Energy scales in a holographic black hole and conductivity at finite momentum

TL;DR

This paper investigates the low-temperature behavior of gauge field correlators with finite momentum in a holographic black hole background, revealing a frequency gap and exponential decay of conductivity with temperature, and contrasting properties at extremal black holes.

Contribution

It provides an analytic study of the temperature dependence of conductivity at finite momentum in holographic black holes, highlighting differences between non-extremal and extremal cases.

Findings

Conductivity exhibits a frequency gap proportional to momentum.

Real part of conductivity decays exponentially with inverse temperature.

Extremal black holes show no sharp gap, with power-law decay of conductivity.

Abstract

In this work we discuss the low temperature ($T$) behavior of gauge field correlators with finite momentum (k) in a $AdS^4$ black hole background. At low temperature, a substantial non-zero conductivity is only possible for a frequency range $\omega>\omega_g=k$. This tallies with the simple fact that at least an amount of energy $\omega_g$ is needed to create an excitation of momentum $k$. Due to the existence of this ``gap'',one may expect that at zero frequency limit the real part of momentum dependent conductivity falls exponentially with $\frac{1}{T}$. Using analytic methods, we found a $\exp(-\frac{\omega_c}{T})$ falloff of the real part of conductivity with inverse temperature. Interestingly, $\omega_g \neq \omega_c$. From the above results we speculate that the ``degrees of freedoms'', say carriers, different than quasi particle excitation determines conductivity at low temperature and low frequency limit. Here $\omega_c < \omega_g$ and we may calculate their ratios analytically. We also discuss similar issues at a finite chemical potential. Situation is rather different for an extremal blackhole. A zero temperature extremal blackhole does not show a sharp gap for the finite momentum excitations and the real part of conductivity is always non-zero for any non-zero frequency $\omega$. However the real part of conductivity goes to zero at $\omega\to0$ limit. Not surprisingly, we find a powerlaw decay with temperature for the same quantity, as the extremal limit is approached.