Holographic conductivity in the massive gravity with power-law Maxwell field

We obtain a new class of topological black hole solutions in (n + 1)-dimensional massive gravity in the presence of the power-Maxwell electrodynamics. We calculate the conserved and thermodynamic quantities of the system and show that the first law of thermodynamics is satisfied on the horizon. Then, we investigate the holographic conductivity for the four and five dimensional black brane solutions. For completeness, we study the holographic conductivity for both massless (m = 0) and massive (m not equal 0) gravities with power-Maxwell field. The massless gravity enjoys translational symmetry whereas the massive gravity violates it. For massless gravity, we observe that the real part of conductivity, Re[sigma], decreases as charge q increases when frequency omega tends to zero, while the imaginary part of conductivity, Im[sigma], diverges as omega -> 0. For the massive gravity, we find that Im[sigma] is zero at omega = 0 and becomes larger as qincreases (temperature decreases), which is in contrast to the massless gravity. It also has a maximum value for omega not equal 0 which increases with increasing q(with fixed p) or increasing p(with fixed q) for (2 + 1)-dimensional dual system, where pis the power parameter of the power-law Maxwell field. Interestingly, we observe that in contrast to the massless case, Re[sigma] has a maximum value at omega = 0 (known as the Drude peak) for p =(n + 1)/4(conformally invariant electrodynamics) and this maximum increases with increasing q. In this case (m not equal 0) and for different values of p, the real and imaginary parts of the conductivity has a relative extremum for omega not equal 0. Finally, we show that for high frequencies, the real part of the holographic conductivity have the power law behavior in terms of frequency omega(a) where a alpha (n + 1 - 4p). Some similar behaviors for high frequencies in possible dual CFT systems have been reported in experimental observations. (C) 2017 The Author(s). Published by Elsevier B.V.