Holographic charge transport in 2+1 dimensions at finite $N$

We study holographic charge transport in (2+1) dimensions at finite $N$, whose dual gravity background is given by perturbative black hole solution in Einstein theory plus cubic terms of Weyl tensor. We consider the higher derivative corrections to the standard Maxwell action, given by the interacting terms between the Weyl tensor and the field strength. We calculate the DC conductivity by using both the membrane paradigm and the Kubo's formula and find precise agreement. We compute the AC conductivity and find an analog of the crossover from `metal' to `bad metal' in the low frequency limit. Moreover, the conductivity becomes a constant in the large frequency limit. We derive two universal relations for the Green's functions and observe that they are exactly the same as the infinite $N$ counterparts.

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