Localization transition in the Discrete Non-Linear Schrödinger Equation: ensembles inequivalence and negative temperatures
Giacomo Gradenigo, Stefano Iubini, Roberto Livi, Satya N. Majumdar
LLocalization in the Discrete Non-Linear Schr¨odinger Equation:mechanism of a First-Order Transition in the Microcanonical Ensemble
Giacomo Gradenigo
Dipartimento di Fisica, “Sapienza” Universit`a di Roma, Piazzale A. Moro 2, I-00185 Roma, ItalyNANOTEC-CNR, Roma, Piazzale A. Moro 2, I-00185 Roma, Italy
Stefano Iubini
Dipartimento di Fisica e Astronomia, Universit`a di Padova, Via Marzolo 8, I-35131 Padova, ItalyISC-CNR, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
Roberto Livi
Dipartimento di Fisica e Astronomia and CSDC, Universit`a degli Studi di Firenze,via G. Sansone 1, I-50019 Sesto Fiorentino, ItalyINFN Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
Satya N. Majumdar
Laboratoire de Physique Th´eorique et Mod`eles Statistiques, LPTMS, CNRS,Universit´e Paris-Sud, Universit´e Paris-Saclay, F-91405 Orsay, France
When constrained to high energies, the wave function of the non-linear Schr¨odinger equationon a lattice becomes localized. We demonstrate here that localization occours as a first-ordertransition in the microcanonical ensemble, in a region of phase space where the canonical and themicrocanonical ensembles are not equivalent. Quite remarkably, the transition to the localized non-ergodic phase takes place in a region where the microcanonical entropy, S N , grows sub-extensivelywith the system size N , namely S N ∼ N / . This peculiar feature signals a shrinking of the phasespace and exemplifies quite a typical scenario of ”broken ergodicity” phases. In particular, thelocalized phase is characterized by a condensation phenomenon, where a finite fraction of the totalenergy is localized on a few lattice sites. The detailed study of S N ( E ) close to the transition value ofthe energy, E c , reveals that its first-order derivative, ∂S N ( E ) ∂E | E = E c , exhibits a jump, thus naturallyaccounting for a first–order phase transition from a thermalized phase to a localized one, whichis properly assessed by studying the participation ratio of the site energy as an order parameter.Moreover, we show that S N predicts the existence of genuine negative temperatures in the localizedphase. In a physical perspective this indicates that by adding energy to the system it condensatesinto increasingly ordered states. a r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov CONTENTS
I. Introduction 2II. The Non-Linear Discretized Schr¨odinger equation: generalities 4III. The microcanonical partition function 5A. From statistical mechanics to large deviations 5B. The peculiar analytic properties of the partition function 7C.
Matching regime: the non-analytic contribution 9IV. The first-order phase transition from a thermalized phase to localization 11A. The microcanonical entropy in the matching regime 11B. The order parameter: participation ratio 13C. Negative temperatures 15D. Finite-size effects and simulations: the pseudo-condensate phase 16V. Conclusions and perspectives 17VI. Acknowledgements 17VII. Appendices 18A. Derivation of the rate function χ ( ζ ) in the intermediate matching regime 18B. Asymptotic behavior of χ ( ζ ) 19C. Explicit expression of χ ( ζ ) 20D. The critical value ζ c I. INTRODUCTION
Beyond its phenomenological interest for many physical applications, such as Bose-Einstein condensates in opticallattices [1, 2] or light propagating in arrays of optical waveguides [4], in the last decades the Discrete NonlinearSchr¨odinger Equation (DNLSE) has revealed an extremely fruitful testbed for many basic aspects concerning statisticsand dynamics in Hamiltonian models, equipped with additional conserved quantities, other than energy (e.g., see[5] and [6]). In particular, since the pioneering paper by Rasmussen et al. [7], the existence of a high-energy phasecharacterized by the condensation of energy in the form of breathers, i.e. localized nonlinear excitations, has attractedthe attention of many scholars. The phase diagram shown in Fig. 1 summarizes the scenario described in [7]. Froma dynamical point of view, extended numerical investigations have pointed out that in such a phase the Hamiltonianevolution of an isolated chain exhibits long-living multi-breather states, that last over astronomical times [2, 3].Moreover, time averages of suitable non-local quantities entering the definition of temperature T , as, e.g., the inverseof the derivative of the microcanonical entropy S with respect to the energy E (in formulae β = 1 /T = ∂S∂E ) [8],predict that T is negative in this high-energy phase [9]. Relying upon thermodynamic considerations Rumpf andNewell [10] and later Rumpf [11–14] argued that these dynamical states should eventually collapse onto an ”equilibriumstate”, characterized by an extensive background at infinite temperature with a superimposed localized breather,containing the excess of energy initially stored in the system. This argument seems plausible in the light of a grand-canonical description, where entropy has to be maximal at thermodynamic equilibrium, in the presence of fluctuationsdue to heat exchanges with a thermal reservoir. Accordingly, in this perspective long-living multi-breather stateswere interpreted as metastable ones, thus yielding the conclusion that negative temperature states are not genuineequilibrium ones. Due to the practical difficulty of observing the eventual collapse to the equilibrium state predictedby Rumpf even in lattices made of a few tens of sites, some authors have substituted the Hamiltonian dynamics witha stochastic evolution rule, that conserves energy and particle densities. In this framework the breather condensationphenomenon onto a single giant breather emerges as a coarsening process ruled by predictable scaling properties[15, 16].Despite many of the dynamical aspects of the deterministic and stochastic evolutions of the DNLSE have been sat-isfactorily described and understood, still the thermodynamic interpretation of this model remains unclear, because itseems to depend on the choice of the adopted statistical ensemble. In fact, already in [7] the authors point out that anapproach based on the canonical ensemble yields a negative temperature in the high-energy phase, which contradictsthe very existence of a Gibbsian measure. Moreover, they claim that only by introducing a suitable grand-canonicalrepresentation, at the price of transforming the original short-range Hamiltonian model into a long-range one, allowsfor a consistent definition of negative temperatures. Also Rumpf makes use of a grand-canonical ensemble to tacklethis question [13, 14] and reaches the opposite conclusion that negative temperature states are not compatible withthermodynamic equilibrium conditions. More recently, the statistical mechanics of the disordered DNLSE Hamilto-nian has been analyzed making use of the grand-canonical formalism [17]: the authors conclude that for weak disorderthe phase diagram looks like the one of the non-disordered model, while correctly pointing out that their results applyto the microcanonical case, whenever the equivalence between ensembles can be established. In a more recent paperthe thermodynamics of the DNLSE Hamiltonian and of its quantum counterpart, the Bose-Hubbard model, has beenanalyzed in the canonical ensemble [21]: the authors even claim that ”... the Gibbs canonical ensemble is, concep-tually, the most convenient one to study this problem” and conclude that the high-energy phase is characterizedby the presence of non-Gibbs states, that cannot be converted into standard Gibbs states by introducing negativetemperatures. The main consideration emerging from the overall scenario is that, while in the low-energy phase thethermodynamics of the DNLSE model exhibits standard properties that are consistent with any statistical ensemblerepresentation, in the high-energy phase the very equivalence between statistical ensembles is at least questionableand it cannot be ruled out as a matter of taste.In this paper we present a clear scenario about the statistical mechanics of the DNLSE problem by performingexplicit