The classical limit of mean-field quantum spin systems

Journal of Mathematical Physics, Volume 61, Issue 12, December 2020. The theory of strict deformation quantization of the two-sphere [math] is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers [math] and A0 = C(S2) form a continuous bundle of C*-algebras over the base space [math], one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps [math], with [math] being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by [math], where n is some natural number. We give an application regarding spontaneous symmetry breaking, 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 S2.