# Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

## TL;DR

This paper classifies 3D fermionic symmetry-protected topological phases by analyzing Abelian loop braiding statistics in gauge theories, revealing new intrinsic phases supported by specific symmetries and constructing models to realize them.

## Contribution

It identifies and classifies intrinsic 3D fermionic SPT phases with unitary symmetries using Abelian loop braiding statistics and constructs models to realize these phases.

## Key findings

- Existence of Abelian loop braiding statistics unique to fermionic particles.
- Classification of all possible Abelian loop braiding statistics in 3D gauge theories.
- Construction of exactly soluble models generalizing Crane-Yetter and Dijkgraaf-Witten models.

## Abstract

We study Abelian braiding statistics of loop excitations in three-dimensional (3D) gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the the presence of fermionic particles, which correspond to 3D "intrinsic" FSPT phases, i.e., those that do not stem from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D systems and in 3D systems with anti-unitary symmetries, their existence in 3D systems with unitary symmetries was not confirmed previously due to the fact that strong interaction is necessary to realize them. We show that the simplest unitary symmetry to support 3D intrinsic FSPT phases is $\mathbb{Z}_2\times\mathbb{Z}_4$. To establish the results, we first derive a complete set of physical constraints on Abelian loop braiding statistics. Solving the constraints, we obtain all possible Abelian loop braiding statistics in 3D gauge theories, including those that correspond to intrinsic FSPT phases. Then, we construct exactly soluble state-sum models to realize the loop braiding statistics. These state-sum models generalize the well-known Crane-Yetter and Dijkgraaf-Witten models.