# The classical limit of mean-field quantum spin systems

July 07, 2020

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
$C^*$-algebras over the base space $I=\{0\}\cup 1/\mathbb{N}^*\subset[0,1]$,
one can define a strict deformation quantization of $A_0$ where quantization is
specified by certain quantization maps $Q_{1/N}: \tilde{A}_0 \rightarrow
A_{1/N}$, 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
$\omega_0(f):=\lim_{N\to\infty}\langle\psi_N,Q_{1/N}(f)\psi_N\rangle$ exists as
a state on $A_0$ given by $\omega_0(f)=\frac{1}{n}\sum_{i=1}^nf(\Omega_i)$,
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 $S^2$.

Keywords:

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