Ferromagnetic Ordering of Energy Levels

TL;DR

This paper proves a conjecture about ferromagnetic energy level ordering in the 1D spin-1/2 Heisenberg XXX and XXZ models, showing lowest energies correspond to maximal total spin, extending understanding of quantum ferromagnets.

Contribution

The authors establish the ferromagnetic energy level ordering conjecture for 1D spin-1/2 Heisenberg models using algebraic and monotonicity techniques, which was previously unproven.

Findings

Confirmed energy levels are strictly ordered by total spin in 1D ferromagnetic models.

Proved the ground state has maximal total spin in these models.

Showed the first excited state on finite trees has one less than maximal total spin.

Abstract

We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb-Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.