Demonstrating the Affleck-Kennedy-Lieb-Tasaki Spectral Gap on 2D Degree-3 Lattices

We report results on solving a long outstanding problem - whether the two-dimensional spin-3/2 antiferromagnetic valence-bond model of Affleck, Kennedy, Lieb, and Tasaki (AKLT) possesses a nonzero gap above its ground state. We exploit a relation between the anticommutator and sum of two projectors and apply it to ground-space projectors on regions of the honeycomb lattice. After analytically reducing the complexity of the resultant problem, we are able to use a standard Lanczos method to establish the existence of a nonzero gap. This approach is also successfully applied to spin-3/2 AKLT models on other degree-3 semiregular tilings, namely, the square-octagon, star, and cross lattices, where the complexity is low enough that exact diagonalization can be used instead. In addition, we close the previously open cases of hybrid AKLT models on the singly decorated honeycomb and singly decorated square lattices.