Linear Response Theory for Random Schr\"odinger Operators and Noncommutative Integration

TL;DR

This paper develops a systematic approach using noncommutative Lp-spaces to rigorously derive linear response formulas, including the Kubo and Kubo-Streda formulas, for ergodic Schrödinger operators with magnetic fields.

Contribution

It provides a more transparent proof of linear response theory and the Kubo formulas using noncommutative integration techniques.

Findings

Rigorous derivation of Kubo formula for electric conductivity.

Recovery of Kubo-Streda formula in localization regions.

Enhanced clarity through noncommutative Lp-space methods.

Abstract

We consider an ergodic Schr\"odinger operator with magnetic field within the non-interacting particle approximation. Justifying the linear response theory, a rigorous derivation of a Kubo formula for the electric conductivity tensor within this context can be found in a recent work of Bouclet, Germinet, Klein and Schenker. If the Fermi level falls into a region of localization, the well-known Kubo-Streda formula for the quantum Hall conductivity at zero temperature is recovered. In this review we go along the lines of but make a more systematic use of noncommutative Lp-spaces, leading to a somewhat more transparent proof.