Christopher P. Herzog

Building a Holographic Superconductor

We show that a simple gravitational theory can provide a holographically dual description of a superconductor. There is a critical temperature, below which a charged condensate forms via a second order phase transition and the (DC) conductivity becomes infinite. The frequency dependent conductivity develops a gap determined by the condensate. We find evidence that the condensate consists of pairs of quasiparticles.

Lectures on holographic superfluidity and superconductivity

Four lectures on holography and the AdS/CFT correspondence applied to condensed matter systems. The first lecture introduces the concept of a quantum phase transition. The second lecture discusses linear response theory and Ward identities. The third lecture presents transport coefficients derived from AdS/CFT that should be applicable in the quantum critical region associated with a quantum phase transition. The fourth lecture builds in the physics of a superconducting or superfluid phase transition to the simple holographic model of the third lecture.

Schwinger-Keldysh propagators from AdS/CFT correspondence

We demonstrate how to compute real-time Greens functions for a class of finite temperature field theories from their AdS gravity duals. In particular, we reproduce the 2 ? 2 Schwinger-Keldysh matrix propagator from a gravity calculation. Our methods should work also for computing higher point lorentzian signature correlators. We elucidate the boundary condition subtleties which hampered previous efforts to build a lorentzian-signature AdS/CFT correspondence. For two-point correlators, our construction is automatically equivalent to the previously formulated prescription for the retarded propagator.

Heating up Galilean holography

We embed a holographic description of a quantum field theory with Galilean conformal invariance in string theory. The key observation is that such field theories may be realized as conventional superconformal field theories with a known string theory embedding, twisted by the R-symmetry in a light-like direction. Using the Null Melvin Twist, we construct the appropriate dual geometry and its non-extremal generalization. From the nonzero temperature solution we determine the equation of state. We also discuss the hydrodynamic regime of these non-relativistic plasmas and show that the shear viscosity to entropy density ratio takes the universal value �/s = 1/4� typical of strongly interacting field theories with gravity duals.

Quantum critical transport, duality, and M-theory

We consider charge transport properties of 2+1 dimensional conformal field theories at non-zero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For N=8 supersymmetric, SU(N) Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) Rcurrents can be obtained in the large N limit by applying the AdS/CFT correspondence to Mtheory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in 3+1 dimensional anti-de Sitter space. The electromagnetic self-duality of the 3+1 dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in 2+1 dimensional Abelian theories. Thus the 2+1 dimensional, superconformal Yang Mills theory obeys a “holographic self duality” in the large N limit, and perhaps more generally.

Superconductors from Superstrings

We establish that in a large class of strongly coupled (3+1)-dimensional N=1 quiver conformal field theories with gravity duals, adding a chemical potential for the R charge leads to the existence of superfluid states in which a chiral primary operator of the schematic form O=lambdalambda+W condenses. Here lambda is a gluino and W is the superpotential. Our argument is based on the construction of a consistent truncation of type IIB supergravity that includes a U(1) gauge field and a complex scalar.

Holographic model of superfluidity

We study a holographic model of a relativistic quantum system with a global U(1) symmetry, at nonzero temperature and density. When the temperature falls below a critical value, we find a second-order superfluid phase transition with mean-field critical exponents. In the symmetry-broken phase, we determine the speed of second sound as a function of temperature. As the velocity of the superfluid component relative to the normal component increases, the superfluid transition goes through a tricritical point and becomes first order.

Multimatrix models and tri-Sasaki Einstein spaces

Localization methods reduce the path integrals in N{>=}2 supersymmetric Chern-Simons gauge theories on S{sup 3} to multimatrix integrals. A recent evaluation of such a two-matrix integral for the N=6 superconformal U(N)xU(N) Aharony-Bergman-Jafferis-Maldacena theory produced detailed agreement with the AdS/CFT correspondence, explaining, in particular, the N{sup 3/2} scaling of the free energy. We study a class of p-matrix integrals describing N=3 superconformal U(N){sup p} Chern-Simons gauge theories. We present a simple method that allows us to evaluate the eigenvalue densities and the free energies in the large N limit keeping the Chern-Simons levels k{sub i} fixed. The dual M-theory backgrounds are AdS{sub 4}xY, where Y are seven-dimensional tri-Sasaki Einstein spaces specified by the k{sub i}. The gravitational free energy scales inversely with the square root of the volume of Y. We find a general formula for the p-matrix free energies that agrees with the available results for volumes of the tri-Sasaki Einstein spaces Y, thus providing a thorough test of the corresponding AdS{sub 4}/CFT{sub 3} dualities. This formula is consistent with the Seiberg duality conjectured for Chern-Simons gauge theories.

Ohm’s law at strong coupling: S duality and the cyclotron resonance

We calculate the electrical and thermal conductivities and the thermoelectric coefficient of a class of strongly interacting 2+1-dimensional conformal field theories with anti-de Sitter space duals. We obtain these transport coefficients as a function of charge density, background magnetic field, temperature, and frequency. We show that the thermal conductivity and thermoelectric coefficient are determined by the electrical conductivity alone. At small frequency, in the hydrodynamic limit, we are able to provide a number of analytic formulas for the electrical conductivity. A dominant feature of the conductivity is the presence of a cyclotron pole. We show how bulk electromagnetic duality acts on the transport coefficients.

Holographic prediction for the deconfinement temperature.

We argue that deconfinement in anti-de Sitter space models of quantum chromodynamics (AdS/QCD models) occurs via a first order Hawking-Page type phase transition between low temperature thermal AdS and a high temperature black hole. Such a result is consistent with the expected temperature independence, to leading order in 1/Nc, of the meson spectrum and spatial Wilson loops below the deconfinement temperature. As a by-product, we obtain model dependent deconfinement temperatures Tc in the hard- and soft-wall models of AdS/QCD. Our result for Tc in the soft-wall model is close to a recent lattice prediction.