On a bulk gap strategy for quantum lattice models

TL;DR

This paper reviews a bulk gap strategy for quantum lattice models that helps determine the presence of a spectral gap in the thermodynamic limit, especially when edge excitations are low-lying but the bulk remains gapped.

Contribution

It reformulates spectral gap methods within an invariant subspace framework and applies this approach to truncated Haldane pseudopotentials in cylinder geometry.

Findings

The bulk gap strategy effectively separates ground and edge states.

Application to 1/3-filled Haldane pseudopotential confirms the method's utility.

Reformulation broadens the applicability of spectral gap bounding techniques.

Abstract

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [Warzel, Young '22] and [Warzel, Young '23] while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.