2016/09/09 by Dominic J. Williamson, Williamson, Dominic J., Nick Bultinck +5 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics #Strongly Correlated Electrons (cond-mat.str-el) #cond-mat.str-el #quant-ph
paper · pdf · doi:10.48550/arxiv.1609.02897
7 pages, 1 figure; v2 minor corrections, references added. These results are significantly expanded upon in arXiv:1707.00470
openalex publication_date 2016/09/09 · arxiv created 2017/10/15 · arxiv updated 2017/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: H1(G,ℤ2), describing which matrix product operators are of Majorana type, H2(G,ℤ2), describing the fermionic nature of the fusion tensors that arise when two matrix product operators are multiplied, and the supercohomolgy group H3(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.