Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

TL;DR

This paper proves the existence of a spectral gap in the $S=3/2$ AKLT model on the hexagonal lattice, confirming it is in a gapped phase through a combination of mathematical and computational methods.

Contribution

It introduces a finite-size criterion for spectral gaps and verifies it numerically, establishing a positive lower bound for the gap in the hexagonal AKLT model.

Findings

Confirmed a size-independent spectral gap $ extgreater 0.006$ in the hexagonal AKLT model

Developed a finite-size criterion linking subsystem gap to the entire system

Validated the criterion using high-precision DMRG calculations

Abstract

The $S=1$ Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the $S=3/2$ AKLT model on the hexagonal lattice is also in a gapped phase has remained open, despite being a fundamental problem of ongoing relevance to condensed-matter physics and quantum information theory. Here we confirm this conjecture by demonstrating the size-independent lower bound $\Delta >0.006$ on the spectral gap of the hexagonal model with periodic boundary conditions in the thermodynamic limit. Our approach consists of two steps combining mathematical physics and high-precision computational physics. We first prove a mathematical finite-size criterion which gives an analytical, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value. Then we verify the finite-size criterion numerically by performing state-of-the-art DMRG calculations on the subsystem.