High temperature convergence of the KMS boundary conditions: The Bose-Hubbard model on a finite graph
aa r X i v : . [ m a t h - ph ] A p r HIGH TEMPERATURE CONVERGENCE OF THE KMS BOUNDARYCONDITIONS: THE BOSE-HUBBARD MODEL ON A FINITE GRAPH
Z. AMMARI AND A. RATSIMANETRIMANANA
Abstract.
The Kubo-Martin-Schwinger condition is a widely studied fundamental property inquantum statistical mechanics which characterises the thermal equilibrium states of quantumsystems. In the seventies, G. Gallavotti and E. Verboven, proposed an analogue to the KMScondition for classical mechanical systems and highlighted its relationship with the Kirkwood-Salzburg equations and with the Gibbs equilibrium measures. In the present article, we provethat in a certain limiting regime of high temperature the classical KMS condition can be derivedfrom the quantum condition in the simple case of the Bose-Hubbard dynamical system on a finitegraph. The main ingredients of the proof are Golden-Thompson inequality, Bogoliubov inequalityand semiclassical analysis.
Contents
1. Introduction 12. Quantum Hamiltonian on a finite graph 43. Quantum KMS condition 54. Convergence 75. Classical KMS condition 11Appendix A. Number estimates 12Appendix B. Technical estimates 13Acknowledgements 14References 141.
Introduction A W ∗ − dynamical system ( A , τ t ) is a pair of a von Neumann algebra of observables A anda one-parameter group of automorphisms τ t on A . Consider for instance a finite dimensionalHilbert space H then A can be chosen to be the set of all operators B ( H ) and τ t to be theautomorphism group defined by τ t ( A ) = e itH A e − itH for any A ∈ A . The operator H denotes the Hamiltonian of a given quantum system and thecouple ( A , τ t ) describes the dynamics. According to quantum statistical physics such systemadmits a unique thermal equilibrium state ω β at inverse temperature β given by, ω β ( A ) = tr( e − βH A )tr( e − βH ) . (1.1)In general, the simplicity of the above statement have to be nuanced. In fact, the characterisationof thermal equilibrium in statistical mechanics is a nontrivial question particularly for dynamical Date : April 15, 2019.
Key words and phrases.
Bose-Hubbard, KMS property, Golden-Thompson and Bogoliubov inequalities, Wignermeasures, semiclassical analysis. systems which have an infinite number of degrees of freedom, see [9, 26]. One of the importantand most elegant characterisation of equilibrium states was noticed by R. Kubo, P.C. Martin andJ. Schwinger in the late fifties. It is based in the following observations in finite dimension. Infact, one remarks by a simple computation that the Gibbs state ω β in (1.1) satisfies for all t ∈ R and any A, B ∈ A the identity, ω β ( A τ t + iβ ( B )) = ω β ( τ t ( B ) A ) , (1.2)where τ t + iβ ( · ) denotes an analytic extension of the automorphism τ t to complex times given by τ t + iβ ( B ) = e ( − β + it ) H B e ( β − it ) H . More remarkable, if one takes a state ω that satisfies the same condition as (1.2) then ω shouldbe the Gibbs state ω β in (1.1). This indicates that the equation (1.2) singles out the thermalequilibrium states among all possible states of a quantum system. In the late sixties, R. Haag,N.M. Hugenholtz and M. Winnink suggested the identity (1.2) as a criterion for equilibriumstates and they named it the KMS boundary condition after Kubo, Martin and Schwinger [19].The subject of KMS states is bynow deeply studied specially from an algebraic standpoint. Forinstance, various characterisation related to correlation inequalities and to variational principleshave been derived (see e.g. [13, 6, 9]). Other perspectives have also been explored related forinstance to the Tomita-Takasaki theory and to the Heck algebra and number theory (see e.g.[11, 5, 7]).In the seventies, G. Gallavotti and E. Verboven, suggested an analogue to the KMS boundarycondition (1.2) which is suitable for classical mechanical systems and highlighted its relationshipwith the Kirkwood-Salzburg equations and with the Gibbs equilibrium measures, see [18]. Thederivation of such condition is based in the following heuristic argument. Consider a state ω ~ satisfying the KMS boundary condition ω ~ (cid:0) BA (cid:1) = ω ~ (cid:0) A τ i ~ β ( B ) (cid:1) (1.3)at inverse temperature ~ β , where ~ refers to the reduced Planck constant. This relation yields ω ~ (cid:18) AB − BAi ~ (cid:19) = ω ~ (cid:18) A τ i ~ β ( B ) − Bi ~ (cid:19) . (1.4)Assume for the moment that the space H = L ( R d ), so one can consider that the Hamiltonian H and the observables A, B are given by ~ -Weyl-quantized symbols (i.e., H = h W, ~ , A = a W, ~ and B = b W, ~ for some smooth functions a and b defined over the phase-space R d ). Then thesemiclassical theory firstly tell us that AB − BAi ~ −→ ~ → { a, b } , and τ i ~ β ( B ) − Bi ~ −→ ~ → β { h, b } , (1.5)where {· , ·} is the Poisson bracket and h denotes the Hamiltonian of the corresponding classicalsystem. Secondly, the quantum states ω ~ (or at least a subsequence) converge in a weak sense to asemiclassical probability measure µ over R d when ~ →
