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

We study Abelian braiding statistics of loop excitations inthree-dimensional gauge theories with fermionic particles and theclosely related problem of classifying 3D fermionic symmetry-protectedtopological (FSPT) phases with unitary symmetries. It is known that thetwo problems are related by turning FSPT phases into gauge theoriesthrough gauging the global symmetry of the former. We show that thereexist certain types of Abelian loop braiding statistics that are allowedonly in the presence of fermionic particles, which correspond to 3D"intrinsic" FSPT phases, i.e., those that do not stem from bosonic SPTphases. While such intrinsic FSPT phases are ubiquitous in 2D systemsand in 3D systems with antiunitary symmetries, their existence in 3Dsystems with unitary symmetries was not confirmed previously due to thefact that strong interaction is necessary to realize them. We show thatthe simplest unitary symmetry to support 3D intrinsic FSPT phases is Z₂ × Z₄. To establish the results, we firstderive a complete set of physical constraints on Abelian loop braidingstatistics. Solving the constraints, we obtain all possible Abelian loopbraiding statistics in 3D gauge theories, including those thatcorrespond to intrinsic FSPT phases. Then, we construct exactly solublestate-sum models to realize the loop braiding statistics. Thesestate-sum models generalize the well-known Crane-Yetter andDijkgraaf-Witten models.