Anomalous transport: a mathematical framework

TL;DR

This paper introduces a mathematical framework for analyzing anomalous transport in homogeneous solids, focusing on spectral and diffusion exponents that influence conductivity and transport anomalies.

Contribution

It develops formulas to compute spectral and diffusion exponents and applies them to models like Wegner's n-orbital and Anderson in coherent potential approximation.

Findings

Spectral and diffusion exponents characterize anomalous transport.

Formulas for calculating these exponents are provided.

Application to specific models demonstrates the framework's utility.

Abstract

We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo's formula for the conductivity and hence lead to anomalies in Drude's formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner's $n$-orbital model as well as the Anderson model in coherent potential approximation.