A generalization of the injectivity condition for projected entangled pair states

We introduce a family of tensor network states that we term semi-injective Projected Entangled-Pair States (PEPSs). They extend the class of injective PEPSs and include other states, like the ground states of the AKLT and the CZX models in square lattices. We construct parent Hamiltonians for which semi-injective PEPSs are unique ground states. We also determine the necessary and sufficient conditions for two tensors to generate the same family of such states in two spatial dimensions. Using this result, we show that the third cohomology labeling of symmetry protected topological phases extends to semi-injective PEPSs.