Quantum transport in disordered systems under magnetic fields: A study based on operator algebras

TL;DR

This paper develops a finite-volume approximation of the noncommutative Kubo formula for quantum transport in disordered systems under magnetic fields, enabling efficient numerical simulations that converge exponentially to the thermodynamic limit.

Contribution

It introduces a finite real-space torus framework for the noncommutative Kubo formula, facilitating practical numerical computation of quantum transport properties.

Findings

Finite-volume noncommutative Kubo formula converges exponentially fast.

Numerical simulations of 2D disordered lattice gas in magnetic field performed.

Approximate formalism is implementable and accurate for finite temperatures and dissipation.

Abstract

The linear conductivity tensor for generic homogeneous, microscopic quantum models was formulated as a noncommutative Kubo formula in Refs. \cite{BELLISSARD:1994xj,Schulz-Baldes:1998vm,Schulz-Baldes:1998oq}. This formula was derived directly in the thermodynamic limit, within the framework of $C^*$-algebras and noncommutative calculi defined over infinite spaces. As such, the numerical implementation of the formalism encountered fundamental obstacles. The present work defines a $C^*$-algebra and an approximate noncommutative calculus over a finite real-space torus, which naturally leads to an approximate finite-volume noncommutative Kubo formula, amenable on a computer. For finite temperatures and dissipation, it is shown that this approximate formula converges exponentially fast to its thermodynamic limit, which is the exact noncommutative Kubo formula. The approximate noncommutative Kubo formula is then deconstructed to a form that is implementable on a computer and simulations of the quantum transport in a 2-dimensional disordered lattice gas in a magnetic field are presented.