Thermodynamics of Dyonic Lifshitz Black Holes

Black holes with asymptotic anisotropic scaling are conjectured to be gravity duals of condensed matter system close to quantum critical points with non-trivial dynamical exponent z at finite temperature. A holographic renormalization procedure is presented that allows thermodynamic potentials to be defined for objects with both electric and magnetic charge in such a way that standard thermodynamic relations hold. Black holes in asymptotic Lifshitz spacetimes can exhibit paramagnetic behavior at low temperature limit for certain values of the critical exponent z, whereas the behavior of AdS black holes is always diamagnetic.Comment: 26 pages, 4 figure

Black Hole Thermodynamics and Heavy Fermion Metals

Heavy fermion alloys at critical doping typically exhibit non-Fermi-liquid behavior at low temperatures, including a logarithmic or power law rise in the ratio of specific heat to temperature as the temperature is lowered. Anomalous specific heat of this type is also observed in a simple class of gravitational dual models that exhibit anisotropic scaling with dynamical critical exponent z > 1.Comment: 17 pages, 4 figures; v2: added references; v3: matches published versio

Analytic Lifshitz black holes in higher dimensions

We generalize the four-dimensional R^2-corrected z=3/2 Lifshitz black hole to a two-parameter family of black hole solutions for any dynamical exponent z and for any dimension D. For a particular relation between the parameters, we find the first example of an extremal Lifshitz black hole. An asymptotically Lifshitz black hole with a logarithmic decay is also exhibited for a specific critical exponent depending on the dimension. We extend this analysis to the more general quadratic curvature corrections for which we present three new families of higher-dimensional D>=5 analytic Lifshitz black holes for generic z. One of these higher-dimensional families contains as critical limits the z=3 three-dimensional Lifshitz black hole and a new z=6 four-dimensional black hole. The variety of analytic solutions presented here encourages to explore these gravity models within the context of non-relativistic holographic correspondence.Comment: 14 page

Gauss-Bonnet Black Holes and Heavy Fermion Metals

We consider charged black holes in Einstein-Gauss-Bonnet Gravity with Lifshitz boundary conditions. We find that this class of models can reproduce the anomalous specific heat of condensed matter systems exhibiting non-Fermi-liquid behaviour at low temperatures. We find that the temperature dependence of the Sommerfeld ratio is sensitive to the choice of Gauss-Bonnet coupling parameter for a given value of the Lifshitz scaling parameter. We propose that this class of models is dual to a class of models of non-Fermi-liquid systems proposed by Castro-Neto et.al.Comment: 17 pages, 6 figures, pdfLatex; small corrections to figure 10 in this versio

Universal thermal and electrical conductivity from holography

It is known from earlier work of Iqbal, Liu (arXiv:0809.3808) that the boundary transport coefficients such as electrical conductivity (at vanishing chemical potential), shear viscosity etc. at low frequency and finite temperature can be expressed in terms of geometrical quantities evaluated at the horizon. In the case of electrical conductivity, at zero chemical potential gauge field fluctuation and metric fluctuation decouples, resulting in a trivial flow from horizon to boundary. In the presence of chemical potential, the story becomes complicated due to the fact that gauge field and metric fluctuation can no longer be decoupled. This results in a nontrivial flow from horizon to boundary. Though horizon conductivity can be expressed in terms of geometrical quantities evaluated at the horizon, there exist no such neat result for electrical conductivity at the boundary. In this paper we propose an expression for boundary conductivity expressed in terms of geometrical quantities evaluated at the horizon and thermodynamical quantities. We also consider the theory at finite cutoff outside the horizon (arXiv:1006.1902) and give an expression for cutoff dependent electrical conductivity, which interpolates smoothly between horizon conductivity and boundary conductivity . Using the results about the electrical conductivity we gain much insight into the universality of thermal conductivity to viscosity ratio proposed in arXiv:0912.2719.Comment: An appendix added discussing relation between boundary conductivity and universal conductivity of stretched horizon, version to be published in JHE

Black holes and black branes in Lifshitz spacetimes

We construct analytic solutions describing black holes and black branes in asymptotically Lifshitz spacetimes with arbitrary dynamical exponent z and for arbitrary number of dimensions. The model considered consists of Einstein gravity with negative cosmological constant, a scalar, and N U(1) gauge fields with dilatonic-like couplings. We study the phase diagrams and thermodynamic instabilities of the solution, and find qualitative differences between the cases with 12.Comment: 27 pages, 10 figures; v2 references added, minor comments adde

