Nonzero spectral gap in several uniformly spin-2 and hybrid spin-1 and spin-2 AKLT models

Abstract

Recently a few 2D AKLT models have been shown to be gapped, including the one on the hexagonal lattice, whose local spin magnitude is spin-3/2. Here we report the existence of a nonzero spectral gap on several nontrivial AKLT models having either solely spin-2 or mixed spin-2 and spin-1 degrees of freedom. The hybrid models we consider are defined on the 3D diamond and 2D kagome lattices, where lattice sites are spin-2 and one single spin-1 degree of freedom is added on some or all of the edges. Although the spectral gap problem for the uniformly spin-2 AKLT models on the diamond, kagome, and square lattices is still open, we are able to establish the existence of the gap for spin-2 AKLT models on two planar lattices, which we call the inscribed square lattice and the triangle-octagon lattice, respectively. So far these latter two are the only two uniformly spin-2 AKLT models that have a provable nonzero gap above the ground state. We also discuss some attempts to prove the gap existence on both the square and kagome lattices. In addition, we show that if one can show that the gap of a finite-size, weighted AKLT Hamiltonian is larger than a certain threshold, then the original AKLT model on the square lattice is gapped in the thermodynamic limit. The threshold of the gap we obtain scales inversely with the linear size of the problem.