We introduce a new framework for constructing black hole solutions that are holographically dual to strongly coupled field theories with explicitly broken translation invariance. Using a classical gravitational theory with a continuous global symmetry leads to constructions that involve solving ODEs instead of PDEs. We study in detail D = 4 Einstein-Maxwell theory coupled to a complex scalar field with a simple mass term. We construct black holes dual to metallic phases which exhibit a Drude-type peak in the optical conductivity, but there is no evidence of an intermediate scaling that has been reported in other holographic lattice constructions. We also construct black holes dual to insulating phases which exhibit a suppression of spectral weight at low frequencies. We show that the model also admits a novel AdS 3 × R solution.

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An analytic expression for the DC electrical conductivity in terms of black hole horizon data was recently obtained for a class of holographic black holes exhibiting momentum dissipation. We generalise this result to obtain analogous expressions for the DC thermoelectric and thermal conductivities. We illustrate our results using some holographic Q-lattice black holes as well as for some black holes with linear massless axions, in both D = 4 and D = 5 bulk spacetime dimensions, which include both spatially isotropic and anisotropic examples. We show that some recently constructed ground states of holographic Q-lattices, which can be either electrically insulating or metallic, are all thermal insulators.

Using simple holographic models in D = 4 spacetime dimensions we construct black hole solutions dual to d = 3 CFTs at finite charge density with a Q-lattice deformation. At zero temperature we find new ground state solutions, associated with broken translation invariance in either one or both spatial directions, which exhibit insulating or metallic behaviour depending on the parameters of the holographic theory. For low temperatures and small frequencies, the real part of the optical conductivity exhibits a power-law behaviour. We also obtain an expression for the the DC conductivity at finite temperature in terms of horizon data of the black hole solutions.

We discuss new types of instabilities of D = 4 electrically charged AdSReissner-Nordström black branes that involve neutral pseudo-scalars. The instabilities spontaneously break translational invariance and are associated with the dual three-dimensional CFTs, at finite temperature and fixed chemical potential with respect to a global abelian symmetry, acquiring striped phases. We show that such instabilities are present for the infinite class of skew-whiffed AdS 4 ×SE 7 solutions of D = 11 supergravity, albeit at a lower temperature than the known superfluid instabilities.

We consider a general class of electrically charged black holes of EinsteinMaxwell-scalar theory that are holographically dual to conformal field theories at finite charge density which break translation invariance explicitly. We examine the linearised perturbations about the solutions that are associated with the thermoelectric DC conductivity. We show that there is a decoupled sector at the black hole horizon which must solve generalised Stokes equations for a charged fluid. By solving these equations we can obtain the DC conductivity of the dual field theory. For Q-lattices and one-dimensional lattices we solve the fluid equations to obtain closed form expressions for the DC conductivity in terms of the solution at the black hole horizon. We also determine the leading order DC conductivity for lattices that can be expanded as a perturbative series about translationally invariant solutions.

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.Comment: 52 pages, 29 figures, very minor changes, version published in JHE

(2015) 'Navier-Stokes equations on black hole horizons and DC thermoelectric conductivity. ', Physical review D., 92 (12). 121901(R).Further information on publisher's website:http://dx.doi.org/10.1103/PhysRevD.92.121901Publisher's copyright statement:Reprinted with permission from the American Physical Society: Physical Review D 92, 121901(R) c (2015) by the American Physical Society. Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modied, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Within the context of the AdS=CFT correspondence, we show that the DC thermoelectric conductivity can be obtained by solving the linearized, time-independent, and forced Navier-Stokes equations on the black hole horizon for an incompressible and charged fluid.