analytic calculations for the microcanonical ensemble in the high energy limit and, in particular, close tothe β = 0 line (see Fig. 1). We want to point out that the thermodynamics of this model is well-defined over thewhole phase diagram only in the microcanonical ensemble. In particular, we show explicitly why and how the non-equivalence between statistical ensembles naturally emerges as a manifest consequence of the non-analytic structureof the microcanonical partition function in the high-energy phase. In fact, the large-deviation approach allows usto obtain an explicit analytic expression of the normalized distribution function associated to the microcanonicalmeasure. Moreover, we obtain that the β = 0 line corresponds to a first-order phase transition boundary betweena standard extensive phase at low energies and a localized (or condensed) phase at high energies, while includingexplicit predictions about the scaling of the partition function and finite size corrections. We obtain also the ex-plicit form of the microcanonical entropy and we analyze the transition by computing the participation ratio ofthe energy per lattice site as the appropriate order-parameter of the first-order phase transition. Finally, we showthat, in the natural microcanonical representation, temperatures in the localized phase are truly negative: this isjust a consequence of the entropy calculation in the microcanonical ensemble. On the other hand, due to ensemblein-equivalence, this has no counterpart in the canonical and grand-canonical ensembles: any attempt to plug byhand T < β domainof the canonical partition function. Then in Sec. IV we explain how the non-analytic contribution to the partitionfunction gives rise to a discontinuity of the first derivative of the microcanonical entropy and we also compute theorder parameter of the transition, the participation ratio. Moreover, we explain how negative temperatures arisenaturally in this context and finally, in Sec. V, we turn to conclusions. The technical details of the calculations areleft for the appendices. a e β = β = + ∞ FIG. 1. Equilibrium phase diagram in the plane ( a = A/N, e = E/N ) for the DNLSE as obtained in [7]. The infinitetemperature line, β = 0, which corresponds to the parabola e = 2 a , is the border of the region where ensembles are equivalent,and represents also the transition line for the localization phase transition in the N → ∞ limit. The ground state ( β = ∞ ) isidentified by the curve e = a − a . The grey area below this curve is inaccessible. II. THE NON-LINEAR DISCRETIZED SCHR ¨ODINGER EQUATION: GENERALITIES
In this paper we study the one-dimensional DNLSE, namely a scalar complex field z j on a one dimensional latticewith periodic boundary conditions made of N sites, whose Hamiltonian reads: H = N (cid:88) j =1 ( z ∗ j z j +1 + z j z ∗ j +1 ) + N (cid:88) j =1 | z j | . (1)The corresponding Hamiltonian dynamics is expressed by the equations of motion i ˙ z j = − ∂ H ∂z ∗ j = − ( z j +1 + z j − ) − | z j | z j . (2)For what follows, it is crucial to point out that these equations of motion conserve not only the total energy H ,but, due to the quantum origin of the problem, also the squared norm of the “wave-function”, that is A = N (cid:88) j =1 | z j | . (3)This functional can be read also as the total number of particles contained in the model. We attribute to H and A the real values E and A , respectively. Due to the presence of these two conserved quantities the deterministicdynamics of this model exhibits quite peculiar features. The overall scenario is summarized in the phase diagramshown in Fig. 1, where e = E/N and a = A/N : there is a region contained in between the zero temperature line( β = 1 /T = + ∞ ) and the infinite temperature one ( β = 0), where any initial condition evolves to a standardthermodynamic equilibrium state, while above the ( β = 0)-line the dynamics of any finite lattice is characterized bythe birth and death of long-living localized nonlinear excitations, called breathers (e.g. see [1, 2]). When a stochasticversion of the dynamics, still conserving both E and A , has been considered [15, 16], the evolution in the regionabove the ( β = 0)-line has been found to amount to a coarsening process eventually yielding a state where a giantbreather confined in a finite lattice region collects a finite fraction of the total energy and norm. In particular, thecondensate “mass” has been found to increase in time t as t / . One cannot exclude that also the deterministicdynamics in the high energy localized phase might eventually evolve to this single-breather state, although it is amatter of fact that it has never been observed, even in relatively small lattices. Finding an interpretation of thisscenario is related to the understanding of the thermodynamics of the DNLSE model. In fact, various attemptshas been made to study its equilibrium properties in the presence of a localized wave-function [7, 17]. The reasonwhy a study of the thermodynamics deep in the localized phase has been (to some extent) so far elusive is thatthe ( β = 0)-line, where condensation takes place, corresponds also to the breakdown of the equivalence betweenstatistical ensembles, as we are going to show. It is worth at this point to quote a serie of recent papers, focusedon the regression dynamics of large anomalous fluctuations, which also had in the background the idea that isolatedanomalous fluctuations must be somehow related to ensemble inequivalence [18–20]. Nevertheless, the first demon-stration that the physics of a localized phase, which can be indeed seen as an anomalous density fluctuation, findsa fully consistent description only in the microcanonical ensemble, it is a fully original contribution of the present work.The main goal of this paper amounts to compute the microcanonical entropy of the DNLSE Hamiltonian [Eq. (1)]for fixed values of E and A close to the ( β = 0)-line, shown in Fig. 1. In general, the microcanonical entropy is definedas S N ( A, E ) = log Ω ON ( A, E ) , (4)where the Boltzmann constant k B is set to unit and Ω ON ( A, E ) is the microcanonical partition function:Ω ON ( A, E ) = (cid:90) N (cid:89) j =1 dµ ( z j ) δ A − N (cid:88) j =1 | z j | δ E − N (cid:88) j =1 ( z ∗ j z j +1 + z i z ∗ j +1 ) + N (cid:88) i =1 | z j | . (5)Here dµ ( z j ) is a shorthand notation for d [ (cid:60) ( z j )] d [ (cid:61) ( z j )]. Since the main interesting features of the DNLSE modeloccur in the vicinity of the ( β = 0)-line, we can assume that in such conditions the contribution to the total energy E of the bi-linear hopping term in Eq. (1) can be neglected with respect to the local nonlinear term. Let us fix forinstance the specific energy to the value e = E/N , and chose a rescaling of the variables z i as to have the local energyalways of order O (1), i.e., we use ˆ z i = z i /e / . In terms of the new variables the Hamiltonian reads: H = √ e N (cid:88) j =1 (ˆ z ∗ j ˆ z j +1 + ˆ z i ˆ z ∗ j +1 ) + e N (cid:88) i =1 | ˆ z j | . (6)Clearly, in the limit of large e the hopping term is subleading with respect to the local quartic self-interaction andcan be neglected, allowing us to write the microcanonical partition function as follows:Ω N ( A, E ) = (cid:90) N (cid:89) i =1 dµ ( z j ) δ A − N (cid:88) j =1 | z j | δ E − N (cid:88) j =1 | z j | . (7)This approximation is not only proper, but even exact in the large- N limit, as shown by the fact that the veryequation e = 2 a identifying the ( β = 0)-line in Fig.1 (i.e., the boundary of the region for ensemble equivalence) isobtained either by retaining the hopping term in the transfer matrix calculation of [7] or by dropping it, within thesaddle-point approximation adopted here in Sec.IV A to estimate the integral in Eq. (51).We note that N -fold integrals with two global constraints, as in Eq. (7), appeared also in different contexts. Forinstance, in Refs. [26–28], such a kind of integrals describe a generalization of mass conservation models, previouslystudied in [23–25]. III. THE MICROCANONICAL PARTITION FUNCTION
In this Section we report explicit calculations of the approximated microcanonical partition function Ω N ( A, E ) byexploiting a large-deviation technique.
