Topological invariant for generic 1D time reversal symmetric superconductors in class DIII

TL;DR

This paper introduces a gauge-invariant topological invariant for 1D time reversal symmetric superconductors in class DIII, applicable to disordered and interacting systems, and demonstrates its use in classifying phases of a Rashba wire.

Contribution

It provides a new, gauge-invariant method to classify topological phases in DIII class superconductors without additional symmetries, simplifying calculations especially with inversion symmetry.

Findings

The invariant can be computed numerically without fixing phase relations.

Inversion symmetry simplifies the topological invariant calculation.

Application to Rashba wires reveals the phase diagram of competing pairing terms.

Abstract

A one dimensional time reversal symmetric topological superconductor (symmetry class DIII) features a single Kramers pair of Majorana bound states at each of its ends. These holographic quasiparticles are non-Abelian anyons that obey Ising-type braiding statistics. In the special case where an additional U(1) spin rotation symmetry is present, this state can be understood as two copies of a Majorana wire in symmetry class D, one copy for each spin block. We present a manifestly gauge invariant construction of the topological invariant for the generic case, i.e., in the absence of any additional symmetries like spin rotation symmetry. Furthermore, we show how the presence of inversion symmetry simplifies the calcuation of the topological invariant. The proposed scheme is suitable for the classification of both interacting and disordered systems and allows for a straightforward numerical evaluation of the invariant since it does not rely on fixing a continuous phase relation between Bloch functions. Finally, we apply our method to compute the topological phase diagram of a Rashba wire with competing s-wave and p-wave superconducting pairing terms.