Topological invariant for finite systems in the presence of disorder

TL;DR

This paper introduces a new topological invariant for finite disordered systems, constructed via periodic repetition, which accurately indicates topological phases in finite-size nanowires with disorder, improving interpretation of experiments.

Contribution

The paper proposes a rigorously defined topological invariant for finite disordered systems based on periodic repetition, addressing biases in existing indicators.

Findings

The new invariant accurately distinguishes topological from trivial phases.

It provides reliable interpretation of experimental results in disordered nanowires.

The approach is general and applicable to various finite disordered systems.

Abstract

Topological invariants, rigorously defined only in the thermodynamic limit, have been generalized to topological indicators applicable to finite-size disordered systems. However, in many experimentally relevant situations, such as semiconductor-superconductor (SM-SC) hybrid nanowires hosting Majorana zero modes, the interplay between strong disorder and finite-size effects renders these indicators (e.g., the so-called topological visibility) biased and ill-defined, significantly limiting their usefulness. In this paper, we propose the topological invariant rigorously defined for an infinite system constructed by periodically repeating the original finite disordered system, as a topological indicator. Using the one-dimensional SM-SC hybrid nanowire as an example, we show that this general and transparent approach yields faithful topological indicators free from the biases affecting commonly used finite-size indicators, capturing the nature (topological or trivial) of the phase at generic points in parameter space, and providing a reliable tool for interpreting experimental results.