A. From statistical mechanics to large deviations
In order to compute Ω N ( A, E ) it is convenient to express the variables in their polar form z j = ρ j e iφ j , thus makingstraightforward the integration over the angular coordinates φ i :Ω N ( A, E ) = (2 π ) N (cid:90) ∞ N (cid:89) j =1 dρ j ρ j δ A − N (cid:88) j =1 ρ j δ E − N (cid:88) j =1 ρ j . (8)The effective strategy for proceeding in this analytic calculation amounts to releasing the conservation constraint on A by means of a Laplace transform:˜Ω N ( λ, E ) = (cid:90) ∞ dA e − λA Ω N ( A, E ) = (2 π ) N (cid:90) ∞ N (cid:89) j =1 dρ j ρ j e − λ (cid:80) Nj =1 ρ j δ E − N (cid:88) j =1 ρ j . (9)From Eq. (9) the partition function at fixed A can be then obtained by a simple inversion of the Laplace transform:Ω N ( A, E ) = 12 πi (cid:90) λ + i ∞ λ − i ∞ dλ e λA ˜Ω N ( λ, E ) , (10)where integration goes over a Bromwich contour in the complex λ plane. Let us further elaborate Eq. (9) by introducingthe change of variables ρ j = ε j , thus yielding the new expression˜Ω N ( λ, E ) = (cid:16) π (cid:17) N (cid:90) ∞ N (cid:89) j =1 dε j √ ε j e − λ (cid:80) Nj =1 √ ε j δ E − N (cid:88) j =1 ε j . (11)We can introduce the normalized probability distribution f λ ( ε ) = Θ( ε ) λ √ ε exp (cid:0) − λ √ ε (cid:1) , (12)where Θ( ε ) is the Heavyside distribution, and rewrite˜Ω N ( λ, E ) = (cid:16) πλ (cid:17) N Z N ( λ, E ) (13)with Z N ( λ, E ) = (cid:90) ∞ N (cid:89) j =1 dε j f λ ( ε j ) δ E − N (cid:88) j =1 ε j , (14)Written in this form, the calculation of Z N ( λ, E ) amounts to computing the probability distribution of the sum of N i.i.d. random variables ε j , individually described by f λ ( ε j ) and constrained to obey the condition E = (cid:80) Nj =1 ε j . Itis well known from the theory of large deviations [25] that this kind of global constraint on the sum of i.i.d. randomvariables yields a condensation phenomenon, when the individual probability distribution is fat-tailed , i.e. it fulfillsthe bounds exp( − ε ) < f λ ( ε ) < ε . (15)According to Eq. (12), this is precisely the case of the partition function Z N ( λ, E ). In particular, the condensationphenomenon occurs when the total energy E overtakes a threshold value E th , that is equal to the average total energy,in formulae E > E th = N (cid:104) ε (cid:105) , (16)where the average (cid:104) (cid:105) is over the probability distribution f λ ( ε ) [25]. In the localized phase it is more convenient forthe system to condensate the excess energy ∆ E = E − E th (i.e., a macroscopic portion of the total energy) ontoa finite region of the lattice. As clearly explained first in [23, 24], localization simply amounts to the fact that forsufficiently large values of E the behaviour of Z N ( λ, E ) is dominated by the probability distribution f λ ( ε i ) of thesingle variable. The large-deviation procedure yielding these results is going to be sketched in the following subsection.We observe that the integral in Eq. (14), i.e. the sum of N i.i.d. random variables, each one obeying a stretchedexponential distribution, emerged quite recently in a completely different context. While the existence of a conden-sation transition with such a stretched exponential distribution was already known before [25–27], the precise largedeviation form of the distribution of the sum was studied only recently by the two of us in [29], by analyzing the N -steps cumulative position distribution for a run-and-tumble particle in one dimension. Moreover, in Ref. [29], an • β β Γ (+) Γ ( − ) Re( β )Im( β ) • • FIG. 2. Analyticity structure of the function z ( β, λ ) [see Eqns. (19),(20) text] in the complex β plane, λ is fixed to a realpositive number. Wiggled line: branch cut on the negative semiaxis. Dashed (blue) line: Bromwich contour for the calculationof the partition function Z ( λ, E ) when E < E th ( A ), with β indicating the location of the saddle-point. Continuous (red) line:Bromwich contour to compute Z ( λ, E ) when E > E th ( A ), β indicates the new saddle point. The values of energy E th ( A )where the equivalence between ensembles breaks down depend on the value of the wavefunction normalization A . Γ (+) andΓ ( − ) are labels for contour pieces in the positive and negative imaginary semiplanes. interesting first-order transition was identified in the large deviation function describing the tails of such a distribution.The first derivative of the large deviation function is discontinuous at the point where the condensate (i.e., a long runof the order of the total number of steps) forms in the system [29]. In fact, we will see that the analysis of the DNLSmodel in this paper proceeds along the same lines of Ref. [29], although the DNLSE has a very different origin fromthe run-and-tumble particle problem.We conclude this subsection by reporting the values of first two momenta and of the variance of the probabilitydistribution in Eq. (12): (cid:104) ε (cid:105) = 2 λ (cid:104) ε (cid:105) = 24 λ σ = (cid:104) ε (cid:105) − (cid:104) ε (cid:105) = 20 λ (17)It is then worth anticipating the result of the saddle-point calculation in Eq. (51), from which we obtain (cid:104) ε (cid:105) = 2 a .This, in turn, allows us to point out that the condition E = E th in the ( a, e )-plane of Fig.1 coincides with the equationidentifying the ( β = 0)-line, i.e. E = E th ←→ e = (cid:104) ε (cid:105) = 2 a (18) B. The peculiar analytic properties of the partition function
In order to proceed in the calculation of the partition function Z N ( λ, E ) defined in Eq. (14), we can first performits Laplace transform with respect to E , introducing then its conjugate variable β , which is in general a complexnumber, i.e. β ∈ C :˜ Z N ( λ, β ) = (cid:90) ∞ dE e − βE Z N ( λ, E ) = (cid:90) ∞ dε . . . dε N e − β (cid:80) Nj =1 ε j N (cid:89) j =1 f λ ( ε j )= (cid:20)(cid:90) ∞ dε e − βε f λ ( ε ) (cid:21) N = exp { N log[ z ( λ, β )] } , (19)where z ( λ, β ) = (cid:90) ∞ dε e − βε f λ ( ε ) = √ π λ √ β exp (cid:18) λ β (cid:19) Erfc (cid:18) λ √ β (cid:19) , (20)and Erfc (cid:18) λ √ β (cid:19) = 2 √ π (cid:90) ∞ λ/ (2 √ β ) e − t dt, (21)is the complementary error function defined in the complex β plane, with a branch-cut on the negative real semiaxis,see Fig.2. The original microcanonical partition function Z N ( λ, E ) can be then recovered by computing the inverseLaplace transform of ˜ Z N ( λ, β ): Z N ( λ, E ) = (cid:90) β + i ∞ β − i ∞ dβ exp { βE + N log[ z ( λ, β )] } . (22)The integral in Eq. (22) should be evaluated by means of a saddle-point approximation, where β is the real positivesolution of the following equation: EN = − z ( λ, β ) ∂z ( λ, β ) ∂β , (23)If such a real positive value β exists, it can be physically interpreted as an inverse temperature and one could finallywrite Z N ( λ, E ) ≈ e β E + N log[ z ( λ,β )] . (24)What happens in the DNLSE model is that Eq. (23) for E < E th admits a unique real positive solution and theintegration contour is shown by the dashed (blue) line in Fig.2. Conversely, for E > E th a real positive solutionof Eq. (23) does not exist, due to the presence of the branch-cut of z ( λ, β ) on the real negative semiaxis. As aconsequence, the calculation of Z N ( λ, E ) can be performed by considering the analytic continuation of z ( λ, β ) in thecomplex β plane and deforming the integration contour as shown by the continuous (red) line in Fig. 2. Followingvery similar large-deviation calculations, contained in a series of papers [23, 24, 26, 29], one can recover Z N ( λ, E ) for E > E th making use of suitable expansions of z ( λ, β ) around the origin β = 0. In order to evaluate Z N ( λ, E ) for agiven fixed scale ∆ E = E − E th ∼ N γ of the excess energy one needs to retain