Dissecting holographic conductivities

The DC thermoelectric conductivities of holographic systems in which translational symmetry is broken can be efficiently computed in terms of the near-horizon data of the dual black hole. By calculating the frequency dependent conductivities to the first subleading order in the momentum relaxation rate, we give a physical explanation for these conductivities in the simplest such example, in the limit of slow momentum relaxation. Specifically, we decompose each conductivity into the sum of a coherent contribution due to momentum relaxation and an incoherent contribution, due to intrinsic current relaxation. This decomposition is different from those previously proposed, and is consistent with the known hydrodynamic properties in the translationally invariant limit. This is the first step towards constructing a consistent theory of charged hydrodynamics with slow momentum relaxation.Comment: v2: minor edits, matches published version. v1: 26 pages, 1 figur

Einstein-Maxwell-Dilaton theories with a Liouville potential

We find and analyse solutions of Einstein's equations in arbitrary d dimensions and in the presence of a scalar field with a Liouville potential coupled to a Maxwell field. We consider spacetimes of cylindrical symmetry or again subspaces of dimension d-2 with constant curvature and analyse in detail the field equations and manifest their symmetries. The field equations of the full system are shown to reduce to a single or couple of ODE's which can be used to solve analytically or numerically the theory for the symmetry at hand. Further solutions can also be generated by a solution generating technique akin to the EM duality in the absence of a cosmological constant. We then find and analyse explicit solutions including black holes and gravitating solitons for the case of four dimensional relativity and the higher-dimensional oxydised 5-dimensional spacetime. The general solution is obtained for a certain relation between couplings in the case of cylindrical symmetry.Comment: v3, Some typos corrected with respect to the published versio

Momentum dissipation and effective theories of coherent and incoherent transport

We study heat transport in two systems without momentum conservation: a hydrodynamic system, and a holographic system with spatially dependent, massless scalar fields. When momentum dissipates slowly, there is a well-defined, coherent collective excitation in the AC heat conductivity, and a crossover between sound-like and diffusive transport at small and large distance scales. When momentum dissipates quickly, there is no such excitation in the incoherent AC heat conductivity, and diffusion dominates at all distance scales. For a critical value of the momentum dissipation rate, we compute exact expressions for the Green's functions of our holographic system due to an emergent gravitational self-duality, similar to electric/magnetic duality, and SL(2,R) symmetries. We extend the coherent/incoherent classification to examples of charge transport in other holographic systems: probe brane theories and neutral theories with non-Maxwell actions.Comment: v1: 41 pages + appendices, 7 figures. v2: references and clarifications added. v3: reference adde

Beyond Drude transport in hydrodynamic metals

In interacting theories, hydrodynamics describes the universal behavior of states close to local thermal equilibrium at late times and long distances in a gradient expansion. In the hydrodynamic regime of metals, momentum relaxes slowly with a rate $\Gamma$, which formally appears on the right-hand side of the momentum dynamical equation and causes a Drude-like peak in the frequency dependence of the thermoelectric conductivities. Here we study the structure and determine the physical implications of momentum-relaxing gradient corrections beyond Drude, i.e. arising at subleading order in the gradient expansion. We find that they effectively renormalize the weight of the Drude pole in the thermoelectric conductivities, and contribute to the dc conductivities at the same order as previously-known gradient corrections of translation-invariant hydrodynamics. Turning on a magnetic field, extra derivative corrections appear and renormalize the cyclotron frequency and the Hall conductivity. This relaxed hydrodynamics provides a field-theoretic explanation for previous results obtained using gauge-gravity duality. In strongly-coupled metals where quasiparticles are short-lived and which may be close to a hydrodynamic regime, the extra contributions we discuss are essential to interpret experimentally measured magneto-thermoelectric conductivities. They also cause a different dependence on doping in the effective mass deduced from the specific heat and from measurements of the cyclotron frequency, which do not agree in cuprate superconductors. More generally, we expect such terms to be present in any hydrodynamic theory with approximate symmetries, which arise in many physical systems.Comment: v1: 28 page

