Spin Topological Field Theory and Fermionic Matrix Product States

TL;DR

This paper explores the connection between spin topological quantum field theories and fermionic matrix product states, providing insights into classifying fermionic phases with symmetries and revealing subtleties in their group structures.

Contribution

It introduces a state-sum construction for G-equivariant spin-TQFTs and demonstrates their states as generalized Matrix Product States across different sectors, advancing fermionic phase classification.

Findings

States in various sectors are generalized Matrix Product States

Revised classification of fermionic Short-Range-Entangled phases

Identified subtleties in symmetry group extensions involving fermion parity

Abstract

We study state-sum constructions of G-equivariant spin-TQFTs and their relationship to Matrix Product States. We show that in the Neveu-Schwarz, Ramond, and twisted sectors, the states of the theory are generalized Matrix Product States. We apply our results to revisit the classification of fermionic Short-Range-Entangled phases with a unitary symmetry G and determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of G by fermion parity.