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

September 24, 2019

Increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ are
known to give rise to a continuous bundle of $C^*$-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 $X_k$ \`{a} la Rieffel, defined by perfectly natural
quantization maps $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$ (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
$N\to\infty$. 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 $\Psi_N^{(0)}$ of this model as $N\to\infty$, in which the
sequence converges to a probability measure $\mu$ 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.

Keywords:

Deformation quantization, Strict Quantization Map, spontaneous symmetry breaking, Curie-Weiss model