Deformations of the Boundary Theory of the Square Lattice AKLT Model

TL;DR

This paper investigates the boundary theory of the 2D AKLT model on a square lattice, using a classical loop model representation to numerically demonstrate the existence of a spectral gap and identify different phases.

Contribution

It introduces a novel boundary state representation of the 2D AKLT model as a classical loop model and provides numerical evidence for a spectral gap and phase distinctions.

Findings

Evidence of a spectral gap in the 2D AKLT model

Identification of three distinct phases in the loop model

Numerical evaluation of boundary Hamiltonian

Abstract

The 1D AKLT model is a paradigm of antiferromagnetism, and its ground state exhibits symmetry-protected topological order. On a 2D lattice, the AKLT model has recently gained attention because it too displays symmetry-protected topological order, and its ground state can act as a resource state for measurement-based quantum computation. While the 1D model has been shown to be gapped, it remains an open problem to prove the existence of a spectral gap on the 2D square lattice, which would guarantee the robustness of the resource state. Recently, it has been shown that one can deduce this spectral gap by analyzing the model's boundary theory via a tensor network representation of the ground state. In this work, we express the boundary state of the 2D AKLT model in terms of a classical loop model, where loops, vertices, and crossings are each given a weight. We use numerical techniques to sample configurations of loops and subsequently evaluate the boundary state and boundary Hamiltonian on a square lattice. As a result, we evidence a spectral gap in the square lattice AKLT model. In addition, by varying the weights of the loops, vertices, and crossings, we indicate the presence of three distinct phases exhibited by the classical loop model.