Quantum Spin Systems after DLS1978

TL;DR

This paper reviews the significant developments in the study of quantum spin systems, especially the Heisenberg model, since Dyson, Lieb, and Simon's 1978 proof of Ne'el order at finite temperature, highlighting key extensions and applications.

Contribution

It provides a comprehensive overview of the advances in understanding quantum lattice systems following the foundational DLS 1978 results.

Findings

Extension of DLS methods to various quantum models

Proofs of Ne'el order in different dimensions and spin values

Development of new techniques for quantum symmetry breaking

Abstract

In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Ne'el order at positive temperature for the spin-S Heisenberg antiferromagnet on the d-dimensional hypercubic lattice when either S >= 1 and d >= 3 or S=1/2 and d is sufficiently large. This was the first proof of spontaneous breaking of a continuous symmetry in a quantum model at finite temperature. Since then the ideas of DLS have been extended and adapted to a variety of other problems. In this paper I will present an overview of the most important developments in the study of the Heisenberg model and related quantum lattice systems since 1978, including but not restricted to those directly related to the paper by DLS.