# The AKLT model on a hexagonal chain is gapped

## TL;DR

This paper proves that the AKLT model on a hexagonal chain has a spectral gap, using a finite-size criterion, and discusses potential extensions to the full hexagonal lattice.

## Contribution

It introduces a finite-size criterion to establish the spectral gap of the AKLT model on a hexagonal chain, advancing understanding of its spectral properties.

## Key findings

- The AKLT Hamiltonian on a hexagonal chain is gapped.
- Finite-size criterion effectively verifies the spectral gap.
- Method potential for generalization to full hexagonal lattice.

## Abstract

In 1987, Affleck, Kennedy, Lieb, and Tasaki introduced the AKLT spin chain and proved that it has a spectral gap above the ground state. Their concurrent conjecture that the two-dimensional AKLT model on the hexagonal lattice is also gapped remains open. In this paper, we show that the AKLT Hamiltonian restricted to an arbitrarily long chain of hexagons is gapped. The argument is based on explicitly verifying a finite-size criterion which is tailor-made for the system at hand. We also discuss generalizations of the method to the full hexagonal lattice.