Marginal and Irrelevant Disorder in Einstein-Maxwell backgrounds

TL;DR

This paper investigates how weak, zero-mean random chemical potentials affect Einstein-Maxwell backgrounds, revealing conditions under which disorder becomes marginal or irrelevant and analyzing their impact on conductivity at zero and finite temperatures.

Contribution

It demonstrates how disorder correlations can tune the relevance of perturbations and derives the resulting geometric and transport properties in Einstein-Maxwell backgrounds.

Findings

Irrelevant disorder leads to vanishing conductivity correction at zero temperature.

Marginal disorder induces a Lifshitz-like IR geometry and positive conductivity correction.

Finite temperature results show a Drude peak in thermal conductivity and unaffected electric conductivity.

Abstract

We study analytically the effect of a weak random chemical potential of zero average in an Einstein-Maxwell background. For uncorrelated disorder this perturbation is relevant however we show that it can become marginal or even irrelevant by tuning disorder correlations. At zero temperature we find that, to leading order in the disorder strength, the correction to the conductivity for irrelevant perturbations vanishes. In the marginal case, in order to renormalize a logarithmic divergence, we carry out a resummation of the perturbative expansion of the metric that leads to a Lifshitz-like geometry in the infrared. Disorder in this case also induces a positive correction to the conductivity. At finite temperature the black hole acquires an effective charge and the thermal conductivity has the expected Drude peak that signals the breaking of translational invariance. However the electric conductivity is not affected by the random chemical potential to leading order in the disorder strength.