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.