Hydrodynamics of Holographic Superconductors

TL;DR

The paper analyzes the hydrodynamics of a holographic superconductor in the probe-limit AdS$_4$ background by tracking the poles of retarded Green functions through quasinormal modes. It introduces a determinant-based method to compute holographic Green functions for coupled bulk fluctuations, identifies a Goldstone mode at the critical temperature $T_c$ that evolves into two second-sound modes below $T_c$, and uncovers a pseudo-diffusion mode with a gap that vanishes as $T\to T_c$. The authors quantify the second-sound speed $v_s$ and attenuation $\Gamma_s$, and show continuous evolution of higher QNMs across the phase transition. They also discuss gauge-invariance of poles and lay out a path for including backreaction and extending to p-wave superconductors and dynamical critical phenomena.

Abstract

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.

Figures (10)

• Figure 1: The condensates as function of the temperature in the two possible theories.
• Figure 2: Lowest scalar quasinormal frequencies as a function of the temperature and at momentum $k=0$, from $T/T_c=\infty$ to $T/T_c=0.81$ in the $O_2$ theory (right) and to $T/T_c=0.56$ in the $O_1$ theory (left). The dots correspond to the critical point $T/T_c=1$ where the phase transition takes place. Red, blue and green correspond to first, second and third mode respectively.
• Figure 3: Schematic plot of the poles in the coupled system, i.e. in the broken phase at small finite momentum right below $T_c$. These poles are present in each retarded correlation function for the coupled fields $\eta,\,\sigma,\,A_t,\,A_x$, while their residues might vanish for specific fields. Close to the origin we find the (pseudo)diffusion mode and two hydrodynamic second sound modes. In addition two sets of higher (non-hydrodynamic) quasinormal modes are shown. In the unbroken phase these poles originate from the scalar (grey dots). We also expect a tower of purely imaginary poles stemming from the longitudinal vector channel (black dots). The grey area indicates where our present numerical methods break down. In this paper we will be concerned primarily with the hydrodynamic modes and will touch upon the higher quasinormal modes only briefly.
• Figure 4: Movement of the positive frequency sound pole away from $\omega=0$ with increasing spatial momentum. Distinct curves correspond to temperatures below the phase transition $T/T_c=0.999$ (black), $0.97$ (pink), $0.91$ (red), $0.71$ (green), $0.52$ (blue), $0.26$ (light blue). Dots on one curve are separated by $\Delta k=0.05$. All curves start at $k=0.05$ and end at $k=1.00$.
• Figure 5: Fits of the real and imaginary part of the hydrodynamic modes in the broken phase to the lowest order approximation $\omega=v_s k -i \Gamma_s k^2$. The left figure shows the real part and the right one the imaginary part. The thick lines are the numerical results and the thin lines are the linear and quadratic fits. The fit is done for a temperature just below the critical one where the range of the approximation is rather small.