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

TL;DR

This paper develops a formal polymer expansion for the Quantum Heisenberg ferromagnet's wavefunctions, proving convergence at high temperatures and relating the wavefunction to a heat equation approximation.

Contribution

It introduces a convergent polymer expansion for expectations in the Quantum Heisenberg ferromagnet at high temperature, connecting wavefunctions to heat equation solutions.

Findings

Polymer expansion converges for small mu (high temperature).

Wavefunction approximated by a product function related to the heat equation.

Provides a rigorous proof supporting previous numerical conjectures.

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.