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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 existence of various physical phenomena stems from the concept called asymptotic emergence, that is, they seem to be exclusively reserved for certain limiting theories. Important examples are spontaneous symmetry breaking (SSB) and phase transitions: these would only occur in the classical or thermodynamic limit of underlying finite quantum systems, since for finite quantum systems, due to the uniqueness of the relevant states, such phenomena are excluded by Theory. In Nature, however, finite quantum systems describing real materials clearly exhibit such effects. In this paper we discuss these apparently "paradoxical" phenomena and outline various ideas and mechanisms that encompass both theory and reality, from physical and mathematical points of view. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many particle systems and quantization maps. [ABSTRACT FROM AUTHOR]

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space ℝ 2 n . The corresponding quantization maps enable us to take the limit for ℏ → 0 of a suitable sequence of algebraic vector states induced by ℏ -dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on ℝ 2 n , defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of C ∗ algebras) parametrized by ℏ. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in ℝ 2 n depending on the symmetry of the potential. In addition, since this C ∗ -algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off ℏ. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models. [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]