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- W3111554911 abstract "The theory of strict deformation quantization of the two sphere $S^2subsetmathbb{R}^3$ 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 $omega_0(f):=lim_{Ntoinfty}langlepsi_N,Q_{1/N}(f)psi_Nrangle$ 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 $S^2$." @default.
- W3111554911 created "2020-12-21" @default.
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- W3111554911 date "2020-12-01" @default.
- W3111554911 modified "2023-10-16" @default.
- W3111554911 title "The classical limit of mean-field quantum spin systems" @default.
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- W3111554911 doi "https://doi.org/10.1063/5.0021120" @default.
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