Fermionic Matrix Product Operators and Topological Phases of Matter

Abstract

We introduce the concept of fermionic matrix product operators, and show that they provide a natural representation of fermionic fusion tensor categories. This allows for the classification of two dimensional fermionic topological phases in terms of matrix product operator algebras. Using this approach we give a classification of fermionic symmetry protected topological phases with respect to a group $G$ in terms of three cohomology groups: $H^1(G,{\mathbb{Z}}_2)$, describing which matrix product operators are of Majorana type, $H^2(G,{\mathbb{Z}}_2)$, describing the fermionic nature of the fusion tensors that arise when two matrix product operators are multiplied, and the supercohomolgy group ${\bar{H}}^3(G,U(1))$ which corresponds to the associator that changes the order of fusion. We also generalize the tensor network description of the string-net ground states to the fermionic setting, yielding simple representations of a class that includes the fermionic toric code.

7 pages, 1 figure; v2 minor corrections, references added. These results are significantly expanded upon in arXiv:1707.00470