0. Therefore, the expected classical KMScondition that should in principle characterise the statistical equilibrium for classical mechanicalsystems is formally given by µ (cid:0) { a, b } (cid:1) = β µ (cid:0) a { h, b } (cid:1) , (1.6)for any smooth functions a, b on the phase-space R d . Here the notation µ ( f ) = R R d f ( u ) dµ ( u )is used. After the works [18, 1], M. Aizenman et al. showed in [2] that the condition (1.6) singlesout thermal equilibrium states for infinite classical mechanical systems among all probability MS BOUNDARY CONDITIONS 3 measures. In particular, the only measure µ satisfying (1.6) in our example is the Gibbs measuredefined with respect to the Lebesgue measure by the density, µ β = 1 z ( β ) e − βh ( u ) , (1.7)where z ( β ) is a normalisation constant. Note that the above Gibbs measure µ β can also becharacterised as an equilibrium state by means of variational methods and maximum entropyproperties or by correlation inequalities, see [9]. Nevertheless, in this note we focus only in theKMS boundary conditions for classical and quantum systems. In general, the derivation of theclassical KMS boundary condition (1.6) from the quantum one is a non trivial and interestingquestion which depends on the considered dynamical system. In our opinion, the classical KMScondition is an elegant characterisation of statistical equilibrium which deserves more attentionfrom PDE analysts. Although this condition has been studied in some subsequent works (see e.g.[17, 23, 25, 24, 10, 14]), it seems not largely known.Our main purpose in this note, is to provide a rigorous and simple proof for the derivation ofthe classical KMS condition (1.6) as a consequence of the relation (1.2) and the classical limit, ~ →
0, for the Bose-Hubbard dynamical system on a finite graph. The system we consider isgoverned by a typical many-body quantum Hamiltonian which can be written in terms of creationsannihilations operators and which is restricted to a finite volume. Our proof of convergenceis based on the Golden-Thompson inequality, the Bogoliubov inequality and the semiclassicalanalysis in the Fock space. Since the classical phase-space of the system considered here is finitedimensional it is possible by change of representation to convert the problem to a semiclassicalanalysis in a L space. However, we avoid such a change as we lose most of the interestinginsights and structures in our problem. In particular, we will rely on the analysis on the phase-space given in [3]. Our interest in the Bose-Hubbard system is motivated by the establishment ofa strong link between classical and quantum KMS conditions so that it leads to the exchange ofthe thermodynamic and the classical limits for infinite dynamical systems and to the investigationof phase transitions. Also note that from a physical standpoint, the Bose-Hubbard model is aquite relevant model describing ultracold atoms in optical lattices with an observed phenomenonof superfluid-insulator transition. From a wider perspective, the question considered here is alsorelated to the recent trend initiated by M. Lewin, P.T. Nam and N. Rougerie [21, 22] about theGibbs measures for the nonlinear Schr¨odinger equations (see also [16] where these investigationswere continued). In this respect, the KMS boundary conditions could provide an alternativeproof for the convergence of Gibbs states. These questions will be considered elsewhere and herewe will only focus on the Bose-Hubbard model on finite graph which is a much simpler model.The article is organised as follows: • In Section 2, the Bose-Hubbard Hamiltonian on a finite graph is introduced and itsrelationship with the discrete Laplacian is highlighted. • Section 3, is dedicated to the description of the unique KMS state of the Bose-Hubbarddynamical system at inverse temperature ~ β and to the extension of the dynamics tocomplex times. • Section 4, contains our main contribution stated in Theorem 4.2. Indeed, we prove that theKMS states of the Bose-Hubbard system converge, up to subsequences, to semiclassical(Wigner) measures satisfying the classical KMS condition. The analysis is based onsemiclassical methods in the Fock space developed in [3]. • Finally, in Section 5, we remark that any probability measure satisfying the classical KMScondition is indeed the Gibbs equilibrium measure for the Discret nonlinear Schr¨odingerequation. The proof of this fact is borrowed from the work [2].
Z. AMMARI AND A. RATSIMANETRIMANANA Quantum Hamiltonian on a finite graph
The discrete Laplacian:
Consider a finite graph G = ( V, E ) where V is the set of vertices and E is the set of edges. Assume furthermore that G is a simple undirected graph and let deg( x )denotes the degree of each vertices x ∈ V . In the following, we denote the graph equivalently G or V . Consider the Hilbert space of all complex-valued functions on V denoted as ℓ ( G ) andendowed with its natural scalar product and with the orthonormal basis ( e x ) x ∈ V such that e x ( y ) := δ x,y , ∀ x, y ∈ V. Then the discrete Laplacian on the graph G is a non-positive bounded operator on ℓ ( G ) givenby, (∆ G ψ ) ( x ) := − deg ( x ) ψ ( x ) + X y ∈ V,y ∼ x ψ ( y ) , with the above sum running over the nearest neighbours of x and ψ is any function in ℓ ( G ). The Bose-Hubbard Hamiltonian:
Consider the bosonic Fock space, F = C ⊕ ∞ M n =1 ⊗ ns ℓ ( G ) , where ⊗ ns ℓ ( G ) denotes the symmetric n -fold tensor product of ℓ ( G ). So, any ψ ∈ ⊗ ns ℓ ( G ) isa functions ψ : V n → C invariant under any permutation of its variables. Introduce the usualcreation and annihilation operators acting on the bosonic Fock space, a x = a ( e x ) and a ∗ x = a ∗ ( e x ) , then the following canonical commutation relations are satisfied, (cid:2) a x , a ∗ y (cid:3) = δ x,y F and (cid:2) a ∗ x , a ∗ y (cid:3) = [ a x , a y ] = 0 , ∀ x, y ∈ V .
