Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings for the Singlet

TL;DR

This paper demonstrates a violation of the ferromagnetic ordering of energy levels conjecture in even-length spin rings, showing that the lowest energy state at spin zero can be lower than at spin one, challenging previous assumptions.

Contribution

The authors provide numerical evidence and rigorous analysis showing the violation of FOEL in even-length spin rings, including a proof of the unique ground state in the spin zero sector.

Findings

Lowest spin 0 energy is lower than spin 1 energy for even L>4.

Violates the FOEL conjecture's (L/2)-th inequality.

Numerical exact diagonalization up to L=20 supports the findings.

Abstract

We demonstrate a violation of the ``ferromagnetic ordering of energy levels'' conjecture (FOEL) for even length spin rings. The FOEL conjecture was a guess made by Nachtergaele, Spitzer and an author for the Heisenberg model on certain graphs: a family of inequalities, the first of which is the statement that the spectral gap of the Heisenberg model equals the gap of the random walk. That first guess was originally a conjecture of Aldous which was later proved by Caputo, Liggett and Richthammer. We claim that for spin rings of even length $L>4$, the lowest spin $S=0$ energy is lower than the lowest spin $S=1$ energy. This violates the $(L/2)$-th inequality in the FOEL conjecture. Our methodology is largely numerical: we have applied exact diagonalization up to $L=20$. We also rigorously consider the Hamiltonian of the Heisenberg spin ring for even length $L$ projected to the spin $S=0$ sector. We prove that it has a unique ground state. Then, using the single mode approximation the uniqueness explains the energy turn-around. Important insight comes from reconsideration of previous work by Sutherland, using the Bethe ansatz. Especially important is a work of Dhar and Shastry that goes beyond the Bethe ansatz.