Incoherent transport in clean quantum critical metals

In a clean quantum critical metal, and in the absence of umklapp, most d.c. conductivities are formally infinite due to momentum conservation. However, there is a particular combination of the charge and heat currents which has a finite, universal conductivity. In this paper, we describe the physics of this conductivity $\sigma_Q$ in quantum critical metals obtained by charge doping a strongly interacting conformal field theory. We show that it satisfies an Einstein relation and controls the diffusivity of a conserved charge in the metal. We compute $\sigma_Q$ in a class of theories with holographic gravitational duals. Finally, we show how the temperature scaling of $\sigma_Q$ depends on certain critical exponents characterizing the quantum critical metal. The holographic results are found to be reproduced by the scaling analysis, with the charge density operator becoming marginal in the emergent low energy quantum critical theory.Comment: v1: 1 + 16 pages + reference

AdS/Ricci-flat correspondence

We present a comprehensive analysis of the AdS/Ricci-flat correspondence, a map between a class of asymptotically locally AdS spacetimes and a class of Ricci-flat spacetimes. We provide a detailed derivation of the map, discuss a number of extensions and apply it to a number of important examples, such as AdS on a torus, AdS black branes and fluids/gravity metrics. In particular, the correspondence links the hydrodynamic regime of asymptotically flat black $p$-branes or the Rindler fluid with that of AdS. It implies that this class of Ricci-flat spacetimes inherits from AdS a generalized conformal symmetry and has a holographic structure. We initiate the discussion of holography by analyzing how the map acts on boundary conditions and holographic 2-point functions.Comment: v3: Minor edits, references added, matches published versio

Incoherent conductivity of holographic charge density waves

The DC resistivity of charge density waves weakly-pinned by disorder is controlled by diffusive, incoherent processes rather than slow momentum relaxation. The corresponding incoherent conductivity can be computed in the limit of zero disorder. We compute this transport coefficient in holographic spatially modulated breaking translations spontaneously. As a by-product of our analysis, we clarify how the boundary heat current is obtained from a conserved bulk current, de fined as a suitable generalization of the Iyer-Wald Noether current of the appropriate Killing vector.Peer reviewe

Hydrodynamic theory of quantum fluctuating superconductivity

A hydrodynamic theory of transport in quantum mechanically phase-disordered superconductors is possible when supercurrent relaxation can be treated as a slow process. We obtain general results for the frequency-dependent conductivity of such a regime. With time-reversal invariance, the conductivity is characterized by a Drude-like peak, with width given by the supercurrent relaxation rate. Using the memory matrix formalism, we obtain a formula for this width (and hence also the dc resistivity) when the supercurrent is relaxed by short range Coulomb interactions. This leads to a new -- effective field theoretic and fully quantum -- derivation of a classic result on flux flow resistance. With strong breaking of time-reversal invariance, the optical conductivity exhibits what we call a `hydrodynamic supercyclotron' resonance. We obtain the frequency and decay rate of this resonance for the case of supercurrent relaxation due to an emergent Chern-Simons gauge field. The supercurrent decay rate in this `topologically ordered superfluid vortex liquid' is determined by the conductivities of the normal component of the liquid. Our work gives a controlled framework for low temperature metallic phases arising from phase-disordered superconductivity.Comment: 1 + 44 pages. 2 figures. v2 discussion improved in places. v3 sign errors fixed in section

AdS/Ricci-flat correspondence and the Gregory-Laflamme instability

We show that for every asymptotically anti–de Sitter (AdS) solution compactified on a torus there is a corresponding Ricci-flat solution obtained by replacing the torus by a sphere, performing a Weyl rescaling of the metric and appropriately analytically continuing the dimension of the torus/sphere (as in generalized dimensional reduction). In particular, it maps Minkowski spacetime to AdS on a torus, the holographic stress energy tensor of AdS to the stress energy tensor due to a brane localized in the interior of spacetime and AdS black branes to (asymptotically flat) Schwarzschild black branes. Applying it to the known solutions describing the hydrodynamic regime in AdS/CFT, we derive the hydrodynamic stress tensor of asymptotically flat black branes to second order, which is constrained by the parent conformal symmetry. We compute the dispersion relation of the Gregory-Laflamme unstable modes through cubic order in the wave number, finding remarkable agreement with numerical data. In the case of no transverse sphere, AdS black branes are mapped to Rindler spacetime and the second-order transport coefficients of the fluid dual to Rindler spacetime are recovered