Definition 2.1 (Bose-Hubbard Hamiltonian) . For ε ∈ (0 , ¯ ε ), λ > κ <
0, define the ε -dependent Bose-Hubbard Hamiltonian on the bosonic Fock space F by H ε := ε X x,y ∈ V : y ∼ x ( a ∗ x − a ∗ y )( a x − a y ) + ε λ X x ∈ V a ∗ x a ∗ x a x a x − εκ X x ∈ V a ∗ x a x . Here λ is the on-site interaction, κ is the chemical potential and ε is the semiclassical parameter. Remark . The first term of the Hamiltonian H ε is the kinetic part of the system and corre-sponds to the second quantization of the discrete Laplacian. Indeed, one can write12 X x,y ∈ V : y ∼ x ( a ∗ x − a ∗ y )( a x − a y ) = X x ∈ V deg ( x ) a ∗ x a x − X x,y ∈ V,y ∼ x a ∗ x a y = dΓ( − ∆ G ) , where dΓ( · ) is the second quantization operator defined on the bosonic Fock space bydΓ( A ) |⊗ ns ℓ ( G ) = n X j =1 ⊗ · · · ⊗ A ( j ) ⊗ · · · ⊗ , (2.1)for any given operator A ∈ B ( ℓ ( G )) and where A ( j ) means that A acts only in the j -th compo-nent.The following rescaled number operator will be often used, N ε := ε dΓ(1 ℓ ( G ) ) = ε X x ∈ V a ∗ x a x . (2.2) MS BOUNDARY CONDITIONS 5
Therefore, one can rewrite the Bose-Hubbard Hamiltonian as follows H ε = ε dΓ (cid:0) − ∆ G − κ ℓ ( G ) (cid:1) + ε λ I G , with the interaction denoted as I G := X x ∈ V a ∗ x a ∗ x a x a x . Since the discrete Laplacian ∆ G is self-adjoint, it is easy to check that H ε defines an (unbounded)self-adjoint operator on the Fock space F over its natural domain (for more details see e.g. [4,Appendix A]). Remark that the operator − ∆ G − κ ℓ ( G ) is positive since the chemical potential κ is negative. 3. Quantum KMS condition
The Bose-Hubbard Hamiltonian defines a W ∗ -dynamical system ( M , α t ) where M is the vonNeumann algebra of all bounded operators B ( F ) on the Fock space and α t is the one parametergroup of automorphisms defined by α t ( A ) = e i tε H ε A e − i tε H ε , for any A ∈ M . The above group of automorphisms α t admits a generator S : M → M with adomain D ( S ) = { A ∈ M , [ H ε , A ] ∈ M } , and satisfies for any A ∈ D ( S ), S ( A ) = lim t → α t ( A ) − At = iε [ H ε , A ] . The latter convergence is with respect to the σ -weak topology on M . Remark also that thedynamics α t depend on the semiclassical parameter ε .Next, we point out that the dynamical system ( M , α t ) admits a unique KMS state at inversetemperature εβ . Here β > ε -independent, effective inverse temperature. Lemma 3.1 (Partition function) . Since the chemical potential κ is negative then tr F (cid:16) e − βH ε (cid:17) < ∞ . Proof.
It is a consequence of [9, Proposition 5.2.27] and the Golden-Thompson inequality. Thelatter, see [15], says that for any Hermitian matrices A and B one has,tr (cid:0) e A + B (cid:1) ≤ tr (cid:0) e A e B (cid:1) . (3.1) (cid:3) Definition 3.2 (Gibbs state) . The Gibbs equilibrium state of the Bose-Hubbard system on a finite graph is well defined, ac-cording to Lemma 3.2, and it is given by ω ε ( A ) = tr F ( e − βH ε A )tr F ( e − βH ε ) . (3.2)For the sake of completeness, we recall some useful details concerning the KMS states. Onesays that A ∈ M is an entire analytic element of α t if there exists a function f : C → M suchthat f ( t ) = α t ( A ) for all t ∈ R and such that for any trace-class operator ρ ∈ M the function z ∈ C → tr( ρf ( z )) is analytic. Let M α denotes the set of entire analytic elements for α , then it isknown that M α is dense in M with respect to the σ -weak topology. For more details on analytic Z. AMMARI AND A. RATSIMANETRIMANANA elements, see [8, section 2.5.3]. In particular, by [8, Definition 2.5.20], an element A ∈ M is entireanalytic if and only if A ∈ D ( S n ) for all n ∈ N and for any t > ∞ X n =0 t n n ! k S n ( A ) k < ∞ . (3.3)Remark that on the set of entire analytic elements M α , the dynamics α t can be extends to complextimes. Indeed, α z ( A ) is well defined, for any A ∈ M α , by the following absolutely convergentseries, α z ( A ) = ∞ X n =0 z n n ! S n ( A ) , ∀ z ∈ C . We say that a state ω is a ( α t , εβ )-KMS state if and only if ω is normal and for any A, B ∈ M α , ω ( A α iεβ ( B )) = ω ( BA ) . (3.4)Remark that the above identity is known to be equivalent to the condition stated in the intro-duction (1.2). In particular, the KMS states are stationary states with respect to the dynamics. Proposition 3.3.
