Local Topological Quantum Order and Spectral Gap Stability for the AKLT Models on the Hexagonal and Lieb Lattices

TL;DR

This paper proves that AKLT models on hexagonal and Lieb lattices exhibit local topological quantum order and spectral gap stability, using a detailed polymer representation analysis.

Contribution

It establishes LTQO and spectral gap stability for these models by analyzing ground state indistinguishability and boundary effects.

Findings

Ground states are indistinguishable from the infinite volume state with exponential decay.

Spectral gap remains stable under small perturbations with sufficient decay.

Finite volume ground state expectations approximate the infinite volume state effectively.

Abstract

We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.