Classifation of 3D fermionic symmetry-protected topological (fSPT) phases

We study the classification of fermionic SPT (fSPT) phases in 3D via the method called three-loop braiding statistics. There is an interesting fact that each anomaly-free SPT phase can be mapped to a corresponding topological order, by a technic called “gauging the symmetry”. By fully gauging all symmetries, we obtain a topological order without any symmetry, but with topological particle-like and loop-like excitations in 3D. The particle-loop braiding process can be understood in a way analogous to the Aharonov-Bohm effect. While since loop-like excitations are topological (i.e., their mutual braiding statistics is invariant under smooth deformations of the loops), a loop-loop braiding can be equivalently transformed to a particle-loop braiding. Particle-loop and loop-loop braidings give no new information up to linear representations of the symmetry group. However, if we insert a base-loop to the two braiding loops, which forms the so-called three-loop braiding process, a loop cannot be equivalently treated as a particle anymore. And the loops with a base loop inserted now become anyons with nontrivial braiding statistics, which may even be non-Abelian. And we find that studying all three-loop braiding statistics gives a classification of all 3D fSPT phases with on-site finite unitary symmetries. Further, we explore the Majorana zero modes on loop-like excitations. We find the non-Abelian three-loop braiding statistics of them as a characteristic indicator of the Majorana chain decoration.