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Tensor networks for topological quantum matter

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In this talk, I present a new framework for topologically ordered gapped ground states in 2+1 spacetime dimensions (which generalizes to higher dimensions) using tensor networks. We will see that topological order can exist in tensor network states (TNS), if the local tensor satisfies certain axioms which we call MPO (matrix product operator)-injectivity and pulling through. We then continue with examples, and see how renormalization fixed point models in the literature (Levin-Wen models, etc.) can be covered in this framework. If time permits, we finish with a discussion of excitations and the generalization to higher dimensions.