$\mathbb Z_2$~Green's function topology of Majorana wires

TL;DR

This paper introduces a novel Green's function-based method to classify the topological phases of one-dimensional superconductors, including interacting and disordered systems, using a $ ext{Z}_2$ invariant.

Contribution

It formulates a $ ext{Z}_2$ topological invariant via Green's functions, extending topological classification to interacting and disordered Majorana wires.

Findings

Invariant expressed in terms of Green's functions at zero frequency

Method applicable to disordered interacting systems

Provides a rigorous topological classification framework

Abstract

We represent the $\mathbb Z_2$~topological invariant characterizing a one dimensional topological superconductor using a Wess-Zumino-Witten dimensional extension. The invariant is formulated in terms of the single particle Green's function which allows to classify interacting systems. Employing a recently proposed generalized Berry curvature method, the topological invariant is represented independent of the extra dimension requiring only the single particle Green's function at zero frequency of the interacting system. Furthermore, a modified twisted boundary conditions approach is used to rigorously define the topological invariant for disordered interacting systems.