A Multi-Dimensional Lieb-Schultz-Mattis Theorem

TL;DR

This paper proves a multi-dimensional Lieb-Schultz-Mattis theorem for quantum spin models, showing that ground state uniqueness implies low-energy excitations with a specific energy bound, extending previous results to higher dimensions.

Contribution

It provides a rigorous proof of a multi-dimensional Lieb-Schultz-Mattis theorem, establishing a bound on excitation energy for finite-range quantum spin models with half-integer spins.

Findings

Ground state uniqueness implies low-lying excitations.

Excitation energy bound scales as (C log L)/L.

Extends Lieb-Schultz-Mattis theorem to higher dimensions.

Abstract

For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C\log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent result by Hastings.