Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model

Abstract

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. 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 [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). 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 [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-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.

Bulk-boundary asymptotic equivalence of two strict deformation quantizations

“…This led to the general concept of a strict deformation of a Poisson manifold X [13,22], which we here state in the case of interest to us in which X is compact, or more generally in which X is a manifold with stratified boundary [15,19]. In that case, the space I in which takes values cannot be all of [0, 1], but should be a subspace I ⊂ [0, 1] thereof that at least contains 0 as an accumulation point.…”

“…That is, we put = 1/N , where N ∈ N is interpreted as the number of sites of the model; our interest is the limit N → ∞. In the framework of C * -algebraic quantization theory, the analogy between the "classical" limit → 0 in typical examples from mechanics (see, e.g., our first example [10]) and the "thermodynamic" limit N → ∞ in typical quantum spin systems (see, e.g., [15,16]) is developed in detail in [14]. We remark that the limit N → ∞ can be taken in two entirely different ways, which depends on the class of observables one considers, namely either quasi-local observables or macroscopic observables.…”

“…The former are the ones traditionally studied for quantum spin systems, but the latter relate these systems to strict deformation quantization, since macroscopic observables are precisely defined by (quasi-)symmetric sequences which form the continuous cross sections of a continuous bundle of C * -algebras. This continuous bundle of C * -algebras is defined over base space I given by (1.9) with fibers 11) and continuity structure specified by continuous cross sections which are thus given by all quasi-symmetric sequences [15] [14, Ch.10]. 3 We refer to the appendix for some useful definitions or to [15] for a more comprehensive explanation.…”

“…This continuous bundle of C * -algebras is defined over base space I given by (1.9) with fibers 11) and continuity structure specified by continuous cross sections which are thus given by all quasi-symmetric sequences [15] [14, Ch.10]. 3 We refer to the appendix for some useful definitions or to [15] for a more comprehensive explanation. The space X k = S(M k (C)) ⊂ R k 2 −1 has the structure of a compact Poisson manifold with stratified boundary.…”

“…., (1.14) and more generally, they are defined as the unique continuous and linear extensions of the written maps. It has been shown in [15] that the quantization maps Q 1/N satisfy all the axioms of Definition 1.1. 4 These data together imply the existence of a strict deformation quantization of the Poisson manifold X k = S(M k (C)) (see [15,Theorem 3.4] for a detailed proof).…”