# A generalization of the injectivity condition for Projected Entangled Pair States

## TL;DR

This paper introduces semi-injective PEPS, a broader class of tensor network states, extending injective PEPS and encompassing important models like AKLT, with new conditions for state equivalence and topological phase classification.

## Contribution

It generalizes the injectivity condition for PEPS, constructs parent Hamiltonians, and extends SPT phase classification to semi-injective PEPS.

## Key findings

- Semi-injective PEPS include ground states of AKLT and CZX models.
- Necessary and sufficient conditions for tensor equivalence in 2D.
- Extension of third cohomology SPT classification to semi-injective PEPS.

## Abstract

We introduce a family of tensor network states that we term semi-injective Projected Entangled-Pair States (PEPS). They extend the class of injective PEPS 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 PEPS 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 PEPS.

---
Source: https://tomesphere.com/paper/1706.07329