  We study the spectral properties of a family of quantum antiferromagnets on
two-dimensional (2D) lattices. This family of models is obtained by a
deformation of the well-studied 2D quantum antiferromagnetic model of Affleck,
Kennedy, Lieb and Tasaki (AKLT); they are described by two-body,
frustration-free Hamiltonians on a three-colourable lattice of spins. Although
the existence of a spectral gap in the 2D AKLT model remains an open question,
we rigorously prove the existence of a gap for a subset of this family of
quantum antiferromagnets. Along with providing new progress for the gap problem
in AKLT-type antiferromagnets in 2D, this result has implications for the
theory of quantum computation as it provides a family of two-body Hamiltonians
for which the ground state is a resource for universal quantum computation and
for which a spectral gap is proven to exist.
