Holographic Checkerboards

We construct cohomogeneity-three, finite temperature stationary black brane solutions dual to a field theory exhibiting checkerboard order. The checkerboards form a backreacted part of the bulk solution, and are obtained numerically from the coupled Einstein-Maxwell-scalar PDE system. They arise spontaneously and without the inclusion of an explicit lattice. The phase exhibits both charge and global U(1)-current modulation, which are periodic in two spatial directions. The current circulates within each checkerboard plaquette. We explore the competition with striped phases, finding first-order checkerboard to stripe phase transitions. We also detail spatially modulated instabilities of asymptotically AdS black brane backgrounds with neutral scalar profiles, including those with an hyperscaling violating IR geometry at zero temperature.Comment: 26 pages, 11 figures. v2: Published versio

Nonlinear conductivity and the ringdown of currents in metallic holography

We study the electric and heat current response resulting from an electric field quench in a holographic model of momentum relaxation at nonzero charge density. After turning the electric field off, currents return to equilibrium as governed by the vector quasi-normal modes of the dual black brane, whose spectrum depends qualitatively on a parameter controlling the strength of inhomogeneity. We explore the dynamical phase diagram as a function of this parameter, showing that signatures of incoherent transport become identifiable as an oscillatory ringdown of the heat current. We also study nonlinear conductivity by holding the electric field constant. For small electric fields a balance is reached between the driving electric field and the momentum sink -- a steady state described by DC linear response. For large electric fields Joule heating becomes important and the black branes exhibit significant time dependence. In a regime where the rate of temperature increase is small, the nonlinear electrical conductivity is well approximated by the DC linear response calculation at an appropriate effective temperature.Comment: 28 pages, 10 figures. Plot of thermal conductivity added. Version as published in JHE

Short-lived modes from hydrodynamic dispersion relations

We consider the dispersion relation of the shear-diffusion mode in relativistic hydrodynamics, which we generate to high order as a series in spatial momentum q for a holographic model. We demonstrate that the hydrodynamic series can be summed in a way that extends through branch cuts present in the complex q plane, resulting in the accurate description of multiple sheets. Each additional sheet corresponds to the dispersion relation of a different non-hydrodynamic mode. As an example we extract the frequencies of a pair of oscillatory non-hydrodynamic black hole quasinormal modes from the hydrodynamic series. The analytic structure of this model points to the possibility that the complete spectrum of gravitational quasinormal modes may be accessible from the hydrodynamic derivative expansion.Comment: 17 pages, 6 figures. Matches published versio

Robinson-Trautman spacetimes and gauge/gravity duality

We study far-from-equilibrium field theory dynamics using gauge/gravity duality applied to the Robinson-Trautman (RT) class of spacetimes and we present a number of new results. First, we assess the applicability of the hydrodynamic approximation to inhomogeneous plasma dynamics dual to RT spacetimes. We prove that to any order in a late time expansion it is possible to identify variables corresponding to the local energy density and fluid velocity. However, we show using numerical examples that this does not hold at the non-perturbative level; for sufficiently inhomogeneous initial data a local rest frame does not exist. Second, we preset a new class of holographic inhomogeneous plasma flows on the plane. The corresponding spacetimes are not of the RT type but they can be obtained from RT spacetimes with spatially compact boundaries by coordinate transformations which generate Poincar\'e patch-like coordinates with planar boundaries. We demonstrate the application of this procedure using numerical examples.Comment: Proceedings prepared for the "Workshop on Geometry and Physics" in memoriam of Ioannis Bakas, November 2016, Ringberg Castle, Germany. v2: Minor changes, added discussion of isotropisation tim

A gravity derivation of the Tisza-Landau Model in AdS/CFT

We derive the fully backreacted bulk solution dual to a boundary superfluid with finite supercurrent density in AdS/CFT. The non-linear boundary hydrodynamical description of this solution is shown to be governed by a relativistic version of the Tisza-Landau two-fluid model to non-dissipative order. As previously noted, the phase transition can be both first order and second order, but in the strongly-backreacted regime at low charge q we find that the transition remains second order for all allowed fractions of superfluid density.Comment: 27 pages, 6 figures, 1 appendix; version published in PR

