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

TL;DR

This paper develops a many-body framework for topological invariants in fermionic phases protected by antiunitary symmetries, avoiding single-particle wave functions, and introduces a fermionic partial transpose concept.

Contribution

It introduces a fully many-body formulation of topological invariants for fermionic phases with antiunitary symmetry, including a fermionic partial transpose and analysis of pin structures.

Findings

Constructed many-body $ extbf{Z}_2$ invariant for time-reversal symmetric topological insulators.

Linked many-body invariants to fermionic partial transpose with sign structure.

Analyzed pin structures, obstructions, and Dirac quantization conditions.

Abstract

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 $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ 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 anti-commuting 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.