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

Abstract

In this work we discuss the low temperature (T) behavior of gauge field correlators with finite momentum (k) in a AdS4 black hole background. At low temperature, a substantial nonzero conductivity is only possible for a frequency range ω > ωg = k. This tallies with the simple fact that at least an amount of energy ωg is needed to create an excitation of momentum k. Due to the existence of this “gap”, one may expect that at the zero frequency limit the real part of momentum-dependent conductivity falls exponentially with 1/T. Using analytic methods, we found a exp(–ωc/T) falloff of the real part of conductivity with inverse temperature. Interestingly, ωg ≠ ωc. From the above results, we speculate that the “degrees of freedom”, say carriers, different than quasiparticle excitation determines conductivity at the low temperature and the low-frequency limit. Here ωc < ωg, and we may calculate their ratios analytically. We also discuss similar issues at a finite chemical potential. The 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 the conductivity is always nonzero for any nonzero frequency ω. However, the real part of the conductivity goes to zero at the ω → 0 limit. Not surprisingly, we find a power-law decay with temperature for the same quantity, as the extremal limit is approached.