The classical limit of mean-field quantum spin systems

The theory of strict deformation quantization of the two-sphere S2 subset of R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by H-N, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A(0) = C(S-2) form a continuous bundle of C*-algebras over the base space I={0}1/N*subset of [0,1], one can define a strict deformation quantization of A(0) where quantization is specified by certain quantization maps Q1/N:A</mml:mover>0 -> A1/N, with A</mml:mover>0 being 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 -> infinity)psi (N), Q(1/N)(f)psi (N) exists as a state on A(0) given by omega 0(f)=<mml:mfrac>1n</mml:mfrac>Sigma i=1n</mml:msubsup>f<mml:mo stretchy="false">(<mml:msub>Omega i<mml:mo stretchy="false">), 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 S-2.