Linear gravity from conformal symmetry

We perform a unified systematic analysis of $d+1$ dimensional, spin $\ell$ representations of the isometry algebra of the maximally symmetric spacetimes AdS$_{d+1}$, $\mathbb{R}_{1,d}$ and dS$_{d+1}$. This allows us to explicitly construct the effective low-energy bulk equations of motion obeyed by linear fields, as the eigenvalue equation for the quadratic Casimir differential operator. We show that the bulk description of a conformal family is given by the Fierz-Pauli system of equations. For $\ell = 2$ this is a massive gravity theory, while for $\ell = 2$ conserved currents we obtain Einstein gravity and covariant gauge fixing conditions. This analysis provides a direct algebraic derivation of the familiar AdS holographic dictionary at low energies, with analogous results for Minkowski and de Sitter spacetimes.Comment: 21 pages, 1 figur

Self-similar equilibration of strongly interacting systems from holography

We study the equilibration of a class of far-from-equilibrium strongly interacting systems using gauge/gravity duality. The systems we analyse are 2+1 dimensional and have a four dimensional gravitational dual. A prototype example of a system we analyse is the equilibration of a two dimensional fluid which is translational invariant in one direction and is attached to two different heat baths with different temperatures at infinity in the other direction. We realise such setup in gauge/gravity duality by joining two semi-infinite asymptotically Anti-de Sitter (AdS) black branes of different temperatures, which subsequently evolve towards equilibrium by emitting gravitational radiation towards the boundary of AdS. At sufficiently late times the solution converges to a similarity solution, which is only sensitive to the left and right equilibrium states and not to the details of the initial conditions. This attractor solution not only incorporates the growing region of equilibrated plasma but also the outwardly-propagating transition regions, and can be constructed by solving a single ordinary differential equation.Comment: 5 pages 3 figures. Published versio

Drude in D major

We study holographic momentum relaxation in the limit of a large number of spacetime dimensions D. For an axion model we find that momentum conservation is restored as D becomes large. To compensate we scale the strength of the sources with D so that momentum is relaxed even at infinite D. We analytically obtain the quasi-normal modes which control electric and heat transport, and give their frequencies in a 1/D expansion. We also obtain the AC thermal conductivity as an expansion in 1/D, which at leading order takes Drude form. To order 1/D our analytical result provides a reasonable approximation to the AC conductivity even at D=4, establishing large D as a practical method in this context. As a further application, we discuss the signature of the transition from coherent to incoherent behaviour known to exist in the system for finite D.Comment: 19 pages, 2 figure

Black branes dual to striped phases

We construct inhomogeneous charged black branes in AdS, holographically dual to a phase at finite chemical potential with spontaneously broken translation invariance in one direction. These are obtained numerically, solving PDEs for the fully backreacted system. Fixing the periodicity scale, we find a second order phase transition to the inhomogeneous phase. We comment on the properties of the state emerging at low temperatures. For some models we demonstrate the existence of a branch of striped solutions but no continuous phase transition.Comment: 16 pages, 9 figure

Warm p-soup and near extremal black holes

We consider a model of D-dimensional supergravity coupled to elementary p-branes. We use gravitational arguments to deduce the low energy effective theory of N nearly parallel branes. This is a (p+1)-dimensional scalar field theory, where the scalars represent the positions of the branes in their transverse space. We propose that the same theory in a certain temperature regime describes a `soup' of strongly interacting branes, giving a microscopic description of near extremal black p-branes. We use natural approximations to estimate the energy density of this soup as a function of the physical parameters; N, temperature, brane tension and gravitational coupling. We also characterise the horizon radius, measured in the metric natural to the branes, with the thermal vev of the scalars. For both quantities we find agreement with the corresponding supergravity black brane results. Surprisingly, beyond the physical parameters, we are naturally able to reproduce certain irrational factors such as pi's. We comment on how these ideas may explain why black hole thermodynamics arises in gauge theories with holographic duals at finite temperature.Comment: 32 pages, no figure