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

TL;DR

This work classifies Abelian three-loop braiding statistics in 3D gauge theories with fermionic matter and connects them to interacting 3D FSPT phases with unitary symmetry. By deriving a complete set of topological invariants $\\{\\Theta_{ij,k},\\Theta_{i,k}\\}$ and a comprehensive web of physical constraints, it isolates intrinsic fermionic FSPT phases that cannot arise from bosonic SPTs, showing the minimal symmetry $\\Z_2^f\\times\\Z_2\\times\\Z_4$ suffices. The authors construct exactly solvable twisted Crane-Yetter state-sum models that realize all identified statistics, and they illuminate the role of dimensional reduction in relating 3D loop braiding to 2D SET data, including explicit intrinsic-FSPT realizations such as $\\Z_8^f\\times\\Z_2$ and $\\Z_2^f\\times\\Z_2\\times\\Z_4$. They also discuss anomalous SETs and connect their framework to Gu-Wen group-supercohomology, highlighting the broader implications for fermionic topological phases and potential non-Abelian generalizations.

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.

Figures (6)

• Figure 1: A $\mathbb Z_4$ symmetry defect in the $\mathbb Z_2^f\times\mathbb Z_2\times\mathbb Z_4$ intrinsic FSPT phase. There lives a 1D helical Dirac fermion (denoted by red and blue arrows) on the defect. The shaded region represents a branch surface associated with the defect.
• Figure 2: Three-loop braiding process involving vortex loops $\alpha$, $\beta$ and $\gamma$. The blue lines are trajectories swept out by two points on $\alpha$
• Figure 3: Two ways of fusing loops: (a) Fusing $\beta_1$ and $\beta_2$, that are linked to the same base $\gamma$, into a new loop, denoted as $\beta_1\times\beta_2$; (b) Fusing $\beta_1$ and $\beta_2$, that are linked to different bases $\gamma_1$ and $\gamma_2$ and that carry the same amount of flux $\phi_{\beta_1}=\phi_{\beta_2}$, into a new loop, denoted as $\beta_1\circ\beta_2$.
• Figure 4: The first thought experiment.
• Figure 5: The second thought experiment.