  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.
