Nonlinear conductivity of a holographic superconductor under constant electric field

The dynamics of a two-dimensional superconductor under a constant electric field $E$ is studied by using the gauge-gravity correspondence. The pair breaking current induced by $E$ first increases to a peak value and then decreases to a constant value at late times, where the superconducting gap goes to zero, corresponding to a normal conducting phase. The peak value of the current is found to increase linearly with respect to the electric field. Moreover, the nonlinear conductivity, defined as an average of the conductivity in the superconducting phase, scales as $\sim E^{-2/3}$ when the system is close to the critical temperature $T_c$, which agrees with predictions from solving the time-dependent Ginzburg-Landau equation. Away from $T_c$, the $E^{-2/3}$ scaling of the conductivity still holds when $E$ is large.

Figure 1

The initial equilibrium configuration of $|\mathrm{\Phi }(z)|$ and $A_t(z)$ at $T=0.68T_c$. Here, $z=1$ is the horizon and $z\to 0$ is the conformal boundary.

Figure 2

The superconducting gap $⟨O_n⟩$ (upper panel) and the induced current $J$ (lower panel) as a function of time $t$ at different temperatures. The electric field here is $E=1$.

Figure 3

The dynamics of $J(t)$ for different values of $E$. The temperature is fixed at $T=0.68T_c$.

Figure 4

The maximum current $J_{\max }(E)$ as a function of the electric field $E$ at different temperatures.

Figure 5

The nonlinear dc conductivity $\sigma (E)$ as a function of the electric field $E$ at different temperatures. The lines are fits to the data in the large $E$ limit using Eq. (15).