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.
Highlights
Topological phases in three spatial dimensions can support particle and loop excitations [1]
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” fermionic symmetry-protected topological (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
Summary
Topological phases in three spatial dimensions can support particle and loop excitations [1]. Loop braiding statistics obtained by gauging symmetries of these intrinsic FSPT phases are the ones that are allowed only in the presence of fermionic particles. III D, if this FSPT phase stems from a BSPT phase, this three-loop braiding phase can only be 0 or π The essence in this difference is that fermion parity vortex loops play a nontrivial role in the three-loop braiding statistics in the FSPT phase constructed from decorated domain walls. This FSPT must be intrinsically fermionic and the corresponding loop braiding statistics can exist only in the presence of fermionic particles. III can be realized by these models, completing the classification
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