Random translation-invariant Hamiltonians and their spectral gaps

TL;DR

This paper studies random translation-invariant quantum spin Hamiltonians in multiple dimensions, showing they are frustration-free and gapped with positive probability under certain conditions, and applies this to derive a 2D area law for ground states.

Contribution

It extends known 1D results to higher dimensions, demonstrating that such Hamiltonians are frustration-free and gapped with positive probability under a rank constraint.

Findings

Hamiltonians are frustration-free under small rank constraint.

They are gapped with positive probability in all dimensions.

A 2D area law for ground states is established using AGSP methods.

Abstract

We consider random translation-invariant frustration-free quantum spin Hamiltonians on $\mathbb Z^D$ in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.