Classifications of symmetry protected topological phases in interacting boson/fermion systems

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Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations. The SPT phases in free fermion systems, like topological insulators, can be classified by K-theory. However, it is not yet known what SPT phases exist in general interacting systems. In this talk, I will first present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases. Just as group theory allows us to construct 230 crystal structures in three dimensions, we find that group cohomology theory allows us to construct many interacting bosonic SPT phases. In my talk, I shall show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians with explicit ground state wavefunctions. Next, I will discuss the generalization of the classifying scheme to interacting fermionic systems and a new mathematical framework – group supercohomology theory, which predicts a fermionic SPT phase that can neither be realized in free fermionic nor interacting bosonic systems.

Finally, I will briefly mention the deep relationship between SPT phases and chiral anomalies in high energy physics.