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 ($m \neq 0$) 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, $\mathrm{Re}[\sigma]$, decreases as charge $q$ increases when frequency $\omega$ tends to zero, while the imaginary part of conductivity, $\mathrm{Im}[\sigma ]$, diverges as $\omega \rightarrow 0$. For the massive gravity, we find that $\mathrm{Im}[\sigma ]$ is zero at $\omega =0$ and becomes larger as $q$\ increases (temperature decreases), which is in contrast to the massless gravity. Interestingly, we observe that in contrast to the massless case, $\mathrm{Re}[\sigma ]$ has a maximum value at $\omega =0$ (known as the Drude peak) for $p=\left( n+1\right) /4$ (conformally invariant electrodynamics) where $p$ is the power parameter of the power-law Maxwell field and this maximum increases with increasing $q$. 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 $a \propto (n+1-4p)$. Some similar behaviors for high frequencies in possible dual CFT systems have been reported in experimental observations.