The Gibbs state ω ε defined by (3.2) is the unique KMS state of the W ∗ -dynamical system ( M , α t ) at the inverse temperature εβ .Proof. For
A, B ∈ M α , one checks α iεβ ( B ) = e − βH ε Be βH ε . The formula (3.2) for the Gibbs state, gives ω ε ( A α iεβ ( B )) = 1tr F ( e − βH ε ) tr F (cid:16) A e − βH ε B (cid:17) = ω ε ( BA ) . Reciprocally, let ω be a ( α t , εβ )-KMS state. In particular, there exists a density matrix ρ suchthat tr F ( ρ ) = 1 and ω ( A ) = tr F ( ρ A ) , ∀ A ∈ M . Using the KMS condition (3.4) and the cyclicity of the trace, one proves for any A ∈ M ,tr( ρ B A ) = tr( e − βH ε B e βH ε ρ A ) . In particular, for any B ∈ M α , ρ B = e − βH ε B e βH ε ρ . (3.5)Hence, one remarks that ρ commutes with any spectral projection of H ε by taking for instance B = 1 D ( H ε ) in the equation (3.5). Therefore, one concludes that e βH ε ρ B | D ( H ε ) F = B e βH ε ρ | D ( H ε ) F , for any bounded Borel subset D of R and any bounded operator B satisfying B = 1 D ( H ε ) B = B D ( H ε ). So, the operator e βH ε ρ commutes with any bounded operator over the subspaces1 D ( H ε ) F . This implies that ρ = c e − βH ε , and then one concludes with the fact that tr( ρ ) = 1. (cid:3) MS BOUNDARY CONDITIONS 7 Convergence
In this section, we prove that the KMS condition (3.4) converges, in the classical limit, towardsthe classical KMS condition. It is enough to prove such convergence for some specific observables
A, B ∈ M . In fact, consider for f, g ∈ ℓ ( G ), A = W ( f ) , and B = W ( g ) , (4.1)where W ( · ) denotes the Weyl operator defined by, W ( f ) = e i √ ε Φ( f ) , with Φ( f ) = a ∗ ( f ) + a ( f ) √ . (4.2)Let χ ∈ C ∞ ( R ) such that 0 ≤ χ ≤ χ ≡ | x | ≤ / χ ≡ | x | ≥
1. Define, for n ∈ N ,the cut-off functions χ n as χ n ( · ) = χ (cid:0) · n (cid:1) . Then, we are going to consider only the following smoothed observables, A n := χ n ( N ε ) A χ n ( N ε ) , and B n := χ n ( N ε ) B χ n ( N ε ) . (4.3) Lemma 4.1.
