$U(1)$ current from the AdS/CFT: diffusion, conductivity and causality

TL;DR

This paper derives a comprehensive, all-order gradient expansion for a $U(1)$ current in a holographic finite temperature setting, revealing causal diffusion and conductivity properties through analytical and numerical methods.

Contribution

It introduces a resummed gradient expansion for a holographic $U(1)$ current, including transport coefficients and causality analysis, extending previous hydrodynamic results.

Findings

Transport coefficients computed analytically in the hydrodynamic limit.

Numerical evaluation of transport functions for generic momenta.

Diffusion memory function found to be causal due to all-order resummation.

Abstract

For a holographically defined finite temperature theory, we derive an off-shell constitutive relation for a global $U(1)$ current driven by a weak external non-dynamical electromagnetic field. The constitutive relation involves an all order gradient expansion resummed into three momenta-dependent transport coefficient functions: diffusion, electric conductivity, and "magnetic" conductivity. These transport functions are first computed analytically in the hydrodynamic limit, up to third order in the derivative expansion, and then numerically for generic values of momenta. We also compute a diffusion memory function, which, as a result of all order gradient resummation, is found to be causal.