the leading terms only up to a givenscale in the expansion of z ( λ, β ). For instance, as explicitly reported in [29], one can evaluate Z N ( λ, E ) at the energyscale ∆ E = E − E th ∼ N / (Gaussian regime) by expanding z ( λ, β ) around β = 0 up to the order β ∼ N − / , thusobtaining the almost trivial result Z N ( λ, E ) = 1 σ √ πN exp (cid:20) − ( E − E th ) σ N (cid:21) , (25)which is a straightforward consequence of the Central Limit Theorem, the dependence on λ coming through σ [seeEq. (17)]. On the other hand, if one aims at evaluating Z N ( λ, E ) at the order ∆ E = E − E th ∼ N , which in [29] isdenoted as the extreme large deviations regime, one has to retain consistently terms of the expansion of z ( λ, β ) up tothe order β ∼ /N . In this case one obtains: Z N ( λ, E ) ∼ exp (cid:16) − (cid:112) E − E th (cid:17) . (26)As discussed in Sec. IV B, the Gaussian regime and the extreme large deviations regime correspond to a delocalizedand to a localized phase, respectively. Therefore, in order to understand whether the crossover between these twophases occurs as a real thermodynamic phase transition, one has to identify an intermediate matching regime, whichcan be heuristically singled out by the following condition( E − E th ) σ N ≈ (cid:112) E − E th , (27)allowing to recognize the intermediate scale E − E th ∼ N / . (28) C. Matching regime: the non-analytic contribution
The main result of this section is the proof that in the matching regime the partition function splits into the sumof a Gaussian contribution and of an non-analytic one, denoted as C ( λ, ζ ): Z N ( λ, E ) = 1 σ √ πN exp (cid:18) − N / ζ σ (cid:19) + C ( λ, ζ ) , (29)where ζ = E − E th N / (30)is a suitable scaling variable essentially inspired by the matching condition in Eq. (27). The first term on the r.h.s. ofEq. (29) comes from the straight part of the deformed contour in Fig. 2 (continuous red line), while the non-analyticcontribution, C ( λ, ζ ), is due to the non-analyticity at the branch-cut along the negative real axis.Since for E > E th the saddle-point condition in Eq. (23) has no real solution, we have to consider the analyticprolongation of z ( λ, β ) in the complex β plane and to evaluate its expansion around the origin β = 0 separately alongthe upper and lower branch cut, i.e. for (cid:60) ( β ) < δ → z ( λ, β + iδ ) = z ( λ, β + i + )lim δ → z ( λ, β − iδ ) = z ( λ, β + i − ) . (31)The expansion around β = 0 yields the expressions z ( λ, β + i + ) = 1 − (cid:104) ε (cid:105) β + 12 (cid:104) ε (cid:105) β + . . . + (cid:115) π (cid:104) ε (cid:105) β exp (cid:18) (cid:104) ε (cid:105) β (cid:19) z ( λ, β + i − ) = 1 − (cid:104) ε (cid:105) β + 12 (cid:104) ε (cid:105) β , (32)where (cid:104) ε (cid:105) and (cid:104) ε (cid:105) are defined in Eq. (17) in terms of λ . Accordingly, the expansion of the local free energy reads:log[ z ( λ, β + i + )] = −(cid:104) ε (cid:105) β + 12 σ β + O ( β ) + . . . + (cid:115) π (cid:104) ε (cid:105) β exp (cid:18) (cid:104) ε (cid:105) β (cid:19) log[ z ( λ, β + i − )] = −(cid:104) ε (cid:105) β + 12 σ β + O ( β ) , (33)where σ is defined in Eq. (17) in terms of λ . The non-analiticity of z ( λ, β ) at the cut, clearly expressed by thedifference between the first and the second line of Eq. (33), suggests to evaluate the contour integral which defines Z N ( λ, E ) [see Eq. (22)], by splitting it in two parts: Z N ( λ, E ) = I (+) ( λ, E ) + I ( − ) ( λ, E ) . (34)0The two terms I (+) ( λ, E ) and I ( − ) ( λ, E ) in the equation above can be evaluated by introducing explicitly the expan-sions of Eq. (33) into the integral of Eq. (22). Recalling then that E th = N (cid:104) ε (cid:105) one gets I (+) ( λ, E ) = (cid:90) Γ (+) dβ πi exp (cid:40) β ( E − E th ) + N σ β + N O ( β ) + N (cid:115) π (cid:104) ε (cid:105) β exp (cid:18) (cid:104) ε (cid:105) β (cid:19)(cid:41) I ( − ) ( λ, E ) = (cid:90) Γ ( − ) dβ πi exp (cid:26) β ( E − E th ) + N σ β + N O ( β ) (cid:27) , (35)where the integration paths Γ (+) and Γ ( − ) are those shown in Fig.2. A better interpretation of this result can beobtained by introducing the scaling variable ζ defined in Eq. (30): I (+) ( λ, ζ ) = (cid:90) Γ (+) dβ πi exp (cid:40) β ζ N / + N σ β + N O ( β ) + N (cid:115) π (cid:104) ε (cid:105) β exp (cid:18) (cid:104) ε (cid:105) β (cid:19)(cid:41) I ( − ) ( λ, ζ ) = (cid:90) Γ ( − ) dβ πi exp (cid:26) β ζ N / + N σ β + N O ( β ) (cid:27) . (36)Since (cid:60) ( β ) <
0, for asymptotically small values of β one has that the non-analytic term exp[1 / (2 (cid:104) ε (cid:105) β )] is exponentiallysmall and, at leading order in N , we can writeexp (cid:34) N (cid:115) π (cid:104) ε (cid:105) β exp (cid:18) (cid:104) ε (cid:105) β (cid:19)(cid:35) ≈ N (cid:115) π (cid:104) ε (cid:105) β exp (cid:18) (cid:104) ε (cid:105) β (cid:19) . (37)By substituting the above expansion into the integral I (+) ( λ, ζ ) we obtain I (+) ( λ, ζ ) = (cid:90) Γ (+) dβ πi e βζN / + N σ β + N O ( β ) ++ N (cid:115) π (cid:104) ε (cid:105) β (cid:90) Γ (+) dβ πi exp (cid:26) βζN / + N σ β + N O ( β ) + 12 (cid:104) ε (cid:105) β (cid:27) . (38)Taking inspiration from [29] we adopt the following scaling ansatz for ββ → β/N / , (39)which is consistent with the idea that in the matching regime the analytic and non-analytic contributions are of thesame order, i.e. βζN / + N σ β ≈ (cid:104) ε (cid:105) β . (40)Hence, the leading contribution to the integral in Eq. (38) reads I (+) ( λ, ζ ) = 1 N / (cid:90) Γ (+) dβ πi e N / [ βζ + σ β ] ++ N (cid:115) π (cid:104) ε (cid:105) (cid:90) Γ (+) dβ πi (cid:112) N / β exp (cid:26) N / (cid:20) βζ + 12 σ β + 12 (cid:104) ε (cid:105) β (cid:21)(cid:27) . (41)Similarly, the integral in the negative imaginary semiplane has the expression I ( − ) ( λ, ζ ) = 1 N / (cid:90) Γ ( − ) dβ πi e N / [ βζ + σ β ] . (42)By summing these two contributions one finally obtains Z N ( λ, ζ ) = I (+) ( λ, ζ ) + I ( − ) ( λ, ζ ) == 1 N / (cid:90) i ∞− i ∞ dβ πi e N / [ βζ + σ β ] + C ( λ, ζ ) , = 1 σ √ πN e − N / ζ σ + C ( λ, ζ ) (43)1where the non-analytic contribution to the partition function is C ( λ, ζ ) = N (cid:115) π (cid:104) ε (cid:105) (cid:90) Γ (+) dβ πi (cid:112) N / β e N / F ζ ( λ,β ) , (44)with F ζ ( λ, β ) = βζ + 12 σ β + 12 (cid:104) ε (cid:105) β . (45)The decomposition of the partition function in the matching regime as the sum of two contributions is the main resultof this section. It remains to perform the explicit calculation of the integral C ( λ, ζ ). This task can be accomplishedby following the same procedure reported in [29]: the key point of this calculation amounts to finding the solution β ∗ ( ζ ) of the saddle-point equation: ∂F ζ ( λ, β ) ∂β = 0 . (46)Details of the calculations are illustrated in Appendices VII A and VII B. Here we just provide the final result: (cid:90) Γ (+) dβ πi (cid:112) N / β e N / F ζ ( λ,β ) ≈ e − N / χ ( ζ ) , (47)where the explicit form of χ ( ζ ) is discussed in Appendix VII C. Its asymptotic behaviours are: χ ( ζ ) = (cid:16) σ (cid:104) ε (cid:105) (cid:17) / ζ → ζ l (cid:113) (cid:104) ε (cid:105) √ ζ − σ (cid:104) ε (cid:105) ζ + O (cid:16) ζ / (cid:17) , ζ (cid:29) , (48)where ζ l is the spinodal point for the localized phase, that is the smallest value of ζ for which the saddle-point equationEq. (46) admits a real solution, namely ζ l = 32 (cid:18) σ (cid:104) ε (cid:105) (cid:19) / . (49) IV. THE FIRST-ORDER PHASE TRANSITION FROM A THERMALIZED PHASE TOLOCALIZATIONA. The microcanonical entropy in the matching regime
