We study the poles of the retarded Green functions of a holographic superconductor. The model shows a second order phase transition where a charged scalar operator condenses and a U (1) symmetry is spontaneously broken. The poles of the holographic Green functions are the quasinormal modes in an AdS black hole background. We study the spectrum of quasinormal frequencies in the broken phase, where we establish the appearance of a massless or hydrodynamic mode at the critical temperature as expected for a second order phase transition. In the broken phase we find the pole representing second sound. We compute the speed of second sound and its attenuation length as function of the temperature. In addition we find a pseudo diffusion mode, whose frequencies are purely imaginary but with a non-zero gap at zero momentum. This gap goes to zero at the critical temperature. As a technical side result we explain how to calculate holographic Green functions and their quasinormal modes for a set of operators that mix under the RG flow.

In the presence of dense matter quantum anomalies give rise to two new transport phenomena. An anomalous current is generated either by an external magnetic field or through vortices in the fluid carrying the anomalous charge. The associated transport coefficients are the anomalous magnetic and vortical conductivities. Whereas a Kubo formula for the anomalous magnetic conductivity is well known we develop a new Kubo type formula that allows the calculation of the vortical conductivity through a two point function of the anomalous current and the momentum density. We also point out that the anomalous vortical conductivity can be understood as a response to a gravitomagnetic field. We apply these Kubo formulas to a simple Holographic system, the R-charged black hole.Comment: 28 pages, 3 figures. Final version published on JHE

The Goldstone theorem implies the appearance of an ungapped mode whenever a continuous global symmetry is spontaneously broken. In general it does not say anything about the precise form of the dispersion relation nor does it imply that there is one massless mode for each broken symmetry generator. It is a well-established fact that even for relativistic field theories in the presence of a chemical potential Goldstone modes with quadratic dispersion relation, the type II Goldstone bosons, appear in the spectrum. We develop two holographic models that feature type II Goldstone modes as part of the quasinormal mode spectrum. The models are based on simple generalizations with U (2) symmetry of the well-studied holographic s-wave superfluid. Our results include Goldstone modes without broken generators but with unusual realization of symmetries and a frequency dependent conductivity of striking resemblance to the one of Graphene.

Abstract:We revisit the question of stability of holographic superfluids with finite superfluid velocity. Our method is based on applying the Landau criterion to the Quasinormal Mode (QNM) spectrum. In particular we study the QNMs related to the Goldstone modes of spontaneous symmetry breaking with linear and quadratic dispersions. In the linear case we show that the sound velocity becomes negative for large enough superfluid velocity and that the imaginary part of the quasinormal frequency moves to the upper half plane. Since the instability is strongest at finite wavelength, we take this as an indication for the existence of an inhomogeneous or striped condensed phase for large superfluid velocity. In the quadratic case the instability is present for arbitrarily small superfluid velocity.

We continue our investigations on the relation between hydrodynamic and higher quasinormal modes in the AdS black hole background started in arXiv:0710.4458 [hep-th]. As is well known, the quasinormal modes can be interpreted as the poles of the retarded Green functions of the dual N = 4 gauge theory at finite temperature. The response to a generic perturbation is determined by the residues of the poles. We compute these residues numerically for energy-momentum and R-charge correlators. We find that the diffusion modes behave in a similar way: at small wavelengths the residues go over into a form of a damped oscillation and therefore these modes decouple at short distances. The sound mode behaves differently: its residue does not decay and at short wavelengths this mode behaves as the higher quasinormal modes. Applications of our findings include the definition of hydrodynamic length and time scales. We also show that the quasinormal modes, including the hydrodynamic diffusion modes, obey causality.