Strict deformation quantization of the state space of $M_k(\mathbb{C})$ with applications to the Curie-Weiss model

Abstract

Increasing tensor powers of the $k\times k$ matrices $M_k(\mathbb{C})$ are known to give rise to a continuous bundle of $C^{\ast }$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset [0,1]$ with fibers $A_{1/N}=M_k(\mathbb{C}{)}^{\otimes N}$ and $A_0=C(X_k)$, where $X_k=S(M_k(\mathbb{C}))$, the state space of $M_k(\mathbb{C})$, which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of $X_k$ Ã la Rieffel, defined by perfectly natural quantization maps $Q_{1/N}:{\tilde{A}}_0\to A_{1/N}$ (where ${\tilde{A}}_0$ 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 ${\mathbb{Z}}_2$ symmetry is spontaneously broken in the thermodynamic limit $N\raw\infty$. 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 $X_2\cong B^3$ (i.e\ the unit three-ball in ${\mathbb{R}}^3$). 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 $\hat{I}{¨}_N^{(0)}$ of this model as $N\raw\infty$, in which the sequence converges to a probability measure $\hat{I}¼$ on the associated classical phase space $X_2$. This measure is a symmetric convex sum of two Dirac measures related by the underlying ${\mathbb{Z}}_2$-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.

53 pages, 13 figures/tables