Extension of the Lieb-Schultz-Mattis theorem

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to include frustrated systems, excited states, and models lacking discrete symmetry, broadening its applicability to more complex quantum spin systems.

Contribution

It generalizes the LSM theorem to frustrated and nonsymmetric models, and improves proofs for excited state extensions.

Findings

Extended LSM theorem to frustrated systems

Proved LSM for excited states with continuous energy eigenvalues

Established applicability to models without discrete symmetry

Abstract

Lieb, Schultz and Mattis (LSM) studied the S=1/2 XXZ spin chain. Theorems of LSM's paper can be applied to broader models. In the original LSM theorem it was assumed the nonfrustrating system. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers have tried to extend the LSM theorem for excited states. In the cases $S^{z}_{T} = \pm 1,\pm 2 \cdots$, the lowest energy eigenvalues are continuous for wave number $q$. But we find that their proofs are insufficient, and we improve them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that the LSM type theorems are applicable for Dzyaloshinskii-Moriya type interactions or other nonsymmetric models.