Lieb-Schultz-Mattis in Higher Dimensions

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to higher-dimensional spin systems, showing that such systems either exhibit long-range order with gapless modes or possess topological excitations with unique properties.

Contribution

It introduces a generalized theorem for higher dimensions using loop operators and establishes bounds that rule out long-range order, demonstrating the existence of topological excitations.

Findings

Higher-dimensional spin systems have either gapless modes or topological excitations.

Cluster bounds are derived to analyze gapped systems.

Topologically excited states differ in momentum but share local expectation values with ground states.

Abstract

A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.