For all ε > and n ∈ N , the elements A n and B n given by (4.3) are entire analyticfor the dynamics α t .Proof. By functional calculus, remark that 1 [0 ,n ] ( N ε ) χ n ( N ε ) = χ n ( N ε ). Moreover, the numberoperator N ε and the Hamiltonian H ε commute in the strong sense. So, the generator S of thedynamics α t satisfies for k ∈ N , S k ( A n ) = (cid:18) iε (cid:19) k [ H ε , · · · [ H ε , A n ] · · · ] , = (cid:18) iε (cid:19) k [ ˜ H ε , · · · [ ˜ H ε , A n ] · · · ] , with ˜ H ε = 1 [0 ,n ] ( N ε ) H ε a bounded operator. Hence, the estimate (3.3) is satisfied and so A n is aentire analytic element. (cid:3) Recall that the ( α t , εβ )-KMS state ω ε satisfies in particular the condition, ω ε ( A n α iεβ ( B m )) = ω ε ( B m A n ) . A simple computation then leads to the main identity, ω ε (cid:18) A n α iεβ ( B m ) − B m iε (cid:19) = ω ε (cid:18) [ B m , A n ] iε (cid:19) . (4.4)Our aim is to take the classical limit ε → { ω ε } ε ∈ (0 , ¯ ε ) . Recallthat µ a Borel probability measure on the phase-space ℓ ( G ) is a Wigner measure of { ω ε } ε ∈ (0 , ¯ ε ) if there exists a subsequence ( ε k ) k ∈ N such that lim k →∞ ε k = 0 and for any f ∈ ℓ ( G ),lim k →∞ ω ε k ( W ( f )) = Z ℓ ( G ) e i √ ℜ e h f,u i dµ . (4.5)Note that the Weyl operator depends here on the parameter ε k instead of ε as in (4.2). Accordingto [3, Thm. 6.2] and Lemma A.3, the family of KMS states { ω ε } ε ∈ (0 , ¯ ε ) admits a non-void set ofWigner probability measures. Later on, we will prove that this set of measures reduces to asingleton given by the Gibbs equilibrium measure. But for the moment, we will use subsequencesas in the definition (4.5). Z. AMMARI AND A. RATSIMANETRIMANANA
The classical Hamiltonian system related to the Bose-Hubbard model is given by the
DiscreteNonlinear Schr¨odinger equation, see [20]. Its energy functional (or Hamiltonian ) is given by h ( u ) = −h u, ∆ G u i − κ k u k + λ X j ∈ V | u ( j ) | . (4.6)Note that ℓ ( G ) is a complex Hilbert space and so in our framework the Poisson structure isdefined as follows. For F, G smooth functions on ℓ ( G ), the Poisson bracket is given by { F, G } := 1 i ( ∂ u F · ∂ ¯ u G − ∂ u G · ∂ ¯ u F ) . (4.7)Here ∂ u and ∂ ¯ u are the standard differentiation with respect to u or ¯ u .Our main result is stated below. Theorem 4.2 (Classical KMS condition) . Let ω ε by the KMS state of the Bose-Hubbard W ∗ -dynamical system ( A , α t ) at inverse temperature ε β . Then any semiclassical (Wigner) measureof ω ε satisfies the classical KMS condition, i.e., for any F, G smooth functions on ℓ ( G ) , β µ ( { h, G } F ) = µ ( { F, G } ) , (4.8) where the classical Hamiltonian h is given by (4.6) and {· , ·} denotes the Poisson bracket recalledin (4.7) . In order to prove Theorem 4.2, one needs some preliminary steps.
Proposition 4.3.
Let ( ε k ) k ∈ N be a subsequence such that lim k →∞ ε k = 0 . Assume that the familyof KMS states { ω ε k } k ∈ N admits a unique Wigner measure µ . Then for all n, m integers such that m ≥ n , lim k →∞ ω ε k (cid:18) [ B m , A n ] iε k (cid:19) = Z ℓ ( G ) χ n ( h u, u i ) { e √ i ℜ e h g,u i ; e √ i ℜ e h f,u i } dµ (4.9)+ Z ℓ ( G ) χ n ( h u, u i ) { e √ i ℜ e h g,u i ; χ n ( h u, u i ) } e √ i ℜ e h f,u i dµ (4.10)+ Z ℓ ( G ) χ n ( h u, u i ) { χ n ( h u, u i ); e √ i ℜ e h f,u i } e √ i ℜ e h g,u i dµ . (4.11) Proof.
For simplicity, we denote ε instead of ε k and χ m instead of χ m ( N ε ). Using the cyclicityof the trace and the fact that χ n χ m = χ n , one remarks that ω ε ([ B m , A n ]) = ω ε ( χ n ( Bχ n A − Aχ n B )) . A simple computation yields,lim ε → ω ε (cid:18) [ B m , A n ] iε (cid:19) = lim ε → ω ε (cid:18) χ n [ B, A ] iε (cid:19) + lim ε → ω ε (cid:18) χ n [ B, χ n ] iε A (cid:19) + lim ε → ω ε (cid:18) χ n [ χ n , A ] iε B (cid:19) . (4.12)The Weyl commutation relations give,[ B, A ] iε = W ( f + g ) ( ℑ m h f, g i + O ( ε )) . MS BOUNDARY CONDITIONS 9
