Orbital polarization and magnetization for independent particles in disordered media

TL;DR

This paper derives simplified, generalized formulas for orbital polarization and magnetization in disordered media, revealing their geometric nature and implications for quantization and boundary currents in topological systems.

Contribution

It introduces new formulas for polarization and magnetization in disordered systems, extending previous results for Bloch electrons and employing advanced mathematical tools.

Findings

Orbital polarization and magnetization are geometric quantities.

Quantization occurs in periodically driven Piezo effects.

Connections between magnetization derivatives and boundary currents in Chern insulators.

Abstract

Formulas for the contribution of the conduction electrons to the polarization and magnetization are derived for disordered systems and within a one-particle framework. These results generalize known formulas for Bloch electrons and the presented proofs considerably simplify and strengthen prior justifications. The new formulas show that orbital polarization and magnetization are of geometric nature. This leads to quantization for a periodically driven Piezo effect as well as the derivative of the magnetization w.r.t. the chemical potential. It is also shown how the latter is connected to boundary currents in Chern insulators. The main technical tools in the proofs are an adaption of Nenciu's super-adiabatic theory to C$^*$-dynamical systems and Bellissard's Ito derivatives w.r.t. the magnetic field.