Demonstrating the AKLT spectral gap on 2D degree-3 lattices

We establish that the spin-3/2 AKLT model on the honeycomb has a nonzero spectral gap. We use the relation between the anticommutator of two projectors and their sum, and apply it to related AKLT projectors that occupy plaquettes or other extended regions. We analytically reduce the complexity in the resulting eigenvalue problem and use a Lanczos numerical method to show that the required inequality for the nonzero spectral gap holds. This approach is also successfuly applied to several other spin-3/2 AKLT models on degree-3 semiregular tilings, such as the square octagon, star and cross lattices, where the complexity is low enough that exact diagonalization can be used instead of the Lanczos method. In addition, we also close the previously open cases in the singly decorated honeycomb and square lattices.