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

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 $ω>ω_g=k$. This tallies with the simple fact that at least an amount of energy $ω_g$ 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 $\exp(-\frac{ω_c}{T})$ falloff of the real part of conductivity with inverse temperature. Interestingly, $ω_g \neq ω_c$. 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 $ω_c < ω_g$ 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 $ω$. However the real part of conductivity goes to zero at $ω\to0$ limit. Not surprisingly, we find a powerlaw decay with temperature for the same quantity, as the extremal limit is approached.