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