# Gaplessness is not generic for translation-invariant spin chains

## TL;DR

This paper shows that translation-invariant quantum spin chains are more likely to have a spectral gap, especially when interactions are of small rank, contrasting with the generic gaplessness in random Hamiltonians.

## Contribution

It demonstrates that translation-invariant spin chains with small-rank interactions are gapped with positive probability, extending previous results and highlighting the role of frustration-freeness.

## Key findings

- Translation-invariant chains with small-rank interactions are often gapped.
- Randomly constructed Hamiltonians are frustration-free due to rank constraints.
- Gap probability is positive under the small-rank interaction assumption.

## Abstract

The existence of a spectral gap above the ground state has far-reaching consequences for the low-energy physics of a quantum many-body system. A recent work of Movassagh [R. Movassagh, PRL 119 (2017), 220504] shows that a spatially random local quantum Hamiltonian is generically gapless. Here we observe that a gap is more common for translation-invariant quantum spin chains, more specifically, that these are gapped with a positive probability if the interaction is of small rank. This is in line with a previous analysis of the spin-$1/2$ case by Bravyi and Gosset. The Hamiltonians are constructed by selecting a single projection of sufficiently small rank at random, and then translating it across the entire chain. By the rank assumption, the resulting Hamiltonians are automatically frustration-free and this fact plays a key role in our analysis.