Boundary Theory of a Deformed AKLT Model on the Square Lattice

The 1D AKLT model is a paradigm of antiferromagnetism initially devised as a modification to the Heisenberg model, and its ground state is a quintessential example of symmetry protected topological order. On a 2D lattice, the AKLT model is particularly interesting because it also exhibits symmetry protected topological order, and can act as a resource for universal quantum computation. In contrast to the 1D case, the existence of the spectral gap in 2D, which guarantees the robustness of the model, remains an open problem despite extensive analyses. Recently, it has been shown that one can deduce this spectral gap by analyzing its boundary theory via a tensor network representation of the ground state. In this work, we present a method to calculate 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, and also indicate the presence of several different phases by varying the weights of the model.