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

Abstract

We study Abelian braiding statistics of loop excitations in three-dimensional 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 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 antiunitary 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 Z 2 × 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.

Tube algebras, excitations statistics and compactification in gauge models of topological phases

“…Firstly, it provides a definition of the model that is analogous to string-net models. Secondly, we will argue that it can be used to shed light on the fusion and the braiding statistics of loop-like excitations [22,24,25,29,31,[63][64][65]. We assume in this appendix that the group is abelian.…”

Quantum statistics and spacetime topology: Quantum surgery formulas

“…TQFTs for the bosonic Dijkgraaf-Witten theory ( [33] and References therein), to 2. Fermionic version of TQFT gauge theory: Fermionic finite-group gauge theory and spin-TQFT, and their braiding statistics or topological link invariants [35,45,46] and 3. Higher-gauge theory as TQFTs, see some selective examples in [47,48], [37], [33], [49][50][51][52][53].…”

Excitations in strict 2-group higher gauge models of topological phases

“…Recently, gapped phases of matter described by topological theories that have a higher gauge theory have been under scrutiny [10][11][12][13][14][15][16][17][18][19][20][21][22]. Examples of such theories are provided by sigma models whose target spaces are given by Postnikov towers [23] built as fibrations of Eilenberg-MacLane spaces [24,25], and the construction of lattice Hamiltonian realizations of such models was considered in [26][27][28].…”