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- W3038366930 abstract "Since the fibers $A_{1/N}=M_{N+1}(mathbb{C})cong B(text{Sym}^N(mathbb{C}^2))$ 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 $A_0$. In this paper we prove the existence of the classical limit of mean-field quantum spin chains whose Hamiltonians $H_N$ typically satisfy $H_N|_{text{Sym}^N(mathbb{C}^2)}=Q_{1/N}(h_0)+sum_{k=1}^MN^{-k}Q_{1/N}(h_k)$, for some natural number $M$ and real polynomials $h_0$ and $h_k$ on the unit two-sphere $S^2 (k=1,...M)$, where $text{Sym}^N(mathbb{C}^2)$ denotes the symmetric subspace of the Hilbert space $bigotimes_{n=1}^Nmathbb{C}^2$ on which $H_N$ is originally defined. Under some assumptions we show that if a sequence of eigenvectors $psi_N$ of $H_N|_{text{Sym}^N(mathbb{C}^n)}$ is invariant under some group action, then this sequence 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 $C(S^2)$ given by $omega_0(f)=frac{1}{n}sum_{i=1}^nf(Omega_i)$. We give an application regarding spontaneous symmetry breaking (SSB) and moreover we show that the spectrum of a mean field quantum spin system converges to the range of the polynomial $h_0$, in that $lim_{Ntoinfty}text{dist}(sigma(H_N|_{text{Sym}^N(mathbb{C}^2}),text{ran}(h_0))=0$, where dist denotes the distance function." @default.
- W3038366930 created "2020-07-10" @default.
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- W3038366930 date "2020-07-07" @default.
- W3038366930 modified "2023-09-27" @default.
- W3038366930 title "The classical limit of mean-field theories" @default.
- W3038366930 hasPublicationYear "2020" @default.
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