Lieb-Schultz-Mattis theorem for quasi-topological systems

TL;DR

This paper investigates the spectral gap properties of a class of local Hamiltonians with known ground states, revealing conditions for gaplessness and gapped topological phases, and introduces the concept of quasi-topological critical points.

Contribution

It introduces the concept of quasi-topological critical points where local correlations are short-ranged but certain non-local correlations are critical, and analyzes their implications for topological phases.

Findings

Critical points exhibit gapless behavior with quadratic dispersion.

Presence of specific Hamiltonian terms leads to gapped topological phases.

Short-range local correlations coexist with critical non-local correlations.

Abstract

In this paper we address the question of the existence of a spectral gap in a class of local Hamiltonians. These Hamiltonians have the following properties: their ground states are known exactly; all equal-time correlation functions of local operators are short-ranged; and correlation functions of certain non-local operators are critical. A variational argument shows gaplessness with $\omega \propto k^2$ at critical points defined by the absence of certain terms in the Hamiltonian, which is remarkable because equal-time correlation functions of local operators remain short-ranged. We call such critical points, in which spatial and temporal scaling are radically different, quasi-topological. When these terms are present in the Hamiltonian, the models are in gapped topological phases which are of special interest in the context of topological quantum computation.