Extension of the Lieb-Schultz-Mattis and Kolb theorem

TL;DR

This paper extends and simplifies the proof of the Lieb-Schultz-Mattis and Kolb theorem, broadening its applicability to frustrated and non-symmetric quantum spin models.

Contribution

It provides a more straightforward proof of the extended LSMK theorem, removing the assumption of unique ground states in complex frustrated systems.

Findings

Extended the LSMK theorem to frustrated and non-symmetric cases

Simplified the proof of the theorem's continuity argument

Broadened the theorem's applicability to more complex models

Abstract

The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. Independently, Kolb considered the relation between the wave number $q$ and the twisting boundary condition, and he obtained a similar result as LSM. However, in frustrating cases it is known that there exist several exceptions for the assumption of the unique lowest state for the finite size, which is important in the traditional LSM theorem. In our previous paper, without the assumption of the uniqueness, we have extended the LSMK theorem for frustrating and non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work.