A Strict Deformation Quantization Map on the state space of M_k(C) and the Classical Limit of the Curie-Weiss model

Increasing tensor powers of the k×k matrices Mk(C) are known to give rise to a continuous bundle of C∗-algebras over I={0}∪1/N⊂[0,1] with fibers A1/N=Mk(C)⊗N and A0=C(Xk), where Xk=S(Mk(C)), the state space of Mk(C), which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of Xk à la Rieffel, defined by perfectly natural quantization maps Q1/N:Ã0→A1/N (where Ã0 is an equally natural dense Poisson subalgebra of A0). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its Z2 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 X2≅B3 (i.e. the unit three-ball in R3). 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(0)N of this model as N→∞, in which the sequence converges to a probability measure μ on the associated classical phase space X2. This measure is a symmetric convex sum of two Dirac measures related by the underlying Z2-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.