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A continuous bundle of C ∗ -algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the classical limit of the quantum system, i.e. a mathematical formalism that examines the convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold). In this manner, we first prove the existence of the classical limit of Gibbs states illustrated with a class of Schrödinger operators in the regime where Planck's constant ℏ appearing in front of the Laplacian approaches zero. We additionally show that the ensuing limit corresponds to the unique probability measure satisfying the so-called classical or static KMS-condition. Subsequently, we conduct a similar study on the free energy of mean-field quantum spin systems in the regime of large particles, and discuss the existence of the classical limit of the relevant Gibbs states. Finally, a short section is devoted to single site quantum spin systems in the large spin limit. [ABSTRACT FROM AUTHOR]

The theory of strict deformation quantization of the two-sphere S 2 ⊂ R 3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A 1 / N = M N + 1 (C) and A0 = C(S2) form a continuous bundle of C*-algebras over the base space I = { 0 } ∪ 1 / N * ⊂ [ 0 , 1 ] , one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps Q 1 / N : A ̃ 0 → A 1 / N , with A ̃ 0 being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by ω 0 (f) = 1 n ∑ i = 1 n f ( Ω i ) , where n is some natural number. We give an application regarding spontaneous symmetry breaking, and moreover, we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere S2. [ABSTRACT FROM AUTHOR]

The existence of a strict deformation quantization of X k = S (M k (C)) , the state space of the k × k matrices M k (C) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. 10.1142/S0129055X20500312). In fact, since increasing tensor powers of the k × k matrices M k (C) are known to give rise to a continuous bundle of C ∗ -algebras over I = { 0 } ∪ 1 / N ⊂ [ 0 , 1 ] with fibers A 1 / N = M k (C) ⊗ N and A 0 = C (X k) , we were able to define a strict deformation quantization of X k à la Rieffel, specified by quantization maps Q 1 / N : / A ~ 0 → A 1 / N , with A ~ 0 a dense Poisson subalgebra of A 0 . A similar result is known for the symplectic manifold S 2 ⊂ R 3 , for which in this case the fibers A 1 / N ′ = M N + 1 (C) ≅ B (Sym N (C 2)) and A 0 ′ = C (S 2) form a continuous bundle of C ∗ -algebras over the same base space I, and where quantization is specified by (a priori different) quantization maps Q 1 / N ′ : A ~ 0 ′ → A 1 / N ′ . In this paper, we focus on the particular case X 2 ≅ B 3 (i.e., the unit three-ball in R 3 ) and show that for any function f ∈ A ~ 0 one has lim N → ∞ | | (Q 1 / N (f)) | Sym N (C 2) - Q 1 / N ′ (f | S 2 ) | | N = 0 , where Sym N (C 2) denotes the symmetric subspace of (C 2) N ⊗ . Finally, we give an application regarding the (quantum) Curie–Weiss model. [ABSTRACT FROM AUTHOR]

Increasing tensor powers of the k × k matrices M k (ℂ) are known to give rise to a continuous bundle of C ∗ -algebras over I = { 0 } ∪ 1 / ℕ ⊂ [ 0 , 1 ] with fibers A 1 / N = M k (ℂ) ⊗ N and A 0 = C (X k) , where X k = S (M k (ℂ)) , the state space of M k (ℂ) , which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of X k à la Rieffel, defined by perfectly natural quantization maps Q 1 / N : à 0 → A 1 / N (where à 0 is an equally natural dense Poisson subalgebra of A 0 ). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its ℤ 2 symmetry is spontaneously broken in the thermodynamic limit N → ∞. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space X 2 ≅ B 3 (i.e. the unit three-ball in ℝ 3 ). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors Ψ N (0) of this model as N → ∞ , in which the sequence converges to a probability measure μ on the associated classical phase space X 2 . This measure is a symmetric convex sum of two Dirac measures related by the underlying ℤ 2 -symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid. [ABSTRACT FROM AUTHOR]