# A class of two-dimensional AKLT models with a gap

## TL;DR

This paper introduces a family of two-dimensional AKLT models on the hexagonal lattice, decorated with one-dimensional chains, and proves they have a spectral gap for all decorations with length three or more.

## Contribution

The paper proves that a new class of decorated two-dimensional AKLT models on the hexagonal lattice are gapped for all chain lengths n ≥ 3.

## Key findings

- Models are gapped for all n ≥ 3
- Supports the conjecture about the spectral gap in 2D AKLT models
- Provides a method to construct gapped 2D quantum spin systems

## Abstract

The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer $n$, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length $n$. We prove that these decorated models are gapped for all $n \geq 3$.