We are now in the position of retrieving the microcanonical partition function by computing the inverse Laplacetransform of Eq. (10): Ω N ( A, E ) = 12 πi (cid:90) λ + i ∞ λ − i ∞ dλ e λA ˜Ω N ( λ, E )= e N log( π ) πi (cid:90) λ + i ∞ λ − i ∞ dλ e N [ aλ − log( λ )] Z N ( λ, E ) , (50)where the final expression on the r.h.s. of Eq. (50) stems from Eq. (13). In order to point out the presence of a phasetransition we are interested to obtain an analytic estimate of Ω N ( A, ζ ) in the matching regime, where Z N ( λ, ζ ) isgiven by Eqs. (43) and Eq. (44) and ζ = ( E − E th ) /N / ≈ O (1):Ω N ( A, ζ ) = e N log( π ) πi (cid:90) λ + i ∞ λ − i ∞ dλ e N [ aλ − log( λ )] (cid:104) e − N / χ ( ζ ) + e − N / ζ / (2 σ ) (cid:105) . (51)In the thermodynamic limit, N → ∞ , the leading contribution to the integral on the r.h.s. of Eq. (51) is given by theterm N [ aλ − log( λ )], which determines the value of λ as the solution of the saddle-point equation ∂∂λ [ aλ − log( λ )] = 0 −→ λ = 1 /a . (52)2 c Ψ (r) r FIG. 3. Continuous (red) line: behaviour of the sub-leading contribution Ψ( r ) [see Eq. (56)] to the microcanonical entropy as afunction of the adimensional variable r = ζ/ζ l , with ζ l defined in Eq. (49). Making use of this rescaled variable r , the spinodalpoint is located at r = 1. The first order transition is located at r c = ζ c /ζ l ≈ / , where d Ψ( r ) /dr is discontinuous: the valueof ζ c is determined by the argument in App. VII D. The dashed lines draw the function χ ( r ) [see Eqns. (47),(48)] for r < r c and r / (2 σ ) for r > r c . The complete expression of Ω N ( A, ζ ) is thus obtained by replacing the multiplier λ with its actual value λ = 1 /a .Accordingly, in the matching regime the complete partition function readsΩ N ( A, ζ ) ≈ e N [1+log( πa )] (cid:104) e − N / χ ( ζ ) + e − N / ζ / (2 σ ) (cid:105) , (53)where σ = 20 a and (cid:104) ε (cid:105) = 2 a , the latter expression being present in the definition of χ ( ζ ) [see Eq. (48)]. This resultprovides us the expression for the microcanonical entropy at leading and sub-leading order in the thermodynamiclimit: S N ( A, ζ ) = N [1 + log( πa )] − N / Ψ( ζ ) , (54)The leading term is extensive in N and it represents a background entropy , S back N ( a ) = N [1 + log( πa )] = N s back ( a ) , (55)as the contribution of the bulk of the system at infinite temperature, which does not depend on ζ , i.e. on the excessenergy ∆ E = E − E th , which is entirely adsorbed by the condensate. At sub-leading order the contribution to themicrocanonical entropy in the thermodynamic limit is given by the functionΨ( ζ ) = inf ζ (cid:110) χ ( ζ ) , ζ / (2 σ ) (cid:111) , (56)and this allows us to identify the critical value ζ c for localization as the one where the (subleading contribution to)entropy of the localized and that of the delocalized phase have identical magnitude, i.e., by the following matchingcondition χ ( ζ c ) = ζ c / (2 σ ) . (57)The function Ψ( ζ ) is shown in Fig.3: more precisely, we have decided to draw it as a function of the rescaled variable r = ζ/ζ l . From the argument discussed in App. VII D we find that the critical value of r , which does not depend onthe parameters of individual energy distributions on lattice sites, is r c = ζ c ζ l = 2 / . (58)Since the derivative of Ψ( ζ ) is discontinuous at ζ c , we can conclude that we are facing a first–order phase transition,from a thermalized phase to a localized one.3 B. The order parameter: participation ratio
The first order transition can be further characterized by introducing the participation ratio of the energy per site Y = (cid:42) (cid:80) Nj =1 ε j (cid:16)(cid:80) Nj =1 ε j (cid:17) (cid:43) , (59)as a suitable order parameter, where the angular brackets denote the equilibrium average. From the definition inEq. (59), we see that if the energy E ∼ O ( N ) is distributed more or less democratically over all sites, then ε j ∼ O (1)for each j . Hence, the numerator scales as N and the denominator scales as N and consequently Y ∼ /N . Incontrast, if an extensive amount of energy is localised at a single site (i.e., in the presence of a condensate), thenumerator will scale as N while the denominator still scales as N . Consequently, Y ∼ O (1) in the large N limit.Hence, the quantity Y is a good measure to detect the localisation/condensation transition.For our purposes it is enough to analyze the behavior of Y in the proximity of the threshold energy, i.e., at E ∼ E th = N (cid:104) ε (cid:105) [see Sec. III A], where, as discussed in the previous section, the transition point is located. In thisregime the equilibrium joint probability distribution of local energies has, within the microcanonical ensemble, thefollowing expression P ( ε , . . . , ε N ) = e Ns back ( a ) Ω N ( A, E ) N (cid:89) j =1 f a ( ε j ) δ E − N (cid:88) j =1 ε j , (60)where Ω N ( A, E ) = e Ns back ( a ) (cid:90) ∞ N (cid:89) j =1 [ dε j f a ( ε j ) ] δ E − N (cid:88) j =1 ε j (61)is the microcanonical partition function. In fact, as shown in the previous section [see the saddle-point condition inEq. (52)], Ω N ( A, E ) in the matching regime can be written as a function of the probability distributions f a ( ε j ) ofi.i.d. energy variables, where the label λ appearing in Eq. (12) can be replaced by the label a , i.e. the inverse ofthe solution of the saddle-point condition. By plugging the expression of P ( ε , . . . , ε N ) written in Eq. (60) into thedefinition of Y given in Eq. (59), we get: Y ( E ) = e Ns back ( a ) Ω N ( A, E ) (cid:90) ∞ dε . . . dε N N (cid:89) j =1 f a ( ε j ) (cid:80) Nj =1 ε j (cid:16)(cid:80) Nj =1 ε j (cid:17) δ E − N (cid:88) j =1 ε j = N e s back ( a ) E (cid:90) ∞ dε ε f a ( ε ) (cid:104) e ( N − s back ( a ) (cid:82) dε . . . dε N (cid:81) Nj =2 f a ( ε j ) δ (cid:16) E − ε − (cid:80) Nj =2 ε i (cid:17)(cid:105) Ω N ( A, E )= N e s back ( a ) E (cid:90) ∞ dε ε f a ( ε ) Ω N − ( A, E − ε )Ω N ( A, E )= N e s back ( a ) E (cid:90) ∞ dε ε ρ ( ε )= NE e s back ( a ) (cid:104) ε (cid:105) , (62)where it is important to recall that E ∼ N , so that Y ( E ) ∼ (cid:104) ε (cid:105) /N. (63)In Eq. (62) we have then introduced the shorthand notation ρ ( ε ) for the marginal distribution of the energy ε on asingle site, defined as ρ ( ε ) = f a ( ε ) Ω N − ( A, E − ε )Ω N ( A, E ) . (64)As a first step in the study of the behaviour of the order parameter Y we observe that for E < E th the marginaldistribution ρ ( ε ) decays exponentially with ε on an energy scale which is independent of N . The key point is the4calculation of Ω N − ( A, E − ε ), which, for very large values of N , can be approximated in a completely harmless waywith Ω N ( A, E − ε ). Since the value of the energy ε on a single site can be at most ε = E , we have that the domain ofthe variable y = E − ε is y ∈ [0 , E ]. Hence the integral which defines Ω N ( A, y = E − ε ) through its inverse Laplacetransform can be still computed by a saddle-point approximation:Ω N ( A, E − ε ) = (cid:90) β + i ∞ β − i ∞ dβ exp { β ( E − ε ) + N log[ z ( a, β )] } ≈ exp { β ( E − ε ) + N log[ z ( a, β )] } Ω N − ( A, E − ε )Ω N ( A, E ) = e − β ε = ⇒ (cid:104) ε (cid:105) N ∼ N (65)Accordingly, the participation ratio Y vanishes as 1 /N in the thermodynamic limit, i.e. in the thermalized phaseclose to E th localization of energy is, as expected, absent. For E > E th we cannot rely anymore on the saddle-pointapproximation for β to compute the integral in Eq. (65). Still, from the study of Ω N ( A, E ) we known that at thescale E − E th ∼ N / the partition function has a Gaussian shape. Therefore, we have that for values of E up to thescale E − E th ∼ N / the marginal probability distribution of energy is given by the expression ρ ( ε ) = f a ( ε ) exp[ − (∆ E − ε ) / (2 σ N )]exp[ − (∆ E ) / (2 σ N )] , (66)where ∆ E = E − E th . By expanding the square in the numerator we obtain a term which simplifies with thedenominator and we are left with the expression: ρ ( ε ) = f a ( ε ) exp (cid:20) − ε σ N + ε ∆ Eσ N (cid:21) . (67)Recalling that ∆ E ∼ √ N , we can estimate the participation ratio by the relation Y ≈ N (cid:90) ∞ dε ε f a ( ε ) exp (cid:20) − ε σ N + εσ √ N (cid:21) . (68)In the limit of large N this integral can be approximated as Y ≈ N (cid:90) ∞ dε ε f a ( ε ) ∼ N , (69)so that even for