Lifshitz black holes in string theory

We provide the first black hole solutions with Lifshitz asymptotics found in string theory. These are expected to be dual to models enjoying anisotropic scale invariance with dynamical exponent z=2 at finite temperature. We employ a consistent truncation of type IIB supergravity to four dimensions with an arbitrary 5-dimensional Einstein manifold times a circle as internal geometry. New interesting features are found that significantly differ from previous results in phenomenological models. In particular, small black holes are shown to be thermodynamically unstable, analogously to the usual AdS-Schwarzschild black holes, and extremality is never reached. This signals a possible Hawking-Page like phase transition at low temperatures.Comment: 19 pages, 7 figures. v2 references adde

Holography of Charged Dilaton Black Holes

We study charged dilaton black branes in $AdS_4$. Our system involves a dilaton $\phi$ coupled to a Maxwell field $F_{\mu\nu}$ with dilaton-dependent gauge coupling, ${1\over g^2} = f^2(\phi)$. First, we find the solutions for extremal and near extremal branes through a combination of analytical and numerical techniques. The near horizon geometries in the simplest cases, where $f(\phi) = e^{\alpha\phi}$, are Lifshitz-like, with a dynamical exponent $z$ determined by $\alpha$. The black hole thermodynamics varies in an interesting way with $\alpha$, but in all cases the entropy is vanishing and the specific heat is positive for the near extremal solutions. We then compute conductivity in these backgrounds. We find that somewhat surprisingly, the AC conductivity vanishes like $\omega^2$ at T=0 independent of $\alpha$. We also explore the charged black brane physics of several other classes of gauge-coupling functions $f(\phi)$. In addition to possible applications in AdS/CMT, the extremal black branes are of interest from the point of view of the attractor mechanism. The near horizon geometries for these branes are universal, independent of the asymptotic values of the moduli, and describe generic classes of endpoints for attractor flows which are different from $AdS_2\times R^2$.Comment: 33 pages, 3 figures, LaTex; v2, references added; v3, more refs added; v4, refs added, minor correction

A holographic model for the fractional quantum Hall effect

Experimental data for fractional quantum Hall systems can to a large extent be explained by assuming the existence of a modular symmetry group commuting with the renormalization group flow and hence mapping different phases of two-dimensional electron gases into each other. Based on this insight, we construct a phenomenological holographic model which captures many features of the fractional quantum Hall effect. Using an SL(2,Z)-invariant Einstein-Maxwell-axio-dilaton theory capturing the important modular transformation properties of quantum Hall physics, we find dyonic diatonic black hole solutions which are gapped and have a Hall conductivity equal to the filling fraction, as expected for quantum Hall states. We also provide several technical results on the general behavior of the gauge field fluctuations around these dyonic dilatonic black hole solutions: We specify a sufficient criterion for IR normalizability of the fluctuations, demonstrate the preservation of the gap under the SL(2,Z) action, and prove that the singularity of the fluctuation problem in the presence of a magnetic field is an accessory singularity. We finish with a preliminary investigation of the possible IR scaling solutions of our model and some speculations on how they could be important for the observed universality of quantum Hall transitions.Comment: 86 pages, 16 figures; v.2 references added, typos fixed, improved discussion of ref. [39]; v.3 more references added and typos fixed, several statements clarified, v.4 version accepted for publication in JHE

Effective Holographic Theories for low-temperature condensed matter systems

The IR dynamics of effective holographic theories capturing the interplay between charge density and the leading relevant scalar operator at strong coupling are analyzed. Such theories are parameterized by two real exponents $(\gamma,\delta)$ that control the IR dynamics. By studying the thermodynamics, spectra and conductivities of several classes of charged dilatonic black hole solutions that include the charge density back reaction fully, the landscape of such theories in view of condensed matter applications is characterized. Several regions of the $(\gamma,\delta)$ plane can be excluded as the extremal solutions have unacceptable singularities. The classical solutions have generically zero entropy at zero temperature, except when $\gamma=\delta$ where the entropy at extremality is finite. The general scaling of DC resistivity with temperature at low temperature, and AC conductivity at low frequency and temperature across the whole $(\gamma,\delta)$ plane, is found. There is a codimension-one region where the DC resistivity is linear in the temperature. For massive carriers, it is shown that when the scalar operator is not the dilaton, the DC resistivity scales as the heat capacity (and entropy) for planar (3d) systems. Regions are identified where the theory at finite density is a Mott-like insulator at T=0. We also find that at low enough temperatures the entropy due to the charge carriers is generically larger than at zero charge density.Comment: (v3): Added discussion on the UV completion of the solutions, and on extremal spectra in the charged case. Expanded discusion on insulating extremal solutions. Many other refinements and corrections. 126 pages. 48 figure