  In this paper we consider projected entangled pair states (PEPS) on arbitrary
lattices. We construct local parent Hamiltonians for each PEPS and isolate a
condition under which the state is the unique ground state of the Hamiltonian.
This condition, verified by generic PEPS and examples like the AKLT model, is
an injective relation between the boundary and the bulk of any local region.
While it implies the existence of an energy gap in the 1D case we will show
that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian
can be gapless with a critical ground state. To show this we invoke a mapping
between classical and quantum models and prove that in these cases the
injectivity relation between boundary and bulk solely depends on the lattice
geometry.
