  In this paper we address the question of the existence of a spectral gap in a
class of local Hamiltonians. These Hamiltonians have the following properties:
their ground states are known exactly; all equal-time correlation functions of
local operators are short-ranged; and correlation functions of certain
non-local operators are critical. A variational argument shows gaplessness with
$\omega \propto k^2$ at critical points defined by the absence of certain terms
in the Hamiltonian, which is remarkable because equal-time correlation
functions of local operators remain short-ranged. We call such critical points,
in which spatial and temporal scaling are radically different,
quasi-topological. When these terms are present in the Hamiltonian, the models
are in gapped topological phases which are of special interest in the context
of topological quantum computation.
