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The thermoelectric properties of inhomogeneous holographic lattices

We consider inhomogeneous, periodic, holographic lattices of D=4 Einstein-Maxwell theory. We show that the DC thermoelectric conductivity matrix can be expressed analytically in terms of the horizon data of the corresponding black hole solution. We numerically construct such black hole solutions for lattices consisting of one, two and ten wave-numbers. We numerically determine the AC electric conductivity which reveals Drude physics as well as resonances associated with sound modes. No evidence for an intermediate frequency scaling regime is found. All of the monochromatic lattice black holes that we have constructed exhibit scaling behaviour at low temperatures which is consistent with the appearance of $AdS_2\times\mathbb{R}^2$ in the far IR at T=0.