E > E th and E − E th ∼ √ N we have that the participation ratio vanishes asymptotically. The systemis still in a delocalized phase, although the decay of ρ ( ε ), as shown in Fig. 4, is not monotonic. With the same kind ofargument it can be shown that even for E − E th ∼ N / the participation ratio vanishes asymptotically. The systemis delocalized up to E c , the critical value of the total energy (which is of order O ( N )) where the first derivative of theentropy exhibits the discontinuity. Such a value E c , according to Eq. (30), reads as: E c = E th + N / ζ c , (70)where ζ c does not depend on N and is determined by the matching condition in Eq. (57), see also Appendix VII D).The situation is different for the case of extreme large deviations of the total energy, i.e., following the terminologyof [29], for E − E th ∼ N . Also in this case the marginal distribution ρ ( ε ) exhibits a bump (see Fig. 4). But in thiscase the whole ρ ( ε ) is dominated in the large N limit by the contribution of the bump, which, for fluctuations of order∆ E − ε ∼ N / around the bump center, reads as ρ ( ε ) ≈ ρ bump ( ε ) = f a ( ε ) 1 N f (∆ E ) 1 √ N exp[ − (∆ E − ε ) / (2 σ N )] , (71)where we have used the fact that in the extreme large deviations regime the whole partition function is identical to N times the distribution of the single variable (see [23, 26]), in formulaeΩ N ( A, E ) ≈ N f a ( E − E th ) . (72)Then, since we are interested to the estimate of ρ bump ( ε ) for ε ∼ ∆ E , we have that f a ( ε ) /f a (∆ E ) = O (1) in the large N limit, so that ρ bump ( ε ) ≈ N / exp[ − (∆ E − ε ) / (2 σ N )] ≈ N δ (∆ E − ε ) . (73)5Since ∆ E ∼ N , we finally obtain (cid:90) ∞ dε ε ρ bump ( ε ) = (∆ E ) N ∼ N, (74)so that Y = NE (cid:104) ε (cid:105) ∼ const . (75)The finite value of the participation ratio signals the localized phase. We want to point out that this scenario indicatesthat localization of energy is on a single site. In fact, the ratio between the width of the bump, of order N / , andits position, ∆ E ∼ N , vanishes asymptotically.The overall situation can be summarized according to the following scheme, where the critical value of the energy E c is the one defined in Eq. (70) above:(A) E < E th : ρ ( ε ) decays monotonically at large ε and the participation ratio decays asymptotically as Y ( E ) ∼ N . (76)(B) E th < E < E c : decay of ρ ( ε ) at large values of ε is non monotonic and the formation of a secondary bump canbe easily seen in Fig. 4. This notwithstanding, the participation ratio still vanishes asymptotically as Y ( E ) ∼ N ; (77)we call this phase, which has been put in evidence also in another recent paper on constraint-driven condensa-tion [30], the pseudo-condensate one.(C) E > E c : ρ ( ε ) has a bump placed at ε ∗ = ( E − E th ) and the participation ratio goes asymptotically to a constantvalue: Y ( E ) ∼ const . (78) C. Negative temperatures
A straightforward consequence of the results reported in the previous subsections is that the only consistent defini-tion of temperature for values of the energy density e > e th = E th /N is the microcanonical one. Taking into accountGaussian fluctuations around e th , i.e. | e − e th | ∼ /N / , we have that the entropy density reads s ( a, e ) = lim N →∞ N log[Ω( a, e )] = − ( e − e th ) σ (79)so that the microcanonical temperature turns out to be:1 T = ∂S N ( A, E ) ∂E = − σ ( e − e th ) . (80)The microcanonical temperature becomes negative as soon as the equivalence between ensembles is broken, that is,already for values of the specific energy above the threshold, i.e., e > e th , but still lower than the critical value forlocalization, i.e., e < e c . This means that for not too large values of N , which might be for instance the numberof sites in a true optical lattice where an atomic condensate can be trapped, our analysis foresees the possibility todetect delocalized states with negative temperature . The fact that, above the threshold energy, temperature is negativeirrespective of whether the asymptotic value of the order parameter is zero or not, reads off clearly from the expressionof the microcanonical entropy in all the three regimes where e > e th , S N ( A, E ) =
N s back − N [ e − e th ] / (2 σ ) e − e th ∼ N − / N s back − Ψ( ζ ) e − e th ∼ N − / N s back − N / √ e − e th e − e th ∼ , (81)6For instance, from Eq. (81) we have that the inverse temperature of the condensate reads1 T = ∂S N ( A, E ) ∂E = − N / √ e − e th (82)Let us notice here that negative temperatures are not a peculiarity of the non-equivalence between canonical andmicrocanonical ensembles: they can be found and have a perfectly consistent physical meaning also in situationswhere the two esembles are equivalent, provided that the Hamiltonian of the system is a bounded function [31–33].On the other hand, in the case of unbounded Hamiltonian, e.g., the DNLSE studied here, negative temperatureshave a physical meaning only within the microcanonical ensemble, thus making ensemble inequivalence a necessarycondition for the observation of negative temperature states. D. Finite-size effects and simulations: the pseudo-condensate phase ε -6 -5 -4 -3 -2 -1 ρ(ε) e = .
00 2 .
18 2 .
35 2 .
50 2 .
70 3 .
00 3 . FIG. 4. Marginal energy density probability ρ ( ε ) [see Eq. (64)] obtained from numerical simulations for N = 128, a = 1 anddifferent values of the energy density, namely e = 2 . , . , . , . , . , . , .
41. The threshold energy (per d.o.f.) for thedata in figure is e th = 2 while the critical energy is e c = 4 . From the two previous section we have learned that there is an intermediate region of energies that we call the pseudo-condensate regime , e th < e < e c , where a quite interesting phenomenon takes place: the order parameterdecreases as 1 /N and the temperature is negative. Although in the thermodynamic limit N → ∞ this region vanishes[see Eq. (86) below] its existence is very important for numerical simulations and experiments, where the value of N is usually not too large. For instance, let us consider the case N = 128 and a = 1, that we have reproducedin numerical simulations. Since the Hamiltonian dynamics in the non-equivalence regime e > e th suffers a criticalslowing down, we have used the stochastic algorithm introduced in [15] and used also in [26, 27] for the investigationof constraint-driven condensation. In detail, we have considered random updates which guarantee the conservationof the two quantities: A = N (cid:88) j =1 ρ j E = N (cid:88) j =1 ρ j . (83)In particular, we have considered local random updates of triplets of neighbouring sites such that: ρ j − ( t + 1) + ρ j ( t + 1) + ρ j +1 ( t + 1) = ρ j − ( t ) + ρ j ( t ) + ρ j +1 ( t ) ρ j − ( t + 1) + ρ j ( t + 1) + ρ j +1 ( t + 1) = ρ j − ( t ) + ρ j ( t ) + ρ j +1 ( t ) , (84)where the variable t is the discrete time of the algorithm and is measured in numbers of random moves divided by N . After a transient of 2 time units, a stationary state is reached where we have sampled the marginal distribution ρ ( ε ) for times up to t = 2 units.7For a = 1 we have from the definitions in Eq. (17) that (cid:104) ε (cid:105) = 2 and σ = 20, respectively. This yields the followingvalues for the spinodal point of the condensate and for the transition point in the scaling variable ζ , which can beclearly seen in Fig. 3: ζ l ≈ . ζ c ≈ .
