Many-body topological invariants for fermionic symmetry-protected topological phases

TL;DR

This paper introduces a method to compute many-body topological invariants for interacting fermionic symmetry-protected topological phases using path integrals on unoriented spacetime, applicable to phases protected by orientation-reversing symmetries.

Contribution

It defines a new approach to evaluate topological invariants via non-local operations on ground states, advancing understanding of interacting fermionic topological phases.

Findings

Computed invariants for $ ext{Z}_8$ and $ ext{Z}_{16}$ classifications.

Established a link between path integrals on unoriented spacetime and ground state properties.

Demonstrated the method's applicability to 1D and 3D topological superconductors.

Abstract

We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ classification of topological superconductors in one and three dimensions.