The noncommutative Kubo Formula: Applications to Transport in Disordered Topological Insulators with and without Magnetic Fields

TL;DR

This paper advances the numerical evaluation of the non-commutative Kubo formula within a $C^*$-algebraic framework, enabling detailed transport analysis in disordered topological insulators under magnetic fields.

Contribution

It provides a practical numerical implementation of the non-commutative Kubo formula, allowing efficient and accurate transport calculations in complex mesoscopic systems.

Findings

Mapped conductivity as a function of Fermi level, disorder, and temperature.

Derived the phase diagram of a 2D Quantum spin-Hall insulator.

Simulated transport properties at finite magnetic fields.

Abstract

The non-commutative theory of charge transport in mesoscopic aperiodic systems under magnetic fields, developed by Bellissard, Shulz-Baldes and collaborators in the 90's, is complemented with a practical numerical implementation. The scheme, which is developed within a $C^*$-algebraic framework, enable efficient evaluations of the non-commutative Kubo formula, with errors that vanish exponentially fast in the thermodynamic limit. Applications to a model of a 2-dimensional Quantum spin-Hall insulator are given. The conductivity tensor is mapped as function of Fermi level, disorder strength and temperature and the phase diagram in the plane of Fermi level and disorder strength is quantitatively derived from the transport simulations. Simulations at finite magnetic field strength are also presented.