12‘\_=12]{}[rat]{} is expanded with respect to the variable [‘\$=12‘\#=12‘\_=12]{}[ep]{} up to [‘\$=12‘\#=12‘\_=12]{}[ep\^5]{}:
S ep;
CF rat;
Polyratfun rat;
L F = rat(ep + 1,ep^2 + 3*ep + 1);
.sort
Polyratfun rat(expand,ep,5);
Print;
.end
The code yields:
F =
rat(1 - 2*ep + 5*ep^2 - 13*ep^3 + 34*ep^4 - 89*ep^5);
All manipulations performed in the expansion mode will truncate the series quickly. As a result, the expansion mode may be faster than the unexpanded mode if the coefficients become complicated. Additionally, it can avoid the intrinsic [‘\$=12‘\#=12‘\_=12]{}[MaxTermSize]{} restriction of [<span style="font-variant:small-caps;">Form</span>]{}.
Textual manipulation on the output by dictionaries
--------------------------------------------------
Occasionally output from computer algebra systems provides input for another program, which may require suitable translation. In [<span style="font-variant:small-caps;">Form</span>]{}, the [‘\$=12‘\#=12‘\_=12]{}[Format]{} statement can be used for preparing the output in a format of another program, for example, [‘\$=12‘\#=12‘\_=12]{}[Fortran]{}, [‘\$=12‘\#=12‘\_=12]{}[C]{} and so on. [<span style="font-variant:small-caps;">Form</span>]{} version 4.2 gives another way of controlling the output to a certain extent: *dictionaries*. A dictionary is a collection of pairs of a *word* and its translation. The word can be a variable, a number, a function with specific arguments or a special character like the multiplication sign ([‘\$=12‘\#=12‘\_=12]{}[\*]{}). The translation can be any string enclosed in double quotes. For example,
S x1,x2,x3;
L F = (x1+x2+x3)^2;
.sort
#opendictionary texdict
#add x1: "x_1"
#add x2: "x_2"
#add x3: "x_3"
#add *: " "
#closedictionary
#usedictionary texdict
#write "%E",F
.end
prints the expression [‘\$=12‘\#=12‘\_=12]{}[F]{} for the LaTeX math mode:
x_3^2 + 2 x_2 x_3 + x_2^2 + 2 x_1 x_3 + 2 x_1 x_2 + x_1^2
Here [‘\$=12‘\#=12‘\_=12]{}[\#opendictionary texdict]{} $\dots$ [‘\$=12‘\#=12‘\_=12]{}[\#closedictionary]{} defines a dictionary with the name [‘\$=12‘\#=12‘\_=12]{}[texdict]{}. Each [‘\$=12‘\#=12‘\_=12]{}[\#add]{} instruction defines a pair of a word and its translation. The [‘\$=12‘\#=12‘\_=12]{}[\#usedictionary]{} instruction sets the predefined dictionary for the textual manipulation on the output. Then, each word in the dictionary is replaced with its translation in the [‘\$=12‘\#=12‘\_=12]{}[\#write]{} instruction.
Spectators
----------
Sometimes a large number of terms is passed through one or more modules for which the user knows that they will remain unmodified. Processing these irrelevant terms will cause overhead, including a potentially heavy sort cost.
The Spectator system can be used to remove terms from the current expression and to store them in a buffer, called a Spectator file. These terms will be ignored until they are copied back into an active expression. In essence, the Spectator system works as a filter. Below is an example:
S x;
CF f,g;
CreateSpectator FSpec "Fspec.spec";
L F = f(1);
.sort
#do i=1,10000
id f(x?) = theta_(10000-x)*f(x + 1) + g(x); * term blow-up
if (count(g,1)) ToSpectator FSpec; * filter the g terms
.sort:round`i';
#enddo
CopySpectator F1 = FSpec;
.sort
RemoveSpectator FSpec;
.end
The above code takes about 0.25 seconds to run. If the line with [‘\$=12‘\#=12‘\_=12]{}[ToSpectator]{} is commented out, it takes about 13 seconds to run.