05 (85)The threshold energy e th for ensemble inequivalence and the localization energy e c reads off, accordingly, as: e th = 2 a = 2 e c = e th + ζ c N / ≈ . N = 128. Fig. 4 shows that already at energies e < e c , the bump of the condensateis clearly visible, although the system is in the pseudo-localized phase. Fig. 4 is indeed quite instructive: the non-monotonic decay of ρ ( ε ) is not enough to say that the system is in the localized phase. In order to ascertain fromdata whether or not we are in the phase where the order parameter tends asymptotically to a finite value, a finite-sizestudy of ρ ( ε ) is necessary. V. CONCLUSIONS AND PERSPECTIVES
In this manuscript we have shown how to compute the partition function of the Non-Linear Schr¨odinger Hamiltonianin Eq. (1) in the microcanonical ensemble and in the approximation of infinite temperature ( β = 0). This has allowedus to present a clear and coherent scenario of the thermodynamics of this model, also in relation with its dynamicalproperties. In fact, previous approaches (e.g., see [7]) provided less transparent interpretations, due to the use of thegrand-canonical ensemble for describing also the phase above the line β = 0 (see Fig.0). In fact, making use of themicrocanonical approach we have been able to show that this is a condensate phase, where typically a finite fractionof the whole mass and energy are localized in a few lattice sites. According to Ruelle [34] this is exactly one of the twoconditions where equivalence between statistical ensembles does not apply. Moreover, we have shown that the β = 0line corresponds to a first-order phase transition from a thermalized phase to a localized one, characterized by a jumpin the derivative ∂S/∂E of the microcanonical entropy with respect to the energy. In a dynamical perspective, thistransition indicates the passage from a phase where equipartition holds to another phase where ergodicity is brokenby the condensation mechanism.There are further results emerging from our study, that merit to be mentioned. In the microcanonical ensemblethe temperature can be computed by the formula 1 /T = ∂S/∂E . We have shown that for e < a the temperatureis positive, then T → ∞ when e → a and finally T < e > a . We want to point out that the change ofsign of the temperature coincides with the formation of the condensate only in the thermodynamic limit. In fact,we have found that, for any finite value of the lattice size N , the true transition line for the condensate formation isslightly above the β = 0 line. Accordingly, for finite N it exists a region where one can observe negative-temperaturestates, that are not yet localized. This is a particularly important outcome in the perspective of designing specificexperiments of BEC in optical lattices, where such peculiar states can be observed, while avoiding the condensationof a large fraction of atoms onto a single or few sites. VI. ACKNOWLEDGEMENTS
We thank for interesting discussions M. Baiesi, S. Franz, L. Leuzzi, G. Parisi, P. Politi, F. Ricci-Tersenghi, L.Salasnich, A. Scardicchio, F. Seno, A. Vulpiani. We also thank N. Smith for pointing out an algebraic error inAppendices VII A and VII C and for suggesting the argument for ζ c in Appendix VII D. G.G. acknowledges thefinancial support of the Simons Foundation (Grant No. 454949, Giorgio Parisi). S.I. acknowledges support fromProgetto di Ricerca Dipartimentale BIRD173122/17 of the University of Padova.8 VII. APPENDICESA. Derivation of the rate function χ ( ζ ) in the intermediate matching regime In this Appendix we study the leading large N behavior of the integral in Eq. (44): C ( λ, ζ ) = N (cid:90) Γ (+) dβi (cid:112) π (cid:104) ε (cid:105) N / β e N / F ζ ( β ) (87)where ζ ≥ F ζ ( λ, β ) = βζ + 12 σ β + 12 β (cid:104) ε (cid:105) . (88)It is important to recall that the contour Γ (+) is along a vertical axis in the complex β -plane with its real partnegative, i.e. Re( β ) <
0. Thus, we can deform this contour only in the upper left quadrant in the complex β plane(Re( β ) < β ) > β -plane where Re( β ) > β planein the left upper quadrant, with fixed ζ . Hence, we look for solutions for the stationary points of the function F ζ ( λ, β )in Eq. (88). They are given by the zeros of the cubic equation dF ζ ( β ) dβ = ζ + σ β − (cid:104) ε (cid:105) β ≡ λ which appears just as fixed parameter, in what follows we will usethe conventions below to denote partial derivatives with respect to β : F (cid:48) ζ ( β ) = ∂F ζ ( λ, β ) ∂βF (cid:48)(cid:48) ζ ( β ) = ∂ F ζ ( λ, β ) ∂β (90)As ζ ≥ β plane. It turns out that for ζ < ζ l (where ζ l is to bedetermined), there is one positive real root and two complex conjugate roots. For example, when ζ = 0, the threeroots of Eq. (89) are respectively at β = (2 (cid:104) ε (cid:105) σ ) − / e iφ with φ = 0, φ = 2 π/ φ = 4 π/
3. However, for ζ > ζ l ,all the three roots collapse on the real β axis, with β < β < β . The roots β < β < β > F (cid:48) ζ ( β ) in Eq. (89) as a function of real β , for ζ = 12 and λ = 1 (so σ = 20 /λ = 20). One finds, using Mathematica, three roots at β = − . . . . (the lowest root on thenegative side), β = − . . . . and β = 0 . . . . . We can now determine ζ l very easily. As ζ decreases, thetwo negative roots β and β approach each other and become coincident at ζ = ζ l and for ζ < ζ l , they split apart inthe complex β -plane and become complex conjugate of each other, with their real parts identical and negative. When β < β , the function F (cid:48) ζ ( β ) has a maximum at β m with β < β m < β (see Fig. 5). As ζ approaches ζ l , β and β approach each other, and consequently the maximum of F (cid:48) ζ ( β ) between β and β approach the height 0. Now, theheight of the maximum of F (cid:48) ζ ( β ) between β and β can be easily evaluated. The maximum occurs at β = β m where F (cid:48)(cid:48) ζ ( β ) = 0, i.e, at β m = − ( (cid:104) ε (cid:105) σ ) − / . Hence the height of the maximum is given by F (cid:48) ζ ( β = β m ) = ζ + σ β m + 12 β m (cid:104) ε (cid:105) = ζ − (cid:18) σ (cid:104) ε (cid:105) (cid:19) / . (91)Hence, the height of the maximum becomes exactly zero when ζ = ζ l = 32 (cid:18) σ (cid:104) ε (cid:105) (cid:19) / . (92)9 -30-20-10 0 10 20 30-1 -0.5 0 0.5 1 β β β F z ′ ( β ) β FIG. 5. A plot of F (cid:48) ζ ( β ) = ζ + σ β − (cid:104) ε (cid:105) β as a function of β ( β real) for ζ = 12, λ = 1 and σ = 20 /λ = 20. There are threezeros on the real β axis (obtained by Mathematica ) at β ≈ − . β ≈ − .
