Nonequilibrium condensation process of a holographic superconductor in de Rham-Gabadadze-Tolley massive gravity

We study the nonequilibrium condensation process of a holographic s-wave superconductor constructed in the background of de Rham-Gabadadze-Tolley massive gravity theory. When the temperature is lower than a critical value, this model has an asymptotic anti–de Sitter (AdS) bald black hole solution and a black hole solution with nontrivial scalar hair, which can be identified as the normal phase and the superconducting phase in boundary theory, respectively. We consider Gaussian-type perturbation of a scalar field on the background of bald AdS black hole spacetime, and we numerically solve the full nonlinear dynamics of the gravitational system in the bulk. With the full time-dependent solution of the gravitational system, we observe a dynamical process from the perturbed bald AdS black hole to the black hole with scalar hair. According to holographic duality, this process can be regarded as the dynamical phase transition process from normal state to s-wave superconducting state in boundary theory. We also investigate the time evolution of the superconducting condensate operator and clarify how the condensation process from the far-from-equilibrium state proceeds in boundary theory. By fitting the evolution data of the superconducting order parameter at early time, we show that the initial nonequilibrium condensation process can be predicted by linear quasinormal modes of scalar field perturbation on the background of the bald AdS black hole. Finally, we study the time evolution of the event and apparent horizons.

Figure 1

The dynamics of the bulk scalar field for initial black hole temperature $T=0.5T_c$. (Left panel) Amplitude of the complex scalar field $|\psi (v,z)|$ as a function of $vTc$ and $zTc$. It is shown that the amplitude grows exponentially till saturation. (Right panel) In order to focus on the behavior of the wave packet of the initial perturbation, we depict the amplitude $|\psi (v,z)|$ for $0\le vTc\le 0.15$.

Figure 2

(Left panel) Time evolution of the real part of asymptotic expansion coefficient ${\psi }_2(v)$ with $T/Tc=0.5$. (Inset) The evolution at late time, which indicates the oscillating nature of complex scalar field when the final equilibrium state is reached. (Right panel) The time evolution of condensate operator $|⟨\mathcal{O}(v)⟩|$.

Figure 3

The dynamics of the order parameter with different initial black hole temperatures. (Left panel) The solid curves from top to bottom correspond to $T/T_c=0.2$, 0.4, 0.6, and 0.8, while the dashed curves represent the relaxation timescale by fitting the evolution data at early time. (Right panel) The different curves from top to bottom correspond to $T/T_c=1.1$, 1.2, and 1.3.

Figure 4

The time evolution of the area of event and apparent horizons with the initial temperature $T/T_c=0.5$. The solid and dashed curves correspond to the area of apparent and event horizons, respectively.