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In this paper, we construct a new class of topological black hole Lifshitzsolutions in the presence of nonlinear exponential electrodynamics forEinstein-dilaton gravity. We show that the reality of Lifshitz supportingMaxwell matter fields exclude the negative horizon curvature solutions exceptfor the asymptotic AdS case. Calculating the conserved and thermodynamicalquantities, we obtain a Smarr type formula for the mass and confirm thatthermodynamics first law is satisfied on the black hole horizon. Afterward, westudy the thermal stability of our solutions and figure out the effects ofdifferent parameters on the stability of solutions under thermal perturbations.Next, we apply the gauge/gravity duality in order to calculate the ratio ofshear viscosity to entropy for a three-dimensional hydrodynamic system by usingthe pole method. Furthermore, we study the behavior of holographic conductivityfor two-dimensional systems such as graphene. We consider linear Maxwell andnonlinear exponential electrodynamics separately and disclose the effect ofnonlinearity on holographic conductivity. We indicate that holographicconductivity vanishes for $z>3$ in the case of nonlinear electrodynamics whileit does not in the linear Maxwell case. Finally, we solve perturbativeadditional field equations numerically and plot the behaviors of real andimaginary parts of conductivity for asymptotic AdS and Lifshitz cases. Wepresent experimental results match with our numerical ones.

Thermodynamics and gauge/gravity duality for Lifshitz black holes in the presence of exponential electrodynamics