Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries

We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single-particle wave functions. For example, we construct the many-body Z2 topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele Z2 invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermionic partial transpose, which is basically the standard partial transpose equipped with a sign structure to account for anticommuting property of fermion operators. We also report some basic results on various kinds of pin structures - a key concept behind our strategy for constructing many-body topological invariants - such as the obstructions, isomorphism classes, and Dirac quantization conditions.