Classical limit of Gibbs states for quantum spin systems

Abstract

We study the relation between quantum mechanical and classical Gibbs states of spin systems with spin quantum number $s$. It is known that quantum states and observables can be represented by functions defined on the phase space ${\mathcal S}$, which in our case is the $N$-fold product of unit spheres. Therefore, the classical limit $s\to\infty$ of (suitably scaled) quantum Gibbs states can be described as the limit of functions defined on ${\mathcal S}$. We choose to approximate the exponential function of the Hamiltonian by a polynomial of degree $n$ and thus have to deal with the problem of the limit of double sequences (depending on $n$ and $s$) treated in the theorem of Moore-Osgood. The convergence of quantum Gibbs states to classical ones is illustrated by the example of the Heisenberg dimer. We apply our method to the explicit calculation of the phase space function describing spin monomials, and finally add some general remarks on the theory of spin coherent states.