The classical limit of mean-field quantum spin systems

Abstract

The theory of strict deformation quantization of the two sphere $S^2\subset {\mathbb{R}}^3$ is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by $H_N$ and where $N$ indicates the number of sites. Indeed, since the fibers $A_{1/N}=M_{N+1}(\mathbb{C})$ and $A_0=C(S^2)$ form a continuous bundle of $C^{\ast }$-algebras over the base space $I=\{0\}\cup 1/{\mathbb{N}}^{\ast }\subset [0,1]$, one can define a strict deformation quantization of $A_0$ where quantization is specified by certain quantization maps $Q_{1/N}:{\tilde{A}}_0\to A_{1/N}$, with ${\tilde{A}}_0$ a dense Poisson subalgebra of $A_0$. Given now a sequence of such $H_N$, we show that under some assumptions a sequence of eigenvectors ${\psi }_N$ of $H_N$ has a classical limit in the sense that ${\omega }_0(f):={\lim }_{N\to \infty }⟨{\psi }_N,Q_{1/N}(f){\psi }_N⟩$ exists as a state on $A_0$ given by ${\omega }_0(f)=\frac{1}{n}{\sum }_{i=1}^{n}f({\Omega }_i)$, where $n$ is some natural number. We give an application regarding spontaneous symmetry breaking (SSB) and moreover we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere $S^2$.

32 pages