17 and β ≈ . Thus we conclude that for ζ > ζ l , with ζ l given excatly in Eq. (92), the function F (cid:48) ζ ( β ) has three real roots at s = β < β < β >
0, with β being the smallest negative root on the real axis. For ζ < ζ l , the pair ofroots are complex (conjugates). However, it turns out (as will be shown below) that for our purpose, it is sufficient toconsider evaluating the integral in Eq. (87) only in the range ζ > ζ l where the roots are real and evaluating the saddlepoint equations are considerbaly simpler. So, focusing on ζ > ζ l , out of these 3 roots as possible saddle points of theintegrand in Eq. (87), we have to discard β > β plane. Now, we deform our vertical contour Γ (+) by rotating it anticlockwise by π/ β and β , it is easy to see [Fig. (5)] that F (cid:48)(cid:48) z ( β ) > s axis) and F (cid:48)(cid:48) z ( β ) < β for large N , we should choose β tobe the correct root, i.e., the largest among the negative roots of the cubic equation z + σ β − / (2 Eβ ) = 0. Thus,evaluating this saddle point (and discarding preexponential terms) we get for large N C ( λ, ζ ) ≈ exp[ − N / χ ( ζ )] (93)where the rate function χ ( ζ ) is given by χ ( ζ ) = − F ζ ( β = β ) = − β ζ − σ β − β (cid:104) ε (cid:105) (94)The right hand side can be further simplified by using the saddle point equation Eq. (89), i.e., ζ + σ β − / (cid:104) ε (cid:105) β = 0.We finally obtain χ ( ζ ) = − ζβ − (cid:104) ε (cid:105) β . (95) B. Asymptotic behavior of χ ( ζ ) We now determine the asymptotic behavior of the rate function χ ( ζ ) in the range ζ l < ζ < ∞ , where ζ l is given inEq. (92). Essentially, we need to determine β (the largest negative root) as a function of ζ by solving Eq. (89), andsubstitute it in Eq. (95) to determine χ ( ζ ).We first consider the limit ζ → ζ l from above, where ζ l is given in Eq. (92). As ζ → ζ l from above, we have alreadymentioned that the two negative roots β and β approach each other. Finally at ζ = ζ l , we have β = β = β m β m = − ( (cid:104) ε (cid:105) σ ) − / is the location of the maximum between β and β . Hence as ζ → ζ l from above, β → β m = − ( (cid:104) ε (cid:105) σ ) − / . Substituting this value of β in Eq. (95) gives the limiting behavior χ ( ζ ) → (cid:18) σ (cid:104) ε (cid:105) (cid:19) / as ζ → ζ l (96)as announced in the first line of Eq. (48).To derive the large ζ → ∞ behavior of χ ( ζ ) as announced in the second line of Eq. (48), it is first convenient tore-parametrize β and define β = − (cid:112) (cid:104) ε (cid:105) ζ θ ζ . (97)Substituting this in Eq. (89), it is easy to see that θ ζ satisfies the cubic equation − b ( ζ ) θ ζ + θ ζ − , (98)where b ( ζ ) = σ (cid:112) (cid:104) ε (cid:105) ζ / . (99)Note that due to the change of sign in going from β to θ ζ , we now need to determine the largest positive root of θ ζ in Eq. (98). In terms of θ ζ , χ ( ζ ) in Eq. (95) reads χ ( ζ ) = (cid:112) ζ (cid:112) (cid:104) ε (cid:105) θ ζ + 3 θ ζ . (100)The representations in Eqs. (98),(99) and (100) are now particularly suited for the large ζ analysis of χ ( ζ ). FromEq. (98), it follows that as ζ → ∞ , θ ζ →
1. Hence, for large ζ or equivalently small b ( ζ ), we can obtain a perturbativesolution of Eq. (98). To leading order, it is easy to see that θ ζ = 1 + b ( ζ )2 + O (cid:16) b ( ζ ) (cid:17) . (101)with b ( ζ ) given in Eq. (99). Substituting this in Eq. (100) gives the large ζ behavior of χ ( ζ ) χ ( ζ ) = (cid:115) (cid:104) ε (cid:105) (cid:112) ζ − σ (cid:104) ε (cid:105) ζ + O (cid:18) ζ / (cid:19) . (102)as announced in the second line of Eq. (48). C. Explicit expression of χ ( ζ ) While the excercises in the previous subsections were instructive, it is also possible to obtain an explicit expressionfor χ ( ζ ) by solving the cubic equation Eq. (98) with Mathematica . The smallest positive root of Eq. (98), using
Mathematica , reads θ ζ = 13 b ζ + 13 · / b ζ (1 − i √ (cid:16) − b ζ + 3 (cid:113) −
12 + 81 b ζ (cid:17) / + 13 · / b ζ (1 + i √ (cid:16) − b ζ + 3 (cid:113) −
12 + 81 b ζ (cid:17) / (103)where b ζ , used as an abbreviation for b ( ζ ), is given in Eq. (99). Using the expression of ζ l in Eq. (92), we can re-express b ζ conveniently in a dimensionless form b ζ = 12 (cid:18) ζ l ζ (cid:19) . (104)1Consequently, the solution θ ζ in Eq. (103) in terms of the adimensional parameter r = ζ/ζ l ≥ θ ζ ≡ θ ( r ) = √ r / (cid:34) − i √ g ( r ) + (1 + i √ g ( r ) (cid:35) (105)where g ( r ) = 1 r (cid:16) i (cid:112) r − (cid:17) / . (106)By multiplying both numerator and denominator of θ ( r ) by (1 − i √ r − / one ends up, after a little algebra, withthe following expression θ ( r ) = √ r / (cid:20) r (cid:16) ξ u / r + ξ u / r (cid:17)(cid:21) , (107)where ξ and u r denotes, respectively, a complex number and a complex function of the real variable r : ξ = 1 + i √ u r = 1 + i (cid:112) r − , (108)and we have also introduced the related complex conjugated quantities: ξ = 1 − i √ u r = 1 − i (cid:112) r − , (109)We can then write the complex expressions in Eq. (107) both in their polar form, i.e., u r = ρ r e iφ r and ξ = ρe iφ ,with, respectively: ρ r = r / φ r = arctan( (cid:112) r −
1) (110)and ρ = 2 φ = arctan( √
3) = π . (111)Finally, by writing ξ and u r inside Eq. (107) in their polar form and taking advantage of the expressions inEqns. (110),(111) we get: θ ( r ) = √ r / (cid:20) r ρ ρ / r (cid:16) e i ( φ + φ r ) + e − i ( φ + φ r ) (cid:17)(cid:21) == √ r / (cid:20) (cid:18) π (cid:112) r − (cid:19)(cid:21) (112)In order to draw explicitly the function χ ( ζ ), e.g. with the help of Mathematica , one can plug the expression of θ ( r = ζ/ζ l ) from Eq. (112) into the following formula: χ ( ζ ) = √ ζ √ E θ ( ζ/ζ l ) + 3 θ ( ζ/ζ l ) , (113) D. The critical value ζ c We show here how to compute the critical value ζ c at which χ ( ζ ) equals ζ / (2 σ ), i.e., the value at which the twobranches in Fig. 3 cross each other. To make the computations easier, it is convenient to work with dimensionlessvariables. Using ζ l = (3 / σ /E ) / from Eq. (92), we express ζ in units of ζ l , i.e., we define r = ζζ l = ζ (cid:18) Eσ (cid:19) / . (114)2In terms of r , one can rewrite b ( ζ ) in Eq. (99) as [using the shorthand notation b ζ = b ( ζ )]: b ζ = 12 (cid:18) r (cid:19) . (115)Consequently, Eq. (98) reduces to − √ (cid:18) (cid:19) / r − / θ ( r ) + θ ( r ) − , (116)where θ ( r ) = θ ζ = rζ l is dimensionless. Quite remarkably, it turns out that to determine the critical value ζ c , ratherconveniently we do not need to solve the above cubic equation, Eq. (116). Indeed, at ζ = ζ c , i.e., r = r c , equating χ ( ζ c ) = ζ c / σ , we get √ ζ c √ E (cid:20) θ ( r c ) + 3 θ ( r c ) (cid:21) = ζ c σ . (117)Expressing in terms of r c , Eq. (117) simplifies to θ ( r c ) + 3 θ ( r c ) = 3 / r / c . (118)Consider now Eq. (116) evaluated at r = r c . In this equation, we replace r c by its expression in Eq. (118). Thisimmediately gives θ ( r c ) = 3 / θ ( r c ) = (cid:114) . (119)Using this exact θ ( r c ) in Eq. (118) gives r c = ζ c ζ l = 2 / = 1 . . . . (120)It is now straightforward to check that the expression of θ ( r ) written in Eq. (112) is consistent with the result justfound, i.e., from it we retrieve θ ( r c = 2 / ) = (cid:112) /
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