Ordering of Energy Levels in Heisenberg Models and Applications

TL;DR

This paper discusses the ferromagnetic ordering of energy levels in Heisenberg models, proves a key conjecture for chain models, and explores implications for relaxation times in related stochastic processes.

Contribution

It provides a proof of the FOEL conjecture for Heisenberg chains with arbitrary spins and coupling constants, and discusses its implications.

Findings

Proved FOEL conjecture for chain models with arbitrary spins.

Established the equality of relaxation times in certain stochastic processes.

Discussed extensions and implications of the energy level ordering.

Abstract

In a recent paper we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.