We establish that the spin-3/2 AKLT model on the honeycomb lattice 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 squareoctagon, 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.

Abdul -Rahman et al. in arXiv:1901.09297 [1] provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number n of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if n ≥ 4. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for n ≥ 4. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to n ≥ 2, including the establishment of the gap for n = 2 in the decorated triangular and cubic lattices. The latter case is interesting, as this shows the AKLT states on the decorated cubic lattices are not Néel ordered, in contrast to the state on the un-decorated cubic lattice.

We study spin-2 deformed-AKLT models on the square lattice, specifically a two-parameter family of O(2)-symmetric ground-state wavefunctions as defined by Niggemann, Klümper, and Zittartz, who found previously that the phase diagram consists of a Néel-ordered phase and a disordered phase which contains the AKLT point. Using tensor-nework methods, we not only confirm the Néel phase but also find an XY phase with quasi-long-range order and a region adjacent to it, within the AKLT phase, with very large correlation length, and investigate the consequences of a perfectly-factorizable point at the corner of that phase.I.