Improved local spectral gap thresholds for lattices of finite dimension

TL;DR

This paper establishes near-optimal local spectral gap thresholds for frustration-free lattice Hamiltonians in finite dimensions, extending and improving previous one-dimensional results using the detectability lemma and coarse-grained Hamiltonian concepts.

Contribution

It introduces a new local spectral gap threshold for finite-dimensional lattices that is optimal up to a constant factor, generalizing prior one-dimensional findings.

Findings

Provides a local spectral gap threshold that is dimension-dependent and nearly optimal.

Utilizes the detectability lemma framework and coarse-grained Hamiltonian to connect global and local spectral gaps.

Extends spectral gap results from one-dimensional systems to higher-dimensional lattices.

Abstract

Knabe's theorem lower bounds the spectral gap of a one dimensional frustration-free local hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap much scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free hamiltonians on a finite dimensional lattice, that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of coarse-grained hamiltonian (introduced in [Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.