Spin Excitations and Sum Rules in the Heisenberg Antiferromagnet

TL;DR

This paper discusses bounds on collective excitation energies in the Heisenberg antiferromagnet using sum rules, highlighting the limitations of the Feynman approximation and proposing bounds related to the order parameter.

Contribution

It introduces new bounds for excitation energies, compares them with Feynman approximation, and discusses their relation to classical spin wave theory and anisotropic models.

Findings

Feynman approximation overestimates spin velocity by about 30% in the $S=1/2$ square lattice.

A Goldstone-type bound depending on the order parameter is proportional to classical spin wave dispersion.

Rigorous bounds for anisotropic Heisenberg models are established.

Abstract

Various bounds for the energy of collective excitations in the Heisenberg antiferromagnet are presented and discussed using the formalism of sum rules. We show that the Feynman approximation significantly overestimates (by about 30\% in the $S={1\over2}$ square lattice) the spin velocity due to the non negligible contribution of multi magnons to the energy weighted sum rule. We also discuss a different, Goldstone type bound depending explicitly on the order parameter (staggered magnetization). This bound is shown to be proportional to the dispersion of classical spin wave theory with a q-independent normalization factor. Rigorous bounds for the excitation energies in the anisotropic Heisenberg model are also presented.