Hydrodynamics of Holographic Superconductors

27 pages, 10 figures.-- Pre-print archive.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.We thank S. Hartnoll and S. Sachdev for discussions. This work has been supported partially by the Plan Nacional de Altas Energ´ıas FPA- 2006-05485, FPA-2006-05423, EC Commission under grant MRTN-CT-2004-005104 and Comunidad de Madrid HEPHACOS P-ESP-00346. I. A. is supported by grant BES-2007- 16830.Peer reviewe