So, using Lemma B.1,lim ε → ω ε (cid:18) χ n [ B, A ] iε (cid:19) = ℑ m h f, g i lim ε → ω ε (cid:0) χ n W ( f + g ) (cid:1) = ℑ m h f, g i Z ℓ ( G ) χ n ( h u, u i ) e √ i ℜ e h f + g,u i dµ . Checking the Poisson bracket, { e √ i ℜ e h g,u i ; e √ i ℜ e h f,u i } = ℑ m h f, g i e √ i ℜ e h f + g,u i , one obtains the right hand side of (4.9). Consider now the second term in (4.12). One can write[ W ( g ) , χ n ] = Z R ˆ χ n ( s ) [ W ( g ) , e isN ε ] ds √ π , where ˆ χ n denotes the Fourier transform of the function χ n ( · ). Using standard computations inthe Fock space and Taylor expansion,[ W ( g ) , e isN ε ] = e isN ε (cid:0) e − isN ε W ( g ) e isN ε − W ( g ) (cid:1) = ie isN ε Z s e − irN ε [ W ( g ) , N ε ] e irN ε dr = − e isN ε Z s e − irN ε W ( g ) (cid:18) ε Φ( ig ) + ε k g k (cid:19) e irN ε dr . Hence, using the cyclicity of the tracelim ε → ω ε (cid:18) χ n [ B, χ n ] iε A (cid:19) = − Z R s ˆ χ n ( s ) lim ε → ω ε (cid:0) χ n e isN ε W ( g )Φ( ig ) W ( f ) (cid:1) ds √ π . (4.13)Knowing, by Lemma B.1, that the Wigner measure of the sequence { W ( f ) ρ ε χ n ( N ε ) e isN ε W ( g ) } is given by (cid:8) µχ n ( h u, u i ) e is k u k e √ i ℜ e h g + f,u i (cid:9) , then one obtains using [3, Thm. 6.13],lim ε → ω ε (cid:18) χ n [ B, χ n ] iε A (cid:19) = −√ Z R s ˆ χ n ( s ) Z ℓ ( G ) χ n ( h u, u i ) e is k u k ℜ e h u, ig i e √ i ℜ e h g + f,u i dµ ds √ π . Integrating back with respect to the variable s ,lim ε → ω ε (cid:18) χ n [ B, χ n ] iε A (cid:19) = √ i Z ℓ ( G ) χ ′ n ( k u k ) χ n ( k u k ) ℑ m h g, u i e √ i ℜ e h g + f,u i dµ . Then checking the Poisson bracket (cid:8) e √ i ℜ e h g,u i ; χ n ( h u, u i ) (cid:9) = √ iχ ′ n ( k u k ) ℑ m h g, u i e √ i ℜ e h g,u i , yields the right hand side of (4.10). The third term in the right side of (4.12) is similar to theabove one. (cid:3) The next step is to prove the convergence of the left hand side of (4.4).
Lemma 4.4. lim k →∞ ω ε k (cid:18) A n α iε k β ( B m ) − B m iε k (cid:19) = β lim k →∞ ω ε k (cid:18) A n [ B m , H ε k ] iε k (cid:19) . (4.14) Proof.
For simplicity, we use ε instead of ε k . According to Lemma 4.1, B m is a entire analyticelement for the dynamics α t . Hence, by Taylor expansion, ω ε (cid:18) A n α iεβ ( B m ) − B m iε (cid:19) = β Z ω ε (cid:18) A n [ α isεβ ( B m ) , H ε ] iε (cid:19) ds . Using the cyclicity of the trace and the fact that A n , B m are entire analytic elements, ω ε (cid:18) A n [ α isεβ ( B m ) , H ε ] iε (cid:19) = ω ε (cid:18) e sβH ε A n e − sβH ε [ B m , H ε ] iε (cid:19) . A second Taylor expansion yields, ω ε (cid:18) A n [ α isεβ ( B m ) , H ε ] iε (cid:19) = ω ε (cid:18) A n [ B m , H ε ] iε (cid:19) + β Z s ω ε (cid:18) e rβH ε [ H ε , A n ] e − rβH ε [ B m , H ε ] iε (cid:19) dr . So, the equality (4.14) is proved sincelim ε → Z ds Z s dr ω ε (cid:18) [ H ε , α − isεβ ( A n )] [ B m , H ε ] iε (cid:19) = 0 , thanks to the Lemma B.2 in the Appendix. (cid:3) Proposition 4.5.
Let ( ε k ) k ∈ N be a subsequence such that lim k →∞ ε k = 0 . Assume that the familyof KMS states { ω ε k } k ∈ N admits a unique Wigner measure µ . Then for all n, m integers such that m ≥ n , lim k →∞ ω ε k (cid:18) A n α iεβ ( B m ) − B m iε (cid:19) = β Z ℓ ( G ) χ n ( h u, u i ) { e √ i ℜ e h g,u i ; h ( u ) } e √ i ℜ e h f,u i dµ . (4.15) Proof.
The previous Lemma 4.4 allowed to get rid of the dynamics at complex times. So, it isenough to show the limit,lim k →∞ ω ε k (cid:18) A n [ B m , H ε k ] iε k (cid:19) = Z ℓ ( G ) χ n ( h u, u i ) { e √ i ℜ e h g,u i ; h ( u ) } e √ i ℜ e h f,u i dµ . For simplicity, we denote ε instead of ε k and χ m instead of χ m ( N ε ). Since m ≥ n then χ n χ m = χ n and one notices that χ n Aχ n [ χ m Bχ m , H ε ] = χ n Aχ n [ B, H ε ] χ m = χ n Aχ n [ W ( g ) , H ε ] χ m . Standard computations on the Fock space yield, (see e.g. [3, Proposition 2.10]), iε [ B, H ε ] = iε ( W ( g ) H ε W ( g ) ∗ − H ) W ( g )= iε (cid:18) h ( · − iε √ g ) − h ( u ) (cid:19) W ick W ( g )= {√ ℜ e h g, u i , h ( u ) } | {z } C Wick
W ick + R ( ε ) W ick W ( g ) . The subscript
Wick refers to the Wick quantization, see [3, section 2]. The remainder R ( ε ) W ick can be explicitly computed and satisfies the uniform estimate k χ n ( N ε ) R ( ε ) W ick k ≤ c ε ,
MS BOUNDARY CONDITIONS 11 which can be easily proved using [3, Lemma 2.5]. Therefore, by using Lemma B.2 one showslim k →∞ ω ε k (cid:18) A n α iεβ ( B m ) − B m iε (cid:19) = β lim k →∞ ω ε k (cid:16) χ n Aχ n C W ick B (cid:17) = β lim k →∞ ω ε k (cid:16) χ n A C
W ick B (cid:17) . Knowing, by Lemma B.1, that the Wigner measure of the sequence { W ( g ) ρ ε χ n ( N ε ) W ( f ) } isgiven by (cid:8) µe √ i ℜ e h f + g,u i χ n ( k u k ) (cid:9) , one concludes by [3, Thm. 6.13],lim ε → ω ε (cid:18) A n α iεβ ( B m ) − B m iε (cid:19) = Z ℓ ( G ) χ n ( k u k ) e √ i ℜ e h f + g,u i C ( u ) dµ . (cid:3) Corollary 4.6.
