Ward Identity and Homes' Law in a Holographic Superconductor with Momentum Relaxation

TL;DR

This paper explores conductivities, Ward identities, and universal laws in a holographic superconductor model with momentum relaxation, revealing constraints, the validity of Uemura's law, and the nature of scalar instabilities.

Contribution

It analytically derives conductivity constraints from Ward identities and examines the applicability of Homes' and Uemura's laws in a holographic superconductor model.

Findings

Conductivity constraints are confirmed both analytically and numerically.

Uemura's law holds at small momentum relaxation, but Homes' law does not.

DC conductivity is finite for neutral scalar instability, infinite for complex scalar instability.

Abstract

We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we find that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding diffeomorphism from field theory perspective. We confirm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes' law and Uemura's law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the superfluid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes' law does not hold but the Uemura's law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is finite for a neutral scalar instability while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected.