For the Quantum Heisenberg Ferromagnet, a Polymer Expansion and its High T Convergence

Abstract

We let Psi_0 be a wave function for the Quantum Heisenberg ferromagnet sharp i sigma_zi and Psi_mu = exp(-mu*H)Psi_0. We study expectations similar to the form <Psi_mu,(sigma_zi)Psi_mu>/<Psi_mu,Psi_mu> for which we present a formal polymer expansion, whose convergence we prove for sufficiently small mu. The approach of the paper is to relate the wavefunction Psi_mu to an approximation to it that is a product function. In the jth spot of the product approximation the upper component is phi_mu(j), and the lower component is (1-phi_mu(j)), where phi satisfies the lattice heat equation. This is shown via a cluster or polymer expansion. The present work began in a previous paper, primarily a numerical study, and provides a proof of results related to Conjecture 3 of this previous paper.