Zero-dimensional sparse tables as pure functions
------------------------------------------------
High-level programming languages are mainly classified into two groups: *imperative* languages and *declarative* languages. Modern programming languages, however, tend to have aspects of both the groups and evolve towards multi-paradigm languages, which makes some programming tasks easier or more intuitive. [<span style="font-variant:small-caps;">Form</span>]{}, which is clearly imperative, is not an exception. The users are now allowed to declare tables without any table indices. Such zero-dimensional (sparse) tables can have function arguments with wildcarding, which effectively leads to user-defined *pure functions* and opens a way of functional programming up to some extent. For example,
S n;
Table fac(n?pos0_);
Fill fac = delta_(n) + theta_(n) * n * fac(n-1);
L F = fac(5);
Print;
.end
implements a function [‘\$=12‘\#=12‘\_=12]{}[fac]{} as $${\bgroup\catcode`\$=12\catcode`\#=12\catcode`\_=12\relax
{\textbf{\ttfamily }}\egroup}{fac}(n) =
\begin{cases}
1 & n = 0, \\
n \times {\bgroup\catcode`\$=12\catcode`\#=12\catcode`\_=12\relax
{\textbf{\ttfamily }}\egroup}{fac}(n-1) & n \ge 1 .
\end{cases}$$ for a non-negative integer $n$, and yields
F =
120;
Of course in this case one can use the more efficient built-in [‘\$=12‘\#=12‘\_=12]{}[fac\_]{} function.[^1]
A more complicated example is given below:
S n,n1,n2;
Table fib(n?int_);
Table fibimpl(n?int_,n1?,n2?);
Fill fib = theta_(-1-n) * sign_(n+1) * fib(-n)
+ theta_(n-1) * fibimpl(n-2,1,1);
Fill fibimpl = theta_(-n) * n2
+ thetap_(n) * fibimpl(n-1,n2,n1+n2);
L F = fib(100);
Print;
.end
This computes the 100th Fibonacci number:
F =
354224848179261915075;
Expression database using
--------------------------
[<span style="font-variant:small-caps;">Form</span>]{} has tables that map integers (keys) to expressions (values). An arbitrary expression cannot be a key of a table in a direct way. However, the new command [‘\$=12‘\#=12‘\_=12]{}[ArgToExtraSymbol]{} with the [‘\$=12‘\#=12‘\_=12]{}[tonumber]{} option gives a way to map an arbitrary expression into an integer, leading to the use of an expression as a key to a table.
Below we give an example where we effectively build a table storing $$\begin{aligned}
{\bgroup\catcode`\$=12\catcode`\#=12\catcode`\_=12\relax
{\textbf{\ttfamily }}\egroup}{g(1)*g(2)} &\Longrightarrow {\bgroup\catcode`\$=12\catcode`\#=12\catcode`\_=12\relax
{\textbf{\ttfamily }}\egroup}{100} , \\
{\bgroup\catcode`\$=12\catcode`\#=12\catcode`\_=12\relax
{\textbf{\ttfamily }}\egroup}{g(1)*g(2)*g(3)} &\Longrightarrow {\bgroup\catcode`\$=12\catcode`\#=12\catcode`\_=12\relax
{\textbf{\ttfamily }}\egroup}{200} ,\end{aligned}$$ by using the following code:
S n;
CF f,g;
L F = f(g(1)*g(2))*100 + f(g(1)*g(2)*g(3))*200;
argtoextrasymbol tonumber,f;
B f;
Print;
.sort
CTable sparse,values(1);
FillExpression values = F(f); * store values in table
Drop F;
.sort
L G = f(g(1)*g(2)) + f(g(1)*g(2)*g(3)) + f(g(1)*g(2)*g(3)*g(4));
argtoextrasymbol tonumber,f;
id f(n?) = f(extrasymbol_(n))*values(n); * read values in table
Print +s;
.end
The first [‘\$=12‘\#=12‘\_=12]{}[Print]{} statement gives:
F =
+ f(1) * ( 100 )
+ f(2) * ( 200 );
and the final output is
G =
+ 100*f(g(1)*g(2))
+ 200*f(g(1)*g(2)*g(3))
+ f(g(1)*g(2)*g(3)*g(4))*values(3)
;
Partition function
------------------
We have added several new convenient statements and functions to [<span style="font-variant:small-caps;">Form</span>]{} 4.2, such as [‘\$=12‘\#=12‘\_=12]{}[Transform dedup/addargs/mulargs]{}, [‘\$=12‘\#=12‘\_=12]{}[id\_]{}, [‘\$=12‘\#=12‘\_=12]{}[perm\_]{} and the [‘\$=12‘\#=12‘\_=12]{}[occurs]{} condition in the [‘\$=12‘\#=12‘\_=12]{}[If]{} statement. Here we showcase the partitions function, which generates all partitions of a list of arguments into $n$ parts. Each part consists of a function name and a size. This function exploits symmetries of the arguments to make sure that no partition is generated twice. Instead, a combinatorial prefactor is computed.
