# The non-commutative topology of two-dimensional dirty superconductors

## TL;DR

This paper applies non-commutative analysis to two-dimensional dirty superconductors, establishing topological invariants and analyzing conductance and edge currents, extending techniques from quantum Hall effect studies.

## Contribution

It introduces a non-commutative topological framework for dirty superconductors and derives Kubo formulas for various Hall conductances and edge currents.

## Key findings

- Topological invariants for superconductors are characterized.
- Kubo formulas for thermal, thermoelectric, and spin Hall conductance are derived.
- Edge currents are analyzed within the non-commutative framework.

## Abstract

Non-commutative analysis tools have successfully been applied to the integer quantum Hall effect, in particular for a proof of the stability of the Hall conductance in an Anderson localization regime and of the bulk-boundary correspondence. In this work, these techniques are implemented to study two-dimensional dirty superconductors described by Bogoliubov-de Gennes Hamiltonians. After a thorough presentation of the basic framework and the topological invariants, Kubo formulas for the thermal, thermoelectric and spin Hall conductance are analyzed together with the corresponding edge currents.