On Mott's formula for the ac-conductivity in the Anderson model

TL;DR

This paper investigates the ac-conductivity in the Anderson model within the localization regime, providing bounds that compare to Mott's formula, and highlights the subtle differences in the predicted leading terms.

Contribution

The paper proves a bound on ac-conductivity in the Anderson model that refines the understanding of its frequency dependence in the localization regime.

Findings

Bound on ac-conductivity: $C u^2 ( ext{log} rac{1}{ u})^{d+2}$

Comparison with Mott's formula: leading term predicted as $C u^2 ( ext{log} rac{1}{ u})^{d+1}$

Results suggest a subtle discrepancy in the logarithmic correction term.

Abstract

We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schr\"odinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the ac-conductivity is bounded by $C \nu^2 (\log \frac 1 \nu)^{d+2}$ at small frequencies $\nu$. This is to be compared to Mott's formula, which predicts the leading term to be $C \nu^2 (\log \frac 1 \nu)^{d+1}$.