Any Wigner measure of the ( α t , εβ ) -KMS family of states ω ε satisfies for all f, g ∈ ℓ ( G ) , β Z ℓ ( G ) { e i ℜ e h g,u i ; h ( u ) } e i ℜ e h f,u i dµ = Z ℓ ( G ) { e i ℜ e h g,u i ; e i ℜ e h f,u i } dµ . (4.16) Proof.
It is a consequence of Proposition 4.3, Proposition 4.5 and dominated convergence whiletaking n, m → ∞ . (cid:3) Thus, we come to the following conclusion.
Proof of Theorem 4.2.
The phase-space ℓ ( G ) is a d -euclidean space. Let F, G be two smoothfunctions in C ∞ ( ℓ ( G )). The inverse Fourier transform gives, F ( u ) = Z ℓ ( G ) e i ℜ e h f,u i ˆ F ( f ) dL ( f )(2 π ) d/ , and G ( u ) = Z ℓ ( G ) e i ℜ e h g,u i ˆ G ( g ) dL ( g )(2 π ) d/ , where ˆ F , ˆ G denote the Fourier transforms of F and G respectively. Multiplying the equation(4.16) by ˆ F ( f ) ˆ G ( g ) and integrating with respect to the Lebesgue measure in the variables f and g , one obtains β Z ℓ ( G ) { G ( u ) , h ( u ) } F ( u ) dµ = Z ℓ ( G ) { G ( u ) , F ( u ) } dµ . This proves the classical KMS condition (4.8). (cid:3) Classical KMS condition
In this section, we point out that the only probability measure satisfying the classical KMScondition is the Gibbs equilibrium measure. This is a known fact and we provide here a shortproof only for reader’s convenience. The argument used below is borrowed from the work ofM. Aizenman, S. Goldstein, C. Grubber, J. Lebowitz and P.A. Martin [2].
Proposition 5.1 (Gibbs measure) . Suppose that µ is a Borel probability measure on ℓ ( G ) sat-isfying the classical KMS condition (4.8) . Then µ is the Gibbs equilibrium measure, i.e., dµdL = e − β h ( u ) z ( β ) , and z ( β ) = Z ℓ ( G ) e − β h ( u ) dL ( u ) , with h ( · ) is the classical Hamiltonian of the Discrete Nonlinear Schr¨odinger equation given by (4.6) and dL is the Lebesgue measure on ℓ ( G ) . Proof.
Consider the Borel probability measure ν = e βh ( u ) µ , so that for any Borel set B , ν ( B ) = Z B e βh ( u ) dµ . Note that, for any
F, G ∈ C ∞ ( ℓ ( G )), the Poisson bracket satisfies (cid:8) F e − βh ( u ) , G (cid:9) = (cid:8) F, G (cid:9) e − βh ( u ) − β (cid:8) h, G (cid:9) F ( u ) e − βh ( u ) . Hence, the classical KMS condition (4.8) can be written as µ (cid:16) e βh ( u ) (cid:8) F e − βh ( u ) , G (cid:9)(cid:17) = 0 , or equivalently for any F, G ∈ C ∞ ( ℓ ( G )), ν (cid:16)(cid:8) F e − βh ( u ) , G (cid:9)(cid:17) = 0 . Remark that the classical Hamiltonian h is a smooth C ∞ ( ℓ ( G )) function. Hence, the measure ν satisfies for any F, G ∈ C ∞ ( ℓ ( G )), ν (cid:0)(cid:8) F, G (cid:9)(cid:1) = 0 . This condition implies that ν is a multiple of the Lebesgue measure. Indeed, take g ( · ) = h e j , ·i ϕ ( · )with ϕ ∈ C ∞ ( ℓ ( G )) being equal to 1 on the support of f . Then the Poisson bracket gives, { f, g } = − i∂ j f . So, in a distributional sense the derivatives of the measure ν are null and therefore dν = c dL forsome constant c . Using the normalisation requirement for µ , one concludes that dν = z ( β ) dL . (cid:3) Appendix A. Number estimates
Consider the quasi free state ω ε ( · ) given by, ω ε ( · ) = tr (cid:0) · e βε dΓ(∆ G + κ (cid:1) tr (cid:0) e βε dΓ(∆ G + κ (cid:1) . The following uniform number of particles estimates are well know. Here we recall them forreader’s convenience. For more details on quasi free states and such inequalities, see e.g. [9, 21, 16].Remember that the rescaled number operator is given by, N ε := ε dΓ (cid:0) ℓ ( G ) (cid:1) = ε X x ∈ V a ∗ x a x . Lemma A.1.
