  Using exact diagonalization and tensor network techniques we compute the gap
for the AKLT Hamiltonian in 1D and 2D spatial dimensions. Tensor Network
methods are used to extract physical properties directly in the thermodynamic
limit, and we support these results using finite-size scalings from exact
diagonalization. Studying the AKLT Hamiltonian perturbed by an external field,
we show how to obtain an accurate value of the gap of the original AKLT
Hamiltonian from the field value at which the ground state verifies e_0<0,
which is a quantum critical point. With the Tensor Network Renormalization
Group methods we provide evidence of a finite gap in the thermodynamic limit
for the AKLT models in the 1D chain and 2D hexagonal and square lattices. This
method can be applied generally to Hamiltonians with rotational symmetry, and
we also show results beyond the AKLT model.
