Anomalous symmetry protected topological states in interacting fermion systems*

The classification and construction of symmetry protected topological(SPT) phases have been intensively studied in interacting systems recently. To our surprise, in interacting fermion systems, there even exists a new class of the so-called anomalous SPT(ASPT) states which are only well defined on the boundary of a trivial fermionic bulk. The physical reason is that certain symmetry action might change the fermion parity locally, but is conserved if we introduce a bulk. We demonstrate the essential idea by considering anomalous topological superconductor with time reversal symmetry $T^2=1$. Finally, we also discuss the classification scheme of anoumalous SPT states for interacting fermions with a total symmetry $G_f=G_b\times\Zf$ and relevant experimental implications.