Abstract
Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for 3D interacting fermion systems. Most surprisingly, we discover new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with emergent fermionic particles). On the other hand, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also gives rise to an alternative way to classify FSPT phases. We further compare the classification results for FSPT phases with arbitrary Abelian unitary total symmetry Gf and find systematical agreement with previous studies.
Highlights
Fractional statistics is one of the most intriguing features of topological phases in 2D
The non-Abelian nature of the new type three-loop braiding process we discovered can be understood as the two-fold degeneracy carried by a pair of linked flux lines, and the braiding statistics between two loops that linked with a third loop should be characterized by a unitary 2 by 2 matrix instead of a simple U(1) phase factor
We obtain the classification of 3D fermionic symmetryprotected topological (FSPT) phases with arbitrary finite unitary Abelian total symmetry Gf, by gauging the symmetry and studying the topological invariants {Θμ,σ, Θμν,σ, Θμνλ,σ} defined through the braiding statistics of loop-like excitations in certain three-loop braiding processes and solving the corresponding constraints for these topological invariants
Summary
Fractional statistics is one of the most intriguing features of topological phases in 2D. Such kind of braiding process can not be reduced to the particle-loop braiding due to the linking with a third loop. Majorana chain onto a pair of linked flux lines (Z2 and Z8 unit flux lines for the former case and two different Z4 unit flux lines for the latter case)
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