For example, to partition eight elements into three partitions, where the first part is a function [‘\$=12‘\#=12‘\_=12]{}[f1]{} with three arguments, the second part a function [‘\$=12‘\#=12‘\_=12]{}[f1]{} with two arguments, and the third part a function [‘\$=12‘\#=12‘\_=12]{}[f2]{} with three arguments, we write:
S x1,x2,x3;
CF f1,f2;
L F = partitions_(3,f1,3,f1,2,f2,3,x1,x1,x1,x1,x2,x2,x2,x3);
Print +s;
.end
F =
+ 18*f1(x1,x1)*f1(x1,x1,x2)*f2(x2,x2,x3)
+ 6*f1(x1,x1)*f1(x1,x1,x3)*f2(x2,x2,x2)
+ 36*f1(x1,x1)*f1(x1,x2,x2)*f2(x1,x2,x3)
+ 36*f1(x1,x1)*f1(x1,x2,x3)*f2(x1,x2,x2)
+ 6*f1(x1,x1)*f1(x2,x2,x2)*f2(x1,x1,x3)
+ 18*f1(x1,x1)*f1(x2,x2,x3)*f2(x1,x1,x2)
+ 12*f1(x1,x1,x1)*f1(x1,x2)*f2(x2,x2,x3)
+ 4*f1(x1,x1,x1)*f1(x1,x3)*f2(x2,x2,x2)
+ 12*f1(x1,x1,x1)*f1(x2,x2)*f2(x1,x2,x3)
+ 12*f1(x1,x1,x1)*f1(x2,x3)*f2(x1,x2,x2)
+ 72*f1(x1,x1,x2)*f1(x1,x2)*f2(x1,x2,x3)
+ 36*f1(x1,x1,x2)*f1(x1,x3)*f2(x1,x2,x2)
+ 18*f1(x1,x1,x2)*f1(x2,x2)*f2(x1,x1,x3)
+ 36*f1(x1,x1,x2)*f1(x2,x3)*f2(x1,x1,x2)
+ 36*f1(x1,x1,x3)*f1(x1,x2)*f2(x1,x2,x2)
+ 18*f1(x1,x1,x3)*f1(x2,x2)*f2(x1,x1,x2)
+ 36*f1(x1,x2)*f1(x1,x2,x2)*f2(x1,x1,x3)
+ 72*f1(x1,x2)*f1(x1,x2,x3)*f2(x1,x1,x2)
+ 12*f1(x1,x2)*f1(x2,x2,x3)*f2(x1,x1,x1)
+ 36*f1(x1,x2,x2)*f1(x1,x3)*f2(x1,x1,x2)
+ 12*f1(x1,x2,x2)*f1(x2,x3)*f2(x1,x1,x1)
+ 12*f1(x1,x2,x3)*f1(x2,x2)*f2(x1,x1,x1)
+ 4*f1(x1,x3)*f1(x2,x2,x2)*f2(x1,x1,x1)
;
All options for [‘\$=12‘\#=12‘\_=12]{}[partitions\_]{} can be found in the manual [@Manual].
Summary {#sec:summary}
=======
We have shown some new features of [<span style="font-variant:small-caps;">Form</span>]{} 4.2. With these features we hope to make the usage of [<span style="font-variant:small-caps;">Form</span>]{} easier and to have improved performance. This version of [<span style="font-variant:small-caps;">Form</span>]{} is the version used by the <span style="font-variant:small-caps;">Forcer</span> [@FORCER] package and its supporting software.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank everyone who has given feedback and reported bugs, in particular Joshua Davies. This work is supported by the ERC Advanced Grant no. 320651, “HEPGAME”.
[^1]: In addition to the performance issue, the user-defined function suffers from a restriction with the current implementation of [<span style="font-variant:small-caps;">Form</span>]{}: the recursion depth is limited by the stack size.
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