For any k ∈ N , there exists a positive constant c k such that ω ε ( N kε ) ≤ c k , uniformly with respect to ε ∈ (0 , ¯ ε ) . Lemma A.2.
There exists a positive constant c such that tr( e βε dΓ(∆ G + κ )tr( e − βH ε ) ≤ c , uniformly with respect to ε ∈ (0 , ¯ ε ) . MS BOUNDARY CONDITIONS 13
Proof.
By using a Bogoliubov type inequality, see [27, Appendix D], one has thatln(tr( e βε dΓ(∆ G + κ )) − ln(tr( e − βH ε )) ≤ β tr (cid:0) ε λ I G e βε dΓ(∆ G + κ (cid:1) tr (cid:0) e βε dΓ(∆ G + κ (cid:1) . According to Definition 2.1, recall that I G = X x ∈ V a ∗ x a ∗ x a x a x . Therefore, there exists c > e βε dΓ(∆ G + κ )) − ln(tr( e − βH ε )) ≤ c (cid:0) ω ε ( N ε ) + ω ε ( N ε ) (cid:1) . Using Lemma A.1, one proves the inequality. (cid:3)
Lemma A.3.
For any k ∈ N , there exists a positive constant c k such that ω ε ( N kε ) ≤ c k , uniformly with respect to ε ∈ (0 , ¯ ε ) .Proof. A direct consequence of Lemma A.1, Lemma A.2 and the Golden-Thompson inequality. (cid:3)
Appendix B. Technical estimates
We refer the reader to [3] for more details in the semiclassical analysis on the Fock space. Here,we only sketch some useful technical results based in the above work. Remember that the KMSstates ω ε , given by (3.2), are normal and so we denote, ω ε ( · ) = tr F ( ρ ε · ) . Furthermore, assume for a subsequence ( ε k ) k ∈ N , such that lim k →∞ ε k = 0, that the set { ρ ε k } k ∈ N admits a unique Wigner measure µ . Then the following result holds true. Lemma B.1.
For any χ ∈ C ∞ ( R ) and f, g ∈ ℓ ( G ) , the set { W ( f ) ρ ε k χ ( N ε k ) W ( g ) } k ∈ N admitsa unique Wigner measure given by (cid:8) µ e √ i ℜ e h f + g,u i χ ( k u k ) (cid:9) . Proof.
For simplicity, we denote ε instead of ε k . It is enough to prove that the set of Wignermeasures for the density matrices { ρ ε χ ( N ε ) } is the singleton { µ χ ( k u k ) } . In fact, using the Weyl commutation relations, one checks according to (4.5),lim ε → tr F ( W ( f ) ρ ε χ ( N ε ) W ( g ) W ( η )) = Z ℓ ( G ) e i √ ℜ e h f + g + η,u i dν , where ν is a Wigner measure of the set of density matrices { ρ ε χ ( N ε ) } . Now, using Pseudo-differential calculus, χ ( N ε ) = (cid:0) χ ( k u k ) (cid:1) W eyl + O ( ε ) , where the subscript refers to the Weyl ε -quantization and the difference between the right andleft operators is of order ε in norm (see e.g. [12, Thm. 8.7]). Then [3, Thm. 6.13] with LemmaA.3, gives ν = µ χ ( k u k ) . (cid:3) Lemma B.2.
For any χ ∈ C ∞ ( R ) and f ∈ ℓ ( G ) , there exists c > such that for all ε ∈ (0 , ¯ ε ) , k χ ( N ε ) [ N ε , W ( f )] χ ( N ε ) k ≤ c ε , and k χ ( N ε ) [ H ε , W ( f )] χ ( N ε ) k ≤ c ε . Proof.
The proof of the two inequalities are similar. We sketch the second one. Using standardcomputation in the Fock space (see e.g. [3, Proposition 2.10]),[ H ε , W ( f )] = W ( f ) (cid:18) h ( · + i ε √ f ) − h ( · ) (cid:19) W ick , where the subscript refers to the Wick quantization, see [3, Section 2], and h is the classicalHamiltonian in (4.6). By Taylor expansion, one writes h ( u + i ε √ f ) − h ( u ) = ε C ε ( u ) , where C ε ( u ) is a polynomial in u which can be computed explicitly. Using the number estimatein [3, Lemma 2.5], one proves the inequality. (cid:3) Acknowledgements
The authors are grateful to Jean-Bernard Bru, Sylvain Gol´enia and Vedran Sohinger for helpfuldiscussions.
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Univ Rennes, [UR1], CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
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