Spectral gaps of frustration-free spin systems with boundary

TL;DR

This paper develops finite-size criteria for spectral gaps in frustration-free quantum spin systems with boundaries, showing how bulk and edge gaps determine the presence of a spectral gap in the thermodynamic limit.

Contribution

It extends existing spectral gap criteria to systems with boundaries, incorporating edge effects and applicable across various dimensions and lattice types.

Findings

Spectral gaps at size n exceeding n^{-3/2} imply a gapped infinite system.

The criterion accounts for both bulk and edge gaps explicitly.

Application to 2D models supports the absence of chiral edge modes in frustration-free systems.

Abstract

In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss "finite-size" criteria for having a spectral gap in frustration-free spin systems and their applications. We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size $n$ exceed an explicit threshold of order $n^{-3/2}$, then the whole system is gapped. The criterion takes into account both "bulk gaps" and "edge gaps" of the finite system in a precise way. The $n^{-3/2}$ scaling is robust: it holds in 1D and 2D systems, on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like $n^{-1}$).