# 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.

open access link

Reviews in Mathematical Physics (2020) in press

@article{,
key = "1756317",
author = "Landsman, Klaas and Moretti, Valter and van de Ven,
Christiaan J. F.",
title = "{Strict deformation quantization of the state space of
$M_k(\mathbb{C})$ with applications to the Curie-Weiss
model}",
eprint = "1909.10947",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1909.10947;%%"
}

Keywords:

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