Partial transpose and many-body invariants in fermionic symmetry protected topological phases

We developed a fully many-body formulation of non-local order parameters which detect various topological phases of fermions protected by antiunitary symmetry. Our formulation does not refer to single particle wave functions. An important ingredient for the construction of non-local order parameters is a fermionic counter part of partial transpose, which I give the detail in this talk. As an application, I present the many-body extension of the Kane-Mele Z2 topological invariant for time-reversal symmetric topological insulators in two spatial dimensions.