Generic Local Hamiltonians are Gapless

TL;DR

This paper proves that generic local quantum Hamiltonians are inherently gapless, with a continuous density of states above the ground state, applicable across various lattice dimensions and graph structures, including disordered and random models.

Contribution

It establishes that generic local Hamiltonians are gapless and provides explicit calculations of the gap scaling for Gaussian random ensembles, excluding translationally invariant cases.

Findings

Generic local Hamiltonians are gapless with a continuous density of states.

Finite size partitions can approximate ground states as product states.

Ground state degeneracies occur when local eigenvalue distributions are discrete.

Abstract

We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian $\beta-$orthogonal random matrix ensemble. As a corollary there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.