We study the gravitational Dirichlet problem in AdS spacetimes with a view to understanding the boundary CFT interpretation. We define the problem as bulk Einstein's equations with Dirichlet boundary conditions on fixed timelike cut-off hypersurface. Using the fluid/gravity correspondence, we argue that one can determine non-linear solutions to this problem in the long wavelength regime. On the boundary we find a conformal fluid with Dirichlet constitutive relations, viz., the fluid propagates on a 'dynamical' background metric which depends on the local fluid velocities and temperature. This boundary fluid can be re-expressed as an emergent hypersurface fluid which is non-conformal but has the same value of the shear viscosity as the boundary fluid. The hypersurface dynamics arises as a collective effect, wherein effects of the background are transmuted into the fluid degrees of freedom. Furthermore, we demonstrate that this collective fluid is forced to be non-relativistic below a critical cut-off radius in AdS to avoid acausal sound propagation with respect to the hypersurface metric. We further go on to show how one can use this set-up to embed the recent constructions of flat spacetime duals to non-relativistic fluid dynamics into the AdS/CFT correspondence, arguing that a version of the membrane paradigm arises naturally when the boundary fluid lives on a background Galilean manifold.

In this paper we study the dynamical properties of charged systems immersed in an external magnetic field and perturbed by a set of scalar operators breaking translations either spontaneously or pseudo-spontaneously. By combining hydrodynamic and quantum field theory arguments we provide analytic expressions for all the hydrodynamic transport coefficients relevant for the diffusive regime in terms of thermodynamic quantities and DC thermo-electric conductivities. This includes the momentum dissipation rate. We shed light on the role of the momentum dissipation rate in the transition between the pseudo-spontaneous and the purely explicit regimes in this class of systems. Finally, we clarify several relations between the hydrodynamic transport coefficients which have been observed in the holographic literature of charge density wave models.

We consider magnetohydrodynamics with an external magnetic field. We find that in general one must allow for a non-zero incoherent Hall conductivity to correctly describe the DC longitudinal and Hall thermal conductivities beyond order zero in the magnetic field expansion. We apply our result to the dyonic black hole, determining the incoherent Hall conductivity in that case, and additionally prove that the existence of this transport coefficient leads to a significantly better match between the hydrodynamic and AC thermo-electric correlators.

Abstract:We use holography to study N = 4 supersymmetric SU (N c ) Yang-Mills theory in the large-N c and large-coupling limits coupled to a number N f N c of (n + 1)-dimensional massless supersymmetric hypermultiplets in the fundamental representation of SU (N c ), with n = 2, 3. We introduce a temperature T , a baryon number chemical potential µ, and a baryon number magnetic field B, and work in a regime with µ T, √ B. We study the collective excitations of these holographic quantum liquids by computing the poles in the retarded Green's function of the baryon number charge density operator and the associated peaks in the spectral function. We focus on the evolution of the collective excitations as we increase the frequency relative to T , i.e. the hydrodynamic/collisionless crossover. We find that for all B, at low frequencies the tallest peak in the spectral function is associated with hydrodynamic charge diffusion. At high frequencies the tallest peak is associated with a sound mode similar to the zero sound mode in the collisionless regime of a Landau Fermi liquid. The sound mode has a gap proportional to B, and as a result for intermediate frequencies and for B sufficiently large compared to T the spectral weight is strongly suppressed. We find that the hydrodynamic/collisionless crossover occurs at a frequency that is approximately B-independent.

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