Universality of DC Electrical Conductivity from Holography

TL;DR

This paper introduces a universal holographic formula relating dc electrical conductivity to black hole properties, valid in systems with broken symmetries, and demonstrates its applicability through various anisotropic and isotropic models.

Contribution

It proposes a universal holographic formula for dc conductivity in symmetry-breaking systems and validates it with explicit anisotropic and isotropic black brane solutions.

Findings

Universal ratio of dc conductivity determinant to black hole area density.

Validation through anisotropic black brane with linear-T resistivity.

Validation through isotropic black brane with linear and quadratic T resistivity.

Abstract

We propose a universal formula of dc electrical conductivity in rotational- and translational- symmetries breaking systems via the holographic duality. This formula states that the ratio of the determinant of the dc electrical conductivities along any spatial directions to the black hole area density in zero-charge limit has a universal value. As explicit illustrations, we give several examples elucidating the validation of this formula: We construct an anisotropic black brane solution, which yields linear in temperature for the in-plane resistivity and insulating behavior for the out-of-plane resistivity; We also construct a spatially isotropic black brane solution that both the linear-T and quadratic-T contributions to the resistivity can be realized.