Counterexamples to Ferromagnetic Ordering of Energy Levels

TL;DR

This paper presents counterexamples to the ferromagnetic ordering of energy levels in the Heisenberg model, showing that the expected monotonic energy behavior does not always hold, especially in certain ring configurations.

Contribution

It provides the first known counterexamples to FOEL in the Heisenberg ferromagnet, challenging previous assumptions and extending the understanding of energy level ordering.

Findings

Counterexamples in spin rings of length 4 to 16.

Violations of FOEL in specific graph configurations.

Implications for the Aldous ordering in symmetric exclusion processes.

Abstract

The Heisenberg ferromagnet has symmetry group ${\rm SU}(2)$. The property known as ferromagnetic ordering of energy levels (FOEL) states that the minimum energy eigenvalue among eigenvectors with total spin $s$ is monotone decreasing as a function of $s$. While this property holds for certain graphs such as open chains, in this note we demonstrate some counterexamples. We consider the spin 1/2 model on rings of length $2n$ for $n=2,3,...,8$, and show that the minimum energy among all spin singlets is less than or equal to the minimum energy among all spin triplets, which violates FOEL. This also shows some counterexamples to the "Aldous ordering" for the symmetric exclusion process. We also review some of the literature related to these examples.