Title: MPO-injectivity and Topologically Ordered PEPS

Abstract:

It is well known that many RG-fixed point states with topological order such as the quantum double models of Kitaev and the string-net models of Levin and Wen can exactly be represented as tensor network states, also known as projected entangled pair states (PEPS). It is also clear that the corresponding PEPS tensors have a very peculiar structure, although this structure has so far only be defined precisely for the quantum doubles. In that case, it gives rise to the concept of G-injectivity, as first introduced by Schuch et al, where the tensors are invariant under the action of the discrete group G at the virtual level. The property of G-injectivity also survives away from the fixed point and was recently generalized to models where the symmetry group is twisted by a 3-cocylce by Buerschaper.

The goal of this talk is to find a minimal set of axioms which are required for a PEPS to exhibit topological order, without requiring that there is an underlying structure such as fusion. Doing so, we introduce a more general notion of injectivity for PEPS which applies to the general Levin-Wen string net models but potentially also to wider classes. We provide an outlook towards the recently discovered chiral PEPS, how to understand excitations and how we can compute the modular S and T matrices in our formalism.