# AKLT models on decorated square lattices are gapped

## TL;DR

This paper proves that AKLT models on decorated square lattices are gapped for certain decorations, extends numerical methods to establish gaps on various lattices, and shows these states are not Ne9el ordered.

## Contribution

The paper analytically proves the spectral gap for decorated square and hybrid lattices and introduces advanced numerical techniques to determine gaps on additional lattices.

## Key findings

- AKLT models on decorated square lattices are gapped for na0a0 4
- Numerical methods establish gaps for na0a0 2 on triangular and cubic lattices
- Decorated cubic lattice AKLT states are not Ne9el ordered

## Abstract

The nonzero spectral gap of the original two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) models has remained unproven for more than three decades. Recently, Abdul-Rahman et al. [arXiv:1901.09297] provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number $n$ of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if $n\ge 4$. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for $n\ge 4$. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to $n\ge 2$, including the establishment of the gap for $n=2$ in the decorated triangular and cubic lattices. The latter case is interesting, as this shows the AKLT states on the decorated cubic lattices are not N\'eel ordered, in contrast to the state on the undecorated cubic lattice.