Existence of a spectral gap in the AKLT model on the hexagonal lattice

Marius Lemm, Anders W. Sandvik, Ling Wang

Published 2019-10-25Version 1

The $S=1$ 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 question 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 a rigorous, 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 by performing state-of-the-art DMRG calculations on the subsystem.