# Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory

## TL;DR

This paper advances the classification of 3D fermionic symmetry protected topological phases by constructing general fixed point wavefunctions and extending group supercohomology theory, achieving a more complete understanding of these complex phases.

## Contribution

It introduces a comprehensive classification framework for 3D interacting fermion SPT phases using fermionic local unitary transformations and extends group supercohomology theory.

## Key findings

- Reproduces partial classifications from existing supercohomology theory.
- Achieves a complete classification with an additional cohomology subgroup.
- Proposes a general procedure for deriving supercohomology in arbitrary dimensions.

## Abstract

Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.