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Recently various 2D AKLT models have been shown to be gapped, including theone on the hexagonal lattice. Here we report on a non-trivial 3D AKLT modelwhich consists of spin-2 entities on the diamond lattice sites and one singlespin-1 entity between every neighboring spin-2 site. Although the nonzero gapproblem for the uniformly spin-2 AKLT models on the diamond and square latticesis still open, we are able to establish the existence of the gap for two planarlattices, which we call the inscribed square lattice and the triangle-octagonlattice, respectively. So far, these latter two models are the only twouniformly spin-2 AKLT models that have a provable nonzero gap above the groundstate. We also discuss some attempts in proving the gap existence on both thesquare and kagome lattices. In addition, we show that if one can solve afinite-size problem of a weighted AKLT Hamiltonian and if the gap is largerthan certain threshold, then the model on the square lattice is gapped in thethermodynamic limit. The threshold of the gap scales inversely with the linearsize of the finite-size problem.

The AKLT models on the singly decorated diamond lattice and two degree-4 planar lattices are gapped