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

Abstract

Here, the authors introduce a set of quantities to diagnose symmetry-protected topological phases of fermions protected by antiunitary symmetries. These quantities, which can be written as expectation values of nonlocal operators, effectively simulate the partition function on nonorientable spacetime manifolds in the operator formalism. The important observation is that the proposed quantities are complex valued for topological states, where the complex phase is the many-body topological invariant. Numerous symmetry classes and dimensionalities are studied in detail. The key ingredient of all quantities is the ``fermionic partial transpose''. For example, the ${Z}_{2}$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions is given.